Searching for warm dark matter down
on Earth and among the stars
Dissertation submitted for the award of the title
“Doctor of Natural Sciences”
to the faculty of Physics, Mathematics and Computer science of
Johannes Gutenberg-Universität in Mainz
Sven Baumholzer
born in Worms
Mainz, November 29th, 2021
First examiner: Prof. Dr. Pedro Schwaller
Date of submission: 29.11.2021
Date of defense: 18.03.2022
Abstract
There are experimental and observational hints calling for new physics beyond the standard
model (SM), among them the intriguing question of the nature of dark matter (DM). In
this thesis we study the phenomenology of models featuring warm DM.
First, we consider the scotogenic model, which features a radiative generation of neutrino
masses, and explore its light dark matter phenomenology. In particular, we focus on keV-
scale DM which can be produced either via a freeze-in mechanism through the decays
of newly introduced scalars or the decays of the next-to-lightest fermionic particle in the
spectrum. The latter possibility is required to be suppressed as it typically produces a
hot DM component. Constraints from big bang nucleosynthesis (BBN) and the number
of non-photonic relativistic particle species, Neff , are also considered, and in combination
with couplings needed to produce enough DM, the DM candidate is required to be light.
To estimate the discovery potential for this scenario we consider collider analyses at the
high luminosity upgrade of the Large Hadron Collider (HL-LHC) and proposed future
hadron and lepton colliders, namely FCC-hh and CLIC, focusing on final states with two
leptons and missing transverse energy. Taking into account the cosmological bounds, we
show that they already probe parts of the HL-LHC discovery region for this scenario, while
future colliders access an even larger region of parameter space.
Second, we derive structure formation limits on DM composed of keV-scale axion-like
particles (ALPs), produced via freeze-in through their interaction with photons and SM
fermions. We employ results from Lyman-α forest data sets as well as the observed number
of Milky Way (MW) subhalos. We compare the momentum distribution function obtained
using Maxwell-Boltzmann and quantum statistics for describing the SM thermal bath.
It should be emphasized that the presence of logarithmic divergences complicates the
calculation of the production rate.
The results obtained in this way, in combination with gamma-ray bounds, exclude the
possibility for a photophilic “frozen-in” ALP DM with mass below ∼ 19 keV. For the
photophobic ALP scenario, in which DM couples primarily to SM fermions, the ALP DM
distribution function is peaked at lower momentum and hence results in weaker limits on
the DM mass. Future facilities, such as the upcoming Vera C. Rubin observatory, will
significantly improve the current bounds up to ∼ 80 keV.
Lastly, we generalize DM production via multiple mechanisms by introducing a model-
independent framework and assess whether its consistency with structure formation ob-
servations. We simulate the matter power spectrum for DM scenarios characterized by at
least two temperatures stemming from the different production mechanisms, and derive
the suppression of structures at small scales and the expected number of MW subhalos.
This allows us to obtain constraints on the parameter space of non-thermally produced
DM. We propose a simple parameterization for non-thermal DM momentum distributions,
present a fitting procedure that can be used to adapt our results to other models, and
demonstrate via some toy models how our results can be applied to other non-thermal
DM models.
Ewig leuchten die Sterne
Und unaufhörlich fällt der Schnee. . .
Der Blick nach oben eröffnet mir
Die Klarheit, Einfachheit und zugleich unendliche Komplexität des Seins
Und ewig leuchten die Sterne. . .
Ewig leuchten die Sterne - Paysage d’Hiver, Das Tor
Then Varda went forth from the council, and she looked out from the height of Taniquetil, and
beheld the darkness of Middle-earth beneath the innumerable stars, faint and far. Then she began
a great labour, greatest of all the works of the Valar since their coming into Arda. She took the
silver dews from the vats of Telperion, and therewith she made new stars and brighter against the
coming of the Firstborn; (..)
John Ronald Reuel Tolkien, Christopher Tolkien (ed.), The Silmarillion, 1977
List of publications
This thesis is based on the following publications and research work. We highlight the
author contribution to each of them.
ˆ S. Baumholzer, V. Brdar, P. Schwaller and A. Segner, Shining Light on the Scotogenic
Model: Interplay of Colliders and Cosmology, JHEP 09 (2020) 136, [1912.08215],
[1]: The foundation of this work was laid in Ref. [2] and thus the introduction
into Chapter III is influenced by chapter V of this reference. All co-authors of [1]
contributed to the text of the manuscript, while the author particularly contributed
the Monte Carlo simulations for the HL-LHC projections and all calculations for
the combined DM and cosmology part of the project. Results for the future collider
studies were derived by Alexander Segner and cross checked by the author. All
plots have been derived by the author, while the exclusion plots based on CLIC and
FCC are based on data from Alexander Segner. The publication is the basis for
Chapters III and IV.
ˆ S. Baumholzer, V. Brdar and E. Morgante, Structure Formation Limits on Axion-
Like Dark Matter JCAP 05 (2021) 004, [2012.09181], [3]: The author performed
all numerical simulations regarding structure formation probes of the model and the
corresponding plots are either made by himself or based on his data. The deriva-
tion of the momentum distribution function for the ALP DM was done in joint
collaboration with Dr. Enrico Morgante and the author crosschecked the results.
All co-authors contributed to the text of the manuscript. Figures not made by the
author are marked as such. Chapter V of this thesis is based on this publication.
ˆ S. Baumholzer and P. Schwaller, Probing non-thermal light DM with structure for-
mation and Neff : All calculations, parameter scans, numerical studies, the corre-
sponding plots and the text were done by the author with advice and corrections
from Prof. Dr. Pedro Schwaller. It is the basis for Chapter VI.
The project is in final preparation for publication.
If not mentioned otherwise, all figures shown in this thesis are done by the author. Figures
produced from co-authors are indicated as such and some of them are modified by the
author.
Contents
Abstract i
List of publications iii
Prelude 1
I Introduction 3
II Theoretical background 7
II.1 The standard model and beyond . . . . . . . . . . . . . . . . . . . . . . . . 7
II.1.1 What we know: the standard model . . . . . . . . . . . . . . . . . . 7
II.1.2 Neutrino masses and mixings . . . . . . . . . . . . . . . . . . . . . . 9
II.1.3 What we don’t know: dark matter . . . . . . . . . . . . . . . . . . . 11
II.2 Cosmology and structure formation . . . . . . . . . . . . . . . . . . . . . . 18
II.2.1 A brief thermal history of the Universe . . . . . . . . . . . . . . . . . 18
II.2.2 Particles in equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 19
II.2.3 Out of equilibrium thermodynamics . . . . . . . . . . . . . . . . . . 20
II.2.4 Cosmological linear perturbation theory . . . . . . . . . . . . . . . . 21
II.2.5 The matter power spectrum . . . . . . . . . . . . . . . . . . . . . . . 22
II.2.6 Impact of warm dark matter on structure formation . . . . . . . . . 23
II.2.6.1 Half-mode analysis . . . . . . . . . . . . . . . . . . . . . . . 25
II.2.6.2 Constraints from the Lyman-α forest . . . . . . . . . . . . 26
II.2.6.3 Counting the number of Milky Way satellites . . . . . . . . 27
II.3 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Main part I: dark matter at current and future colliders 31
III Collider studies of the scotogenic model 33
III.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
III.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
III.3 The model and neutrino mass generation . . . . . . . . . . . . . . . . . . . 36
III.4 Projections for the HL-LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
III.4.1 Di-tau+ET signature . . . . . . . . . . . . . . . . . . . . . . . . . . 39
III.4.2 Di-lepton + ET signature . . . . . . . . . . . . . . . . . . . . . . . . 42
III.5 Projections for future colliders . . . . . . . . . . . . . . . . . . . . . . . . . 44
iv
CONTENTS
III.5.1 FCC-hh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
III.5.2 CLIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
III.6 Summary of Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Main part II: warm dark matter in cosmology: from models to a full
picture 51
IV The scotogenic model and cosmology 53
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
IV.2 Light dark matter in the scotogenic model . . . . . . . . . . . . . . . . . . . 53
IV.2.1 Dark matter production mechanisms . . . . . . . . . . . . . . . . . . 54
IV.3 Cosmological constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
IV.3.1 Constraints from structure formation . . . . . . . . . . . . . . . . . . 56
IV.3.2 Constraints from additional radiation contribution . . . . . . . . . . 60
IV.3.3 Constrains from big bang nucleosynthesis . . . . . . . . . . . . . . . 63
IV.4 Combining collider and cosmological limits . . . . . . . . . . . . . . . . . . 64
IV.5 Summary of Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
V The cosmology of frozen-in axion-like particle dark matter 69
V.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
V.2 The model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
V.2.1 Calculation of the momentum distribution function . . . . . . . . . . 71
V.2.2 Axion-like particle properties and production . . . . . . . . . . . . . 76
V.3 Structure formation probes: Lyman-α forests and Milky Way satellites . . 78
V.4 Suppressing production from misalignment and topological defects . . . . . 81
V.5 Structure formation limits on the model parameter space . . . . . . . . . . 83
V.5.1 Results for the photophilic scenario . . . . . . . . . . . . . . . . . . . 83
V.5.2 Results for the photophobic scenario . . . . . . . . . . . . . . . . . . 85
V.6 Summary of Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
V.7 Appendix of Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
V.7.1 Amplitudes for dark matter production channels . . . . . . . . . . . 91
V.7.1.1 Process A: f γ → f a . . . . . . . . . . . . . . . . . . . . . 91
V.7.1.2 Process B: f f̄ → γ a . . . . . . . . . . . . . . . . . . . . . 92
V.7.2 Calculating the cross section . . . . . . . . . . . . . . . . . . . . . . 93
V.7.2.1 Cross section for process A . . . . . . . . . . . . . . . . . . 95
V.7.2.2 Cross section for process B . . . . . . . . . . . . . . . . . . 95
V.7.3 Impact of Nsub on the mass limits . . . . . . . . . . . . . . . . . . . 96
VI Two temperature dark matter: a general picture 99
VI.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
VI.2 Setup of the framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
VI.2.1 Decay parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 101
VI.2.2 Decay modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
v
CONTENTS
VI.3 Constraints on the model parameter space . . . . . . . . . . . . . . . . . . 104
VI.3.1 Limits from additional radiation . . . . . . . . . . . . . . . . . . . . 104
VI.3.2 Limits from structure formation . . . . . . . . . . . . . . . . . . . . 105
VI.3.2.1 Lyman-α forest data . . . . . . . . . . . . . . . . . . . . . . 107
VI.3.2.2 Number of Milky Way satellites . . . . . . . . . . . . . . . 108
VI.4 Detailed study of the parametrization . . . . . . . . . . . . . . . . . . . . . 108
VI.4.1 Non-thermalized parent particles . . . . . . . . . . . . . . . . . . . . 109
VI.4.2 Impact of a variation in gs∗(T ) during dark matter production . . . . 110
VI.4.3 Three-body decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
VI.5 Analytical fitting of the exclusion limits . . . . . . . . . . . . . . . . . . . . 113
VI.6 Application to sample models . . . . . . . . . . . . . . . . . . . . . . . . . . 115
VI.6.1 Model I: thermalized + out of equilibrium parents, constant gs∗(T ) . 115
VI.6.2 Model II: thermalized + out of equilibrium parents . . . . . . . . . . 115
VI.6.3 Model III: thermalized + frozen-in parents . . . . . . . . . . . . . . 119
VI.6.4 Exploring the cold limit of the model parametrization . . . . . . . . 120
VI.7 Summary of Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
VI.8 Appendix of Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
VI.8.1 Momentum distribution function of out of equilibrium parents . . . 123
VI.8.2 Momentum distribution function of never thermalized parents . . . . 123
VI.8.3 Change in 〈x〉 due to gs∗(T ) . . . . . . . . . . . . . . . . . . . . . . . 123
VI.8.4 Details on fitting procedure . . . . . . . . . . . . . . . . . . . . . . . 123
VI.8.5 Impact of Nsub on the parameter limits . . . . . . . . . . . . . . . . 125
Aftermath 127
VII Summary and conclusion 129
List of figures 140
List of tables 142
List of abbreviations 143
List of experiments and surveys 144
References 165
vi
Prelude
1

Chapter I
Introduction
The standard model (SM) of particle physics is one of the most successful theories ever
developed in the field of physics and is the framework for a wide variety of phenomena.
It was developed and established in the 1960s [4–7] and contains all known particles and
their interactions via the strong, weak and electromagnetic force. One crucial corner-
stone is the Higgs boson [8–12]; a massive neutral scalar responsible for the generation
of particle masses via its spontaneous symmetry breaking. It was a longstanding predic-
tion of the SM and its final discovery at the ATLAS and CMS experiments a the LHC in
2012 [13, 14] marks a triumph of the predictive power of the SM. Besides this, the SM
was further successfully tested in a wide variety of processes at the LHC and other colliders.
On the contrary, despite the confirmation of many predictions of the SM, there are several
pressing open questions whose answers require physics beyond the existing framework,
known as beyond the SM (BSM) physics. On the one hand, these problems arise from
theoretical viewpoints and considerations, such as the hierarchy or naturalness problem
of the different scales within the SM or the question why the strong interaction violates
CP only minimally. On the other hand, there are experimental anomalies and obser-
vations that are not in agreement with the predictions of the SM. Examples include
the observation of neutrino oscillations due to non-vanishing mass squared differences,
the existence of an asymmetry between matter and antimatter, evidence for the exis-
tence of new physics through differences in decays of b-quarks (commonly referred to as
b-anomalies), and last but not least, very strong cosmological evidence for the existence
of dark matter (DM), which is one of the most pressing open questions in physics today.
So far, a wide range of different models has been proposed to extend the SM and address
one or several of these open questions. The discussion ranges from simple SM extensions
with gauge singlet particles to the introduction of entire new sectors with their own parti-
cles and interactions. A popular example for the latter case ar supersymmetric extensions
(SUSY) of the SM. In this thesis we explore the phenomenology of BSM models using the
scotogenic model (ScM) and an axion-like particle (ALP) extension of the SM. In partic-
ular, we investigate the nature of the associated DM candidates. In Chapters III to V, we
focus on these specific SM extensions, while in Chapter VI we present a model-independent
parametrization to study non-thermally produced DM in a more general context.
We will use two complementary probes to test and constrain these BSM models. In the
first direction, collider experiments for both p p and e+e− facilities will be used to search
for signatures of the ScM in final states with missing transverse energy. The second ap-
proach, on the other hand, uses cosmological observations to study the influence of DM
on the evolution of the early Universe and formation of structures.
Following the first avenue, particle colliders are a powerful tool to explore phenomenon
in elementary particle physics at high energies. Colliding highly energetic particles essen-
3
CHAPTER I. INTRODUCTION
tially creates a small “Big Bang” and the observation of the collision remnants allows us
to unravel the physics at high energy scales which are otherwise difficult to access. These
facilities, including the LHC, allow the SM to be measured very accurately and pre-
cisely and thus contribute significantly to our current understanding of particle physics
by revealing numerous particles. Finally, these colliders open a window to the potential
discovery of new physics, because collisions of highly energetic particles allow for a sub-
stantial production of new heavy states. Due to their large masses these new particles
are in general only short-lived, i.e. they decay promptly into SM particles after their pro-
duction. Either the decay products such as leptons or quarks are directly traceable, or
can be only seen indirectly, if they are neutrinos or DM candidates. In addition, there
are other possible decay topologies, for example new particles may travel a macroscopic
distance from the interaction point before they decay.
However, this assumes that a BSM model couples sufficiently strong to the SM and its
particle masses are at least couples of GeV in order to be detectable. If DM is light
and/or only weakly coupled to the SM, or even worse, interacts only gravitationally, it
cannot be discovered at colliders. In fact, despite their great potential, colliders have not
reported any clear observation of new physics and complementary probes are needed to
explore the phenomenology of BSM models. Instead of colliding highly-energetic parti-
cles, we can simply look at the stars instead: cosmological and astrophysical observations
allow us to test DM scenarios which are not accessible at colliders. According to our
current understanding, after its creation, the Universe underwent a phase of exponen-
tial expansion during which it increased dramatically in size. This phase of inflation left
a nearly homogenous and flat Universe which was reheated after inflation ended. After-
wards, it kept expanding and its temperature decreased while it evolved in time. During
its cooling, the Universe went through several important stages. In its early stage, these
are the electroweak symmetry breaking and the quantum chromodynamics (QCD) phase
transitions, as well as the formation of light nuclei. Later, the age of recombination fol-
lows, at the end of which the Universe becomes translucent; even today, we can perceive
this afterglow in the form of the cosmic microwave background (CMB). Finally, the age
of structure formation of galaxies and clusters follows.
Light DM present during some of these formative stages can leave an imprint which is
observable today. For instance, the formation of light nuclei limits the number of rela-
tivistic particle species, as does the CMB which furthermore measures the total amount
of DM. Surveys dedicated to unraveling the distribution of matter can be used to place
constraints on the suppression of structure formation. This includes, for instance, the
number of accompanying Milky Way (MW) dwarf galaxies.
In summary, there is an intriguing interplay between laboratory and cosmological probes
of BSM models, as they offer complementary ways to probe the parameter space of new
physics. Experiments at collider and accelerators can detect new particles either directly
via resonances or indirectly via their effect on observables. Cosmology, on the other hand,
constraints model features in a more general way, such as the mass of a new light DM
candidate or the associated averaged momentum of its production mechanism.
This thesis is organized as follows. We start by briefly reviewing the SM and its open
problems and present an introduction to the standard model of cosmology, the ΛCDM,
and galactic structure formation in Chapter II. Then, in the first part of the thesis, we
present dedicated collider searches. Focusing on two leptons (e, µ or τ) and missing trans-
verse energy in the final state, we derive limits on the ScM in Chapter III. Afterwards, we
turn towards the cosmological impact of light DM. Again, we consider the ScM, but derive
cosmological limits based on the effective number of relativistic particle species, lifetime
4
bounds stemming from the abundance of light nuclei and suppression of small scale struc-
tures. We overlap them with the collider analysis results in Chapter IV. Chapter V is
dedicated to warm axion-like particle DM produced in 2→ 2 scattering and annihilation
processes. We derive two complementary structure formation bounds for the case where
the ALP couples either to photons or fermions only. We generalize our findings in Chap-
ter VI where we develop a model-independent parametrization to consider the impact of
DM produced at different times on the formation of galactic structures. This allows to eas-
ily adapt our approach for more specific models of light and “warm” DM. We summarize
and conclude our results in Chapter VII.
5

Chapter II
Theoretical background
II.1 The standard model and beyond
Our journey begins with a brief review of the standard model of particle physics (SM).
For a more comprehensive study we would like to point the reader to existing literature
[15–18] and references therein. Following this chapter, we present another short review on
the cosmological standard model and proceed with discussing open questions beyond the
standard model (BSM), which are the main motivation behind this thesis.
II.1.1 What we know: the standard model
The standard model of particle physics, or in short standard model1, emerged in the 70s
after a stormy decade full of discoveries and newly found particles. It was established
by the groundbreaking work of many people in the physics community [4–7] in order to
structure and organize the “zoo of particles” which puzzled particle physicists for decades.
Following its first formulation nearly all of its forecasts have been confirmed so far in a
wide range of experiments and observations. The end of this road is marked by the im-
portant discovery of the so-called Higgs particle, the last cornerstone of the SM, by the
two experimental collaborations ATLAS [13] and CMS [14] at the LHC in CERN.
The SM is a renormalizable quantum field theory which describes all known particle species
and the three fundamental interactions among them: strong, weak and electromagnetic
force, while gravity is not included in this framework. The backbone of the SM is the
gauge group SU(3)C×SU(2)L×U(1)Y , which contains a SU(3)C color group of quantum
chromodynamics (QCD) and the unification of the electroweak force in terms of a SU(2)L
subgroup for the weak isospin and a hypercharge U(1)Y subgroup. The fermionic particle
content of the SM is given by the following gauge group representations and charges:
SU(3)C SU(2)L U(1)Y
LH quarks QiL 3 2
1
6
RH up-type quarks uiR 3 1
2
3
RH up-type quarks ui 3 1 −1R 3
LH leptons `iL 1 2 −12
RH charged leptons eiR 1 1 −1
Here, i = 1, 2, 3 labels the three copies of each representation. The SM is a chiral gauge
theory because left- and right-handed fermions are living in different gauge representations
1 Not to be confused with the cosmological standard model which will be explained in the next section.
7
CHAPTER II. THEORETICAL BACKGROUND
of the SU(2)L × U(1)Y group. The right-handed (RH) fermions are singlets, while the
left-handed (LH) particles are grouped in doublets, Q = (ui , di TL L L) and `L = (ν
i , eiL L)
T .
Besides the fermions, the SM contains four additional bosons which acts as the interaction
mediators. The strong force is mediated by eight color carrying gluons, while the elec-
troweak force is carried by a charged W± boson and 2 neutral bosons, the Z boson and
the photon γ.
Finally, the SM contains a SU(2)L doublet scalar field H, which transforms as (1, 2, 1/2)
under the SM gauge group and is commonly called the Higgs doublet. Its existence makes
life much more interesting, because it acquires a vacuum expectation value (VEV) and
spontaneously breaks the group SU(2)L×U(1)Y down to electromagnetism (EM) U(1)EM.
This mechanism is the so-called Higgs mechanism which is crucial for the generation of
mass terms for the otherwise massless SM particles [8–12]. The Higgs boson acts as an
ultraviolet (UV)-completion of the electroweak theory, because it guarantees renormaliz-
ibility at energies E & 1 TeV. The co(mplex Higgs double)t can be decomposed as following,
G+
H = √1 , (II.1)(h+ v + iG )
2 0
where v ≈ 246 GeV is the VEV of the Higgs, h the physical real Higgs scalar and G+, G0
are massless Goldstone bosons which become the longitudinal modes of the gauge bosons
W± and Z after electroweak symmetry breaking (EWSB). They acquire a mass term
induced by their derivative coupling to the Higgs boson after EWSB, while the photon is
the only remaining massless boson.
Additionally, the introduction of the Higgs doublet allows us to write down gauge invariant
Yukawa interactions between LH and RH fermions,
LYuk = −(ye) ¯̀iij LHejR − (yu) iijQ̄LH̃u
j
R − (yd)ijQ̄i Hd
j
L R + h.c. , (II.2)
where the (yk)ij are in general complex 3 × 3 Yukawa matrices. For the up-type quark
mass term, the quantity H̃ ≡ iσ H∗2 , where σ2 is the second Pauli matrix has to be used
due to project out the correct component of the QiL doublet for up-type quarks. After the
Higgs gains a VEV, fermions acquire a mass via above Yukawa interactions. Substituting
the SU(2)L doublets, the following mass terms can be derived:
LYuk ⊃ −(M ) ēi je ij LeR − (M iu)ij ūLu
j
R − (Md)ij d̄LdR + h.c. , (II.3)
√
where mass matrices Mf = yfv/ 2, (f = u, d, e) are introduced. They can be diagonalized
using a singular value decomposition by independently rotating the LH and RH fields by
unitary 3×3 matrices and one is left with nine real mass eigenstates (six quarks and three
charged leptons) whereas the three neutrinos, or neutral leptons, stay massless in the SM.
In the quark sector, the mass matrices can be diagonalized by transforming to the basis
(u′i ′iL dL),
u′i = U ijuj , d′i = U ijdjL u L L d L . (II.4)
In this basis, all SM quark currents except the W± boson current, JµW , remain unchanged:
it transforms to
µ √1
( )
J = ūi γµdi
1 ′i µ † ′i 1 ′i µ ′i
W L L ⇒ √ ūLγ UuUd dL ≡ √ ūLγ VijdL, (II.5)2 2 2
where the unitary mixing matrix Vij has been introduced, it is the Cabibbo-Kobayashi-
Maskawa (CKM) matrix [19, 20]. Its off-diagonal entries are responsible for transitions
between different quark families mediated by charged W bosons.
8
II.1. THE STANDARD MODEL AND BEYOND
As a unitary matrix, V has 32 = 9 real parameters: six phases and three rotation angles.
However, of these five phases can be absorbed by rotating the quark fields by global phases
and only one CP violating phase is left.
On the contrary, the lepton sector does not contain such a mixing matrix, because there is
no mass term for the neutrinos νiL and this additional freedom allows us to transform the
LH neutrinos in the same way as eiL and so the would-be mixing matrix simply gives an
identity matrix. Only after the νiL acquire a mass term via some mechanism beyond the
SM is a leptonic mixing matrix, the so-called Pontecorvo-Maki-Nakagawa-Sakata matrix
(PMNS) generated [21–23]. We will discuss ways to achieve a mass term for the neutrinos
in Section II.1.2.
II.1.2 Neutrino masses and mixings
As pointed out in the previous section, there is no way to generate a gauge invariant mass
term for the neutrinos in the SM at renormalizable level and they are considered to be
massless particles. However, measurements of the electron neutrino flux from the sun
already done in the late 1960’s observed a flux deficit compared to the SM prediction [24].
This observation was later attributed to oscillations among the three neutrino states and
confirmed by an independent measurement of atmospheric neutrinos [25]. While these
experiments established that there are tiny, but non-zero, mass differences between the
individual neutrino states, the absolute mass scale is still unknown and only a upper
mass limit exist. Recent measurements of the end point of the energy spectrum of β-
decay electrons by KATRIN using kinematic methods have brought the limit down to
mν < 1.1 eV (90% CL) [26]. Further, observations of the cosmic microwave background
(CMB) power spectrum and baryonic acoustic oscillations b∑y the Planck collaboration
point to even lower values on the sum of neutrino masses,2 mν < 0.12 eV (95% CL)
[27]. Another experimental method is to search for neutrinoless double-beta decays, which
places a limit on the effective mass down to mββ ≡ |U2 ieimν | < 0.11 eV (see Ref. [28] for an
overview), where U is the PMNS matrix.
Another open question is the ordering of neutrino masses. Since only the absolute size of
two mass differences are known at the moment, there are two ways to arrange them. One
is called the normal ordering where the two lighter neutrinos are close in mass and the
heavy neutrino is separate. Flipping the mass spectrum such that the two heavier states
are nearly mass degenerate is referred to as inverted ordering. At the moment, there is
only a mild preference for the normal ordering scenario [29], but the exact nature is still
under debate, and future experiments as the long baseline experiment DUNE are expected
to settle the ordering discussion [30].
Neutrinos do not carry a charge under the SM gauge group and hence they are the only
known particles which could have in principle a Majorana nature and this would allow for
a Majorana mass term of the form (dropping flavor indices):
LMm ⊃ −
1
m cLν̄LνL , (II.6)2
where the charge conjugation matrix C = iγ2 has been used and νc ≡ CνTL L . However, such
an expression would break the global total lepton number of the SM by two units. There
are ongoing efforts to look for neutrinoless double beta decays to look into the nature of
neutrinos (see Ref. [31] for an overview).
Although there is no renormalizable expression to generate neutrino masses within the SM,
2 Although this mass limit is one order of magnitude stronger than the KATRIN limit, it is not com-
pletely model independent but relies on assumptions about the cosmological history. In that sense the
kinematically derived mass limit is more robust.
9
CHAPTER II. THEORETICAL BACKGROUND
they can be created by a unique non-renormalizable dimension five operator, the Weinberg
operator [32], by combining two bilinears `H in a gauge invariant way, Λ−1(¯̀cLH̃
∗)(H̃†`L).
The scale Λ corresponds to a scale where new physical degrees of freedom have to be
incorporated in the theory as well.3 A simple way to explicitly realize the Weinberg
operator at tree-level is the introduction of SM gauge singlets or RH neutrinos NαR, α =
(1, . . . n). This allows mass terms analogous to the Higgs mechanism by coupling the SM
neutrinos with these new states by a new Yukawa coupling yν ,
L ⊃ −(y ) `i H̃NαBSM ν iα L R. (II.7)
Similar to charged leptons, afte√r electroweak symmetry breaking, this term gives rise to a
Dirac mass (mD)iα ≡ (yν)iαv/ 2 for the νiL, but the smallness of the neutrino masses of
O(1 eV) requires puzzingly small couplings (y ) ≈ 10−11ν iα , way smaller than the electron
Yukawa couplings.
Including Majorana mass terms for the RH neutrino states paves the way to explain the
tiny neutrino masses via the so-called (type-I) seesaw mechanism [33–35]. In this scenario,
the SM Lagrangian is extended by the following terms
LBSM ⊃ −
1
(Yν)
i
iα`LH̃N
α
R − (MR) α β cαβN̄R(NR) + h.c. , (II.8)2
where MR is a n × n Majorana mass matrix for the NαR. In a basis (νc , N )TL R the mass
terms can be re(written i)n(a compact)fo(rm a)s1 0 mD νc 1 ( ) ( )νcLm ⊃ − ν̄ N̄ c LL R T ≡ − ν̄ c Lm M N L N̄2 2 R M + h.c. . (II.9)D R R NR
For the sake of simplification flavor indices have been omitted. Diagonalizing this mass
matrix gives rise to 3+n Majorana neutrinos and its off-diagonal element mD introduces a
mixing among them. Now, the smallness of the νL can be explained by “seesawing” them
with a large right-handed neutrino mass MR. In fact, in the seesaw limit, if one sets mD
to be at the weak scale while MR is much larger, there are 3 light neutrinos with masses
m ' −m M−1mTν D R D and n heavy neutrinos with masses ∼ MR, while the mixing among
the light and heavy states are governed by a mixing angle θ ' m −1DMR  1. In order
to reproduce the experimental observed neutrino mass squared differences, at least two
right-handed neutrinos are needed with masses of Mα 14R ∼ 10 GeV, but apart from that
there is no upper limit on their quantity n. Another attractive feature is the possibility
to explain further problems of the SM within this framework. For example, the heavy
right-handed neutrinos can be used to explain the observed matter-antimatter asymmetry
in the Universe by the so-called leptogenesis mechanism [36, 37], a topic to which we will
return briefly in Chapter III.
Besides the type-I seesaw, there are other possibilities to realize the Weinberg operator.
For instance, in a similar spirit as the type-I, one can add a scalar SU(2)L triplet instead
of SM gauge singlets to generate neutrino masses as in the type-II seesaw [34, 38, 39] or
employ a type-III seesaw scenario which comes with new fermion SU(2)L triplets [40, 41].
Usually these tree-level realizations require heavy new states and/or small couplings to
the SM, a fact which makes it hard to look for these new particles in experiments. An
interesting alternative is to add new particles and interactions such that neutrino masses
are generated radiatively at one-loop or even higher (see Ref. [42] for an overview). Because
of additional loop suppression the new particle masses do not have to be extremely large
and hence one can exploit already existing experiments to test this class of neutrino mass
models. As an example we will discuss the so-called scotogenic model in Section V.2.
3 In a similar way, the β-decay can be described by an effective four-fermion operator at low scales by
integrating out the weak scale particles.
10
II.1. THE STANDARD MODEL AND BEYOND
II.1.3 What we don’t know: dark matter
Another important, if not the most important, question unanswered in the SM is the ex-
istence of dark matter (DM), and answering this puzzle is one of the cornerstones of this
thesis.
The nature of DM is a long standing open question with a long history spanning nearly a
century (for a review on this topic we refer to Ref. [43]). Hints or indications of DM could
be found on a wide range of length scales [44]. There are ideas that the observed phe-
nomena can be explained by a modification of general relativity (GR). The most famous
of these theories is the Modified Newtonian Dynamics model (MOND): it is an alteration
of standard GR at large scales, such that the observed galactic rotation curves can be ex-
plained without the existence of DM. Currently, this theory is still lacking a conclusive
general picture4 and it is not easy to generalize this theory to incorporate relativistic ef-
fects.
Therefore, it is established that DM is a non-luminous type of matter which interacts
at least gravitationally. Furthermore, it cannot be of baryonic origin, as gravitational
microlensing searches for faint massive compact halo objects (MACHOs)5 composed of
ordinary matter found that only a small part of the observed DM amount could be in
form of MACHOs [45, 46]. Another possible DM candidate are primordial black holes
(PBH), which could explain the observed DM abundance, only if they are very light or
very massive (see Ref. [47] for a recent overview). We know that DM make up 27% of to-
day’s energy budget of the Universe, whereas only 5% is visible matter and the rest is
composed of “dark energy”, an even more mysterious component responsible for the ac-
celerated expansion of the Universe. Taken together, dark energy expressed in terms of
a cosmological constant Λ, cold dark matter (CDM), ordinary matter and radation form
the building blocks of the cosmological standard model, the ΛCDM.
First evidence for DM had been found by Zwicky in 1933 when he measured the veloc-
ity dispersion of the Coma cluster and found higher values than expected from luminous
matter only [48]. On smaller galactic scales, the indication for DM was found in 1970
by Rubin and Ford who measured the circular velocity profile of stars and gas clouds,
or rotation curve, of the Andromeda galaxy [49]. They observed an unex√pected behav-
ior of galactic rotation curves at large radii: instead of the expected 1/ r, the curves
tend to stay flat for a wide range of radii, indicating more matter than visible in the
outer parts of galaxies. Further evidence was gathered in observations of the Bullet clus-
ter which consists of two colliding galaxies [50, 51]. While the luminous parts of both
galaxies were heavily affected and slowed down by the collision, gravitational lensing in-
dicated that the bulk mass remained unaffected. Lastly, DM leaves an imprint on the
largest scales as well, as can be inferred from anisotropies in the power spectrum of the
cosmological microwave background (CMB). A study of these anisotropies allows to de-
termine the total amount of cold dark matter in the Universe, Ω h2DM = 0.120 ± 0.001
[52]. Here “cold” means that the DM becomes non-relativistic before the epoch of struc-
ture formation, hence allowing for a fast collapse of DM clumps and birthing seeds for
the latter formation of baryonic structures. If the DM stays relativistic at the begin-
ning or during the early formation of structures, it will has finite free-streaming length
below which small scale formation is delayed or completely erased. Such DM candi-
dates are called “warm” or even “hot” DM and structure formation observables can
place constraints on the “warmness” of DM as we will discuss in Section II.2.6. For
example SM neutrinos can be considered as “hot” DM, but it turned out there are
4 Fitting a variety of different rotation curves does not give one universal parameter, but rather one unique
parameter for each galaxy observed.
5 Examples include black holes, brown dwarfs, neutron stars and old white dwarfs.
11
CHAPTER II. THEORETICAL BACKGROUND
too few of them to explain Ω h2DM and furthermore it is not even possible to have
neutrinos as DM, because this scenario would radically alter the formation of struc-
tures of small scales and hence light SM neutrinos can only constitute a subfraction of DM.
Although we have surprisingly exact knowledge about how much DM there is, we do not
know what it is. We only know that DM has a gravitational interaction, is stable or has a
lifetime larger than the age of the Universe, and is electrically neutral or at most, it could
only carry a small fraction of the electric charge. There exists a large plethora of different
DM models, each with its own set of DM candidate(s) and production mechanisms to
reproduce the observed DM abundance and match the cosmological and astrophysical
observations. In fact, the mass range of potential DM candidates spans several orders of
magnitude. An upper bound of mDM < 10
18M [47], where M ≈ 2 × 1030 kg denotes
a solar mass, can be applied for DM in form of PBHs. From a particle physics point
of view the upper mass bound for a DM particle is the Planck scale ∼ 1019 GeV. On
the other hand, lower mass bounds depend on the type of particles. Bosonic DM can be
ultralight as long as the corresponding De Broglie wavelength λ ∼ 1/mDM is smaller than
the size of dwarf galaxies. Observations of structure formation and the Lyman-α forest
indicate a bound of mDM ≥ 10−21 eV [53]. On the other hand, the mass of fermionic
DM is more severely constrained due to the Pauli exclusion principle blocking phase space
occupation numbers for fermions, the so-called Tremaine-Gunn bound limit [54]. Analysis
using Lyman-α data from dwarf galaxies yields mDM ≥ 70 eV [55]. Even stronger mass
limits, based on Lyman-α observations, can be derived for thermal DM whose mass must
not fall below mDM ≥ 5.3 keV [56], otherwise the structure formation on small scales will
be too strongly affected. We will explain in particular this last mass limit in Section VI.3.2
in more detail.
In the following we briefly present a selection of possible DM candidates which are of
importance for the considered mechanisms and models in this thesis.
ˆ WIMPs
One simple model to answer the DM puzzle introduces a weakly interacting massive par-
ticle (WIMP) as a DM candidate which couples to some SM particles. Within this frame-
work, DM thermalizes in the early Universe, because its coupling with SM particles keeps
DM number density changing processes in equilibrium with each other. At some later
point these processes become inefficient when their respective interaction rates are becom-
ing smaller than the expansion rate of the Universe, characterized by the Hubble rate.
Finally, the DM decouples chemically from the thermal plasma and its comoving num-
ber density stays constant until today. This DM production mechanism is called thermal
freeze-out and it will play a role for DM production in Chapters IV and VI. Boltzmann
equations are used to track the time evolution of the corresponding DM number density
[57] which evaluates approximately to [5(8, 59]× − − )6 10 28 cm3 s 1
Ω h2DM ' 0.12 〈 〉 xf , (II.10)σv
where 〈σv〉 denotes the corresponding thermally averaged cross section of the number
density changing processes and xf = mχ/Tf is the time at which DM drops out of equi-
librium, or “freeze-out”; for WIMPs it evaluates to xf ' 20. This numerical result for the
DM abundance is the origin of the so-called “WIMP miracle”: WIMPs with masses be-
tween 10 GeV and a few TeV and approximately weak scale cross sections reproduce the
observed DM abundance and are ideal candidates for CDM. For example, in supersym-
metry (SUSY) models the lightest supersymmetric particle can constitute a WIMP DM
candidate if it does not carry an EM charge.
12
II.1. THE STANDARD MODEL AND BEYOND
In terms of experimental evidence, WIMPs have the pleasant property that there is a
close relationship between the process responsible for thermal freeze-out and other open
avenues to detect these kind of particles in experimental setups or observational surveys,
and in the last few decades there had been tremendous experimental progress to search
for WIMPs. The three different ways to detect WIMP DM are sketched in Fig. II.1 and
they can be formulated flatly as “shake it, make it or break it”. The first process refers
to direct detection by elastic or inelastic DM scattering off heavy atomic nuclei or elec-
trons inside detector materials (see Ref. [60] for an overview). The recoiling scattering
partners would than leave a trace inside the detector which can be observed [61]. At the
moment, leading limits are set by the XENON1T experiment which uses liquid Xenon as
their target medium. It can probe WIMP DM with masses between a few GeV up to
1 TeV and excludes spin independent nucleon cross sections down to σSI & 4× 10−47 cm2
[62]. Probing WIMP DM at smaller masses require different experimental setups and
analysis techniques. For example, doing a S2 only analysis,6 masses down to 60 MeV can
be probed by XENON1T measuring electronic recoils [63]. Even lower masses can be
reached by searches for electron recoils in cryogenic solid-state detectors and experiments
like CRESST-III [64] and SuperCDMS [65] may reach masses down to mDM ' 1 MeV. Fi-
nally, the sub-MeV mass range can be explored by experiments like SENSEI, which uses
a skipper-charged-coupled-device to search for low energy electron recoils [66].
Indirect detection of DM searches for SM remnants, like photons, neutrinos, e± or protons
and antiprotons, produced in regions with a high DM concentration by the same anni-
hilation processes that also fix the DM relic density. These annihilation channels could
produce distinctive signatures such as highly energetic neutrinos produced in the sun, or
gamma-rays from the center of the Milky Way (MW). For instance, IceCube searched
for high energy neutrinos coming from DM annihilation processes in the sun [67] or the
galactic core [68]. Dedicated DM annihilation gamma-ray searches were performed by
Fermi-LAT, H.E.S.S. and MAGIC, observing dwarf spheroidal galaxies of the MW [69, 70]
or its galactic center [71].
Lastly, inverting the annihilation process, DM can be pair-produced by accelerating and
colliding highly energetic SM particles. After their production, the DM particles escape
the detectors without any visible trace, leaving a missing energy signature which can be
measured. Hence, dedicated DM searches at colliders exploit recoiling SM particles such
as photons [72, 73] or jets [74, 75].
However, despite all the experimental efforts so far, no conclusive positive WIMP DM
Direct detection
  
Figure II.1: The trinity of DM detection: production of DM with highly energetic beams
at colliders, direct detection of DM scattering events inside detectors and searches for
decay products from DM annihilation processes. The circle in the middle represents an
unspecified effective coupling between two DM particles χ and SM particles.
6 Here, S2 refers to a secondary scintillation signal, in contrast to the prompt signal S1. Most analysis use
both signals for a better discrimination between DM candidates and background events.
13
Indirect detection Production
CHAPTER II. THEORETICAL BACKGROUND
signals had been found and there exist strong exclusion limits on the masses and interac-
tions of this DM candidate. Although it does not mean that WIMP DM can be considered
“dead” or ruled out, it is definitely challenging the WIMP paradigm and new DM mod-
els and production mechanisms are studied going beyond the thermal freeze-out scenario.
Interesting alternatives include for example a non-thermal DM production by decays or
scatterings of thermal bath particles [76], production by decaying topological effects [77]
or the introduction of entire dark sectors with new number-changing processes [78, 79].
Some of these beyond-WIMP avenues will be explored in this thesis.
ˆ Sterile neutrinos
As already pointed out in Section II.1.2, right-handed neutrinos are a well-motivated
extension of the SM. They are gauge singlets and therefore do not weakly interact, hence
they are often called “sterile” neutrinos in contrast to the “active” LH SM neutrinos. In
addition to their role in neutrino mass generation, they may also serve as an interesting
DM candidate, an idea which was already put forward over two decades ago [80] (see
Refs. [81, 82] for an overview).
A simple sterile neutrino production is the so-called Dodelson-Widrow mechanism [80]. It
is a freeze-in mechanism, i.e. compared to the previously discussed WIMP production, this
DM candidate is only weakly coupled and does not thermalize with the SM plasma in the
early Universe. Its abundance is not determined by a decoupling from the plasma but is
instead produced via a small mixing ϑ between the sterile and the active neutrino states.
Thus larger couplings yields larger DM abundances, as opposed to a freeze-out, where
stronger couplings keeps DM longer in thermal equilibrium and deplete the abundance.
If a SM neutrino flavor eigenstate is produced in the early Universe, it will contain an
admixture of a sterile state proportional to ϑ. As it travels, it can scatter coherently with
weakly charged particles and by doing so it acquires an effective mass and thus an effective
mixing angle based on the vacuum angle ϑ arises [83],
∆22 (p) sin
2(2ϑ)
sin (2ϑm) = 2 − − . (II.11)∆2(p) sin (2ϑ) + (∆(p) cos(2ϑ) V D V T (p))2
Here, ∆(p) = ∆m2/(2p) is set by the vacuum mass splitting between neutrino mass
eigenstates, and matter effects are separated into finite density and temperature potentials,
V D and V T . If the SM neutrino experiences a energy or momentum changing scattering,
i.e. a measurement, the sterile neutrino eigenstate will be projected out with a probability
∝ sin2(2ϑm). If there is no resonant enhancement, the resulting sterile neutrino abundance
will be set by the relation [83] ( )
2 ≈ sin
2(2ϑ ) ( )m ms 2
ΩDMh 0.3 . (II.12)
10−10 100 keV
In order to explain the relic density with not too large mixing angles, sterile neutrinos
need a mass of at least O(keV) which is also required by the aforementioned Tremaine-
Gunn bound.7 The same mixing angles also controls back-reactions of sterile neutrinos.
Precisely, they decay mainly into three neutrinos at tree level, νS → ννν, and their lifetime
τ is given by [82] ( )
10 keV 5 1
τ = Γ(ν −1 14s → 3ν) = 1.5× 10 s , (II.13)
m 2s ϑ̄
7 An anomalous neutrino flux excess in short baseline experiments may hint to an additional sterile neutrino
state with mass O(eV) [84]. Although these might be a potential sign for new physics, the best fit
parameters for this sterile neutrino are in conflict with cosmological observations, i.e. they give a too
large mixing angle. New measurement results published by the MicroBooNE experiment, however, could
not verify this anomalous excess [85].
14
II.1. THE STANDARD MODEL AND BEYOND
∑
where ϑ̄2 ≡ α |ϑα|2. Requiring it to be larger than the lifetime of the Universe, τU ≈
4.3 × 1017 s, places an upper bound on the mixing angle. Stronger constraints can be
derived from sub-dominant radiative decays, νs (→ ν γ, wh)ose decay rates are given by[86–88]
→ × −29 −1 sin
2(2ϑ) ( m )s 5
Γ(νs γ ν) = 1.38 10 s − . (II.14)10 10 keV
It would leave a potentially visible peak in the gamma-ray spectrum at energies half the
DM mass [89]. Dedicated surveys have been done to look for signals in the energy range
between keV and MeV, such as XMM-Newton [90], Suzaku [91], Fermi-LAT [92] and NuS-
tar [88]. Interestingly, several experiments have found an unidentified line feature at an
energy of ∼ 3.55 keV in the gamma-ray spectra of galaxy clusters [93], the Andromeda
[94] and the MW galaxy [95] with a formal statistical significance above 5σ [94]. This
sparked some interest in keV-scale neutrino DM, because it can be attributed to decays
of sterile neutrinos with mass 7.1 keV and an accordingly chosen mixing angle. However,
this line could not be found in all surveys looking at DM-dominated regions of space and
up to date there is no conclusive interpretation, since this gamma-ray line could origin
from atomic transitions in gas clouds instead (for a detailed discussion see Ref. [96] and
Refs. [97, 98]). The next generation of gamma-ray experiments will definitely shed more
light onto this question.
Nevertheless, ongoing and past gamma-ray surveys together with bounds from structure
formation have placed strong constraints on the mass-mixing angle plane and the sim-
ple Dodelson-Widrow mechanism is now completely ruled out in the favored mass range
[88, 99]. One way to circumvent this gamma-ray bounds is to introduce an effective back-
ground caused by a lepton asymmetry to allow for an resonantly enhanced production
rate even with small mixing angles [100], similarly to the Mikheyev-Smirnov-Wolfenstein
(MSW) [101, 102] seen in neutrino oscillations inside matter. If sterile neutrinos are added
with an entire dark sector, there is the possibility to protect them from decaying into vis-
ible gamma-rays by charging them and the other dark sector particles under a discrete
symmetry, for instance a Z2 [2, 103]. An interesting model is the neutrino minimal stan-
dard model (νMSM) [104, 105]. It extends the SM by three additional RH neutrinos and
two of them explain the observed mass squared differences of the active neutrinos and pro-
vide an explanation for a non-vanishing lepton asymmetry, while the third one serves as
a keV-scale DM candidate.
A non-thermal production of sterile neutrino DM is also possible via decays of parent par-
ticle, either while they are in thermal equilibrium or at later times and already frozen-out.
Compared to the production via active-sterile mixing, a decay production allows for larger
sterile neutrinos masses even above the electroweak scale. For instance they are produced
by decays of SM particles, such as pions [106], the Higgs boson [107] or vector bosons
[108], but these production modes are subdominant compared to the freeze-in via afore-
mentioned mixing angle. Enhancing this idea further, new models with particles coupled
to sterile neutrinos have been proposed in order to achieve a sizeable production via de-
cays. Examples include a scalar singlet coupled via a Yukawa interaction to νs [109–111]
or an additional Higgs doublet [112, 113]. We will explore and discuss an example for the
latter possibility in more details in Chapter III.
If DM is made up by keV-scale sterile neutrinos, this would indicate a deviation from
ΛCDM, because the DM is not necessarily cold anymore and would be called “warm” DM
(WDM) (see for example [80, 100, 114]). In contrast to CDM, WDM is relativistic when it
is produced, for instance by decays of thermal bath particles, but becomes non-relativistic
before matter-radiation equality. This property changes the formation of structures in the
early Universe, as relativistic WDM do not clump together at these early times as com-
pared to non-relativistic CDM and hence structures are washed out below a non-vanishing
15
CHAPTER II. THEORETICAL BACKGROUND
free-streaming length [58, 115],
∫t ′ ( )
′ v(t ) ≈ 1 keV TDMλFS = dt 1 Mpc , (II.15)
a(t′) ms T
ti
where TDM is the corresponding WDM temperature.
8 There are several options to search
for the impact of WDM on the formation of structures: first, a measurement of the matter
power spectrum at different scales, for instance using the so-called Lyman-α forest or the
21 cm absorption line spectroscopy, counting the number of observed collapsed structures
such as dwarf galaxies (or quasars) and compare them against WDM predictions. Last
but not least, one can use the matter distribution inside DM dominated objects to place
constraints. We will elaborate further on these techniques in Section II.2.
ˆ Axions and axion-like particles
Another way to realize DM is via axion-like particles (ALP). Since they are neutral scalars
or pseudo-scalars, the Tremaine-Gunn bound does not apply for them and they can ex-
plain the DM relic density with even lighter masses compared to sterile neutrinos. They
can be associated to a spontaneous breaking of a global U(1) symmetry. It gives rise
to massless Nambu-Goldstone bosons (NGB) and if there is a small explicit symmetry
breaking, they acquire a mass and are called pseudo-NGBs. Their coupling to the SM is
inversely proportional to a decay constant f which in turn is linked to the scale of symme-
try breaking. Usually the symmetry breaking takes place at very high scales and therefore
these light states couple only weakly to the SM. Usually, the expression “axion” is used
for the pseudo-NGB associated to the spontaneously broken Peccei-Quinn U(1)PQ sym-
metry which is a possible solution to the strong CP problem in QCD. This problem is
a matter of naturalness: the QCD Lagrangian allows to define a CP violating term, the
so-called θ term, of the following form,
2
L g= θ s G G̃µνaθ µν,a , (II.16)
32π2
where Gµν/G̃µν is the field strength tensor and its dual. Although the term can be
written as a total derivative, it has to be kept in the Lagrangian, because it is related
to the QCD vacuum structure and corresponding instanton effects. Due to the chiral
anomaly, the quark fields can be rotated q → eiαγ5q and this changes the θ value into
θ̄ = θ + arg det(Mq); a quantity which can be physically measured in the electric dipole
moment (EDM) of the neutron [116, 117]. A recent measurement of the EDM indicates
that θ̄ < 5× 10−12 (90% CL) [118], a puzzingly small number compared to the technical
natural choice θ ≈ O(1). A solution is to introduce the axion, a dynamical pseudo-scalar
field φa characterized by its decay constant fa. It features an additional coupling to GG̃
such that Eq. (II.16) is modified to ( )
2
L gs φaθ = θ̄ + G µν,aµν,aG̃ . (II.17)
32π2 fa
The axion(field φa i)s the NGB of a broken U(1)PQ symmetry [119, 120]. Including theeffects of the QCD anomaly, there is an effective potential for the axion field given by
V ∼ cos θ̄ + 〈φa〉eff f [121], that means the effective potential is minimized if the axiona
8In principle, for scenarios with only a single DM production mechanism, the free-streaming length can be
used to quantify the “warmness” of the DM candidate. For example, DM with λFS < 0.01 Mpc can be
designated as “hot” and with lengths above as “warm”. In the following, however, we will not make use
of such a definition because the models we will study feature multiple DM production mechanisms.
16
II.1. THE STANDARD MODEL AND BEYOND
acquires a VEV that cancels the θ̄ exactly. This explains the observed smallness of the
CP violation in the strong sector.
The corresponding axion mass, ma, calculat(ed in chira)l perturbation theory is given by[122, 123]
≈ 10
9 GeV
ma 6 meV , (II.18)
fa/C
where the integer valued C is the color anomaly of the PQ symmetry. The decay constant
and mass of a QCD axion are related to each other and fixed by Eq. (II.18). On the
contrary, generalized ALPs do not have to fulfill this relation and they have a larger
available parameter space for fa and ma which can be tested and constrained by a plethora
of various laboratory, astrophysical and cosmological probes (see Ref. [124] for an overview)
by exploiting that ALPs couple to SM particles via the following Lagrangian
L −caXX µν caff= X X̃ φ µa µν a + ∂µφaf̄γ γ5f , (II.19)
4fa fa
where f denotes a SM fermion and Xµν is an arbitrary field strength tensor. For exam-
ple, there are dedicated collider analysis searching for heavy ALPs decaying into a pair
of SM particles or missing energy signatures due to ALPs in the final state [125]. “Light
shining through a wall” experiments such as the Any Light Particle Search (ALPS) [126]
use strong magnetic fields to convert photons into axions which then travel freely through
a wall and are converted back into photons. Another set of tools are helioscopes such as
CAST [127]. They rely on axion production inside the sun and converting them into a pho-
ton in a magnetic field at the earth. Recently, low-energy electronic recoil data released
by XENON1T has shown an excess of recoil events at energies between 2 − 3 keV [128].
One explanation are solar axions produced in the sun, traveling to the detector and recoil-
ing against electrons inside it [128, 129], but the best fit parameters for this models are
severly in tension with stellar constraints [130] and the excess may be caused by a contam-
ination of the target material with unaccounted radioactive sources. Other astrophysical
bounds stem from the ratio of horizontal branch (HB) stars to red giants in galactic glob-
ular clusters [131]. Axion conversion inside star cores would lead to an energy loss and an
observable lifetime of HB stars. All these examples do not rely on the assumption that
all of DM consists of ALPs. If DM consists of an ALP, there will be more experimen-
tal constraints on the parameter space, for example from haloscope searches [132]. They
use a cavity to resonantly enhance the conversion rate of an axion into a photon which
can then be detected. Similarly to sterile neutrinos, ALP DM can leave a detectable sig-
nal in gamma-ray spectra from decays φa → γγ. Finally, analyzing the acoustic peaks
and anisotropies of the CMB matter power spectrum constrain ALPs at really low masses,
m . 10−25a eV [133, 134].
The details of ALP DM production, for instance if it is produced during or after inflation,
depends on the specific model, but a convenient way to produce ALP DM non-thermally
is via the misalignment mechanism [135]. In the early Universe, the axion field is elon-
gated at some initial value φia and frozen due to strong Hubble friction. Once the Hubble
rate become comparable to the axion mass, it starts to roll down its potential and os-
cillates around its minimum. This oscillations give the same behavior as non-relativistic
matter and hence, despite its low mass, this axion DM can be considered CDM. If the PQ
symmetry is broken before the end(of infla)tion,(the axion )abundance is given by [136]1/2 i
Ωah
2 ≈ ma faφ0.12 a F(T1) , (II.20)
54 eV 1011 GeV
where F(T1) is a smooth function of O(1) evaluated at temperature T1 set by the criterion
3H(T1) ≡ ma. Other non-thermal mechanisms to produce ALP DM will be discussed in
Chapter V.
17
CHAPTER II. THEORETICAL BACKGROUND
II.2 Cosmology and structure formation
Over the past decades there had been a tremendous progress in understanding the evolu-
tion of our Universe from its hot epoch shorty after the big bang until its much cooler state
today. All this information is condensed in one single model, which can explains several
key observations such as the existence of the cosmological microwave background (CMB),
the formation and abundance of light nuclei (like deuterium, Helium and Lithium), the
observed large scale galactic structure or distribution of matter and the accelerated ex-
pansion of the Universe. This model is the ΛCDM model, and it is the simplest model
to account for these observations. This model describes the Universe as almost flat, ho-
mogeneous and expanding by a Friedmann-Robertson-Walker (FRW) metric. Today, it
consists mainly of collisionless cold dark matter, a small amount of baryonic matter and
an even smaller radiation contribution. Besides that it also contains dark energy which is
thought to be responsible for a non-vanishing cosmological constant Λ.
