UNIVERSIDAD NACIONAL AUTÓNOMA
DE MÉXICO
Posgrado en Ciencias F́ısicas
Instituto de F́ısica
F́ısica de Altas Enerǵıas, F́ısica Nuclear, Gravitación y
F́ısica Matemática
DARK MATTER PARTICLE PRODUCTION 
DURING INFLATIONARY REHEATING
T E S I S
que para OPTAR POR el t́ıtulo de
Maestro en Ciencias (F́ısica)
PRESENTA:
Francisco Barreto Basave
TUTOR PRINCIPAL:
Dr. Marcos Alejandro Garćıa Garćıa
Instituto de F́ısica, UNAM
MIEMBROS DEL COMITÉ TUTOR:
Dra. Mariana Vargas Magaña
Instituto de F́ısica, UNAM
Dr. Juan Carlos Hidalgo Cuéllar
Instituto de Ciencias F́ısicas, UNAM
Ciudad Universitaria, México, Noviembre 2024
 
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PROTESTA UNIVERSITARIA DE INTEGRIDAD Y 
HONESTIDAD ACADÉMICA Y PROFESIONAL 
(Graduación con trabajo escrito) 
De conformidad con lo dispuesto en los artículos 87, fracción V, del Estatuto General, 68, primer 
párrafo, del Reglamento General de Estudios Universitarios y 26, fracción I, y 35 del Reglamento 
General de Exámenes, me comprometo en todo tiempo a honrar a la Institución y a cumplir con los 
principios establecidos en el Código de Ética de la Universidad Nacional Autónoma de México, 
especialmente con los de integridad y honestidad académica. 
De  acuerdo  con  lo  anterior,  manifiesto  que  el   trabajo  escrito  titulado
que presenté para obtener el grado de es original, de mi autoría y lo realicé con 
el rigor metodológico exigido por mi programa de posgrado, citando las fuentes de 
ideas, textos, imágenes, gráficos u otro tipo de obras empleadas para su desarrollo. 
En consecuencia, acepto que la falta de cumplimiento de las disposiciones reglamentarias y 
normativas de la Universidad, en particular las ya referidas en el Código de Ética, llevará a la 
nulidad de los actos de carácter académico administrativo del proceso de graduación. 
Atentamente 
(Nombre, firma y Número de cuenta de la persona alumna) 
Dark Matter Particle production during Inflationary Reheating
Francisco Barreto Basave
523011352
-----Maestria-----
f
i

C O O RDINA C IÓ N G E NE RA L DE  E STUDIO S DE  PO SG RA DO  
CARTA AVAL PARA DAR INICIO A LOS TRÁMITES DE GRADUACIÓN 
Univers idad  Nacional A u tónoma de México  
Secretaría G eneral 
C oord inación G eneral de E studios  de Posgrado  
Dr. A lberto  G ü ijosa Hidalgo  
Programa de Posgrado  en  C iencias  F ís icas  
Presen te 
Quien suscribe,     , tutor(a) 
principal de     , con 
número de cuenta            , integrante  del alumnado 
de  de ese programa, manifiesto 
bajo protesta de decir verdad que conozco el trabajo escrito de graduación 
elaborado por dicha persona, cuyo título es: 
, así como el reporte que contiene el resultado emitido por la herramienta 
tecnológica de identificación de coincidencias y similitudes con la que se 
analizó ese trabajo, para la prevención de faltas de integridad académica. 
De esta manera, con fundamento en lo previsto por los artículos 96, 
fracción III del Estatuto General de la UNAM; 21, primero y segundo 
párrafos, 32, 33 y 34 del Reglamento General de Exámenes y; 22, 49, 
primer párrafo y 52, fracción II del Reglamento General de Estudios de 
Posgrado, A VA LO  que el trabajo de graduación presentado se envíe al 
jurado para su revisión y emisión de votos, por considerar que cumple con 
las exigencias de rigurosidad académica previstas en la legislación 
universitaria. 
Protesto lo necesario, 
Ciudad Universitaria, Cd. Mx., a  de de 202 
Tutor(a) principal 
Dr. Marcos Alejandro García García
Francisco Barreto Basave
523011352
Maestría en Ciencias (Física)
Dark Matter particle production during inflationary
reheating
Dr. Marcos Alejandro García García
11 octubre 4
iii

To my parents
¡Gracias!

vii
Acknowledgments
First of all, I would like to express my gratitude to my advisor, Dr. Marcos, for his
invaluable guidance, support, and encouragement throughout the past years I have had
the privilege of working alongside him. I am especially thankful for the patience, insight,
and the countless hours he has dedicated to teaching me and I sincerely hope to continue
working together in the future as I further pursue my academic aspirations.
I also want to extend my appreciation to my tutor committee: Dra. Mariana, for her
essential role in my academic formation; and to Dr. Hidalgo, through whom I met my
current advisor, for his invaluable suggestions during my master’s program and for always
supporting me with a recommendation letter whenever I needed them.
To the academics members of the jury that finally evaluated this work. Dra. Myriam, for
her valuable remarks, insightful questions and constructive suggestions. Dr. Manfred, for
his meticulous corrections and attentions to detail, which greatly improved the clarity of
this work while challenging me to refine my arguments. Dr. Tonatiuh for his questions and
suggestions done. Finally, but by no means less important, Dr. Alberto, for his thoughtful
feedback and valuable corrections.
I am deeply greatful to the Consejo Nacional de Humanidades, Ciencias y Tecnoloǵıas
(CONAHCYT) for their financial support.
My work was also supported by the DGAPA-PAPIIT grant IA103123 at UNAM. As well
as by the CONAHCYT ’Ciencia de Frontera’ grant CF-2023-I-17.
Without any of them this thesis would not have been possible.

ix
Resumen en español
Estudiamos la producción de Part́ıculas Masivas Débilmente Interactuantes (por sus siglas
en inglés, WIMPs), part́ıculas candidatas a ser materia oscura (DM), durante el proceso
denominado como recalentamiento inflacionario. El recalentamiento sucede mientras el
campo del inflatón, ϕ, oscila alredor del mı́nimo de su potencial. Durante estas oscila-
ciones, los acoplamientos del inflatón con los campos del modelo estándar (SM) se vuelven
relevantes, permitiendo decaimientos del inflatón a part́ıculas del SM, creando aśı el baño
térmico que eventualmente lleva al universo a un época de dominación por radiación. Du-
rante esta fase de recalentamiento, la producción de DM sucede a través de mecanismos
de dispersión involucrando dos part́ıculas del SM que se aniquilan en un par de WIMPs
(y viceversa). Los WIMPs son part́ıculas hipotéticas con una escala de interacción con
los campos del modelo estándar del orden de la escala electrodébil. Estos acoplamien-
tos permiten a los WIMPs estar en equilibrio térmico con el resto de part́ıculas, lo cual
se le conoce como materia oscura térmica. Cuando la expansión del universo se vuelve
mayor que la interacción DM-SM, los WIMPs salen del equilibrio térmico, dicho proceso
se le conoce como freeze-out. La dinámica del freeze-out está descrita por la ecuación de
Boltzmann, la cual resolvemos en la aproximación de onda s.
Consideramos un escenario en el que el proceso de recalentamiento sucede en un potencial
inflacionario el cual, cerca de su mı́nimo, toma la forma V ∼ ϕ4. En estas circuntancias,
las inhomogeneidades espaciales en el campo del inflatón, producidas debido a autoint-
eracciones, crecen exponencialmente rápido, llevando al campo del inflatón a un estado
fragmentado. Analizamos el impacto de la fragmentación en el proceso de recalentamiento
y en la producción de WIMPs, para esto último calculamos la densidad de reliquia de DM
resultante, ΩDMh
2.

xi
Abstract
We study the production of Weakly Interacting Massive Particles (WIMPs), which are
considered candidates for Dark Matter (DM), during the epoch known as inflationary
reheating. Reheating takes place as the inflaton field, ϕ, oscillates about the minimum
of its potential. During these oscillations, the couplings of the inflaton become relevant
allowing for decays into Standard Model (SM) particles, eventually creating the thermal
bath that leads the universe to a radiation domination epoch. During this period, DM
particles are produced through scattering processes where two SM particles annihilated
into two DM particles, and vice versa. WIMPs are hypothetical particles with a coupling
to SM fields in the range of the electroweak scale, this coupling would allow the WIMPs to
remain in thermal equilibrium with the rest of the SM particles; this is known as thermal
dark matter. When the expansion rate of the universe is larger than this interaction rate,
the WIMPs will drop out of thermal equilibrium, or as it is most commonly refer to,
DM freezes-out. The dynamics of the freeze-out are described by the Boltzmann equation
which we solve in the s-wave approximation.
We consider a scenario in which the reheating process occurs in an inflationary potential
which nears its minimum takes the form V ∼ ϕ4. In this set-up, small inhomogeneities
on the inflaton field, arising from self-interactions, grow exponentially fast, leading the
inflaton to a fragmentated state. We analyze how fragmentation affects the reheating
process and WIMP production, for this last one, we compute the resulting relic abundance,
ΩDM,0h
2.

xiii
Contents
Acknowledgments vii
Resumen en español ix
Abstract xi
1 Modern Cosmology 1
1.1 Elements of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 The homogeneous and isotropic metric . . . . . . . . . . . . . . . . 9
1.4 The Cosmological equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Standard Cosmological Solutions . . . . . . . . . . . . . . . . . . . 11
1.4.2 Density parameters Ωi . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.3 Redshift z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 The standard model of cosmology . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Thermodynamics in an expanding universe . . . . . . . . . . . . . . . . . . 17
1.6.1 The adiabatic expansion of the universe . . . . . . . . . . . . . . . 19
1.7 Thermal history of the universe . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Dark Matter 27
2.1 Evidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Dark Matter in galaxy clusters . . . . . . . . . . . . . . . . . . . . . 28
2.1.2 Rotation curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
xiv
2.1.3 Stabilization of Large Scale Structures . . . . . . . . . . . . . . . . 30
2.1.4 Large Scale Structure formation . . . . . . . . . . . . . . . . . . . . 30
2.1.5 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.6 Cosmic Microwave Background anisotropies . . . . . . . . . . . . . 32
2.1.7 Alternatives to Dark Matter . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Current status of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Wino status as the Lightest Supersymmetric Particle . . . . . . . . 39
2.3 Weakly Interacting Massive Particles . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 Freeze-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 Radiation domination era . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.4 Solutions to the Boltzmann equation . . . . . . . . . . . . . . . . . 47
2.3.5 Dark Matter relic density computation . . . . . . . . . . . . . . . . 49
3 Inflation 51
3.1 Limitations of the Standard Model of Cosmology . . . . . . . . . . . . . . 51
3.2 Cosmological Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Continuity equation of the inflaton . . . . . . . . . . . . . . . . . . 54
3.2.2 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.3 The Slow-roll Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Primordial Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Problems revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.1 Alternatives to inflation . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Inflationary potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.1 T-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Reheating 69
4.1 Coherent oscillations regime . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1 Reheating in a minimum V ∼ ϕk . . . . . . . . . . . . . . . . . . . 70
xv
4.1.2 Fermionic Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.3 Bosonic Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Boltzmann equation during reheating . . . . . . . . . . . . . . . . . . . . . 78
5 Fragmentation 83
5.1 Inhomogeneities during oscillations . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Decay rate of the fragmentated inflaton . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Decay of the coherent oscillations . . . . . . . . . . . . . . . . . . . 92
5.2.2 Decay of the fragmentated inflaton . . . . . . . . . . . . . . . . . . 92
5.2.3 Total decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 Dark Matter relic abundance density 95
6.1 Reheating dynamics with fragmentation . . . . . . . . . . . . . . . . . . . 96
6.1.1 Temperature evolution with Fragmentation . . . . . . . . . . . . . . 98
6.2 Wino Relic Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Conclusion 104
Appendices 107
A Friedmann metric detailed 107
A.1 Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2 Ricci’s tensor and scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3 Friedman equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.4 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B Boltzman Equation 115
B.1 Non relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B.2 Relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.3 In a FLRW universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
References 121

1
Chapter 1
Modern Cosmology
1.1 Elements of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The Cosmological equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 The standard model of cosmology . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Thermodynamics in an expanding universe . . . . . . . . . . . . . . . . . . 17
1.7 Thermal history of the universe . . . . . . . . . . . . . . . . . . . . . . . . 21
The starting assumption for cosmology, as in most areas of physics, is that the naturalness
of the universe operates under some set of rules that we can discover by careful observation.
One of the first observations about our universe was that at distances larger than 300
million light years, it appears the same in all directions around us [1], based on this, we
supposed that our universe is isotropic at such distances [2]. It would be complicated, to
say the least, to assume that we are in any special position in the universe thereby the
most logical conclusion is that the universe should also appear isotropic to most observers1
throughout the universe. This means that our universe should be spatially homogeneous
[3]. Both of these assumptions form what we have come to define as the cosmological
principle, this principle ensures that observations made from our single vantage point are
representative of the universe and therefore established cosmology as a scientific area.
The word principle in physics is used to refer to, what at the time are, wild intuitive
guesses2; and for most of the twentieth century, the cosmological principle, had been
1The observers for whom this is true are called comoving observers and will be defined later in this
chapter.
2In contrasts, a law in physics refers to experimental established facts.
2 1. Modern Cosmology
taken as an assumption and remained that way until firm empirical data confirmed large
scale homogeneity and isotropy [4], though, it is still referred to as a principle.
To describe a universe that is both homogeneous and isotropic, we use the Friedmann-
Lemâıtre-Robertson-Walker (FLRW) metric3 that is written as
ds2 = dt2 − a2(t)
(
dr2
1−Kr2 + r2dθ2 + r2 sin2 θdϕ2
)
, (1.1)
where a(t) is known as the scale factor, K = {−1, 0, 1} is the curvature parameter and
{t, r, θ, ϕ} are comoving coordinates. Substituting this metric into the Einstein field equa-
tions, leads us to the Friedmann equation
H2 ≡
(
ȧ
a
)2
=
8πGN
3
ρ− K
a2
+
1
3
Λ, (1.2)
where H is known as the Hubble parameter, GN is Newton’s gravitational constant, ρ is the
energy density and Λ is the cosmological constant. This equation, along with a continuity
equation and an equation of state, dictates how the the whole universe, a(t), behaves
depending on its energy content, ρ.
In this chapter we will introduce the cosmological model Λ-CDM ; where Λ is the cosmo-
logical constant (that appears in the Friedmann equation) and CDM stands for Cold Dark
Matter. CDM is a form of matter that does not produce (nor absorb) light and it is a
fundamental part of the present thesis, thereby, chapter 2 is completely dedicated to it.
1.1 Elements of General Relativity
The Theory of General Relativity (GR) was formulated at the beginning of the twentieth
century (1915) by the famous physicist Albert Einstein. GR is a generalization of the
Special Theory of Relativity (or simply Special Relativity, SR), which was also introduced
by him earlier in 1905.
1.1.1 Special Relativity
The Theory of SR is based upon on two key postulates
Principle of relativity: The laws of physics remain the same in any inertial frame4.
The speed of light in vacuum is constant for all inertial observers.
3Setting the speed of light in vacuum, c, to be one.
4An inertial frame is a reference frame with constant velocity.
1. Modern Cosmology 3
Combining both postulates, we get that the line element (the distance between two space-
time points, events), ds2, is invariant for all inertial observes. The line element is expressed
as
ds2 = ηµνdx
µdxν , (1.3)
where ηµν is called the metric. We use greek letters, such as µ, to be spacetime indices.
Spacetime indices take values from zero to three, µ = {0, 1, 2, 3} = {0, i}, where the zero
component represents the time component, meanwhile, latin letters such as i, j, k, . . . are
spatial indices and take on values from one to three, i = {1, 2, 3}. In addition, we are using
Einstein notation convention (or Einstein summation convention) over repeated indices;
this operations is known as a contraction (of indices).
SR is a particular case of GR in which there is no gravity, in this scenario, spacetime is
described by a Minkowski metric, denoted by ηµν . As in most books on particle physics
[5, 6], we introduce the Minkowski metric in the mostly minus convention as
ηµν ≡




1
−1
−1
−1




= diag (1,−1,−1,−1) , (1.4)
where all of the entries left empty outside the diagonal are exactly zero.
We define a contravariant four-vector position as
xµ ≡
(
x0, xi
)
= (ct,x). (1.5)
where c is the speed of light in vacuum.
We use the metric, ηµν (and the inverse metric, ηµν) to lower (raise) indices, as follows
xµ = ηµνx
ν , (xµ = ηµνxν). (1.6)
Quantities with all lower (upper) spacetime indices, such as xµ (xµ), receive a special
name, covariant (contravariant) quantities, in this example, a covariant (contravariant)
position four-vector. It is straightforward to verify that this new quantity, ηµν , is in fact
the inverse of the metric ηµν , that is, it satisfies the relation
ηµαηαν = δµν =
{
1, µ = ν
0, µ ̸= ν
(1.7)
where δµν is the Kronecker delta.
4 1. Modern Cosmology
The most general transformations between two inertial frames that preserve the Minkowski
interval, i.e., ds2 = ηµνdx
µdxν = ds′2, (where the ′ refers to quantities measured in the sec-
ondary inertial reference frame) are called Poincaré transformations. A Poincaré trans-
formation is written as
x′µ = Λµνx
ν + aµ, (1.8)
where Λµν is a Lorentz transformation and aµ is a constant four-vector that represents
spacetime translations. By direct computation, we verify that the Lorentz transformations
form a group that we called the SO(3, 1) group, that means, a group of Special (deter-
minant equal to one) Orthogonal (their transpose is equal to their inverse) matrices of
3+1 dimensions. In the case of Lorentz transformations, the quantity invariant is not the
identity matrix, but rather, the Minkowski metric. The relation satisfied by the Lorentz
matrices, Λµν , is
ΛµαηµνΛ
ν
β = ηαβ. (1.9)
1.1.2 General Relativity
From here on out, unless stated otherwise, we will be using natural units. This set of units
is defined in such a way that
c = ℏ = kB = 1, (1.10)
where ℏ is the reduced Planck constant and kB is the Boltzmann constant. These units
are used for convenience because they simplify equations by eliminating unnecessary con-
stant factors, allowing expressions to be written in terms of purely physical quantities.
Ultimately, results can always be converted back to common units (e.g., the international
system of units) by multiplying each physical quantity by the appropriate conversion fac-
tor.
GR is constructed on three main ideas
General Covariance principle: The laws of physics remain the same in all references
frame5.
Equivalence principle: Locally, the effects of gravity are indistinguishable from ac-
celeration.
Einstein’s equations: The spacetime curvature tells matter how to move while matter
tells spacetime how to curve.
5Inertial and non-inertial frames.
1. Modern Cosmology 5
In order to introduce the Einstein’s equations, we need to introduce the (generalized)
metric tensor, gµν , which is a generalization of the Minkowski metric ηµν
6,
ηµν → gµν = gµν(x
α). (1.11)
This generalization of the metric tensor also relates the invariant line element as follows
ds2 = gµνdx
µdxν . (1.12)
If we are interested in studying the movement of a test particle7 under GR theory, we
must apply the following principle
Principle of least action:
The action is stationary under small variations with respect to the fields of the theory.
In GR the action of a particle is given by
S = −m
∫
dλ
√
−gµν
dxµ
dλ
dxν
dλ
, (1.13)
where λ parametrize the points along the path of movement and it is known as the affine
parameter of the trajectory. Applying the principle of least action
δS = 0, (1.14)
with a variation with respect to the metric gµν , we get that, the path, x
µ, that extremizes
the action satisfies the geodesic equation. The geodesic equation reads as follows
d2xµ
dλ2
+ Γµαβ
dxα
dλ
dxβ
dλ
= 0, (1.15)
where Γµαβ are the Christoffel symbols. The Christoffel symbols are related to the metric
tensor by the following expression
Γµαβ =
1
2
gµρ (∂αgρβ + ∂βgαρ − ∂ρgαβ) , (1.16)
where ∂µ ≡ ∂/∂xµ. Using expression (1.16), we verify that the Christoffels symbols are
symmetric under the exchange of the two lower indices, that is Γµαβ = Γµβα. As in SR, gµρ
is the inverse of the metric tensor gρν , and satisfies gµρgρν = δµν .
6We continue using the mostly minus signature (convention).
7A test particle is an idealized (massive) particle that feels the effects of the metric (gravity) but does
not affects it.
6 1. Modern Cosmology
1.2 Einstein’s field equations
So far we have seen that given a metric gµν , a test particle will follow a path described by
the geodesic equation (1.15). However, we do not yet have a clear method for determining
the metric that describes a particular spacetime. As it turns out, Einstein’s field equations
provide us with an expression regarding the evolution of the metric.
Einstein equations arise directly from the exact same principle of least action as the
geodesic equation, though, we need a completely different action. The action that leads to
the Einstein equations is known as the Einstein-Hilbert (EH) action, SEH , and it is given
as
SEH =
∫
d4x
√−gLEH = − 1
16πGN
∫
d4x
√−gR, (1.17)
where g ≡ det (gµν), LEH = −R/(16πGN) is the Einstein-Hilbert Lagrangian density with
R being the Ricci scalar, that we will be introduce next.
The Ricci scalar or as it is sometimes referred to, the curvature scalar, can be computed
as a function of the Riemann tensor, Rρ
λµν . The Riemann tensor is constructed over
contractions of the Christoffel symbols as follows
Rρ
λµν ≡ ∂µΓ
ρ
λν − ∂νΓ
ρ
λµ + ΓσλνΓ
ρ
σµ − ΓσλµΓ
ρ
σν . (1.18)
The definition of this tensor depends on the signature used (mostly minus), and so will
do the subsequent quantities derived from it, such as the Ricci tensor. The Ricci tensor,
denoted by Rµν , is the contraction over two indices of the Riemman tensor,
Rµν ≡ Rρ
µρν . (1.19)
Finally, the Ricci scalar, R, is the trace of the Ricci tensor,
R ≡ Rµ
µ. (1.20)
If we want to include the cosmological constant Λ into our problem, then we must add it
to the action, and we achieve that by simply writing
SEHΛ =
∫
d4x
√−gLEHΛ = − 1
16πGN
∫
d4x (R + 2Λ) , (1.21)
where now we have the Einstein-Hilbert, with cosmological constant, action8, SEHΛ. To
simplify our convention, we denote this by simply, SEH .
8The factor 2, on the cosmological constant, Λ, is introduced by hand to simplify the factors on the
resulting equations.
1. Modern Cosmology 7
To describe a scenario in the presence of matter, we write
S = SEH + Sm, (1.22)
where Sm is the matter action, without making any assumptions about it, we write
Sm =
∫
d4x
√−gLm, (1.23)
for a given lagrangian density of matter, Lm, for instance, this could be the lagrangian of
a perfect fluid (as in cosmology), a real scalar field (as in inflation), the electromagnetic
field, or else.
Now we may apply the principle of least action. Varying the action, S, with respect to
the metric and asking for this to be stationary,
δS = δSEH + δSm = 0, (1.24)
where the variation of each action with respect to the metric is given by
δSEH =
1
16πGN
∫
d4x
√−g
(
Rµν −
1
2
Rgµν − Λgµν
)
δgµν (1.25a)
δSM = −1
2
∫
d4x
√−gTµνδgµν , (1.25b)
here, we have introduced a new term called the stress-energy or energy-momentum tensor,
Tµν . This tensor is defined as a functional derivative of the action with respect to the
inverse metric, that is
Tµν ≡
2√−g
δSm
δgµν
(1.26)
Inserting both expressions, (1.25a) and (1.25b) back into our principle of least action
expressed as equation (1.26), we get the Einstein field equations,
Rµν −
1
2
Rgµν ≡ Gµν = 8πGNTµν + Λgµν . (1.27)
Additionally, the energy-momentum tensor defined in equation (1.26) comes with a con-
servation equation associated to it, this conservation equation readas as follows
∇µT
µν ≡ ∂µT
µν + ΓνµσT
µσ + ΓµµσT
σν = 0, (1.28)
where ∇µ is the covariant derivative.
8 1. Modern Cosmology
1.3 The Cosmological Principle
To describe the dynamics of our universe one must solve the Einstein’s equations given by
expression (1.27), i.e.,
Rµν −
1
2
Rgµν ≡ Gµν = 8πGNTµν + Λgµν ,
for the whole universe. It important to notice that the independent variables are the
metric, gµν , and the stress-energy tensor, Tµν ; meanwhile, the Ricci tensor, Rµν , and Ricci
scalar, R, are no linearly independent, they are determined by the metric. Therefore, we
must find a metric and an energy-momentum tensor suitables to accurately describe the
observations of our universe.
In fact Einstein himself attempted to do this [7]. To simplify his computations, he assumed
that the universe was smooth and static, such that the energy content was conserved, that
is, he assumed ρ̇ = 0, where the dot means derivative with respect to time. In order to get
a non-changing universe filled with matter (stars, planets, and so on) under gravitational
forces, he needed something to counteract the pull of gravity. To achieve this, Einstein
introduced a Λ term that would produce a negative pressure, in his model, a negative
pressure just enough to counter act the gravitational forces. Later, this came the be
known as the cosmological constant.
In 1929, Edwin Hubble measured that distant galaxies were receding away from us, but
not only that, Hubble also noticed that the galaxies that are farther away from us, are
receding faster than those galaxies that are closer to us [8]. Nowadays, this has been called
as the Hubble law, and it may be expressed as
v = H0rphys, (1.29)
where v and rphys are the relative velocity and physical position9 of the observed object,
respectively, and H0 is called the Hubble constant. The Hubble constant is the value of the
Hubble parameter, H, today. In his original paper, he estimated a value of H0 ∼ 500 km
s−1 Mpc−1, redefined techniques today have allowed us for better measurements of it. For
example, the Planck Collaboration in 2018 [9] constrained the value to H0 = (67.66±0.42)
km s−1 Mpc−1. The precise value of this constant is an ongoing problem today and it
known as the Hubble tension [10, 11]. Without attempting to address the complicated
issues in measuring it, we parametrize our lack of precise knowledge of H0 in little h, by
writting
H0 = 100 h km s−1 Mpc−1, (1.30)
and according to Planck’s 2018 results, h ∼ 0.6766± 0.0042.
9A physical quantity is affected by the expansion of the universe.
1. Modern Cosmology 9
Our current model of cosmology is based on the following principle
The Cosmological Principle
For an observer on the Hubble flow 10, the distribution of matter and radiation in the
universe on sufficiently large scale is homogeneous and isotropic.
While this does not assure us that the entire universe is smooth, it does imply that a
region at least as large as our present Hubble volume is smooth. The Hubble volume is
defined by the Hubble radius ; the Hubble radius is the region where all of the galaxies are
receding away faster than the speed of light. Inside the Hubble volume the velocities of
galaxies are smaller or equal to the speed of light.
The cosmological principle is just an extension of the Copernican principle., that is, our
place in the solar system is not special, so neither is our position in the Milky Way nor
is the position of our galaxy in the entire universe. Based on current evidence, we know
that the universe looks the same in all directions, that is, the universe is isotropic, and we
assume that it is also spatially homogeneous, the same wherever we are, or else, we would
be in the center of it.
1.3.1 The homogeneous and isotropic metric
The metric suitable to our Cosmological Principle is known as the Friedmann-Lemâıtre-
Robertson-Walker (FLRW) metric, in which the line element is written as
ds2 = gµνdx
µdxν = dt2 − a2(t)
(
dr2
1−Kr2 + r2dθ2 + r2 sin2 θdϕ2
)
, (1.31)
where a(t) is the scale factor, {t, r, θ, ϕ} are comoving coordinates, t is usually refer to
as the cosmic time and K is the curvature parameter, which, with a proper re-scaling of
coordinates can be chosen to be K = {+1, 0,−1}, each one of these describes a spatially
closed, flat and open universe, respectively. Notice here that our cosmological principle
simplified the dynamics of the whole universe to only the dynamics of an homogeneous
scale factor a(t).
A comoving quantity has been scaled out such that it remains constant with the expansion
of the universe, for example, in the comoving coordinates, a galaxy moving along with the
expansion of the universe (in the Hubble flow) will remained at a fixed position, even
though its physical distance is increasing. To get the physical distance, we use the scale
factor a(t),
rphys(t) = a(t)r. (1.32)
10The Hubble flow is the flow that galaxies follow as a result of the expansion of universe, it is also refer
to as the cosmic rest frame.
10 1. Modern Cosmology
It is interesting to notice that this solution by Friedmann (and LRW) breaks Lorentz
invariance. Observers moving with respect to the cosmic rest frame will perceive the
universe different, therefore, there is a preferred time direction, τ , the proper time of an
observer on the Hubble flow. Thus, we found a specific GR solution that is not Lorentz
invariant, nevertheless, on small scales, like a laboratory, this is negligible.
It is known that any other homogeneous and isotropic metric, may be rewritten as the
FLRW metric, with a suitable coordinate transformation, as discussed by Weinberg [12];
therefore this metric is the most general (explicitly) homogeneous and isotropic metric.
In order for the right-hand side of the Einstein equations to be compatible with our
cosmological principle, the energy-momentum tensor Tµν must be diagonal, the simplest
realization of such tensor is the one of an ideal fluid,
Tµν = (ρ+ p) uµvν − pgµν , (1.33)
where p is the pressure and ρ is the energy density, both of them isotropic, ρ = ρ(t) and
p = p(t). We choose uµ = (1, 0, 0, 0) to be the four-velocity vector in a comoving frame
and using our Friedmann metric we get
Tµν =