In the following we assume that the reader is at least familiar with the basic principles of
cosmology and we only very briefly repeat some details. Instead we focus on an introduc-
tion into the topic of formation of structures within linear perturbation theory, because
this paves the road for some methods used in Chapter III. We refer the interested reader
to [58, 59, 137] for an introduction into cosmology and the formation of structures.
II.2.1 A brief thermal history of the Universe
The origin of our Universe we live in is still one of (if not the) biggest mysteries of physics.
It is believed it had been formed in a big bang out of a single singularity. Although
our current picture does not offer an explanation for its starting point, extrapolating our
knowledge back allows for a surprisingly accurate understanding of the early epoch directly
after the big bang. At these times the Universe was totally different compared to today:
it consisted of a hot and dense thermal plasma of rapidly interacting particles without any
visible structures at all. Very simplistic the Universe can be understood as an expanding
fluid which cools down over time.
Its composition in terms of energy densities is expressed in the first Friedmann equation:
H2(R) ρX
= Ω R−4r + Ω R
−3
m + Ω R
−2
k + ΩΛ , where ΩX ≡ . (II.21)
h2 ρcrit
Here, R is the scale factor, a quantity which can be used to track the time evolution of
the Universe,9 H(R) is the Hubble factor H(R) ≡ Ṙ/R and h is the Hubble constant.
Moreover, we introduced dimensionless density parameters Ωi with respect to the critical
energy density ρc for radiation, matter, curvature
10 and dark energy. Since each of the
individual components scale differently with R, there are times in the Universe where only
one of the components dominate. In the early epoch, radiation is dominating, but since
it falls off quicker compared to the other ingredients, matter starts to take over after a
certain time (often called matter-radiation equality). Finally, the dark energy contribution
will dominate over matter in the latter stage of the Universe until today.11
9 Another convenient time variable is the redshift z. It stems from observations of distant objects and acts
as a distance measure in that sense. It is related to a by 1 + z = 1/R.
10 We included this expression for the sake of completeness only. In the following we assume that Ωk = 0.
11 At the moment it is not clear, whether ΩΛ will stay constant forever, or may show a slow dynamics. Its
behavior will govern the future evolution of our Universe
18
II.2. COSMOLOGY AND STRUCTURE FORMATION
II.2.2 Particles in equilibrium
The early Universe is characterized by high temperatures and densities, and all SM parti-
cle species are in thermal, i.e. chemical and kinetic, equilibrium with each other. Together
they form a thermal plasma. Any new BSM particle will be part of it if its interaction
with the plasma, specified by an interaction rate Γ, is large enough.
If Γ is larger than H, particles will interact very rapidly compared to the expansion time
scale and thus are kept in thermal equilibrium. As the Universe expands throughout time,
it cools down and Γ decrease faster than H. At the point where Γ ∼ H the interaction
rates are not fast enough anymore to keep these particle species in equilibrium and they
decouple from the thermal plasma. In general, all particle species decouple at different
times, characterized by the size of Γ, and weakly coupled particles will decouple earlier.
To describe the macroscopic behavior of microscopic particles in the early Universe, meth-
ods and techniques of statistical mechanics are used. A dilute, weakly interacting gas of
particles is characterized by its number density n, energy density ρ and pressure P . These
quantities depend on the distribution of particles in momentum space, as encoded in the
corresponding momentum (or phase space) distribution function f(|p|, t).12 For a given
particle distribution function, f(p, t), its p∫roperties are given by:
g
n(t) = ∫ d3p f(p, t) , (II.22)(2π)3
g
ρ(t) = ∫ d3pE(p)f(p, t) , (II.23)(2π)3
g |p|2
P (t) = d3p f(p, t) , (II.24)
(2π)3 3E(p)
where g denotes the internal degrees of freedom (DOF) of the particle in question. If
particles exchange energy and momentum effectively via scattering off the thermal plasma,
they will be in kinetic equilibrium and in that case f(p, t) will be given by
1 lo⇒w Tf(p, t) = f(p, t) ≈ e−(E(p)−µ)/T− ± , (II.25)exp[(E(p) µ)/T ] 1
with a Fermi-Dirac (upper sign) for fermions or a Bose-Einstein distribution (lower sign)
for bosons. Here, µ is the chemical potential of the particle species. If particles are in
chemical equilibrium, i.e. they equally react back and forth, their chemical potentials will
be related to each other. However, at early times they are negligible compared to E,
µ  T . If particles are in thermal equilibrium, they will share a common temperature,
usually called “temperature of the Universe”.13
Considering the total energy density of the Universe, it is a good approximation to in-
clude relativistic particles only, because ρ is exponentially suppressed for non-relativistic
particles. In that case, the sum over∑all relativistic species yieldsπ2
ρr = ρi ≡ g∗(T )T 4, (II.26)
30
i
where T is the photon temperature and g∗(T ) is the effective number of relativistic DOF,
defined as ∑ ( )4 ∑ ( )Ti 7 T 4i
g∗(T ) = gi + gi . (II.27)
T 8 T
bosons fermions
12 Since the Universe is isotropic and spatially homogenous, it only depends on the absolute value of the
momentum, |p| = p.
13 Often, this is called photon temperature instead, since photons are the only relativistic thermal bath
particles left in Universe today, whereas all other particles have become non-relativistic way earlier.
19
CHAPTER II. THEORETICAL BACKGROUND
Ti is the temperature for each species which can be different from the photon temperature
if the species is not coupled anymore to the thermal bath. We will discuss examples for
this in Chapter VI. At high temperatures all SM particles are in thermal equilibrium and
g∗(T ) = 106.75. As temperature drops, various particle species become non-relativistic
and g∗(T ) decreases.
In a radiation dominated Universe, Eq. (II.26) allows to relate temperature and time via
the Friedmann equation for H(t), √
1 8π √ T 2
H(t) = = ρr ' 1.66 g∗(T ) , (II.28)
2t 3M2Pl MPl
where MPl ' 1.22 × 1019 GeV is the Planck mass. Another important quantity is the
entropy of the Universe. Since the Universe expands approximately adiabatic, the entropy
stays constant and one can define an entropy density s by
∑ ρ 2i + pi
s = ≡ 2π gs∗(T )T 3 , (II.29)Ti 45i
where gs∗(T ) is the effective number of entropic DOF. In our subsequent analysis in Chap-
ters IV to VI we will u(se an a)nalytical[expression f(or the DOF (see A)p]pendix of Ref. [135]):∑5
s T T − cig∗ = exp c + ci0 1 1 + tanh 2 , (II.30)GeV ci
i=1 3
c0 = 1.36,
c1 = (0.498, 0.327, 0.579, 0.140, 0.109) ,
c2 = (−8.74, −2.89, 1.79, −0.102, 3.82) ,
c3 = (0.693, 1.01, 0.155, 0.963, 0.907) .
Entropy conservation implies sR3 ∝ gs∗(T )T 3R3 = const. That means, away from particle
thresholds, the thermal plasma cools down as T ∝ R−1. On the contrary, if particles drop
out of thermal equilibrium, conservation of entropy implies a reheating of the remaining
bath, due to a decrease in gs∗(T ). For example, SM neutrinos have a slightly colder
temperature compared to the photon bath, because they had decoupled before the entropy
in electron-positron pairs has been transferred to the remaining photons.
In the latter chapters, it will be convenient to calculate the DM abundance Ω h2DM by
integrating the corresponding momentum distribution function f(p, t) over all momenta
and normalize it by the entropy density [58], ∫∞ 
2 s0mDMΩDMh =  45 dp p2 f(p, t0) , (II.31)
ρ /h2 4π4 3c T g∗(Tprod)
0
where f(p, t0) is evaluated today; the entropy density is s0 = 2891.2 cm
−3 [138] and the
critical density is given by ρ = 1.054× 10−2 MeV cm−3 h2c [138].
II.2.3 Out of equilibrium thermodynamics
If the Universe had been in thermal equilibrium from its beginning until present days, it
essentially would have been an empty place, since there would have been no formation of
structures. Deviations from equilibrium, i.e. decouplings from thermal plasma indicated
by H  Γ, are thus a necessity.
20
II.2. COSMOLOGY AND STRUCTURE FORMATION
In order to track the evolution of a particle’s momentum distribution function f(E, t) a
Boltzmann equation has to be defined. In its most general form it is given by
L̂[f ] = Ĉ[f ] . (II.32)
L̂[f ] is the Liouville operator, decribing the evolution of f due to gravitational forces
and the collision operator Ĉ[f ] inhibits particle number changing processes. For a FRW
metric, it can be rewritten in terms of the numb∫er density as
g d3p
ṅ+ 3Hn = Ĉ[f ] . (II.33)
(2π)3 E
In case of no number-changing collision processes the r.h.s is zero and n is simply propor-
tional to R−3 as the Universe expands.
The exact form of the collision operator is determined by the particle physics model and
depending on the size of the interactions; there are a lot of different possibilities how n(t)
evolves throughout time. For example, as explained in Section II.1.3, WIMPs are cou-
pled to the thermal plasma in the early Universe but freeze-out at a later time. On the
other hand, we will see other possibilities in Chapters IV to VI, where particles are never
in thermal equilibrium and their abundance will freeze-in instead [76].
II.2.4 Cosmological linear perturbation theory
The CMB is remarkably isotropic and only small anisotropies have been observed, i.e. the
Universe had been nearly smooth at these times. Today this is still true at scales larger
than galactic clusters, i.e. above 100 Mpc, but at the scale below this is certainly not true
anymore as there are large differences in the observed energy densities. The question is
how this density fluctuations arose between the CMB epoch and today.
It can be explained by the phenomenon of gravitational instability: as matter is attracted
by regions of high-density it starts to collapse and increase the density of these regions even
further. To illustrate this effect in the following, we restrict our discussion on Newtonian
dynamics to highlight some of the key features of the growth of structures.
A perfect fluid is described by seven characteristics: energy density ρ, pressure P , entropy
S, gravitational potential Φ and the three-dimensional local fluid velocity v. Considering
only slight and adiabatic, i.e. spatially constant, perturbations of these quantities, a linear
expansion in these disturbances can be done. It is convenient to define a fractional density
fluctuation with respect to an averaged density ρ̄ as
δρ(x) ρ(x)− ρ̄
δ(x) ≡ = , where δ(x) ρ̄ . (II.34)
ρ̄ ρ̄
A Fourier expansion of the averaged density∫ yields in the continuum limit
V
δ(x) = d3k δk e
−ix·k , (II.35)
(2π∫)3 V
1
δ = d3x δ(x) eix·kk . (II.36)
V
V
For the time-dependent Fourier modes(δk ≡ δk(t) the)following set of independent equa-tions can be derived:
c2k2
δ̈k + 2Hδ̇k +
s − 4πGρ̄ δk = 0 , (II.37)
R2
21
CHAPTER II. THEORETICAL BACKGROUND
where cs is the speed of sound in the thermal plasma. This set of equation describes the
time evolution of density fluctuations in an expanding Universe whose characteristics are
determined by their wavenumber k (and their corresponding length scale λ ∝ k−1). There
is an associated critical wavenumber, the Jea√ns wavenumber kJ defined by the relationR
kJ = 4πGρ̄ . (II.38)
cs
If the wavenumber is small, i.e. k < kJ , the density fluctuations will be sound waves
traveling through the plasma. On the other hand, if the wavenumber is larger than the
Jeans wavenumber, there will be an power-law growth of these density fluctuations. The
scaling is governed by the dominating energy contribution during the time of growth: dur-
ing the radiation-dominated era, the density fluctuations evolve like δ ∝ C + C lnR.141 2
On the other hand, for a matter-dominated epoch, δ −3/2k ∝ C1R + C2 a and finally
δ ∝ C + C R−2k 1 2 , for a Λ-dominated Universe. It is worth noting, that the growth of
δk is only efficient during matter-domination. If radiation is dominating, fluctuations will
only grow slowly, i.e. formation of structures are effectively taking place only after matter-
radiation equality. In the last stage, when dark energy is dominating, it counter-balances
gravitational effects and hence the fluctuations stop growing. In summary, since density
fluctuations only grows substantially during matter-domination, the initial density per-
turbations have to be large enough and the CDM contribution has to take over radiation
early enough such that the observed structures in the Universe can form.
However, the Newtonian limit only holds for subhorizon modes, i.e. k < H. The treat-
ment of modes which are larger than the Hubble horizon at a given time needs a fully
general-relativistic treatment, as does the inclusion of relativistic fluids. In the relativistic
treatment, one solves for the gravitational potential which in turns determine the evolu-
tion of density fluctuations. At subhorizon scales this treatment shows the same behavior
as the Newtonian limit: during radiation domination the density fluctuations only grow
slow and the graviational potential decays due to radiation pressure, while a stronger grow-
ing shows up in matter-dominated phase. On the other hand, at superhorizon scales, the
fluctuations are frozen as the potential stays constant.
II.2.5 The matter power spectrum
The degree of clustering, i.e. the excess probability, can be analyzed by using the two-point
correlation (or auto-correlation) function ξ(r) of density fluctuations δ(x),
ξ(r) = 〈δ(x)δ(x− r)〉 , (II.39)
where 〈. . . 〉 denotes an ensemble averaging. In Fourier space, it can be expressed by the
matter power spectrum P (k) defined by
P (k) = 〈|δ(k)|2〉 . (II.40)
Isotropy demands that it only depend∫s on k = |k|. Consequently, P (k) and ξ(r) are relatedby
sin(k r)
ξ(r) = dk P (k) k2 . (II.41)
k r
Because the growth rate of fluctuation modes is time- and scale-dependent, the power
spectrum at late times15 can be decomposed in the following form:
P (k,R) ∝ T 2(k)D2(R) kns . (II.42)
14 Here and in the following, C1,2 denotes some arbitrary coefficients.
15 This assumption is valid, because observations of the power spectrum take place at z → 0.
22
II.2. COSMOLOGY AND STRUCTURE FORMATION
Here, T (k) is a transfer function which describes the growth of modes at horizon crossing
and the transition from radiation to matter-domination (characterized by REQ). The
growth factorD(R) encapsulates the growing of modes in time, as explained in the previous
section. Finally, a convenient choice is to set the spectral index of P (k) to ns = 1 [139,
140].
This allows us to describe the asymptotic behavior of the matter power spectrum. It
can be qualitatively understood from the different growing behavior of modes which enter
the horizon at different times. Modes with large k enter the horizon well before matter-
radiation equality REQ and thus the associated gravitational potential decays, leading to
a decrease of T (k) which is proportional to k−2. On the other hand, small k modes enter
the horizon after REQ and experience no suppression, i.e. T (k) ∝ 1 for these modes. For
these reasons, the power spectrum scale{s as:
∝ k, k < kEQ ,P (k) (II.43)
k−3, k > kEQ .
The wavenumber kEQ reenters the horizon at matter-radiation equality; it further de-
termines the turnover in P (k) as it transits from small to large k. An example for the
matter power spectrum at z = 0 is shown in Fig. II.2. It clearly features a peak at
kEQ ∼ 10−2 h/Mpc. Qualitatively, the mode at this wavenumber is the largest in the
spectrum, because it is not suppressed due to radiation and has the longest time to grow.
Smaller wavenumbers show a smaller grow due to reentering the horizon in radiation dom-
ination, while larger wavenumbers have less time to grow.
The small bumps at wavenumbers k ' 0.1h/Mpc shown in Fig. II.2 stem from baryonic
acoustic oscillations (BAO): in contrast to CDM, baryons are electromagnetically coupled
to photons. After recombination the photon mean free path dramatically increases and
this creates a baryon drag out of gravitational wells formed by CDM. After some time
they bounce back into these regions and this leads to the BAO. The horizontal lines refers
to k-ranges which can be probed by different observations and surveys. The red line refers
to precise measurements of different CMB modes [27], the blue line are galaxy count sur-
veys [141], the orange one indicates observations of the Lyman-α forest [142] and lastly,
the cyan line is weak lensing data due to cosmic shear effects of underlying matter on
far-traveling photons [143, 144]. All of them allow to probe the matter power spectrum
across a broad range of scales.
II.2.6 Impact of warm dark matter on structure formation
The ΛCDM explains the observed behavior of our Universe to a very good degree, incor-
porating DM which became non-relativistic at pretty early times, therefore called “cold”
DM and whose density fluctuations acts as gravitational wells for the baryonic matter,
hence it is crucial for the formation of structures over a broad range of scales. However,
as already pointed out in Section II.1.3, there is the intriguing idea to deviate from this
picture and allow for DM candidates which stay relativistic for longer times. Such par-
ticles are referred to as “warm” DM and they influence the formation of structures in
the Universe, because they come with a non-vanishing free-streaming length. This length
is associated to the time period where the DM was relativistic, i.e. particles which be-
come non-relativistic at later times thus have a larger free-streaming length. An example
are SM neutrinos: in principle they are a good DM candidate, because they carry no
electric charge and interact only very weakly with matter. However, as we have seen in
Section II.1.2 neutrinos are extremely light and are relativistic until late times. If neutri-
nos constituted all of DM, we would face the following problem: they are simple moving
too fast to allow for clumping of baryonic matter, because there would be no gravitational
23
CHAPTER II. THEORETICAL BACKGROUND
105
CMB
galaxy count
Lyman-  forest
weak lensing
104
103
102
101
10 4 10 3 10 2 10 1 100 101
wavenumber k [h/Mpc]
Figure II.2: The matter power spectrum P (k) derived using CLASS [145, 146] and for
cosmological parameters specified in Ref. [27]. At small wavenumbers it is increasing ∝ k,
whereas it drops ∝ k−3 after it reaches a peak at k ≈ 10−2 h/Mpc. The different colored
horizontal lines indicate ranges of the spectrum which can be experimentally verified by
observations and which are explained in more detail in the text. This figure is inspired by
Fig. 1 in Ref. [147].
wells as neutrinos would exit such high density regions of space with the speed of light.
In conclusion, neutrinos as DM were already excluded decades ago as they are essentially
too “hot”.
Now one may ask, whether there is a DM species with non-vanishing free-streaming length
but smaller than that for neutrinos. In that case, a similar but weaker effect would take
place. The escape of this DM would erase gravitational sinks below a certain scale and
hinder the formation of structures smaller than this given size. Such a candidate would
leave us with some fingerprints in the matter power spectrum, as shown in Fig. II.3. We
assumed a warm DM with a Fermi-Dirac distribution similar to neutrinos, but with masses
O(keV). It is clearly visible, that WDM with smaller masses give rise to a matter power
spectrum which deviates from ΛCDM at smaller wavenumbers, or inversely, on larger
scales.
The deviation for a given WDM model can be quantified by the ratio of the corresponding
power spectra, called linear squared transfer function T (k):
2 ≡ PWDM(k)T (k) . (II.44)
PCDM(k)
The transfer function for a WDM model can be expressed via the following analytical
expression [148(]: )−5/β
T (k) = 1 + (αk)2β ( , ) ( ) ( ) (II.45)m 0.11 1.22WDM −1.11 ΩWDM h
where β = 1.12, α = 0.049 h−1 Mpc . (II.46)
1 keV 0.25 0.7
As shown in Fig. II.2 there are surveys and observational techniques which can probe P (k)
at these scales and search for deviations from ΛCDM. These allow us to place constraints
24
matter power spectrum P(k) [(h/Mpc)3]
II.2. COSMOLOGY AND STRUCTURE FORMATION
105
mWDM = 0.5 keV
104 mWDM = 1.0 keV
mWDM = 2.0 keV
3 m10 WDM
= 4.0 keV
CDM
102
101
100
10 1
10 2
10 310 4 10 3 10 2 10 1 100 101 102
wavenumber k [h/Mpc]
Figure II.3: The corresponding matter power spectrum for a WDM with mass in range
mWDM = 0.5 keV–4 keV. In general, a suppression of the power spectrum compared to
ΛCDM can be observed and the scale at which it starts to deviate is related to the mass
of the WDM candidate.
on the parameters of a WDM model. Usually, one has to rely on extensive hydrodynam-
ical N -body simulations for a given WDM model and compare their predictions against
observations. However, there are shortcuts to this procedure and we will explain in the
following how to probe the matter spectrum using three different approaches which were
used to constrain our model setups given in Refs. [1, 3].
II.2.6.1 Half-mode analysis
Conventionally, structure formation analyses quote their limits on WDM in terms of a
thermal relic WDM candidate with mass mTR as a reference. It is assumed that it was
thermalized early in the Universe and thus maintained a Fermi-Dirac distribution before
it decoupled at some point at temperature TD. It allows for an adaption of structure
formation limits for other similar models, although it is not a “physical” model, because it
would require too much entropy dilution to reproduce the observed DM relic abundance,
Ω 2DMh , with O(keV) DM masses [149],( )
gs≈ ∗(T )
( )
ν mTR
ΩDM 10 , (II.47)
gs∗(TD) keV
where Tν is the temperature of neutrino decoupling. Consequently, g
s
∗(TD) ≈ 900 would
be required to satisfy Ω 2DMh = 0.12.
For ΛCDM, T 2(k) = 1 ∀k, whereas for a given WDM setup, it is clear that lim 2k→0 T (k) =
1 because at large scales it is indistinguishable from ΛCDM. On the other hand, at small
scales, lim 2k→∞ T (k) < 1, because of the suppression of structures due to WDM. The
simplest approach is to calculate T 2(k) for a WDM model and compare it against a
reference model by inserting a specific limit on the TR mass as mWDM in Eq. (II.46), this
defines a reference transfer function T 2 2 2lim(k). If T (k) ≤ Tlim(k) ∀k, the WDM will not be
in conflict with observations, because its suppression is less than observed. However, the
WDM momentum distribution does not necessarily have to be of thermal origin as it is
25
matter power spectrum P(k) [(h/Mpc)3]
CHAPTER II. THEORETICAL BACKGROUND
the case for the reference model. In general, the corresponding transfer function can have
a different slope compared to T 2lim(k) and both might intersect each other. Therefore, it
is convenient to define the half-mode khalf by demanding that [111]
T 2
! 1
(khalf) = . (II.48)
2
Based on this wavenumber, we can use the following criteria whether a given WDM model
is allowed if and only if the following criterion is fulfilled:
T 2(k) ≤ T 2lim(k) ∀k < khalf . (II.49)
This analysis technique will be used to constrain freeze-in ALP DM in Chapter V and an
adapted version will be used in Chapter IV to place limits on light DM in the context of
the scotogenic model.
II.2.6.2 Constraints from the Lyman-α forest
Another way to constrain WDM models uses the so-called Lyman-α forest. As shown in
Fig. II.2, it can be used to probe the smallest scales of the matter power spectrum and to
measure potential deviations from ΛCDM stemming from WDM16. The Lyman-α forest
method examines the intergalactic medium (IGM) by looking at absorption lines from
neutral hydrogen along the line-of-sight of highly redshifted quasars. The most prominent
one of these absorption lines is the Lyman-α, hence the obvious name for this observation
method. Each point along one line-of-sight corresponds to different redshifts and thus
the respective absorption lines are observed at different wavelengths. A combination of
various line-of-sight measurements allow to estimate the hydrogen density, which in turns
acts as a tracer for the underlying DM density.
The Lyman-α forest surveys probe the absorption line spectra of quasars at redshifts
3.5 < z < 5 and the flux power spectrum, PF (z, kν), at these redshifts is derived for
different wavenumbers in velocity-space, kν . In order to derive constraints for a specific
WDM model, the one-dimensional power spectrum P1D(k) (see for instance Refs. [99,
150, 151]) is derived by integrating the matter power spectrum along the observed k-
range. The one-dimensional spectrum is related to the measured flux spectrum by a bias
function, b2(k) ≡ PF (kν , z)/P 1D(k), which allows us to translate bounds from surveys
into a quantity which can be accessed easily for a given WDM model by computing its
matter power spectrum. The conversion factor between velocity-space wavenumbers and
H(z)
inverse comoving length scales used in the power spectrum is given by k = kν . The
1 + z
one-dimensional power spectrum is an integral of the P (k) evaluated at z = 0 and given
by ∫∞
P 1D
1
(k) = dk′ k′P (k′) . (II.50)
2π
k
A suppression of this spectrum can be quantified by defining a one-dimensional transfer
function φ(k) in a similar way as T (k),
P 1D
φ(k) = WDM
(k)
, (II.51)
P 1DΛCDM(k)
16 Strictly speaking, possible deviations do not necessarily have to be caused by DM; they can also be
indications of unknown processes in structure formation at these scales. This is one of the reasons why
there are sometimes significant differences in the limits for mTR quoted in the literature.
26
II.2. COSMOLOGY AND STRUCTURE FORMATION
where P 1DΛCDM(k) is the one-dimensional power spectrum of ΛCDM. The next step is to
quantify how much a given WDM model differs from a ΛCDM scenario. We integrate
Eq. (II.51) over all scales typically probed by Lyman-α observations given in the range
(kmin, kmax):
k∫max
A = dk φ(k) , (II.52)
kmin
and this integral can be used to approximate the amount of suppression for a given WDM
model by defining the estimator
AΛCDM −A
δA ≡ , (II.53)
AΛCDM
where AΛCDM = kmax − kmin, whose value is set by the respective observational data set
of Lyman-α surveys.
In the final step, we have to calculate a reference value δAref first, using a thermal relic
with mass mTR, where mTR is an observational limit from Lyman-α surveys. Now, WDM
models which give rise to δA > δAref are excluded because their small scale suppression
is too large.
II.2.6.3 Counting the number of Milky Way satellites
An additional way to search for deviations from ΛCDM at galactic scales is to estimate
the number of accompanying satellites of the Milky Way (MW). Generally, WDM models
predict fewer satellites for MW-like galaxies compared to vanilla ΛCDM, because they
tend to suppress the mass distribution function of the subhalos [148, 149, 152–154]. This
can be understood by defining an effective Jeans mass Mj(t) based on Eq. (II.38) which
is given by [155] ( )
4π π 3
MJ(t) = ρm(t) , (II.54)
3 kJ
and perturbations are damped for M < MJ . Using observational data to count the
accompanying satellites of the MW, we can place limits on WDM models by calculating
their predicted number of subhalos.17
The starting point to estimate the number of subhalos accompanying the MW is to use
an extension of the Press-Schechter formalism [162] to determine the number density of
halos. Using linear perturbation theory for the density fluctuations, the probability for
the collapse of a fluctuation is related to the mass of the respective halo. Based on this, we
define a conditional mass function which takes the merging history of halos into account
[154, 163],
dn(M |M0) −M0 dS= f (δc, S|δc,0, S0) . (II.55)
d log(M) M d log(M)
Here, δc is the critical density below which fluctuations collapse and form compact struc-
tures; it is a time dependent quantity and can be seen as a time variable. S is the variance
of the matter power spec∫trum smoothed by a sharp-k window function W (k,R),
dk k2
S(R) = P (k)W 2(k,R) , W (k,R) = θ(1− kR) , (II.56)
2π2
17 Associated to this is the “missing satellite” debate: simulations of galaxy formation using only ΛCDM
predicted too many accompanying dwarf galaxies compared to the observed number of satellites of the
MW [156–158] which sparked particular interest in WDM models. However, recent detections of ultra
faint dwarfes [159, 160] and more detailed investigations of the halo profile [161] indicate that this problem
can also be solved within the ΛCDM paradigm.
27
CHAPTER II. THEORETICAL BACKGROUND
4π
where the filter scale R is related to an enclosed mass M by M = Ωmρc(cR)
3, where
3
c = 2.5 stems from a matching to simulations [99], and the matter density is given by
Ωm = 0.315. Finally, f (δc, S|δc,0, S0) is a probability distribution which encodes the
merging history of a halo forming at δc with variance S up to present time δc,0 with
S0(R0), where R0 is the filter scale associated to the host galaxy with mass Mh. Assuming
a spherical collapse, it is given by [163] [
δc − δc,0 (δc − 2
]
δc,0)
f (δc, S|δc,0, S0) = √ exp − . (II.57)
2π(S(R)− S0(R0))3 2(S(R)− S0(R0))
Integrating Eq. (II.55) over time gives the number of subhalos Nsub:
∫∞
dNsub 1 dn(M |Mh) 1 1 Mh √P (1/R)= dδc = , (II.58)
d log(M) C d log(M) C 6π2 M R3 2π(S − S0)
δc,0
where C is a normalization constant used to match with N -body simulations and depends
on the definition of the host halo. In our case, the boundary of the host halo is set by the
criterion that its density is 200 times the critical density ρc of the Universe and hence we
use C = 34 [99] in the following.
Finally, Eq. (II.58) gives an estimate for the numbers of subhalos with mass M around
the host galaxy, thus integrating Eq. (II.58) from M = 108 h−1min M to Mh yields the
total amount of subhalos ∫Mh
dNsub
Nsub = dM . (II.59)
d log(M)
Mmin
There are two uncertain numbers in the following discussion: first, the observed number
of MW subhalos and second the mass of the host galaxy, i.e. the MW in our case.
Addressing the counting of subhalos we follow the approach outlined in Refs. [99, 151]:
there exist 11 “classical” satellites which are combined with 15 ultra-faint satellites
found by the Sloan Digital Sky Survey (SDSS) (taken from the seventh data release,
see Refs. [164, 165] for more details). Assuming that the satellites found by SDSS are dis-
tributed isotropically over the sky, the latter number can be multiplied by a factor of 7/2
because SDSS has a sky coverage of only 28.3%; this yields in total Nsub = 64.
18 How-
ever, as pointed out in Ref. [99] we should decrease this number by ∼ 10% to take the
history of halo formation into account. Further, the assumption that the satellites found
by SDSS are isotropically distributed might lead to an overestimation of Nsub, for exam-
ple, the classical satellites are arranged in a plane rather than isotropically [166].
One the other hand, in addition to SDSS, several more ultra-faint satellites or candidates
have been reported by other surveys (see for instance Ref. [167]) and future searches could
reveal even more satellites, so we regard Nsub = 64 as a conservative estimate for the
number of MW companions. In fact, simulations predict O(100) subhalos [168, 169] which
could be detected with future observations and therefore open new possibilities to test
WDM models even further.
Since we make different assumptions for the MW mass in our subsequent analysis, we re-
fer the reader to Chapters V and VI for further discussion, where we use Nsub to constrain
warm ALP DM and a more general WDM framework respectively.
18 Assuming Poisson statistics, the error of this number is ≈ 13. Nonetheless, we will mostly refer to the
mean value of Nsub, but comment on the dependency in the respective chapters.
28
II.3. PRELUDE
II.3 Prelude
During this research, programs, libraries and other tools were used in addition to those
explicitly mentioned in the relevant chapters. For example Mathematica 11+12 [170] was
used for numerical calculations. Furthermore, Python 2.7 [171] and Python 3.6 [172]
was used in combination with the libraries NumPy [173], Matplotlib [174], SymPy [175],
SciPy [176] and the IPython [177] environment. All Feynman diagrams were created with
the TikZ-Feynman package [178]. The Pandas [179] package was used for parts of data
manipulation.
29

Main part I: dark matter at
current and future colliders
31
DARK MATTER AT CURRENT AND FUTURE COLLIDERS
The first part of this thesis is dedicated to collider probes of extensions of the standard
model. In the following, we are going to consider the well motivated scotogenic model
(ScM) as an exemplary extension.
The current energy frontier is set by the largest collider experiment, the Large Hadron
Collider (LHC) located at CE√RN. It delivers and collides protons or heavy ions with
center-of-mass energies up to s = 13 TeV for four different experiments: ATLAS, CMS,
LHCb and ALICE. For our analysis the first two will be the most relevant, since they al-
low for a dedicated search for new particles produced in collisions and decaying promptly,
leaving recognizable signatures in the ou√tward detector layers.
Data collection started in 2011/2012 at s = 7–8 TeV respectively and an integrated lu-
minosity √of roughly 20–25 fb
−1 was reached. During the second run, in which data was
taken at s = 13 TeV from 2015 until the end of 2018, the LHC delivered additionally
∼ 150 fb−1 to both, ATLAS and CMS. After a long shutdo√wn, the experiment is planned
to resume in 2022 with Run-3 at an increased energy of s = 14 TeV and collecting at
least ∼ 300 fb−1 of p p collision data until the end of 2024.
After this run, it is foreseen to upgrade the existing LHC collider into the high-luminosity
LHC (HL-LHC) and increase the number of p p co√llisions dramatically. This will allow to
reach an integrated luminosity up to 4000 fb−1 at s = 14 TeV in a decade starting from
2027.
At the moment, possible experiments for the post-LHC era are being discussed and their
potential to measure the SM at high precision and search for new physics is being eval-
uated. One idea is to reuse th√e LHC infrastructure and upgrade the machine such that
it can deliver p p collisions at s = 27 TeV, the h√igh-energy LHC (HE-LHC). Going even
beyond, options for a circular proton collider at s = 100 TeV denoted by Future Circu-
lar Collider (FCC-hh) have been considered.
On the other hand, new electron-positron colliders are discussed as well. This includes
circular colliders similar to proton colliders such as the Circular Electron Positron
C√ollider (CEPC) in China or the FCC-ee at CERN, both running at energy setuprs
s ∼ 100–350 GeV. Since e+e− collisions offer a cleaner experimental condition com-
pared to p p collisions, this machines would allow for precise measurements of observables.
However, energy losses due to Bremsstrahlung limit the acceleration of electrons inside
circular colliders and thus linear colliders have been proposed as well. Future facilities
like the International L√inear Collider (IL√C) or the Compact Linear Collider (CLIC) would
deliver energies up to s = 1 TeV and s = 3 TeV, respectively, together with high lu-
minosities.
Current and future colliders allow to probe new physics beyond the SM (BSM) in dif-
ferent ways: there are direct searches looking for distinctive signatures caused by new
particles, such as corresponding large missing-energy signatures. A plethora of studies
have been conducted by ATLAS and CMS and results are often quoted in a specific su-
persymmetric model (SUSY) framework. Limits on BSM models are extracted from these
studies by recasting and adapting these searches using Monte Carlo simulations. Alterna-
tively, BSM models can have an indirect effect on SM processes. For instance electroweak
precision measurements allow to place bounds on new physics extensions.
In the following chapter, we will present an example for the first approach. We recast
existing searches for pair production of the electroweak SUSY partners to limit the ScM
parameter space in Chapter III. Further we present search strategies to test this model at
future colliders as well.
32
Chapter III
Collider studies of the scotogenic
model
III.1 Introduction
While more than two decades have passed since the groundbreaking discovery of neutrino
oscillations, which unambiguously established that the most elusive standard model (SM)
particles are massive, the origin of neutrino mass still remains unknown. In spite of the
viable scenario in which, by supplementing SM left-handed neutrino fields with right-
handed (RH) components, neutrino masses are generated in the same way as for all the
other fermions, the smallness of Yukawa couplings required for generating eV-scale masses
has led to a much greater interest in Majorana mass models. The most famous realization
of the latter possiblity is the type-I seesaw model [33, 35, 180, 181] in which neutrino
masses are generated at tree-level in the presence of at least two generations of heavy
neutral leptons. For “natural” O(1) Yukawa couplings, this model implies that the heavy
lepton mass scale is ' 1013 GeV, unreachable at any terrestrial experiment. In contrast,
radiative neutrino mass models can lower the scale of new physics by several orders of
magnitude.
Among radiative neutrino models, one of the simplest realizations is the so-called sco-
togenic model (ScM) [103]. It imposes a Z2 symmetry in order to forbid the tree-level
neutrino mass generation, hence the lightest among the newly introduced particles can be
a viable dark matter (DM) candidate [182–186]. The success of thermal leptogenesis in this
model has also been demonstrated in Refs. [187–193]. In addition, recently it was shown
by previous work of the author that light DM and leptogenesis via the Akhmedov-Rubakov-
Smirnov (ARS) mechanism[194], i.e. the dynamical creation of a baryon asymmetry via
CP-violating oscillations among RH neutrinos (RHN), can be embedded simultaneously
in this framework [2]. It was found that the spectrum of new particles can be below the
TeV-scale, while bounds from cosmology require the mass of the light DM to be below
O(10 keV). This raises the question whether we can test this model setup at present and
future colliders, where the accessible energy range exceeds the masses of all newly intro-
duced particles. Moreover, the light DM candidate can lead to interesting consequences
for the early Universe, which will be discussed in the second part of this thesis.
In this chapter we will focus on collider searches at the (HL-)LHC and the future facilities
FCC-hh and CLIC to project exclusion bounds on the model parameter space. For this
purpose we are going to investigate pair production of the charged new scalar, which de-
cays further either into e or µ and missing transverse energy, ET , or into tau leptons and
ET as the designated final state for the experiments under consideration.
33
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
This chapter is organized as follows: we briefly motivate our model spectrum in Sec-
tion III.2 before we introduce the model in Section III.3. Section III.4 is dedicated to a
projected HL-LHC analysis based on existing searches. Going beyond, we discuss the re-
sults for the future colliders FCC-hh and CLIC in Section III.5. We summarize our results
in Section III.6.
III.2 Motivation
The idea to setup a dedicated search for this specific model was sparked by previous re-
search results presented in Ref. [2]. In this paper it has been shown that it is possible to
combine an explanation for neutrino masses and a light RHN N1 as a DM candidate with
a dynamical generation of the observed baryon asymmetry of the Universe (BAU) by ap-
plying a process known as leptogenesis. Generally, in this framework, CP violating decays
or oscillations of heavy RHN are used to create an asymmetry in the SM lepton sector
at high temperatures early in the Universe which is than transferred via non-perturbative
processes, so-called sphalerons, to the baryon sector before the electroweak symmetry
breaking takes place at T ' 160 GeV. More precisely, the considered production relies on
a combination of the ARS mechanism and thermally enhanced CP violating decays of the
Σ scalar (see Ref. [195]). Unlike standard thermal leptogenesis, the asymmetry is gener-
ated at lower scales of a few TeV instead of high scales of ∼ 1012 GeV. This is possible due
to an enhancement effect stemming from a tiny mass splitting between the heavy RHN
N2 and N3 which is quantified by the parameter mN3 −mN2 ≡ δM  1.19
One of the challenges was to find regions in the parameter space for which the observed
values of DM relic abundance and the baryon asymmetry are simultaneously reached. Gen-
erally, these two mechanisms have conflicting requirements on the strength of the Yukawa
coupling which can be set by the parameter η (see Eq. (III.6)). In order to not overproduce
DM from decays of the heavier RHN, sufficiently large Yukawa interactions are required.
On the other hand, leptogenesis relies on weak interactions or otherwise washout effects
would easily destroy any generated lepton asymmetry. This tension can be cured by im-
posing coannihilations between the RHN and the scalar particles by choosing their masses
to be of similar size. Such a regime opens up new scalar annihilation channels which do
not rely on the strength of the second and third generation Yukawa couplings. There-
fore, a huge suppression of the relic density can be achieved for Yukawa couplings set low
enough to generate a significant lepton asymmetry.
Employed with this mass spectrum we calculated the DM relic density for different scalar
19 Besides this concrete model, there are other intriguing possibilites to generate the baryon asymmetry via
degenerate RH neutrinos and intertwine it with other BSM physics explanations as well. For instance,
the model considered in Ref. [196] delivers an explanation to the observed discrepancies in anomalies in
b decays, the so-called b-anomalies (see Ref. [197] for an overview). By extending the SM group to a
SU(4) × SU(3) × SU(2) × U(1) and incorporating a scalar leptoquark which couples third generation
quarks and leptons, it explains the observed ∼ 3σ discrepancy in the b→ ` ` s [198, 199] and b→ c τ ντ
[200] decay rate. It features an inverse seesaw mechanism [201, 202] to explain the smallness of neutrino
masses, i.e. it introduces in total six particles (3 RHNs and 3 gauge singlets), two of which form a nearly
mass degenerate pseudo Dirac pair (PDP) with masses MR and a coupling M
D
ν = Yν v to the SM leptons.
Their mass difference is characterized by a small lepton number violating (LNV) parameter µ. They are
related to the light neutrino mass matrix Ml via Ml ≈ MD −1ν MR µ(M
T )−1(MD)TR ν . Here, the smallness
of the LNV parameter guarantees an adaption of the ARS mechanism [203]. Assuming the mass of
the lightest PDP to be of MR,1 ' O(MeV) and choosing appropriately small couplings Yν , we find a
space of parameter region which successfully explains the BAU. It can be enlarged by demanding further
couplings between the gauge singlets and leptons. We checked that the corresponding PMNS mixing
matrix does not violate unitarity bounds and further that the lightest PDP does not thermalize due to
mixing with the heavier states. The later requirement is crucial for the success of the ARS mechanism;
it relies on a freeze-in of the PDP in order to generate an asymmetry via oscillations among the mass
degenerate states.
34
III.2. MOTIVATION
mass choices and further evaluated the corresponding baryon asymmetry in a range of the
heavy RHN mass, mN2 , and the strength of the Yukawa coupling. The results can be seen
in Fig. III.1, where we have fixed the DM mass mN1 to be 6 keV and δ = 10
−11
M . The
blue line indicates the region which gives a too small baryon asymmetry to explain for
the observed value.20 On the other hand, the red shaded region is ruled out because this
parameter choice would spoil the formation of light nuclei due to late-time production of
highly energetic DM. Here we adopted the limits on leptonic decays from [204]. There are
several things we would like to point out. First, the final baryon asymmetry only mildly
depends on the involved particle masses and in general larger mass scales will only lead
to a slight decrease as can be seen from the slope of the BAU line in Fig. III.1. In con-
trast, choosing a higher mass scale slightly weakens the BBN bounds and broadens up the
available parameter region. On the other hand, allowing for a slightly larger degeneracy
between right-handed neutrinos, such as δ = 10−10M , would correspond to a shift of the
blue exclusion curve in Fig. III.1 towards the BBN bound at the left. For δM ≥ 10−10
and DM masses of ∼ 6 keV the corresponding parameter space is completely constrained
either by BBN or BAU requirements.
ƞ ƞ
Figure III.1: Allowed region in the η–mN2 parameter space for a charged scalar mass
m± ' 590 GeV (left) and m± ' 795 GeV (right). The BBN exclusion limits are shown in
red, while the blue shaded region does not produce a sufficiently large baryon asymmetry.
We observe that there is a region consistent with BBN limits in which the correct amounts
of DM and baryon asymmetry can be obtained.
We have also observed the dependence of BBN limits on the maximum allowed DM mass.
For scalar masses in the range 300–1000 GeV, we found a maximal value of the DM mass
of 9.4 keV. Generally, choosing higher degeneracies will open up the available parameter
space to some extent, thus allowing for larger DM masses.21 However, even in such cases
we estimated the maximal allowed DM mass to be at most O(10 keV).
In summary, we identified the parameter space in which the produced DM abundance and
BAU are in accord with the observed values. We have seen that the most stringent con-
straints arise from BBN considerations, which can be relaxed by employing coannihilation
20 In principle the observed asymmetry is only reproduced on the blue line. However, since we maximized
the CP violating parameters, it is possible to adjust them such that the correct asymmetry can be
achieved for the parameter region left to the blue line.
21 Although smaller mass differences provide an increase in asymmetry, this effect breaks off at a certain
value of the difference, and the asymmetry cannot be increased further. In our case this is δ −13M ≈ 10
35
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
processes between RH neutrinos and scalars which effectively put an upper bound on the
allowed mass for the DM. We found that the DM production in our model is mainly driven
by the freeze-in. These results sparked the question of whether this scenario could poten-
tially be tested by searches at collider experiments, given that the masses of the particles
involved are below the TeV scale.
III.3 The model and neutrino mass generation
In addition to the SM field content, the ScM contains one scalar doublet Σ = (σ+, σ0)T
as well as three generations of RHNs Ni (i = 1, 2, 3). In addition to these new fields, the
model requires a discrete Z2 symmetry under which the new particle fields have an odd
charge. The part (of the Lagrangian containing newl)y introduced fields is
L ⊃ i 1N̄i ∂/Ni − yiα N̄i Σ̃†Lα + mNiN̄ N ci i + h.c. + (DµΣ)†(DµΣ)− V (Φ,Σ) , (III.1)2 2
where yiα is the Yukawa coupling between a RHN Ni, Σ and a SM lepton doublet Lα =
(να, α
−)T, (α = e, u, τ), mNi is the mass of i-th RHN, Dµ is the covariant derivative,
Φ = (φ+, φ0)T is the SM Higgs doublet, and V (Φ,Σ) represents the scalar potential
1
V (Φ,Σ) =µ2 Φ†Φ + µ2 Σ†Σ + λ ((Φ†Φ)2
1
1 2 1 + λ2)(Σ†Σ)2 + λ3 (Φ†Φ)(Σ†Σ)2 2λ5
+ λ †4 (Φ Σ)(Σ
†Φ) + (Φ†Σ)2 + h.c. . (III.2)
2
The couplings in the scalar sector are constrained by vacuum stability requirements, i.e.
the potential sh√ould not diverge for large field valu√es [183, 205],
λ3 > − λ1λ2 , λ3 + λ4 − |λ5| > − λ1λ2 , λ1,2 > 0 . (III.3)
From Eq. (III.2), we can directly infer the masses of novel scalar degrees of freedom after
electroweak symmetry breaking (EWSB):
m2 2 2± = µ2 + λ3v ,
m2 2 2S = µ2 + (λ3 + λ4 + λ5) v ,
m2A = µ
2
2 + (λ3 + λ4 − λ 25) v , (III.4)
√
where v = 246/ 2 is the vacuum expectation value (VEV) of the Higgs field. The mass
of the charged scalar is given in the first line of Eq. (III.4), whereas the latter two masses
corre√spond to the CP-even (S) and CP-odd (A) neutral scalars, defined as σ
0 = (S +
iA)/ 2.
Since the exact Z2 symmetry forbids a generation of neutrino masses at tree-level, they
are realized radiatively with the following expression obtained by calculating self-energy
corrections to the neutrino propagator from the exchange of neutral scalar fields S and A
[103, 206, 207],
∑ [ ( ) ( )]y 2 2 2 2iαyiβmN
(m ) = i
mS mS mA mA
ν αβ ∑ 2 − 2 ln − ln (III.5)32π2 mS mN m2N m2 −m2 m2i i i A Ni Ni
≡ yiαyiβ Λi .
i
Here, the summation index runs over the RHN generations and in the last equality we
abbreviated this formula with Λi. In the model spectrum considered in the following, the
36
III.3. THE MODEL AND NEUTRINO MASS GENERATION
mass of the lightest RHN is O(10 keV) with y1α ' O(10−8) and this state effectively does
not participate in the neutrino mass generation. This makes the lightest active neutrino
effectively massless, which is a viable scenario, consistent with the data from neutrino
oscillation experiments that are probing only mass squared differences. In this case, N1
is decoupled from the mass generation and consequently only the elements of a 2 × 3
submatrix of y, which we denote by y′, enter in Eq. (III.5).
In order to properly account for low-energy neutrino data in the analysis, we employ
the Casas-Ibarra parametrization which imposes the following expression for the Yukawa
sub-matrix [208] (√ ) √
′ −1y = i Λdiag R mdiagν U
†
PMNS , (III.6)
where Λdiag = diag(Λ2,Λ3), and R is an orthogonal matrix parametrized with a complex
angle θ = ω
(
− i η )
 0 cos θ − sin θ( , for normal neutrino mass ordering (NO) , 0 sin θ cos θR =  ) (III.7)cos θ − sin θ 0 , for inverted neutrino mass ordering (IO) .
sin θ cos θ 0
The remaining ingredients in Eq(. (II√I.6) are√the neu)trino massesdiag(0√, m
2
sol√, m2atm , ) for NO ,mdiagν =  (III.8)diag m2 2 2atm, msol +matm, 0 , for IO ,
where m2sol and m
2
atm are solar and atmospheric mass squared differences, and the leptonic
mixing matrix, UPMNS, which is parameterized as in Ref. [209]. The relevant parameters
for us are one Dirac phase, δ, and two Majorana CP phases, α1 and α2. While the mixing
angles are relatively precisely determined, the value of the Dirac CP phase is practically
unconstrained, and Majorana phases are not testable at neutrino oscillation facilities. We
use the values from Ref. [210] in the following:
m2sol = 7.39× 10−23 GeV2, m2 −21atm = 2.525× 10 GeV2,
θ12 = 33.62
◦, θ23 = 47.2
◦, θ13 = 8.54
◦. (III.9)
The size of the elements of the Yukawa sub-matrix y′ are constrained from above due to
non-observation of lepton flavor violating processes (LFV) such as ` → `′ γ and ` → 3`′
where ` and `′ denote different species of charged leptons. The upper bounds on the
branching ratios (BR) for these types of decays are given in Ref. [209] and also compiled in
Table 1 in Ref. [2]. While we have implemented all available constraints from LFV decays
it is worthwhile pointing out that the dominant effect arises from the lack of observation
of µ → e γ process. The upper bound on the BR for this process is 4.2 × 10−13 which
converts to [206], ∣∣∣ ∑ ∣∣∣ ( )∗ − m± 2yiµ yie . 4.3× 10 3 , (III.10)1 TeV
i=2,3
for mN2,3 ' 0.1 TeV.
Finally, the neutrino mass matrix in Eq. (III.5) depends on λ5 which enters in the ex-
pression for mS and mA (see again Eq. (III.4)). This formula actually features a linear
37
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
dependence between the neutrino masses and λ5 for small values of λ5 [103]. The param-
eter λ5 and the entries of y
′ depend on each other and jointly set the scale for neutrino
mass as ∼ 0.1 eV. This means that there is a lower bound λ5 & O(10−7), calculated for
m± = O(TeV). In general, the Z2-odd scalar doublet can affect electroweak precision ob-
servables, but constraints arising from electroweak precision data [211] are not competitive
to the above discussed ones and thus we do not include them in our discussion.
III.4 Projections for the HL-LHC
In this section we explore the di-lepton and di-tau signatures with missing transverse
energy (denoted as ET ) due to non-detectable Ni from σ
± decays in the final state (see
Fig. III.2):22 {
±
→ + − → σ → `
±Ni (` = e
± orµ±) ,
p p σ σ
σ± → τ±
(III.11)
Ni .
For these event topologies, ET∣∑is calc∣ulatemomenta, ∣∣ ∣∣ ∣∣∣∑
d b√y summing ove∣r the observed transverse∣
ET ≡ pi = (piT x)2 + (pi 2y) ∣ . (III.12)
vis vis
The former signature, for instance, was already applied to the ScM in Ref. [212], where
the authors used data from LHC Run-1 to set their limits. Our aim is to extend this
search by calculating the projected sensitivities for the HL-LHC using the same analysis
techniques as presented in recent ATLAS publications, [213] and [214] for the di-lepton
and di-tau channel, respectively. In pa√rticular, we are using the results from Run-2 with an
integrated luminosity of 36.1 fb−1 at s = 13 TeV.23. Furthermore, we will present both
optimal and realistic projections; the first one is defined such that the BR for a charged
scalar decaying into a RHN and a charged lepton considered in the search is set to 1.
This case is, however, not feasible in our model, due to the requirement to reproduce the
observed neutrino mass differences and mixing angles. Therefore, we also define realistic
projections by maximizing the respective Yukawa couplings. This will lead to BRs smaller
than one.
The model files were created with FeynRules [216]. The signal processes, as shown in
Fig. III.2, were simulated at leading order with MadGraph5 aMC@NLO 2.6.3.2 [217] inter-
faced with Pythia8 [218] for showering the events and Delphes 3.4.1 [219] was used for
a fast detector simulation. By using these tools, we were able to reproduce the results
given in Refs. [213, 214].
In Fig. III.3 we show the expected cross section for p p → σ± σ∓ pair production, σprod,
for different center-of-mass energies and masses. One can already conclude that large
luminosities are needed to see a significant number of events inside the detector. For
definiteness we have fixed the portal couplings to
λ = 0.3 , λ = 0.5 , λ = 10−43 4 5 . (III.13)
For us, the most relevant parameter in the scalar sector is the physical mass of the charged
scalar, m±, and it is this quantity that will appear in all our sensitivity projections.
22 We did not consider the additional A0 and S0 production, because they are sub-dominant compared to
σ± pair production.
23 Meanwhile, the ATLAS collaboration published new results for the di-lepton final state based on a search
at 139 fb−1 [215]. Although this analysis can place stronger limits on the parameter range studied, we
have not explicitly included it because we are primarily focused on future experiments. With an in-depth
analysis, it may be possible to test at least a small portion of the parameter range of our model with
available data.
38
III.4. PROJECTIONS FOR THE HL-LHC
g Nl q Nl
`∓ `∓
σ∓
j ∓ jσ
h γ/Z
σ± σ±
`± `±i i
g Nk q Nk
Figure III.2: Production channels for the `i `j +ET process at the LHC. Pair produced
charged scalars decay into heavy leptons N2,3 and charged SM leptons (e, µ or τ).
103 s = 13 TeV
101 s = 100 TeV102
101
100 100
10 1
10 1 10 2
20 40 60 80 100 200 400 600 800 1000
s  [TeV] m ±  [GeV]
Figure III.3: Cross section σprod for pair production of charged scalars σ±. The left panel
shows the increase of σprod for different center-of-mass energies and fixed scalar mass,
m± = 400 GeV. The black (blue) vertical lines indicate the energy range of the LHC
(FCC-hh) and dashed lines indicate to the respective cross section of the experiment. In
the right panel we show σprod for different scalar masses and fixed energy. By increasing
m± the cross section drops significantly.
We assume that mN2 = mN3 in the following, but in general we could take a hierarchical
spectrum as well, i.e. mN2 < mN3 or vice versa. A hierarchical spectrum would, however,
weaken the search strategy because in this case decays σ± → `±N3 are more likely to yield
soft leptons due to the smaller mass gap between σ± and N3. On the other hand, this
could give rise to interesting event topologies, because N3 can decay inside of the detector
into two leptons and N2 and this would possibly create multi-lepton + ET signatures.
However, we are going to focus on the di-tau + ET and di-lepton + ET searches within
the degenerate spectrum.
III.4.1 Di-tau+ET signature
Following the procedure outlined in Ref. [214], the following cuts were applied after event
reconstruction: events shall contain no b-jet but at least two tau leptons with opposite
charges. The invariant mass of every tau pair has to be larger than 12 GeV and must be
10 GeV away from the mean visible Z boson mass, set at 79 GeV.24 Then, two different
trigger setups are defined. The asymmetric trigger requires p1T > 85 GeV and p
2
T > 50 GeV
for the first two pT ordered tau leptons. Second, there is theET trigger set by p
1
T > 35 GeV,
p2T > 25 GeV and ET > 50 GeV. For further discrimination from SM background, the so-
24 The mean visible mass of the Z boson is smaller than its physical mass, because is reconstructed from
a sample of Z → τ+τ− events [214].