ρ
a2
1−Kr2p
a2r2p
a2r2 sin2 θp






. (1.34)
1.4 The Cosmological equations
To get the Friedmann equations, we substitute into the Einstein’s equations (1.27), both
the FLRW metric (1.31), and the energy-momentum tensor (1.34). The results are what
we called the cosmological equations, and the first of these equations, is known as the
Friedmann equation,
H2 ≡
(
ȧ
a
)2
=
1
3M2
p
ρ− 1
a2
K +
1
3
Λ, (1.35)
where we define the Hubble parameter
H ≡ ȧ
a
, (1.36)
and we also defined the Reduced Planck Mass, Mp in terms of the Planck mass, Mpl
1. Modern Cosmology 11
M2
p ≡
1
8π
M2
pl ≡
1
8πGN
. (1.37)
The second equation, of said cosmological equations, is the continuity equation. The
continuity equation is obtained from the time component of the conservation equation
(1.28). For a FLRW universe this conservation equation leads us to
ρ̇+ 3H (ρ+ p) = 0. (1.38)
and the third and final equation is the equation of state
p = wρ, (1.39)
where w is known as the equation of state parameter, it is important to clarify that this
third equation is not derived but rather it is assumed from thermodynamics properties.
Therefore, our universe is described by the following system of (three) equations known
as the cosmological equations11
H2 ≡
(
ȧ
a
)2
=
1
3M2
p
ρ− 1
a2
K +
1
3
Λ,
ρ̇ = −3H (ρ+ p) ,
p = wρ.
(1.40)
Cosmology is based on solving these equations and they can be found in any standard
book on cosmology [13].
Sometimes, it might be useful to use the second Friedmann equation, which is a linear
combination of both the Friedmann and the continuity equations, and it is written as
ä
a
= − 1
6M2
p
(1 + 3w)ρ+
1
3
Λ. (1.41)
1.4.1 Standard Cosmological Solutions
There are simplified scenarios where we might be able to find exact solutions to the
cosmological equations. Assuming a time independent equation of state parameter, i.e.,
w = constant. The continuity equation (1.38) can be integrated analytically,
ρ(a) = ρ(a0)
(a0
a
)3(1+w)
= ρ0a
−3(1+w), (1.42)
11A detailed deduction of these equations can be found in appendix A.
12 1. Modern Cosmology
where ρ(a0) = ρ0 and a0 respectively denote the value of the energy density and the
scale factor at present time, that is, today. Since the FLRW metric is invariant under
a rescaling of coordinates, we have freedom to set the value of the scale factor today a0
to be a0 = 1. Substituting this energy density as a function of the scale factor into the
Friedmann equation (1.35); and assuming that both the curvature K and the cosmological
constant Λ are zero, we may solve for the scale factor as a function of time and we get
a(t) =
[
3
2
√
1
3M2
p
ρ0(1 + w)t
]
2
3
1
1+w
. (1.43)
In any case, we get the following behavior for the energy density and the scale factor
ρ(a) ∼ a−3(1+w), → H ∼ ρ
1
2 ∼ a−
3
2
(1+w),
a(t) ∼ t
2
3
1
1+w , → H ≡ ȧ
a
=
2
3
1
1 + w
t−1,
(1.44)
where the corresponding factors are time-independent, therefore, the value of the Hubble
parameter H as a function of the cosmic time t is exact. We distinguished some important
scenarios for different values of w.
Matter: This is known as a dust universe, a universe with only non-relativistic
matter, and it is determined by pm = 0, so that wm = 0.
Radiation: Massless particles such that p = (1/3)ρ, that is the parameter of the
equation of state is wr = 1/3.
Cosmological constant: A universe dominated by the vacuum Λ ̸= 0, with a
curvature parameterK, pressure p and energy density of matter {pm, ρm} negligibles,
we get
H2 =
1
3
Λ ≡ 1
3M2
p
ρΛ, (1.45)
where we have defined a vacuum energy density, ρΛ as
ρΛ ≡M2
pΛ, (1.46)
with an equation of state parameter wΛ = −1.
Summarizing these results, we get that our fundamental quantities evolve as:
Matter: wm = 0, ρ ∼ a−3, a ∼ t
2
3 , H =
2
3t
;
Radiation: wr =
1
3
, ρ ∼ a−4, a ∼ t
1
2 , H =
1
2t
;
Vacuum Energy: wΛ = −1, ρΛ = const, a ∼ e
√
Λ
3
t, H =
√
Λ/3.
(1.47)
1. Modern Cosmology 13
1.4.2 Density parameters Ωi
Using Friedmann equation (1.41), we define a critical density, ρc, as
ρc ≡ 3M2
pH
2, (1.48)
which corresponds to the energy density required to get a spatially flat universe, given by
K = 0. The value of this critical density today, using Planck 2018 results [9], ρc,0, is
ρc,0 = 3H2
0M
2
p = 1.878× 10−29 h2g cm−3, (1.49)
which the energy density needed to make our universe spatially flat today. Using the
critical density, we also define the density parameter as
Ωi ≡
ρi
ρc
, (1.50)
which is the ratio of the actual energy density of the universe to the critical density.
In analogy with the vacuum energy density defined previously on equation (1.47), we
define the curvature energy density by
ρK ≡ −
1
a2
3M2
pK. (1.51)
Thus, the Friedmann equation may be expressed as
H2(a) =
1
3M2
p
∑
i ̸=K
ρi(a) = H2
0
∑
i ̸=K
Ωi,0a
−3(1+wi)
= H2
0
[
Ωr,0a
−4 + Ωm,0a
−3 + ΩK,0a
−2 + ΩΛ,0
]
,
(1.52)
where we identify the parameter of the equation of state for a universe dominated by
curvature, wK , as wK = −1
3
, and defined the density parameter of curvature, ΩK,0, as
ΩK,0 ≡
ρK,0
ρc,0
= − K
H2
0
, where we made used of a0 = 1.
This form of the Friedmann equation given by exression (1.52) explicitly shows the evolu-
tion of the Hubble parameter as a function of the scale factor, a = a(t), depending on the
dominant component of the universe: matter, radiation, vacuum constant or curvature.
In this form, it is clear that the energy density associated to a cosmological constant ΩΛ,0
is not affected by the scale factor. That means that, if there existed such a quantity today
different from zero, independently of the matter, radiation and curvature content of the
universe, it will inevitable lead to an universe in exponential expansion. As we will discuss
in the subsequent section, this is the current state of our universe.
14 1. Modern Cosmology
1.4.3 Redshift z
All we know about our universe today, including the values of the density parameters
today, Ωi,0, has been inferred from the light we received from distant objects, and this
light can be interpreted as either
Classically: Propagating electromagnetic waves.
Quantum mechanically: Freely propagating photons.
Say we consider this light in the Quantum Mechanical approach, this means, let us consider
it as freely-propagating photons. Then, we need to solve the geodesic equation (1.15),
d2xµ
dτ 2
+ Γµαβ
dxα
dτ
dxβ
dτ
= 0,
for a massless particle (photon) in a Friedman universe. The time component, µ = 0, tells
us that the energy, E, of the photons at any time, in terms of the energy at the moment
of emission, Eemitted, is given as
E = Eemitted
(
a
aemitted
)−1
, → E ∼ a−1. (1.53)
Using Einstein’s relation for photons, E = ω = 2πν =
2π
λ
, where ω is the angular fre-
quency, ν is the frequency and λ is the wavelength of the photon. We get that the wave-
length, λ, at a any time t, is given by
λ = λemitted
(
a
aemitted
)
, → λ ∼ a. (1.54)
The energy (wavelength) of the photons measured today is smaller (larger) than the en-
ergy they were emitted with, which is the exact same result we would have gotten using
a classical approach. Therefore, we define the redshift as the fractional shift in the wave-
length
z ≡ λ− λemitted
λemitted
, (1.55)
substituting into equation (1.54), we get the following expression
1 + z =
1
aemitted
. (1.56)
so that z = 0 corresponds to today.
1. Modern Cosmology 15
1.5 The standard model of cosmology
The total energy density of the universe is given by
Ωtot =
∑
i ̸=K
Ωi = Ωm + Ωr + ΩΛ = 1− ΩK , (1.57)
If the universe is spatially flat, ΩK = 0, and then Ωtot = 1. From the Planck 2018 results,
we get that the values of these parameters today, and they are
Ωm,0h
2 = 0.3111± 0.0056 =
{
Ωb,0h
2 = 0.02242± 0.00014,
ΩDM,0h
2 = 0.11933± 0.00091,
ΩΛ,0h
2 = 0.3147± 0.0034,
ΩK,0h
2 = 0.00032± 0.0008.
(1.58)
where Ωb,0 refers to the contribution of baryonic matter (the matter that we are made of)
and ΩDM,0 is the contribution of (Cold) Dark matter12, and the contribution of relativistic
species today, Ωr,0, is less than 0.001 percent of the total amount of energy of the universe
and it is conform by
Cosmic Microwave Background (CMB) photons: These photons correspond
to a density parameter today equal to Ωγ,0h
2 = 2.47×10−5, according to the Particle
Data Group (PDG) 2024 [14].
Neutrinos: The value of the masses of these particles are still unknown [15, 16],
however, according to the PDG, the density parameter of neutrinos (of mass in the
range of 5× 10−4 eV to 1 MeV) today, Ων,0, is given in good approximation by
Ων,0h
2 =
∑
mν
93.12 eV
, (1.59)
where the sum is carried over all families. Results of atmospheric and Solar neutrino
oscillations [17] suggests a lower limit of
∑
mν = 0.06 eV which means Ων,0h
2 ∼ 10−4.
Higher masses range require a more sophisticated calculation.
We can see from equation (1.58) that ΩK,0h
2 ≪ 1, which means that our universe today
is almost perfectly flat. Therefore, adding all these density parameters would lead us
to Ωtot,0h
2 = (1− ΩK,0)h
2 ≈ h2, where little h was introduced in equation (1.30), and
according to the Planck collaboration has a value of h ≈ 0.6766± 0.0042.
Plotting these density parameters today we get figure 1.1. From this figure it is clear
that our universe today is mostly dominated by the cosmological constant, Λ. Nowadays,
12The evidences we supporting the existence of this form of matter will be discussed in chapter 2.
16 1. Modern Cosmology
Figure 1.1: Components of universe according to the Planck collaboration 2018 results
[9]. We can see that most of our universe is made of Dark Energy and the second most
predominant component is Cold Dark Matter. Illustration taken from reference [18].
we call the cosmological constant as dark energy [19] and it is the responsible for the
accelerated expansion of the universe that Hubble measured on his original work. The
value of this constant, the origin of it and much more insights about it, remain all open
questions, and, on this work, we will not attempt to deal with them. However, the second
most predominant component in our universe today is DM, the indirect evidences to this
form of matter will be discussed in chapter 2 and a possible origin to it will be studied in
chapter 6. Finally, let us introduce
Λ-CDM cosmological model
A cosmological principle, from which we derived the cosmological equations (1.41),
Friedmann equation: H2 =
1
3M2
p
ρ+
1
3
Λ− 1
a2
K
= H2
0
[
Ωr,0a
−4 + Ωm,0a
−3 + ΩK,0a
−2 + ΩΛ,0
]
,
Continuity equation: ρ̇ = −3H(1 + ω)ρ,
Equation of state: p = ωρ
(1.60)
Cold Dark Matter, with density parameter today ΩDM,0, strictly speaking13 this com-
ponent is constant in time, ΩDM, and it is known as the DM relic density abundance.
A cosmological constant, Λ, with density parameter today ΩΛ,0.
Baryonic matter, with density parameter today Ωb,0
Radiation, composed of CMB photons, Ωγ,0, and neutrinos, Ων,0.
13See chapter 2 and 6 for details.
1. Modern Cosmology 17
1.6 Thermodynamics in an expanding universe
Today, we know that the radiation in our universe is composed by the microwave photons
with a temperature T = 2.75 K, and the 3 cosmic seas of relic neutrinos with a temperature
T = 1.96 K. However we know that in the past the radiation was constituted by these plus
the rest of SM particles with a temperature T ≫ mi, where mi represents the mass of the
i-species. To study the evolution of species through the thermal history of the universe,
we need to introduce the phase space distribution function, fi = fi(p, t), for species i. We
are assuming that the distribution is homogeneous14.
Using the distribution function, we can compute the number density ni, energy density ρi
and pressure pi of each species as
ni = gi
∫
d3pi
(2π)3
fi(pi, t), (1.61a)
ρi = gi
∫
d3pi
(2π)3
Ei(pi)fi(pi, t), (1.61b)
pi = gi
∫
d3pi
(2π)3
|pi|2
3Ei
fi(pi, t), (1.61c)
where gi and Ei are the internal degrees of freedom and the energy, of each i-species,
respectively. Notice here that the energy, Ei, satisfies E
2
i = |pi|2 +m2
i which is known as
the Einstein dispersion relation. It is known that for a species in kinetic equilibrium their
distribution function is given by
fi(Ei, t) =
[
exp
(
Ei − µi
T
)
± 1
]−1
, (1.62)
where µi is the chemical potential and the plus (minus) sign corresponds to fermions
(bosons) and it is known as the Fermi-Dirac (Bose-Einstein) distribution function. As-
suming our species of interests are in thermal equilibrium, we may integrate analytically
equations (1.61a)-(1.61c) in some regimes.
The Relativistic limit: In this limit mi, µi ≪ T , therefore
ni(T ) =
ζ(3)
π2
T 3



gB Bosons.
3
4
gF Fermions,
(1.63a)
ρi(T ) =
π2
30
T 4



gB, Bosons,
7
8
gF , Fermions.
(1.63b)
pi(T ) ≈
1
3
gi
∫
d3pi
(2π)3
Ei(pi)fi(pi, t) =
1
3
ρi(T ) (1.63c)
14To be compatible with our Cosmological Principle, otherwise, we would have had fi = fi(r,p, t)
18 1. Modern Cosmology
10-6 0.001 1 1000
0
20
40
60
80
100
Figure 1.2: Evolution of the effective relativistic degrees of freedom in the Standard Model,
g⋆, as a function of the temperature, T .
In equation (1.63a) we have, ζ(3) which is the Riemann-zeta function of three, and
on equation (1.63a)-(1.63c) we have gB/F which are the degrees of freedom associated
to bosons (fermions), and they are given by
spin s = 0, → gB = 1,
spin s =
1
2
, → gF = 2,
spin s = 1, → gB = 3, (g = 2 for massless particles).
(1.64)
We also confirmed that, for a relativistic species, the pressure is one third of the
energy density, that is wr = 1/3, as we said before on equation (1.47).
For a system composed of several relativistic species,
ρtot =
∑
i
ρi =
π2
30
g⋆(T )T
4; (1.65)
where we defined the effective degrees of freedom (in thermal equilibrium), g⋆(T ) as
g⋆(T ) =
∑
B
gB +
7
8
∑
F
gF . (1.66)
1. Modern Cosmology 19
Using these effective degrees of freedom we may simplify expressions (1.63a)-(1.63b),
ni(T ) =
ζ(3)
π2
g⋆(T )T
3, (1.67a)
ρi(T ) =
π2
30
g⋆(T )T
4, (1.67b)
pi(T ) =
1
3
ρi(T ), (1.67c)
which are the density number, energy density a pressure for relativistic species
(mi, µi ≪ T ). The evolution of the the effective degrees of freedom, g⋆, as a function
of the temperature, T , is shown in figure 1.2.
Non-relativistic limit: In this limit, the masses of species, mi, are much bigger
than the temperature, T , i.e., mi ≫ T . The Einstein dispersion relation, reduces to
Ei ≈ mi + p2/2mi, and the distribution function can be approximated as
fi =
[
exp
(
Ei − µi
T
)
± 1
]−1
≈ exp
(
−mi − µi
T
)
exp
(
− p2
2mT
)
. (1.68)
Integration of this distribution function leads to
ni(T ) = gi
(
miT
2π
) 3
2
exp
(
−mi − µi
T
)
, (1.69a)
ρi(T ) = mini(T ), (1.69b)
pi(T ) = ni(T )T. (1.69c)
1.6.1 The adiabatic expansion of the universe
Another physical quantity of relevancy to us is the entropy, S. First, let us rewrite the
continuity equation (1.60) as
d
(
ρa3
)
= −pd
(
a3
)
. (1.70)
This means that the change in energy in a comoving volume element, d
(
ρa3
)
, is equal to
minus the pressure times the change in volume, −pd
(
a3
)
. Alternatively, the first law of
thermodynamics describes how the internal energy U of a system changes in response to
changes in volume V and the exchange of heat, Q, with its surroundings, and it reads
dU = −pdV + dQ = −pdV + TdS, (1.71)
where p is the pressure.
20 1. Modern Cosmology
It is clear then that this form of the first law of thermodynamics (1.71) with dQ = 0 is
equivalent to the continuity equation given by (1.70). A process with no exchange of heat
with its surroundings, dQ = 0, is known as an adiabatic process. That means that the
expansion of our universe is an adiabatic process15.
Alternatively, using the fact that the second partial derivatives of a continuous function
are symmetric, i.e.,
∂2S
∂V ∂T
=
∂2S
∂T∂V
, it is easy to prove that the entropy density S(T ) is
conserved. To do this, we introduce the comoving entropy density s(T ) defined as
s(T ) ≡ S(T )
a3
=
(1 + w)ρ
T
. (1.72)
and using the continuity equation, we verify that in fact
ρ̇+ 3H(1 + w)ρ =
T
a3
d
dt
[
a3(1 + ω)ρ
T
]
= 0, → dS
dt
=
d
dt
(
sa3
)
= 0. (1.73)
Therefore, we have shown that the comoving entropy S is conserved, and the entropy
density, s, redshifts as s ∼ a−3. In the case of radiation domination, we get
s =
4
3
1
T
ρ(T ) =
2π2
45
g⋆(T )T
3. (1.74)
which relates the density of entropy, s (entropy per comoving volume), to the temperature,
T . For radiation, we get that
d
dt
(
sa3
)
= 0 → g⋆(T )T
3a3 ∼ const. → a3 ∼ 1
g⋆(T )T 3
∼ 1
T 3
, (1.75)
where we have assumed the effective degrees of freedom have a mild dependence on the
temperature, that is, we assume that the changes with respect to the temperature are
small, which is a good approximation during radiation domination. Therefore, during
radiation domination, the temperature redshifts as
T ∼ a−1. (1.76)
which is a result of the photons losing energy E ∼ a−1.
15This is valid as long as the continuity equation holds true, which will not be the case during reheating,
detailed in chapter 4.
1. Modern Cosmology 21
1.7 Thermal history of the universe
Finally we would like to review a few important stages of the evolution of the universe
and the thermal evolution we are considering in our present work.
Inflation ( E ∼ 1015 GeV ) [Chapter 3]:
This epoch corresponds to the beginning of our universe and it is a theoretical model
that would be a solution to both the horizon16 and the flatness problems17. We solve
both this issues at once, if, at some point in time, we meet the condition
d
dt
1
aH
< 0. (1.77)
One of the simplest realization of this era is obtained by introducing a real scalar field,
namely the inflaton, ϕ, which slowly rolls down its own potential to the minimum
of it, causing a period of exponential expansion of the universe.
Fragmentation (E ∼ 1011 GeV) [Chapter 5]:
After inflation has been completed, that is, after the slow-roll regime has ended, the
inflaton continues with a period of coherent oscillations about the minimum of its
potential. During these oscillations, quantum fluctuations due to self-interactions
would arise on the condensate creating small inhomogeneities, which, at the linear
order, may be expressed as
ϕ(t,x) = ϕcl(t) + δϕ(x, t), (1.78)
where ϕcl(t) corresponds to the explicitly classical and homogeneous component
of the inflaton, and δϕ(x, t) represent the small fluctuations. It turns out that,
depending on the shape inflationary potential nears its minimum, these fluctuations
on the inflaton field can grow up exponentially fast, driving the inflaton field into
the non-linear regime, process that is known as fragmentation. In the scenario of
interests to us, where the potential takes on the form
V (ϕ) = λϕ4, (1.79)
the process of fragmentation takes place at an energy of the order E ∼ 1011 GeV.
Dark Matter (DM) freeze-out (E ∼ 103 GeV) [Chapter 6]:
Assuming DM is made of Weakly Interacting Massive Particles (WIMPs) whose
interactions with the Standard Model (SM) particles, ΓDM↔SM, are of the order
16The horizon problem means why two points in Cosmic Microwave Background have the almost the
exact same temperature, given that at the moment of emission they were casually disconnected.
17The flatness problem refers to attempting to answer the question: why is our universe today almost
perfectly flat?. Remember that according the Planck collaboration, ΩK,0 ∼ 0.0007± 0.0019.
22 1. Modern Cosmology
of 100 GeV, similar to the electroweak scale. These interaction rates, would allow
the WIMPs to remain in thermal equilibrium with the SM thermal bath. As the
universe expands, approximately when the Hubble rate, H, becomes larger than the
interaction rate, that is, when H > ΓDM↔SM , DM annihilation (and creation) will
become significantly less common. When this happens, the WIMPs will start to
drop out of thermal equilibrium, eventually, the number of DM Particles will remain
constant in time, freezing-out of thermal equilibrium. This process is described by
the (Relativistic) Boltzmann equation that may be written as
ṅ+ 3Hn = −⟨σv⟩
(
n2 − n2
eq
)
, (1.80)
where n (neq) is the density number of DM particles per comoving volume (in ther-
mal equilibrium), ⟨σv⟩ is known as the thermally averaged (DM) annihilation cross
section that will be defined next in chapter 2. In our scenario, the departure from
thermal equilibrium occurs at an energy of around E ∼ 103 GeV, which is well before
reheating has been completed.
Electroweak phase transition (E ∼ 200 GeV):
All of the SM particles, except for photons, become massive. Therefore the interac-
tions are mediated by massive bosons W± and Z.
Reheating (E ⩽ 0.1 GeV) [Chapter 4]:
Reheating is the epoch of transition of an universe in an almost empty and cold
state, as result of inflation, to a radiation dominated state. The simplest realization
of such process is achieved during the coherent oscillations epoch, at this point the
interactions between the inflaton and the SM fields become relevant.
The dynamics of the reheating process are governed by a continuity equation for the
energy density of the inflaton, ρϕ, and one for radiation, ρR, said equations are
ρ̇R + 4HρR = R(t),
ρ̇ϕ + 3H (1 + wϕ) ρϕ = −R(t),
(1.81)
where the Hubble parameter is given by the Friedmann equation
H2 =
1
3M2
p
(ρR + ρϕ) , (1.82)
and R(t) will depend on the allowed couplings between the inflaton and SM fields,
in addition, it also depends on if there is fragmentation or no,
Without: R(t) = (1 + wϕ) Γϕρϕ
Fragmentated: R(t) = (1 + wϕ) Γϕρ̄ϕ + Γδϕmϕnδϕ,
(1.83)
1. Modern Cosmology 23
where ρ̄ϕ is the fraction18 of the energy density that is in the spatially homogeneous
condensate which is obtained by computing the spatial average of the inflaton field,
ϕ̄, and its time-derivative, ¯̇ϕ, that is, ρ̄ϕ =
1
2
¯̇ϕ2 + V (ϕ̄); Γϕ (Γδϕ) represents the decay
rate of the classical condensate (inhomogeneities), mϕ is the inflaton mass associated
to the second derivative of its potential and nδϕ is the comoving density number of
inhomogeneities.
If we allow for a coupling between the inflaton, ϕ(t,x), and two SM fermions (fermion
ψ and anti-fermion ψ̄) through a Yukawa coupling,
Lint= −yϕ(t,x)ψ̄ψ
= −yϕcl(t)ψ̄ψ − yδϕ(t,x)ψ̄ψ,
(1.84)
where the coupling constang y is known as a Yukawa coupling constant. Then the
decay rate Γϕ will be given by
Γϕ→ψ̄ψ = α2 y
2
8π
mϕ,
Γδϕ→ψ̄ψ =
y2
8π
mϕ,
(1.85)
where α2y2 ≡ y2eff is the effective Yukawa coupling which is obtained after averaging
over each oscillation of the classical field19.
As the energy of the inflaton field is transferred to relativistic SM fermions, these
fermions would interact with themselves producing the rest of the SM particles. At
some point, the energy budget of radiation will be the same as the remaining energy
of the inflaton field (including the energy density of the inhomogeneities). This
condition, denotes the end of reheating and it is denoted by
ρreh ≡ ρreh(areh) = ρR(areh) =
π2
30
grehT
4
reh, (1.86)
where areh, greh and Treh referred to the values of the scale factor, the relativistic
degrees of freedom and the temperature at the end of reheating. After this condition
is achieved, radiation will dominate over the inflaton field. In our present work, we
are considering low reheating temperatures scenarios, in particular we are setting
the reheating temperature to Treh = 0.1 GeV.
QCD phase transition (E ∼ 150 MeV):
Quarks and gluons bind together to form baryons and mesons, becoming the relevant
degrees of freedom.
18In scenarios without fragmentation, the homogeneous condensate is the complete energy budget of
the inflaton field.
19See chapter 5 for the details.
24 1. Modern Cosmology
0.23
0.24
0.25
0.26
Y
P
=
4
Y
H
e
1
5
10
50
D
/H
3
H
e
/H
(x
1
0
5
)
5.8 6. 6.2 6.4
2.3
2.5
2.7
10
10η
D
/H
(x
1
0
5
)
P
la
n
c
k
1 2 3 4 5 6 7 8 910
1
2
5
10
20
10
10η
7
L
i/H
(x
1
0
1
0
)
Figure 1.3: Relative abundance of light nuclei (in blue) with respect to hydrogen as a
function of the baryon-to-photon density ratio η as predicted by Big-Bang-Nucleosynthesis
computations compared to the observed values (in green). Illustration taken from reference
[20].
Neutrino decoupling (E ∼ 1 MeV):
Due to the fact that the interaction rate between neutrinos and electrons is smaller
than the Hubble expansion rate (as it happened with the WIMPs), neutrinos decou-
ple from the thermal bath while still being relativistic.
Electron-positron annihilation (E ∼ 500 keV):
At this temperature, electrons and positrons become non-relativistic while annihilat-
ing to photons. After this, photons and neutrinos are the only relativistic (radiation)
contribution to the universe.
Big Bang Nucleosynthesis (BBN) (E ∼ 100 keV):
Formation of light nuclei such as Deuterium (2H), Tritium (3H), Helium-3 (3He),
Helium-4 (4He), and Lithium-7 (7Li) sets in. The first process that takes place is
1. Modern Cosmology 25
the deuterium 2H though the only possible two-body process
n+ p↔ 2H+ γ. (1.87)
This process is highly sensitive to the initial relative baryon to photon abundance η,
as shown on figure 1.320. The baryon to proton abundance is given by
η ≡ ρb
ργ
≪ 1. (1.88)
At some point, deuterium is sufficiently produced to start producing heavier isotopes.
Matter-radiation equality (E ∼ 0.7 eV):
Non-relativistic matter, Ωm, dominates the energy budget of the universe (over ra-
diation, Ωr). This can be seen on equation (1.74), since a(t) is growing with time,
at some point radiation will become negligible, because a−4 < a−3, and matter will
dictate the evolution of the universe. At this point, the universe is composed out of
light nuclei, relativistic electrons (and positrons) and neutrinos.
Recombination and photon decoupling (E ∼ 0.3 eV):
Electrons and protons recombine to form neutral Hydrogen, causing the universe to
become transparent. The photons released at this moment travel freely in all direc-
tions, and as the universe expands, their wavelength redshifts (as seen in equation
(1.53)). These photons that we received today at the microwave range, create the
image of Cosmic Microwave Background (CMB), which is the earliest stage of the
thermal history visible to us.
Some people prefer to use redshift rather than energy; the redshift associated to the
CMB is zCMB = 1, 100, or a time tCMB ∼ 380, 000 years.
Reionization (E ∼ 5 meV):
The first stars and galaxies are formed.
Λ−matter equality (E ∼ 0.3 meV):
The energy density of the universe is dominated by a cosmological constant Λ, caus-
ing an accelerating expansion of the universe.
Present time (E ∼ 0.24 meV):
Today.
20The expected abundance and the observed value are equal except for Lithium-7, this discrepancy is
known as the lithium problem [21, 22].