39
prod [fb]
prod [fb]
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
called stransverse mass [220, 221], mT2 , is introduced. It is of particular interest in event
topologies where pair produced particles decay into a directly visible final state and an
invisible one. It is defined by [ { }]
m2T ≡√min max m2T (p 22 T`− ,p1),mp +p =p T (pT`+ ,p2) (III.14)1 2 T
with mT (pT ,qT ) = 2(pT qT − pT · qT ),
where we have to minimize the larger of the two transverse masses mT using the two-
dimensional transverse momenta p1 and p2 such that their sum reproduces the observed
missing momentum pT . Further, pT`− and pT`+ are the transverse momenta of the visible
charged leptons.
Finally, two signal regions based on mT2 andET cuts were defined as shown in Table III.1.
They are chosen such that different mass gaps between σ± and N2,3 can be covered, due
SR-lowMass SR-highMass
mT2 > 70 GeV mT2 > 70 GeV & m(τ1, τ2) > 110 GeV
ET trigger ET trigger asymmetric trigger
ET > 150 GeV ET > 150 GeV ET > 110 GeV
p1T > 50 GeV p
1
T > 80 GeV p
1
T > 95 GeV
p2T > 40 GeV p
2
T > 40 GeV p
2
T > 65 GeV
Table III.1: Signal regions with the corresponding cuts on final state momenta used in the
di-tau analysis. The definition of the respective triggers are explained in the text.
to the different cuts employed on the pT of the tau final states. The 95% CL upper limits
on the cross sections are summarized in Table III.2.
signal region Nobs obs. σ
95
vis [fb]
SR-lowMass 10 0.26
SR-highMass 5 0.20
Table III.2: 95% CL limits on the non-SM cross section for the di-tau + ET analysis.
The final cross section in our model is determined by the product
σ(p p→ `±`∓N N ) = σ(p p→ σ±σ∓i j k l )× BR(σ± → `±N ∓ ∓i k)× BR(σ → `j Nl). (III.15)
The cross section can be increased by maximizing the respective BRs. The latter can be
achieved by making use of unconstrained parameters. In particular, we took the complex
angle θ = ω − iη (see Eq. (III.7)), the Majorana phase α2 and the CP phase δ as free
parameters25 and maximized the expression
y2τ (ω,∑η, α∑, δ)22 + y3τ (ω, η, α 22, δ) . (III.16)
yki(ω, η, α2, δ)2
k=2,3 i
The parameter values that correspond to the extremum are in what follows denoted as
ω0, η0, α
0
2, δ0. We have performed this procedure for both NO and IO neutrino masses and
the corresponding minimization results are given in Table III.3.
For calculating the sensitivity curves we fixed the portal couplings λi (see Eq. (III.13))
25 We allowed δ to float in the range (135◦, 366◦) which corresponds to a 3σ range from recent fits. We
found that the second Majorana phase α1 does not affect the minimization.
40
III.4. PROJECTIONS FOR THE HL-LHC
ω η α00 0 2 δ0 BR(σ
± → τ±Nk)
NO 1.61 > 2 π 2π 38.26 %
IO 1.31 > 2 −π π 27.30 %
Table III.3: Largest possible BRs for the decay of σ± into τ and N2,3. Above the given
value for η0, the BRs are to a good approximation independent of this parameter. As can
be seen, the IO gives rise to smaller BRs compared to NO.
such that σ± is the lightest Z2-odd scalar, forbidding possible decays into the other scalars.
We scanned over the charged scalar mass as well as the heavy lepton masses mN2,3 ; our
grid spans m± ∈ (150 GeV, 600 GeV) and mN2,3 ∈ (10 GeV, m±) and we simulated 104
events for each point.
We compared the simulation with the ATLAS results derived at 36.1 fb−1 and found that
the corresponding sensitivities are not strong enough to place limits. The reason is twofold:
first, the cross section for the pair production in the model is significantly smaller than the
one in the simplified model used in the ATLAS analysis. Second, the analysis uses specific
cuts on kinematic variables which do suppress SM background but unfortunately also cut
away a significant portion of signal events. For instance, the “best case” benchmark point,
where we set the BR into tau leptons equal to one) features only a small surviving cross
section:
mN2,3 = 10 GeV, m± = 200 GeV ⇒ SR-highMass: σvis = (0.15± 0.06) fb.
Taking the Casas-Ibarra parametrization into account and inserting model-dependent
BRs, the situation gets even worse as the cross section is further reduced due to
non-maximal BR into tau leptons.
This finding motivated us to go beyond the current experimental results and consider a
similar search at the foreseen HL-√LHC facility at CERN which will deliver a final integrated
luminosity of up to 4000 fb−1 at s = 14 TeV [222]. This would lead to a huge increase in
potential signal events. To estimate the potential of HL-LHC to test the ScM we conduct
a similar analysis as in Ref. [214] but use a projected sensitivity,
S = √ S , (III.17)
S +B
instead, where S and B represent signal and background events, respectively. This formula
is derived in the limit S/B  1 fro√m t(he general e(xpressio)n)for the case of exclusion limits,
S S1 = 2 S −B log 1 + , (III.18)
B
obtained using the procedure described in Ref. [223].
By using the same signal regions as in the previous analysis and assuming a similar scaling
of signal and background for the increased center of mass energies and luminosities we can
now redo the cut and count analysis for the increased event rates. As can be seen in
Fig. III.4, where we show the corresponding exclusion limits as solid lines, this allows us
to significantly enhance the testable parameter space. For such high luminosities, scalar
masses of up to 420 GeV and respective RHN masses of 170 GeV can be tested and there
is even a potential discovery region for scalar masses between 200 and 300 GeV. The sharp
drop for large masses is due to a decrease in the pair production cross section. The cuts
on the kinematic variables, such as transverse momentum pT and stransverse mass mT2,
41
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
need a sufficiently large mass gap between σ± and mN2,3 which bounds the accessible
parameter space from above and also from the left, because charged scalar cannot not be
too light, as in this regions leptons are too soft. We also show in dashed the corresponding
sensitivity curves for the optimal case where the BR into tau is equal to one. As expected,
the sensitivities improve in this case, however we have not found such scenario in our
numerical procedure (see Table III.3); the BRs for the decay into pair of tau leptons can
be at most around 40%.
250
S=2, BR=1
S=5, BR=1
S=2, CI-BR
200 S=5, CI-BR
150
100
50
200 300 400 500
m± [GeV]
Figure III.4: Projected sensitivities for HL-LHC using L = 4000 fb−1 and the same analysis
techniques as in Ref. [214]. Given in blue are exclusion limits, S = 2, and shown in red
are discovery limits, where S = 5. The dashed lines correspond to a 100% BR into tau
leptons using the Casas-Ibarra parametrization, whereas the solid lines represent the case
in which the maximized BR for NO (shown in Table III.3) is employed.
III.4.2 Di-lepton + ET signature
Now we turn our attention to the di-lepton +ET channel: following the procedure outlined
in Ref. [213], the following cuts were applied after event reconstruction and preselection:
the invariant di-lepton mass m`` should be larger than 40 GeV. Events should not contain
any b-jet with pT > 20 GeV nor a jet with pT > 60 GeV. In the 2` + 0jets channel, six
different signal regions were defined: four are aiming for different flavor (DF) leptons in the
final states and two for leptons with the same flavor (SF). All regions are inclusively defined
and mainly separated by increasing cuts on the invariant mass of the lepton pair and mT2,
ranging from m`` > 110 GeV to m`` > 300 GeV and mT2 > 100 GeV to mT2 > 300 GeV.
The 95% CL upper limits on the cross sections are summarized in Table III.4.
In contrast to Section III.4.1 we now want to minimize the expression given in Eq. (III.16)
in order to maximize the possible BRs into leptons. Under the same assumptions as
before and taking both, NO and IO regimes into account we obtain the results given in
Table III.5. From the respective BRs we can calculate the suppression factor B of the pair
production cross section a∑ccording to ∑
B ≡ BR(Nk`i)2 + BR(Nk`i)BR(Nl`j). (III.19)
k=2,3 k,l=2,3
i=e,µ i,j=e,µ
k 6=l∨i 6=j
42
mN [GeV]2,3
III.4. PROJECTIONS FOR THE HL-LHC
signal region m`` [GeV] mT2 [GeV] N
95
obs obs. σvis [fb]
SF-loose > 100 > 111 153 2.02
SF-tight > 130 > 300 9 0.29
DF-100 > 100 > 111 78 0.88
DF-150 > 150 > 111 11 0.32
DF-200 > 200 > 111 6 0.33
DF-300 > 300 > 111 2 0.18
Table III.4: 95% CL limits on the non-SM cross section for the di-lepton + ET analysis.
ω0 η
0
0 α2 δ0 BR(σ → `iNk)
NO 2.37 < −2 π 2π 86.42 %
IO 3.07 < −2 −π π 99.75 %
Table III.5: Largest possible BRs for the decay of σ± into e±, u±. Below the given value for
η0, the BRs are to a good approximation independent of this parameter. Interestingly, the
IO regime can feature a situation with a very small BR into tau’s, implying approximate
zeros in the third column of the Yukawa matrix.
The first sum corresponds to SF lepton channels, whereas the second term sums over
DF leptons as final state particles. We introduced the shorthand notation BR(Nk`i) ≡
BR(σ± → `±i Nk). Inserting for instance the BR for NO (Table III.5), the final suppression
factor is B ≈ 75%. While IO yields larger B values, we present our results for NO, as
we did in Section III.4.1, since NO is favored by roughly 3σ by a global fit analysis of
neutrino oscillation data [210]. The sensitivities are shown in Fig. III.5 and by comparing
S=2, BR=1
400 S=5, BR=1
S=2, CI-BR
S=5, CI-BR
300
200
100
200 300 400 500 600 700
m±[GeV]
Figure III.5: Projected sensitivities for HL-LHC using L = 4000 fb−1 and the same analysis
techniques as [213]. Given in blue are exclusion limits, S = 2, and shown in red are
discovery limits, where S = 5. The dashed lines correspond to optimal BR into leptons,
whereas to obtain the solid lines we used the value for NO given in Table III.5.
them to the results from Fig. III.4 it can be seen that the di-lepton search offers stronger
sensitivities for the parameter space. This is a result of the much cleaner signature resulting
43
mN [GeV]2,3
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
from light leptons in the final state. Unlike taus, they do not decay hadronically and can
be reconstructed more easily, which prevents misidentification of events.
The di-lepton search constrains scalar masses up to 650 GeV and it is interesting to go
beyond the energy range of the HL-LHC and consider proposed future hadron and lepton
colliders to check how much of the parameter space can be probed in the future. In the
following, we will stick to the di-lepton search only, as the results from this section clearly
indicate stronger sensitivities compared to the di-tau analysis.
III.5 Projections for future colliders
The HL-LHC projections presented in Section III.4 allow us to test a sizeable region of
parameter space below O(few 100 GeV) for scalar masses, but to reach the TeV-scale
and go even beyond requires future hadron and lepton colliders. We will discuss the
corresponding projections in the following section.
III.5.1 FCC-hh
We start by discussing the ScM in the context of a future circular hadron collider, dubbed
FCC-hh [224]. For this purpose, we follow Ref. [225], where an opposite-sign-di-lepton
(OSDL) final state with missing energy is discussed with the goal of finding TeV-scale
winos and light binos. In contrast to the previous analyses, the following variables were
used in the analysis to define cuts: first, Meff, which is the scalar sum of the pT of leptons,
jets and missing transverse energy, ∑ ∑
Meff = pT + pT +ET . (III.20)
leptons jets
Based on this variable, M ′eff = Meff− pT (`1) is introduced, where pT (`1) is the pT value of
the harder of the two final state leptons. Further, the invariant mass of the same flavor
opposite sign (SFOS) lepton pair, mSFOS, is used and, finally, the transverse mass mT .
We simulated signal events for different mass parameters using the same pipeline as in
Section III.4. To ensure that our simulations are comparable to those in Ref. [225], the
most dominant backgrounds from di-boson intermediate states, WW and WZ, were also
simulated and compared to the cut flow given in Ref. [225]. Our results are presented in
Table III.6. Besides the cuts explicitly mentioned, the Baseline cut requires additionally
that SFOS di-leptons have mSFOS> 12 GeV and that the mSFOS closest to the Z mass,
mZ , denoted by mSFOS(Z), fulfills |mSFOS − mZ | > 30 GeV. Further, the transverse
momentum of the lepton pair, pT (``), should be larger than 30 GeV. Finally, it is required
that either ET > 100 GeV or that pT (`1) > 100 GeV.
Since we generally find that the number of events for signal and background are of the
same order, the significance given in Eq. (III.17) is not a good approximation and, hence,
for exclusions we use Eq. (III.1√8),(while for disc(overies w)e emp)loy [223]
S S0 = 2 (S +B) log 1 + − S . (III.21)
B
We summarize our results on the parameter space exclusion capability at FCC-hh with
two different luminosities, L = 3 ab−1 and L = 30 ab−1 in Fig. III.6. Both contours
corresponding to S1 = 2 and S0 = 5 are derived for the maximal BRs into electrons and
muons for the case of NO (see Table III.5). This scenario is dubbed “best case” in Fig. III.6
and it can be inferred that for such couplings FCC-hh will provide the possibility to probe
a large portion of parameter space. In the final luminosity stage, it will be possible to test
scalar masses up to 2 TeV and RHN masses of 1.4 TeV.
44
III.5. PROJECTIONS FOR FUTURE COLLIDERS
Cut S B S0 S1
Baseline 351 5.9× 105 0.5 0.5
M ′eff > 1100 GeV 90 625 3.5 3.4
mT (ET , l1 + l2) > 1100 GeV 90 234 5.5 5.3
ET /Meff > 0.36 33 63 3.9 3.6
pT (l2)/pT (l1) > 0.24 19 18 3.8 3.4
Table III.6: Cuts made for distinguishing signal and background at FCC-hh with a lu-
minosity of 3 ab−1. We show the number of signal and background events for mN2,3 =
500 GeV and m± = 1 TeV. In contrast to Eq. (III.13), λ3 = −0.27 was used in this
analysis. No systematic errors on the background were assumed for this analysis.
1400 S1=
-1
2@3ab
= @ -1S0 5 3ab
-1
S1=2@30ab
= @ -11200 S0 5 30ab
1000
800
600
400
200
500 1000 1500 2000
m±[GeV]
Figure III.6: Sensitivity of FCC-hh with L = 3 ab−1 (30 ab−1) is shown with solid (dashed)
lines. The analysis is based on a proposed search for a supersymmetric (SUSY) model
presented in Ref. [225]. The red and blue curves correspond to the “best case” scenario
with maximized couplings to e, µ, indicating that a significant portion of the parameter
space can be probed at FCC-hh. The situation further improves for larger luminosity.
The thin black line corresponds to mN2,3 = m±.
III.5.2 CLIC
The Compact Linear Collider (CLIC) [226] is a proposed e+e−√ collider that will operate
in three stages with s= 380 GeV, 1.5 TeV, and 3 TeV, respectively [226, 227]. In the
following, however, we restrict ourselves to the latter case as it offers the possibility to
test the largest parameter space of Z2-odd particle masses in comparison to the two other
stages. As for the FCC-hh analysis, we consider the di-lepton + ET signal.
At e+e− colliders there are two complementary processes to produce σ± in our model
as shown in Fig. III.7: one is through the exchange of a Z boson or a photon in the
s-channel;26 another possibility is via RHNs in the t-channel. In the latter case, the
26 In this case, the s-channel Higgs exchange diagram is negligible, because of the small Higgs-electron
Yukawa couplings.
45
mN [GeV]2,3
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
production cross section is proportional to the corresponding Yukawa coupling ym1 and
we have checked that this production channel is subdominant for the size of Yukawa
couplings employed in this work.
e+
∓
Nl e+ `j
σ∓
`∓
σ∓
j
Nl
Nm
γ/Z
± Nkσ
`±i
− σ
±
e−
±
Nk e `i
Figure III.7: Production channels for the `i `j +ET process at CLIC. The left process is
similar to hadron colliders, except for electrons in the initial state. Additionally, produc-
tion via e+e− collision can happen via a RHN mediated t-channel diagram, although it is
suppressed due to Yukawa couplings y.
The most outstanding advantage that lepton colliders offer with respect to hadron colliders
is the clean signal; without parton distribution functions to be considered, missing energy
and momentum can be reconstructed to a large precision and the distributions of the
corresponding kinematic variables are not as smeared out as for parton collision where
a wide range of energies is involved. Furthermore, background events due to quantum
chromodynamics (QCD) processes are reduced compared to pure electroweak processes
which is particularly relevant for our search analysis.
For general background rejection, we make the following preselection cuts: to remove QCD
processes, we require the final state to contain no jets. Furthermore, we require exactly
two leptons of opposite charge and that ET exceeds 100 GeV. The last cut suppresses
most of the e+e− → `+ `− background.
The dominant background after the preselection cuts is e+e− → `+`−ν ν̄ and in the
following this is the only background process we consider. In the case of τ leptons in the
final state, only those events in which τ decay leptonically are considered; thus in this
case there are four final state neutrinos. We have checked that any other SM background
gives a negligible contribution after the above preselection cuts. A very promising result
for the exclusion limits was found by choosing cuts as given in Table III.7.
The variables mSFOS and Meff, already employed for our FCC-hh analysis, were
reused for CLIC because of their great potential for discriminating the ScM, with its
comparatively large masses of Z2-odd particles, against the SM. Their cuts are pa-
rameter dependent, because they are derived by maximizing S1 for different parameter
choices in the considered mass range. Furthermore, the pseudorapidity of the first lep-
ton, η(`1), turns out to be a very useful variable; it peaks at large values for the SM
background while most of the signal events are more central in this variable. It is de-
fined in terms of the angle φ between lepton track and beam pipe by η(`1) = − ln tan(φ/2).
A further suppression of the background can be made by utilizing the kinematics of this
process. In the di-lepton search at CLIC, we can reconstruct the four-momenta√of the two
RHNs (denoted by pµ3,4 in what follows) by using only the initial state energy s and the
known four-momenta of the two charged leptons (p∑µ1,2). Four ou√t of these eight unknowns
are fixed by four-momentum conservation, namely 4 µi=1 pi = ( s , 0 , 0 , 0). Furthermore,
46
III.5. PROJECTIONS FOR FUTURE COLLIDERS
SR “best case”
Cut S B S0 S1
preselection 4101 1.0× 106 4.1 4.1
Meff > 0.5µ2 − 1.2mN2,3 + 1000 GeV 3442 2.7× 105 6.6 6.6
mSFOS > −0.1µ2 − 0.3mN2,3 + 530 GeV 3260 2.2× 105 6.9 6.8
|η(l1)| < 0.6 2502 2.1× 104 16.9 16.6
kinematics 2136 908 55.6 45.6
SR “worst case”
Cut S B S0 S1
preselection 1188 1.0× 106 1.2 1.2
mSFOS > −0.3mN2,3 + 130 GeV 1153 9.7× 105 1.2 1.2
|η(l1)| < 0.6 800 1.2× 105 2.3 2.3
kinematics 386 2806 7.1 7.0
Table III.7: Signal regions used for an analysis at CLIC with a center of mass energy of
3 TeV and a luminosity of 5 ab−1. We show the number of signal and background events
together with the corresponding sensitivities for a benchmark point mN2,3 = 500 GeV and
m± = 1 TeV. The results are shown both for “best” and “worst” case scenarios which
correspond to maximizing couplings to e, µ and τ lepton, respectively.
imposing that all intermediate and final state particles are on-shell yields the following
relations:
(p1 + p3)
2 = m2± = (p2 + p4)
2 , p23 = p
2
4 = m
2
N . (III.22)2,3
In total, we end up with a solvable system of equations. For separating signal and back-
ground events we use that the latter ones typically have different kinematic properties,
since the mass of the intermediate particle is for instance set by the W boson mass. Af-
ter requiring that there is a physical solution to the aforementioned equations, i.e. we
demand the momenta of the invisible particles to be real valued and that there is no real
valued solution if m2± and m
2
N in Eq. (III.22) get replaced with the W boson and active2,3
neutrino mass, respectively, we reach the following effect: the number of signal events
typically decreases only by ' 30%, whereas background is strongly reduced, at most only
a few percent of such events survive this cut. In reality, the four-momenta in Eq. (III.22)
are only known to a finite precision. So it is crucial to include the finite detector res-
olution by using the respective Delphes card for the CLIC detector which is based on
Ref. [228]. For this reason, the simulated pT values experience a smearing with respect to
the detector resolution. This last step allows for an efficient background suppression as
can be seen in Table III.7.
We wish to stress that the cuts on mSFOS and Meff are not optimal for each parameter
point since we only used a simple function of the model parameters. Still, the resulting
exclusion limits, shown in Fig. III.8, already indicate the great potential for testing this
model setup at CLIC. For this case, S1 = 2 and S0 = 5 curves reach similar limits for
these two cases. This is a consequence of the aforementioned use of kinematics: due to the
suppression of background events, we find large sensitivity values across the parameter
space. Although the “worst´ case” features fewer event rates due to smaller BRs, it still
yields a sizable sensitivity after cutting the background, in contrast to FCC sensitivities.
Even for the “worst case” with a large number of surviving background events, system-
atic uncertainties are negligible, because according to Ref. [226], they amount only to 0.3%.
47
CHAPTER III. COLLIDER STUDIES OF THE SCOTOGENIC MODEL
S1=21200
S0=5
S1=2, Di-τ
S0=5, Di-τ
1000
800
600
400
200
200 400 600 800 1000 1200 1400 1600
m±[GeV]
Figure III.8: CLIC sensitivity for the di-lepton search. Using maximized couplings to e, µ
we obtained the red solid contour that corresponds to a 5σ discovery and the blue one
that represents 2σ exclusion. The corresponding dashed contours are for the case where
τ couplings are maximized. The thin black line indicates mN2,3 = m±.
In comparison with FCC-hh sensitivities at 3 ab−1, CLIC can test for higher RHN masses
while being less sensitive to σ± masses larger than 1400 GeV. However, with a total center
of mass energy of 3000 GeV this is expected. Overall, CLIC offers a testable parameter
space which is comparable to the FCC-hh result. While the reach at CLIC is not as large
in comparison to FCC-hh with 30 ab−1, we note that, unlike FCC-hh, CLIC can nearly
close the available kinematic window in the vicinity of mN2,3 = m±.
In conclusion, we have shown in this section that both lepton and hadron colliders offer
promising and complementary ways to look for the considered ScM spectrum.
48
mN [GeV]2,3
III.6. SUMMARY OF CHAPTER III
III.6 Summary of Chapter III
The ScM is a very popular extension of the Standard Model which can explain both neu-
trino masses and the origin of DM. In this chapter we focused on the case of keV-scale
fermionic DM with the mass of remaining Z2-odd fermion and scalar degrees of freedom
at O(100) GeV and explored the collider phenomenology of this model setup. In partic-
ular we studied p p → σ±σ∓ → `±`∓ +ET channels. We demonstrated that testing this
model at colliders strongly relies on forthcoming stages of the LHC as well as future col-
liders since we were not able to extract robust bounds by using 36.1 fb−1 data. Although
the LHC delivered now 139 fb−1 of data at the en√d of its second run, we were consider-
ing HL-LH√C with a final luminosity of 4 ab
−1 at s = 14 TeV, FCC-hh with 3 ab−1 and
30 ab−1 at s = 100 TeV and CLIC with 5 ab−1 at a center-of-mass energy of 3 TeV to set
parameter limits as large as possible. For the HL-LHC, we have found that the testable
region covers a range up to 650 GeV for scalar and up to 350 GeV for RHN masses in the
di-lepton + ET analysis which gives stronger constraints compared to the di-tau + ET
signature. Designing our own analysis setup to extract constraints based on the future
experiment CLIC and FCC-hh we were able to probe a large region of parameter space.
The sensitivity reach of CLIC exceeds 1 TeV for both RHN and charged scalar masses
and FCC-hh with 3 ab−1 would reach similar limits for the scalar and even larger RHN
masses, and this further improves for 30 ab−1 where RHN masses up to 2 TeV would be
probed. In particular, we found that CLIC offers the potential to test a compressed ScM
spectrum, which, as we discussed in Section III.2, is of particular interest to address sev-
eral open questions of the SM and explain them in a single BSM framework.
Despite the great discovery and exclusion potential of these experiments, they are nearly
independent of the underlying light DM properties. As discussed in Section V.2, N1 effec-
tively decouples from the spectrum and thus the chance to observe it directly at a collider
are virtually absent, as long as the respective BR of σ± → N `±1 α is negligibly small. We
will see in the next section, that this is guaranteed by the very small Yukawa couplings
required to generate the observed DM abundance. Only by inverting the mass spectrum
such that m± < mN2,3 , or for DM with masses of O(MeV), there is the possibility to have
σ± particles which travel a few mm before decaying, hence offering the potential for ded-
icated displaced vertex studies. On the other hand, DM can be pair produced at e+e−
colliders, but again the smallness of the corresponding N1 coupling would make a suc-
cessful ET +X search, where X can be for instance a photon or a jet, very unlikely.
Luckily, such light DM candidate with masses O(10 keV) leaves an imprint in the history
of the early Universe and the formation of structures and we are will show in the next
chapter how we can use cosmological observations to constrain the model parameter space
further, taking the nature of DM into account as well.
49

Main part II: warm dark matter in
cosmology: from models to
a full picture
51
MAIN PART II
In the first part of the thesis we have seen that it can be quite challenging to observe light
and only weakly coupled DM at collider experiments. In the context of the scotogenic
model (ScM), collider searches allowed to place limits only on the heavy particles in the
model spectrum up to 2 TeV. Direct experimental validation of such light DM candidates
requires different setups and search strategies. For example, there are dedicated experi-
ments at the high-luminosity frontier that do not aim for the highest energies, but aim to
achieve as many particle collisions as possible to detect rare interactions and elusive par-
ticles. Another setup are beam-dump experiments, where highly energetic protons are
dumped into dense material to absorb their remnants as much as possible and filter out
long-lived stable or at least sufficiently long lived particles which are detected at a dis-
tant detector.
Nevertheless, these searches require that a potential light DM candidate has at least
some coupling to the SM in order to detect it. If it interacts only gravitationally or
evades detection due to too weak couplings, it will still be possible to observe its effects
in cosmological observations (if light enough). In the Standard Model of cosmology, the
ΛCDM, DM is supposed to be non-relativistic throughout the history of the early Universe
and thus it plays a crucial role as the dominant driving force responsible for the formation
of structures. On the other hand, light DM can stay relativistic during a significant
amount of time in the early Universe which in turn leads to a suppression of structures
below a certain scale. Observations targeting the matter profile of the Universe across
a wide range of scales allow to search for deviations from the standard scenario due to
light DM. Conversely, such surveys can be used to obtain limits on the allowed light DM
parameter space; in particular, this can be used to exclude models that would induce more
suppression of structures as observed.
Chapter IV is dedicated to explore the phenomenology of light DM in the context of the
ScM. We study the impact of late-time produced keV-scale right-handed neutrinos on the
formation of light nuclei during Big Bang Nucleosynthesis (BBN) and the suppression
of the corresponding matter power spectrum. Further, the contribution to the effective
number of relativistic particle species, Neff is calculated and we overlap our findings with
the results from Chapter III. In Chapter V we consider a light axion-like particle (ALP) DM
candidate produced via a freeze-in mechanism in two different scenarios: either the ALP is
coupled to photons or SM fermions. Using a half-mode analysis and counting the number
of MW subhalos we can place limits on the allowed ALP mass. Finally, in Chapter VI
we are following a model-independent as possible approach to analyze DM models, where
different production mechanisms can lead to DM subsets with two associated temperatures.
The results allow to adapt our limits derived from a MW subhalo count and Lyman-α data
for a wide setup of such models.
52
Chapter IV
The scotogenic model and cosmology
IV.1 Introduction
In the previous chapter, we have shown that future collider facilities are a promising way
to test the scotogenic model (ScM) which is a phenomenological interesting extension of
the SM. Although these searches will be able to constrain the masses of the new scalars
and right-handed neutrinos (RHN), they do not offer a way to probe the lightest particle,
N1 which is considered to have a mass of O(10 keV). In fact, from the point of view of ex-
perimental validation by these experiments, this particle is effectively detached from the
spectrum and the obtained collider limits are essentially independent of the DM proper-
ties.
An intriguing and complementary way to constrain DM in the mass window we consider
is to turn the gaze away from Earth-based experiments and towards the stars. Informa-
tion on the distribution of matter in the Universe obtained from various surveys, such as
Lyman-α forest probes allow to gather information on structures at different scales and
constrain the light DM parameter space. Additionally, light DM which stays relativistic
for sufficiently long times can contribute to the effective number of relativistic particle
species, Neff . We have to consider two different DM production mechanisms: first, we
can produce N1 via a freeze-in of thermalized Z2-odd scalars and, second, via late-time
three-body decays of heavy RHNs N2,3. The latter turns out to create highly energetic, i.e.
“hot” DM and thus can have a drastic impact on the considered cosmological quantities.
Further, these decays have to happen before the formation of light nuclei takes place dur-
ing the epoch of big bang nucleosynthesis (BBN) starting at T ≈ 1 MeV, or otherwise the
abundances of newly formed nuclei are altered by the presence of relativistic DM particles.
This chapter is organized as follows: in Section IV.2 we will review the DM production
and explain the constraints from cosmology in Section IV.3. Afterwards, we combine
our results from collider searches and limits on the parameter space from cosmological
observations in Section IV.4. We summarize our results in Section IV.5.
IV.2 Light dark matter in the scotogenic model
Since the lightest of the newly introduced particles is stable, it is natural to consider
whether it can account for the observed amount of DM in the Universe. In the ScM
there are neutral particles both in the fermionic and scalar sector, making them poten-
tial candidates. Motivated by the results in the previous chapter, we consider fermionic
DM and keep the scalars heavier than the RHNs. It was shown in Refs. [2, 182] that in
the ScM with fermionic DM with mass O(100 GeV), the relic abundance from freeze-out
generally strongly exceeds the measured values. There are, however, options how to solve
53
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
this problem: first, if additional processes, namely coannihilations of heavy DM with the
other new particles, are involved, DM can stay longer in the thermal equilibrium and
freeze-out with a much smaller abundance (see for instance Fig. 2 in Ref. [2]). The coan-
nihilations are only effective if the splitting between the DM and scalar mass is tiny.
However, considering searches with two leptons in the final state stemming from decays
of pair-produced new scalars (see Sections III.4 and III.5), such a scenario would yield
soft leptons which are hard to reconstruct and hence freeze-out of O(100 GeV) DM is not
compatible with the signatures at hadron colliders that were studied in Chapter III. This
conclusion changes for the case of the future lepton collider CLIC, for which we have
shown in Section III.5 that this regime can be probed in principle.
Second, the overproduction problem can be solved by considering light, non-thermally
produced DM. Such DM can be produced either via freeze-in [76, 229] via decays of neu-
tral and charged scalars in the Σ doublet or by decays of frozen-out next-to-lightest RHN,
i.e. N .272 However, the latter mechanism is also constrained by requiring N2 to decay be-
fore the time of BBN. Namely, if N2 is too long lived, the abundances of light nuclei will
be altered. This production mechanism also leads to too hot momentum distributions
and therefore needs to be subdominant with respect to the scalar decay contribution. We
elaborate on this in the present section.
Generally, in the freeze-in scenario there is some freedom to choose the DM mass. On the
contrary, in this model there is an upper bound from demanding that the DM production
has stopped before BBN takes place, as will be discussed in the last part of Section IV.3.
From the point of view of freeze-in DM production, a RHN N1 with masses up to few MeV
is a perfectly viable DM candidate, as there are no direct upper mass limits. However,
in the following we will consider the DM mass to be O(10 keV). This particular choice is
motivated by our findings in the previous chapter, where we identified an interesting open
window in parameter space to successfully incorporate a resonantly enhanced leptogenesis
mechanism as well. At the end of Section IV.5 we will come back to the question of DM
mass in the context of interpreting our cosmological limits.
IV.2.1 Dark matter production mechanisms
The processes through which keV-scale N1 are frozen-in are (see left figure of Fig. IV.1),
A,S → N ± ±1 να , σ → N1 lα . (IV.1)
The corresponding Boltzmann equation for the DM yield, YFI, which is the ratio of DM
number density and entropy density(, reads [2, 76], )
dY 2FI 135M0 |y1|
= r3 2K (r) + 3 K ( r) + 3 K ( r) . (IV.2)
dr 64π5 s
1 1 A 1 S
(g∗)
3/2m A S±
Here, M0 ' 7.35 × 1018 GeV, and the number of effective entropic degrees of freedom
(DOF) across relevant temperatures is fixed to gs s∗(T ) ≡ g∗ = 114.25, taking new particle
DOF into account. For simplicity, we assumed that the Yukawa couplings of N1 are flavor
universal, i.e. y1α ≡ y1. Furthermore, the abbreviations A ≡ mA/m± and S ≡ mS/m±
are introduced to account for all three scalar decay production channels.28 K1(r) is the
modified Bessel function of the second kind and a dimensionless temperature r = m±/T
is introduced. In the computation, we use Maxwell-Boltzmann (MB) distributions for all
27 A similar scenario was already outlined and discussed in Ref. [182], where the authors presented the
available parameter space for freeze-in production.
28 In this chapter we fix the scalar couplings to the same values as given in Eq. (III.13).
54
IV.2. LIGHT DARK MATTER IN THE SCOTOGENIC MODEL
thermalized Σ particles. Finally, we assume the initial DM number density to vanish and
this simplifies the computation of the DM abundance, because, instead of solving a differ-
ential equation, a straightforward integration is possible.
We can obtain the expression for YFI by simply integrating Eq. (IV.2) between the tem-
perature at the end of inflation and the present one, where the former is associated to
the reheating temperature which is assumed to be larger than all particle masses in the
model. Practically, this allows us to use x = 0 and x = ∞ as the respective integration
boundaries. We obtain
405M |y |20 1 2AS + S + A
YFI = . (IV.3)
128π4 (gs)3/2∗ m± A S
Taking λ4 = λ5 = 0 for a moment allows to simplify above expression by S = A =
1 and by using the relation between DM yield and relic abundance, Ω h2FI = 2.742 ×
102 (mN1/keV) YFI, we arrive at (the analytical)es(timate f)or(the DM) relic abundance,2
Ω h2 ≈ |y1| mN 1 TeVFI 0.12 1× − , (IV.4)2.36 10 8 1 keV m±
from which we infer that in order to have scalar decays as a dominant DM production
mechanism, the required DM Yukawa couplings need to be O(10−8) for O(TeV) masses
of new scalars and keV-scale N1.
In addition to the described freeze-in mechanism, DM in this model can be produced
from the decays of next-to-lightest Z2-odd particle, N2 (see right figure of Fig. IV.1). The
Yukawa couplings y2α, that are required for the successful generation of neutrino masses
via the mechanism described in Section III.3, are sufficiently strong to put this particle
in thermal equilibrium with the SM bath. Hence, N2 will freeze-out at r
′ ≡ mN2/T .
15. Additionally, the Z2-odd scalars are also in thermal equilibrium due to their gauge
interactions with SM particles. Still, all heavy Z2-odd particles will eventually decay into
N2 and hence one effectively needs to solve a single Boltzmann equation for the yield of
N2 [57]: √
dY sN2 πg∗ MPlmN= 2 2 2
dr′ 45 r′
〈σeffv〉 (Y2 EQ − YN ) , (IV.5)2
where 〈σeffv〉 accounts for the annihilations and coannihilations in the Z2-odd sector, and
MPl ' 1.22 × 1019 GeV is the Planck mass. The equilibrium yield for a thermalized
fermionic species, Y , is given by Y = 45 r′2K (r′)/(2π4 sEQ EQ 2 g∗). We use micrOMEGAs 5.1
[230] to evaluate Eq. (IV.5) numerically. We refer the reader to Ref. [2] for a detailed
description and derivation of Eq. (IV.5); in particular, the suppression of the N2 freeze-out
abundance due to possible coannihilation channels with Z2-odd scalars has been outlined.
After freeze-out, N2 decays into N1 and a pair of charged or neutral leptons with the rate
[182],
5 ( )m
Γ(N → ` ¯̀ N ) = N2 22 α β 1 |y1| |y 2 2 22α| + |y1| |y2β| , (IV.6)
6144π3M4
where M stands for the mass of the scalar particle that is exchanged in the process and
α and β denote the flavor of final state leptons. The decay of N2 gives a contribution to
the total DM abundance of the form¸ ( )
100 GeV ( m )
Ω 2 −7
N1 2
N2→N1h = 10 ΩN2h , (IV.7)mN2 10 keV
where ΩN2h
2 is the freeze-out abundance of N2 which is set by the corresponding yield,
YN2 , calculated by solving Eq. (IV.5).
55
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
N1
y2α Σ y1
N2 N1
Σ y1
`α `
¯̀
α β
Figure IV.1: Production of the light DM N1 either via two-body decays of heavy scalars
(left figure) or three-body decays of heavy N2,3 (right figure). Here, Σ stands for all scalar
particles: σ± and σ0.
Even though these N2 decays give an extra source of DM, this production mechanism
actually has two side effects:
ˆ N2 decays occur after freeze-out, at temperatures much lower than mN2 and this
leads to the production of DM particles with a hot momentum distribution, i.e.
they feature large averaged momenta. In particular, it occurs for our considered
parameter choices, although it is not a generic property of the model. As explained
in Section II.2.6 this can drastically suppress the formation of structures at galactic
scales and may not be compatible with observations.
ˆ The decays of N2 should be fast enough in order not to violate BBN predictions
for the abundance of light nuclei such as D, 3He, 4He and 7Li. For N2 decays
to tau leptons, which dominantly decay hadronically, the decay time needs to be
τN2→N1 . 1 s, whereas decays into the first and second generation leptons lead to
less stringent limits, requiring only τN2→N1 . 100 s [204].
IV.3 Cosmological constraints
IV.3.1 Constraints from structure formation
In this section we consider whether DM produced by the two previously described mecha-
nisms is compatible with structure formation observations. The structure formation limits
on keV-scale sterile neutrino DM are commonly derived for non-resonant production, so-
called Dodelson-Widrow production [80], for which non-zero mixing between active and
sterile states is required. Currently, the most stringent structure formation limit on the
mass of non-resonantly produced particles, mNRP, arises from Lyman-α forest data and
yields mNRP & 28.8 keV [231]. However, this limit may be too constraining, because the
Lyman-α forest absorption spectra can be altered by effects stemming from gas dynam-
ics in the intergalactic medium [232]. On the other hand, constraints arising from Milky
Way satellite counts give mNRP & 10 keV [233].
In order to derive constraints from these observations for our model, we evaluate the DM
momentum distribution function fN1(x, r) which is calculated as a function of the dimen-
sionless variables x ≡ p/T and r ≡ mP /T . Here, mP stands for the mass of a parent
particle, which is either a heavy scalar in the case of freeze-in or N2 in late-time next-to-
lightest particle decays. For scalar decays we are following the discussion in Ref. [234],
whereas for the case of N2 decays we employ the procedure outlined in Ref. [235]. The
total distribution function is hence given as a sum of the two contributions,
f Σ N2N1(x, r) = fN (x, r) + f1 N (x, r) , (IV.8)1
56
IV.3. COSMOLOGICAL CONSTRAINTS
where we indicate the production mechanisms of N1 with corresponding superscripts. In
the following we discuss the calculation of both components. The general expression for
fΣN (z, r) assuming aMB distribut[ion is] given by [234, 231 √ ( ) 
5]
e−x πErf √r √r2
Σ 4x −x +1 r πfN (x, r) = 4C
Σ √ − e 4x2 Γ ⇒ 4CΣΓ e−x, (IV.9)1 2 x 2x r→∞ x
where CΣΓ = M0 ΓΣ/m
2
± is the effective decay width, originating from rescaled two-body
decays of the Σ particles, ( )
Σ M
2 2 2
0 6|y1| m± 3|y1| mS 3|y1| mA
CΓ = + + , (IV.10)gs∗(T )m
2
± 16π 16π 16π
An important property for a production mechanism with an associated momentum distri-
bution function is the averaged momentum∫ :∞ 3
〈 〉prod ∫0 dxx f(x, r)x = ∞ . (IV.11)
dxx20 f(x, r)
For fΣN (x, r) the averaged DM momentum is given by 〈x〉
prod
FI ≈ 2.5. This result, together1
with the information that the production dominantly occurs at temperatures T ∼ mΣ/3
(see for instance Fig. 1 in Ref. [2]), allows(us to estim)ate the limit on mN1 by using [234]〈p/T 〉prod 10.75 1/3
mN1 = mNRP , (IV.12)3.15 gs∗(Tprod)
and assuming that the freeze-in DM production dominates. Here, the entropy dilu-
tion factor (10.75/gs∗(T
1/3
prod)) takes into account that the DM production happens at
early times where gs∗(Tprod) = 106.75. Taking the aforementioned limits mNRP & 10 keV
and mNRP & 28.8 keV, we obtain mN1 > 3.7 keV and mN1 > 10 keV, respectively.
Hence we are going to set mN1 = 6 keV as our limit on the DM mass in the following.
29
The combination of this limit and Eq. (IV.4) sets the upper bound on the magnitude of y1.
If decays of Z2-odd scalars were the only source of DM production, our structure formation
analysis would end here. However, decays of N2 significantly complicate the picture. To
calculate the DM distribution function for the production via N2 decays, we apply the
procedure from [235] and evaluate the equation30
∫r ∞2 ∫
fN
r̂ x̂
2
N (x, r) = dr̂ C
N2
Γ dx̂√ fN (x̂, r̂) . (IV.13)1 x2 2 2x̂ + r̂2
rFO |x−r̂2/(4x)|
Typically, the freeze-out temperature rFO ranges between 8–16 and C
N2
Γ is the effective
decay width given by CN2 = M Γ/(gsΓ 0 ∗(T )m
2
N ), where Γ is the decay width of N2 into N2 1
29 Compared to the updated analysis using a MW subhalo count and Lyman-α forest data done in Sec-
tions V.3 and VI.3.2 this limit is more conservative.
30 Actually this equation was derived for two-body decay kinematics, but we are dealing with three-body
decays instead. Consequently, this treatment gives rise to momentum distribution functions with larger
averaged momenta 〈x〉 compared to actual three-body decays. However, our discussion of a more general
setup in Chapter VI shows that this simplified ansatz still yields robust limits: while 〈x〉 would decrease
in the case of three-body kinematics (see Section VI.4.3), this is compensated for by taking into account
the time dependence of gs∗(T ) which shifts the averaged momentum to higher values (see Section VI.4.2).
We will discuss and compare both approaches in Section VI.6.2.
57
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
and a pair of leptons given in Eq. (IV.6). The expression for the distribution function of
N2 after freeze-out is [235]
[ ( )] √  N C 2Γ x2/2− 2 2 r + r2 + x2fN2(x, r) = exp x + r FO √ ×
[ ( √rFO + r2 2FO +√x )]
× exp −CN2Γ /2 r x2 + r2 − r 2FO x + r2FO , (IV.14)
where a MB distribution for N2 is assumed.
In Fig. IV.2 we show fN1(x, r)x
2 for two N2 masses, namely mN2 = 100 GeV and
400 GeV, while the charged scalar mass is set to m± = 600 GeV. The red curves rep-
resent fN2 2N (x, r)x , obtained by solving Eq. (IV.13) and fixing r to sufficiently large values1
in order to capture the effect of decaying N2. For comparison, we also show the distribu-
tion function corresponding to the production via freeze-in (blue), taking r →∞. Clearly,
the peak of fN2N (x, r)x
2 is shifted to very large values of x indicating that N2 decays yield1
a “hot” DM component. However, we also see from the figure that its amplitude is greatly
suppressed with respect to fΣ 2N (x, r)x , implying that this component is subdominant for1
the selected benchmark point. Quantitatively, the distribution shown in the left panel
yields Ω 2N2→N1h = 0.03ΩDMh
2, whereas for mN2 = 400 GeV it follows that less than 1
per mille of the observed DM abundance is produced in N2 decays. Further, the param-
eter choice in the left figure features a more prominent peak for fN2N (x, r) compared to1
the other case, because N2 decays at later time where g
s
∗(T ) are decreasing, thus affecting
the effective decay width CN2Γ . In the following we are using an approach based on the
0.100 C Σ=7.61×10-3 0.100 C Σ=7.84×10-3
Σ decays Γ Σ decays Γ
C N2Γ =4.65×10
-11 C N2Γ =3.78×10
-9
0.001 0.001
10-5 10-5
10-7 10-7
10-9 10-9 y
s
ays de
ca
dec N2
N2
10-11 10-11
0.1 1 10 100 1000 104 105 0.1 1 10 100 1000 104 105
p p
x= x=
T T
Figure IV.2: In the left (right) panel we show the DM momentum distribution function
fN1(x, r)x
2 including both DM production mechanisms, taking m± = 600 GeV and mN2 =
100 GeV (mN2 = 400 GeV). The blue and red curve correspond to early Σ decays and N2
late-time decays, respectively.
half-mode analysis explained in Section II.2.6.1 to constrain our model. We evaluate the
transfer function T (k), given by the power spectrum ratio,
T 2
P (k)
(k) = , (IV.15)
P (k)ΛCDM
where P (k) is the matter power spectrum calculated from fN1(x, r) using CLASS [145,
146] and P (k)ΛCDM is the corresponding ΛCDM power spectrum. The transfer function
indicates at which scales non-cold DM will lead to deviations in comparison to cosmological
58
x2 fN (x,r)
x2 fN (x,r)
IV.3. COSMOLOGICAL CONSTRAINTS
observations. The temperature of the DM species, TN1 , relative to the photon temperature,
Tγ , is relevant for the evaluation of the matter power spectrum: since we have two different
mechanisms for DM production in the model, we are left with two independent dark sector
temperatures.
The smaller one of the temperatures is set by the time when N1 is produced via a freeze-
in mechanism from the decays of heavy scalars. These processes occur when the heavy
scalars are still in thermal equilibrium implying that DM particles are produced with
temperatures identical to those of the SM sector. After production, N1 is decoupled and
does not experience reheating when SM DOF drop out of equilibrium. The temperature
ratio, governed by the entr(opy dilutio)n factor, y(ields )
freeze-in ≈ g
s(T ) 1/3 3.94 1/3
T ∗
0
N Tγ = Tγ ≈ 0.33Tγ , (IV.16)1 gs∗(Tprod) 114.25
where Tprod roughly corresponds to Z2-odd scalar masses and T0 is the temperature of the
Universe today.
Additionally, to evaluate the temperature of DM produced from out of equilibrium N2
decays, we estimate the temperature when these decays are taking place. We assume
an instantaneous decay at τ = 1/Γ and make use of the time-temperature relation for a
radiation-dominated Universe,
√ ( )1 1 MeV 2t = 2.42 s , (IV.17)
gs∗(T ) T
which allows us to obtain an expression( for the temperatu)re at which N2 particles decay,1/2
T = (gs
Γ
Γ ∗(T
−1/4
Γ)) MeV . (IV.18)
2.72× 10−25 GeV
For the benchmark point already used for presenting momentum distributions in the right
panel of Fig. IV.2, we obtain
Γ = 1.52× 10−22 GeV , ΩN2h2 = 5.61× 103 , TΓ = 13 MeV . (IV.19)
At T ∼ TΓ, RHNs are at rest and each decay product has an energy E ≈ mN2/3 ≈
O(100 GeV). Note that by dividing this energy with TΓ in Eq. (IV.19) we obtain x '
104 and this explains the approximate position of the N2 decay peak in the momentum
distribution (see Fig. IV.2). Finally, we have to take into account that the SM bath
is reheated when the epoch of electron-positron annihilation occurs. In summary, the
temperature of this DM contribution is (given)by
m 1/3N
T decay ' 2 4N Tγ ≈ 7300Tγ , (IV.20)1 3TΓ 11
where the temperature is evaluated for the aforementioned benchmark point.
In order to assess the cosmological viability of particular benchmark points, we compare
the calculated T 2(k) against the function corresponding to the constraint stemming from
Lyman-α forests. For the latter, we adopt an analytical fit for the transfer function [148]
(see Eq. (II.45)), taking mNRP ' 10 keV which corresponds to a thermal relic mass of
mTR = 2 keV; the mass relation between non-resonantly produced particles and a thermal
relic is given by m = 4.35 (m / keV)4/3NRP TR [234]. We are using an adapted version of the
half-mode criterion (see Eq. (II.48)): since we are dealing with a momentum distribution
with two peaks, we are expecting two associated wavenumbers at which P (k) < P (k)ΛCDM.
59
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
This is illustrated in Fig. IV.2 by the red line that deviates from 1 early on, but reaches
a plateau at intermediate wavenumbers, before it drops to 0 at large k. To take this
early deviation into account (and not excluding too much of our parameter space), we
define a new reference P (k)ΛCDM where Neff is larger than the SM value; precisely, we
use bounds stemming from the epoch of BBN. As we will discuss in Section IV.3.2, they
are stronger than corresponding Neff limits from measurements of the cosmic microwave
background (CMB). Than we calculate T 2(k) based on this new reference matter power
spectrum and compare it against a limiting transfer function, T 2lim(k) (see Section II.2.6.1).
Consequently, a parameter choice is allowed if and only if
T 2(k)− T 2lim(k) > 0 for k ∈ {0.02, 1} h Mpc−1, (IV.21)
is fulfilled. This analysis technique may seem somewhat unconventional compared to the
one used in Eq. (II.49), but our method is motivated by two observations: first, some
transfer functions feature small bumps at wavenumbers k ' 0.1h/Mpc in T 2(k), as seen
in Fig. IV.2 and we want to avoid this region when comparing the corresponding T 2(k)
against each other. Second, in this analysis we are not necessarily interested in a lower
mass limit for the DM, but want to primarily study the influence of the hot DM compo-
nent produced by late-time N2 decays on wavenumbers k < 1. At the end, we are going
to present two separate exclusion limits based on this transfer function formalism.
In Fig. IV.3 we show in red (green) the calculated transfer function for fN1(z, r) with
m± = 600 GeV and mN2 = 100 GeV (mN2 = 400 GeV); these are identical benchmark
points as those from Fig. IV.2. If a given curve lies below the Lyman-α limit (blue curve)
the corresponding parameter point is disfavored. We observe that the scenario with
lighter mass of N2 is excluded since the abundance of hot DM is too large in this case
and hence larger cosmological scales than observed are affected. In fact for these bench-
mark point the DM carries so much energy, it effectively acts as dark radiation. On the
other hand, the green curve is in agreement with observational data; it seems to “over-
shoot” the ΛCDM reference, because the transfer function is defined with respect to a
different reference P (k)ΛCDM with Neff = 3.046 + 0.28. One should note that both curves
drop to zero at roughly the same point, because this wavenumber is set by the temper-
ature of the dominant, frozen-in, DM component. If N1 would freeze-in at later times
the curves would shift to the left. For this analysis and in what follows, we choose the
Yukawa couplings y2i to be as large as possible due to LFV constraints (see Eq. (III.10)).
Smaller couplings would give rise to stronger constraints due to DM production via N2
decays at later times.
In addition to the published version, we determine similar limits on the parameter range
using the δA method (see Section II.2.6.2 for a detailed explanation). For this we use
the conservative limit mTR = 2 keV to define a reference WDM model and choose k =
100h/Mpc as an upper limit for Eq. (II.50). Further, for typical surveys the k-range is
between kmin = 0.5h/Mpc and kmax = 10h/Mpc. Using all these ingredients we find a
reference suppression factor, δAref = 0.45, as the exclusion criterion. We are going to
compare and discuss the results of both analysis techniques in the next section.
IV.3.2 Constraints from additional radiation contribution
When discussing possible implications on the formation of structures in the early Universe,
we should also take into account that keV-scale DM could change the number of relativistic
non-photonic DOF, Neff. This number enters in the expression for the radiation density
60
IV.3. COSMOLOGICAL CONSTRAINTS
1.0
0.8
0.6
0.4
Lyman-α (mTR=2keV)
0.2 mN =400GeV2
mN =100GeV2
ΛCDM
0.0
0.1 1 10 100
k [h/Mpc]
Figure IV.3: Transfer function T 2(k) for the same benchmark points as in Fig. IV.2. The
constraint from structure formation, using Eq. (II.45) with mTR = 2 keV is shown in
blue. RHN masses of around 100 GeV clearly violate this constraint while the green line,
corresponding to mN2 = 400 GeV, is consistent with the data. The black dashed line
represents ΛCDM.
ρrad, which is, after electron-positro[n annih(ilati)on, given]by
7 4 4/3
ρrad = 1 + Neff ργ , (IV.22)
8 11
where ργ represents the energy density of photons. In the SM, Neff = 3.046 [236] and thus
we denote contributions from additional relativistic species as ∆Neff = Neff − 3.046. The
contribution to ∆Neff from N1 can be estimated by comparing its energy density against
the one corresponding to a fully relativistic ne√utrino with temperature Tν [235]:∫∞ ( )
60 gs(Tν) m xT
2
N ν
∆Neff(Tν) =
∗ 1 dx  1 + − 1 x2fN (x, Tν)×
7π4{gs∗(T 1prod) Tν mN10
× 1(, ) if Tν > 1 MeV
11 4/3
. (IV.23)
4 , if Tν < 1 MeV
Using as an example the benchmark points employed in Fig. IV.2, we can derive the
following values:
mN2 = 100 GeV→ ∆Neff ∼ 263 , mN2 = 400 GeV→ ∆Neff ∼ 0.07 . (IV.24)
Clearly, large mass gaps between N2 and σ
± are disfavored. The reason is that such cases
would lead to larger abundances of N2 and therefore the hot DM component becomes
more prominent in the spectrum.
Current measurements by the Planck collaboration allow for an upper limit of ∆Neff = 0.28
(95% CL) (TT, TE, EE+lowE+lensing+BAO), which we refer to as “CMB strong”. In-
cluding the present tension in the measurement of the Hubble constant, this value in-
creases to ∆Neff = 0.52 (95% CL) (TT, TE, EE+lowE+lensing+BAO+R18) [52], thus
called “CMB weak”. Another bound can be derived from the BBN epoch, at which
∆Neff = 0.344 (95% CL) [237]. In contrast to the analysis done in Ref. [1] where we did
not include the entropy dilution factor in Eq. (IV.23), we are going to use the later bound
on ∆Neff. Since BBN takes place way before the CMB epoch in the early Universe, the
61
2
T (k)
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
500 500
ΔNeff CMB weak new ΔNeff
ΔNeff CMB strong old ΔNeff
SF+CMB weak δA analysis
400 SF+CMB strong
2
400 Half-mode 0.3%ΩDMh
300 0.5%
2 2
ΩDMh 300 0.5%ΩDMh
200 200
100 100
200 400 600 800 1000 1200 200 400 600 800 1000 1200
m± [GeV] m± [GeV]
Figure IV.4: Left figure: constraints from structure formation (red curves) confronted
with Neff limits (blue curves) derived using Eq. (IV.23). The solid curves correspond to
the “CMB weak” and dashed ones to the “CMB strong” choice of ∆Neff. Note that our
structure formation limits also indirectly depend on ∆Neff as it is an input parameter
for CLASS. Clearly, Neff yields much stronger limits in comparison to those arising from
structure formation. Shown in black solid is the curve for Ω 2 −3N2→N1h = 0.6× 10 . Right
figure: same as left figure, whose contours are shown faded. The blue line indicates the
new updated bound coming from the BBN limit on ∆Neff while the red line stems from
a δA structure formation analysis as explained in Section IV.3.1. Additionally, the black
dashed line indicates Ω 2N2→N1h = 0.36× 10−3.