27
Chapter 2
Dark Matter
2.1 Evidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Current status of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Weakly Interacting Massive Particles . . . . . . . . . . . . . . . . . . . . . 40
So far we have seen that our universe at large scales is homogeneous and isotropic for a
set of observers on the Hubble flow1. These two facts about our universe are known as
the Cosmological principle, see section 1.3. Combining this principle with a cosmological
constant Λ (responsible for the accelerated expansion of the universe today) and a compo-
nent of Cold Dark Matter (CDM), conforms our current model of cosmology, the Λ-CDM
model.
In this chapter we will present an overview of the evidences for the component of (Cold)
Dark Matter (DM). To do so, we will describe the gravitational evidences supporting
its existence, these arise from the discrepancies between theory and observations [23].
Subsequent to that, in section 2.2, there is a brief discussion of the current status of
DM searches in all three cases which correspond to direct and indirect detection, and
production at colliders. The direct detection corresponds to measuring a signal coming
from an interaction between a DM and a SM particle [24]. Indirect detection refers to
a signal coming from two DM particles annihilating into SM particles [25]. Finally, we
could produce DM particles by annihilating (colliding) SM particles into DM particles; for
example, at particle colliders [26].
At the end, in section 2.3, we will study what is called the WIMP paradigm [27]. Intro-
ducing a particle candidate to be DM in the form of Weakly Interacting Massive Particles
(WIMPs). An attractive quality of such hypothetical particles is that the annihilation
cross section, σ, of these WIMPs, while being small, could still allow direct detection [28].
1Observers who sees themselves at the center of the expanding universe, they move as a result of to
the expansion of the universe.
28 2. Dark Matter
2.1 Evidences
In section 1.5, we reviewed the current estimates by the Planck collaboration over the
density parameters today (strictly speaking, as of 2018), given by equation (1.58). The
measurement over DM set ΩDM,0h
2 = 0.11933± 0.00091. This means that close to 27% of
the total content of our visible universe is DM2, moreover, it is also predominant over the
ordinary (baryonic) matter which has a density parameter of Ωb,0h
2 = 0.02242± 0.00014
which is close to one fifth of the DM relic density. In the present section we will discuss
briefly the main observations that lead us to the idea of the existence of such weird form
of matter we have come to call Dark.
2.1.1 Dark Matter in galaxy clusters
The physicist Fritz Zwicky is regarded by most physicists as the pioneer of Dark Matter.
In 1933, he was studying the redshifts galaxies in the Coma Cluster [29] and concluded
that the missing matter in the cluster could be problematic. As one might have suspected,
many physicists before him had also notice that, in some galaxies, there was missing matter
needed to fit the theoretical predictions to the observational measurements. However,
given that apparently, the magnitude of missing matter was smaller than the observed
one, most of them considered that the missing matter would be in the form of dead
stars, asteroids, black holes, or dust. This problem was known as the problem of missing
luminosity.
When measuring the velocities of individual galaxies in the Coma Cluster, Zwicky noticed
that, each of the galaxies was following a movement that seemed rather fast compared to
the movement of closer galaxies. The velocity of individual galaxies measured by him were
much larger than the cluster mean velocity; a value much greater than those expected by
the total visible mass of the cluster. Since planets orbit stars, stars within galaxies (some
in spiral patterns), and galaxies move in clusters, and we know that for planetary motion,
the velocities are balanced by the gravity as the virial theorem dictates. Zwicky decided
to implement the virial theorem to galaxies in the Cluster. By doing this, he inferred that
amount of matter that was not seen corresponded to about 400 times the mass of the
luminous matter, much more than previous estimates.
2.1.2 Rotation curves
The most convincing evidence for DM at the galactic scale comes from the observations
of the rotation curves (of stars in galaxies), [30] that had been done before Zwicky. We
can determine the angular velocity of stars at different distances from the galactic center,
r, and plot the corresponding rotation curve, v = v(r), as seen in figure 2.1. Numerous
observations in spiral galaxies showed that the outer regions of galaxies, stars exhibited
2Using h ∼ 0.6766± 0.0042, we get ΩDM,0 ∼ 0.11933/h2 ∼ 26.8 percent.
2. Dark Matter 29
Figure 2.1: Rotation curve of a galaxy. The expected evolution (considering only the
visible light) correspond to the dark blue line, while the light blue shows the observed
velocity of galaxies. Additionally, in light dashed gray the velocity of stars if there was
only DM, a Disk or bulge light. Illustration retrieved from reference [18].
an anomaly in their velocity.
Let us consider the whole galaxy as a spherical distribution of matter, M(r), and apply
Newtonian dynamics to an individual star of mass m inside the galaxy. We know that
stars in the galaxy are under the action of a gravitational force, F (r) = FGravity, and
Newton’s dynamics dictates that this force is proportional to the objects (the star in
question) acceleration. For spiral shape galaxies, the stars are in a circular motion, so the
acceleration is the centripetal acceleration, that is
acentripetal =
v2
r
=
GNM(r)
r2
→ v =
√
GNM(r)
r
, (2.1)
where M(r) is the mass of the matter density contained in a sphere of radius r, that is
M(r) = 4π
∫ r
0
ρ(x)x2dx, where ρ(x) is the spherical distribution of mass of the galaxy. Out-
side of the optical disk, by definition, the luminous matter becomes constant,M(r)
∣
∣
r>rod
∼
constant, where rod is the radius of the optical disk. Therefore, the velocity of galaxies
30 2. Dark Matter
outside of the optical disk should be proportional to v ∼ r−
1
2 . However that is not what
was measured, what people measured was that
Outside the optical disk:
{
v(r) ∼ r−
1
2 , theory,
v(r) ∼ constant, observation.
(2.2)
To reconcile observations and theory, without modifying gravity, we assume the existence
of some form of matter halo around the visible disk such that M(r) ∼ r. If this was the
case, the tangent velocity (of individual galaxies outside the optical disk) will evolve as
v(r) ∼ (M(r)/r)
1
2 ∼ constant, as observed. These halos would be composed of DM, that
is, matter which we do not see directly but rather we measure its gravitational effects over
ordinary matter (in this case, the stars orbiting the galaxy).
The history of the measurements at the galactic scale dates back to 1912, when Vesto
Slipher observed the Andromeda galaxy [31]. Slipher noticed that the movement of stars
around the galaxy bulge corresponded to those of a disc spinning around it, that is, the
galaxy exhibited a spiral behavior. Naturally people started measuring the angular velocity
of individual stars on others. Later in 1939, Horace Babcock [32], extended this studies
and notice that the angular velocity of individual stars, at large distances from the center
of their corresponding galaxy, was constant, as seen in equation (2.2).
2.1.3 Stabilization of Large Scale Structures
In the 1970, as computing power was getting better, physicists like Frank Hohl [33], carried
out N-body simulations (for N ∼ 105 stars) to test the stability of spinning galactic
structures. Hohl, and others physicists, noticed that a disk of particles supported entirely
by rotation and gravity, could only hold up the spiral shape for very few revolutions (less
than three rotations) before elongating. Some astrophysicists, thinking the structure was
losing energy, attempted to stabilize it by artificially introducing energy into the system;
nonetheless, the simulations kept collapsing after a few revolutions.
After multiple attempts, they realized that the spiral shape could be stabilized if they
introduced into the simulations a matter halo around the galaxy [34, 35]. Therefore,
structure stabilization will only by possible if these halos of matter surrounding the spiral
galaxy existed, since we do not see such structures, they would made of DM [36].
2.1.4 Large Scale Structure formation
It is important to explain that DM is not only relevant for the stabilization of spiral
galaxies; it also plays an important role explaining the origin of them [37]. In a perfectly
homogeneous universe, no clustering of galaxies or large-scale structures would have been
possible. However, in the real universe, small primordial density fluctuations served as
seeds for structure formation [38]. DM being non-relativistic began to cluster around
2. Dark Matter 31
these perturbations creating gravitational wells, and after radiation and matter decoupling,
baryonic matter would have fallen into these wells; eventually clustering into large scale
(stable) structures [39].
2.1.5 Gravitational lensing
As we saw in chapter 1, particles in a curved spacetime move accordingly to the geodesic
equation (1.15), needless to say, this includes massless particles, i.e., photons. Conse-
quently, there exists an effect known as gravitational lensing. A photon near a gravita-
tional field moves as if it had a mass (because it has energy, it gravitates), therefore it
bends its path when it is near a massive object. Thus, we conclude that massive objects
can act as gravitational lenses, bending and focusing light from more distant sources.
Figure 2.2: The Bullet Cluster photo in x-ray (the pink region) represents visible matter,
superimposed with the gravitational lensing (blue area, which corresponds to the region
were astronomers find the most mass), figured retrieved from reference [40].
Depending on the deflection angle, one can estimate the size (the mass) of the object that
caused such effect. If we measure light with some deflection angle but we do not see the
object causing it, we could say that the bending is explained by some other source of
matter. Certainly, this could be easily confused with common dead stars or Black Holes.
32 2. Dark Matter
The most convincing evidence of DM producing gravitational lensing effects is seen on
the Bullet cluster [40], figure 2.2. The bullet cluster consists of two colliding clusters of
galaxies. Gravitational lensing studies of this cluster [41] reveal that most of the ordinary
(visible) matter (pink regions) in the cluster is clearly separate from the matter responsible
of the gravitational lensing (blue regions). The reason for the separation is that normal
matter slowed down during the collision while the DM did not interact, hence, it moved
freely to the sides of the collision. That being said, whatever it is the (source of) matter
causing the effect of gravitational lensing, is not normal matter, thereby, we conclude that
nearly all of the matter in the bullet cluster is dark. For this reason the bullet cluster is
considered one of the best evidences supporting DM.
We can observed that the galaxies in the background of image, particularly those behind
the blue region, appear fainted and slightly elongated compared to those galaxies in the
corners of the image, this elongation is a result of the gravitational lensing. In practice,
astronomers use statistical techniques and computer methods to study how the shapes
differ from what is expected in the absence of lensing.
2.1.6 Cosmic Microwave Background anisotropies
As we saw in section 1.7, the Cosmic Microwave Background (CMB) is composed by the
photons that were emitted at the moment of recombination. The most accurate photo of
the CMB available today was taken by the Planck collaboration in 2015 [42], updated in
2018 [43].
We know that the CMB follows with extraordinary precision the black-body spectrum with
a temperature TCMB,0 = 2.726 K. However there are some temperature fluctuations δT
that deviate from the CMB temperature T = TCMB + δT , by δT/T ∼ 10−3 − 10−5, these
deviations can be expressed as a function of the usual spherical harmonics,
δT
T
=
∞
∑
ℓ=2
ℓ
∑
m=−ℓ
aℓmYℓm(θ, ϕ), (2.3)
where the degree m describes the angular orientation of a fluctuation mode, and ℓ deter-
mines the characteristic angular size. In a universe with no preferred directions (isotropic),
the power spectrum should be independent of m, and, in an homogeneous universe, it
should be independent of ℓ. Finally, aℓm allows us to compute the power spectrum
Cℓ = ⟨aℓma∗ℓ′m′⟩ = 1
2ℓ+ 1
ℓ
∑
m=−ℓ
|aℓm|2 . (2.4)
The fact that Cℓ only depends on one ℓ reflects the fact that there is no covariance between
different values of ℓ. The power spectrum Cℓ tell us that, if for some ℓ, Cℓ is large, the
2. Dark Matter 33
0
1000
2000
3000
4000
5000
6000
D
T
T
ℓ
[µ
K
2
]
30 500 1000 1500 2000 2500
ℓ
-60
-30
0
30
60
∆
D
T
T
ℓ
2 10
-600
-300
0
300
600
Figure 2.3: Planck 2018 CMB power spectrum, compared with base Λ-CDM best fit to
Planck (TT,TE,EE+LowE+Lensing) data (blue line). The upper panel shows the power
spectrum while the lower panel the residuals. In the horizontal axis, the scale changes
from logarithmic to linear at the scale ℓ = 29. On the y-axis, the upper panel has
Dℓ = ℓ(ℓ + 1)Cℓ/2π. For the residuals, the vertical axis scale changes to the difference
between best fit and prediction. Figure retrieved from reference [43].
correlation function3 is large, for two points in the sky which are separated by an angle
approximately equal to π/ℓ.
The power spectrum measured by Planck in 2018 is shown in the upper panel of figure
2.3. We have Dℓ = ℓ(ℓ + 1)Cℓ/2π as a function of ℓ; after removing the dipole anisotropy
(ℓ = 1). The dipole anisotropy arises due to the motion of the observer (e.g., us) relative
to the CMB rest frame. As a result of this relative motion, the photons receive in the
(opposite) direction of motion are blueshifted (redshifted). While, in the lower panel, we
have the residuals which show the difference between the observed data and a theoretical
model to fit. According to this plot, the biggest CMB hot spot make an angle of 1 degree
or ℓ ∼ 200. The position of the first peak tells us that, if we pick a point in the sky that
3Also known as the covariance function, this function answers the question: suppose we are given two
points in the sky which are separated by an angle θ. If we measure the temperature fluctuation at point
1, what do we expect the temperature fluctuation at point 2 to be?.
34 2. Dark Matter
has a large value of δT , then we are likely to find other points separated by π/200 ∼ 1
degree which also have large values for δT .
One can give an interpretation to the peaks in the CMB power spectrum. The fist of them,
the one located around ℓ = 200, arises from regions in the early universe where matter
and radiation were most compressed during acoustic oscillations in the baryon-photon
fluid. These (gravity driven) accoustic-like oscillations are the result of the back-and-
forth motion generated as gravitational collapse pulled matter into overdense regions,
compressing the baryons and photons together. However, as the gas is compressed its
temperature increases, resisting the collapse and creating an oscillatory behavior. The
first peaks corresponds to overdense regions in the early universe that had undergone one
full oscillation. Higher peaks have gone through more oscillations, as a result of this, they
are damped.
The first peak is sensitive to the total matter content, but it does not directly differentiates
between baryonic and DM. The ratio of the first to second peak heights provide the clearest
evidence for dark matter. If there were no dark matter, the first peak would have been
much lower (relative to the second peak), given that, without DM the gravitational wells
would have been weaker; thus, the baryons and photons would not have been compressed
as strongly. This would have lower significantly the height of the first peak.
2.1.7 Alternatives to Dark Matter
This description would not be complete if we did not mention alternatives to the dark
matter scenario. All of the evidences provided here rely heavily on the assumption that
we know the rules of gravity at all scales of energy. However, some authors are working
on cosmological scenarios without DM [44]. It is possible that, by modifying Einstein’s
gravity, we could avoid the necessity for DM [45, 46]. Among these, we find
MoNDs.
The theory of Modified Newtoninan Dynamics (MoNDs) proposed by Mordehai Mil-
grom [47] implies abandoning GR, which could be the case as physics has taught us
over the years4. MoNDs have been proven to be effective reproducing the observa-
tional data at the scale of galaxies [48]. Nevertheless, there is evidence for DM at
multiples scales, such as at Large Scale Structure formation (clustering) or the CMB
anisotropies, where MoNDs struggle [49].
TeVeS.
A relativistic generalization of MoNDs, was develop primarily by J. Bekenstein [50],
where he introduced tensor-vector-scalar gravity (TeVeS) [51]. Such TeVeS could be
considered as an effective field for DM as discussed in reference [52].
4For example when we move from classical mechanics in favor of Quantum Mechanics, we went from
deterministic theory to a probabilistic one. Or when we abandon the idea of an static universe proposed
by Einstein.
2. Dark Matter 35
Dark Matter
Particle
Dark Matter
Particle
Standard Model
Particle
Standard Model
Particle
Annihilation/Dispersion
Production (Colliders)
Annihilation (Indirect Detection)
S
ca
tt
er
in
g
(D
ir
ec
t
D
et
ec
ti
on
)
Figure 2.4: Possible interactions between dark matter (DM) and standard model (SM)
particles. The top red arrow represents a process of production (of DM), the bottom blue
arrow represents a process of indirect detection while the right orange arrow represents
scattering of DM and SM particles, direct detection.
Emergent Gravity.
A theoretical framework proposed by E. Verlinde [53] that suggest that gravity is
not a fundamental force but an emergent phenomenon arising from properties of
spacetime. This theory predicts deviations from GR at very large scales producing
additional effects which mimic the effects attributed to DM [54].
f(R) Gravity.
Is a modification of GR introducing a term f(R)5, into the Einstein-Hilbert action
[55], where f(R) can take various forms [56]. This additional term introduces cor-
rections to the Einstein field equations which become significant at low regions of
curvature (cosmic scales) [57, 58] or high curvature (near a black hole) [59]. How-
ever, the choice of f(R) is somewhat arbitrary and different functions lead to different
predictions, making this theory less constrained.
Based on the extensive experimental validations of GR, for example, the detection of
gravitational waves by LIGO [60] or the photo of the (shadow of the) supermassive black
hole at the center of galaxy M87 by the Event Horizon Telescope [61], among others, we
consider that GR is correct, hence, we need a new form of matter.
A most detailed description on some of the evidences on DM can be seen in the work done
by Bertone [62, 63], or the review by Garret and Gintaras [64]. We would also like to
5Where R is the Ricci scalar
36 2. Dark Matter
Figure 2.5: Current status for direct WIMPs searches, x-axis corresponds to multiple
values for the mass of the DM candidate and the y-axis show the values for the annihilation
WIMP-Nuclei cross section ⟨σv⟩. Credit to the Super Cryogenic DM Search [68].
mention the good review that goes beyond DM and into the history of Modern Cosmology
written by the Nobel prize winner physicists, P.J.E. Peebles [65].
2.2 Current status of Dark Matter
Despite all the gravitational evidence; there is still not a convincing signal of the existence
of DM particles [66]. If we introduced the so called Weakly Interacting Massive Particles
(WIMPs) as a candidate to be DM. The cross-sections for scattering between DM and SM
particles would, in theory, be large enough for direct detection [67]. In figure 2.3, we have
a visual representation of what we mean by direct detection.
Direct detection: Corresponds to DM + SM scattering into DM + SM. Orange
arrow on the right hand side of the diagram.
Indirect detection: Corresponds to DM + DM into SM + SM. Blue arrow at the
bottom of the image.
Production: Corresponds to SM + SM annihilating into DM + DM. Red arrow at
the top of the figure.
2. Dark Matter 37
Figure 2.6: Constraints on the spin-dependent part of the WIMP scattering cross section
versus its mass It also includes the ANTARES experiment expected allow region of detec-
tion. Illutration taken from reference [66].
2.2.1 Direct detection
Direct detection involves a scattering process
DM+ SM→ DM+ SM.
The experimental restrictions for direct detection by current and future gas noble based
experiments are shown in figure 2.5, which was taken from the Super Cryogenic Dark Mat-
ter Search collaboration [68]. The y-axis corresponds to different annihilation cross-section
⟨σv⟩ values, the lower the cross-section, the more difficult it is to detect. Meanwhile, the
x-axis shows the mass range of the candidate. The colored lines on the top correspond
to different experiments sensitivities; thus, the gray area above these colored lines corre-
sponds to the region of parameters where WIMPs have not been directly detected by said
experiments. Moreover, the yellow region at the bottom of the plot represents the neutrino
background. This is the region from which background signals coming from neutrinos be-
come indistinguishable from potential WIMP signals. Therefore if our WIMP had values
inside this region, it could not be detected directly. Consequently, the region of allowed
parameters for possible WIMP-like DM particles (that could be detected directly), lies in
the region below the experimental sensitivities and above the neutrino floor.
38 2. Dark Matter
Figure 2.7: Scatter plot of models for the EWkino scan in the Ωh2 on the Y -axis and
the mass m̃0
χ mass on the x-axis colored with RGB for (wino, bino, higgsino) respectively.
The horizontal black dotted line corresponds to the relic density abundance ΩDM,0h
2 =
0.11933± 0.00091. Illustration taken from reference [69].
2.2.2 Indirect detection
The fundamental nature of particle DM can be probed indirectly, without detecting the
DM particle itself, for example, through the process of DM annihilating or decaying into
SM particles, that is:
Decay: DM decaying into two SM particles, DM → SM + SM.
Annihilation: A pair of DM particles annihilating into two SM particles, that
means, DM + DM → SM + SM.
2. Dark Matter 39
In fact, none of the processes listed above is bound to necessarily occur in any model.
For example, in most models, DM is supposed to be absolutely stable, or at least with
a life-time comparable to life of the universe and annihilation processes could be highly
suppressed depending on the couplings. However there are still reasons to be optimistic,
in some extensions of the SM including a χ candidate, as some Minimal Supersymmetric
Standard Model (MMSM) extensions, there would be a non-vanishing coupling between χ
and SM that would allow for annihilation.
DM+DM→ SM + SM.
In figure 2.5, taken from reference [66], we show the constraints related to indirect detec-
tion of DM through gamma-ray observations. Where he have the expected detection area
of the Astronomy with Neutrino Telescope and Abyss enviromental RESearch (ANTARES)
experiment. This experiment aims to detect neutrinos by observing the Cherenkov radia-
tion6. The Cherenkov radiation that ANTARES is detecting is being produced by particles
created when neutrinos interacted with other particles like electrons, photons or neutrons.
The ANTARES collaboration expects to detect neutrinos with specific energy spectra and
directional patterns that can provide evidence for the presence of dark matter.
2.2.3 Wino status as the Lightest Supersymmetric Particle
There are some Minimal Supersymmetric Standard Models (MSSM), extensions of the SM
of particles, where the Lightest Supersymmetric Particle (LSP) is the Wino [70], as it has
also been suggested by Howard Baer [71]. A wino corresponds to the super-partner of the
W boson.
Due to R-parity conservation [72, 73], the LSP of most MSSM theories is a stable particle
and hence, an attractive candidate to be DM. This R parity, for example, implies that
supersymmetric (susy) particles are always produced (by interactions of SM particles) in
pairs (production at colliders). Alternatively, if we start with a susy particle at the initial
state, the allowed final states must always have one (or odd numbers off) susy particle(s)
and whatever number of SM particles allow by the rest of quantum numbers. Hence, if
the initial state is the LSP, this particle could not decay, because it requires at least one
susy particle in the final state, and by definition, the rest of the susy particles have larger
masses and a particle can not decay into itself. Therefore, the LSP of most MSSM models
is an attractive candidate to play the role of the WIMP.
Directly from the Atlas collaboration [69] we get figure 2.6. In this figure we have scattered
points that correspond to different masses (x-axis) of particles and their corresponding
relic density abundance Ωh2 (Y -axis). The horizontal black dashed line corresponds to
the relic density ΩDM,0h
2 = 0.11933 ± 0.00091, above the dashed line, the models are
excluded because of over production.
6The Cherenkov radiation is produced when high energy particles electrically charged move faster than
the speed of light in water.
40 2. Dark Matter
In this figure, they are considering three possible LSP: wino (s-partner of W boson, red
dots), higgsino (s-partner of the Higgs green) and Bino (s-partner of the photon, blue). We
can see that all three colored dots reach the black line, that means all three of them could
account for the total amount of DM relic density. One might say that there is a wider
range of masses for higgsino and bino like particles (compared to wino) that could be DM,
given that fewer red dots reach the dashed line. The reason as to why we are considering
winos over binos or higgsinos will be discussed later in the s-wave approximation.
2.3 Weakly Interacting Massive Particles
As we have seen, observational evidences support the idea of the existence of a form
of matter that is not visible, DM, that plays a fundamental role explaining the CMB
anisotropies measured today (among others); therefore, this form of matter must have been
present at the moment of recombination, see section 1.7. Based on this, we must study
the production of Dark Matter, at least, as early as the epoch of radiation domination.
During this era all particle species interacted with one another (including themselves),
and for most of their history, they remained in thermal equilibrium. However, there have
been a number of very notable departures from said equilibrium, for example, QCD and
Electroweak phase transition, neutrino decoupling, primordial BBN and recombination to
mention some. The departures from equilibrium have led to important relics such as the
light elements, the neutrino backgrounds, a net baryon number and so on.
In the hypothetical scenario where DM is thermal7, as a result of the universe expansion,
eventually, DM would also depart from thermal equilibrium. Example of thermal DM are
models based on Weakly Interacting Massive Particles (WIMPs), however, WIMPs is a