DM has less time to cool down and thus it gives rise to stronger constraints.
By using Eq. (IV.23), we can estimate which parameter choices lead to ∆Neff values that
exceed the bound. We have performed a scan and have deduced the following condition:
Ω 2N2→N1h . 0.4% , CN2 −10
Ω h2 Γ
& 5× 10 GeV , (IV.25)
DM
necessary for consistency with cosmology, which is further indicated by the black curve
in Fig. IV.4.
Due to the above mentioned correction for Neff compared to the published version, we
compare the two results from Ref. [1] with the new single limit resulting from the BBN
limit on ∆Neff . As can be seen, the BBN limit on ∆Neff provides slightly stronger
constraints on mN2,3 compared to the results quoted in Ref. [1]. We have updated all cor-
responding plots with this new limit. Furthermore, in Fig. IV.4 we compare the results
from the two different structure formation analyses: the faded red lines correspond to the
adapted half-mode analysis, while the thick red line corresponds to a δA analysis with
δAref = 0.45. As indicated by the red arrow, the latter yields weaker limits on the neu-
trino mass, which is ultimately due to our conservative choice for mTR.
In Fig. IV.3 and Eq. (IV.24) we demonstrate that one of the chosen benchmark points is
excluded by both structure formation and Neff limits. Comparing both probes in Fig. IV.4
we conclude that the BBN ∆Neff bound generally leads to stronger exclusion limits. Hence,
in Section IV.4 we will compare regions in parameter space that are accessible at colliders,
with this ∆Neff limit. For instance, taking m± = 1 TeV, the lower bound on the heavy
lepton mass is mN2 & 340 GeV.
31
31 There is a caveat as we have a freedom to choose the couplings between N2,3 and the charged leptons.
Throughout this section we assume couplings to tau leptons to be subdominant.
62
mN [GeV]2,3
mN [GeV]2,3
IV.3. COSMOLOGICAL CONSTRAINTS
IV.3.3 Constrains from big bang nucleosynthesis
As we have seen in the previous section, N2 decays produce SM leptons with large mo-
menta, which can inject a lot of energy into the plasma, thus affecting the primordial
abundances of light nuclei. Specifically, we need to ensure that N2 decays are fast enough
such that these highly energetic particles can thermalize with the plasma and thus the
abundances of light nuclei essentially remain unaffected during the BBN epoch. The cor-
responding decay rate of N2 into N1 and a pair of leptons is proportional to two powers
of the small Yukawa coupling y1 (see Eq. (IV.6)).
In order to obtain the BBN limits in the considered scenario, we adopt the results from
[204] where the authors studied the impact of decaying hidden sector particles to the
abundances of light nuclei. The channels of our interest are those containing charged
leptons. Decays of N2 into electrons and muons can take as long as O(100 s), since
they mainly induce electromagnetic cascades, which affect BBN at later times only. In
contrary, tau leptons decay mostly hadronically and this can significantly alter the abun-
dances of light nuclei due to a change in the neutron-proton ratio. Hence, unless the
abundance of N2 is strongly suppressed, the N2 decay time has to be . 1 s.
0.1
0.1 LFV excluded
0.05
0.05
N
e
LFV excluded ff Neff
DM fromN DM
2 de frc oa my
0.01 s
N
on 2ly 0.01 decays only
0.1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.4 0.6 0.8 1
mN mi Ni
m± m±
Figure IV.5: BBN constraints for the case where N2 dominantly decays into electrons and
muons (left panel) and taus (right panel) are disfavoring parameter space below the black
lines. The regions excluded by the BBN epoch limit on ∆Neff are shown in red. The solid
green curve indicates the parameter space for a limiting case in which all of the DM is
produced by N2 decays. The regions in blue represent constraints from LFV experiments.
Finally, the region to the left of the vertical blue dashed line is favored by our collider
analysis in the sense that the mass gap between σ± and N2,3 is sufficiently large. The value
of the charged scalar mass in both panels is fixed to m± = 600 GeV√. ∑On the y-axis we show
the average Yukawa coupling of N and N , defined as ȳ + ȳ ≡ (|y |22 3 2 3 2α + |y3α|2)/3.
α
To be conservative, we impose the decay time to be shorter than 1 s for all N2 decay
channels.32 The corresponding results for different Yukawa coupling strengths and RHN
masses are shown in Fig. IV.5, where black lines indicate BBN exclusion limits for a rep-
resentative value of m± = 600 GeV. The left panel corresponds to the dominant decay
32 Additionally, this ensures that SM neutrinos produced in these decays thermalize with the SM bath.
For temperatures T < 1 MeV their interaction with the plasma is not strong enough such that they reach
thermal equilibrium.
63
y2 + y3
y2 + y3
ed
vo
r
a
dis
f
BB
N
ore
d
av
dis
f
BB
N
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
into e±/µ± and in the right panel the case where N2 decays prominently into τ
± pairs
is shown. These channels are motivated by the di-lepton and di-tau searches at collid-
ers which were presented and discussed in Chapter III. In addition, the red shaded areas
are excluded by the ∆Neff < 0.344 limit, which generally limits the parameter space the
most. Decreasing the magnitude of the corresponding Yukawa coupling, both, BBN and
∆Neff yield stronger constraints on mN2,3 with respect to the charged scalar mass, be-
cause according to Eq. (III.5) smaller couplings have to be compensated by larger RHN
masses in order to get the observed SM neutrino masses. Further, by increasing m±,
larger y1 are required for DM production through Σ decays and hence the BBN bounds
get weaker. LFV bounds (blue regions) are then also relaxed, see Eq. (III.10). Moreover,
the LFV bounds are weaker for maximized coupling to τ , because the most stringent lim-
its stem from transitions between second and first family leptons.
The thick green solid line indicates a parameter space corresponding to the DM production
only through N2 decays and such a scenario is clearly excluded. Finally, in the parameter
region to the left of the blue vertical dashed line, energies of would-be final state lep-
tons at colliders, arising from σ± → N2 `±α are below 100 GeV, making them hard to detect.
Here we have shown that there are regions of parameter space, in particular the region
m± → mNi , unconstrained by collider searches but which can still be excluded by Neff and
BBN limits. They start be competitive at small sizes of the involved Yukawa couplings,
because the collider searches are effectively independent on these parameters.
IV.4 Combining collider and cosmological limits
Having examined the impact of the corresponding light DM on cosmology in this chapter
and explored the capability of the HL-LHC, as well as future hadron (FCC-hh) and lepton
(CLIC) colliders for testing the ScM in Chapter III, we are going to summarize the limits
in the following. We start to discuss the implications for the parameter space available
at the HL-LHC. The left figure of Fig. IV.6 contains the sensitivity limits from the di-tau
search at the HL-LHC overlapped with bounds from BBN and ∆Neff and the right figure
is the corresponding di-lepton case. Considering the di-tau scenario, it turns out that the
discovery region is in tension with cosmological probes. Only a small part of the parameter
space around mN2,3 ' 100 GeV is not excluded for the optimal case while the sensitivity
for the realistic case is disfavored by cosmological limits. However, a certain portion of
the potential exclusion parameter region around mN2,3 ' 100–150 GeV and m± ' 200–
400 GeV is not constrained by cosmological data.
In the case of the di-lepton scenario, cosmology disfavors even more significant parts of
the available parameter space. It can be seen by comparing both plots of Fig. IV.6 that
Neff limits are stronger for the di-lepton case. This is due to the fact, that we can have
larger couplings in the di-tau case, because LFV processes yield less stringent bounds on
the Yukawa couplings for the third lepton generation. Hence, Neff bounds are weakened
in such a case since larger interaction rates give rise too a smaller freeze-out abundance
of N2, suppressing the hot DM component. This is the same reason why the BBN limit is
stronger for the di-lepton scenario, although, in contrast to the di-tau parametrization, it
cannot compete with the ∆Neff bound. The fact that the projected HL-LHC results are
covered by current cosmological observations is another motivation to consider facilities
with higher energies, as the most dominant Neff bound starts to flatten for m± > 600 GeV.
64
IV.4. COMBINING COLLIDER AND COSMOLOGICAL LIMITS
250
S=2, BR=1 S=2, BR=1
S=5, BR=1 400 S=5, BR=1
S=2, CI-BR S=2, CI-BR
200 S=5, CI-BR S=5, CI-BR Neff
300
150
200
Neff
100 BBN
BBN
100
50
200 300 400 500 200 300 400 500 600 700
m± [GeV] m±[GeV]
Figure IV.6: Same figures as shown in Figs. III.4 and III.5, but now overlapped with BBN
constraints discussed in Section IV.3.3 and Neff limits discussed in Section IV.3.1. For the
di-lepton scenario (right picture), cosmological constraints already exclude most of the
potential exclusion region while the discovery region is completely covered for both cases.
A compilation of all derived constraints is shown in Fig. IV.7. The figure contains all
sensitivity curves already presented in Figs. III.4 to III.6 and III.8 as well as BBN and
∆Neff limits indicated by a black and gray line, respectively. While a potential discovery
at HL-LHC is less likely due to the tension with cosmological limits, the future colliders
offer a more promising situation in which large portions of the parameter space can be
tested.
1400 CLIC
-1
FCC (3 ab )
( -1FCC 30 ab )
1200 HL-LHC
1000 N f (100×m
)
N1
ef
800
600
BBN (100×mN )1
400 Neff
BBN
200
500 1000 1500 2000
m±[GeV]
Figure IV.7: Summarized sensitivity curves for S0 = 2 as discussed in Sections III.4.2,
III.5.1 and III.5.2 for the HL-LHC (red shaded region) with 4 ab−1 (red shaded region),
CLIC with 5 ab−1 and (blue shaded region) and FCC-hh with 3 ab−1 and 30 ab−1 luminos-
ity (orange and green shaded regions). Again, the thin black line indicates mN2,3 = m±,
whereas the thick black line shows BBN constraints and the gray solid curve represent
Neff constraints for mN1 = 6 keV. The corresponding dashed contours indicate BBN and
Neff constraints for a DM particle with mN1 = 600 keV. We assumed a maximal coupling
to e and µ in this case.
65
mN [GeV]2,3
mN [GeV]2,3
mN [GeV]2,3
CHAPTER IV. THE SCOTOGENIC MODEL AND COSMOLOGY
On the contrary, the non-collider limits shown in Fig. IV.7 depend on the magnitude of
the involved coupling yiα and the DM mass mN1 , while the collider searches do not de-
pend on them, as long as the BRs are staying the same. First, the BBN bound depends
on the lifetime τ of N2 given by τ ∝ Γ−1 ∝ (y y )21 2β (using Eq. (IV.6)). Thus, if smaller
couplings than anticipated are used, this bound will strengthen accordingly. It further
features an implicit DM mass dependence, because it is related to τ via y21 ∝ m−1N due to1
the DM abundance given by Eq. (IV.4).
Second, the ∆Neff bound depends on the DM temperature T
dec
N ∝ (y y )−1T1 1 2β γ (see
Eq. (IV.20)) and the DM mass. However, for large T decN , Eq. (IV.23) can be significantly1
simplified and the explicit DM mass dependence vanishes if ∆Neff is evaluated at the
BBN epoch and thus the limit is expected to scale similarly to the BBN bound.
Only the results of the half-mode analysis do dependent on mN1 : again, T
dec
N is impor-1
tant for evaluating the suppression of the corresponding matter power spectrum P (k)
and the impact of different y2α is rather simple, while the influence of mN1 is more subtle.
On the one hand, an increase in the DM temperature due to smaller y1 is compensated
by larger DM masses.33 On the other hand, having larger DM masses increases the N2
decay contribution (see Eq. (IV.7)) and hence the hot subcomponent becomes more im-
portant. In summary, also this bound gives stronger constraints, and the cosmological
bounds will persist even when N1 is heavier than we assumed in our analysis. Although
substantial larger DM masses are not directly motivated by our findings in Section III.2,
we consider a more general model setup and illustrate this effect by calculating the corre-
sponding limits for mN1 = 600 keV and show the results in Fig. IV.7 as dashed lines. For
this DM mass choice, it can be observed that the cosmological bounds increase by up to
a factor ∼ 2.5 compared to mN1 = 6 keV, since the corresponding decrease in y1 has to
be compensated for by a smaller abundance of DM produced in N2 decays.
In the following, we list upcoming cosmological and terrestrial searches which will provide
stronger limits on the above mentioned parameters, allowing for a complementary probe
of the parameter regions covered by future collider experiments.
ˆ New searches for LFV processes can lower the bounds on Yukawa couplings y2α and
y3α. For instance the MEG II experiment [238] features a projected sensitivity of
BR(µ→ e γ) < 6×10−14; an improvement of about an order of magnitude compared
to the previous bound, which in turn would give rise to even later DM production
via N2 decays due to smaller Yukawa couplings.
ˆ New observations of small scale structures in combination with detailed simulations
of warm DM will push mNRP to larger values and this in turn requires smaller y1
to produce the observed DM abundance via freeze-in, which will also impact BBN
limits. Some of these updated mass limits will be discussed in the next sections.
ˆ The upcoming CMB-S4 experiment will measure Neff to a precision of ∆Neff = 0.06
[239] which leaves less room for a hot DM subcomponent.
To summarize, these rather complementary searches would probe the parameter space
up to even smaller mass ratios between σ± and RHNs. They directly or indirectly set a
stricter upper limit on the abundance of N2 which is crucial for cosmology. Hence, in the
near future these experiments will offer novel relations between collider searches and cos-
mological observations.
In particular, we want to point out, that this is the kinematic window in which coannihila-
tions are very effective and this can strongly suppress the “hot” DM component, relaxing
33 A DM species with mass mDM and temperate TDM has the same matter power spectrum as another DM
species with larger mass αmDM and higher temperature αTDM, where α > 1.
66
IV.5. SUMMARY OF CHAPTER IV
the ∆Neff limits, while BBN limits still play a role. We wish to stress that such small
splitting between Z2-odd fermions and scalars at mass scales of ' 1 TeV is exactly the
setup that the observed amounts of DM and baryon asymmetry of the Universe can be si-
multaneously explained within the ScM, as shown in [2]. It is very intriguing that future
lepton colliders will offer the potential to search fo such scenarios.
IV.5 Summary of Chapter IV
Compared to the first chapter of this thesis, dedicated to collider searches, we focused in
this chapter on the cosmological implications of keV-scale fermionic DM N1 within the
ScM. In this setup there are two distinct DM production mechanisms: freeze-in through
the decays of heavy scalars and the production from the decay of the next-to-lightest
Z2-odd particle N2, which itself is produced via a freeze-out. The large mass gap between
the DM and N2 generally allows for a sufficient suppression of the abundance arising
from the latter mechanism. This is crucial, because this DM subcomponent features a
corresponding momentum distribution function with large momenta and could hence
lead to washout of structures at small scales. We showed that even stronger constraints
arise from the contribution of such hot DM to the effective number of relativistic species,
∆Neff at the epoch of BBN. We also derived BBN bounds from the requirement that N2
particles decay within ∼ 1 second, or otherwise they would hinder the formation of light
nuclei during this epoch.
For a DM mass of 6 keV and Yukawa couplings allowed by LFV experiments, we found
that the limit ∆Neff < 0.344 constrains the parameter space the most. In particular,
hot DM from N2 decays can contribute at most ' 0.1% to the total DM density. We
overlapped these cosmological limits with the projected collider exclusion results, de-
rived in the previous chapter, and were able to show that both approaches allow for an
complementary probe of the parameter space. In fact, these cosmological limits already
constrain a significant part of the collider accessible parameter space, in particular for the
HL-LHC. Further, we discussed how these cosmological bounds strengthen with larger
mN1 and/or smaller couplings y2, as these cases lead to an even later production of DM
particles and hence a further suppression of this subcomponent is necessary. The same
does not hold for the collider limits; they are effectively independent of the associated N1
DM phenomenology.
In summary, we have shown that if the scotogenic model is realized in Nature, there is
a number of complementary tests, stemming from terrestrial experiments to cosmological
surveys, indicating a rich model phenomenology and a very promising discovery potential
in the near future.
67

Chapter V
The cosmology of frozen-in axion-like
particle dark matter
V.1 Motivation
Strong experimental limits severely constraint a weakly interacting massive particle
(WIMP) as a viable dark matter (DM) candidate [240] and this shifts DM research to-
wards alternative realizations. For instance, non-thermally produced light DM in form of
sterile neutrinos [80, 100, 241–243], fuzzy DM [244, 245], hidden photons [246, 247] and
ALPs [136, 248, 249] is not only receiving a significant attention with regards to exist-
ing and forthcoming terrestrial experiments [250], but also has the potential to resolve
discrepancies between observations and simulations, as for instance for properties of halo
formation [251]. Additionally, non-thermal DM candidates can be related to new solu-
tions of other open questions in the standard model (SM), such as the hierarchy problem
in the case of relaxion DM [252–254].
In this chapter, we focus on ALP DM with O(1–100 keV) mass, which is the scale moti-
vated by hints in gamma-ray data [93, 94] as well as the recent measurement of an excess
in electron recoil spectrum performed by the XENON1T collaboration [128]. In fact, the
former measurement can be explained by an ALP-photon coupling [248], while for the
latter, an ALP-fermion coupling, in particular to electrons, suffices [253]. In light of these
hints, we study ALP production via freeze-in through feeble interactions with SM gauge
bosons and fermions. Our main goal is to compute the structure formation limits that
have not been derived to date for keV-scale ALPs. For this purpose we use recent data
from Lyman-α forests as well as the observed number of MW subhalos. These limits are
widely scrutinized for light sterile neutrino DM [99, 233]. While using the results from
these studies would allow us to get a very rough estimate on the structure formation lim-
its for ALP DM, we find it valuable, especially in light of aforementioned experiments
that may start to detect DM, to perform a dedicated study and determine with large
precision a viable parameter space for the considered model. We also point out that
for bosonic DM such bounds are important irrespective of their strength, because the
Tremaine-Gunn bound on the DM mass does not apply, in contrast to fermionic DM [54].
This chapter is organized as follows: in Section V.2 we introduce the model and discuss
production channels for ALP DM. Following this, we calculate the DM momentum distri-
bution in V.2.1, where we use Maxwell-Boltzmann (MB) statistics to describe SM bath
particles and discuss briefly the case when quantum statistics are used instead. Then, in
Section V.3 we discuss experimental observations that allow us to constrain the ALP DM
parameter space. In Section V.4 we discuss theoretical aspects of ALP DM in the early
Universe, focusing on the interplay between misalignment and freeze-in production. In
Section V.5 our main results are presented and summarize in Section V.6.
69
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
V.2 The model setup
The part of the Lagrangian relevant for the subsequent analysis is given by
L 1 µ 1 1= ∂µa ∂ a+ m2 2aa + f̄(i∂/−mf )f − F µνµνF2 2 4
− c/ aγγ caffq e f̄A f + aF F̃µνµν + ∂µa f̄γµγ5f . (V.1)
4fa fa
The first two terms are the kinetic and mass term for the pseudoscalar ALP field a. The
following three terms describe quantum electrodynamics (QED), namely a SM fermion
f with an electric charge q interacting with photon field Aµ whose corresponding (dual)
field strength tensor is denoted by Fµν (F̃µν). Finally, the last two terms describe the in-
teraction of an ALP with Aµ and f , respectively. We will focus on analyzing the ALP
coupling to photons and fermions separately, i.e. working under the assumption that ei-
ther caff or caγγ vanishes. These scenarios are dubbed as photophilic and photophobic,
respectively.
Regarding the photophilic case, at temperatures above ∼ 160 GeV the electroweak (EW)
symmetry SU(2)L×U(1)Y is restored, and we have to work in the {Bµ,W aµ} basis instead,
where Bµ is the hypercharge gauge boson, whereas the W
a
µ are the three corresponding
SU(2)L gauge bosons. In the following, we assume for simplicity that the ALP couples
only to the U(1)Y gauge field via the corresponding field strength tensor Bµν ,
caBB
aBµνB̃
µν . (V.2)
4fa
The caBB coupling is related to the ALP-photon coupling via the weak mixing angle θW ,
caBB cos(θW ) = caγγ , where cos θW ∼ 0.88.
The inclusion of the W aµ and gluon fields would increase the production rate of ALPs,
but for the moment we consider only a coupling to photons, which is the scenario typi-
cally considered in the literature for heavy ALP searches [255–258]. We will come back
to this case when discussing our results in Section V.5.
In the photophobic case, using the equations of motion, we can replace the last term in
Eq. (V.1) with the term proportional to the fermion mass [125], (2mf/fa) a f̄γ5f . This
does not imply that such coupling is only present once EW symmetry is broken. After
a phase redefinition of the quark fields f and Higgs field H, a term iyf (cfa/fa)Q̄LHqR
is generated [259], where QL is a left-handed (LH) quark doublet, qR is a right-handed
(RH) quark field and cf is a free dimensionless parameter. In general, such coupling
exists for all SM quarks, but since it is proportional to the Yukawa coupling of the re-
spective quark, all contributions except the top quark are negligible due to their small
coupling strength. However, thanks to the large top Yukawa, the term iyt(cta/fa)Q̄3HtR
contributes very efficiently to the ALP production, and it can be the dominant source
when all ALP couplings to SM particles have the same order of magnitude.
In this work, however, we limit ourselves to sub-TeV values of the reheating temper-
ature in the photophobic scenario, namely TRH < 160 GeV. In that regime, the top
quark is Boltzmann suppressed and does not contribute to ALP production. Since the
ALP-fermion coupling is proportional to the fermion mass mf , relevant for ALP produc-
tion are the heaviest fermions after the top quark, namely bottom and charm quarks as
well as tau leptons.
The kinetic and interaction terms in Eq. (V.1) enable ALP production from thermalized
fermions and vector bosons V (the photon, a gluon, or the U(1)Y gauge field) through
70
V.2. THE MODEL SETUP
photophilic photophobic
caγγ 6= 0, caff = 0 caγγ = 0, caff 6= 0
B a f a
γ/g a
f V → f a
f f
f f γ/g f̄
f a f̄ a
f a
f f̄ → V a
f̄ B
f̄ γ/g f γ/g
Figure V.1: Tree level Feynman diagrams for the two processes of interest, fV → fa (top)
and f̄f → V a (bottom), in the photophilic (left) and photophobic (right) cases. Image
credits: Enrico Morgante
V f → af and f̄f → V a channels. For both of these processes there are one photon- and
two f -mediated Feynman diagrams contributing at tree level (see Fig. V.1). We note
that for mixed scenarios where caff ∼ caγγ , our results turn out to be driven by couplings
to photons, namely they effectively match the photophilic scenario. This is because in the
photophobic case, the ALP interaction operator is proportional to the fermion mass, and
thus the production rate is suppressed with respect to the photophilic case by a factor
mf/TRH.
Independently of the scenario under consideration, the production of keV-scale ALP DM
through the processes depicted in Fig. V.1 should occur via “freeze-in” [76], because for
ma & 100 eV the freeze-out would lead to DM relic abundance that greatly exceeds present
measurements [260]. The relic abundance of non-thermalize∣d ALP DM is given by [248]
2 ' ma 106.75 ΓΩALPh 0.12 s ∣∣∣∣ , (V.3)154 eV g∗(TRH) H TRH
and a freeze-in scenario requires that Γ < H during reheating, or otherwise the ALP DM
would thermalize instead.
V.2.1 Calculation of the momentum distribution function
Equipped with the model Lagrangian we will derive the ALP DM momentum distribution
function f(pa, t) in this section. It is a key ingredient needed to calculate the properties
of ALP DM and its influence on structure formation and can be obtained by solving the
Boltzmann equation, which,[for the process]1 + 2→ 3 + a, reads:
∂ − ∂Hpa f(pa, t) = C(pa) . (V.4)
∂t ∂pa
71
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
The collision term on∫ the r.h.s depends on the DM production mechanism and is given by
1 d3 3C p1 d p d
3
2 p3
(pa) = (2π)
4δ4(P1 + P2 − P3 − Pa)×
2E (2π)32E1 (2π)32E2 (2π)32E3
× |M|2f1(E1, T )f2(E2, T )(1− f3(E3, T )) . (V.5)
Here, |MA|2 is the averaged squared amplitude for the considered process. In what fol-
lows, we present the computation of the ALP DM distribution function for the case of MB
statistics describing the SM bath and highlight some differences compared to a quantum
statistical treatment.
Using MB statistics and assuming fi  1 for all particles involved, simplifies the Boltz-
mann equation considerably [234]. Indeed, we can approximate f1(E1)f2(E2)(1−f3(E3)) ≈
exp[−(E1 + E2)/T ] = exp(−P0/T ), where P = P1 +P2 = P3 +P4 and P0 is the first com-
ponent of the four-vector P∫. The collision term can then be factorized as
1 d4P e−P0/TC(pa) =∫ (2π) δ(E4 3 + Ea − P0)×2E (2π) 2E3
× d
3p d32 p3
(2π)4 δ4(P1 + P2 − P ) |M |2A . (V.6)
(2π)32E 32 (2π) 2E3
Neglecting any CP violation, the second line is nothing but the reduced cross section, σ̂,
of the inverse process 3 + a→ 1 + 2, multiplied by a phase space factor λ(s,m2,m2)1/21 2 /s.
It is related to the usual cross section through σ̂ = 2 [λ(s,m2,m21 2)/s]σ, where s is a
Mandelstam variable and λ√is the Källén function. This function is approximated by
λ ≈ s2 for masses ma,mf . s ∼ T which is the case for the relevant epoch in the early
Universe associated to ALP DM production. The second line of Eq. (V.6) is invariant
under longitudinal boosts, which allows us to compute it in the center-of-mass frame.
We calculate σ̂ analytically at tree level considering the diagrams shown in Fig. V.1. For
QED with one fermion of charge(±1, we ob)ta√in
2 c2 m2 4m2e aγγ f f
σ̂f̄f→γa = s 1 + 2
(12π f2a s) 
1−
s √ 
c2
2
2 c c s− 2m − s 1− 4m2/s− 2 2 aff aff aγγ f fe mf 2 + log 2 , (V.7)π f2a f2a 2mf
and ( )2 [ ( ) ]
2 c2 m2 (s−m2)2 2 4e aγγ − f f
mf mf
σ̂fγ→fa(= s 116π f2 ) 4 log − 3− 2 +s ( )(sm2a γ ( ) s s2 )
e2 c
2 m2 m2 4
− aff caffcaγγ f s f
mf
2 + m2 1− 2 log − 3 + 4 − . (V.8)
2π f2a f
2 f
a s m
2 2
f s s
Details on the derivation of the cross section starting from the matrix amplitudes of the
corresponding Feynman diagrams are given in Section V.7. Here, mγ ≈ eT/3 is the plas-
mon mass that serves as a regulator for the diagram including t-channel photon exchange.
The expressions in Eqs. (V.7) and (V.8) are obtained for general cases of both caγγ and
caff 6= 0 and have well defined limits if any of these two couplings vanishes. We note that
Eq. (V.8) holds for both f and f̄ scattering off photons and thus it must be summed over
twice.
72
V.2. THE MODEL SETUP
The expressions in Eqs. (V.7) and (V.8) must be generalized to account for the presence
of SM fermions. Additionally, in the photophilic scenario, in which the ALP population
is gen∑erated at T  160 GeV, e
2caγγ must be replaced by g
′2c ′2 2aBB = g caγγ/ cos θW ,
and the result must be multiplied by the sum over the hypercharge of SM fermions,
gY ≡ Y 2L ncnfnL = 10, where the sum runs over all LH doublets (nL = 2) and RH sin-
glets (nL = 1). The plasmon mass for a B gauge boson is given by (11/12)
1/2g′T [259].
On the contrary, in the photophobic scenario, the electric charge e2 should be multi-
plied by q2nc for leptons, and replaced with q
2n e2c + 4g
2
s for quarks, and finally summed
over the SM fermions involved (typically b, c, τ). Here, nc is the number of colors of
the fermion f and q its electric charge in units of e. For quarks, we take the diagrams
with an external gluon into account as well. The diagrams involving an external gluon
can be derived from the QED diagrams by replacing ieγµ with igsγ
µT a where T a are
SU(3) generators. Summing over initial and final states, we have to evaluate the trace
Tr [T aT a] which gives the factor 4 in front of the second term of the substitution for quarks.
It is convenient to rewrite the Boltzmann equation (Eq. (V.6)) by defining dimensionless
quantities, r = mH/T and x = p/T , where mH is a reference mass which we fix to be
the Higgs mass; its actual value is irrelevant as mH cancels in the calculation of physical
quantities. Thus, r acts as a time variable, while x denotes the comoving ALP momentum.
With this redefinition we obtain the expression given in Eq. (11) of [234] for an ALP DM
distribution function, ∫rf ∫∞ (
M m2
)
0 H y
[
− y
]
f(x) = dr dy σ̂ exp x− , (V.9)
√ 16π
2m x2 r2H 4x
ri y∗ √
where M0 = MPl 45/(4π3gs∗(T )) and MPl = 1/ GN ≈ 1.22 × 1019 GeV is the Planck
mass; ri,f are the lower and upper integration boundaries set by mH/TRH and mH/mf ,
respectively; y∗ = 4rm2f/m
2
H for the fermion annihilation process and y
∗ = rm2f/m
2
H for
fermion scattering. We would like to point out, that f(x) has no explicit time dependence,
because it is evaluated at times rf . Using this approach, the change of the number of
effective entropic degrees of freedom (DOF), gs∗(T ), is not taken into account but in the
relevant high-temperature range (that is above the QCD phase transition at T ∼ 100 MeV)
such approximation is justified.
In Eq. (V.9) and in the following, we expand the expressions up to the first non-zero order
in mf , i.e. we set mf = 0 everywhere apart from the cross section where we keep the
factor m2f multiplying the c
2
aff terms. We also keep only the first non-zero mf -dependent
term inside the logarithms (see Eqs. (V.7) and (V.8)). The same holds for the integration
limits rf and y
∗, which can be sent, respectively, to infinity and to zero in the photophilic
ALP case. Finally, we drop the interference terms ∝ caγγcaff , as we are going to consider
separately the cases caγγ = 0 and caff = 0. Note that the interference terms in Eqs. (V.7)
and (V.8) are proportional to the corresponding c2aff terms, and hence, even if considered,
they would not lead to any major effect, in particular the energy dependence of the cross
section would be unaltered.
The distribution function can thus be integ{rated a[nalytically. We o(btain′ 2 )]}g 2 caγγ 3 48x
f(x) = gYM T e
−x
0 RH 1 + 1− 4γE + 4 log , (V.10)
12π3 cos2 θ 2W fa 2 11g
′2
in the photoph∑ilic and 2 − {[ ( ) ( )]1 c e xaff √ 1 1
f(x) = M0 κf 2 πx (1 + log 2) erf √ + 2Γ 0,
2π3 f2
f a
x x x
73
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
[√ ( ) ( )]}1 1
+ πx erf √ + 2Γ 0, , (V.11)
2 x 4x
in the photophobic scenario. In these expressions, erf(x) is the error and Γ(a, x) is the
incomplete Gamma function and the factor κ is equal to m n q2e2f f c for leptons and to
m n q2e2f c + 4m g
2
f s for quarks. The strong gauge coupling, gs, is fixed to 1.31 which is
obtained by averaging αs over the relevant energy range between the b and Z mass,∫mZ
1
αs ≡ dµαs(µ) ≈ 0.137, (V.12)
mZ −mb
mb
where the running of αs(µ) is given by the corresponding renormalization
group equation [261]. Finally, the parameter κf takes the approximate values
{0.16, 8.97, 29.0, 38.1, 38.9}GeV for tau, charm, bottom, the sum over these three, and
the sum over all SM fermions (excluding the top), respectively.
In both, Eq. (V.10) and Eq. (V.11), the first term in the curly brackets comes from the
fermion annihilation process ff̄ → V a, while the second arises from scattering fV → fa.
-1
-1 10
10
-2
-2 10
10
-3
-3 10
10
-4
-4
10 10
- -55
10 10
-6 -6
10 10
-7 -7
10 10
-8 -8
10 10
-9 -9
10 10
-10 -10
10 10
-1 0 1 -3 -2 -1 0 1
10 10 10 10 10 10 10 10
Figure V.2: The ALP momentum distribution function plotted as x2f(x). Left figure: In
blue we show the photophilic case for MB statistics; black lines represent the photophilic
case with quantum statistics and in green we show the photophobic case for MB statistics.
For each, the dashed (dot-dashed) line represents f̄f → V a (fV → fa) and the solid line
is their sum. The vertical dashed line marks p = g′T , left of which thermal corrections
should be added [262]. Right figure: Photophilic scenario for MB and quantum statistics in
QED, namely, including only a single fermion carrying unit charge. Image credits: Enrico
Morgante (adapted by author)
The left panel of Fig. V.2 shows the momentum distribution multiplied with x2 obtained
for the photophilic (blue) and photophobic (green) ALP DM, in units of M 2 20 TRH ca/fa ,
where ca denotes caγγ or caff , respectively.
The dashed lines represent the contribution from fermion annihilation, the dot-dashed ones
correspond to fermion scattering, and solid lines are the sum. We rescaled the photophobic
ALP lines by TRH/κf , as otherwise, for caγγ ∼ caff choice, green lines would not be visible
on the same panel next to those corresponding to photophilic ALP. Moreover, since for
the photophobic ALP, f(x) ∝ mf q2nc, with such rescaling the computed green curve is
74
V.2. THE MODEL SETUP
independent of the fermion choice. It can be seen from the plot that, if both caγγ and
caff are present, the contribution from caff is negligible as long as caff . caγγTRH/mf .
Finally, the black lines are the results for f(x) when using quantum statistics for the
involved particles, and we will discuss this case briefly at the end of this section.
Importantly, at p/T ' 4 × 10−2 the dot-dashed blue line turns negative and hence the
momentum distribution becomes unphysical for smaller values of p/T . The reason for that
is a simplified treatment of the infrared (IR) divergence in the t-channel photon exchange
diagram where we use the plasmon mass mγ as a momentum cutoff. For the purpose of
our computations, we make a sharp cut at the location where the function turns negative.
In the case of a MB distribution, we infer this condition to be
m2γ
p/T = exp [−1/4 + γE ] ' 0.04 . (V.13)
4T 2
The same condition holds also for QED, with appropriately adjusted plasmon mass mγ .
Such a single fermion scenario is shown in the right panel of Fig. V.2 and we can observe
the shift of the cutoff toward an order of magnitude smaller values of p/T ; this effect only
arises due to different values for the plasmon mass in the full model and QED, respec-
tively. In any case, for both panels, the cutoff occurs at a value of p/T at least two orders
of magnitude smaller than the expected mean, 〈p/T 〉. If the cutoff was at p/T ' O(1),
the further calculation of structure formation based on this f(x) would not be justified.
Coming back to the left panel of Fig. V.2, we can infer that in the photophilic case, the
annihilation process is subdominant, and as such it is often neglected in the literature
(see for example Refs. [136, 260, 263]). On the other hand, in the photophobic case it
yields a contribution comparable to scattering (see also Eqs. (V.21) and (V.22)). For the
average momentum we obtain 〈p/T 〉 = 3.24 for a photophilic ALP and 〈p/T 〉 = 2.36 for
a photophobic one. Up to higher order corrections, this result is independent of mf .
The procedure described above is only approximate, and it fails to capture two features
that may a priori be important. First, it cannot account correctly for the phase space
regions where pi . T , as in these regions MB statistics deviates from a quantum statistics
treatment. Second, thermal field theory effects are important for pi . eT , and must
be correctly resummed. This is especially true for the process fB → fa, which has
a logarithmic IR divergence when |pγ − pa| → 0. In that context, it is interesting to
consider the question whether the usage of quantum statistics could in principle cure
the occurrence of a negative momentum distribution function at low p/T . Following the
procedure discussed in Ref. [263], we isolate the momentum k flowing in the t-channel
propagator and calculated the hard part of the process, by imposing an IR cutoff in
the integration, |k| > kcut, which diverges for kcut → 0. By adding a soft term which is
extracted from the ALP self-(en√ergy evaluated with a resummed photon propagator at finitetemperature, the sum is finite since the )cutoff drops out and only the thermal photon mass
remains, log(T/mγ) = − log 11/12g′ . More details on the calculation can be found in
Ref. [3] and we are going to highlight two important consequences in the following: first,
these expressions are only accurate for p > g′a T , but they correctly account for the case
p ′1,2,3 < g T . Below this momentum, the ALP-photon vertex should be thermally corrected
as well [262]. Second, obtaining the momentum distribution from the collision term by
solving the Boltzmann equation (see Eq. (II.33)), which is most conveniently written in
terms of already introduced dimensionless quantities
m2H ∂ fQS(x, r) = C(x, r) . (V.14)
M0r ∂r
we find that the corresponding distribution function is similar in shape compared to the
one obtained with MB statistics (see again Fig. V.2). The overall normalization differs
75
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
∫ ∫
by a factor d3 pafMB/ d
3 pafQS ∼ 5, which is not relevant for the purpose of deriving
structure formation limits. More importantly, the average momentum is very similar be-
tween the two cases; we obtain 〈p/T 〉 = 3.18 for quantum statistics that is to be compared
with 3.24 from MB yielding only a ≈ 2% difference.
Due to practically identical results on 〈p/T 〉 that stem from the two approaches and the
fact that the cutoff to be imposed is at values of p/T that do not effectively impact 〈p/T 〉,
we proceed with the classical statistics description that allows us to work with fully ana-
lytical expressions when deriving limits from structure formation in Section V.5.
V.2.2 Axion-like particle properties and production
In the following, we are going to use the ALP DM momentum distribution function f(x)
given in Eqs. (V.10) and (V.11) to derive important quantities of the DM species such as
its abundance for instance.
The first step is to verify, that the ALP does not thermalize in the early Universe. This
implies that the reheating temperature, TRH, should not exceed the temperature at which
ALP decouples from the thermal plasma. This decoupling temperature, Tdec, can be esti-
mated by equating the scattering rate Γ and the Hubble rate H(T ) (see Eq. (II.28)). The
scattering rate can be calculated from the momentum integrated collision term, C(x, r),
which can be derived from the production cross section by comparing Eq. (V.9) and
Eq. (V.14), and the number density of SM fermions, nf = 4(6 + 3 · 6)× 3ζ(3)T 3/(4π2) =
72ζ(3)T 3/π2, where we sum over all lepton and colored quark families and counting par-
ticles/antiparticles. The number density is derived from Eq. (II.22) using a Fermi-Dirac
distribution in∫the highly relativistic limit. In the photophilic case we find for Γ,
d3p C(x, r)
Γ =
(2π3∫) n∞4 ∫f∞T ( ) [ ]
= dx dy σ̂ T 2 y exp − − yx
8π3nf ( 4x0 y∗
g′
( ))
2 T 3 c2 ′2aγγ 11g
= gY 23− 24γE − 12 log , (V.15)
1728π3ζ(3) cos2 θ f2W a 48
where γE is the Euler-Mascheroni constant.√For the numerical values of the QED and
U(1) ′Y coupling constants, e and g , we use 4π/137 and 0.35, respectively and neglect
the running of these parameters.
On the other hand, the requirement that the correct DM abundance is reproduced can be
used to fix the reheating temperature in terms of caγγ[(see Eq. (II.31)), (
gs(T ) 4π4 ρ 2 3 2 2 ′
)]
2 −1
∗ RH c 12faπ cos θw fT = a 2 − 11gRH ′ Ω h 23 24γ + 12 log .m 45 s g 2 2 DMa 0 M0 gY caγγ 48
(V.16)
Comparing both temperatures gives the following ratio, where most prefactors dropped
out,34 ( )( )
T sRH ' × −4 g∗(T ) 10 keV2 10 . (V.17)
Tdec 106.75 ma
Hence, for ma ∼ O(10 keV) it is guaranteed that the ALP DM never thermalizes.
For the photophobic scenario, Γ/H(T ) is largest for temperatures at T ' mf : at higher
34 This ratio slightly differs from the published version. Although it is roughly two times larger, it does
not significantly affect the interpretation of the results based on it.
76
V.2. THE MODEL SETUP
temperatures, the Hubble rate is stronger than Γ, while at T < mf the scattering rate
drops because of the Boltzmann suppression of the respective fermion. However, as we
will show next, demanding that Ω h2 is matched, we require c /f . 10−9DM aff a GeV
−1 for
ma = O(10 keV) and with such feeble couplings it is guaranteed that photophobic ALP
DM does not thermalize with SM species and thus freeze-in is viable.
Doing a momentum integration of the distribution function given in Eq. (V.10), we find
for the contribution fr(om f̄f → B)a in the photophilic scenario to the DM abundance,
106.75 3/2
( )2 ( )( )
f̄f→Ba 2 ≈ caγγ/fa mDM TRHΩDM h 0.12 gs∗(T ) 10−
,
17
prod GeV−1 10 keV 6.7× 1016 GeV
(V.18)
and the fB → fa pro(cess yields)3/2( )2 ( )( )
ΩfB→fa 2 ≈ 106.75 caγγ/fa mDM TRHDM h 0.12 .gs∗(T ) 10−17prod GeV−1 10 keV 2.7× 1015 GeV
(V.19)
Including both co(ntributions)we (get ) ( )
2 ≈ 106.75
3/2 c 2 ( )aγγ/fa mDM TRH
ΩDMh 0.12 . (V.20)
gs∗(Tprod) 10−17 GeV
−1 10 keV 2.6× 1015 GeV
We keep gs∗(Tprod) fixed at the time of ALP DM production, i.e. the reheating temperature
TRH. Clearly, the scattering process fB → fa is the dominant production channel in the
photophilic scenario and hence will also be the more important process in the determina-
tion of structure formation limits. The reason for that lies in the logarithmic enhancement
arising due to regularizing the t-channel B-mediated process with a thermal gauge boson
mass which is absent for the production via fermion annihilation.
The equivalent equati(ons to Eqs.)(V.18) and (V(.19)∑for the)p(hotophobic scenario )read3/2 ( )
f̄f→V a κ
2
Ω h2DM ≈
80 mDM f f caff/fa
0.12 ,
gs∗(Tprod) 10 keV 38.9 GeV 9.6× 10−11 GeV−1
(V.21)
and ( ) ∑
80 3/2 ( ( )( )fV→fa 2 ≈ m ) 2DM f κf caff/faΩDM h 0.12 ,gs∗(Tprod) 10 keV 38.9 GeV 1.2× 10−10 GeV−1
(V.22)
while the sum giv(es ) ( ∑ )( )
2 ≈ 80
3/2 ( m )DM f κf c 2aff/fa
ΩDMh 0.12 . (V.23)
gs∗(Tprod) 10 keV 38.9 GeV 7.6× 10−11 GeV−1
One can infer from Eqs. (V.21) and (V.22) that both f̄f → V a and fV → fa are of
similar strength in the photophobic scenario and thus both contributions are important
when deriving limits from structure formation.
77
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
V.3 Structure formation probes: Lyman-α forests and
Milky Way satellites
As already discussed in Section II.2.6, a DM candidate which features a non-vanishing
distribution function for rather large values of p/T can be considered to be “warm”. It
starts to wash out structures at small scales and this can be quantified by the suppression
of the matter power spectrum at such scales. The respective DM momentum distribution
function f(x, r) is essential in the computation of the matter power spectrum, P (k) at
wavenumbers k. Hence, computing the matter power spectrum is a starting point and the
essence for assessing the structure formation limits. In the following, we are going to use
the tools developed in Sections II.2.6.1 and II.2.6.3 to calculate the suppression effect on
small scale structures for this warm ALP DM model, i.e we are going to use a half-mode
analysis and counting the number of Milky Way (MW) subhalos to limit the ALP DM pa-
rameter space.
Qualitatively, the suppression of small scale structures due to ALP DM can be quanti-
fied by the average momentum 〈p/T 〉 of the respective momentum distribution function
corresponding to both production c∫hannels for which we find:∞
0 dxx
3ff̄f→γa(x)〈p/T 〉f̄f→V a = ∫∫∞ = 3.0 (2.31) ,0 dxx2ff̄f→γa(x)∫∞ dxx3〈 〉 0 ffγ→fa(x)p/T fV→fa = ∞ ≈ 3.24 (2.44) ,
0 dxx
2ffγ→fa(x)
⇒ 〈p/T 〉 ≈ 3.24 (2.36) , (V.24)
where f(x) is evaluated at rf = mH/mf for the photophobic and at r → 0 for the
photophilic scenario. The numbers in each line correspond to the photophilic and photo-
phobic scenario, the latter given in brackets. The last line refers to the sum of the two
process, and it is the one which is physically relevant. Based on the averaged momentum
the photophilic scenario is slightly “warmer” than the photophobic case.
We start by repeating some details for the respective analysis techniques. The starting
point of a half-mode analysis is the transf√er function T (k),
T (k) ≡ P (k) , (V.25)
P (k)ΛCDM
where the subscript ΛCDM in the denominator denotes the reference matter power spec-
trum and the expression in the numerator is the corresponding power spectrum for ALP
DM. In the following we calculate it numerically by providing analytical expressions for
f(x) to CLASS [145, 146]. Furthermore, we have to take into account entropy dilution ef-
fects due to a change in gs∗(T ) between the time of production and today. This effect is
quantified via an effective DM temperature TDM which is related to the photon tempera-
ture Tγ by ( ) ( )
gs∗(T0)
1/3 3.94 1/3
TDM = Ts γ = Tγ . (V.26)g s∗(Tprod) g∗(Tprod)
As can be seen this leads to a “cooling” of the ALP DM and the strength of this effect
depends on the time of production Tprod. Above the electroweak phase transition we have
gs∗(T ) = 106.75 for the SM.
As explained in Section II.2.6.1, the computed transfer function is compared to an analyti-
cal fit of the transfer function of a warm thermal relic (TR) with mass mTR and abundance
78
V.3. STRUCTURE FORMATION PROBES: LYMAN-α FORESTS AND MILKY
WAY SATELLITES
ΩTR [148] (see Eqs. (II.45) and (II.46) for further details on the analytical expression). In
the following, we use for the Hubble constant, h, and ΩTR the values quoted in Ref. [52].
We can relate the masses ma and mTR by equalizing the free-streaming length, λFS, for
the ALP and TR model. In the thermal relic case, λFS ∼ 0.22 Mpc (keV/mTR)4/3, whereas
for the ALP [234], ( )( )
keV gs∼ ∗(Tν)
1/3
λFS Mpc . (V.27)
ma gs∗(Tprod
Since we deal with large production temperatures, Tprod, we have g(Tprod) = 106.75,
whereas the thermal relics decouple at a later time Tν , where g
s
∗(Tν) = 10.75. This allows
us to derive the following relation between the t(wo masses:m )TR 4/3
(ma)min ' 2.1 keV . (V.28)
1 keV
Our averaged ALP DM momentum is similar to the result quoted in Ref. [264].
A better comparison can be done by utilizing the half-mode criterion (see Eqs. (II.48)
and (II.49)). The reference transfer function, Tlim, is based on Eq. (II.45) and uses mTR
adopted from dedicated searches using limits from Lyman-α forest observations. The
lower bounds on the mass of thermal relic, mTR, span the range between mTR = 1.9 and
5.58 keV (see Refs. [56, 148, 231, 265, 266]). When showing our limits on the ALP DM
parameter space, we will present both weak and strong bounds, corresponding to this
mass range of mTR. In Fig. V.3 we show the transfer function of ALP DM for a selection
of different masses ma as solid, dotted and dashed blue lines and compare them against
the limits which exclude the red and light red shaded regions. One can infer that the
more stringent limit (mTR = 5.58 keV) is essentially excluding ALP masses ma . 15 keV,
whereas by taking the weaker limit of mTR = 1.9 keV, only ma . 5 keV are disfavored.
Using the corresponding half-mode wavenumber khalf , we can deduce a relation similar
to Eq. (V.28). For the ALP model we derived khalf numerically from the matter power
spectrum for different ma and(fitted) the result with an exponential to get the relationp
kALP
ma 2
half = p1 , with p1 = 0.87 , p2 = 3.56 . (V.29)keV
In the case of a thermal relic, we can solve the equation T (k )2 = 1/2 using Eq. (II.45)
directly to find that ( ) half
ν/10 − 1/(2ν)
kTR
2 1
half = , (V.30)α
where α depends on mTR. Equating Eqs. (V.29) and (V.30) allows to derive a numerical
relation between mTR and our ALP DM mass, ( m )TR 1.27
(ma)min ' 2.1 keV , (V.31)
1 keV
which can be used to map limits from structure formation quoted in mTR directly onto a
limit (ma)min. Assuming a different value for g
s
∗(Tprod) we have to modify this relation by
taking the entropy dilution (Eq. (V.26)) into account. Consequently, the previous equation
is modified by an additional factor
( m ) ( )1.27 106.75 1/3TR
(ma)min ' 2.1 keV . (V.32)
1 keV gs∗(Tprod)
Therefore, smaller values of gs∗(Tprod) have to be compensated by larger ALP DM masses.
We will come back to this when discussing limits for the photophobic scenario.
79
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
1.0
0.8
0.6
0.4 ALP, ma = 1 keV
ALP, ma = 5 keV
0.2 ALP, ma = 10 keV
analytic, mTR = 1.9 keV
analytic, mTR = 5.58 keV
0.0
10 1 100 101 102
k [hMpc 1]
Figure V.3: Transfer function T (k) shown as blue lines for photophilic ALP DM with
different masses compared to thermal relic limits presented as an analytic fit given in
Eq. (II.45). Shaded in red is the weak limit whereas the light red shaded region is excluded
by the strong limit. Hence, the weakest mass bound stemming from this analysis is
ma ' 5 keV, as indicated by the dotted blue line.
As far as the number of MW satellites is concerned, ALP DM with a momentum distri-
bution function which yields an averaged momentum as given in Eq. (V.24) would cause
the occurrence of smaller number of subhalos compared to ΛCDM. The predicted number
of satellites has an intriguing connection with the type of DM and its production and we
are able to set limits on ma by requiring that the number of satellites is not smaller that
what was observed. As explained in Section II.2.6.3, we are going to set Nsub = 64 as the
number of observed subhalos and thus ALP DM scenarios which predict less subhalos will
be excluded. However, there is some uncertainty associated to this number and therefore
we will examine the influence of Nsub on our results in more detail in Section V.7.3.
A detailed discussion on the derivation of Nsub is given in Section II.2.6.3. In the fol-
lowing, we are going to calculate Nsub via the integral over the subhalo mass range (see
Eqs. (II.58) and (II.59) for more details)
M∫MW
1 1 MMW √P (1/R)Nsub = dM , (V.33)
34 6π2 M R3 2π(S − S0)
Mmin
which depends explicitly and implicitly via the variance S on the matter power spectrum
P (k) of the underlying ALP DM model. Here, the integration boundaries are given by,
M = 108min M and the MW mass, MMW. Decreasing ma leads to smaller variances Si
as well as a suppression of P (k) which then features an earlier drop. The latter effect is,
however, stronger and therefore less subhalos Nsub are predicted.
We further note, that the prediction for Nsub clearly depends on the MW mass, MMW. The
precise value of this quantity is still under investigation and, depending on the analysis,
the reported values range roughly between 1 × 1012M < MMW < 1.5 × 1012M (see
Refs. [267–271] and references therein), using recent data from the GAIA survey. In
the following, we will refer to the lower mass as strong, while the higher mass bound is
80
T(k)
V.4. SUPPRESSING PRODUCTION FROM MISALIGNMENT AND
TOPOLOGICAL DEFECTS
dubbed as weak, because a heavier MW mass allow for a larger number of subhalos in
the considered ALP DM model. We want to stress the following two subtleties: first, the
quoted MW masses are defined with respect to densities 200 times larger than the critical
density, the so-called virial mass of the MW. For this reason, the constant C of Eq. (II.59)
is given by C = 34. Second, Eq. (V.33) uses MW masses in units of M/h as an input
and we have to take this additional factor into account.
V.4 Suppressing production from misalignment and topo-
logical defects
An important assumption of our discussion is that the ALP DM is predominantly pro-
duced through a freeze-in process. This means that a contribution to the ALP abundance
by the misalignment, as well as any ALP population produced from the decay of topolog-
ical defects such as cosmic strings, must be suppressed. As we are going to show in what
follows, this is not trivial for the photophilic ALP, as the requirement of a small misalign-
ment energy density implies bounds on TRH and fa which are in contrast with what is
needed in order to match the observed DM abundance with ALP production via freeze-in.
If ALP DM features a significant contribution from misalignment, limits from structure
formation will be weakened, because, in contrast to ALP DM produced by a freeze-in, this
ALP behaves like “cold” DM (CDM) after the axion field relaxed to the potential min-
imum. For this reason, in order to justify the assumption that the misalignment energy
density is suppressed, we will need some specific assumption about the axion potential
and the specific symmetry breaking pattern through which it develops in the early Uni-
verse. This statement holds only for the photophilic axion, as we will discuss below.
The energy density from ALP misalignmen(t, ass)umi(ng a consta)nt ALP mass, is [136]
− ma 1/22 × 2 ai
2
Ωah = 1.61 10 . (V.34)
eV 1011 GeV
Requiring that this value does not exceed t(he obser)ved DM abundance yields1/4
a < 2.6× 1010 10 keVi GeV , (V.35)
ma
where ai is the displacement of the ALP from the potential minimum when it starts
to oscillate. It√is related to the misalignment angle θi via ai = θi fa. If the ALP ispresent during inflation,35 the minimal misalignment is set by quantum fluctuations during
inflation, ai > NeHI/2π [58]. Here, Ne is the number of e-folds
36 of inflation and HI
is the Hubble rate during the inflation epoch. This, combined with Eq. (V.35), results in
a rather stringent upper limit for HI . Recalling that the maximal reheating temperature
is obtained under the assumption of instantaneous reheating, i.e. the case in which the
entire energy density of inflation is converted into radiation, we obtain a bound on the
reheating temperature, ( ) ( ) ( )
× 14 106.75
1/4 10 keV 1/8 60 1/4
TRH < 1.2 10 GeV . (V.36)
gs∗(Tprod) ma Ne
35 If the ALP is the pseudo-Nambu-Goldstone boson (PNGB) of a spontaneously broken global symmetry,
this corresponds to the scenario where the symmetry is broken before or during inflation, with HI < fa.
In any case, this assumption is not crucial for our discussion.
36 One e-fold is the time interval in which a patch of the Universe is grown by a factor of e. Hence, it acts
as a time variable in inflation scenarios.
81
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
With such values of TRH in the photophilic scenario, matching the DM abundance would
typically be only achievable for large caγγ/fa couplings, disfavored by gamma-ray con-
straints for ma & 2 keV (see TRH = 1014 GeV line in Fig. V.4). For the photophobic
scenario instead, this bound is always satisfied in our considered parameter space.