rather generic name to refer to all thermal candidates. As we discussed before, in the
present work we are considering it to be the Wino, which is the super-partner of the W
boson, and could be the LSP in some MSSM models.
2.3.1 Freeze-out
We have seen, in section 1.6, that a particle species in thermal equilibrium will have a
distribution function dictated by either the Fermi-Dirac (FD) or the Bose-Einstein (BE)
function. As the universe cools down, particles become non-relativistic (mi ≫ T ), if
these particles remain in thermal equilibrium, their density number becomes exponentially
suppressed by the temperature, equation (1.69a), i.e.,
ni = gi
(
miT
2π
) 3
2
exp
(
−mi − µi
T
)
. (2.5)
7Thermal DM, refers to DM that is composed of particles that have a sufficiently large interaction rate
with the SM particles to have remained in thermal equilibrium.
2. Dark Matter 41
In order for such particles to survive until the present time, they must have dropped out
of thermal equilibrium before mi/T becomes much larger than unity, or else we would not
see them today, ni ∼ 0. This decoupling occurs roughly when the interaction rate of the
particles ΓDM↔SM becomes smaller than the expansion rate H, that is
ΓDM↔SM ≳ H (coupled),
ΓDM↔SM < H (decoupled).
(2.6)
Once a species totally decouples from the plasma its evolution is very simple: since there
are no new particles being created or annihilated, the comoving density number of particles
Ni = nia
3 remains constant, that means that the density number ni decreases as a
−3; and
the particle momenta pi decreases as ∼ a−1, that is,
Decoupled species i: ni ∼ a−3, and pi ∼ a−1, (2.7)
the issue lies in describing the exact moment in which the freeze-out occurs.
2.3.2 Boltzmann equation
While equation (2.6) is surprisingly a very accurate approximation, in order to properly
treat decoupling one must follow the microscopic evolution of the particle’s phase space
distribution function, f = f(pµ, xµ, t), over time. When the particle in question is in a non-
equilibrium state, the evolution of the distribution function is governed by the Boltzmann
equation [74], which reads as
L̂[f ] = Ĉ[f ], (2.8)
where L̂ and Ĉ are the Liouville and the collision operators, respectively. The covariant
relativistic generalization of the Liouville operator is given by
L̂ = pµ
∂
∂xµ
− Γµαβp
αpβ
∂
∂pµ
, (2.9)
now let us simplify this equation using our cosmological principle.
Distribution function: In an homogeneous and isotropic universe, the phase space
distribution function would also be homogeneous and isotropic, that is,
f = f(|p|, t) = f(E, t). (2.10)
42 2. Dark Matter
Liouville operator: In the case of a spacetime described by the FLRW metric, the
Liouville operator readas as
L̂ = E
∂
∂t
−H |p|2 ∂
∂E
, (2.11)
where |p|2 is the physical momentum.
Collision operator: The collision term depends on all of the allowed interactions
between the species and other particles (including itself) that may alter the phase-
space distribution.
Assuming we are describing particle χ in a general process
χ+ a+ b+ · · · ←→ i+ j + · · · ,
then, the most general collision term is given by
Ĉ[fχ] = −
1
2
∫
dΠadΠb · · · dΠidΠj · · · (2π)4δ(4) (pχ + pa + · · · − pi − pj − · · · )
×
[
|M|2χ+a+b+···→i+j+··· fχfafb · · · (1± fi)(1± fj) · · ·
− |M|2i+j+···→χ+a+b+··· fifj · · · (1± fχ)(1± fa)(1± fb) · · ·
]
,
(2.12)
where fχ, fa, fb, . . . , fi, fj, . . . are the phase space distributions of χ, a, b, . . . , i, j, . . .
respectively, the (+,−) terms respectively apply to bosons and fermions and are
referred to as the blocking and stimulated emission factors; the matrix element
squared |M|2χ+a+b+···→i+j+··· for the process χ+a+b+· · · → i+j+· · · is averaged over
initial spins and summed over the final spins, and includes the appropriate symmetry
factors for identical particles in the initial or final states and the 4-dimensional delta
function enforces energy conservation,.
We also introduced the Lorentz invariant phase space element (LIPS element), dΠn,
defined as
dΠn ≡ gn
d3pn
(2π)3
1
2En
, n = {χ, a, b, i, j, . . . }, (2.13)
where gn measures the internal degrees of freedom of the n’th species.
After we substitute all of these terms into the Boltzmann equation, we multiply both sides
of the equation by 2dΠχ and integrate over all momenta d3pχ
8, we obtain
8The details are shown in appendix B.
2. Dark Matter 43
dn
dt
+ 3Hn = gχ
∫
d3pχ
(2π)3
1
Eχ
Ĉ[fχ].
= −
∫
dΠχdΠadΠb · · · dΠidΠj · · · (2π)4δ(4) (pχ + pa + · · · − pi − · · · )
×
[
|M|2χ+a+b+···→i+j+··· fψfafb · · · (1± fi)(1± fj) · · ·
− |M|2i+j+···→χ+a+b+··· fifj · · · (1± fχ)(1± fa)(1± fb) · · ·
]
,
(2.14)
on the left hand side we have used the definition of the number density nχ equation (1.61a),
nχ(t) ≡ n = gχ
∫
d3pχ
(2π)3
fχ(E, t). (2.15)
In principle, to solve equation (2.14), we would need to solve the corresponding Boltzmann
equations to get each and every single distribution function, fn = {fa, fb, . . . , fifj, . . . },
that appear in the collision term. This would form a set of integro-partial differential
equations known as Boltzmann hierarchy problem.
Fortunately, in the problem of interest, most species will be described by a phase space
distribution function in equilibrium (FD or BE), except for the particle of interests to us,
χ. This simplifies our problem to a single equation, given by (2.14), before attemting to
solve this equation, let us consider the following scenarios:
Absence of collisions: Let us consider the case of a species in the absence of
interactions, Ĉ = 0,
dn
dt
+ 3Hn =
1
a3
d
dt
(
na3
)
= 0. → n ∼ a−3, (2.16)
we get back our predicted result on equation (2.7).
Collision term simplifications: If we take a look at the right-hand side of Boltz-
mann equation (2.14) it looks rather complicated. There are a two well motivated
assumptions to make our computation simpler, those are:
CPT Invariance: The first is the assumption that our process is invariant
under Charge conjugation, Parity and Time reversal, which implies
|M|2χ+a+b+···→i+j+··· = |M|
2
i+j+···→χ+a+b+··· ≡ |M|
2. (2.17)
which is true for most processes known in physics, except, perhaps, in very
specific scenarios such as baryogenesis [75, 76].
44 2. Dark Matter
Maxwell-Boltzmann (MB) statistics: The next simplification is that we
can use MB statistics for all species in thermal equilibrium,
fn(En) ≡ Fn(En) exp
(
−En
T
)
, n = {χ, a, b, i, j, . . . }, (2.18)
while also assuming that at high energies, the chemical potential of all species
are negligible, En ≫ µn. In other words, we are assuming the absence of
Bose condensation and Fermi degeneracy, so that the blocking and stimulated
emission factors, (1± fn) in the collision operator, can be ignored,
1± fn ≈ 1, n = {χ, a, b, i, j, . . . }. (2.19)
That is, we are considering that the quantum statistical effects are negligible
for all the species, which is a good approximation when occupation numbers
are small.
Under these two assumptions, the Boltzmann equation is simplifies to
dn
dt
+ 3Hn = −
∫
dΠχdΠadΠb · · · dΠidΠj · · · [fχfafb · · · − fifj · · · ]
× (2π)4δ(4)(pχ + pa + pb + · · · − pi − pj − · · · ) |M|2 .
(2.20)
2.3.3 Radiation domination era
Let us solve the Boltzmann equation (2.20) for DM during the epoch of radiation dom-
ination. To do so, suppose that our DM particle, χ, is stable, so that only annihilation
and inverse annihilation processes are predominant, that is
χ̄χ↔ ψ̄ψ, (2.21)
here ψ denotes all species into which all the χ’s, can annihilate. In addition we assume
there is no asymmetry between particles and antiparticles so that the degrees of freedom
are equal, gχ̄ = gχ, and gψ̄ = gψ, therefore equation (2.20) simplifies to
ṅ+ 3Hn = −
∫
d3k1
(2π)3
gχ
2k01
d3k2
(2π)3
gχ
2k02
d3p1
(2π)3
gψ
2p01
d3p2
(2π)3
gψ
2p02
(2π)4
× δ(4)(p1 + p2 − k1 − k2)
[
fχ(p1)fχ̄(p2)− fψ̄(k1)fψ(k2)
]
|M|2χ̄χ↔ψψ .
(2.22)
Now, assuming that the deviation from thermal equilibrium of the DM is momentum-
independent, as equation (2.18), fχ(p1) ≡ Fχ exp
(
−p01/T
)
.
2. Dark Matter 45
Using energy-momentum conservartion, we get
fχ(p1)fχ(p2) = Fχf
(eq)
χ (p01)Fχf
(eq)
χ (p02)
= F 2
χ exp
(
−p
0
1 + p02
T
)
= Fχfψ(k
0
1)Fχfψ(k
0
2),
where we used energy momentum conservation, and we get Fχ by using nχ = Fχn
(eq)
χ ,
substituting into expression (2.22),
ṅ+ 3Hn = − 1
(neq)2
∫
d3p1
(2π)3
gχ
2p01
d3p2
(2π)3
gχ
2p02
d3k1
(2π)3
gψ
2k01
d3k2
(2π)3
gψ
2k02
× (2π)4δ(4)(p1 + p2 − k1 − k2) |M|2χ̄χ↔ψψ f
eq
χ (p1)f
eq
χ (p2)
[
n2 − n2
eq
]
,
(2.23)
where the density number in equilibrium, neq, is simply
neq ≡ gχ
∫
d3p
(2π)3
fχ,eq(E, t). (2.24)
Defining the thermally-averaged annihilation cross section, ⟨σχ̄χ→γγ|v|⟩,
⟨σχ̄χ→ψψ|v|⟩ ≡
1
(neq)2
∫
d3p1
(2π)3
gχ
2p01
d3p2
(2π)3
gχ
2p02
d3k1
(2π)3
gψ
2k01
d3k2
(2π)3
gψ
2k02
× (2π)4δ(4)(p1 + p2 − k1 − k2) |M|2χ̄χ↔ψψ f
(eq)
χ (p1)f
(eq)
χ (p2).
(2.25)
In terms of the thermally averaged annihilation cross section, Boltzmann equation reads
ṅ+ 3Hn = −⟨σχ̄χ→ψψ|v|⟩
[
n2 − n2
eq
]
. (2.26)
It is useful to scale out the effect of the expansion of the Universe by considering the
evolution of the number of particles in a comoving volume. We know that after they
decouple, they will redshift as n ∼ a−3, equation (2.16), and we also know that the
entropy density also evolves as s ∼ T 3 ∼ a−3, equation (1.80). Combining these two
quantities we define the Yield DM, Ỹ by,
Ỹ ≡ n
s
∼ a−3
a−3
∼ constant. (2.27)
Using entropy conservation, sa3 = constant, it follows that ṡ = −3Hs, applying this
condition to the left-hand side of equation 2.25, we get, ṅ+ 3Hn = s ˙̃Y .
46 2. Dark Matter
Furthermore, it is useful to introduce the dimensionless variable, x, as
x ≡ mχ
T
, (2.28)
where mχ is any convenient mass scale, in this case taken as the mass of the particle of
interest, χ. During the radiation-dominated epoch x and t are related by
t = 0.301g
− 1
2
⋆
Mp
T 2
= 0.301g
− 1
2
⋆
Mp
m2
χ
x2. (2.29)
where Mp is the reduced Planck mass that appears in Friedmann equation (1.60) and g⋆
are the relativistic degrees of freedom available. By the chain rule we get we get that,
d
dt
= xH(x)
d
dx
, (2.30)
the Hubble parameter H(x) is given by H2(x) = H2(mχ)/x
4. In terms of these new
variables the Boltzmann equation is written as
dỸ
dx
= − λ̃
x2
[
Ỹ 2 − Ỹ 2
eq
]
, (2.31)
where we have defined
λ̃
x2
≡ xs
H(mχ)
⟨σχ̄χ→ψψ|v|⟩, (2.32)
and H(mχ) =
√
gρ/(30π2)m2
χ/Mp. Note that the thermally averaged cross section, in
general, depends on the temperature, thus, one can express λ̃ as
λ̃ = λ̃0x
−n. (2.33)
Therefore equation (2.31) can be written as
dỸ
dx
= −λ̃0x−(n+2)
[
Ỹ 2 − Ỹ 2
eq
]
. (2.34)
which is the Boltzmann equation for the Yield DM, Ỹ = n/s, during radiation domination,
and Ỹeq is obtained by substituting equation (2.24) into expression (2.27), i.e.,
Ỹeq ≡
neq
s
=
gχ
s
∫
d3p
(2π)3
fχ,eq(E, t). (2.35)
2. Dark Matter 47
2.3.4 Solutions to the Boltzmann equation
We are looking for solutions to the Bolzmann equation given by expression (2.34). In the
case of wino DM, these winos would be fermion like particles, so in equilibrium they will
be described by a Fermi-Dirac distribution function
fχ,eq = fFD =
[
exp
(
Eχ
T
)
+ 1
]−1
, (2.36)
in terms of our dimensionless variable x = mDM/T , we get the equilibrium yield as
Ỹeq(x) =
45
4π4
gχ
g⋆
∫ ∞
0
dy
y2
exp
(
√
y2 + x2
)
+ 1
, (2.37)
where g⋆ are the effective relativistic degrees of freedom contributing to entropy, those
shown in figure 1.2.
After freeze-out: The number density, Ỹ , is much larger than the equilibrium one,
Ỹeq, so equation (2.34) simplifies to
dỸχ
dx
≈ −λ̃0x−(n+2)Y 2
χ , (2.38)
this equation can be integrated with respect to x, from the time of freeze-out, xfo,
to later times, x =∞, we get
1
Ỹfo
− 1
Y∞
= − λ̃0
n+ 1
x
−(n+1)
fo . (2.39)
where Ỹfo is the yield at freeze-out and by Ỹ∞ represents the yield at much later
times, and it is given by
Y∞ ≈
π2
ζ(3)
(
π2g⋆
90
)
1
2 (n+ 1)xf
Mpmχ⟨σχ̄χ↔ψ̄ψ⟩
. (2.40)
In general we may expand the thermally averaged annihilation cross section ⟨σχ̄χ→ψψ|v|⟩
as a power series [77, 78]
⟨σχ̄χ→ψψ|v|⟩ = σs + σpv
2 +O(v4), (2.41)
48 2. Dark Matter
0.01 0.10 1 10 100
10-6
10-5
10-4
0.001
0.010
Figure 2.8: Numerical solutions to Boltzmann equation (2.42). In the vertical axis we
have the dimensionless variable x = mχ/T and on the vertical axis the Yield, Ỹ = n/s.
The black dashed line the corresponds to the Yield in thermal equilibrium while the lines
in blue, red and purple correspond to different values of of the constant λs .
In the s-wave approximation9, we get that that ⟨σχ̄χ→ψψ|v|⟩ ≈ σs, that is, it is constant
with respect to the relative velocity. Using equation 2.32 we get
⟨σχ̄χ→ψψ|v|⟩ =
1
s
λ̃0x
−(n+3)H(mχ) = constant, (2.42)
in order for this cross section to be constant, assuming the relativistic degrees of freedom
does not changes much, g⋆(T ) ≈ constant. We get that n = 2, in our series expansion
(2.33), corresponds to the s-wave approximation.
Therefore, the s-wave approximation for our Boltzmann equation is expressed as
dỸ
dx
= − λ̃s
x2
[
Ỹ 2 − Ỹ 2
eq
]
, (2.43)
The solutions to different values of the constant λ̃s are shown in figure 2.8. In the horizontal
axis we have the dimensionless variable x = mχ/T , and on the horizontal axis we have the
9The name s-wave comes from quantum mechanics. When solving the Schrödinger equation for a par-
ticle moving in a central potential, V (r) (the hydrogen atom), the quantum state with angular momentum
ℓ = 0 is called S orbital, while ℓ = 1 (spherically distributed) corresponds to P orbital, higher orbitals are
D, F,. . . , thereby, the name s-wave, p-wave and so on.
2. Dark Matter 49
yield Ỹ = n/s. The black dotted line corresponds to the yield in equilibrium, Ỹeq, which
is being exponentially suppressed, see equation (2.5). The remaining three lines in blue,
red and purple are numerical solutions to the Boltzmann equation with different values of
the thermally averaged annihilation cross section, the values used correspond to
s-wave =