In the following, we present two alternative ways in which the misalignment contribution
can be suppressed without implying a low reheating temperature as in Eq. (V.36).
For instance, if the Peccei-Quinn (PQ) symmetry breaks after inflation, the Universe
consists of many patches with different values for ai. Thus, neglecting anharmonicities in
the potential, the initial misalignm∫ent angle is given by by the root mean average overπ
all possible values for θ , 〈θ2〉 = dθ̄ θ̄2 2i i −π i i /2π = π /3. After inserting this value into
Eq. (V.34) and imposing Ω h2a < ΩDMh
2 we obtain
( )
10 keV 1/4
fa < 1.4× 1010 GeV . (V.37)
ma
In this case we have no upper bound on the reheating temperature, because 〈ai〉 is fixed
from the beginning. We only need to assume that the ALP-SM couplings are suppressed
compared to fa, i.e. caγγ , caff  α/(2π), the r.h.s being the reference value of this cou-
pling in typical axion models. Further, we need to make sure that the contribution from
the decay of cosmic string is negligible. For the typical example of a PNGB from the
breaking of a global U(1) symmetry, such as the standard QCD axion, the axion abun-
dance from the decay of topological defects is similar in magnitude to the one from
misalignment [124].37 Assuming this is the case, we expect both contributions to be sub-
dominant when Eq. (V.37) is satisfied.
The second way of suppressing the misalignment energy density is to have a very small
angle from the start. This is achieved if the PQ symmetry is broken already during in-
flation and the axion is heavy, ma > HI , hence suppressing quantum fluctuations. If the
axion mass is constant, this would imply a too strong upper bound on TRH. Still, the
mass of the axion could have been much larger during inflation than today, caused for in-
stance by time evolving PQ breaking dynamics (see e.g. Refs. [272, 273] and references
therein for some explicit realizations). In particular, for ma ≈ 1016 GeV, the reheating
temperature can be as large as 1017 GeV during inflation. If, after inflation, the ALP is
light again, thermal fluctuations could in principle increase the energy density from the
misalignment mechanism. This is, however, not the case for the photophilic scenario un-
der our consideration as ALPs do not thermalize with the SM plasma.
We again stress that, for the photophobic ALP, the requirement of a vanishing misalign-
ment energy density is easier to satisfy. If the axion is present during inflation, Eq. (V.36)
applies, which is always satisfied for our choice TRH . O(100 GeV). If, instead, the PQ
symmetry is broken after inflation, fa should satisfy Eq. (V.37). Comparing this require-
ment with the preferred value of caff/fa shown in Fig. V.6, obtained by imposing that
the freeze-in abundance matches the observed DM one, results in caff . O(1).
Nonetheless, we will briefly discuss the case of non-negligible misalignment production of
ALP DM when discussing constraints from the MW subhalo count in the next section.
37 So far, it is still not exactly known how large this additional abundance contribution is, but it is estimated
to be at least of similar size as the contribution from axion misalignment production.
82
V.5. STRUCTURE FORMATION LIMITS ON THE MODEL PARAMETER SPACE
V.5 Structure formation limits on the model parameter
space
We use the distribution functions in Eqs. (V.10) and (V.11) to calculate the matter power
spectrum using CLASS, derive the corresponding transfer function and the number of MW
subhalos and finally compare against results from observations following the strategy out-
lined in Section V.3. As long as the observed DM abundance is matched, the result does
not depend on the couplings caγγ and caff . Therefore, we performed a scan over ma to
derive limits.
V.5.1 Results for the photophilic scenario
The results for the Lyman-α forest and the MW subhalo count for the photophilic scenario
are shown in Fig. V.4. For both analysis techniques, we show weak and strong bounds as
discussed in Section V.3. The explicit values for the lower limits on the DM mass ma are
also quoted in Table V.1. For this scenario we set gs∗(Tprod) = 106.75 for all parameter
choices because of the high reheating scale involved.
For the photophilic scenario we also show limits from gamma-ray searches which severely
narrow the viable parameter space in the ma & 17 keV region where structure formation
constraints fade away. We have calculated gamma-ray limits on ALP DM by utilizing
existing ones on keV-scale sterile neutrino DM [81] and matching them with the corre-
sponding ALP decays into photons.
In the photophilic scenario, an ALP can decay directly into two photons and the decay
width is given by [274]
→ 1
c2aγγ
Γ(a γ γ) = m3. (V.38)
64π f2 aa
Usually available limits from gamma-ray searches are quoted in the parameter space for
sterile neutrinos, which decay into a photon and a SM neutrino. Using Eq. (II.14), we
match existing bounds on the sterile mixing angle ϑ onto the ALP-photon coupling by
equating the corresponding decay widths, Γ(a→ γ γ) ≡ Γ(νs → γ ν)/2. Here, the factor 2
compensates that νs decays only into one γ. This yields a limit c
lim
aγγ for the ALP-photon
coupling, ( )
clim ( ) 2 lim 1/2aγγ ma sin (2ϑ )
= 9.55× 10−18 GeV−1
f keV 10−
, (V.39)
8
a
where ϑlim denotes a mixing angle limit from a given gamma-ray survey. We use limits
derived from INTEGRAL [275] (dark blue), NuSTAR [276] (orange) and M31 [277] (red
shaded region) to constrain the ALP DM parameter space and show the results together
with our own limits derived from structure formation indicated by the blue shaded regions
in Fig. V.4. As already pointed out, the limits from structure formation are independent
from the coupling caγγ and constrain only ma. The respective lower mass limits are given
in Table V.1. Last but not least, the red line is a limit based on estimations of the
sensitivity for an upcoming survey. We will briefly comment about this at the end of this
section.
weak strong
Lyman-α 4.9 keV 19.1 keV
MW subhalo 10.3 keV 17.4 keV
Table V.1: Structure formation limits on the ALP mass ma in the photophilic scenario.
83
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
We want to point out one important thing: as explained in Eq. (V.16), TRH is fixed
by requiring the ALP to be abundant enough. This relation is indicated for the pho-
tophilic ALP by the gray lines in Fig. V.4, where we present three different choices:
T 14RH = 10 GeV (solid), T
16 −1/2
RH = 10 GeV (dotted) and TRH = mPl ≡ MPl(8π)
(dashed) based on the reduced Planck mass. The values of TRH in the region below the
solid gray line are violating the bound from Eq. (V.36). Although a substantial pro-
duction from misalignment would invalidate such parameter space, we have shown in
Section V.4 several ways to remedy such situation and suppress “cold” ALP DM pro-
duction via misalignment. Such requirements are not even necessary in the region to the
right of T = 1014RH GeV where the misalignment production is suppressed.
The other lines correspond to different upper limits on the reheating temperature:
the limit on the tensor-to-scalar power ratio r obtained by Planck sets the bound
T . 1016RH GeV [278] (dotted gray line). There can be inflationary scenarios in which
this limit does not apply; in these cases the maximum allowed reheating temperature is
set by mPl. That means, on the left of the TRH = mPl dashed gray line, the value of the
reheating temperature necessary to obtain Ω 2DMh = 0.12 exceeds the reduced Planck
scale making this parameter space theoretically implausible. In summary, it is interest-
ing to note, that limits from current gamma-ray surveys are able to probe most of the
accessible photophilic ALP DM parameter region for ma & 2 keV, if we consider only
ALP-photon couplings.
On the other hand, taking more production channels into account, we can alleviate this
bound on TRH. For this purpose we are going to assume that the ALP couples not only
to B aµ bosons, but also to the SU(2) gauge bosons, Wµ , via caWW = caγγ sin θW . In
that case, we can derive similar diagrams as shown in Fig. V.1, but they give rise to a
larger contribution to the DM abundance. Compared to the cross section stemming from
ALP-Bµ boson coupling, σ
aBB, the(cros)s se(ction inc)ludi∑ng W
a couplings, σaWWµ is given
by38
σaWW ∝ nc nf + nf g
2
2 cos θ
2
W a tr(σ
aσa) ' 22 . (V.40)
σaBB gY g′ sin θW 4
The first expression takes into account, that the W a boson couple only to LH quark and
lepton doublets; the second factor includes the SU(2) gauge coupling, g2 ' 0.65, and the
third the corresponding relations to caγγ ; the last expression is a trace over two SU(2)
generators where σa are the Pauli matrice√s. Including this additional production chan-
nel requires smaller TRH by a factor of ' 22. We indicate the combination of Bµ and
W aµ couplings by the corresponding light gray lines in Fig. V.4. For this case, structure
formation can rule out ALP DM parameter space not covered by current gamma-ray
searches. Future surveys such as the THESEUS mission [279] will be able to provide a
complementary probe of this interesting parameter region. Nonetheless, we observe that
for ma . 2 keV the structure formation limits are clearly leading, even taking only Bµ
couplings into account.
In light of our results, we want to briefly comment on the previously reported unidenti-
fied line at ∼ 3.55 keV in a gamma-ray data spectrum [93, 94]. It sparked lots of interest
as a potential signal of decaying DM and one of the favored explanations is the decay
of keV-scale sterile neutrinos, but a 7.1 keV ALP DM was discussed as well [248]. Our
strong structure formation limits are, however, clearly disfavoring such an interpretation,
while the weak ones are marginally consistent with it. We should still stress that our
38 Since the W aµ have a larger thermal mass, m2 = (11/12)
1/2g2T compared to the Bµ boson, the corre-
sponding momentum distribution functions features a slightly larger averaged momentum 〈p/T 〉 ∼ 3.33,
but we neglect this effect here to allow for a simple rescaling.
84
V.5. STRUCTURE FORMATION LIMITS ON THE MODEL PARAMETER SPACE
findings are not in general disfavoring a DM interpretation of a 3.5 keV line and hold only
for the freeze-in production of ALPs. For instance, scenarios where ALPs are dominantly
produced via the misalignment mechanism are still viable in this regard.
As mentioned before, above reheating temperatures T 14RH = 10 GeV, we have to sup-
press ALP production by the misalignment mechanism. Keeping this in mind, we address
the case for a photophilic ALP, where the production via scattering processes only con-
tributes a fraction of “warm” ALP DM, F ≡ Ωa/ΩDM, to the total abundance, while the
rest is “cold” ALP DM generated by the misalignment mechanism as described in Sec-
tion V.4. In this case ALP DM consists of a mixture of two components with different
temperatures and the limits on the ALP parameter space change accordingly.
The results are shown in Fig. V.5 where we vary F between 0.05 and 1 and derive the cor-
responding limits on ma based on the MW subhalo count using the strong mass choice.
39
In particular, for F < 0.2, i.e. only a small “warm” ALP DM fraction, the mass limit
starts to drop and effectively vanishes for very small fractions . 0.05 since the larger
“cold” ALP DM component dominates the matter power spectrum in this case. That
means, if misalignment is not significantly suppressed, the mass limits in Fig. V.4 have to
be rescaled and become significantly weaker.
V.5.2 Results for the photophobic scenario
In this section, we apply the same techniques as above to the photophobic scenario. In
that case we derived Eq. (V.11) by expanding Eq. (V.9) for small mf assuming that
TRH  mf . By doing so, the dependence on the reheating temperature actually drops out.
We implicitly assumed that TRH . 160 GeV. However, for smaller reheating temperatures
we have to keep the full expression and an explicit TRH dependence is recovered, as we
will discuss later in this section.
Using Eq. (V.23), the accessible parameter space is then represented by a linear relation
between ma and caff , ( ) ( )
caff ' × −11 10 keV
1/2 38.9 GeV 1/2
7.49 10 GeV−1 . (V.41)
fa ma κf
The results are shown in Fig. V.6 where the green diagonal lines denotes the viable pa-
rameter space of the model assuming couplings to different fermions. The upper line
corresponds to a tau lepton contribution only, while the lower edge shows the sum of tau
lepton, c and b quark contributions with equal coefficients, and the two lines in between
indicates an exclusive coupling to either c or b quark.
The regions shown in blue are disfavored using Lyman-α forest limits and the MW subhalo
counts. The strongest limit on the photophobic ALP arises from Lyman-α data (see also
Table V.2) and reads ma ≈ 15 keV. The derived constraints are slightly weaker compared
to the photophilic case, because ALP DM produced in the photophobic scenario is slightly
“cooler” compared to the other case (see Section V.2).40 However, in the photophobic
scenario we assumed TRH . 160 GeV and so we have to take a change in gs∗(Tprod) into
account. The results were derived using gs∗(Tprod) = 106.75 and so the limits on the ALP
39 The half-mode analysis is not particularly suited to constrain ma for this setup, since the presence of
the CDM component gives rise to a plateau in the transfer function and it is not clear how to define
khalf properly. See Section VI.3.2 for further comments on this issue.
40 A recent study including an updated number of MW satellites quotes stronger results in terms of
mTR > 6.5 keV [280]. This allows us to use the ALP DM transfer function directly to derive mass
bounds. After doing so we find ma > 23.0 keV for the photophilic and ma > 17.5 keV for the photophobic
scenario, which extends even beyond the strong bounds derived in our own analysis.
85
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
10-15
M31 NuSTAR
1 14
-16 0 Ge INTEGRAL10 V
-17 101610 GeV
10-18 TRH=m
Pl (caBB)
TRH=m
10-19
Pl (caBB+caWW)
10-20
1 5 10 50 100
ma [keV]
Figure V.4: The blue shaded regions are structure formation limits on the photophilic ALP
parameter space derived using a half-mode analysis technique as well as a MW subhalo
count. The former (latter) are denoted as Lyman-α (MW). For both, weak and strong
limits are shown (see Section V.3 for further analysis details). Additionally, gamma-ray
limits from INTEGRAL (dark blue shaded) [275], NuSTAR (orange shaded) [276] and
M31 [277] (red shaded) are shown. The diagonal dashed lines indicate specific values of
TRH, calculated by the requirement of producing the observed amount of DM. The dark
gray line assumes only couplings between ALP and B bosons, whereas the light gray lines
takes an additional coupling caWW into account. Finally, the red line at ma ∼ 80 keV is
the projected sensitivity from the forthcoming Vera C. Rubin observatory. Image credits:
Vedran Brdar (adapted by author)
mass have to be rescaled according to Eq. (V.32). For instance, if gs∗(Tprod) = 80, all mass
limits are increased by a factor ' 1.1 compared to the results shown in Table V.2.
weak strong
Lyman-α 3.7 keV 15.5 keV
MW subhalo 7.8 keV 13.3 keV
Table V.2: Structure formation limits on the ALP mass ma in the photophobic scenario
under the assumption that gs∗(Tprod) = 106.75.
If the ALP has flavor universal couplings or if the strength of the ALP-fermion coupling
is at least comparable for several flavors including electrons, further constraints have to
be applied on the parameter space. First, if there is an ALP-electron coupling, ALPs
are produced via Bremsstrahlung processes inside red giants (RG), would leave the core
and lead to additional energy emission of RGs which can be detected. Recent bounds
on such cooling effects during the Helium ignition phase of RGs, place a limit on the
dimensionless coupling g < 1.2×10−13ae [281]. Converting it to our parametrization gives
86
c -1γγ/f [GeV ]
Ly-α weak
MW weak
MW strong
Ly-α strong
Vera C. Rubin
Observatory
V.5. STRUCTURE FORMATION LIMITS ON THE MODEL PARAMETER SPACE
10
5
1
0.50
0.10
0.05 Excluded by MW strong
0.01
0.05 0.10 0.50 1
F
Figure V.5: Limits for the photophilic ALP F − ma parameter plane stemming from a
MW subhalo count assuming the strong MW mass choice. We considered that freeze-in
contributes only a fraction F < 1 to the total ALP DM abundance. The blue shaded region
is excluded because these parameters yield Nsub < 64. As the cold ALP DM contribution
increases, the limit on ma decreases correspondingly and effectively vanishes for F . 0.05.
(see Ref. [282]),
caee gae
= < 1.2× 10−10 GeV−1. (V.42)
fa 2me
However, ALPs can only be produced inside RG, if they are light enough; assuming a
RG core temperature of 108 K, the limits applies only for ALPs with mass ma . 9 keV.
Second, recent electron recoil measurements from the XENON1T collaboration limits the
ALP-electron coupling even further [128]. The latter is clearly more relevant for a larger
span of ALP masses. Consequently, the XENON1T line is chiefly below the green band in
the region ma . 10 keV, disfavoring the flavor universal ALP coupling scenario for such
parameter choice.
Although a tree-level ALP-photon is absent in the photophobic scenario, there is still an
effective coupling c′aγγ to photons due to fermion loops [125]. In the limit of ma → 0 it is
given by
c′ 2 2 ∑ n Q2aγγ ' −mae c f caff
f 24π2 m2
, (V.43)
a f faf
where we have to sum over all fermions f involved with their respective electric charges
Qf in units of e. Inserting the effective coupling into Eq. (V.38) allows to evaluate con-
straints from gamma-ray searches for the photophic scenario as well. We have, however,
found that such gamma-ray limits are only relevant if ALP couples to electrons, because
the loop induced coupling to photons is suppressed by the mass ratio m2 2a/mf . In this
case, even parameter space above ma ∼ 10 keV, unexcluded by XENON1T, would be
disfavored in a flavor universal scenario. We note, however, that it was recently shown
that by adding more new physics, destructive interference between loop-level diagrams
contributing to decays into SM photons can be achieved; hence the limit can be relaxed
substantially [283], particularly for higher values of ma.
As already pointed out, keeping higher order terms of mf in f(x) as opposed to Eq. (V.41)
restores a TRH dependence for the photophobic scenario as well and the viable parame-
87
ma [keV]
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
10-9
10-9.5
τ
Red Giants
10-10
c
10-10.5
c b+b+τ
10-11
1 5 10 50 100
ma [keV]
Figure V.6: As before, blue shaded regions are structure formation limits on the photo-
phobic ALP parameter space derived using a half-mode analysis technique as well as a
MW subhalo count. The former (latter) are denoted as Lyman-α (MW). For both, weak
and strong limits are shown (see Section V.3 for further analysis details). The green lines
indicate the viable parameter space based on Eq. (V.41). If the ALP couples only to tau
leptons, the ALP needs large couplings indicated by the uppermost line, whereas the other
lines correspond to couplings to c or b quark and the sum over all three fermions. If the
ALP has a flavor universal coupling, additional bounds from red giant cooling and lim-
its from XENON1T has to be taken into account. Finally, the red line at ma ∼ 60 keV is
the projected sensitivity from the forthcoming Vera C. Rubin observatory. Image credits:
Vedran Brdar (adapted by author)
ter space, i.e. the green lines in Fig. V.6, does depend on the choice of the reheating
temperature. The dependency is shown in Fig. V.7 where we have varied TRH between
160 GeV and 10 GeV (shown as green dashed lines) for a photophobic ALP which couples
exclusively to b quarks. Compared to Fig. V.6, lower reheating temperatures have to be
compensated by slightly larger coupling coefficients caff . Further, we included the same
entropy dilution factor as shown in Eq. (V.32), to take a decrease in gs∗ for smaller reheat-
ing temperatures into account: while the blue shaded regions correspond to mass limits
for gs∗(Tprod) = 106.75, the black lines are bend to higher masses, because the DM tem-
perature increases compared to the thermal bath. For TRH → 10 GeV this factor is set by
' 1.07. We have restricted the lower temperature limit to TRH & 2mf , because at even
lower reheating temperatures the fermion mass starts to become relevant and the b quark
will not necessarily reach a thermal population.
We note here that our results constrain the relaxion41 DM model of [252], in which the
41A relaxion is an ALP with special properties; it is constructed such that it can address the hierarchy
problem of the SM. During the cosmological evolution, it scans the Higgs mass parameter range by
providing a time-dependent effective Higgs mass term. Back-reactions will eventually stop the relaxion,
hence allowing a natural realization of the weak scale Higgs mass.
88
c /f [GeV
-1
 ]
XENON1T
Ly-α weak
MW weak
MW strong
Ly-α strong
Vera C. Rubin
Observatory
V.5. STRUCTURE FORMATION LIMITS ON THE MODEL PARAMETER SPACE
b-quark
10-10
T
RH ≥
160 GeV
T
RH ≤
10 GeV
10-10.5
1 5 10 20 50
ma [keV ]
Figure V.7: Impact of TRH on the viable parameter space of the photophobic ALP, as-
suming that it couples only to b quarks. A larger ALP-fermion coupling is needed to get
the correct DM abundance for lower reheating temperatures. The blue shaded regions are
the same mass limits as in Fig. V.6, where gs∗(Tprod) = 106.75, while the black lines are
computed taking a change in gs∗(Tprod) into account.
relaxion production occurs via freeze-in similar to the scenario considered here. The re-
laxion can constitute 100% of DM only if a large mass is allowed by a large hierarchy in
the relaxion couplings. Even assuming the most conservative bounds from the number of
MW subhalos, the model provides an explanation of the XENON1T anomaly only if the
relaxion is responsible for a fraction of the DM abundance [253].
Finally, we are going to comment on future sensitivity projections. The Vera C. Rubin
observatory is designed to go beyond the results derived from SDSS and among observ-
ing other properties, it can measure the MW mass halo function down to 106M. Hence,
its resolution allows to detect more ultra-faint subhalos and it will improve limits dra-
matically. Using its projected limit on thermal relic DM, mTR > 18 keV [284, 285], we
find that, for the photophilic case, the lower limit on the ALP DM mass would be pushed
to ma > 83 keV. Similarly, we find ma > 62 keV for the photophobic case (both values
are illustrated in Figs. V.4 and V.6 with red lines). It is rather intriguing that for the
case of photophilic ALP, the structure formation limits offer a complementary probe to
gamma-ray limits in the ma ∼ O(100 keV) region.
89
c -1 /f [GeV ]
Ly-α weak
MW weak
MW strong
Ly-α strong
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
V.6 Summary of Chapter V
ALPs are currently one of the most popular extension of the SM, being studied and tested
across a wide range of masses and couplings. In this section we studied keV-scale ALP DM
produced via a freeze-in through feeble interactions with gauge bosons, such as photons,
Bµ bosons and gluons, and SM fermions. The respective momentum distribution function
was calculated assuming MB statistics for the involved particles and we briefly discussed
the implementation of quantum statistics and compared both methods with each other.
Although these approaches did not capture the full picture at small values of p/T , they
turned out to be robust for constraining the parameter space using the matter power spec-
trum derived from the respective momentum distribution function. Using Lyman-α forest
data as well as the observed number of MW companions we derived structure formation
limits that were missing to date. For the photophilic ALP DM, we found the most strin-
gent limits to exclude ALP DM masses below ∼ 19 keV, complementing constraints from
gamma-ray data. In particular, we have found these searches to severely limit our scenario
where only Bµ couplings are considered, because the corresponding reheating temperature
necessary to achieve enough ALP DM abundance is too large. Thus, non-trivial inflation
scenarios or additional production channels via a ALP-W coupling are needed in order to
avoid these astrophysical constraints.
We also discussed how to suppress an additional “cold” ALP DM production via misalign-
ment which would weaken the derived limits on the ALP mass.
For the photophobic ALP, the obtained limits are somewhat milder, because in that case
we found DM to be “cooler”, i.e. the averaged momentum is slightly smaller. Further, if
the photophobic ALP does couple flavor-universal and not only to heavy fermions, the vi-
able parameter space will be in tension with limits from gamma-ray data.
Utilizing measurements from the upcoming Vera Rubin observatory, we found that the
mass bounds will be strongly improved up to ma & 60 keV for both scenarios.
90
V.7. APPENDIX OF CHAPTER V
V.7 Appendix of Chapter V
V.7.1 Amplitudes for dark matter production channels
In the following chapters, we will go into more details regarding the calculation of the
reduced cross sections σ̂ needed to determine the momentum distribution f(x, r) of the
ALP DM. We discuss the ALP production via the scattering process first, before we turn
on the production of ALPs via fermion annihilation. To keep the expressions as simple as
possible, we mention only the coupling to photons in the following. However, the results
can easily be generalized to the case of a B gauge boson or gluons.
V.7.1.1 Process A: f γ → f a
This process, also-called Primakoff process for ALP-photon couplings only, comes with
three separate diagrams: the first one involves a coupling to photons, caγγ , we will refer to
it as A1, whereas the other two are proportional to caff instead. We refer to the u-channel
diagram as A21 and the s-channel as A22. The corresponding production diagrams are
shown in Fig. V.8. Following the procedure outlined in Ref. [234] we calculate the inverse
process f(p1) a(p)→ f(p3)γ(p ).422
a(p) γ(p2) a(p) f(p1)
a(p) γ(p2)
caγγ caff
caff
f(p1) f(p3)
A22
f(p3) f(p1) f(p3) γ(p2)
A1 A21
Figure V.8: Inverse production process of ALP DM via its coupling to photons (A1) and
fermions (A21 and A22). An ALP particle a is converted into a photon γ by scattering
off a fermion f .
The matrix elements for the respectiv(e processes ar)e given by
M − caγγ gρβA1 = e ε∗ν ū(p3)γρu(p1) −i − × 
αβµν(p2 − p)α(−p2µ) , (V.44)
fa (p p )22
M − caff ∗ i(p
3 − p
 +mf )A21 = 2iemf εν ū(p3)γ ν5 − − 2 γ u(p1) , (V.45)fa (p3 p)2 mf
M caff i(p+ p1 +mf )A22 = −2iem ε∗f ν ū(p3)γν 
 
 − 2 γ5u(p1) . (V.46)f (p+ p 2a 1) mf
Here we have used the equations of motion to rewrite the axion-fermion coupling as
caff caff
∂µaf̄γ
µγ5f = −2imf a f̄γ5f , (V.47)
fa fa
and the four-dimensional Levi-Civita tensor arises from the ALP-photon interaction term
caγγ
aF F̃µνµν .
f
42 Since we do not assume any CP violation this process is in fact the same as f(p3)γ(p2)→ f(p1) a(p).
91
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
The Mandelstam variables of this process are:
s = (p+ p1)
2 = m2a +m
2
f + 2p · p1
= (p2 + p3)
2 = m2f + 2p2 · p3, (V.48)
t = (p1 − p3)2 = 2m2f − 2p1 · p3
= (p− p )22 = m2a − 2p · p2, (V.49)
u = (p− p3)2 = m2 +m2a f − 2p · p3
= (p1 − p2)2 = m2f − 2p1 · p2. (V.50)
They are related by
s+ t+ u = m2a + 2m
2
f → u = m2a + 2m2f − s− t. (V.51)
Squaring the sum of Eqs. (V.44) to (V.46), summing over the polarizations of the outgoing
photon and the spins of the initial fermions and setting ma = 0, we find for the amplitude
of process A:
c2aγγ (−4m2fs+ 2m4f + 2s2 + 2st+ t2)|M 2 2A| = − e ( ) (V.52)f2a t
c2 2
− 2 2 aff caγγ caff t8e mf 2 + .f2a f2 2a (mf − s)(s+ t−m2f )
We used Eq. (V.51) to remove the dependence on the Mandelstam variable u.
V.7.1.2 Process B: f f̄ → γ a
This process involves three separate diagrams as well: one s-channel diagram with a
coupling to photons, we will refer to it as B1, whereas the other two involves an ALP-
fermion coupling instead. We refer to the t-channel diagram as B21 and the u-channel as
B22. The corresponding production diagrams are shown in Fig. V.9. The inverse process
we are interested in is γ(p1) a(p)→ f(p3)f̄(p2).
a(p) f(p2) a(p) f(p2)
a(p) f(p2)
caff
c
c affaγγ
γ(p1) f(p3)
B1
γ(p1) f(p3) γ(p1) f(p3)
B21 B22
Figure V.9: Inverse production process of ALP DM via its coupling to photons (B1)
and fermions (B21 and B22). An ALP particle a annihilates with a photon γ into a
fermion-antifermion pair.
The matrix elements for the respect(ive processes a)re given by
M − gaγγB1 = e ū(p2)γρv(p3) −
gρβ
i × αβµν(−p− p1)αp1µεν , (V.53)
fa (p+ p )21
92
V.7. APPENDIX OF CHAPTER V
M − caff 5 i(p= 2iem ε ū(p )γ 
1 − p
3 +mf ) νB21 f ν 2 − − 2 γ v(p3) , (V.54)f 2a (p1 p3) mf
M − caff ν i(p
 − p
3 +mf )= 2iem ε ū(p )γ γ5B22 f ν 2 − − 2 v(p3) . (V.55)fa (p p3)2 mf
As above we have used the equations of motion to rewrite the axion-fermion vertex. The
Mandelstam variables of this process are:
s = (p+ p1)
2 = m2a + 2p · p1
= (p + p )2 = 2m22 3 f + 2p2 · p3, (V.56)
t = (p1 − p3)2 = m2f − 2p1 · p3
= (p− p 2 2 22) = ma +mf − 2p · p2, (V.57)
u = (p− p3)2 = m2 +m2a f − 2p · p3
= (p 2 21 − p2) = mf − 2p1 · p2. (V.58)
They fulfill the same relation as the Mandelstam variables of process A given in Eq. (V.51).
Squaring the sum of Eqs. (V.53) to (V.55), summing over the polarizations of the outgoing
photon and the spins of the initial fermions and setting ma = 0, we find for the amplitude
of process B:
c2 2 4 2aγγ (−4mf t+ 2mf + s + 2st+ 2t2)|M 2 2B| = e (V.59)
f2a ( s )
c2 2
− 2 2 aff caγγ caff s8e mf 2 +f2 f2 (t−m2)(s+ t−m2 .a a f f )
V.7.2 Calculating the cross section
To find the momentum distribution we have to calculate the reduced cross section σ̂ of a
a+ 3→ 1 + 2 process first.∫It is given by [286]
λ(s,m23,m
2
a) d
3p 31 d p2
σ̂ = (2π)4δ4(p 2
3 3 1
+ p2 − p3 − pa)|M|
s (2π) 2E1 (2π) 2E2
≡ λ(s,m
2
3,m
2
a)Φ ,
s
where mi, (i = 1, 2, 3), is the mass of particle with momentum pi and λ(x, y, z) is the
Källén or triangle function,
λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz. (V.60)
For the calculation of the final state phase space integral, Φ, it is convenient to use the
center-of-mass frame and assign the following choices for the particle momenta:
pa = (0, 0, k,
p3 = (0, 0,−k),
p1 = p3 + `, |p1| ≡ q, (V.61)
where ` = (sin θ cosφ, sin θ sinφ, cos θ) `
p2 = −p1.
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CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
The ` integration will be done last in order to isolate possible IR divergences in the t-
channel. Using the Mandelstam variable s = (p1 + p
2
2) = (p3 + pa)
2, we find for the
squared particle momenta, [( ) ]
2 2 2s 2 2
k2 = [ 1−
m3 − ma − m3m4 a] , (V.62)4 ( s s ) s2
s m2 2
2
m m2m2
q2 = 1− 1 − 2 − 4 1 2 . (V.63)
4 s s s2
They are related to ` by
k2 − q2 + `2
cos θ = . (V.64)
2k`
The integration range for ` can be derived by imposing cos θ < 1 and cos θ > −1:
|k − q| < ` < k + q . (V.65)
Finally, the Mandelstam variables t√, u can be√rewritten as
t = m2 +m2 − 2√q2 +m2√k2 +m2 + k2 + q2 − `21 3 1 3 , (V.66)
u = m2 +m2 2 2 2 2 21 a − 2 q +m1 k +ma − k − q2 + `2 , (V.67)
which, in the massless limit (mi → 0), simplifies to
t = −`2, (V.68)
u = `2 − s. (V.69)
Now everything is expressed in terms of the integration variable ` and we can compute the
phase space integral Φ. Using the three-dimensional δ3-distribution including the particle
momenta and s = (p3 + pa∫)2 = (E3 + Ea)2, it is given by
3 1 −√Φ = d p1 δ(E1 + E2 s) |M|2 . (V.70)
16π2E1E2
Changing integration variable from p1 to `, the δ distribution can be rewritten as
√
δ(E1√+ E2 − s) = √
= δ( k2 +(`2 − 2k` cos θ +m21 +) k2 + `2 − 2k` cos θ +m22 −√s)
E1√E
2
2 k − q2 + `2
= δ cos θ − , (V.71)
k` s 2k`
and the int∫egral ∫is given by2π 1 `∫max ( )
2 ` 1 E E
2 2 2
1√2 − k − q + `Φ = dφ d cos θ d` δ cos θ |M|2
2 16π2E1E2 k` s 2k`
0 −1 `min
`∫max
1 1
= d`2 |M|2 . (V.72)
8π ( )1/2λ s,m23,m2a `min
The integration boundaries are fixed by Eq∫. (V.65). Finally, the reduced cross section is`max
1
σ̂ = d`2 |M|2 . (V.73)
8πs
`min
In the next step, we determine σ̂ for the two respective production processes.
94
V.7. APPENDIX OF CHAPTER V
V.7.2.1 Cross section for process A
For this process we have to substitute the masses with m1 = m3 = mf , m2 = 0 and in the
ma → 0 limit, the integration boundaries (E(q. (V.65))) become
2 2mf
0 < `2 < s 1− . (V.74)
s
However, the c2aγγ term from the t-channel diagram diverges logarithmically for `
2 → 0.
For this reason, this expression has to be regularized by introducing a finite thermal photon
mass mγ = eT/3 > 0.
43 The Mandelstam variables are given by
t = −`2,
u = `2 + 2m2f − s. (V.75)
Inserting these into Eq. (V.52) and adapting the ` boundaries from Eq. (V.74) with the
lower cutoff mγ , t(he reduce)d c(ross se(ction is given by
2 2
) )
e2 caγγ m
2 (s−m2)2 m2 4f f f mf
σ̂A = ( s 1−16π f2 s ) 4 log( − 3− 2 +sm)2a γ ( ( )s s2 )
e2 c
2 2 2
aff caffcaγγ mf s mf m
4
− 2 − f2 + mf 1 2 log 2 − 3 + 4 − , (V.76)2π f2 f2 2a a s mf s s
where we kep only the logarithmic dependence on mγ in the caγγ term. Expanding it for
small mf up to second orde(r yield(s the)simpl)er expression:
e2 c2aγγ s
σ̂A = s 4 log − 3
16π f2a ( m2γ )( ( ) ) (V.77)
e2 c
2
2 aff caffcaγγ s+ mf 2 + 2 log − 3 +O(m3) .2π f2a f2a m2 ff
V.7.2.2 Cross section for process B
For the second process, the masses are m1 = m2 = mf , m3 = 0 and for ma → 0, Eq. (V.65)
becomes
 √   √ 1 2m2 4m2 2m2 4m2s 1− f − − f 1 f f1 < `2 < s1− + 1−  (V.78)
2 s s 2 s s
and no cutoff has to be specified, because the finite fermion mass already serves as an IR
regulator in the t-channel. The corresponding Mandelstam variables for process B are the
same ones as in Eq. (V(.75). The p)h√ase space integration can be straightforward evaluated:
e2 c2
2
aγγ mf 4m
2
f
σ̂B = s 1 + 2 1−
12π f2a ( s )s  √ 
c2 s− 2m22 c − s 1− 4m2/s− 2 2 aff affcaγγ  f fe m 2 + log f 2 . (V.79)π f2a f2a 2mf
43 The more formal treatment is to regularize this expression by an IR cutoff `cut for the ` integration and
include the thermal two-point function of the photon. At the end, a logarithmic dependence on mγ is
left, while the `cut terms cancel with each other (see Ref. [263] for more details).
95
CHAPTER V. THE COSMOLOGY OF FROZEN-IN AXION-LIKE PARTICLE
DARK MATTER
At second order in mf , it reduces to( ) ( )
e2 c2aγγ 2 c
2
2 2 aff caffcaγγ sσ̂B = s+ e m 2 + log +O(m3) .
12π f2 π f f2 2 2 fa a fa mf
V.7.3 Impact of Nsub on the mass limits
Regarding the MW subhalo count, we have already pointed out associated uncertainties
in Section V.3. We have presented our limits on ma for different choices of the MW mass
(see Tables V.1 and V.2) but kept the requirement that Nsub ≤ 64. As explained in
Section II.2.6.3, this value is derive assuming an isotropic distribution of MW satellites
seen by SDSS which may be too simplistic. Therefore, we examine how the ALP mass
limits change for different numbers of MW subhalos in Fig. V.10. To illustrate this effect,
we choose the photophilic ALP case and compare the corresponding limits for Nsub = 50
(red lines) and Nsub = 80 (blue lines) against the limits derived in Section V.5 shown as
black lines. The lower part shows the results for the weak MW mass choice, whereas the
upper part are the limits for a strong MW mass choice. Especially for the latter case, the
mass limits clearly dependent on the choice of Nsub: for Nsub = 80, ALP masses up to
' 48 keV would be excluded, while the same requirement yields only a limit of ma ' 14
for the weak MW mass choice. The exact results are quoted in Table V.3.
Nsub 50 64 80
MW
weak 8.8 keV 10.3 keV 13.6 keV
strong 12.3 keV 17.4 keV 47.6 keV
Table V.3: Limits on the ALP mass ma for different choices of the observed amount
of subhalos Nsub compared to the case Nsub = 64 as used in this section, assuming a
photophilic scenario.
96
V.7. APPENDIX OF CHAPTER V
This highlights how discoveries of new ultra-faint subhalos by future surveys will be able
to probe the parameter space of freeze-in ALP DM and WDM in general. However, a
precise measurement of the MW mass is additionally needed, as the interpretation of the
results depends not only on the amount of observed subhalos, but on the properties of the
MW as well.
photophilic
MW strong
Nsub=50
Nsub=64
MW weak Nsub=80
5 10 20 50
ma [keV]
Figure V.10: Comparison of ALP DM mass limits for different choices of Nsub and MW
masses in the photophilic scenario. The upper part is the strong MW mass and the lower
the weak mass choice. The blue lines are ma limits for Nsub = 80 and the red lines are
using Nsub = 50 while the black lines are the results given in Tables V.1 and V.2.
97

Chapter VI
Two temperature dark matter: a
general picture
VI.1 Motivation
While the WIMP paradigm is strictly speaking not ruled out yet, it is under pressure
from current direct detection results [62] and hence the last several years have seen a rise
in interest of other avenues to explain the DM puzzle (see for instance Refs. [81, 148, 240,
287]). Specifically, the focus has shifted to DM candidates of lighter masses. While these
particles allow for potential new experimental probes, they may have an impact on the
cosmology of the early Universe as well. In fact, if DM is light enough such that it is still
relativistic at sufficiently late times it has a non-vanishing free-streaming length and will
thereby change the formation and the properties of galaxies compared to the standard
ΛCDM paradigm [288] below this length scale.
Initially, the effect of such “hot” DM has been studied in the context of neutrino DM,
which turned out to feature a too large free-streaming length, effectively erasing struc-
tures at far too large scales. Thus, attention has shifted to DM models which can be
considered to be “warm” instead. Interestingly, N -body simulations of structure for-
mation in the ΛCDM regime revealed small scale structures which are in tension with
observations (for an overview, see Ref. [289]). These simulations predict too large num-
bers of accompanying galaxies of the Milky Way (MW) [156, 290], which is usually called
the “missing satellite” problem. Further, the shape of galactic cores do not match with
observations [291] (the cusp versus core problem) and lastly, the too big too fail problem
[292, 293] addresses a mismatch in the dynamics of the brightest MW satellites. It is still
under debate if these issues can be alleviated in the ΛCDM paradigm by including bary-
onic feedback in the simulations (see for instance [294–296]).
On the other hand, going beyond the ΛCDM paradigm, aforementioned tensions can be
cured by invoking “warm” DM models which alter the small scale structures but agree
with cold DM (CDM) at large scales. Of particular interest with respect to above men-
tioned questions, are DM mixtures of “cold” and “warm” DM species. Often, the “cold”
component is considered to be of standard ΛCDM origin, while the “warm” component
is made of something else: popular extensions involve sterile neutrinos [152, 154, 233], ul-
tralight particles [150, 245], axions [251], fuzzy DM [297, 298] or non cold thermal relics
[299, 300]. Some of the first models invoked neutrinos as a “hot” DM candidate mixed
with a “cold” component (see for instance [301–305]).
In the following, we are mainly concerned with scenarios with only a single DM species,
which, however, features a non-thermal momentum distribution, for example due to dif-
ferent production mechanisms contributing to its relic abundance.
Such scenarios of DM with a non-thermally produced warm admixture have been consid-
ered for instance in Refs. [1, 2, 234, 306–309] and we discussed examples in Chapters IV
99
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
and V, where we analyzed the scotogenic model (ScM) framework and axion-like particle
(ALP) DM. Precisely, we will have a subset of DM characterized by a higher temperature
such that it can be considered to be “warm” or even “hot”. The aim is to quantify our
results in a way, that they can be matched for a wide variety of models which might fea-
ture other ways to produce DM at different times. A similar idea was done in Ref. [310],
where the authors trained an emulator using the matter power spectrum of a mixed cold
and warm DM model setup.
In particular, this chapter extends and refines the analysis of Chapter IV, where we used
simplifying assumptions which are going to be relaxed in the following sections, espe-
cially with respect to the treatment of the effective number of entropic degrees of freedom
(DOF), gs∗(T ).
We introduce a model-independent parametrization of the DM momentum distribution
function and use current observational limits on observables related to the matter power
spectrum to set bounds on the allowed parameter space. In the following, we will refer to
our framework as the two temperature dark matter (2TDM).
This chapter is organized as follows: in Section VI.2 we discuss how we parameterize the
2TDM and explain the assumptions we made when modeling the production mechanism.
Section VI.3 consists of a discussion of constraints on the parameter space derived from
cosmological observations. In Section VI.4 we discuss some modifications of the simplified
description. Results are shown in Section VI.5 and an application of the matching to some
sample models is given in Section VI.6. Lastly, we summarize our findings in Section VI.7.
VI.2 Setup of the framework
We consider a framework where the observed DM density is explained via one particle
species which is produced in two different production modes at separate times. Therefore,
an intrinsic temperature can be assigned to both production possibilities. Consequently,
the final DM abundance is made up by two shares of a single particle species each with
its own temperature. For convenience we are going to refer to the earlier produced part
as the first subset, while the other one is the second subset. A schematic representation
of this is shown in Fig. VI.1. Crucially, the second DM subset whose production happens
at a later time t2 features a significantly larger temperature than the first DM set which
is produced earlier on at time t1, that means we take that T2 > T1.
44 A scenario with
these assumptions has important consequences for the behavior of DM. In fact, while the
first subset might be cold, the second one will be warm or even hot DM. The question
which arises is two-fold: how large can the temperature of the second subset be, and how
much of it can be produced? To assess these questions, we will study the impact of the
2TDM on the structure formation at galactic scales. To be precise, as will be explained
in Section VI.3.2, we are adopting observations of the Lyman-α forest and the observed
number of MW subhalos to constrain the 2TDM.
A visualization of the parameters we are constraining is given in Fig. VI.2 (similar to the
benchmark points presented Fig. IV.2 for the ScM), where we show an example of the
DM momentum distribution functions weighted by the comoving momentum, x = p/T ,
squared, x2 f(x), which is a characteristic quantity for a given DM model. The spectrum
features two distinctive peaks, which can be characterized by two quantities: first, the
44 This does not have to be necessarily true for all cases. Depending on the properties of the parent
species, one can have model setups, where the temperature of the latter produced DM subset is smaller
compared to the temperature of the DM produced at earlier times. However, in that case one can change
the naming 1↔ 2 for both DM subsets.
100
VI.2. SETUP OF THE FRAMEWORK
T
DM,1
T2 DM,2
T1
t1 t2
t
Figure VI.1: Schematic representation of our setup: given in black is the background
temperature Tγ of the universe, while the blue and red curve represents the time evolution
of the temperature of the first and second DM set, respectively. The first DM subset is
produced at time t1 with a temperature similar to Tγ . After a while, the second DM subset
is produced at t2 but it has a higher temperature T2 compared to T1. The bumps in the
solid black line mimics entropy dilution due to particle freeze-out in the SM thermal bath.
position of their peaks which is related to the respective DM temperature and second, their
contribution to the total DM density Ω h2DM which is set by the area under the respective
curve. As can be seen, in this example the first share is the dominant contribution which
contributes a part A1 to the total DM density (blue shaded region), while the hotter DM
part contributes A2 (red shaded region).
VI.2.1 Decay parametrization
The momentum distribution shown in Fig. VI.2 corresponds to DM freeze-in production
via decays of thermalised (blue curve) and non-thermal particles (red curve). Following
this, we are going to assume that DM in the 2TDM is produced via decays of heavy parent
particles irrespective whether they were previously thermalized or not. Specifically, we are
interested in the following two parameters:
T2
Temperature ratio: ξ ≡ > 1,
T1
Abundance : A2 ∈ (0, 1] .
The abundance A1 of the first DM subset is fixed by the requirement A1 = 1 − A2 such
that the DM relic abundance is achieved.
It is important to note that we assume the temperature between both DM subsets to stay
constant once produced during the subsequent time evolution of the universe. In principle,
we could have entropy dilution in the dark sector similar to the SM thermal bath, but in
the following we want to neglect such dilution effects for the dark sector.
VI.2.2 Decay modeling
In the following, we model the production of the individual DM subsets by decays of
particles P , which decay into a DM particle X via P → XX and whose mass is mP . More
details on the DM production via decaying scalars can be found for instance in Refs. [109,
101
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
0.100
0.001
10-5
Comp I
10-7
10-9
10-11 Comp II
0.1 1 10 100 1000 104 105
p
x=
T
Figure VI.2: Example x2 f(x) spectrum for a specific 2TDM scenario with two observable
peaks calculated for times when DM production is finished. The blue area corresponds
to a dominant subset whereas the read area refers to a subdominant DM share but with
higher temperature with respect to the first set.
111, 235, 241]. In the following we want to focus solely on analytical expressions derived in
Ref. [235], similar to our procedure in Section IV.3.1, which corresponds to the following
types of production:
ˆ The parent particle decays while it is still in thermal equilibrium.
ˆ For sufficiently small decay widths, the scalar particle freeze-out before it decays
into DM.
ˆ If the scalar is only weakly coupled to the SM, it never thermalizes before or during
its decays, but freezes in instead.
Solving the Boltzmann equation for the DM momentum distribution yields [235]
∫r ∞
C s2
∫
Γ x̃
f(x, r) = 2 ds dx̃√ fP (x̃, s) . (VI.1)
gs∗(Tprod) x
2 x̃2 + s2
0 x̃min
Here, CΓ denotes an effective DM production rate via decays of P . Consequently, it is
related to the decay width45 Γ of P by C 2Γ = M0 Γ/mP , where M
18
0 = 7.35 × 10 GeV.
The factor of two in front of the expression has to be dropped if we look at processes
P → X S instead. Although it is implicitly assumed for this derivation that the number
of entropic DOF, gs∗(T ), are not changing at all, a dimensionless time variable r = mP /T
is introduced to track changes in gs∗(T ) during DM production; for a general discussion
we refer to Section VI.4.2.
In the following, we present the result for parent particles decaying while maintaining
thermal equilibrium, whereas the other two cases are given in Sections VI.8.1 and VI.8.2.
45 We assume that there is only one decay channel for P and so Γ is a total decay width.
102
x2f(x)
VI.2. SETUP OF THE FRAMEWORK
Assu(mi√ng a Max)well-Boltzmann (MB) distribution46 for the parent particle, fP (x, r) =
exp − r2 + x2 , we can derive the following expression for the DM momentum distri-
bution [235] ( √ [ ])
f(x, r) = 8C e−x
r −r2/4x 1 π r
Γ e + Erf √ . (VI.2)
2x 2 x 4x
In the limit r →∞, Eq. (VI.2) reduces to √
π
f(x,∞) ≡ f(x) = 4C e−xΓ . (VI.3)
x
Inserting this expression into Eq. (II.31), we can derive the DM abundance
mDM g
ΩDMh
2 = CΓK , (VI.4)
GeV gs∗(Tprod)
where K ' 3 × 108 and g denotes the internal DOF of the DM species. It may seem
counterintuitive at first that the DM density is directly proportional to the decay width
of P , but under this approximation the parents do not deviate from a MB distribution
and hence DM can only be efficiently produced before the parent particles experience a
Boltzmann suppression at temperatures T ≈ mP . The corresponding abundances for the
other two production processes do not depend on the size of CΓ.
Above formulas can be straightforward generalized to a DM scenario composed of i subsets,
each with a temperature Ti. In that case we are defining a reference temperature T1
which we take to be given by the lowest temperature of the DM species and so the final
momentum dist∑ribution for th∑e DM,√fDM(xi), is given by∑the follo√wing sum:π π
fDM(x1) = fi(xi) = 4 C
−xi −x1T1/Ti
Γ,i e = 4√ CΓ,i Tie . (VI.5)
xi x Ti i 1 1 i
Here each subspecies has its own decay width CΓ,i and temperature xi = p/Ti. For the
2TDM we find that √
π ( √ )
f −x1 −x1/ξDM(x1) = 4 CΓ(,1e + CΓ,2 ξe , ) (VI.6)x1
2 mDM CΓ,1 CΓ,2ΩDMh = K g + ξ
3
GeV gs∗(Tprod,1) g
s
∗(Tprod,2)
≡ Ω 2DMh (A1 +A2) . (VI.7)
Thus, Eq. (VI.7) allows to relate A1 and A2 to the decay width CΓ,i and demanding
that the DM relic abundance is generated constrains CΓ,1, CΓ,2 and ξ. However, above
expressions are only valid if the respective parent particles are in thermal equilibrium
during their decay into DM.
In the following we will match our prescription onto DM produced by long-lived particles
by defining a relation between the production rate CΓ and the temperature ratio of the
2TDM. We start by comparing the averaged momentum 〈x2〉 for the late-time produced
particles against 〈x1〉 stemming from DM production of thermalized parent particles. The
respective averaged momenta a∫re given by∞
〈 〉 ∫0 dx1 x31f(x1/ξ)x2 = ∞ = 2.5 ξ = ξ〈x1〉 , (VI.8)2
0 dx1 x1f(x1/ξ)
46 This assumption is crucial to derive analytic results, while the inclusion of a Bose-Einstein or Fermi-
Dirac distribution only marginally changes the result. Hence our discussion holds for all types of parent
particles.
103
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
where the averaged value 〈x2〉 is shifted by a factor ξ compared to 〈x1〉 = 2.5. This is im-
portant for matching the temperature ratio ξ to a specific decay width of long-lived parent
particles.
For this matching we assume the parent particles to be at rest when decaying and pro-
ducing the DM. Further we assume an instantaneous decay at time τ = t1/2 = log 2/Γ set
by the half-time of the parent particle, which can be converted into a temperature by the
relation
T ( )1/2
= 1.55 gs(T )−1/4
sec
MeV ∗ τ √
' mP8.5× 105 gs∗(T )−1/4 CΓ , (VI.9)TeV
where we used Γ = m2P CΓ/M0 in the last step. This temperature has to be compared to
the energy of the DM which is roughly set by E ≈ p = mP /2.
Finally, we can equate p/T and Eq. (VI.8) to derive a relation between our model param-
eter ξ and the DM production rate CΓ,
mP −1/2
ξ = =⇒ ξ ' 0.24C gs(T 1/4Γ ∗ prod) . (VI.10)5T (CΓ)
The last expression is only valid, if gs∗(T ) ≡ gs∗(Tprod) is constant during DM production.
As expected smaller decay widths give rise to a hotter second DM subset. The reason why
we use the half-life t1/2 instead of 1/Γ, stems from the matching between a production
assuming a shifted MB distribution and an explicit long-lived particle production mecha-
nism, which will be explained in Section VI.4.1. Therefore, such long-lived decays are well
approximated by a suitable choice of ξ based on Eq. (VI.10) and we can use it as an input
parameter for our simulations.
VI.3 Constraints on the model parameter space
Depending on the temperature ratio ξ, the subdominant production mechanism may lead
to a “warm” or even “hot” DM subset which could lead to a significant contribution to
the effective number of relativistic species, Neff, or alter small scale structures. Therefore
the 2TDM can be constrained by cosmological and astrophysical observations and mea-
surements. Stringent constraints arise from flux spectra analyses of the Lyman-α forest
and the number of dwarf galaxies of the MW as well as the measured value for Neff.
VI.3.1 Limits from additional radiation
Similar to Section IV.3.2, there are parts of the parameter space, where the second DM
subset effectively acts as radiation in the early universe and hence increases Neff by an
amount ∆Neff. For the SM, Neff = 3.046 [236] while measurements by the Planck collabo-
ration yield Neff = 2.99
+0.34
−0.33 (95% CL) (TT, TE, EE+lowE+lensing+BAO) [52] from the
cosmic microwave background (CMB) whereas at the onset of big bang nucleosynthesis
(BBN), Neff = 2.88 ± 0.52 (95% CL) [237]. Since BBN takes place at much earlier times
the latter bound is more relevant for us, because the warm DM subset has more time to
cool down until the CMB epoch.
We follow the procedure outlined in Section IV.3.2 and estimate ∆Neff by comparing the
kinetic energy of the DM species with temperature T2 = T1ξ to the energy density of a
massless Dirac fermion with a temperature equal to the neutrino temperature Tν , given
104
VI.3. CONSTRAINTS ON THE MODEL PARAMETER SPACE
by: ρ = 7π2ferm /120T
4 47
ν ,
≡ ρ(T1)−
( )
n(T )m 60 T 4 m gs1 DM 1 DM (T1)
∆Neff =
∗ ×
∫ 2ρ √ ( 7π4 )T ( T g)sferm ν 1 ∗(mP )∞
× 2

gs∗(T )
2/3 z 21 1T1
dz1 z 1 1 + − 1 f(zs 1/ξ) , (VI.11)( ) g∗(mP ) mDM0
gs(mP )
1/3
where z1 =x
∗
1 ,
gs∗(mP /r)
is a redefinition of the comoving momentum x1 including g
s
∗(T ), which was neglected in the
corresponding discussion of Section IV.3.2. The dependence on the parameters A2 and ξ
are encoded in f(z1/ξ). The prefactor (T /T )
4
1 ν evaluates to (11/4)
4/3 below temperatures
of 1 MeV and can be dropped for temperatures above. In the temperature range we are
interested in, T1 ≈ 1 MeV, we can simplify Eq. (VI.11) by expanding the square root for
small arguments. Using the (expression)for f(z1/ξ)(given in Eq. (VI).3) we find in this case
' 180 g
s
∗(T1) mDM 5T
s
1 g∗(T
1/3
1)
∆Neff C
4
Γ,2 ξ . (VI.12)
7π3 gs∗(mP ) T1 2mDM g
s
∗(mP )
This expression has to be evaluated at temperatures T1 for given choices of ξ and A2 which
are defined in CΓ,2. Inserting Eq. (VI.7) we find an expression for ∆Neff only in terms of
our model parameters(: )( )( )( ) ( )
gs 2' × −4 ∗(T1) ΩDMh 10 keV g
s
∗(T1)/10.75
1/3 2
∆Neff 8.1 10 A ξ .
10.75 0.12 m gs
2
DM ∗(mP )/106.75 g
(VI.13)
In Fig. VI.3, ∆Neff bounds from BBN and CMB are compared for two different DM
masses: the solid and dashed line represents BBN limits, i.e. we can use Eq. (VI.13)
evaluated at T1 = 1 MeV, for mDM = 10 keV and mDM = 100 keV respectively, while the
dotted and dashed-dotted line correspond to CMB limits for the same masses. The last
two are derived by evaluating Eq. (VI.11) at T1 = 0.24 eV. As already pointed out, the
limits from BBN are in general stronger than the respective ∆Neff results from CMB,
especially for smaller ξ values. The largest possible temperature T2 for a 10 keV DM
particle is ∼ 240T1, assuming g = 1, i.e. a scalar DM species. For fermions or vector
particles this bound has to be rescaled accordingly. Moreover, we observe that the limits
from CMB scales differently compared to BBN bounds at smaller temperatures. In that
regime, expanding the square root in Eq. (VI.11) leads to additional powers of T1 and
hence a larger temperature sensitivity.
VI.3.2 Limits from structure formation
Generally, a detailed study for a given “warm” DM or mixed “warm”/“hot” and CDM
model (in the following we will generally refer to these as WDM) would require hydrody-
namical N -body simulations to infer their impact on the formation of cosmological struc-
tures. However, the influence of a specific model on small scales can usually be understood
by comparing its corresponding matter power spectrum with the associated power spec-
trum of ΛCDM. Based on this comparison, conclusions can be drawn whether a given
WDM model features a too large suppression of structure formation at small scales. We
47 The additional factor of two in the denominator takes into account that f(x, r) includes both particles
and anti-particles, whereas Neff is defined with regards to families of relativistic particle species.
105
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
107
106
105
104
103 Neff BBN, mDM = 10 keV
Neff BBN, mDM = 100 keV
Neff CMB, mDM = 10 keV
Neff CMB, mDM = 100 keV
10210 5 10 4 10 3 10 2 10 1 100
A2
Figure VI.3: ∆Neff bounds derived from Eq. (VI.13) assuming mP = 1 TeV. The black
solid line corresponds to mDM = 10 keV and the dashed line to mDM = 100 keV using
the BBN bound ∆Neff < 0.344. The dotted and dashed-dotted line are the corresponding
bounds for mDM = 10 keV and 100 keV from CMB measurements, ∆Neff < 0.28. For
A2 = 1 temperature ratios ξ & 240 are excluded.
are using the public code CLASS [145, 146] to derive the matter power spectrum for the
2TDM model.
As explained in Section II.2.6, the suppression features of WDM models can be parame-
terized in terms of the transfer function T (k),
PWDM
T (k)2 = . (VI.14)
PΛCDM
It can be used to employ a half-mode analysis (see Section II.2.6.1) to match limits from
structure formation with specific WDM models.
On the contrary, this procedure is not suitable for the 2TDM model. Similar to mixed
DM models we are dealing with a plateau in the transfer function [152, 311] and as such a
simple half-mode analysis does not capture the whole picture of this model. Some examples
are shown in Fig. VI.4 where transfer functions for three different parameter choices with
mDM = 30 keV are compared to a thermal relic with mass mTR = 2 keV, which is shown
in blue. The solid and dashed red curves correspond to A2 = 0.2 and ξ = 25 or 125
respectively and the green curve has A2 = 0.05 and ξ = 25. While the transfer functions
shown in red and green are generally smaller than the thermal reference below some
wavenumber k due to the warmer DM subset, they still cross the blue line because the
larger first DM subset features a milder suppression of galactic structures. Using a half-
mode analysis would therefore exclude all three parameter choices and even a pretty small
deviation for A2  1 would be disfavored by such analysis. However, the parameter choice
shown in green is still allowed by limits on structure formation observables. Consequently,
we are going to use the matter power spectrum directly to extract limits on the model
parameter space. A similar conclusion for the ScM can be drawn from Fig. IV.4, where
an adapted half-mode analysis gives rise to stronger constraints compared to a matter
106
VI.3. CONSTRAINTS ON THE MODEL PARAMETER SPACE
1.0
larger T2 smaller A2
0.8
0.6
T2/T1 = 25, A2 = 0.2
T2/T1 = 25, A2 = 0.05
T2/T1 = 125, A2 = 0.2
mTR = 2 keV
0.4 100 101
k [h/Mpc]
Figure VI.4: Transfer function T (k) for a 2TDM model where mDM = 30 keV, A2 = 0.2
and ξ = 25, 125 shown in red and dashed red respectively, while the blue line is derived
from Eq. (II.45) for a thermal relic mass mTR = 2 keV indicating a potential limit from
an analysis on structure formation. The green line correspond to ξ = 25 and a smaller
A2 = 0.05. Applying a half-mode analysis, all three parameter choices would be excluded,
but limits from observables on the matter power spectrum only exclude the two red lines,
while the green line is not in conflict.
power spectrum analysis similar to the one discussed in the next subsection, but in that
case these constraints were subdominant only. On the other hand, the half-mode analysis
done in Section V.3 is reliable even though we considered two different DM production
mechanisms. Note that for this case the corresponding averaged momenta are nearly
identical and, further, for the photophilic ALP, one channel is clearly dominating.
VI.3.2.1 Lyman-α forest data
The first tool we use to constrain the 2TDM framework is based on Section II.2.6.2. It uses
a ratio δA, calculated from the momentum integrated P (k) of 2TDM parameter choices
and compare it to ΛCDM, to approximate the amount of suppression by
AΛCDM −A
δA ≡ , (VI.15)
AΛCDM
where AΛCDM = kmax − kmin. The results we are taken from an analysis examining
the combination of the MIKE/HIRES and the XQ-100 datasets [56]. MIKE/HIRES ob-
served quasars with redshifts z = 4.2–5.4, while XQ-100 measured between z = 3–4.2.
Both sets combined span a range in kν-space from 0.003–0.08 s km
−1. Hence, we will set
kmin = 0.5h/Mpc and kmax = 10h/Mpc in the following.
To derive limits on the 2TDM, we have to define a reference WDM model with a cor-
responding δAref value first. The analysis in Ref. [56] yields a lower bound for thermal
WDM given by mWDM = 3.5 keV (at 95% CL) considering a conservative thermal history
of the universe. Under the assumption of a power-law evolution this bound strengthens
to mWDM = 5.3 keV. Using these masses as input parameters for a thermal WDM model
107
T(k)
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
we derive the following values:48
mWDM = 3.5 keV ⇒ δAref,1 = 0.30, (VI.16)
mWDM = 5.3 keV ⇒ δAref,2 = 0.20. (VI.17)
That means all parameter points in our scenario which have δA > δAref are excluded since
their small scale suppression is too strong.
VI.3.2.2 Number of Milky Way satellites
To predict the number of MW satellites or subhalos for the 2TDM we follow the procedure
already explained in Sections II.2.6.3 and V.3. We calculate Nsub via the integral over the
subhalo mass range (see Eqs. (II.58) and (II.59) for more details),
M∫MW
1 1 M0 √P (1/R)Nsub = dM , (VI.18)
34 6π2 M R3 2π(S − S0)
Mmin
which depends explicitly and implicitly via the variance S on the matter power spectrum
P (k) of the underlying 2TDM. Here, the integration boundaries are given by, Mmin =
108M and the MW mass, MMW.
Compared to Section V.3, we refine the limits on the MW mass. With the second data
release of the GAIA mission, several works have calculated the MW mass using different
analysis techniques (see Refs. [268, 269, 313–321] and [271]) and a compilation of these
results is shown in Fig. VI.5. Combining every measurement and following Ref. [322] how
to combine data points with asymmetric errors, we calculate for the MW mass,
M = 1.180.16 × 1012MW −0.15 M (95% CL). (VI.19)
In the following we will set these limits as a lower (i.e. light) and upper (i.e. heavy) MW
mass bound and reject parameter points if they have Nsub < 64.
As a side remark, we comment briefly on the MW mass dependence of this procedure.
We matched it to the prediction of the Aquarius simulation [324], taking Msub > 10
8M,
which is Nsub = 158 by calculating P (k) for ΛCDM and a larger MW mass, MMW '
2 × 1012M. In contrast to the predicted number of this specific simulation, using the
mass choices of Eq. (VI.19), we have found the number of subhalos to be only ' 100 in
the ΛCDM case.49
VI.4 Detailed study of the parametrization
So far, we were assuming that the parent particles are thermalized when decaying, but in
general this assumption does not hold for rather long-lived or weakly coupled particles.
In the following, we show that a shifted MB distribution for f(x, r) can be used to describe
the momentum distribution of DM produced from the decay of non-thermal parent parti-
cles, whose distribution function is set by a freeze-in or freeze-out mechanism. Further, we
study the impact of a temperature dependent gs∗(T ) on the DM momentum distribution.
48 As pointed out in Refs. [309, 312], these reference values are not clearly defined as they depend on the
examined k-range and the wavenumber cutoff for the one-dimensional power spectrum. We use the cutoff
k = 200h/Mpc when evaluating Eq. (II.50) for our analysis.
49 This observation was already pointed out in Ref. [325] as a possible explanation for the missing satellites
problem.
108
VI.4. DETAILED STUDY OF THE PARAMETRIZATION
[323]
[321]
[320]
[319]
[318]
[317]
[316]
[315]
[314]
[313]
[269]
[268]
0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
MMW [1012 M ]
Figure VI.5: Compilation of different MW mass analyses using the second data release
from the GAIA survey. Further information on the results can be found in Refs. [268, 269,
313–321, 323]. The black dotted line is the combination of all measurements and the green
shaded region represents a 2σ error range. On the contrary, blue errorbars correspond to
68% CL limits of the respective analyses.
VI.4.1 Non-thermalized parent particles
In Section VI.2.2 we suggested to use a shifted MB distribution to model late-time decays
of parent particles. In the following, we are going to verify that this is a good approxima-
tion for the cases where fP (x, r) is determined by a freeze-out, if the particle is sufficiently
coupled or, if not, by a freeze-in.
To compare this approximation and the two late-time regimes we make use of Eq. (VI.10)
to mock these decays, which are governed by CΓ, with a shifted MB distribution and cor-
responding temperature ratio ξ (see Eq. (VI.5)). This will guarantee that DM is produced
at approximately the same time. For the case of the frozen-in or frozen-out parent particle,
we calculate f(x, r) numerically by inserting the corresponding fP (x, r) into Eq. (VI.1).
As an illustrative example we choose CΓ ' 5.2 × 10−4, which corresponds to ξ ≈ 40 and
set A2 = 0.5 for each production mechanism, to compare the results for the correspond-
ing transfer functions for the case of a shifted MB distribution with the other two cases in
Fig. VI.6. Shown in green is the result using a shifted MB distribution, while the trans-
fer function for DM production by decays of parent particles after they are frozen-out or
frozen-in are shown in blue and gray, respectively. We observe that the corresponding
matter power spectra feature a similar scale where they deviate from ΛCDM. Only de-
cays of frozen-in parent particles give rise to a slightly earlier drop in T (k). Overall, the
deviation between an appropriately shifted MB distribution and a freeze-in or freeze-out
parent is only marginal; this allows us to model late-time decays using our simpler analytic
expressions.
To highlight implications for structure formation even more, we calculate Nsub using the
lighter MW mass and δA for fixed mDM = 50 keV and different choices of A2. The respec-
tive results are shown in Table VI.1.
In summary, our findings indicate that we can model late-time decays to a good approx-
imation by a temperature shifted momentum distribution assuming thermalized parents
109
Refs.
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
1.0
0.8
0.6
0.4
freeze-out
0.2 freeze-in
shifted MB
0.010 2 10 1 100 101 102
k [h/Mpc]
Figure VI.6: Comparison of the transfer function T (k) for a shifted MB distribution and
a freeze-in/freeze-out scenario where the temperature ratio is set to ξ = 40 and A2 = 0.5,
while mDM = 10 keV. We see that the analytical shifted MB distribution (red curve)
is a good tool to approximate the numerical results for decays of frozen-out or frozen-
in parents, shown in blue and gray, respectively. Hence we will use it in the following
analysis to deduce constraints from structure formation. The horizontal black dashed line
corresponds to a pure ΛCDM scenario.
shifted MB freeze-in freeze-out
A2 Nsub δA Nsub δA Nsub δA
0.1 76 0.335 76 0.341 76 0.334
0.3 37 0.627 36 0.638 37 0.630
0.5 14 0.787 14 0.798 14 0.791
Table VI.1: Comparison of the corresponding structure formation observables using a
shifted MB distribution for parent particles and the respective momentum distributions
for non-thermal parent particles. The DM mass is set to 50 keV and only the abundance
A2 is varied. The predictions for Nsub and δA are nearly identical.
only.
As a final remark, we use above mentioned methods to place absolute lower mass bounds
on the DM mass, mlimDM, by assuming ξ = 1. These values act as a guideline for the al-
lowed parameter choices for the 2TDM and the respective limits from structure formation
are summarized in Table VI.2.
Nsub Lyman-α
light MW heavy MW δAref,1 δAref,2
mlimDM [ keV] 12.8 9.0 12.7 7.7
Table VI.2: Lower DM mass limit mlimDM using constraints from structure formation as-
suming ξ = 1, i.e. all of DM has a common temperature T2 = T1.
VI.4.2 Impact of a variation in gs∗(T ) during dark matter production
In the previous sections we have treated the number of entropic DOF gs∗(T ) as a fixed
quantity. This assumption is only well justified for high decoupling temperatures, Tdec &
160 GeV, where gs∗(T ) = 106.75 is constant (neglecting non-SM DOF). This simplification
may be applicable for the first DM subset, but this simplifying assumption does not hold
necessarily for the second warmer DM subset. Of course, the impact of a varying gs∗(T )
110
T(k)
VI.4. DETAILED STUDY OF THE PARAMETRIZATION
depends on the production time of the second DM subset, which is related to the mass of
its respective parent particle.50 We will use Eq. (II.30) to include the dynamics of gs∗(T )
in our analysis. Introducing a new va(riable z for )the comoving momentum,
gs∗(m )
1/3
P
z = x , (VI.20)
gs∗(mP /r)
we rewrite Eq. (VI.1) to derive the more general DM momentum distribution function
[111] ( ) ( )
∂f(z, r) √ C sΓ − r ∂rg∗(mP /r) r2 gs∗(mP ) 2/3= 2 1
∂r ∫gs(m /r) 3gs∗ P ∗(mP /r) z2 gs∗(mP /r)∞
× √ r2dy fP (y, r) , (VI.21)
( r2 + y2ymin ) ( )
gs 1/3 2 s∗(mP /r) − r g∗(m )
1/3
P
where ymin = z .
gs∗(mP ) 4z g
s
∗(mP /r)
To outline the impact of a variation in gs∗(T ), we insert the momentum distribution from
Eq. (VI.3) into Eq. (VI.21) and vary mP . Assuming ξ = 1, the results for z
2f(z) are
shown in Fig. VI.7. The mass scale mP sets the time of the DM production, i.e. DM is
produced earliest for mP√= 1 TeV. For a better comparison we rescale the momentum
distribution with a factor gs∗(mP ), to compensate for the decrease in g
s
∗(T ) at late times.
Besides from this overall change in magnitude, the shape of the distributions are going
to change when DM is produced during periods of time where gs∗(T ) is rapidly changing:
while the curves derived for mP = 10
−5 GeV, 10 GeV and 1 TeV are nearly identical, the
other two curves, where mP = 10
−1 GeV and 1 GeV clearly deviate, because the QCD
phase transition leads to a rapid change in gs∗(T ) at T ≈ 200 MeV. In particular, the
momentum distribution gets shifted to larger z values when DM is produced during this
period of time, as can be seen from the blue curve, for which mP = 1 GeV.
To summarize, we have to be careful when defining a proper temperature ratio ξ, because
its definition is done by using 〈z〉 = 2.5 assuming constant gs∗(T ).51 That means we have
to include a shift in the averaged momentum due to a change in gs∗(T ) when comparing
against our results shown in the next section. For this purpose we compared the result for
〈z〉 of Eq. (VI.21) to the reference case Eq. (VI.1) in Fig. VI.15 for mP between 1 MeV and
roughly 1 TeV and ξ up to ' 6000; a maximum deviation of ≈ 2.5 can be observed at large
ξ values and for mP ' 1 MeV. For a given parent particle mass, we extract a function
h(ξ) from this contour plot and rescale the temperature ratio accordingly, ξ′ = ξ/h(ξ).
A change in the degrees of freedom also leads to a heating of the photon plasma compared
to the decoupled DM temperature. This does not effect the ratio T2/T1 but for the
derivation of the matter power spectrum the DM temperature has to be defined with
respect to the photon temperature Tγ . Compared to this reference temperature, the DM
temperatures evolve as: ( ) ( )
T gs1 ∗(T )
1/3 T gs 1/3γ 2 (Tγ)
= , = ξ ∗ . (VI.22)
T gsγ ∗(Tprod,1) Tγ g
s
∗(Tprod,2)
Late times of production will come with a decrease in the number of entropic DOF,
gs∗(T
s
prod,2) < g∗(Tprod,1) and compared to the photon temperature, T2 is increased and ξ is
50 As discussed in Chapter IV, freeze-in is most dominant at temperatures r ≈ 3, i.e. T ≈ mP /3.
51 In the case gs∗(T ) = const, the dimensionless momentum variables are identical, z ≡ x.
111
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
m = 10 5 GeV
100
P
mP = 0.1 GeV
mP = 1 GeV
mP = 10 GeV
mP = 1000 GeV
100 101
z
Figure VI.7: Numerical results for z2f(z) as defined in Eq. (VI.21) for different masses of
the parent particle, ranging from mP = 10
−5 GeV to mP = 1 TeV and using Eq. (VI.3)
for f(z) as an illustration. Masses of 0.1 a√nd 1 GeV lead to the biggest impact on theshape of the spectrum, while the distributions of the other masses have similar shapes.
All distributions are rescaled with a factor gs∗(mP ) for an easier comparison.
larger by a factor (gs(T s 1/3∗ prod,1)/g∗(Tprod,2)) . Again, as for a shift in 〈z〉 this can be taken
into account by defining a ξ′ = ξ (gs∗(Tprod,1)/g
s
∗(T
−1/3
prod,2)) to include the reheating
effect of the thermal plasma.
We will explain these rescaling procedures of ξ in more details in Section VI.6 where we
apply it to specific models and extract limits on the allowed temperature ratio.
VI.4.3 Three-body decays
Compared to two-body decays, decays involving three or more particles are more likely to
feature small decay widths, because they can be suppressed by powers of small couplings,
heavy off-shell intermediate particles or large mass ratios. In case of three-body decays
we can have production of a DM particle X via the processes P → S S X, P → S X X
or P → XXX. Similarly to the previously discussed two-body decays we can derive an
analytic expression for the DM momentum distribution, assuming a thermalized parent
particle [312],
f(x) ∝ x−1.2 exp (−1.11x) . ∫ (VI.23)
The prefactor of this function is fixed by demanding A2 = dx
2
1(x1) f(x2). In contrast
to two-body decays, the energy of the parent particle is distributed among three parti-
cles. This has two consequences for the interpretation of our results in the next section.
First, the averaged momentum should be smaller by a factor of 2/3, in fact, we found
that 〈x〉 = 1√.62 using Eq. (VI.23). Further, the same factor has to be used when map-ping the assumed decay width to the temperature ratio, in that case that relation is given
by ξ ' 0.16/ C s 1/4Γg∗(Tprod) for the case of constant gs∗(T ) during DM production. We
have checked that both, using the momentum distribution given in Eq. (VI.3) with a spe-
cific choice for ξ and Eq. (VI.23) with an appropriately rescaled ratio, give rise to nearly
identical matter power spectra.
Although we do not present analytical results for three-body decays of frozen-out or frozen-
in parent particles as for the two-body decays, we are confident that one can make use of
112
z2f(z)
VI.5. ANALYTICAL FITTING OF THE EXCLUSION LIMITS
our procedure to extract limits for the case when the second subset is produced via late
three-body decays with appropriately chosen values for the temperature ratio ξ. Since the
deviations in the matter power spectrum are not very drastic, we expect that the results
from Fig. VI.15 holds for three-body decays to a good degree and this allows to derive
limits on production via three-body decays by applying our findings.
VI.5 Analytical fitting of the exclusion limits
In the following we are using the tools discussed in Section VI.3 to answer the question how
large and how hot the second DM subset can be. We are going to present our results in
terms of the A2–ξ parameter space of the 2TDM. Results are derived for different choices
of mDM as higher DM masses give rise to weaker constraints.
As an example we show the constraints on the parameter plane in Fig. VI.8 where we set
mDM = 20 keV and keep g
s
∗(T ) = 106.75 fixed until all of the DM production has been
completed. As can be seen, the limits from structure formation place strong constraints
on the temperature ratio ξ in the range 10−2 < A2 < 1.0 while the ∆Neff bound from the
BBN epoch starts to become relevant at rather large temperature ratios, ξ > 104. Above
this value, the bounds from structure formation become less reliable, because the hot DM
subset starts to act like dark radiation instead of matter, an effect not captured in the
calculation of P (k). We already observed in Chapter IV that the ∆Neff bound exceeds
structure formation limits for DM subsets with very high temperatures.
100000
10000
1000
100
Aref, 1
10 A Lref, 2 yman-
light MW
heavy MW
Neff BBN
110 3 10 2 10 1 100
A2
Figure VI.8: Limits from structure formation and the number of effective entropic DOF
for m sDM = 20 keV and fixed g∗(T ) = 106.75. The blue shaded region is disfavored by
the MW subhalo count and the green shaded region by limits from Lyman-α surveys,
respectively. The solid lines are the corresponding stronger limits, whereas the weaker
constraints are shown as dashed lines. The red shaded region in the upper right corner is
disfavored by a too large ∆Neff value.
In the following, our aim is to provide our results in a model-independent way such that
they can be applied to a variety of scenarios. For this(reason we fit th)e respective exclusion
limits with an exponential of the form ξ(A ) = exp p ·A−p12 0 2 + p2 , which we found to
113
alo
s
 h
W s
ub
M
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
be generally suitable. Furthermore, one parameter can be removed, because we know that
the curve endpoint, ξ(A2 = 1), scales linearly with the DM mass starting from m
lim
DM as
given in Table VI.2. By eliminating p2, the ex[po(nential can)]be reduced to the expressionmDM
ξ(A −p12) = exp p A − 1 . (VI.24)
mlim
0 2
DM
This enables us to use either the abundance A2 or the temperature ξ as an input parameter
and derive constraints on the other variable. The resulting fitting parameter for all four
exclusion contours for mDM = 20 keV are shown in Table VI.3.
Lyman-α Nsub
δAref,1 δAref,2 light MW heavy MW
p0 0.464 0.546 0.141 0.196
p1 0.581 0.672 1.11 1.29
Table VI.3: Fit results for the respective exclusion limits based on Eq. (VI.24) using
mDM = 20 keV.
Since only the limit endpoint scales linearly with DM mass, while the exclusion curve
for A2 < 1 changes non-trivially with it, we extend this fitting procedure for other DM
masses and simulate the exclusion limits for mDM between 20 keV and 1000 keV. Then,
the respective pa(rameter)s pi are ext(racted)and fitte(d using) the f(ollowin)g polynomial:mDM mDM −1 mDM mDM 2
pi = ai + bi + ci + di , (VI.25)
keV keV keV keV
to derive a final fit function which takes A2 and mDM as input parameters to give the
allowed temperature ratio ξ ≡ ξ(mDM, A2). The results for all eight fit parameters
can be found in Section VI.8.4 in the appendix. We explicitly compare this analyti-
cal fit against numerical simulations in the context of toy model examples in Section VI.6.1.
However, the assumption of constant gs∗(T ) obviously does not hold in general and we
might have to drop at least some or all of above mentioned simplifying assumptions. We
will explain in the following how to adapt our results beyond the simple picture.
The starting point is to collect the A2–ξ relation given in Eq. (VI.24), the expression for
pi(mDM) (see Eq. (VI.25)) and the corresponding fit parameters given in Table VI.5. As
mentioned, this gives a first approximation of the exclusion limits, under the assumption
gs∗(T ) = 106.75 = const during DM production. If this does not apply for the second
subset, because its production happens at times T sprod,2 where g∗(T ) is changing, it will
lead in general to a warmer DM subset as compared to the case where gs∗(T ) is constant.
Two corrections have to be done: first, the exclusion limit on ξ has to be divided by
(gs s 1/3∗(Tprod,1)/g∗(Tprod,2)) . Second, we have to take the change in 〈x〉 into account by
extracting a correction function h(ξ) for the corresponding mP from Fig. VI.15. This
gives a rescaled version of the temperature rat(io, ξ
′,
′ ( )ξ(mDM, A2) ) gs∗(T 1/3prod,2)ξ = . (VI.26)
h ξ(mDM, A2) 106.75
Additionally, if the first subset features 〈x〉 different from 2.5 or has gs∗(Tprod,1) < 106.75,
which gives rise to a higher DM temperature T1 relative to the photon bath, one further
step has to be done before the corresponding limits on ξ–A2 can be extracted. This change
of A1 can be quantified by multiplying 〈x〉 with a factor α, which is either given by the ratio
114
VI.6. APPLICATION TO SAMPLE MODELS
between the averaged momentum and our reference case, α = 〈x〉/2.5, or by the entropy
dilution factor, α = (106.75/gs∗(T
1/3
prod,1)) . Regarding its matter power spectrum, a DM
with temperature αT1 and mass mDM has the same properties as a DM with temperature
T1 and mass mDM/α. However, changing the mass by 1/α in Eq. (VI.25), we have to
rescale the outcome for ξ′ by multiplying it with α, because the second DM subset is not
affected. Taking all these corrections into accou(nt, the temp)erature limit is set by
ξ′
ξ
= α ((m /α,A )) gsDM 2 ∗(T 1/3prod,2) . (VI.27)
h ξ(mDM, A2) 106.75
In the next section we will explain in more detail how to incorporate these effects using
several toy models as examples. A special emphasis will be put on the proper extraction
of the limits on ξ when gs∗(T ) is not fixed.
VI.6 Application to sample models
To explain the matching between our parametrization and “real” DM models we consider
several example models where the DM is produced in different ways. Furthermore, we
compare the limits obtained using our fit functions with those obtained from explicit
simulations, where the matter power spectrum for particular choices of the momentum
distribution function is calculated. Finally, we match our findings to the exemplar DM
models, which we discussed in Chapters IV and V.
VI.6.1 Model I: thermalized + out of equilibrium parents, constant gs∗(T )
We start by discussing two thermalized parent particles, S and P , with masses mS =
mP = 50 TeV which are producing a DM species with mass mDM = 50 keV with their
respective decays. Due to their large mass the number of entropic DOF can be treated as
constant until all of them decayed into DM. Only afterwards, the dilution of gs∗(T ) has to
be taken into account. We assume that S decays rapidly, while remaining in equilibrium
with the thermal plasma. This is going to produce an amount A1 < 1 of DM particles with
averaged momenta given by 〈x〉 = 2.5. Additionally, P is going to decay at late times,
after it is already frozen-out and produces an amount A2 of DM, such that A1 +A2 = 1.
Now, because these decays are taking place at later times compared to production via S
decays, we end up with DM which is highly energetic compared to the thermal plasma.
The difference in the respective DM temperatures gives the ratio ξ.
The interesting question now is how hot the DM share A2 can be, without being in
conflict with observations of structure formation. Following the procedure outlined in the
previous section, we can neglect modifications stemming from a change in gs∗(T ) during
DM production and extract the limits on ξ directly from Eq. (VI.24).
In Fig. VI.9 we compare our fitted exclusion limits for a DM mass of 50 keV against
a numerical simulation. The left figure shows results from Lyman-α data (green shaded
regions) and the right one from a MW subhalo count (blue shaded regions). In both figures,
simulated results are shown as black lines and the stronger/weaker bounds are given by
solid or dashed lines, respectively. As one can see, our fit gives a good approximation of
the simulated results.
VI.6.2 Model II: thermalized + out of equilibrium parents
In the previous example we illustrated how to interpret and extract the corresponding lim-
its on A2 and ξ using our fit procedure under the assumption of constant g
s
∗(T ) during the
production of DM. Based on this, we consider now a similar setup as before: again, we
115
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
1000 1000
100 100
10 10
fit for Aref, 1 fit for light MW
fit for Aref, 2 fit for heavy MW
Aref, 1, simulated light MW, simulated
Aref, 2, simulated heavy MW, simulated
1 10 1 100 1 10 1 100
A2 A2
Figure VI.9: Structure formation limits given by shaded regions and derived from our
analytic fit, Eqs. (VI.24) and (VI.25), using δA in the left and Nsub in the right figure.
The DM mass is given by mDM = 50 keV and we assume g
s
∗(T ) = const during DM
production. Shown as black lines are corresponding limits from a numerical simulation
and solid/dashed lines correspond to stronger or weaker bounds, respectively.
have two thermalized parent particles, S and P whose decays will produce a DM species
with mass mDM = 35 keV. However, now the parent masses are given by mS = 1 TeV
and mP = 10 GeV. As before, S is producing a DM amount A1 while it remains in equi-
librium with the thermal plasma, whereas an additional DM subset, A2, is produced by
late-time decays of P after it is frozen-out.
Since DM production now happens at times where gs∗(T ) is changing, we have to compen-
sate for this effect by using Eq. (VI.26). Here we assume that the first subset is produced
at early times where all SM particles are still part of the thermal bath, while the function
h(ξ) can be read off Fig. VI.15 for the particle masses involved. Now we can take ξ′ as the
constraint on the temperature ratio for this toy model. Limits derived using δAref,1 and
light MW masses are shown in the left and right figure of Fig. VI.10 as green and blue
solid lines respectively, while the respective weaker bounds are shown as dashed lines. The
black line and the red shaded region above it gives ∆Neff > 0.344 during the BBN epoch;
we note that this limit exceeds bounds from the MW subhalo count even for ξ < 1000, in
particular the heavy MW mass choice. Compared to Fig. VI.9 we observe that the exclu-
sion bands feature a kink around A2 = 0.2 and therefore smaller ξ
′ values are excluded in
this region. The reason is that at this point, P particles decay around a temperature of
T ' 1 GeV where gs∗(T ) is rapidly changing and hence it gives rise ro a larger averaged
momentum 〈z〉, as illustrated in Fig. VI.7.
To demonstrate the usefulness of our approach, we simulate the combined matter power
spectrum for this toy model with two different DM production channels for the benchmark
points ξ = 10 and 40. We explicitly insert the out of equilibrium momentum distribu-
tion for S (see Eq. (VI.34)) taking changes in gs∗(T ) into account as well. The limits
on A2 for these choices are indicated by a star and a diamond in both plots: here, the
slightly grayed out symbols indicate the weaker limits. These benchmark points have to be
matched onto ξ′ as well, by multiplying them with a factor (106.75/gs(T ))1/3∗ prod,2 , which
evaluates roughly to 0.5 for ξ = 40 and 0.8 for ξ = 10.
Further, it is noticeable that the benchmark points, indicated by dark blue stars or dia-
monds, seem to yield weaker constraints compared to our fit. For the scenario in mind,
this can be explained by a DM fraction which is already produced while the parent par-
ticle is still in thermal equilibrium, giving rise to a peak at smaller momenta similar to
the case of thermalized parent particle decays. Whereas in our parametrization we as-
sume the second subset to be fully produced via late time decays. Consequently, at large
ξ this fraction is only marginal, but becomes more dominant for smaller ξ. We calculated
116
VI.6. APPLICATION TO SAMPLE MODELS
1000 1000
100 100
10 10
Aref, 1, fit light MW, fit
Aref, 2, fit heavy MW, fit
1 10 1 100 1 10 1 100
A2 A2
Figure VI.10: Structure formation limits using the δAref criterion in the left and a MW
subhalo count in the right figure for the second toy model. Everything indicated by the
green or blue shaded region is constrained by structure formation and solid lines correspond
to strong and dashed lines to weak limits. For comparison, full numerical simulations to
extract limits are run for ξ = 10 and 40 and are indicated by the stars and diamonds in
both plots. The difference between the dark blue and black symbols indicated by an arrow
are due to specifics of this model choice and are further explained in the text. The red
shaded region yields a too large ∆Neff stemming from the second subset.
this fraction explicitly and numerically extracted the updated bound on A2 shown as the
black diamonds and stars in the plot. The difference between both is indicated by a black
arrow and it is obvious, that the latter, more careful treatment fits better with our ana-
lytical result.
Furthermore, we show how the constraints on A2–ξ can be matched onto specific param-
eters for a concrete freeze-out model. The abundance A2, produced by late-time decays,
is fixed by the abundance of P which in turn is set by the time of its freeze-out, rFO.
Integrating Eq. (VI.34)(over x, t)h(e yield YDM )=(nDM/s is give)n byΩ h2DM ' m 2DM 106.75 rFOK2(rFO) ( m )DMA2 = . (VI.28)
0.12 10 keV gs∗(Tprod,2) 49.5 20 keV
The relation between the decay width of P and ξ is already discussed in Eq. (VI.10).
Now we have all the ingredients to match between this specific model and our A2–ξ
parametrization.
This toy model is similar to the ScM setup we discussed in Chapter IV, although we used
a more conservative DM mass limit in Section IV.3.1 to extract limits from an adapted
half-mode analysis. As we have shown, stronger bounds from Lyman-α and MW sub-
halo counts require mDM & 13 keV, which would already exclude the scenario we consid-
ered in Chapter IV. Choosing as an example a fourth times larger mass for the lightest
right-handed neutrino N1, mN1 = 24 keV, we use our result derived in this chapter to
reevaluate the corresponding limits on the m±–mN2,3 parameter space. For this, we con-
sider the parent particle N2 to have a mass mN2 ' 300 GeV and extract the corresponding
h(ξ) function. Further, we take the three-body kinematics of N2 decays into account. We
show the corresponding limits in the left plot of Fig. VI.11. As before, the green shaded
region is excluded by δAref,1 and the blue shaded one by a subhalo count using the light
MW mass.
These results have to be matched onto the ScM; this is shown in the right figure of
Fig. VI.11, where we fix the mass of the charged Z2 scalar to be m± = 1200 GeV and vary
the mass of N2. We can neglect the complication of the matching of A2 which we dis-
cussed above, because we are interested in the large ξ region where the second DM subset
117
′
′
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
is produced at sufficiently late-times. Compared to the 2TDM framework, there is less
freedom to choose parameters; consequently, the black line indicates the temperature ra-
tio ξ for the second DM subset, while the purple line shows the amount A2 of this subset.
The green and blue lines are the corresponding lower limits on mN2 using Eq. (VI.26).
Using δAref,1 limits mN2 & 316 GeV, thus excluding A
′
2 & 0.5% and ξ & 18000, while the
subhalo count analysis yields a weaker limit, mN2 & 190 GeV. Both limits clearly exceed
the results shown in Fig. IV.4 assuming a conservative mTR = 2 keV limit. The bound on
∆Neff gives an even stronger constraint, mN2 & 435 GeV, hence excluding ξ
′ & 11000 and
A2 & 0.2%, which is in accordance with the results we found in Section IV.3.2.
In conclusion, we can quickly and reliably answer the question of how much DM, origi-
nating from a subdominant production mechanism, is possible, and how hot it is allowed
to be in the process.
100000 100 105
10 1
10 2
1000
10 3
100
10 4
104
10 5
fit for A 10ref, 1 Abundance N2 decays
fit for light MW Temperature ratio
6
110 3 10 2 10 1 100 10 102 103
A2 mN2 [GeV]
Figure VI.11: Left figure: Application of our fit results to the ScM with m± = 1200 GeV
and mN2 ' 300 GeV. The green shaded region is the δAref,1 limit and the blue one the light
MW subhalo count. Again, the red shaded region is excluded by the ∆Neff bound. Right
figure: The corresponding abundance A2 (purple line) and DM temperature ratio ξ
′ (black
line) for the same ScM setup, where Yukawa interactions are chosen as large as possible.
The green/blue shaded regions are excluded by the respective structure formation limits
and the red shaded regions by ∆Neff . For both plots, the mass of the DM candidate is
mN1 = 24 keV.
Finally, we want to comment on the potential issue of late-time decays of heavy particles
which may happen during the epoch of BBN and hence can spoil the abundance of light
nuclei by injecting highly energetic particles into the thermal plasma [204, 326, 327].
However, this danger does not appear for our model setup, because we assume that the
parent particle decays exclusively into DM via P → XX. There is no heating of the
SM plasma due to these decays, because the coupling between X and the SM plasma is
assumed to be zero. On the contrary, for models which feature decays into SM particles
besides DM, P → X S, these decays tend to be dangerous when decaying at temperatures
T . 1 MeV. Using Eqs. (VI.9) and (VI.10) this can be translated into a bound on ξ,
T 200 mP
= . (VI.29)
MeV ξ GeV
For our sample model this bound evaluates to ξ ≤ 2000 if P decays dominantly into
hadrons. On the contrary, especially for mainly leptonic decays, the condition that T &
1 MeV can be relaxed substantially, if A2  1 as shown in Ref. [1].
118
′
A2
′
VI.6. APPLICATION TO SAMPLE MODELS
VI.6.3 Model III: thermalized + frozen-in parents
A a last toy model we assume that mDM = 125 keV, again an amount A1 of DM is produced
via thermalized parent decays with mS = 1 TeV, but A2 stems from a frozen-in parent
with mass mP = 80 GeV. Similarly to the previous toy model, we have to take a change
in gs∗(T ) into account, although it will impact the limits on ξ at higher values, since mP
is larger in this case and DM is produced at earlier times compared to the previous case.
The results of the fit are shown in Fig. VI.12 where we use the same color coding as before.
Since the DM in this model is heavier than in the first toy model, the exclusion limits are
shifted to larger ξ values. Further, the bound from ∆Neff is only marginally visible in this
plot due to the rather large DM mass.
As before, we compare our analytical exclusion limits against some benchmark points
using a full numerical simulation, but in this case we choose ξ = 40 and 300. Compared to
the previous example, the benchmark point are closer to the fitted curve here. This is to
be expected, as the momentum distribution function f(x, r) (see Eq. (VI.35)) arising from
decays of frozen-in parents only features one distinct peak, because the parent particles
are never thermalized and so no early decays are taking place. Hence, rescaling ξ by an
appropriate factor, as explained around Eq. (VI.26), can be safely done even for A2 →
1. One can now match these limits onto model parameters describing frozen-in parent
1000 1000
100 100
10 10
Aref, 1, fit light MW, fit
Aref, 2, fit heavy MW, fit
1 10 1 100 1 10 1 100
A2 A2
Figure VI.12: Structure formation limits using the δAref criterion in the left and a MW
subhalo count in the right figure for the third toy model. Everything indicated by the green
or blue shaded region is constrained by structure formation and solid lines correspond to
strong and dashed lines to weak limits. For comparison a full numerical simulation to
extract limits was run for ξ = 40 and 300 and is indicated by the stars and diamonds in
both plots. The ∆Neff bound is absent, because it is weaker compared to Fig. VI.10 due
to the larger DM mass.
particles. Assuming that P itself is produced via an effective coupling CP , the DM yield
for this mechanism is given by
135 1
YDM = C(P , (VI.30)64π2 gs∗((Tprod,2) )( )
' m
)
DM 106.75 CP
A2 . (VI.31)
125 keV gs∗(Tprod,2) 1.7× 10−3
When discussing freeze-in scenarios one might want to construct a model where all particles
are decoupled from the thermal bath and therefore all of DM is produced by frozen-in
parent particles. However, this situation requires some modifications, because the averaged
momentum for the first subset generally differs from the case 〈z〉 = 2.5. As pointed out
in Section VI.5 this different setup can be handled by introducing a shift 〈z〉 = 2.5α. If
α > 1, the DM carries a larger averaged momentum and hence stronger bounds are set
119
′
′
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
1000 1000
= 0.6, simulation = 0.6, simulation
= 1.4, simulation = 1.4, simulation
= 0.6, fit = 0.6, fit
= 1.4, fit = 1.4, fit
100 100
10 Aref, 101
110 2 10 1 100 110 2 10 1 100
A2 A2
Figure VI.13: Effect of changing 〈z〉 for the A1 set of the DM on the excluded region in the
A2–ξ plane based on the δAref,1 criterion. The dashed lines correspond to α = 0.6, while
the dotted curves represents α = 1.4. The black curves stem from a numerical analysis
and the green ones from the adapted analytical fit as explained in Section VI.5. The left
figure is derived for mDM = 20 keV and the right one for mDM = 60 keV.
on the parameter space. On the other hand, α < 1 corresponds to a DM with smaller
averaged momenta and weaker structure formation constraints. It can be understood in
terms of the temperature T1 as well: lower temperatures leads to a smaller value of 〈p/T 〉,
which gives rise to less stringent exclusion limits and vice versa for larger temperatures.
This effect is shown in Fig. VI.13 where we choose α = 0.6 and 1.4 for two different DM
mass choices and compare the corresponding δAref,1 exclusion bound to the reference case
α = 1, but do not change A2. Further, these numerical results shown as the black curves
are compared against limits we derived using our analytical fit prescription (green curves),
adapted as explained at the end of Section VI.2. As can be seen both approaches agree
to a good approximation. Only for rather small DM masses and α > 1 both curves differ
from each other. However, this is not unexpected, as the structure formation observables
are quickly changing in this region of parameter space, because for mDM = 20 keV and
α = 1.4 the absolute mass limit given in Table VI.2 is reached. Finally, as expected, the
impact on the bounds gets weaker if the DM mass is increasing and it becomes negligible
for mDM & 100 keV.
A similar effect appears when gs∗(Tprod,1) is smaller compared to our assumption where all
SM particles are still in the thermal bath. In that case, we would find a larger temperature
T1 compared to the photon bath due to smaller reheating effects and accordingly the
exclusion curves have to be corrected similar to the case of larger 〈z〉 values shown in
Fig. VI.13. For this case, α is defined as the increase in T1, which is given by the ratio
α = (106.75/gs∗(T
1/3
prod,1)) and for sufficiently late production times the temperature can
be twice as large as compared to early decays.
VI.6.4 Exploring the cold limit of the model parametrization
Motivated by our results in Section V.4 where we discussed that a warm freeze-in ALP DM
may be accompanied by a fraction of “cold” ALP DM stemming from the misalignment
mechanism, we briefly want to discuss in this section the case where the first DM subset
can be considered CDM instead. Formally, this corresponds to the case T1 → 0, whereas
the model parameter ξ has to be interpreted now in relation to our reference choice in the
previous chapters, ξ = T2/2.5.
As before, we are going to use Eq. (VI.24) to fit the corresponding exclusion limits based
on our structure formation analysis. The parameters derived for mDM = 20 keV are shown
in Table VI.4.
120
VI.6. APPLICATION TO SAMPLE MODELS
Lyman-α Nsub
δAref,1 δAref,2 light MW heavy MW
p0 0.899 0.823 0.285 0.333
p1 0.564 0.652 1.04 1.22
Table VI.4: Fit results for the respective exclusion limits based on Eq. (VI.24) using
mDM = 20 keV and assuming that the first subset is composed of CDM.
Compared to the 2TDM, this scenario allows for a simple generalization for all mDM
choices. The effect of the CDM contribution A1 on structure formation is essentially
independent of the DM mass; it only matters for A2. For that reason, the fitted exclusion
limits can be easily rescaled to accommodate heavier DM. To illustrate this procedure, we
consider the photophilic ALP DM scenario discussed in Section V.5 consisting of “warm”
ALP DM produced via scatterings with 〈p/T 〉 ≈ 3.24 and a potential subfraction of
“cold” ALP DM produced via the misalignment mechanism. Using the values quoted
in Table VI.4, correcting for the warmer averaged momentum by α ∼ 1.3 and using
mDM = 17.4 keV (stemming from the corresponding strong MW mass limit, i.e. the lighter
mass), we are able to reproduce the limits as shown in Fig. VI.14 to a good extent. The
blue shaded region is excluded by a subhalo count for this ALP DM setup, while the red
curve gives the exclusion contour based on our fitting approach. It gives slightly weaker
limits compared to the numerical study, because the lower MW mass limit employed in
this chapter is marginally larger compared to Chapter V.
10
5
1
0.50
0.10
0.05 Excluded by MW agg
0.01
0.05 0.10 0.50 1
F
Figure VI.14: Similar to Fig. V.5, where limits for the photophilic ALP F −ma parameter
plane stemming from a MW subhalo count assuming the lighter MW mass choice are
derived. Freeze-in contributes only a fraction F < 1 of warm DM to the total ALP DM
abundance, while the rest is made up by a CDM component. The blue shaded region is
excluded because these parameters yield Nsub < 64. The red line is the limit derived using
our analytic expressions for the light MW mass choice.
121
ma [keV]
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
VI.7 Summary of Chapter VI
Many extensions of the SM introduce entire dark sectors with several new particles and
interactions among them. Therefore, it is natural to ask the question what happens if the
DM is produced via different production channels, leading to DM composed of two sub-
sets, each characterized by its own temperature. Depending on the size and the nature
of the involved couplings and particles, these decays can easily take place at late times in
the early Universe and give rise to an increased DM temperature. We examined such a
scenario as model-independently as possible to allow for an easy comparison with specific
WDM models. For this purpose, we assumed that the DM is produced by two different
decay channels. One is due to decays of thermalized parents at rather early times, while
the second contribution stems from decays happening at later times. Our setup (referred
to as 2TDM) is parameterized by two key parameters: the abundance A2 of the second
subset produced at later times and the temperature ratio between both DM subsets, de-
noted by ξ.
The impact of such a model setup on the formation of structures in the universe was eval-
uated. Specifically we derived predictions for the number of MW subhalos and the flux
power spectrum and compared them against observations. Based on these, limits on the
parameters ξ and A2 were derived for DM masses between 20–1000 keV. For ξ = 1, i.e. a
single DM temperature setup, our limits on the DM mass are up to 11 keV using Lyman-α
measurements and up to 13 keV counting the observed MW subhalos and using the MW
mass derived from recent GAIA measurements. Typically, we could probe and constrain
parameters for A2 between 1 and 0.01 for temperature ratios up to 1000 and in general,
DM with a high temperature T2 can only make up a few percent of the total DM number
density.
We presented an analytical fit for the respective exclusion limit and discussed further steps
how to extract limits on specific model realizations. One focus was the incorporation of
a change in the number of effective entropic DOF during the time of production of the
DM species, as this impacted the interpretation of the fitted results. As an example we
considered different examples and compared our analytical prediction against numerical
results, where we made direct use of appropriate momentum distribution functions. Our
procedure showed a good agreement between the analytical fit and actual results. Hence,
it allows to predict limits on the temperature of a warmer DM fraction and its abundance
without extensive simulations.
Further, we commented on the treatment of DM production via three-body decays inside
our framework and how we calculated exclusion limits for these cases applying rescaled
results. As an example we derived limits for a benchmark point of the ScM setup consid-
ered in Chapter IV. We also discussed the limit where T1 → 0, i.e. the first DM subset can
be considered to be CDM instead; this particular case allowed us to present a simpler fit-
ting method, since the corresponding results scale directly with mDM. As an example, we
matched this result to the corresponding numerical results for the photophilic ALP DM
model with an admixture of misalignment DM, discussed in Section V.5 of Chapter V.
As a final remark, we want to point out that it is also possible to reinterpret our model
in terms of a mixture of two DM species with different masses mDM,1 and mDM,2, by
adapting the model parameter ξ accordingly: ξ ≡ T2/T1mDM,1/mDM,2.
122
VI.8. APPENDIX OF CHAPTER VI
VI.8 Appendix of Chapter VI
VI.8.1 Momentum distribution function of out of equilibrium parents
If the parent particle is sufficiently long-lived and has a sizeable coupling CP , it thermalizes
and its decay will happen after it drops out of the thermal bath. After the time of freeze-
out, rfo, the momentum distribution function of the parent is given by
fP (x, r) = fEQ(x, r), r < rFO , (VI.32)√ CΓx2/2
r + r2 + x2
fP (x, r) = fEQ(x, rFO) √  × (VI.33)
r + r2 2FO
√ FO
+ x
√
× e−CΓ(r r2+x2−rFO r2 2FO+x )/2, r > rFO . (VI.34)
VI.8.2 Momentum distribution function of never thermalized parents
Weakly coupled parent particles with CP  1 never reach thermal equilibrium, but rather
freeze-in before they start to decay. Their momentum distribution can be derived as∫r √
exp(√− ρ2 + x2)  √ ( √ )x2/2 CΓ/2eρ ρ2+x2 ρ+ ρ2 + x2fP (r, x) = CP dρ ρK1(ρ) √ √  .
ρ2 + x2 er r2+x2 r + r2 + x2
0
(VI.35)
VI.8.3 Change in 〈x〉 due to gs∗(T )
As explained in Section VI.4.2, the time-dependence of the number of effective entropic
DOF, gs∗(T ), generally shifts the DM momentum distribution function towards larger
momenta. To examine this effect for the case of out of equilibrium decays, the average
value 〈z〉 is calculated, taking a change in gs∗(T ) into account, and compared against 〈x〉,
where gs∗(T ) are kept fixed in the calculation. The result is shown in Fig. VI.15, where we
varied the mass of the parent particle between 1 MeV and roughly 10 TeV for ξ between
1 and 6000. As can be seen, at rather small ξ the effect of a change in gs∗(T ) starts to be
dominant at mP < 10 GeV; below this mass the averaged momentum can be even twice
as large as compared to the case where gs∗(T ) are treated as constant. However, increasing
ξ will give rise to a shift in 〈z〉 even for rather large parent particle masses. This behavior
is expected, because if DM is produced at sufficiently early times by decays of very heavy
parent particles, gs∗(T ) stays approximately constant. However, demanding that this DM
should have a large ξ as well, requires that it is produced at later times in the temperature
and the effect of a change in gs∗(T ) becomes relevant.
VI.8.4 Details on fitting procedure
As explained in Section VI.5, we fit the respective exclusion contours using Eq. (VI.24) and
derive the fit parameters pi for mDM between 20–1000 keV. The mass-dependent pi are
than fitted using Eq. (VI.25). The results for all four structure formation limits are shown
in Fig. VI.16 respectively. The numerical results of a MW subhalo count analysis using a
light/heavy MW mass are shown as blue/dark blue crosses, whereas the δAref limits are
shown in light and dark green. We compare these with the corresponding results of the
aforementioned polynomial fit, which are presented in the same color scheme as a solid and
dashed line, respectively. In summary, our fit describes the numerical results accurately,
only for DM masses above 700 keV slightly larger deviations are visible, which is why we
123
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
ratio of x
3.5 2.2
3.0 2.0
2.5 1.8
2.0
1.6
1.5
1.4
1.0
0.5 1.2
0.0 3 2 1 0 1 2 3 4 1.0
log(mP/GeV)
Figure VI.15: Ratio of 〈x〉 derived comparing Eq. (VI.21) against Eq. (VI.1), i.e. com-
paring a time-dependent gs∗(T ) against constant case g
s
∗(T ) = 106.75. As can be seen, the
averaged value is shifted to larger values, especially for mP ∼ 1 GeV and large values of ξ,
because in that region gs∗(T ) is changing rapidly. However, the effect becomes less promi-
nent for heavier or lighter parent particles. It can also be seen that 〈x〉 stays constant if
both, mP and ξ are increased. The black shaded region indicates decays which would take
place after BBN.
1.5 heavy MW
light MW
1.0 Aref, 2Aref, 1
0.5
102 103
1.5 heavy MW
light MW
Aref, 2
1.0 Aref, 1
0.5
102 103
mDM [GeV]
Figure VI.16: Comparison of the fit parameters pi for different DM masses and structure
formation limits shown as crosses, and the corresponding fourth-order polynomial fit (see
Eq. (VI.24). Stronger limits are shown as solid and weaker ones as dashed lines.
only examine DM masses up to 1000 keV. An even larger mass range would require an
extended fit equation with more parameters. Therefore, we are using eight fit parameters
in Eq. (VI.24) for our final fit in total and the results are summarized in Table VI.5. We
124
p1(mDM) p0(mDM)
log( )
VI.8. APPENDIX OF CHAPTER VI
δAref,1 δAref,2 light MW mass heavy MW mass
a0 1.16 1.03 0.383 0.428
b0 −13.7 −8.83 −4.28 −3.77
c 6.08× 10−4 2.64× 10−4 −5.07× 10−4 −6.60× 10−40
d0 −1.95× 10−7 −0.02× 10−7 5.70× 10−7 4.70× 10−7
a1 0.504 0.589 0.942 1.07
b1 1.47 1.39 2.80 3.63
c1 −1.82× 10−4 −1.23× 10−4 6.16× 10−4 12.2× 10−4
d 0.800× 10−7 0.288× 10−71 −7.32× 10−7 −8.50× 10−7
Table VI.5: Final fit parameters for all four exclusion contours, according to Eq. (VI.25).
The DM mass ranges between 20–1000 keV.
want to point out, that we do not assign an error on these results. Although we can
estimate a statistical uncertainty stemming from a finite simulation grid size, systematical
uncertainties of the fitted structure formation limits indicated by the strong/weak bound
are clearly dominating.
VI.8.5 Impact of Nsub on the parameter limits
We already discussed the uncertainties involved when calculating Nsub. While we derived
an upper and lower MW mass bound and presented our results for both mass choices, we
kept the requirement that our 2TDM should fulfill Nsub ≤ 64 throughout this section. In
the same way as done in Section V.7.3 we illustrate how the exclusion contours depend on
the choice of the number of sub halos in Fig. VI.17 for the 2TDM with mDM = 50 keV.
The left figure shows the result for the light MW mass and the right one for the heavy MW
mass. Shown in blue is the exclusion limit for Nsub = 80, the red line is for Nsub = 50 and
the black line, finally corresponds to the standard choice. As in the case of ALP DM we
observe that results derived for the light MW mass choice is more sensitive on the value
of Nsub, although its impact is less compared to our results in Section V.7.3, because the
lighter MW mass choice is slightly heavier.
103 103
Nsub = 50 Nsub = 50
Nsub = 64 Nsub = 64
Nsub = 80 Nsub = 80
102 102
101 101
100 10 1 10
0
10 1
A2 A2
Figure VI.17: Comparison of mass limits for different choices of Nsub and MW masses for
the 2TDM model with mDM = 50 keV. The left figure shows results for the light MW mass
and the right one for the conservative choice. The blue lines are ma limits for Nsub = 80
and the red lines are using Nsub = 50 while the black lines are results for Nsub = 64.
125
CHAPTER VI. TWO TEMPERATURE DARK MATTER: A GENERAL PICTURE
In addition we have evaluated the absolute mass limits mlimDM for the different cases similar
to the procedure done to derive Table VI.2 and the results are shown in Table VI.6. This
Nsub 50 64 80
MW
light mass 8.8 keV 12.8 keV 23.5 keV
heavy mass 7.2 keV 9.0 keV 11.4 keV
Table VI.6: Lower DM mass limit mlimDM for different choices of the observed amount of
subhalos Nsub compared to the case Nsub = 64 as used in this section. They were derived
assuming ξ = 1, i.e. all of DM has a common temperature T2 = T1.
highlights again, that potentially new discoveries of subhalos offer the intriguing possibility
of further constraining our framework.
126
Aftermath
127

Chapter VII
Summary and conclusion
In this thesis, we studied the phenomenology of certain light dark matter (DM) models.
In Chapter III we studied the prospects to discover light DM embedded in a dark sector
at present and future particle colliders, whereas in Chapters IV to VI the impact on the
early Universe and structure formation was studied in the context of these and further
models.
Our first scenario consisted of the scotogenic model (ScM), which extends the standard
model (SM) with three right-handed neutrinos (RHN) and a new scalar doublet, all charged
under a discrete Z2 symmetry. It features a rich phenomenology and allows for a simul-
taneous explanation of various open problems: the smallness of neutrino masses, DM and
the observed baryon asymmetry of the Universe introducing TeV scale new physics. This
motivated our search for this particular model setup at particle colliders which has been
done in Chapter III.
For this purpose, we studied the pair production of the new charged scalar σ± which de-
cays into two different missing transverse energy (ET ) signatures: first, we studied a two
tau + ET (di-tau) signature based on the ATLAS search on 36.1 fb
−1 LHC data [214].
The other signature consists of two e or µ and ET (di-lepton) and was outlined again by
ATLAS using 36.1 fb−1 of collected collision data [213]. We recasted these analyses and
used the existing limits on the cross section to place bounds on our model. We found
that the current limits do not constrain the model in consideration significantly and only
a few parameter choices can be ruled out. Hence, we estimated the projected sensitiv-
ities for the high luminosity phase of the LHC (HL-LHC) based on the current analysis
tec√hniques, assuming a total luminosity of 4000 fb
−1 collected at a center-of-mass energy
of s = 14 TeV. We presented our results for the di-tau and di-lepton signatures for dif-
ferent branching ratios and masses of the scalar, m± and heavy RHN, mN2 = mN3 . The
corresponding couplings for the decays of σ± are fixed by the Casas-Ibarra parametriza-
tion in order to satisfy the observed SM neutrino masses and mixings. Hence, we
optimized these couplings to maximize the decays into e and µ or tau leptons and com-
pared them against the case where σ decays exclusively into one of the two final state
signatures.
Our projected HL-LHC sensitivity curves allow to probe a significant part of the param-
eter space, ranging up to m± ' 650 GeV for the scalar mass and up to mN2 ' 350 GeV
for the RHN in the case of a di-lepton signature. The projected results for the di-tau case
are weaker since tau final states are not as clean as e or µ tracks, because tau leptons de-
cay hadronically as well.
Then, we presented results for future collider experiments, considering proton and elec-
tron colliders. For the√former, we consider the FCC-hh with the luminosity goals of
√3 ab
−1 and 30 ab−1 at s = 100 TeV, and studied the electron linear collider CLIC at
s = 3 TeV and a total luminosity of 5 ab−1. Extending and adapting the analysis we
129
SUMMARY AND CONCLUSION
used for HL-LHC, we projected sensitivities for both experiments using the di-lepton sig-
nature. We found that they can probe a wide range of parameter space, extending up
to m± ' 2 TeV and mN2 ' 1.4 TeV at FCC-hh. On the other hand, due to cleaner ex-
perimental signatures at lepton colliders, CLIC can probe the parameter region where
mN2 . m±; this is of particular interest when considering a resonant leptogenesis sce-
nario to generate the observed baryon asymmetry.
Although it is possible to explore the heavy matter sector of the scotogenic model, we
have seen that it is hard to detect its light DM component with mN1 = 6 keV at collid-
ers. Therefore, we turned our attention in Chapter IV to study the light DM imprint
on the cosmology of the early Universe and the formation of structures. In this frame-
work, DM can be produced via prompt decays of thermalized σ± scalars, or late-time
decays of the next-to-lightest Z2-odd particle in the spectrum. Of particular interest
is the second production mechanism, because it can give rise to highly energetic light
DM which is relativistic for a significant time period. Thus, it contributes to the effec-
tive number of relativistic species, Neff , with an amount ∆Neff and alters the formation
of structures due to its non-vanishing free-streaming length. We employed the bound
∆Neff < 0.344 during the time of big bang nucleosynthesis (BBN) and calculated the
matter power spectrum P (k) of the DM to employ a half-mode analysis to put limits on
the allowed model parameters; it turned out that the ∆Neff bound is superior. Further,
all DM production has to happen before BBN takes place, or otherwise the abundance of
light nuclei might be altered. Demanding that the production terminates before t = 1 s
restricts the parameter space further. In order to satisfy these bounds, the hot DM
component can contribute only at the sub-percent level to the total DM abundance. Fi-
nally, we overlapped our findings with the projected sensitivities of the collider analysis
and were able to show that cosmology already constrains parameter regions which can-
not be accessed by current collider experiments. Further, we discussed how the bounds
from cosmology will change when considering other coupling strengths and heavier DM N1.
The model we studied in Chapter V extends the SM by an axion-light particle (ALP)
a which can couple to photons and fermions. Assuming small couplings to these SM
particles, the ALP DM can be produced via a freeze-in process by 2→ 2 scatterings or an-
nihilations. We have studied two different cases: either the ALP couples only to photons
(photophilic) or fermions (photophobic). We calculated the ALP DM momentum distri-
bution function f from the corresponding cross sections assuming Maxwell-Boltzmann
distributions for the involved particles and derived the power spectrum P (k) for both
cases. The photophilic scenario features an infrared (IR) singularity in the cross section
due to a virtual gauge boson, which has to be regularized by an IR cutoff. As a result
the momentum distribution function turns unphysical below a certain momentum. We
redid the analysis using quantum statistics for the in- and out-going particles and regu-
larized the IR singularity by the respective thermal gauge boson mass. Even in this case,
we found the momentum distribution function to become negative below a certain mo-
mentum.
However, for structure formation the most relevant quantity is the averaged momentum
〈p/T 〉 ' 3 and f turns only unphysical at rather low momenta. For this reason, we could
still use it to extract limits on the parameter space. We found that besides an overall
factor, the distribution function for both approaches are nearly identical and we used
the analytic results assuming Maxwell-Boltzmann statistics for the analysis. Using a
half-mode analysis assuming a weak and strong limit from Lyman-α forest observations,
we can place limits on the ALP DM mass ma . 19 keV. Additionally, as a complemen-
tary probe, we calculated the predicted number of Milky Way (MW) subhalos for the
130
SUMMARY AND CONCLUSION
ALP DM and compared it against observations. Demanding that enough subhalos even-
tually form, the limit ma . 17 keV was derived, depending on the MW mass. Our limits
do not dependent on the coupling, as long as the ALP does not thermalize, but we con-
sidered gamma-ray searches to constrain these couplings due to a→ γ γ decays.
For the photophobic case there is no infrared-singularity because fermion masses act as a
cutoff, and we found the limits ma . 16 keV from a half-mode analysis and ma . 13 keV
based on a subhalo count. Compared to the photophilic case, at lowest order, the results
do not depend on the assumed reheating temperature. If the ALP-fermion couplings are
flavor-universal, additional bounds from XENON1T, red giant cooling and loop-induced
photonic decays have to be taken into account. Finally, we considered the future Vera C.
Rubin observatory which allows us to place strong mass limits up to ma . 80 keV in the
photophilic case.
Inspired and motivated by these findings, we constructed a model-independent framework
with two distinctive DM production mechanisms, each producing a subset of DM featuring
its own temperature. We considered the first subset to be produced in decays of thermal-
ized parent particles, while the second one stems from non-thermal and long-lived parents.
The parameter space of this framework, which we refereed to as the two temperature dark
matter (2TDM) model, is spanned by the abundance A2 of the hotter subset to the total
DM abundance, the temperature ratio ξ between both subsets, and the DM mass mDM.
We derived constraints on these parameters based on the number of MW subhalos, limits
from the flux power spectrum of Lyman-α forest surveys and an additional contribution
to Neff . Further, we presented a fit equation which reproduces the corresponding exclu-
sion limits for DM masses between 20 keV and 1 MeV to a very good approximation. We
discussed how to adapt our result for the more general case when the relativistic degrees
of freedom are not constant during DM production, employed our findings for exemplary
toy models, and compared them against numerical simulations. Finally, we commented
on the possible inclusion of three-body decays, a generalization of the first production
method, and discussed the case where one subset can be considered “cold” DM instead.
In particular, we matched these results to the aforementioned models.
131
132
List of Figures
Chapter II
II.1 The trinity of DM detection: production of DM with highly energetic beams
at colliders, direct detection of DM scattering events inside detectors and
searches for decay products from DM annihilation processes. The circle in the
middle represents an unspecified effective coupling between two DM particles
χ and SM particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II.2 The matter power spectrum P (k) derived using CLASS [145, 146] and
for cosmological parameters specified in Ref. [27]. At small wavenumbers
it is increasing ∝ k, whereas it drops ∝ k−3 after it reaches a peak at
k ≈ 10−2 h/Mpc. The different colored horizontal lines indicate ranges of
the spectrum which can be experimentally verified by observations and which
are explained in more detail in the text. This figure is inspired by Fig. 1 in
Ref. [147]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
II.3 The corresponding matter power spectrum for a WDM with mass in range
mWDM = 0.5 keV–4 keV. In general, a suppression of the power spectrum
compared to ΛCDM can be observed and the scale at which it starts to
deviate is related to the mass of the WDM candidate. . . . . . . . . . . . . . 25
Chapter III
III.1 Allowed region in the η–mN2 parameter space for a charged scalar mass m± '
590 GeV (left) and m± ' 795 GeV (right). The BBN exclusion limits are
shown in red, while the blue shaded region does not produce a sufficiently
large baryon asymmetry. We observe that there is a region consistent with
BBN limits in which the correct amounts of DM and baryon asymmetry can
be obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
III.2 Production channels for the `i `j +ET process at the LHC. Pair produced
charged scalars decay into heavy leptons N2,3 and charged SM leptons (e, µ
or τ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
133
LIST OF FIGURES
III.3 Cross section σprod for pair production of charged scalars σ±. The left panel
shows the increase of σprod for different center-of-mass energies and fixed
scalar mass, m± = 400 GeV. The black (blue) vertical lines indicate the
energy range of the LHC (FCC-hh) and dashed lines indicate to the respective
cross section of the experiment. In the right panel we show σprod for different
scalar masses and fixed energy. By increasing m± the cross section drops
significantly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
III.4 Projected sensitivities for HL-LHC using L = 4000 fb−1 and the same anal-
ysis techniques as in Ref. [214]. Given in blue are exclusion limits, S = 2,
and shown in red are discovery limits, where S = 5. The dashed lines corre-
spond to a 100% BR into tau leptons using the Casas-Ibarra parametrization,
whereas the solid lines represent the case in which the maximized BR for NO
(shown in Table III.3) is employed. . . . . . . . . . . . . . . . . . . . . . . . 42
III.5 Projected sensitivities for HL-LHC using L = 4000 fb−1 and the same analysis
techniques as [213]. Given in blue are exclusion limits, S = 2, and shown in
red are discovery limits, where S = 5. The dashed lines correspond to optimal
BR into leptons, whereas to obtain the solid lines we used the value for NO
given in Table III.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
III.6 Sensitivity of FCC-hh with L = 3 ab−1 (30 ab−1) is shown with solid (dashed)
lines. The analysis is based on a proposed search for a supersymmetric
(SUSY) model presented in Ref. [225]. The red and blue curves correspond to
the “best case” scenario with maximized couplings to e, µ, indicating that a
significant portion of the parameter space can be probed at FCC-hh. The sit-
uation further improves for larger luminosity. The thin black line corresponds
to mN2,3 = m±. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
III.7 Production channels for the `i `j +ET process at CLIC. The left process is
similar to hadron colliders, except for electrons in the initial state. Addition-
ally, production via e+e− collision can happen via a RHN mediated t-channel
diagram, although it is suppressed due to Yukawa couplings y. . . . . . . . . 46
III.8 CLIC sensitivity for the di-lepton search. Using maximized couplings to e, µ
we obtained the red solid contour that corresponds to a 5σ discovery and the
blue one that represents 2σ exclusion. The corresponding dashed contours are
for the case where τ couplings are maximized. The thin black line indicates
mN2,3 = m±. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter IV
IV.1 Production of the light DM N1 either via two-body decays of heavy scalars
(left figure) or three-body decays of heavy N2,3 (right figure). Here, Σ stands
for all scalar particles: σ± and σ0. . . . . . . . . . . . . . . . . . . . . . . . . 56
IV.2 In the left (right) panel we show the DM momentum distribution func-
tion fN1(x, r)x
2 including both DM production mechanisms, taking m± =
600 GeV and mN2 = 100 GeV (mN2 = 400 GeV). The blue and red curve
correspond to early Σ decays and N2 late-time decays, respectively. . . . . . . 58
134
LIST OF FIGURES
IV.3 Transfer function T 2(k) for the same benchmark points as in Fig. IV.2. The
constraint from structure formation, using Eq. (II.45) with mTR = 2 keV is
shown in blue. RHN masses of around 100 GeV clearly violate this constraint
while the green line, corresponding to mN2 = 400 GeV, is consistent with the
data. The black dashed line represents ΛCDM. . . . . . . . . . . . . . . . . . 61
IV.4 Left figure: constraints from structure formation (red curves) confronted with
Neff limits (blue curves) derived using Eq. (IV.23). The solid curves corre-
spond to the “CMB weak” and dashed ones to the “CMB strong” choice
of ∆Neff. Note that our structure formation limits also indirectly depend on
∆Neff as it is an input parameter for CLASS. Clearly, Neff yields much stronger
limits in comparison to those arising from structure formation. Shown in
black solid is the curve for Ω h2 = 0.6 × 10−3N2→N1 . Right figure: same
as left figure, whose contours are shown faded. The blue line indicates the
new updated bound coming from the BBN limit on ∆Neff while the red line
stems from a δA structure formation analysis as explained in Section IV.3.1.
Additionally, the black dashed line indicates Ω 2N2→N1h = 0.36× 10−3. . . . . 62
IV.5 BBN constraints for the case where N2 dominantly decays into electrons and
muons (left panel) and taus (right panel) are disfavoring parameter space
below the black lines. The regions excluded by the BBN epoch limit on ∆Neff
are shown in red. The solid green curve indicates the parameter space for a
limiting case in which all of the DM is produced by N2 decays. The regions
in blue represent constraints from LFV experiments. Finally, the region to
the left of the vertical blue dashed line is favored by our collider analysis in
the sense that the mass gap between σ± and N2,3 is sufficiently large. The
value of the c√harged scalar mass in both panels is fixed to m± = 600 GeV.On the y-axis w∑e show the average Yukawa coupling of N2 and N3, defined
as ȳ2 + ȳ3 ≡ (|y2α|2 + |y 23α| )/3. . . . . . . . . . . . . . . . . . . . . . . . 63
α
IV.6 Same figures as shown in Figs. III.4 and III.5, but now overlapped with
BBN constraints discussed in Section IV.3.3 and Neff limits discussed in
Section IV.3.1. For the di-lepton scenario (right picture), cosmological con-
straints already exclude most of the potential exclusion region while the dis-
covery region is completely covered for both cases. . . . . . . . . . . . . . . . 65
IV.7 Summarized sensitivity curves for S0 = 2 as discussed in Sections III.4.2,
III.5.1 and III.5.2 for the HL-LHC (red shaded region) with 4 ab−1 (red
shaded region), CLIC with 5 ab−1 and (blue shaded region) and FCC-hh with
3 ab−1 and 30 ab−1 luminosity (orange and green shaded regions). Again,
the thin black line indicates mN2,3 = m±, whereas the thick black line
shows BBN constraints and the gray solid curve represent Neff constraints
for mN1 = 6 keV. The corresponding dashed contours indicate BBN and Neff
constraints for a DM particle with mN1 = 600 keV. We assumed a maximal
coupling to e and µ in this case. . . . . . . . . . . . . . . . . . . . . . . . . . 65
V.1 Tree level Feynman diagrams for the two processes of interest, fV → fa
(top) and f̄f → V a (bottom), in the photophilic (left) and photophobic
(right) cases. Image credits: Enrico Morgante . . . . . . . . . . . . . . . . . . 71
135
LIST OF FIGURES
Chapter V
V.2 The ALP momentum distribution function plotted as x2f(x). Left figure: In
blue we show the photophilic case for MB statistics; black lines represent the
photophilic case with quantum statistics and in green we show the photopho-
bic case for MB statistics. For each, the dashed (dot-dashed) line represents
f̄f → V a (fV → fa) and the solid line is their sum. The vertical dashed
line marks p = g′T , left of which thermal corrections should be added [262].
Right figure: Photophilic scenario for MB and quantum statistics in QED,
namely, including only a single fermion carrying unit charge. Image credits:
Enrico Morgante (adapted by author) . . . . . . . . . . . . . . . . . . . . . . 74
V.3 Transfer function T (k) shown as blue lines for photophilic ALP DM with
different masses compared to thermal relic limits presented as an analytic
fit given in Eq. (II.45). Shaded in red is the weak limit whereas the light
red shaded region is excluded by the strong limit. Hence, the weakest mass
bound stemming from this analysis is ma ' 5 keV, as indicated by the dotted
blue line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
V.4 The blue shaded regions are structure formation limits on the photophilic
ALP parameter space derived using a half-mode analysis technique as well as
a MW subhalo count. The former (latter) are denoted as Lyman-α (MW).
For both, weak and strong limits are shown (see Section V.3 for further anal-
ysis details). Additionally, gamma-ray limits from INTEGRAL (dark blue
shaded) [275], NuSTAR (orange shaded) [276] and M31 [277] (red shaded)
are shown. The diagonal dashed lines indicate specific values of TRH, cal-
culated by the requirement of producing the observed amount of DM. The
dark gray line assumes only couplings between ALP and B bosons, whereas
the light gray lines takes an additional coupling caWW into account. Finally,
the red line at ma ∼ 80 keV is the projected sensitivity from the forthcoming
Vera C. Rubin observatory. Image credits: Vedran Brdar (adapted by author) 86
V.5 Limits for the photophilic ALP F − ma parameter plane stemming from a
MW subhalo count assuming the strong MW mass choice. We considered that
freeze-in contributes only a fraction F < 1 to the total ALP DM abundance.
The blue shaded region is excluded because these parameters yield Nsub <
64. As the cold ALP DM contribution increases, the limit on ma decreases
correspondingly and effectively vanishes for F . 0.05. . . . . . . . . . . . . . 87
V.6 As before, blue shaded regions are structure formation limits on the photo-
phobic ALP parameter space derived using a half-mode analysis technique as
well as a MW subhalo count. The former (latter) are denoted as Lyman-α
(MW). For both, weak and strong limits are shown (see Section V.3 for fur-
ther analysis details). The green lines indicate the viable parameter space
based on Eq. (V.41). If the ALP couples only to tau leptons, the ALP needs
large couplings indicated by the uppermost line, whereas the other lines cor-
respond to couplings to c or b quark and the sum over all three fermions. If
the ALP has a flavor universal coupling, additional bounds from red giant
cooling and limits from XENON1T has to be taken into account. Finally,
the red line at ma ∼ 60 keV is the projected sensitivity from the forthcoming
Vera C. Rubin observatory. Image credits: Vedran Brdar (adapted by author) 88
136
LIST OF FIGURES
V.7 Impact of TRH on the viable parameter space of the photophobic ALP, as-
suming that it couples only to b quarks. A larger ALP-fermion coupling is
needed to get the correct DM abundance for lower reheating temperatures.
The blue shaded regions are the same mass limits as in Fig. V.6, where
gs∗(Tprod) = 106.75, while the black lines are computed taking a change in
gs∗(Tprod) into account. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
V.8 Inverse production process of ALP DM via its coupling to photons (A1) and
fermions (A21 and A22). An ALP particle a is converted into a photon γ by
scattering off a fermion f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
V.9 Inverse production process of ALP DM via its coupling to photons (B1) and
fermions (B21 and B22). An ALP particle a annihilates with a photon γ into
a fermion-antifermion pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
V.10 Comparison of ALP DM mass limits for different choices of Nsub and MW
masses in the photophilic scenario. The upper part is the strong MW mass
and the lower the weak mass choice. The blue lines arema limits forNsub = 80
and the red lines are using Nsub = 50 while the black lines are the results
given in Tables V.1 and V.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Chapter VI
VI.1 Schematic representation of our setup: given in black is the background tem-
perature Tγ of the universe, while the blue and red curve represents the time
evolution of the temperature of the first and second DM set, respectively.
The first DM subset is produced at time t1 with a temperature similar to Tγ .
After a while, the second DM subset is produced at t2 but it has a higher
temperature T2 compared to T1. The bumps in the solid black line mimics
entropy dilution due to particle freeze-out in the SM thermal bath. . . . . . . 101
VI.2 Example x2 f(x) spectrum for a specific 2TDM scenario with two observable
peaks calculated for times when DM production is finished. The blue area
corresponds to a dominant subset whereas the read area refers to a subdom-
inant DM share but with higher temperature with respect to the first set.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
VI.3 ∆Neff bounds derived from Eq. (VI.13) assuming mP = 1 TeV. The black
solid line corresponds to mDM = 10 keV and the dashed line to mDM =
100 keV using the BBN bound ∆Neff < 0.344. The dotted and dashed-dotted
line are the corresponding bounds for mDM = 10 keV and 100 keV from CMB
measurements, ∆Neff < 0.28. For A2 = 1 temperature ratios ξ & 240 are
excluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
137
LIST OF FIGURES
VI.4 Transfer function T (k) for a 2TDM model where mDM = 30 keV, A2 = 0.2
and ξ = 25, 125 shown in red and dashed red respectively, while the blue line
is derived from Eq. (II.45) for a thermal relic mass mTR = 2 keV indicating
a potential limit from an analysis on structure formation. The green line
correspond to ξ = 25 and a smaller A2 = 0.05. Applying a half-mode analysis,
all three parameter choices would be excluded, but limits from observables
on the matter power spectrum only exclude the two red lines, while the green
line is not in conflict. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
VI.5 Compilation of different MW mass analyses using the second data release
from the GAIA survey. Further information on the results can be found in
Refs. [268, 269, 313–321, 323]. The black dotted line is the combination of all
measurements and the green shaded region represents a 2σ error range. On
the contrary, blue errorbars correspond to 68% CL limits of the respective
analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
VI.6 Comparison of the transfer function T (k) for a shifted MB distribution and
a freeze-in/freeze-out scenario where the temperature ratio is set to ξ = 40
and A2 = 0.5, while mDM = 10 keV. We see that the analytical shifted MB
distribution (red curve) is a good tool to approximate the numerical results for
decays of frozen-out or frozen-in parents, shown in blue and gray, respectively.
Hence we will use it in the following analysis to deduce constraints from
structure formation. The horizontal black dashed line corresponds to a pure
ΛCDM scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
VI.7 Numerical results for z2f(z) as defined in Eq. (VI.21) for different masses of
the parent particle, ranging from mP = 10
−5 GeV to mP = 1 TeV and using
Eq. (VI.3) for f(z) as an illustration. Masses of 0.1 and 1 GeV lead to the
√biggest impact on the shape of the spectrum, while the distributions of theother masses have similar shapes. All distributions are rescaled with a factor
gs∗(mP ) for an easier comparison. . . . . . . . . . . . . . . . . . . . . . . . . 112
VI.8 Limits from structure formation and the number of effective entropic DOF
for mDM = 20 keV and fixed g
s
∗(T ) = 106.75. The blue shaded region is
disfavored by the MW subhalo count and the green shaded region by limits
from Lyman-α surveys, respectively. The solid lines are the corresponding
stronger limits, whereas the weaker constraints are shown as dashed lines.
The red shaded region in the upper right corner is disfavored by a too large
∆Neff value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
VI.9 Structure formation limits given by shaded regions and derived from our
analytic fit, Eqs. (VI.24) and (VI.25), using δA in the left andNsub in the right
figure. The DM mass is given by mDM = 50 keV and we assume g
s
∗(T ) = const
during DM production. Shown as black lines are corresponding limits from a
numerical simulation and solid/dashed lines correspond to stronger or weaker
bounds, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
138
LIST OF FIGURES
VI.10 Structure formation limits using the δAref criterion in the left and a MW sub-
halo count in the right figure for the second toy model. Everything indicated
by the green or blue shaded region is constrained by structure formation and
solid lines correspond to strong and dashed lines to weak limits. For compar-
ison, full numerical simulations to extract limits are run for ξ = 10 and 40
and are indicated by the stars and diamonds in both plots. The difference
between the dark blue and black symbols indicated by an arrow are due to
specifics of this model choice and are further explained in the text. The red
shaded region yields a too large ∆Neff stemming from the second subset. . . . 117
VI.11 Left figure: Application of our fit results to the ScM with m± = 1200 GeV
and mN2 ' 300 GeV. The green shaded region is the δAref,1 limit and the blue
one the light MW subhalo count. Again, the red shaded region is excluded
by the ∆Neff bound. Right figure: The corresponding abundance A2 (purple
line) and DM temperature ratio ξ′ (black line) for the same ScM setup, where
Yukawa interactions are chosen as large as possible. The green/blue shaded
regions are excluded by the respective structure formation limits and the red
shaded regions by ∆Neff . For both plots, the mass of the DM candidate is
mN1 = 24 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
VI.12 Structure formation limits using the δAref criterion in the left and a MW
subhalo count in the right figure for the third toy model. Everything indicated
by the green or blue shaded region is constrained by structure formation
and solid lines correspond to strong and dashed lines to weak limits. For
comparison a full numerical simulation to extract limits was run for ξ = 40
and 300 and is indicated by the stars and diamonds in both plots. The ∆Neff
bound is absent, because it is weaker compared to Fig. VI.10 due to the larger
DM mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
VI.13 Effect of changing 〈z〉 for the A1 set of the DM on the excluded region in the
A2–ξ plane based on the δAref,1 criterion. The dashed lines correspond to
α = 0.6, while the dotted curves represents α = 1.4. The black curves stem
from a numerical analysis and the green ones from the adapted analytical fit
as explained in Section VI.5. The left figure is derived for mDM = 20 keV and
the right one for mDM = 60 keV. . . . . . . . . . . . . . . . . . . . . . . . . . 120
VI.14 Similar to Fig. V.5, where limits for the photophilic ALP F −ma parameter
plane stemming from a MW subhalo count assuming the lighter MW mass
choice are derived. Freeze-in contributes only a fraction F < 1 of warm DM
to the total ALP DM abundance, while the rest is made up by a CDM com-
ponent. The blue shaded region is excluded because these parameters yield
Nsub < 64. The red line is the limit derived using our analytic expressions
for the light MW mass choice. . . . . . . . . . . . . . . . . . . . . . . . . . . 121
VI.15 Ratio of 〈x〉 derived comparing Eq. (VI.21) against Eq. (VI.1), i.e. comparing
a time-dependent gs∗(T ) against constant case g
s
∗(T ) = 106.75. As can be seen,
the averaged value is shifted to larger values, especially for mP ∼ 1 GeV and
large values of ξ, because in that region gs∗(T ) is changing rapidly. However,
the effect becomes less prominent for heavier or lighter parent particles. It
can also be seen that 〈x〉 stays constant if both, mP and ξ are increased. The
black shaded region indicates decays which would take place after BBN. . . . 124
139
LIST OF FIGURES
VI.16 Comparison of the fit parameters pi for different DM masses and structure
formation limits shown as crosses, and the corresponding fourth-order poly-
nomial fit (see Eq. (VI.24). Stronger limits are shown as solid and weaker
ones as dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
VI.17 Comparison of mass limits for different choices of Nsub and MW masses for
the 2TDM model with mDM = 50 keV. The left figure shows results for the
light MW mass and the right one for the conservative choice. The blue lines
are ma limits for Nsub = 80 and the red lines are using Nsub = 50 while the
black lines are results for Nsub = 64. . . . . . . . . . . . . . . . . . . . . . . . 125
140
List of Tables
III.1 Signal regions with the corresponding cuts on final state momenta used in
the di-tau analysis. The definition of the respective triggers are explained in
the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
III.2 95% CL limits on the non-SM cross section for the di-tau + ET analysis. . . 40
III.3 Largest possible BRs for the decay of σ± into τ and N2,3. Above the given
value for η0, the BRs are to a good approximation independent of this pa-
rameter. As can be seen, the IO gives rise to smaller BRs compared to NO. . 41
III.4 95% CL limits on the non-SM cross section for the di-lepton + ET analysis. . 43
III.5 Largest possible BRs for the decay of σ± into e±, u±. Below the given value
for η0, the BRs are to a good approximation independent of this parameter.
Interestingly, the IO regime can feature a situation with a very small BR into
tau’s, implying approximate zeros in the third column of the Yukawa matrix. 43
III.6 Cuts made for distinguishing signal and background at FCC-hh with a lumi-
nosity of 3 ab−1. We show the number of signal and background events for
mN2,3 = 500 GeV and m± = 1 TeV. In contrast to Eq. (III.13), λ3 = −0.27
was used in this analysis. No systematic errors on the background were as-
sumed for this analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
III.7 Signal regions used for an analysis at CLIC with a center of mass energy of
3 TeV and a luminosity of 5 ab−1. We show the number of signal and back-
ground events together with the corresponding sensitivities for a benchmark
point mN2,3 = 500 GeV and m± = 1 TeV. The results are shown both for
“best” and “worst” case scenarios which correspond to maximizing couplings
to e, µ and τ lepton, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 47
V.1 Structure formation limits on the ALP mass ma in the photophilic scenario. 83
V.2 Structure formation limits on the ALP mass ma in the photophobic scenario
under the assumption that gs∗(Tprod) = 106.75. . . . . . . . . . . . . . . . . . 86
V.3 Limits on the ALP mass ma for different choices of the observed amount
of subhalos Nsub compared to the case Nsub = 64 as used in this section,
assuming a photophilic scenario. . . . . . . . . . . . . . . . . . . . . . . . . . 96
141
LIST OF TABLES
VI.1 Comparison of the corresponding structure formation observables using a
shifted MB distribution for parent particles and the respective momentum
distributions for non-thermal parent particles. The DM mass is set to 50 keV
and only the abundance A2 is varied. The predictions for Nsub and δA are
nearly identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
VI.2 Lower DM mass limit mlimDM using constraints from structure formation as-
suming ξ = 1, i.e. all of DM has a common temperature T2 = T1. . . . . . . . 110
VI.3 Fit results for the respective exclusion limits based on Eq. (VI.24) using
mDM = 20 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
VI.4 Fit results for the respective exclusion limits based on Eq. (VI.24) using
mDM = 20 keV and assuming that the first subset is composed of CDM. . . . 121
VI.5 Final fit parameters for all four exclusion contours, according to Eq. (VI.25).
The DM mass ranges between 20–1000 keV. . . . . . . . . . . . . . . . . . . . 125
VI.6 Lower DM mass limit mlimDM for different choices of the observed amount of
subhalos Nsub compared to the case Nsub = 64 as used in this section. They
were derived assuming ξ = 1, i.e. all of DM has a common temperature
T2 = T1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
142
List of abbreviations
νMSM — neutrino minimal standard lowE — low-multipole CMB data
model MACHO — massive compact halo object
ΛCDM — standard model of cosmology MB — Maxwell-Boltzmann
ALP — axion-like particle MET — missing transverse energy (also
ARS — Akhmedov-Rubakov-Smirnov called ET )
mechanism MW — Milky Way
BAO — baryonic acoustic oscillation MSW — Mikheyev-Smirnov-Wolfenstein
BAU — baryon asymmetry of the Universe NGB — Nambu-Goldstone boson
BBN — big bang nucleosynthesis NO — normal ordering of neutrino masses
BR — branching ratio PBH — primordial black hole
BSM — beyond the Standard Model PDP — pseudo Dirac pair
CDM — cold dark matter PMNS — Pontecorvo-Maki-Nakagawa-
CKM — Cabibbo-Kobayashi-Maskawa ma- Sakata matrix
trix PNGB — pseudo Nambu-Goldstone boson
CL — confidence level PQ — Peccei-Quinn
CMB — cosmic microwave background QCD — quantum chromodynamics
DM — dark matter QED — quantum electrodynamics
DOF — degrees of freedom RH — right-handed
EDM — electric dipole moment RHN — right-handed neutrino
EE — CMB E mode polarization power SM — standard model
spectrum ScM — scotogenic model
EM — electromagnetism SUSY — supersymmetry
EW — electroweak TT — CMB temperature polarization
EWSB — electroweak symmetry breaking power spectrum
FRW — Friedmann-Robertson-Walker TE — CMB temperature-E polarization
HB — horizontal branch cross-power spectrum
IGM — intergalactic medium TR — thermal relic
IO — inverted ordering of neutrino masses VEV — vacuum expectation value
IR — infrared WDM — warm dark matter
LFV — lepton flavor violation WIMP — weakly interacting massive par-
LH — left-handed ticle
LNV — lepton number violating
List of experiments and surveys
Collider experiments and facilities
ˆ CLIC — Compact Linear Collider
ˆ CEPC — Circular Electron Positron Collider
ˆ FCC — Future Circular Collider, as lepton (FCC-ee) or hadron (FCC-hh) collider
ˆ LHC — Large Hadron Collider
– ALICE — A Large Ion Collider Experiment
– ATLAS — A Toroidal LHC ApparatuS
– CMS — Compact Muon Solenoid
– LHCb — LHC beauty
ˆ HE-LHC — high-energy LHC
ˆ HL-LHC — high-luminosity LHC
ˆ ILC — International Linear Collider
Neutrino experiments
ˆ DUNE — Deep Underground Neutrino Experiment
ˆ KATRIN — Karlsruhe Tritium Neutrino Experiment
ˆ IceCube — IceCube Neutrino Observatory
ˆ MicroBooNE — liquid argon detector for neutrino beamline
Direct DM detection
ˆ CREST-III — Third stage of Cryogenic Rare Event Search with Superconducting
Thermometers
ˆ SENSEI — Sub-Electron-Noise Skipper-CCD Experimental Instrument
ˆ SuperCDMS — Super Cryogenic Dark Matter Search
ˆ XENON1T — liquid xenon detector
Gamma-ray spectra surveys and indirect DM detection
ˆ Fermi-LAT — Fermi Large Area Telescope
ˆ H.E.S.S. — High Energy Stereoscopic System
ˆ INTEGRAL — International Gamma-Ray Astrophysics Laboratory
ˆ MAGIC — Major Atmospheric Gamma Imaging Cherenkov Telescopes
ˆ M31 — reference to data taken by the Chandra X-ray Observatory
ˆ NUSTAR — Nuclear Spectroscopic Telescope Array
LIST OF EXPERIMENTS AND SURVEYS
ˆ Suzaku — Suzaku satellite
ˆ THESEUS — Transient High Energy Sky and Early Universe Surveyor
ˆ XMM-Newton — X-ray Multi-Mirror Mission
Cosmological and astrophysical surveys
ˆ CMB-S4 — Stage-4 ground based cosmic microwave background experiment
ˆ GAIA — Global Astrometric Interferometer for Astrophysics
ˆ HIRES — Keck High Resolution Echelle Spectrometer
ˆ MIKE — Magellan Inamori Kyocera Echelle spectrograph
ˆ PLANCK — Planck satellite
ˆ R18 — reference to Supernova H0 for the Equation of State (SH0ES)
ˆ SDSS — Sloan Digital Sky Survey
ˆ XQ-100 — dataset from the XSHOOTER spectrograph on the Very Large Telescope
(VLT)
Other terrestrial experiments
ˆ ALPS — Any Light Particle Search
ˆ CAST — CERN Axion Solar Telescope
ˆ MEG II — Mu to E Gamma II
145

References
[1] Sven Baumholzer et al. “Shining Light on the Scotogenic Model: Interplay of Col-
liders and Cosmology”. In: JHEP 09 (2020), p. 136. arXiv: 1912.08215 [hep-ph].
[2] Sven Baumholzer, Vedran Brdar, and Pedro Schwaller. “The New νMSM (ννMSM):
Radiative Neutrino Masses, keV-Scale Dark Matter and Viable Leptogenesis with
sub-TeV New Physics”. In: JHEP 08 (2018), p. 067. arXiv: 1806.06864 [hep-ph].
[3] Sven Baumholzer, Vedran Brdar, and Enrico Morgante. “Structure Formation
Limits on Axion-Like Dark Matter”. In: JCAP 05 (2021), p. 004. arXiv: 2012.
09181 [hep-ph].
[4] S. L. Glashow. “Partial Symmetries of Weak Interactions”. In: Nucl. Phys. 22
(1961), pp. 579–588.
[5] Steven Weinberg. “A Model of Leptons”. In: Phys. Rev. Lett. 19 (21 Nov. 1967),
pp. 1264–1266.
[6] Abdus Salam. “Weak and Electromagnetic Interactions”. In: Conf. Proc. C 680519
(1968), pp. 367–377.
[7] Gerard ’t Hooft and M. J. G. Veltman. “Regularization and Renormalization of
Gauge Fields”. In: Nucl. Phys. B 44 (1972), pp. 189–213.
[8] F. Englert and R. Brout. “Broken Symmetry and the Mass of Gauge Vector
Mesons”. In: Phys. Rev. Lett. 13 (9 Aug. 1964), pp. 321–323.
[9] Peter W. Higgs. “Broken Symmetries and the Masses of Gauge Bosons”. In: Phys.
Rev. Lett. 13 (16 Oct. 1964), pp. 508–509.
[10] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble. “Global Conservation Laws
and Massless Particles”. In: Phys. Rev. Lett. 13 (20 Nov. 1964), pp. 585–587.
[11] Peter W. Higgs. “Spontaneous Symmetry Breakdown without Massless Bosons”.
In: Phys. Rev. 145 (4 May 1966), pp. 1156–1163.
[12] T. W. B. Kibble. “Symmetry Breaking in Non-Abelian Gauge Theories”. In: Phys.
Rev. 155 (5 Mar. 1967), pp. 1554–1561.
[13] Georges Aad et al. “Observation of a new particle in the search for the Standard
Model Higgs boson with the ATLAS detector at the LHC”. In: Phys. Lett. B 716
(2012), pp. 1–29. arXiv: 1207.7214 [hep-ex].
[14] Serguei Chatrchyan et al. “Observation of a New Boson at a Mass of 125 GeV with
the CMS Experiment at the LHC”. In: Phys. Lett. B 716 (2012), pp. 30–61. arXiv:
1207.7235 [hep-ex].
[15] Michael E. Peskin and Daniel V. Schroeder. An Introduction to quantum field
theory. Reading, USA: Addison-Wesley, 1995.
147
REFERENCES
[16] Matthew D. Schwartz. Quantum Field Theory and the Standard Model. Cambridge
University Press, Mar. 2014.
[17] Steven Weinberg. The Quantum theory of fields. Vol. 1: Foundations. Cambridge
University Press, June 2005.
[18] Steven Weinberg. The quantum theory of fields. Vol. 2: Modern applications. Cam-
bridge University Press, Aug. 2013.
[19] Nicola Cabibbo. “Unitary Symmetry and Leptonic Decays”. In: Phys. Rev. Lett.
10 (12 June 1963), pp. 531–533.
[20] Makoto Kobayashi and Toshihide Maskawa. “CP Violation in the Renormalizable
Theory of Weak Interaction”. In: Prog. Theor. Phys. 49 (1973), pp. 652–657.
[21] B. Pontecorvo. “Mesonium and anti-mesonium”. In: Sov. Phys. JETP 6 (1957),
p. 429.
[22] B. Pontecorvo. “Inverse beta processes and nonconservation of lepton charge”. In:
Zh. Eksp. Teor. Fiz. 34 (1957), p. 247.
[23] Ziro Maki, Masami Nakagawa, and Shoichi Sakata. “Remarks on the unified model
of elementary particles”. In: Prog. Theor. Phys. 28 (1962), pp. 870–880.
[24] Raymond Davis, Don S. Harmer, and Kenneth C. Hoffman. “Search for Neutrinos
from the Sun”. In: Phys. Rev. Lett. 20 (21 May 1968), pp. 1205–1209.
[25] Y. Fukuda et al. “Evidence for oscillation of atmospheric neutrinos”. In: Phys.
Rev. Lett. 81 (1998), pp. 1562–1567. arXiv: hep-ex/9807003.
[26] M. Aker et al. “Improved Upper Limit on the Neutrino Mass from a Direct Kine-
matic Method by KATRIN”. In: Phys. Rev. Lett. 123.22 (2019), p. 221802. arXiv:
1909.06048 [hep-ex].
[27] N. Aghanim et al. “Planck 2018 results. VI. Cosmological parameters”. In: Astron.
Astrophys. 641 (2020). [Erratum: Astron.Astrophys. 652, C4 (2021)], A6. arXiv:
1807.06209 [astro-ph.CO].
[28] Michelle J. Dolinski, Alan W. P. Poon, and Werner Rodejohann. “Neutrinoless
Double-Beta Decay: Status and Prospects”. In: Ann. Rev. Nucl. Part. Sci. 69
(2019), pp. 219–251. arXiv: 1902.04097 [nucl-ex].
[29] Ivan Esteban et al. “The fate of hints: updated global analysis of three-flavor
neutrino oscillations”. In: JHEP 09 (2020), p. 178. arXiv: 2007.14792 [hep-ph].
[30] B. Abi et al. The DUNE Far Detector Interim Design Report Volume 1: Physics,
Technology and Strategies. arXiv: 1807.10334 [physics.ins-det].
[31] Stefano Dell’Oro et al. “Neutrinoless double beta decay: 2015 review”. In: Adv.
High Energy Phys. 2016 (2016), p. 2162659. arXiv: 1601.07512 [hep-ph].
[32] Steven Weinberg. “Baryon and Lepton Nonconserving Processes”. In: Phys. Rev.
Lett. 43 (1979), pp. 1566–1570.
[33] Peter Minkowski. “µ→ eγ at a Rate of One Out of 109 Muon Decays?” In: Phys.
Lett. B 67 (1977), pp. 421–428.
[34] Rabindra N. Mohapatra and Goran Senjanovi ć. “Neutrino Mass and Spontaneous
Parity Nonconservation”. In: Phys. Rev. Lett. 44 (14 Apr. 1980), pp. 912–915.
[35] Tsutomu Yanagida. “Horizontal Symmetry and Masses of Neutrinos”. In: Conf.
Proc. C7902131 (1979), pp. 95–99.
[36] M. Fukugita and T. Yanagida. “Baryogenesis Without Grand Unification”. In:
Phys. Lett. B 174 (1986), pp. 45–47.
148
REFERENCES
[37] Sacha Davidson, Enrico Nardi, and Yosef Nir. “Leptogenesis”. In: Phys. Rept. 466
(2008), pp. 105–177. arXiv: 0802.2962 [hep-ph].
[38] J. Schechter and J. W. F. Valle. “Neutrino Masses in SU(2) x U(1) Theories”. In:
Phys. Rev. D 22 (1980), p. 2227.
[39] George Lazarides, Q. Shafi, and C. Wetterich. “Proton Lifetime and Fermion
Masses in an SO(10) Model”. In: Nucl. Phys. B 181 (1981), pp. 287–300.
[40] Robert Foot et al. “Seesaw Neutrino Masses Induced by a Triplet of Leptons”. In:
Z. Phys. C 44 (1989), p. 441.
[41] Ernest Ma. “Pathways to naturally small neutrino masses”. In: Phys. Rev. Lett.
81 (1998), pp. 1171–1174. arXiv: hep-ph/9805219.
[42] Yi Cai et al. “From the trees to the forest: a review of radiative neutrino mass
models”. In: Front. in Phys. 5 (2017), p. 63. arXiv: 1706.08524 [hep-ph].
[43] Gianfranco Bertone and Dan Hooper. “History of dark matter”. In: Rev. Mod.
Phys. 90.4 (2018), p. 045002. arXiv: 1605.04909 [astro-ph.CO].
[44] Gianfranco Bertone, Dan Hooper, and Joseph Silk. “Particle dark matter: Evidence,
candidates and constraints”. In: Phys. Rept. 405 (2005), pp. 279–390. arXiv: hep-
ph/0404175.
[45] C. Alcock et al. “The MACHO project: Microlensing results from 5.7 years of
LMC observations”. In: Astrophys. J. 542 (2000), pp. 281–307. arXiv: astro-
ph/0001272.
[46] L. Wyrzykowski et al. “The OGLE view of microlensing towards the Magellanic
Clouds - IV. OGLE-III SMC data and final conclusions on MACHOs”. In: Monthly
Notices of the Royal Astronomical Society 416.4 (Aug. 2011), pp. 2949–2961.
[47] Bernard Carr and Florian Kuhnel. “Primordial Black Holes as Dark Matter Can-
didates”. In: Les Houches summer school on Dark Matter. Oct. 2021. arXiv:
2110.02821 [astro-ph.CO].
[48] F. Zwicky. “Die Rotverschiebung von extragalaktischen Nebeln”. In: Helv. Phys.
Acta 6 (1933), pp. 110–127.
[49] Vera C. Rubin and W. Kent Ford Jr. “Rotation of the Andromeda Nebula from a
Spectroscopic Survey of Emission Regions”. In: Astrophys. J. 159 (1970), pp. 379–
403.
[50] Maxim Markevitch et al. “Direct constraints on the dark matter self-interaction
cross-section from the merging galaxy cluster 1E0657-56”. In: Astrophys. J. 606
(2004), pp. 819–824. arXiv: astro-ph/0309303.
[51] Douglas Clowe, Anthony Gonzalez, and Maxim Markevitch. “Weak lensing mass
reconstruction of the interacting cluster 1E0657-558: Direct evidence for the exis-
tence of dark matter”. In: Astrophys. J. 604 (2004), pp. 596–603. arXiv: astro-
ph/0312273.
[52] N. Aghanim et al. “Planck 2018 results. VI. Cosmological parameters”. In: Astron.
Astrophys. 641 (2020), A6. arXiv: 1807.06209 [astro-ph.CO].
[53] Matteo Nori et al. “Lyman α forest and non-linear structure characterization in
Fuzzy Dark Matter cosmologies”. In: Mon. Not. Roy. Astron. Soc. 482.3 (2019),
pp. 3227–3243. arXiv: 1809.09619 [astro-ph.CO].
[54] Scott Tremaine and James E. Gunn. “Dynamical Role of Light Neutral Leptons in
Cosmology”. In: Phys. Rev. Lett. 42 (6 Feb. 1979), pp. 407–410.
149
REFERENCES
[55] Lisa Randall, Jakub Scholtz, and James Unwin. “Cores in Dwarf Galaxies from
Fermi Repulsion”. In: Mon. Not. Roy. Astron. Soc. 467.2 (2017), pp. 1515–1525.
arXiv: 1611.04590 [astro-ph.GA].
[56] Vid Iršič et al. “New Constraints on the free-streaming of warm dark matter from
intermediate and small scale Lyman-α forest data”. In: Phys. Rev. D 96.2 (2017),
p. 023522. arXiv: 1702.01764 [astro-ph.CO].
[57] Paolo Gondolo and Graciela Gelmini. “Cosmic abundances of stable particles:
Improved analysis”. In: Nucl. Phys. B 360 (1991), pp. 145–179.
[58] Edward W. Kolb and Michael S. Turner. The Early Universe. Vol. 69. 1990.
[59] V. Mukhanov. Physical Foundations of Cosmology. Oxford: Cambridge University
Press, 2005.
[60] P. Cushman et al. “Working Group Report: WIMP Dark Matter Direct Detection”.
In: Community Summer Study 2013: Snowmass on the Mississippi. Oct. 2013.
arXiv: 1310.8327 [hep-ex].
[61] Mark W. Goodman and Edward Witten. “Detectability of Certain Dark Matter
Candidates”. In: Phys. Rev. D 31 (1985). Ed. by M. A. Srednicki, p. 3059.
[62] E. Aprile et al. “Dark Matter Search Results from a One Ton-Year Exposure of
XENON1T”. In: Phys. Rev. Lett. 121.11 (2018), p. 111302. arXiv: 1805.12562
[astro-ph.CO].
[63] E. Aprile et al. “Light Dark Matter Search with Ionization Signals in XENON1T”.
In: Phys. Rev. Lett. 123.25 (2019), p. 251801. arXiv: 1907.11485 [hep-ex].
[64] A. H. Abdelhameed et al. “First results from the CRESST-III low-mass dark
matter program”. In: Phys. Rev. D 100.10 (2019), p. 102002. arXiv: 1904.00498
[astro-ph.CO].
[65] R. Agnese et al. “First Dark Matter Constraints from a SuperCDMS Single-Charge
Sensitive Detector”. In: Phys. Rev. Lett. 121.5 (2018). [Erratum: Phys.Rev.Lett.
122, 069901 (2019)], p. 051301. arXiv: 1804.10697 [hep-ex].
[66] Orr Abramoff et al. “SENSEI: Direct-Detection Constraints on Sub-GeV Dark
Matter from a Shallow Underground Run Using a Prototype Skipper-CCD”. In:
Phys. Rev. Lett. 122.16 (2019), p. 161801. arXiv: 1901.10478 [hep-ex].
[67] M. G. Aartsen et al. “Search for annihilating dark matter in the Sun with 3 years
of IceCube data”. In: Eur. Phys. J. C 77.3 (2017). [Erratum: Eur.Phys.J.C 79, 214
(2019)], p. 146. arXiv: 1612.05949 [astro-ph.HE].
[68] M. G. Aartsen et al. “Search for Neutrinos from Dark Matter Self-Annihilations in
the center of the Milky Way with 3 years of IceCube/DeepCore”. In: Eur. Phys.
J. C 77.9 (2017), p. 627. arXiv: 1705.08103 [hep-ex].
[69] M. L. Ahnen et al. “Limits to Dark Matter Annihilation Cross-Section from a
Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf Satellite
Galaxies”. In: JCAP 02 (2016), p. 039. arXiv: 1601.06590 [astro-ph.HE].
[70] Sebastian Hoof, Alex Geringer-Sameth, and Roberto Trotta. “A Global Anal-
ysis of Dark Matter Signals from 27 Dwarf Spheroidal Galaxies using 11 Years
of Fermi-LAT Observations”. In: JCAP 02 (2020), p. 012. arXiv: 1812.06986
[astro-ph.CO].
[71] H. Abdallah et al. “Search for dark matter annihilations towards the inner Galactic
halo from 10 years of observations with H.E.S.S”. In: Phys. Rev. Lett. 117.11 (2016),
p. 111301. arXiv: 1607.08142 [astro-ph.HE].
150
REFERENCES
[72] Albert M Sirunyan et al. “Search for new physics in final states with√a single photon
and missing transverse momentum in proton-proton collisions at s = 13 TeV”.
In: JHEP 02 (2019), p. 074. arXiv: 1810.00196 [hep-ex].
[73] Georges Aad et al.√“Search for dark matter in association with an energetic photon
in pp collisions at s = 13 TeV with the ATLAS detector”. In: JHEP 02 (2021),
p. 226. arXiv: 2011.05259 [hep-ex].
[74] G. Aad et al. “Search for new phenomena in ev√ents with an energetic jet and
missing transverse momentum in pp collisions at s = 13 TeV with the ATLAS
detector”. In: Phys. Rev. D 103 (11 June 2021), p. 112006.
[75] Armen Tumasyan et al. “Search for new particles in events with en√ergetic jets and
large missing transverse momentum in proton-proton collisions at s = 13 TeV”.
In: JHEP 11 (2021), p. 153. arXiv: 2107.13021 [hep-ex].
[76] Lawrence J. Hall et al. “Freeze-In Production of FIMP Dark Matter”. In: JHEP
03 (2010), p. 080. arXiv: 0911.1120 [hep-ph].
[77] Mark Hindmarsh, Russell Kirk, and Stephen M. West. “Dark Matter from Decaying
Topological Defects”. In: JCAP 03 (2014), p. 037. arXiv: 1311.1637 [hep-ph].
[78] Yonit Hochberg et al. “Mechanism for Thermal Relic Dark Matter of Strongly
Interacting Massive Particles”. In: Phys. Rev. Lett. 113 (2014), p. 171301. arXiv:
1402.5143 [hep-ph].
[79] Duccio Pappadopulo, Joshua T. Ruderman, and Gabriele Trevisan. “Dark matter
freeze-out in a nonrelativistic sector”. In: Phys. Rev. D 94.3 (2016), p. 035005.
arXiv: 1602.04219 [hep-ph].
[80] Scott Dodelson and Lawrence M. Widrow. “Sterile-neutrinos as dark matter”. In:
Phys. Rev. Lett. 72 (1994), pp. 17–20. arXiv: hep-ph/9303287 [hep-ph].
[81] M. Drewes et al. “A White Paper on keV Sterile Neutrino Dark Matter”. In: JCAP
1701.01 (2017), p. 025. arXiv: 1602.04816 [hep-ph].
[82] A. Boyarsky et al. “Sterile neutrino Dark Matter”. In: Prog. Part. Nucl. Phys. 104
(2019), pp. 1–45. arXiv: 1807.07938 [hep-ph].
[83] Kevork Abazajian, George M. Fuller, and Mitesh Patel. “Sterile neutrino hot,
warm, and cold dark matter”. In: Phys. Rev. D64 (2001), p. 023501. arXiv: astro-
ph/0101524 [astro-ph].
[84] Mona Dentler et al. “Updated Global Analysis of Neutrino Oscillations in the
Presence of eV-Scale Sterile Neutrinos”. In: JHEP 08 (2018), p. 010. arXiv: 1803.
10661 [hep-ph].
[85] P. Abratenko et al. Search for an anomalous excess of charged-current νe inter-
actions without pions in the final state with the MicroBooNE experiment. arXiv:
2110.14065 [hep-ex].
[86] R. Shrock. “Decay L0 → νlγ in gauge theories of weak and electromagnetic inter-
actions”. In: Phys. Rev. D 9 (1974), pp. 743–748.
[87] Palash B. Pal and Lincoln Wolfenstein. “Radiative Decays of Massive Neutrinos”.
In: Phys. Rev. D 25 (1982), p. 766.
[88] Kerstin Perez et al. “Almost closing the νMSM sterile neutrino dark matter window
with NuSTAR”. In: Phys. Rev. D95.12 (2017), p. 123002. arXiv: 1609.00667
[astro-ph.HE].
151
REFERENCES
[89] Kevork Abazajian, George M. Fuller, and Wallace H. Tucker. “Direct detection of
warm dark matter in the X-ray”. In: Astrophys. J. 562 (2001), pp. 593–604. arXiv:
astro-ph/0106002.
[90] Alexey Boyarsky et al. “Constraints on sterile neutrino as a dark matter candidate
from the diffuse x-ray background”. In: Mon. Not. Roy. Astron. Soc. 370 (2006),
pp. 213–218. arXiv: astro-ph/0512509.
[91] O. Urban et al. “A Suzaku Search for Dark Matter Emission Lines in the X-
ray Brightest Galaxy Clusters”. In: Mon. Not. Roy. Astron. Soc. 451.3 (2015),
pp. 2447–2461. arXiv: 1411.0050 [astro-ph.CO].
[92] Kenny C. Y. Ng et al. “Improved Limits on Sterile Neutrino Dark Matter using
Full-Sky Fermi Gamma-Ray Burst Monitor Data”. In: Phys. Rev. D 92.4 (2015),
p. 043503. arXiv: 1504.04027 [astro-ph.CO].
[93] Esra Bulbul et al. “Detection of An Unidentified Emission Line in the Stacked
X-ray spectrum of Galaxy Clusters”. In: Astrophys. J. 789 (2014), p. 13. arXiv:
1402.2301 [astro-ph.CO].
[94] Alexey Boyarsky et al. “Unidentified Line in X-Ray Spectra of the Andromeda
Galaxy and Perseus Galaxy Cluster”. In: Phys. Rev. Lett. 113 (2014), p. 251301.
arXiv: 1402.4119 [astro-ph.CO].
[95] Alexey Boyarsky et al. “Checking the Dark Matter Origin of a 3.53 keV Line
with the Milky Way Center”. In: Phys. Rev. Lett. 115 (2015), p. 161301. arXiv:
1408.2503 [astro-ph.CO].
[96] Tesla E. Jeltema and Stefano Profumo. “Discovery of a 3.5 keV line in the Galactic
Centre and a critical look at the origin of the line across astronomical targets”.
In: Mon. Not. Roy. Astron. Soc. 450.2 (2015), pp. 2143–2152. arXiv: 1408.1699
[astro-ph.HE].
[97] A. Boyarsky et al. Comment on the paper ”Dark matter searches going bananas:
the contribution of Potassium (and Chlorine) to the 3.5 keV line” by T. Jeltema
and S. Profumo. arXiv: 1408.4388 [astro-ph.CO].
[98] Esra Bulbul et al. Comment on ”Dark matter searches going bananas: the con-
tribution of Potassium (and Chlorine) to the 3.5 keV line”. arXiv: 1409.4143
[astro-ph.HE].
[99] Aurel Schneider. “Astrophysical constraints on resonantly produced sterile neutrino
dark matter”. In: JCAP 04 (2016), p. 059. arXiv: 1601.07553 [astro-ph.CO].
[100] Xiang-Dong Shi and George M. Fuller. “A New dark matter candidate: Nonthermal
sterile neutrinos”. In: Phys. Rev. Lett. 82 (1999), pp. 2832–2835. arXiv: astro-
ph/9810076 [astro-ph].
[101] L. Wolfenstein. “Neutrino Oscillations in Matter”. In: Phys. Rev. D 17 (1978),
pp. 2369–2374.
[102] S. P. Mikheyev and A. Yu. Smirnov. “Resonance Amplification of Oscillations in
Matter and Spectroscopy of Solar Neutrinos”. In: Sov. J. Nucl. Phys. 42 (1985),
pp. 913–917.
[103] Ernest Ma. “Verifiable radiative seesaw mechanism of neutrino mass and dark
matter”. In: Phys. Rev. D73 (2006), p. 077301. arXiv: hep-ph/0601225 [hep-ph].
[104] Takehiko Asaka, Steve Blanchet, and Mikhail Shaposhnikov. “The nuMSM, dark
matter and neutrino masses”. In: Phys. Lett. B631 (2005), pp. 151–156. arXiv:
hep-ph/0503065 [hep-ph].
152
REFERENCES
[105] Takehiko Asaka and Mikhail Shaposhnikov. “The nuMSM, dark matter and baryon
asymmetry of the universe”. In: Phys. Lett. B620 (2005), pp. 17–26. arXiv: hep-
ph/0505013 [hep-ph].
[106] Louis Lello and Daniel Boyanovsky. “Cosmological Implications of Light Sterile
Neutrinos produced after the QCD Phase Transition”. In: Phys. Rev. D 91 (2015),
p. 063502. arXiv: 1411.2690 [astro-ph.CO].
[107] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov. “On initial conditions for the
Hot Big Bang”. In: JCAP 06 (2009), p. 029. arXiv: 0812.3622 [hep-ph].
[108] Louis Lello, Daniel Boyanovsky, and Robert D. Pisarski. “Production of heavy
sterile neutrinos from vector boson decay at electroweak temperatures”. In: Phys.
Rev. D 95.4 (2017), p. 043524. arXiv: 1609.07647 [hep-ph].
[109] Kalliopi Petraki and Alexander Kusenko. “Dark-matter sterile neutrinos in models
with a gauge singlet in the Higgs sector”. In: Phys. Rev. D 77 (2008), p. 065014.
arXiv: 0711.4646 [hep-ph].
[110] Bibhushan Shakya. “Sterile Neutrino Dark Matter from Freeze-In”. In: Mod. Phys.
Lett. A 31.06 (2016), p. 1630005. arXiv: 1512.02751 [hep-ph].
[111] Johannes König, Alexander Merle, and Maximilian Totzauer. “keV Sterile Neutrino
Dark Matter from Singlet Scalar Decays: The Most General Case”. In: JCAP 1611
(2016), p. 038. arXiv: 1609.01289 [hep-ph].
[112] Adisorn Adulpravitchai and Michael A. Schmidt. “Sterile Neutrino Dark Matter
Production in the Neutrino-phillic Two Higgs Doublet Model”. In: JHEP 12 (2015),
p. 023. arXiv: 1507.05694 [hep-ph].
[113] Marco Drewes and Jin U Kang. “Sterile neutrino Dark Matter production from
scalar decay in a thermal bath”. In: JHEP 05 (2016), p. 051. arXiv: 1510.05646
[hep-ph].
[114] A. D. Dolgov and S. H. Hansen. “Massive sterile neutrinos as warm dark matter”.
In: Astropart. Phys. 16 (2002), pp. 339–344. arXiv: hep-ph/0009083.
[115] J. R. Bond, G. Efstathiou, and J. Silk. “Massive Neutrinos and the Large Scale
Structure of the Universe”. In: Phys. Rev. Lett. 45 (1980). Ed. by M. A. Srednicki,
pp. 1980–1984.
[116] R. J. Crewther et al. “Chiral Estimate of the Electric Dipole Moment of the
Neutron in Quantum Chromodynamics”. In: Phys. Lett. B 88 (1979). [Erratum:
Phys.Lett.B 91, 487 (1980)], p. 123.
[117] Maxim Pospelov and Adam Ritz. “Electric dipole moments as probes of new
physics”. In: Annals Phys. 318 (2005), pp. 119–169. arXiv: hep-ph/0504231.
[118] C. Abel et al. “Measurement of the permanent electric dipole moment of the neu-
tron”. In: Phys. Rev. Lett. 124.8 (2020), p. 081803. arXiv: 2001.11966 [hep-ex].
[119] R. D. Peccei and Helen R. Quinn. “CP Conservation in the Presence of Instantons”.
In: Phys. Rev. Lett. 38 (1977), pp. 1440–1443.
[120] R. D. Peccei and Helen R. Quinn. “Constraints Imposed by CP Conservation in
the Presence of Instantons”. In: Phys. Rev. D 16 (1977), pp. 1791–1797.
[121] R. D. Peccei. “The Strong CP problem and axions”. In: Lect. Notes Phys. 741
(2008). Ed. by Markus Kuster, Georg Raffelt, and Berta Beltran, pp. 3–17. arXiv:
hep-ph/0607268.
[122] Frank Wilczek. “Problem of Strong P and T Invariance in the Presence of Instan-
tons”. In: Phys. Rev. Lett. 40 (1978), pp. 279–282.
153
REFERENCES
[123] Steven Weinberg. “A New Light Boson?” In: Phys. Rev. Lett. 40 (1978), pp. 223–
226.
[124] David J. E. Marsh. “Axion Cosmology”. In: Phys. Rept. 643 (2016), pp. 1–79.
arXiv: 1510.07633 [astro-ph.CO].
[125] Martin Bauer, Matthias Neubert, and Andrea Thamm. “Collider Probes of Axion-
Like Particles”. In: JHEP 12 (2017), p. 044. arXiv: 1708.00443 [hep-ph].
[126] Klaus Ehret et al. “New ALPS Results on Hidden-Sector Lightweights”. In: Phys.
Lett. B 689 (2010), pp. 149–155. arXiv: 1004.1313 [hep-ex].
[127] S. Andriamonje et al. “An Improved limit on the axion-photon coupling from the
CAST experiment”. In: JCAP 04 (2007), p. 010. arXiv: hep-ex/0702006.
[128] E. Aprile et al. “Excess electronic recoil events in XENON1T”. In: Phys. Rev. D
102.7 (2020), p. 072004. arXiv: 2006.09721 [hep-ex].
[129] Christina Gao et al. “Reexamining the Solar Axion Explanation for the XENON1T
Excess”. In: Phys. Rev. Lett. 125.13 (2020), p. 131806. arXiv: 2006 . 14598
[hep-ph].
[130] Luca Di Luzio et al. “Solar axions cannot explain the XENON1T excess”. In: Phys.
Rev. Lett. 125.13 (2020), p. 131804. arXiv: 2006.12487 [hep-ph].
[131] Pierluca Carenza et al. “Constraints on the coupling with photons of heavy axion-
like-particles from Globular Clusters”. In: Phys. Lett. B 809 (2020), p. 135709.
arXiv: 2004.08399 [hep-ph].
[132] P. Sikivie. “Experimental Tests of the Invisible Axion”. In: Phys. Rev. Lett. 51
(1983). Ed. by M. A. Srednicki. [Erratum: Phys.Rev.Lett. 52, 695 (1984)], pp. 1415–
1417.
[133] Renée Hlozek et al. “A search for ultralight axions using precision cosmological
data”. In: Phys. Rev. D 91.10 (2015), p. 103512. arXiv: 1410.2896 [astro-ph.CO].
[134] Vivian Poulin et al. “Cosmological implications of ultralight axionlike fields”. In:
Phys. Rev. D 98.8 (2018), p. 083525. arXiv: 1806.10608 [astro-ph.CO].
[135] Olivier Wantz and E.P.S. Shellard. “Axion Cosmology Revisited”. In: Phys. Rev.
D 82 (2010), p. 123508. arXiv: 0910.1066 [astro-ph.CO].
[136] Paola Arias et al. “WISPy Cold Dark Matter”. In: JCAP 06 (2012), p. 013. arXiv:
1201.5902 [hep-ph].
[137] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the
General Theory of Relativity. New York: John Wiley and Sons, 1972.
[138] P.A. Zyla et al. “Review of Particle Physics”. In: PTEP 2020.8 (2020), p. 083C01.
[139] Edward R. Harrison. “Fluctuations at the threshold of classical cosmology”. In:
Phys. Rev. D 1 (1970), pp. 2726–2730.
[140] Ya. B. Zeldovich. “A Hypothesis, unifying the structure and the entropy of the
universe”. In: Mon. Not. Roy. Astron. Soc. 160 (1972), 1P–3P.
[141] Max Tegmark et al. “The 3-D power spectrum of galaxies from the SDSS”. In:
Astrophys. J. 606 (2004), pp. 702–740. arXiv: astro-ph/0310725.
[142] Héctor Gil-Marın et al. “The clustering of galaxies in the SDSS-III Baryon Os-
cillation Spectroscopic Survey: RSD measurement from the power spectrum and
bispectrum of the DR12 BOSS galaxies”. In: Mon. Not. Roy. Astron. Soc. 465.2
(2017), pp. 1757–1788. arXiv: 1606.00439 [astro-ph.CO].
154
REFERENCES
[143] Chiaki Hikage et al. “Cosmology from cosmic shear power spectra with Subaru
Hyper Suprime-Cam first-year data”. In: Publ. Astron. Soc. Jap. 71.2 (2019), p. 43.
arXiv: 1809.09148 [astro-ph.CO].
[144] M. Gatti et al. “Dark energy survey year 3 results: weak lensing shape catalogue”.
In: Mon. Not. Roy. Astron. Soc. 504.3 (2021), pp. 4312–4336. arXiv: 2011.03408
[astro-ph.CO].
[145] Julien Lesgourgues and Thomas Tram. “The Cosmic Linear Anisotropy Solving
System (CLASS) IV: efficient implementation of non-cold relics”. In: JCAP 1109
(2011), p. 032. arXiv: 1104.2935 [astro-ph.CO].
[146] Julien Lesgourgues. The Cosmic Linear Anisotropy Solving System (CLASS) I:
Overview. arXiv: 1104.2932 [astro-ph.IM].
[147] Max Tegmark and Matias Zaldarriaga. “Separating the early universe from the
late universe: Cosmological parameter estimation beyond the black box”. In: Phys.
Rev. D 66 (2002), p. 103508. arXiv: astro-ph/0207047.
[148] Matteo Viel et al. “Constraining warm dark matter candidates including sterile
neutrinos and light gravitinos with WMAP and the Lyman-alpha forest”. In: Phys.
Rev. D71 (2005), p. 063534. arXiv: astro-ph/0501562 [astro-ph].
[149] Stephane Colombi, Scott Dodelson, and Lawrence M. Widrow. “Large scale struc-
ture tests of warm dark matter”. In: Astrophys. J. 458 (1996), p. 1. arXiv: astro-
ph/9505029.
[150] Takeshi Kobayashi et al. “Lyman-α constraints on ultralight scalar dark mat-
ter: Implications for the early and late universe”. In: Phys. Rev. D 96.12 (2017),
p. 123514. arXiv: 1708.00015 [astro-ph.CO].
[151] Riccardo Murgia et al. “”Non-cold” dark matter at small scales: a general ap-
proach”. In: JCAP 11 (2017), p. 046. arXiv: 1704.07838 [astro-ph.CO].
[152] Alexey Boyarsky et al. “Lyman-alpha constraints on warm and on warm-plus-cold
dark matter models”. In: JCAP 0905 (2009), p. 012. arXiv: 0812.0010 [astro-ph].
[153] Andrew J. Benson et al. “Dark matter halo merger histories beyond cold dark
matter – I. Methods and application to warm dark matter”. In: Monthly Notices
of the Royal Astronomical Society 428.2 (Nov. 2012), pp. 1774–1789.
[154] Aurel Schneider. “Structure formation with suppressed small-scale perturbations”.
In: Mon. Not. Roy. Astron. Soc. 451.3 (2015), pp. 3117–3130. arXiv: 1412.2133
[astro-ph.CO].
[155] Aurel Schneider, Robert E. Smith, and Darren Reed. “Halo Mass Function and
the Free Streaming Scale”. In: Mon. Not. Roy. Astron. Soc. 433 (2013), p. 1573.
arXiv: 1303.0839 [astro-ph.CO].
[156] Anatoly A. Klypin et al. “Where are the missing Galactic satellites?” In: Astrophys.
J. 522 (1999), pp. 82–92. arXiv: astro-ph/9901240.
[157] Ben Moore et al. “Cold collapse and the core catastrophe”. In: Mon. Not. Roy.
Astron. Soc. 310 (1999), pp. 1147–1152. arXiv: astro-ph/9903164.
[158] A.M. Nierenberg et al. “The Cosmic Evolution of Faint Satellite Galaxies as a Test
of Galaxy Formation and the Nature of Dark Matter”. In: Astrophys. J. 772 (2013),
p. 146. arXiv: 1302.3243 [astro-ph.CO].
[159] Daisuke Homma et al. “Searches for New Milky Way Satellites from the First Two
Years of Data of the Subaru/Hyper Suprime-Cam Survey: Discovery of Cetus˜III”.
In: Publ. Astron. Soc. Jap. 70 (2018), p. 18. arXiv: 1704.05977 [astro-ph.GA].
155
REFERENCES
[160] Daisuke Homma et al. “Boötes. IV. A new Milky Way satellite discovered in the
Subaru Hyper Suprime-Cam Survey and implications for the missing satellite prob-
lem”. In: Publ. Astron. Soc. Jap. 71.5 (2019). arXiv: 1906.07332 [astro-ph.GA].
[161] Stacy Y. Kim, Annika H. G. Peter, and Jonathan R. Hargis. “Missing Satellites
Problem: Completeness Corrections to the Number of Satellite Galaxies in the
Milky Way are Consistent with Cold Dark Matter Predictions”. In: Phys. Rev.
Lett. 121.21 (2018), p. 211302. arXiv: 1711.06267 [astro-ph.CO].
[162] William H. Press and Paul Schechter. “Formation of galaxies and clusters of galaxies
by selfsimilar gravitational condensation”. In: Astrophys. J. 187 (1974), pp. 425–
438.
[163] Cedric G. Lacey and Shaun Cole. “Merger rates in hierarchical models of galaxy
formation”. In: Mon. Not. Roy. Astron. Soc. 262 (1993), pp. 627–649.
[164] Emil Polisensky and Massimo Ricotti. “Constraints on the Dark Matter Particle
Mass from the Number of Milky Way Satellites”. In: Phys. Rev. D 83 (2011),
p. 043506. arXiv: 1004.1459 [astro-ph.CO].
[165] Christopher P. Ahn et al. “The Tenth Data Release of the Sloan Digital Sky
Survey: First Spectroscopic Data from the SDSS-III Apache Point Observatory
Galactic Evolution Experiment”. In: Astrophys. J. Suppl. 211 (2014), p. 17. arXiv:
1307.7735 [astro-ph.IM].
[166] Noam I. Libeskind et al. “The Distribution of satellite galaxies: The Great pan-
cake”. In: Mon. Not. Roy. Astron. Soc. 363 (2005), pp. 146–152. arXiv: astro-
ph/0503400.
[167] A. Drlica-Wagner et al. “Milky Way Satellite Census. I. The Observational Selec-
tion Function for Milky Way Satellites in DES Y3 and Pan-STARRS DR1”. In:
Astrophys. J. 893 (2020), p. 1. arXiv: 1912.03302 [astro-ph.GA].
[168] Oliver Newton et al. “Constraints on the properties of warm dark matter using
the satellite galaxies of the Milky Way”. In: JCAP 08 (2021), p. 062. arXiv:
2011.08865 [astro-ph.CO].
[169] Oliver Newton et al. “The total satellite population of the Milky Way”. In: Monthly
Notices of the Royal Astronomical Society 479.3 (May 2018), pp. 2853–2870. eprint:
https://academic.oup.com/mnras/article- pdf/479/3/2853/25149561/
sty1085.pdf.
[170] Inc. Wolfram Research. Mathematica, Version 11+12. Champaign, IL, 2021.
[171] Guido Van Rossum and Fred L Drake Jr. Python reference manual. Centrum voor
Wiskunde en Informatica Amsterdam, 1995.
[172] Guido Van Rossum and Fred L. Drake. Python 3 Reference Manual. Scotts Valley,
CA: CreateSpace, 2009.
[173] Charles R. Harris et al. “Array programming with NumPy”. In: Nature 585.7825
(2020), pp. 357–362. arXiv: 2006.10256 [cs.MS].
[174] John D. Hunter. “Matplotlib: A 2D Graphics Environment”. In: Comput. Sci. Eng.
9.3 (2007), pp. 90–95.
[175] Aaron Meurer et al. “SymPy: symbolic computing in Python”. In: PeerJ Comput.
Sci. 3 (2017), e103.
[176] Pauli Virtanen et al. “SciPy 1.0–Fundamental Algorithms for Scientific Computing
in Python”. In: Nature Meth. 17 (2020), p. 261. arXiv: 1907.10121 [cs.MS].
156
REFERENCES
[177] Fernando Perez and Brian E. Granger. “IPython: A System for Interactive Scientific
Computing”. In: Comput. Sci. Eng. 9.3 (2007), pp. 21–29.
[178] Joshua Ellis. “TikZ-Feynman: Feynman diagrams with TikZ”. In: Comput. Phys.
Commun. 210 (2017), pp. 103–123. arXiv: 1601.05437 [hep-ph].
[179] Wes McKinney. “Data Structures for Statistical Computing in Python”. In: Pro-
ceedings of the 9th Python in Science Conference. Ed. by Stéfan van der Walt and
Jarrod Millman. 2010, pp. 56–61.
[180] Rabindra N. Mohapatra and Goran Senjanovi ć. “Neutrino Mass and Spontaneous
Parity Nonconservation”. In: Phys. Rev. Lett. 44 (14 Apr. 1980), pp. 912–915.
[181] Murray Gell-Mann, Pierre Ramond, and Richard Slansky. “Complex Spinors and
Unified Theories”. In: Conf. Proc. C790927 (1979), pp. 315–321. arXiv: 1306.4669
[hep-th].
[182] Emiliano Molinaro, Carlos E. Yaguna, and Oscar Zapata. “FIMP realization of the
scotogenic model”. In: JCAP 1407 (2014), p. 015. arXiv: 1405.1259 [hep-ph].
[183] Manfred Lindner et al. “Fermionic WIMPs and vacuum stability in the scotogenic
model”. In: Phys. Rev. D94.11 (2016), p. 115027. arXiv: 1608.00577 [hep-ph].
[184] Kresimir Kumericki, Ivica Picek, and Branimir Radovcic. “TeV-scale Seesaw with
Quintuplet Fermions”. In: Phys. Rev. D86 (2012), p. 013006. arXiv: 1204.6599
[hep-ph].
[185] Vedran Brdar, Ivica Picek, and Branimir Radovcic. “Radiative Neutrino Mass
with Scotogenic Scalar Triplet”. In: Phys. Lett. B728 (2014), pp. 198–201. arXiv:
1310.3183 [hep-ph].
[186] Ivania M. Ávila et al. “Phenomenology of scotogenic scalar dark matter”. In: Eur.
Phys. J. C 80.10 (2020), p. 908. arXiv: 1910.08422 [hep-ph].
[187] Shoichi Kashiwase and Daijiro Suematsu. “Baryon number asymmetry and dark
matter in the neutrino mass model with an inert doublet”. In: Phys. Rev. D86
(2012), p. 053001. arXiv: 1207.2594 [hep-ph].
[188] Daijiro Suematsu. “Thermal Leptogenesis in a TeV Scale Model for Neutrino
Masses”. In: Eur. Phys. J. C72 (2012), p. 1951. arXiv: 1103.0857 [hep-ph].
[189] Thomas Hugle, Moritz Platscher, and Kai Schmitz. “Low-Scale Leptogenesis in the
Scotogenic Neutrino Mass Model”. In: Phys. Rev. D 98.2 (2018), p. 023020. arXiv:
1804.09660 [hep-ph].
[190] J. Racker. “Mass bounds for baryogenesis from particle decays and the inert doublet
model”. In: JCAP 1403 (2014), p. 025. arXiv: 1308.1840 [hep-ph].
[191] Devabrat Mahanta and Debasish Borah. “Fermion Dark Matter with N2 Lepto-
genesis in Minimal Scotogenic Model”. In: JCAP 1911.11 (2019), p. 021. arXiv:
1906.03577 [hep-ph].
[192] Debasish Borah, P. S. Bhupal Dev, and Abhass Kumar. “TeV scale leptogenesis,
inflaton dark matter and neutrino mass in a scotogenic model”. In: Phys. Rev.
D99.5 (2019), p. 055012. arXiv: 1810.03645 [hep-ph].
[193] Devabrat Mahanta and Debasish Borah. “TeV Scale Leptogenesis with Dark Matter
in Non-standard Cosmology”. In: JCAP 04.04 (2020), p. 032. arXiv: 1912.09726
[hep-ph].
[194] Evgeny K. Akhmedov, V. A. Rubakov, and A. Yu. Smirnov. “Baryogenesis via
neutrino oscillations”. In: Phys. Rev. Lett. 81 (1998), pp. 1359–1362. arXiv: hep-
ph/9803255 [hep-ph].
157
REFERENCES
[195] Thomas Hambye and Daniele Teresi. “Higgs doublet decay as the origin of the
baryon asymmetry”. In: Phys. Rev. Lett. 117.9 (2016), p. 091801. arXiv: 1606.
00017 [hep-ph].
[196] Admir Greljo and Ben A. Stefanek. “Third family quark–lepton unification at
the TeV scale”. In: Phys. Lett. B 782 (2018), pp. 131–138. arXiv: 1802.04274
[hep-ph].
[197] Dario Buttazzo et al. “B-physics anomalies: a guide to combined explanations”.
In: JHEP 11 (2017), p. 044. arXiv: 1706.07808 [hep-ph].
[198] Roel Aaij et al. “Search for lepton-universality violation in B+ → K+`+`− decays”.
In: Phys. Rev. Lett. 122.19 (2019), p. 191801. arXiv: 1903.09252 [hep-ex].
[199] A. Abdesselam et al. “Test of Lepton-Flavor Universality in B → K∗`+`− Decays at
Belle”. In: Phys. Rev. Lett. 126.16 (2021), p. 161801. arXiv: 1904.02440 [hep-ex].
[200] A. Abdesselam et al. Measurement of R(D) and R(D∗) with a semileptonic tagging
method. arXiv: 1904.08794 [hep-ex].
[201] R. N. Mohapatra and J. W. F. Valle. “Neutrino mass and baryon-number noncon-
servation in superstring models”. In: Phys. Rev. D 34 (5 Sept. 1986), p. 1642.
[202] Rabindra N. Mohapatra. “Mechanism for understanding small neutrino mass in
superstring theories”. In: Phys. Rev. Lett. 56 (6 Feb. 1986), p. 561.
[203] Asmaa Abada et al. “Neutrino masses, leptogenesis and dark matter from small lep-
ton number violation?” In: JCAP 12 (2017), p. 024. arXiv: 1709.00415 [hep-ph].
[204] Masahiro Kawasaki et al. “Revisiting Big-Bang Nucleosynthesis Constraints on
Long-Lived Decaying Particles”. In: Phys. Rev. D97.2 (2018), p. 023502. arXiv:
1709.01211 [hep-ph].
[205] G. C. Branco et al. “Theory and phenomenology of two-Higgs-doublet models”.
In: Phys. Rept. 516 (2012), pp. 1–102. arXiv: 1106.0034 [hep-ph].
[206] Takashi Toma and Avelino Vicente. “Lepton Flavor Violation in the Scotogenic
Model”. In: JHEP 01 (2014), p. 160. arXiv: 1312.2840.
[207] Alexander Merle and Moritz Platscher. “Running of radiative neutrino masses: the
scotogenic model — revisited”. In: JHEP 11 (2015), p. 148. arXiv: 1507.06314
[hep-ph].
[208] J. A. Casas and A. Ibarra. “Oscillating neutrinos and muon —¿ e, gamma”. In:
Nucl. Phys. B618 (2001), pp. 171–204. arXiv: hep-ph/0103065 [hep-ph].
[209] C. Patrignani et al. “Review of Particle Physics”. In: Chin. Phys. C40.10 (2016),
p. 100001.
[210] Ivan Esteban et al. “Global analysis of three-flavour neutrino oscillations: synergies
and tensions in the determination of θ23, δCP , and the mass ordering”. In: JHEP
01 (2019), p. 106. arXiv: 1811.05487 [hep-ph].
[211] Amine Ahriche, Adil Jueid, and Salah Nasri. “Radiative neutrino mass and Ma-
jorana dark matter within an inert Higgs doublet model”. In: Phys. Rev. D97.9
(2018), p. 095012. arXiv: 1710.03824 [hep-ph].
[212] Andre G. Hessler et al. “Probing the scotogenic FIMP at the LHC”. In: JHEP 01
(2017), p. 100. arXiv: 1611.09540 [hep-ph].
[213] M. Aaboud et al. “Search for electroweak√production of supersymmetric particles
in final states with two or three leptons at s = 13 TeV with the ATLAS detector”.
In: Eur. Phys. J. C78.12 (2018), p. 995. arXiv: 1803.02762 [hep-ex].
158
REFERENCES
[214] Morad Aaboud et al. “Search for the√direct production of charginos and neutralinos
in final states with tau leptons in s = 13 TeV pp collisions with the ATLAS
detector”. In: Eur. Phys. J. C78.2 (2018), p. 154. arXiv: 1708.07875 [hep-ex].
[215] Georges Aad et al. “Search for electroweak production of charginos and sleptons
√decaying into final states with two leptons and missing transverse momentum in
s = 13 TeV pp collisions using the ATLAS detector”. In: Eur. Phys. J. C 80.2
(2020), p. 123. arXiv: 1908.08215 [hep-ex].
[216] Adam Alloul et al. “FeynRules 2.0 - A complete toolbox for tree-level phenomenol-
ogy”. In: Comput. Phys. Commun. 185 (2014), pp. 2250–2300. arXiv: 1310.1921
[hep-ph].
[217] J. Alwall et al. “The automated computation of tree-level and next-to-leading order
differential cross sections, and their matching to parton shower simulations”. In:
JHEP 07 (2014), p. 079. arXiv: 1405.0301 [hep-ph].
[218] Torbjorn Sjostrand, Stephen Mrenna, and Peter Z. Skands. “A Brief Introduction
to PYTHIA 8.1”. In: Comput. Phys. Commun. 178 (2008), pp. 852–867. arXiv:
0710.3820 [hep-ph].
[219] J. de Favereau et al. “DELPHES 3, A modular framework for fast simulation of
a generic collider experiment”. In: JHEP 02 (2014), p. 057. arXiv: 1307.6346
[hep-ex].
[220] C. G. Lester and D. J. Summers. “Measuring masses of semiinvisibly decaying
particles pair produced at hadron colliders”. In: Phys. Lett. B 463 (1999), pp. 99–
103. arXiv: hep-ph/9906349.
[221] Alan Barr, Christopher Lester, and P. Stephens. “m(T2): The Truth behind the
glamour”. In: J. Phys. G29 (2003), pp. 2343–2363. arXiv: hep - ph / 0304226
[hep-ph].
[222] G. Apollinari et al. “High Luminosity Large Hadron Collider HL-LHC”. In: CERN
Yellow Rep. 5 (2015), pp. 1–19. arXiv: 1705.08830 [physics.acc-ph].
[223] Glen Cowan et al. “Asymptotic formulae for likelihood-based tests of new physics”.
In: Eur. Phys. J. C71 (2011). [Erratum: Eur. Phys. J.C73,2501(2013)], p. 1554.
arXiv: 1007.1727 [physics.data-an].
[224] A. Abada et al. “FCC-hh: The Hadron Collider”. In: Eur. Phys. J. ST 228.4
(2019), pp. 755–1107.
[225] Stefania Gori et al. “Prospects for Electroweakino Discovery at a 100 TeV Hadron
Collider”. In: JHEP 12 (2014), p. 108. arXiv: 1410.6287 [hep-ph].
[226] J. de Blas et al. The CLIC Potential for New Physics. arXiv: 1812.02093 [hep-ph].
[227] M J Boland et al. Updated baseline for a staged Compact Linear Collider. Ed. by
P Lebrun et al. arXiv: 1608.07537 [physics.acc-ph].
[228] Dominik Arominski et al. A detector for CLIC: main parameters and performance.
arXiv: 1812.07337 [physics.ins-det].
[229] Alexander Kusenko. “Sterile neutrinos, dark matter, and the pulsar velocities in
models with a Higgs singlet”. In: Phys. Rev. Lett. 97 (2006), p. 241301. arXiv:
hep-ph/0609081.
[230] D. Barducci et al. “Collider limits on new physics within micrOMEGAs 4.3”. In:
Comput. Phys. Commun. 222 (2018), pp. 327–338. arXiv: 1606.03834 [hep-ph].
159
REFERENCES
[231] Christophe Yèche et al. “Constraints on neutrino masses from Lyman-alpha forest
power spectrum with BOSS and XQ-100”. In: JCAP 1706.06 (2017), p. 047. arXiv:
1702.03314 [astro-ph.CO].
[232] Girish Kulkarni et al. “Characterizing the Pressure Smoothing Scale of the In-
tergalactic Medium”. In: Astrophys. J. 812 (2015), p. 30. arXiv: 1504.00366
[astro-ph.CO].
[233] Alexander Merle, Aurel Schneider, and Maximilian Totzauer. “Dodelson-Widrow
Production of Sterile Neutrino Dark Matter with Non-Trivial Initial Abundance”.
In: JCAP 1604.04 (2016), p. 003. arXiv: 1512.05369 [hep-ph].
[234] Julian Heeck and Daniele Teresi. “Cold keV dark matter from decays and scatter-
ings”. In: Phys. Rev. D96.3 (2017), p. 035018. arXiv: 1706.09909 [hep-ph].
[235] Alexander Merle and Maximilian Totzauer. “keV Sterile Neutrino Dark Matter
from Singlet Scalar Decays: Basic Concepts and Subtle Features”. In: JCAP 1506
(2015), p. 011. arXiv: 1502.01011 [hep-ph].
[236] Pablo F. de Salas and Sergio Pastor. “Relic neutrino decoupling with flavour oscil-
lations revisited”. In: JCAP 07 (2016), p. 051. arXiv: 1606.06986 [hep-ph].
[237] Cyril Pitrou et al. “Precision big bang nucleosynthesis with improved Helium-4 pre-
dictions”. In: Phys. Rept. 754 (2018), pp. 1–66. arXiv: 1801.08023 [astro-ph.CO].
[238] A. M. Baldini et al. “The design of the MEG II experiment”. In: Eur. Phys. J.
C78.5 (2018), p. 380. arXiv: 1801.04688 [physics.ins-det].
[239] Kevork Abazajian et al. CMB-S4 Science Case, Reference Design, and Project
Plan. arXiv: 1907.04473 [astro-ph.IM].
[240] Giorgio Arcadi et al. “The waning of the WIMP? A review of models, searches, and
constraints”. In: Eur. Phys. J. C78.3 (2018), p. 203. arXiv: 1703.07364 [hep-ph].
[241] Alexander Merle, Viviana Niro, and Daniel Schmidt. “New Production Mechanism
for keV Sterile Neutrino Dark Matter by Decays of Frozen-In Scalars”. In: JCAP
03 (2014), p. 028. arXiv: 1306.3996 [hep-ph].
[242] Vedran Brdar et al. “X-Ray Lines from Dark Matter Annihilation at the keV
Scale”. In: Phys. Rev. Lett. 120.6 (2018), p. 061301. arXiv: 1710.02146 [hep-ph].
[243] André De Gouvêa et al. “Dodelson-Widrow Mechanism in the Presence of Self-
Interacting Neutrinos”. In: Phys. Rev. Lett. 124.8 (2020), p. 081802. arXiv: 1910.
04901 [hep-ph].
[244] Jae-Weon Lee. “Brief History of Ultra-light Scalar Dark Matter Models”. In:
EPJ Web Conf. 168 (2018). Ed. by B. Gwak et al., p. 06005. arXiv: 1704.05057
[astro-ph.CO].
[245] Lam Hui et al. “Ultralight scalars as cosmological dark matter”. In: Phys. Rev. D
95.4 (2017), p. 043541. arXiv: 1610.08297 [astro-ph.CO].
[246] Javier Redondo and Marieke Postma. “Massive hidden photons as lukewarm dark
matter”. In: JCAP 02 (2009), p. 005. arXiv: 0811.0326 [hep-ph].
[247] Thomas Hambye et al. “Dark matter from dark photons: a taxonomy of dark
matter production”. In: Phys. Rev. D 100.9 (2019), p. 095018. arXiv: 1908.09864
[hep-ph].
[248] Joerg Jaeckel, Javier Redondo, and Andreas Ringwald. “3.55 keV hint for decaying
axionlike particle dark matter”. In: Phys. Rev. D 89 (2014), p. 103511. arXiv:
1402.7335 [hep-ph].
160
REFERENCES
[249] Sang Hui Im and Kwang Sik Jeong. “Freeze-in Axion-like Dark Matter”. In: Phys.
Lett. B 799 (2019), p. 135044. arXiv: 1907.07383 [hep-ph].
[250] Jonathan L. Ouellet et al. “First Results from ABRACADABRA-10 cm: A Search
for Sub-µeV Axion Dark Matter”. In: Phys. Rev. Lett. 122.12 (2019), p. 121802.
arXiv: 1810.12257 [hep-ex].
[251] David J. E. Marsh and Joe Silk. “A Model For Halo Formation With Axion Mixed
Dark Matter”. In: Mon. Not. Roy. Astron. Soc. 437.3 (2014), pp. 2652–2663. arXiv:
1307.1705 [astro-ph.CO].
[252] Nayara Fonseca and Enrico Morgante. “Relaxion Dark Matter”. In: Phys. Rev. D
100.5 (2019), p. 055010. arXiv: 1809.04534 [hep-ph].
[253] Nayara Fonseca and Enrico Morgante. “Probing photophobic axion and relaxion
dark matter”. In: Phys. Rev. D 103.1 (2021), p. 015011. arXiv: 2009.10974
[hep-ph].
[254] Abhishek Banerjee, Hyungjin Kim, and Gilad Perez. “Coherent relaxion dark mat-
ter”. In: Phys. Rev. D 100.11 (2019), p. 115026. arXiv: 1810.01889 [hep-ph].
[255] Jonathan L. Feng et al. “Axionlike particles at FASER: The LHC as a photon beam
dump”. In: Phys. Rev. D 98.5 (2018), p. 055021. arXiv: 1806.02348 [hep-ph].
[256] Matthew J. Dolan et al. “Revised constraints and Belle II sensitivity for visible
and invisible axion-like particles”. In: JHEP 12 (2017), p. 094. arXiv: 1709.00009
[hep-ph].
[257] Sergey Alekhin et al. “A facility to Search for Hidden Particles at the CERN SPS:
the SHiP physics case”. In: Rept. Prog. Phys. 79.12 (2016), p. 124201. arXiv:
1504.04855 [hep-ph].
[258] Vedran Brdar et al. “Axionlike Particles at Future Neutrino Experiments: Closing
the Cosmological Triangle”. In: Phys. Rev. Lett. 126.20 (2021), p. 201801. arXiv:
2011.07054 [hep-ph].
[259] Alberto Salvio, Alessandro Strumia, and Wei Xue. “Thermal axion production”.
In: JCAP 01 (2014), p. 011. arXiv: 1310.6982 [hep-ph].
[260] Davide Cadamuro and Javier Redondo. “Cosmological bounds on pseudo Nambu-
Goldstone bosons”. In: JCAP 02 (2012), p. 032. arXiv: 1110.2895 [hep-ph].
[261] P. A. Zyla et al. “Review of Particle Physics”. In: PTEP 2020.8 (2020), p. 083C01.
[262] Eric Braaten and Tzu Chiang Yuan. “Calculation of screening in a hot plasma”.
In: Phys. Rev. Lett. 66 (1991), pp. 2183–2186.
[263] M. Bolz, A. Brandenburg, and W. Buchmuller. “Thermal production of gravitinos”.
In: Nucl. Phys. B 606 (2001). [Erratum: Nucl.Phys.B 790, 336–337 (2008)], pp. 518–
544. arXiv: hep-ph/0012052.
[264] Kyu Jung Bae et al. “Light axinos from freeze-in: production processes, phase space
distributions, and Ly-α forest constraints”. In: JCAP 01 (2018), p. 054. arXiv:
1707.06418 [hep-ph].
[265] A. Garzilli et al. How warm is too warm? Towards robust Lyman-α forest bounds
on warm dark matter. arXiv: 1912.09397 [astro-ph.CO].
[266] Jen-Wei Hsueh et al. “SHARP – VII. New constraints on the dark matter
free-streaming properties and substructure abundance from gravitationally lensed
quasars”. In: Mon. Not. Roy. Astron. Soc. 492.2 (2020), pp. 3047–3059. arXiv:
1905.04182 [astro-ph.CO].
161
REFERENCES
[267] Wenting Wang et al. “Estimating the dark matter halo mass of our Milky Way
using dynamical tracers”. In: Mon. Not. Roy. Astron. Soc. 453.1 (2015), pp. 377–
400. arXiv: 1502.03477 [astro-ph.GA].
[268] Thomas Callingham et al. The mass of the Milky Way from satellite dynamics.
arXiv: 1808.10456 [astro-ph.GA].
[269] Marius Cautun et al. “The Milky Way total mass profile as inferred from Gaia
DR2”. In: Mon. Not. Roy. Astron. Soc. 494.3 (2020), pp. 4291–4313. arXiv: 1911.
04557 [astro-ph.GA].
[270] E.V. Karukes et al. “A robust estimate of the Milky Way mass from rotation curve
data”. In: JCAP 05 (2020), p. 033. arXiv: 1912.04296 [astro-ph.GA].
[271] Wenting Wang et al. “The mass of our Milky Way”. In: Sci. China Phys. Mech.
Astron. 63.10 (2020), p. 109801. arXiv: 1912.02599 [astro-ph.GA].
[272] G.R. Dvali. Removing the cosmological bound on the axion scale. arXiv: hep-
ph/9505253.
[273] Raymond T. Co, Eric Gonzalez, and Keisuke Harigaya. “Axion Misalignment
Driven to the Bottom”. In: JHEP 05 (2019), p. 162. arXiv: 1812.11186 [hep-ph].
[274] Maxim Pospelov, Adam Ritz, and Mikhail B. Voloshin. “Bosonic super-WIMPs as
keV-scale dark matter”. In: Phys. Rev. D 78 (2008), p. 115012. arXiv: 0807.3279
[hep-ph].
[275] Alexey Boyarsky et al. “Constraining DM properties with SPI”. In: Mon. Not.
Roy. Astron. Soc. 387 (2008), p. 1345. arXiv: 0710.4922 [astro-ph].
[276] Brandon M. Roach et al. “NuSTAR Tests of Sterile-Neutrino Dark Matter: New
Galactic Bulge Observations and Combined Impact”. In: Phys. Rev. D 101.10
(2020), p. 103011. arXiv: 1908.09037 [astro-ph.HE].
[277] Shunsaku Horiuchi et al. “Sterile neutrino dark matter bounds from galaxies of
the Local Group”. In: Phys. Rev. D89.2 (2014), p. 025017. arXiv: 1311.0282
[astro-ph.CO].
[278] Y. Akrami et al. “Planck 2018 results. X. Constraints on inflation”. In: Astron.
Astrophys. 641 (2020), A10. arXiv: 1807.06211 [astro-ph.CO].
[279] L. Amati et al. “The THESEUS space mission concept: science case, design and
expected performances”. In: Adv. Space Res. 62 (2018), pp. 191–244. arXiv: 1710.
04638 [astro-ph.IM].
[280] E. O. Nadler et al. “Milky Way Satellite Census. III. Constraints on Dark Matter
Properties from Observations of Milky Way Satellite Galaxies”. In: Phys. Rev. Lett.
126 (2021), p. 091101. arXiv: 2008.00022 [astro-ph.CO].
[281] Francesco Capozzi and Georg Raffelt. “Axion and neutrino bounds improved with
new calibrations of the tip of the red-giant branch using geometric distance de-
terminations”. In: Phys. Rev. D 102.8 (2020), p. 083007. arXiv: 2007.03694
[astro-ph.SR].
[282] Nicolás Viaux et al. “Neutrino and axion bounds from the globular cluster M5
(NGC 5904)”. In: Phys. Rev. Lett. 111 (2013), p. 231301. arXiv: 1311.1669
[astro-ph.SR].
[283] Cristina Benso et al. “Prospects for Finding Sterile Neutrino Dark Matter at KA-
TRIN”. In: Phys. Rev. D 100.11 (2019), p. 115035. arXiv: 1911.00328 [hep-ph].
[284] Alex Drlica-Wagner et al. Probing the Fundamental Nature of Dark Matter with
the Large Synoptic Survey Telescope. arXiv: 1902.01055 [astro-ph.CO].
162
REFERENCES
[285] Cora Dvorkin, Tongyan Lin, and Katelin Schutz. “Cosmology of Sub-MeV Dark
Matter Freeze-In”. In: Phys. Rev. Lett. 127.11 (2021), p. 111301. arXiv: 2011.08186
[astro-ph.CO].
[286] M. A. Luty. “Baryogenesis via leptogenesis”. In: Phys. Rev. D 45 (1992), pp. 455–
465.
[287] Simon Knapen, Tongyan Lin, and Kathryn M. Zurek. “Light Dark Matter: Models
and Constraints”. In: Phys. Rev. D 96.11 (2017), p. 115021. arXiv: 1709.07882
[hep-ph].
[288] Paul Bode, Jeremiah P. Ostriker, and Neil Turok. “Halo formation in warm dark
matter models”. In: Astrophys. J. 556 (2001), pp. 93–107. arXiv: astro- ph/
0010389.
[289] James S. Bullock and Michael Boylan-Kolchin. “Small-Scale Challenges to the
ΛCDM Paradigm”. In: Ann. Rev. Astron. Astrophys. 55 (2017), pp. 343–387. arXiv:
1707.04256 [astro-ph.CO].
[290] B. Moore et al. “Dark matter substructure within galactic halos”. In: Astrophys.
J. Lett. 524 (1999), pp. L19–L22. arXiv: astro-ph/9907411.
[291] W. J. G. de Blok. “The Core-Cusp Problem”. In: Advances in Astronomy 2010
(2010), pp. 1–14.
[292] Michael Boylan-Kolchin, James S. Bullock, and Manoj Kaplinghat. “Too big to
fail? The puzzling darkness of massive Milky Way subhaloes”. In: Monthly Notices
of the Royal Astronomical Society: Letters 415.1 (June 2011), pp. L40–L44.
[293] Michael Boylan-Kolchin, James S. Bullock, and Manoj Kaplinghat. “The Milky
Way’s bright satellites as an apparent failure of ΛCDM”. In: Monthly Notices of
the Royal Astronomical Society 422.2 (Mar. 2012), pp. 1203–1218.
[294] Adi Zolotov et al. “BARYONS MATTER: WHY LUMINOUS SATELLITE
GALAXIES HAVE REDUCED CENTRAL MASSES”. In: The Astrophysical Jour-
nal 761.1 (Nov. 2012), p. 71.
[295] Aaron A. Dutton et al. “NIHAO V: too big does not fail – reconciling the conflict
between ΛCDM predictions and the circular velocities of nearby field galaxies”.
In: Mon. Not. Roy. Astron. Soc. 457.1 (2016), pp. L74–L78. arXiv: 1512.00453
[astro-ph.GA].
[296] Mark R. Lovell et al. “Addressing the too big to fail problem with baryon physics
and sterile neutrino dark matter”. In: Mon. Not. Roy. Astron. Soc. 468.3 (2017),
pp. 2836–2849. arXiv: 1611.00005 [astro-ph.GA].
[297] Wayne Hu, Rennan Barkana, and Andrei Gruzinov. “Cold and fuzzy dark matter”.
In: Phys. Rev. Lett. 85 (2000), pp. 1158–1161. arXiv: astro-ph/0003365.
[298] Bodo Schwabe et al. “Simulating mixed fuzzy and cold dark matter”. In: Phys.
Rev. D 102.8 (2020), p. 083518. arXiv: 2007.08256 [astro-ph.CO].
[299] Roberta Diamanti et al. “Cold dark matter plus not-so-clumpy dark relics”. In:
JCAP 06 (2017), p. 008. arXiv: 1701.03128 [astro-ph.CO].
[300] D Anderhalden et al. “The galactic halo in mixed dark matter cosmologies”. In:
Journal of Cosmology and Astroparticle Physics 2012.10 (Oct. 2012), pp. 047–047.
[301] Marc Davis, F.J. Summers, and David Schlegel. “Large scale structure in a universe
with mixed hot and cold dark matter”. In: Nature 359 (1992), pp. 393–396.
[302] Anatoly Klypin et al. “Structure formation with cold plus hot dark matter”. In:
Astrophys. J. 416 (1993), pp. 1–16. arXiv: astro-ph/9305011.
163
REFERENCES
[303] Anatoly Klypin et al. “Damped Lyman alpha systems versus Cold + Hot Dark
Matter”. In: Astrophys. J. 444 (1995), p. 1. arXiv: astro-ph/9405003.
[304] Chung-Pei Ma and Edmund Bertschinger. “Do galactic systems form too late in
cold + hot dark matter models?” In: Astrophys. J. Lett. 434 (1994), p. L5. arXiv:
astro-ph/9407085.
[305] Scott Dodelson, Evalyn Gates, and Albert Stebbins. “Cold + hot dark matter and
the cosmic microwave background”. In: Astrophys. J. 467 (1996), pp. 10–18. arXiv:
astro-ph/9509147.
[306] Dan Hooper, Farinaldo S. Queiroz, and Nickolay Y. Gnedin. “Non-Thermal Dark
Matter Mimicking An Additional Neutrino Species In The Early Universe”. In:
Phys. Rev. D 85 (2012), p. 063513. arXiv: 1111.6599 [astro-ph.CO].
[307] Mathias Garny and Jan Heisig. “Interplay of super-WIMP and freeze-in production
of dark matter”. In: Phys. Rev. D 98.9 (2018), p. 095031. arXiv: 1809.10135
[hep-ph].
[308] Jonathan L. Feng, Arvind Rajaraman, and Fumihiro Takayama. “SuperWIMP dark
matter signals from the early universe”. In: Phys. Rev. D 68 (2003), p. 063504.
arXiv: hep-ph/0306024.
[309] Quentin Decant et al. Lyman-α constraints on freeze-in and superWIMPs. arXiv:
2111.09321 [astro-ph.CO].
[310] G. Parimbelli et al. Mixed dark matter: matter power spectrum and halo mass
function. arXiv: 2106.04588 [astro-ph.CO].
[311] Wayne Hu, Daniel J. Eisenstein, and Max Tegmark. “Weighing neutrinos with
galaxy surveys”. In: Phys. Rev. Lett. 80 (1998), pp. 5255–5258. arXiv: astro-
ph/9712057.
[312] Francesco D’Eramo and Alessandro Lenoci. Lower Mass Bounds on FIMPs. arXiv:
2012.01446 [hep-ph].
[313] G. Monari et al. “The escape speed curve of the Galaxy obtained from Gaia DR2
implies a heavy Milky Way”. In: Astronomy & Astrophysics 616 (Aug. 2018), p. L9.
[314] Alis J Deason et al. “The local high-velocity tail and the Galactic escape speed”.
In: Monthly Notices of the Royal Astronomical Society 485.3 (Mar. 2019), pp. 3514–
3526.
[315] Robert J J Grand et al. “The effects of dynamical substructure on Milky Way mass
estimates from the high-velocity tail of the local stellar halo”. In: Monthly Notices
of the Royal Astronomical Society: Letters 487.1 (June 2019), pp. L72–L76.
[316] Laura L. Watkins et al. “Evidence for an Intermediate-mass Milky Way from Gaia
DR2 Halo Globular Cluster Motions”. In: The Astrophysical Journal 873.2 (Mar.
2019), p. 118.
[317] T K Fritz et al. “The mass of our Galaxy from satellite proper motions in the Gaia
era”. In: Monthly Notices of the Royal Astronomical Society 494.4 (Apr. 2020),
pp. 5178–5193.
[318] Lorenzo Posti and Amina Helmi. “Mass and shape of the Milky Way’s dark matter
halo with globular clusters from Gaia and Hubble”. In: Astronomy & Astrophysics
621 (Jan. 2019), A56.
[319] Eugene Vasiliev. “Proper motions and dynamics of the Milky Way globular cluster
system from Gaia DR2”. In: Monthly Notices of the Royal Astronomical Society
484.2 (Jan. 2019), pp. 2832–2850.
164
REFERENCES
[320] Gwendolyn Eadie and Mario Jurić. “The Cumulative Mass Profile of the Milky
Way as Determined by Globular Cluster Kinematics from Gaia DR2”. In: The
Astrophysical Journal 875.2 (Apr. 2019), p. 159.
[321] Zhao-Zhou Li et al. “Constraining the Milky Way Mass Profile with Phase-space
Distribution of Satellite Galaxies”. In: The Astrophysical Journal 894.1 (Apr. 2020),
p. 10.
[322] Roger Barlow. “Asymmetric statistical errors”. In: Statistical Problems in Particle
Physics, Astrophysics and Cosmology. June 2004, pp. 56–59. arXiv: physics/
0406120.
[323] Lina Necib and Tongyan Lin. Substructure at High Speed II: The Local Escape
Velocity and Milky Way Mass with Gaia DR2. arXiv: 2102.02211 [astro-ph.GA].
[324] Mark R. Lovell et al. “The properties of warm dark matter haloes”. In: Mon. Not.
Roy. Astron. Soc. 439 (2014), pp. 300–317. arXiv: 1308.1399 [astro-ph.CO].
[325] Jie Wang et al. “The missing massive satellites of the Milky Way”. In: Monthly
Notices of the Royal Astronomical Society 424.4 (July 2012), pp. 2715–2721.
[326] Marco Hufnagel, Kai Schmidt-Hoberg, and Sebastian Wild. “BBN constraints on
MeV-scale dark sectors. Part I. Sterile decays”. In: JCAP 02 (2018), p. 044. arXiv:
1712.03972 [hep-ph].
[327] Paul Frederik Depta, Marco Hufnagel, and Kai Schmidt-Hoberg. “Updated BBN
constraints on electromagnetic decays of MeV-scale particles”. In: JCAP 04 (2021),
p. 011. arXiv: 2011.06519 [hep-ph].
165
Lebenslauf
Sven Baumholzer
Geburtsdatum: 28. Juni 1993
Anschrift: Westring 257, 55120 Mainz
Telefon (privat): 0170/3303763
E­Mail: svenbaumholzer@gmx.net
Staatsangehörigkeit: deutsch
Ausbildung
06/18 ­ heute Forschender Doktorand, Fachrichtung Physik
Johannes Gutenberg­Universität, Mainz
Titel: “Searching for warm dark matter down on Earth and among the stars”
Betreuer: Prof. Dr. Pedro Schwaller, Bewertung: magna cum laude (1.0)
Abgabe: 30.11.2021
Prüfung: 18.03.2022
04/16 ­ 04/18 Master of Science, Fachrichtung Physik
Johannes Gutenberg­Universität, Mainz
Masterarbeit: “New production mechanism for sterile Neutrinos”,
Betreuer: Prof. Dr. Pedro Schwaller, Gesamtnote: 1.0
10/12 ­ 12/15 Bachelor of Science, Fachrichtung Physik
Johannes Gutenberg­Universität, Mainz
Bachelorarbeit: “Strahlungskorrekturen zum Quark­Formfaktor in
masseloser QCD”,
Betreuer: apl. Prof. Dr. Hubert Spiesberger, Gesamtnote: 1.3
08/09 ­ 04/12 Allgemeine Hochschulreife
Gymnasium am Römerkastell, Alzey
Gesamtnote: 1.2 Bestes Abiturzeugnis
166
Publikationen
2021 S. Baumholzer, P. Schwaller
Probing non­thermal light DM with structure formation and Neff, prepared for submission, E­
Print: 2112.03993
2020 S. Baumholzer, V. Brdar, E. Morgante
Structure Formation Limits on Axion­Like Dark Matter, JCAP 05 (2021) 004, E­Print:
2012.09181
2019 S. Baumholzer, V. Brdar, P. Schwaller, A. Segner
Shining Light on the Scotogenic Model: Interplay of Colliders and Cosmology, JHEP 09 (2020)
134, E­Print: 1912.08215
2018 S. Baumholzer, V. Brdar, P. Schwaller,
The New νMSM (ννMSM): Radiative Neutrino Masses, keV­Scale Dark Matter and Viable
Leptogenesis with sub­TeV New Physics, JHEP 08 (2018) 067, E­Print: 1806.06864
Mainz, 5. April 2022
167