λ̃
(1)
s = 103,
λ̃
(2)
s = 105,
λ̃
(1)
s = 106.
(2.44)
The bigger the strength of the coupling, the longer the species remains in thermal equi-
librium and the more suppressed it gets [79].
2.3.5 Dark Matter relic density computation
Finally, after the species has decoupled from thermal equilibrium, how do we compute the
relic density ΩDM,0 from it. To do this, we only need to use the definition of the density
parameter
ΩDM,0h
2 ≡ ρDM,0
ρc,0
h2 =
mχnχ
ρc,0
h2 =
mχs0Ỹ0
ρc,0
h2. (2.45)
where Ỹ0 is the yield today10 (after freeze-out), s0 is the entropy density today, mainly, the
background radiation of CMB photons TCMB,0 = 2.726 K, ρc is the critical energy density
today given by equation (1.49) and little h was introduced in equation (1.30).
In our three solutions plotted in figure 2.8, we used mχ = 100 GeV. Following up on
expression (2.32) we get
λ =
x3s
H(mχ)
⟨σv⟩ = 2g⋆π
15
√
10
gρ
mχMp⟨σv⟩, (2.46)
from this expression we may determine the value of ⟨σv⟩ used to get the corresponding
values of λ reported in expression (2.44). In order to reproduce the correct relic density
abundance in equation (2.45) we need a thermally averaged ⟨σv⟩ ∼ 10−10 GeV−2.
Finally, the thermally averaged annihilation cross section defined in equation (2.25), using
that the rest of particles are in thermal equilibrium may be written as
⟨σv⟩ =
∫ ∫
d3p1d
3p2(σs + σpv
2)e−E1/T e−E2/T
∫ ∫
d3p1d3pe−E1/T e−E2/T
. (2.47)
10The value of this is obtained from our numerical solutions plotted in figure 2.3.
50 2. Dark Matter
In this expression we used the s-wave and p-wave approximation. We consider winos
because they have been studied to have an s-wave dominated channel of annihilation into
W bosons by T. Moroi and L. Randall [80], where in the non-relativistic limite we can
write
⟨vσχχ̃→W+W−⟩ = g42
2π
1
m2
χ
(1− xW )3/2
(2− xW )2
, (2.48)
where g2 is the gauge coupling constant of SU(2)L, which depends on the energy scale of
the breaking of supersymmetry, and
xW =
m2
W
m2
χ
. (2.49)
Thereby, we consider Winos as our WIMPs because they satisfy the s−wave approxima-
tion.
51
Chapter 3
Inflation
3.1 Limitations of the Standard Model of Cosmology . . . . . . . . . . . . . . 51
3.2 Cosmological Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Primordial Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Problems revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Inflationary potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
The cosmological standard model describes the evolution of the universe from an era of
radiation domination to the formation of galaxies up until today in a consistent manner.
It is impressive how a model with very few hypothesis and parameters can have so many
predictions. However, when putting some numbers, it seems that this model exhibits some
issues in the very early stage of the Universe [81, 82]. In this chapter we will introduce these
shortcomings of that Standard Model of Cosmology and a solution to them in the form
of an epoch of exponential expansion of the universe, preceding the radiation domination
era, known as Inflationary cosmology [83, 84].
3.1 Limitations of the Standard Model of Cosmology
The observations we have today point out to some problems when applying the Standard
Model of Cosmology to very early times in the Universe, these issues are known as:
The Horizon problem.
The first problem that was noticed, is usually referred to as the horizon problem or
the homogenity and isotropy problem. Why is the universe spatially homogeneous
and isotropic at large scales?.
52 3. Inflation
When we look at the CMB photons, we get an almost homogeneous distribution of
the temperature across the entire universe, except for small anisotropies, δT , which
deviate from the CMB temperature by δT/TCMB ∼ 10−3 to 10−5. That means that
the temperature of the entire universe at the epoch of recombination was almost the
same in each and every single region of it, given consideration to, the fluctuations of
the order of micro Kelvin (10−3).
Although, if we compute the maximal distant that light could have traveled in a
FLRW from t = 0 to the moment of recombination, t = tCMB
1, we get that around
∼ 104 regions would have been causally-disconnected regions. Even so, we measure
that they all have the same temperature.
In a simple manner, this means that photons CMB we received from one side of the
universe have the same temperature as the ones coming from the opposite direction.
The Flatness Problem.
This problems can be easily understood if one wonders why Euclidean geometry
is such a good approximation in our universe. Euclidean geometry is the geometry
where the internal angles of all triangles add up to π and parallel lines never intersect
with each other.
This is a problem of fine tuning in the sense that the universe should have the exact
energy density today equal to the critical density, ρc,0, using equation (1.49),
ρc,0 = 3M2
pH
2
0 = 1.878× 10−9h2 g cm−3, (3.1)
and from our density parameters today given by equation (1.58), we have ΩK,0 =
0.0007 ± 0.00192, that means that today we have an almost spatially flat universe,
that is
|1− Ωtot,0| ≤ 0.007. (3.2)
Why should the total density today be equal to one, Ωtot,0 ≈ 1?. Using Friedmann
equation (1.60), when the universe was dominated by radiation and matter, early
times, the density parameter evolved as
1− Ω(t) =
(1− Ωtot,0)a
2
Ωr,0 + Ωm,0a
∼
{
a2 ∼ t Radiation domination,
a ∼ t2/3 Matter domination.
(3.3)
Using the density parameters given in expression (1.58), we may estimate the value
of the density parameter at the moment of radiation-matter equality, Ωrm, or even
at the moment of Big Bang Nucleosynthesis (BBN), ΩBBN and get
|1− Ωrm| ≤ 10−6,
|1− ΩBBN| ≤ 10−16,
(3.4)
1Assuming at t = 0 we had radiation domination.
2Notice that we remove the factor h2 from the value reported on equation (1.58).
3. Inflation 53
where we can clearly see that the problem only gets worse as we go back in time [85].
If we go back to the Planck scale, assuming the universe was radiation dominated,
the scale factor at this Planck scale, aP , will be given by aP = TCMB,0/TP ∼ 10−32,
therefore, |1− ΩP | ∼ 10−62.
It seems that in order to have an spatially flat universe today, we need to fix the
values of universe contents to 1062 digits, at the beginning of the universe. Thus we
need a fine tuning in the initial conditions.
Magnetic Monopole problem.
This is a theoretical issue arising from the lack of magnetic monopoles in the universe.
These magnetic monopoles are hypothetical stable field configurations that carried
a net magnetic dipole moment and they appear as a prediction in some Grand
Unification Theories (GUTs). A GUT is a field theory that attempts to unify the
electromagnetic force, the weak and the nuclear force. Unification of forces has
been a goal for theoretical scientists since the 1870, when James Maxwell unified
electricity and magnetism into the electromagnetic field. About a century later, in
1982, Weinberg, Salam and Glashow [86] successfully demonstrated that at an energy
EEW ∼ 1 TeV, the electromagnetic and the weak force unified in the electroweak
force. Later, when the predictions of the electroweak theory were experimentally
confirmed, they received their Nobel prizes.
Physicists, like Alan Guth [87], estimated a GUT scale EGUT ∼ 1016 GeV, the
breaking of these symmetry could produce magnetic monopoles, though, there is no
convincing evidence for the existence of them [88].
3.2 Cosmological Inflation
All of these problems can be solved at once introducing an epoch known as cosmological
inflation. The idea is that earlier than the CMB photons were emitted, the universe was
in a much smaller size such that most of the universe was causally connected, then, it
expanded exponentially fast flattening all initial perturbations.
The simplest realization of inflationary cosmology [89, 90] is achieved by introducing a
single real classical scalar field, called the inflaton field, denoted by ϕ, that has an action
Lϕ =
1
2
gµν∂µϕ∂νϕ− V (ϕ), (3.5)
with an action given by
Sϕ =
∫
d4x
√−gLϕ =
∫
d4x
√−g
(
1
2
gµν∂µϕ∂νϕ− V (ϕ)
)
, (3.6)
where g = det(gµν).
54 3. Inflation
3.2.1 Continuity equation of the inflaton
Variation of the inflaton action, Sϕ, with respect to metric transformations, leads us to
δgSϕ =
1
2
∫
d4x
√−g (−gµνLϕ + ∂µϕ∂νϕ) δg
µν =
1
2
∫
d4x
√−gT ϕµνδgµν = 0, (3.7)
from this we identify the stress-energy tensor associated with the inflaton as
T ϕµν = ∂µϕ∂νϕ− gµνLϕ, (3.8)
which comes with a the corresponding conservation law given by expression (1.28), i.e.,
∇µTµν = 0. We assume that the stress-energy tensor associated to the inflaton field, Tµν ,
has the general form of a perfect fluid, as equation (1.34). Therefore, the energy density
of the inflaton, ρϕ, is given by
ρϕ =
1
2
ϕ̇2 +
1
2a2
(∇ϕ)2 + V (ϕ), (3.9)
and the pressure of the inflaton field, pϕ given by the (i− i) component, reads as
pϕ =
1
2
ϕ̇2 − 1
6a2
(∇ϕ)2 − V (ϕ). (3.10)
Notice here that if both, the time variations, ϕ̇, and the spatial variation, ∇ϕ, are negligi-
ble, then, ρϕ = V (ϕ) = −pϕ, that is, wϕ = −1, like a cosmological constant. This means
that the scale factor grows exponentially fast, see equation (1.47).
3.2.2 Klein-Gordon Equation
Applying the principle of least action with respect to the field ϕ to the inflaton action,
δϕSϕ =
∫
d4x
[
−∂µ
(√−ggµν∂νϕ
)
− ∂V (ϕ)
∂ϕ
√−g
]
δϕϕ = 0, (3.11)
The path that extremizes the action, substituting the FLRW given by equation (1.31), is
1√−g∂µ
(√−ggµν∂νϕ
)
+
∂V (ϕ)
∂ϕ
= ϕ̈+ 3Hϕ̇− 1
a2
∇2ϕ+
∂V (ϕ)
∂ϕ
= 0. (3.12)
which is known as the Klein-Gordon (K-G or KG) equation.
3. Inflation 55
Alternatively, a shorter approach to deduce the equation of motion is starting with the
conservation equation associated with the energy-momentun tensor, equation (1.60), and
substitute for the inflaton energy density, equation (3.9) (ρ → ρϕ), and the pressure,
equation (3.10) (p→ pϕ). Doing this, as we anticipated, we also get the KG equation
ϕ̈+ 3Hϕ̇− 1
a2
∇2ϕ+ V ′(ϕ) = 0, (3.13)
where V ′(ϕ) = ∂ϕV (ϕ).
If we assume that the inflaton field is spatially homogeneous, then, ∇2ϕ = 0. The KG
equation shows that ϕ behaves like a particle rolling down a potential, V (ϕ), and that is
being slowed down by a frictional term represented by 3Hϕ̇, due to the expansion of the
universe.
Notice also that this equation assumes a stable field ϕ, if, as we will discuss in the following
chapter, the inflaton decays into lighter particles, one needs to add a friction term to the
equation of motion, we do this by simply adding to the right hand side of the equation a
term −Γϕϕ̇, where Γϕ is the width of the inflaton. Thereby, in the most general case of a
decaying inflaton field, the KG equation is expressed as
ϕ̈+ 3Hϕ̇+
1
a2
∇2ϕ+ V ′(ϕ) = −Γϕϕ̇. (3.14)
Equation (3.11) describes how the kinetic, potential and frictional energy of the inflaton
field evolve over time.
One can solve this equation in two different regimes, the slow-roll (section 3.2.3) and the
coherent oscillations regimes (chapter 4).
3.2.3 The Slow-roll Regime
Assuming that the inflaton field is stable, Γϕ = 0, and that it is spatially homogeneous,
∇2ϕ = 0, the slow-roll approximation considers that at the beginning the acceleration,
ϕ̈, is negligible, compared to the potential gradient, V ′(ϕ) and to the friction term, 3Hϕ̇,
that means, ϕ̈≪ −3Hϕ̇ and ϕ̈≪ V ′(ϕ). In this approximation the KG equation (3.14)
simplifies to
3Hϕ̇ ≈ −V ′(ϕ). (3.15)
Remarking that during the slow-roll, ϕ is almost constant (ϕ̇2 ≪ V (ϕ)),
H2 =
1
3M2
p
ρϕ =
1
3M2
p
[
1
2
ϕ̇2 + V (ϕ)
]
≈ 1
3M2
p
V (ϕ). (3.16)
56 3. Inflation
Then, our equation of motion (3.15) can be written as
ϕ̇2 ≈
M2
p
3
V ′(ϕ)2
V (ϕ)
≪ V (ϕ), (3.17)
thereby, we define our first slow roll parameter, ϵV , as
ϵV ≡
M2
p
2
(
V ′(ϕ)
V (ϕ)
)2
, (3.18)
which in a good approximation tell us that the expansion of the universe is accelerated as
long as ϵV ≪ 1. Alternatively, using the chain rule, we may express equation (3.17) as
ϕ̈ =
d
dt
ϕ̇ = −ϕ̇ d
dϕ
V ′(ϕ)
3H
=
V ′′(ϕ)V ′(ϕ)
9H2
≪ V ′ϕ, (3.19)
this leads us to our second slow-roll parameter, namely ηV , defined as
ηV ≡M2
p
V ′′(ϕ)
V (ϕ)
. (3.20)
Therefore, the slow-roll parameters are given by equation (3.18) and (3.20), i.e.,
ϵV ≡
M2
p
2
(
V ′(ϕ)
V (ϕ)
)2
,
ηV ≡M2
p
V ′′(ϕ)
V (ϕ)
.
(3.21)
Using the acceleration equation (the second Friedmann equation given by expression
(1.41)) during the slow-roll regime, and asking for ä > 0,
ä
a
= − 1
6M2
pl
(ρ+ 3p) > 0, (3.22)
during this epoch the inflaton is the dominant source of energy, ρ→ ρϕ and p→ pϕ,
pϕ < −
1
3
ρϕ → wϕ < −
1
3
. (3.23)
That is, inflation is equivalent to
Inflation: ↔ ä > 0, ↔ ωϕ < −
1
3
, ↔ ϵV , ηV ≪ 1. (3.24)
3. Inflation 57
As we know, the end of inflation is determined by the condition ä = 0, which is equivalent
to ϵV = 1. This happens when the kinetic energy becomes comparable to the potential
energy. Using equation (3.19), we get that at end of inflation ρϕ(aend) + 3pϕ(aend) = 0,
where, aend denotes the value of the scale factor at the end of inflation. Using our expres-
sions for both the energy density (3.6) and pressure of the inflaton (3.7), we get
ϕ̇2
end = V (ϕend), (3.25)
therefore, the energy density at the end of inflation is given by
ρend =
1
2
ϕ̇2
end + V (ϕend) =
3
2
V (ϕend). (3.26)
Also, some define the slow-roll parameters as a function of the Hubble [91] as
ϵH ≡ −
Ḣ
H2
,
ηH ≡
ϵ̇H
HϵH
.
(3.27)
and we get inflation as long as ϵH and ηH are smaller than unity. A quick computation
verify us that, during the slow-rollow regime, these parameters are equivalent to ϵV , ηV .
Alternatively, we can verify that during inflation
d
dt
1
aH
= −ϵHH2 < 0, (3.28)
because during inflation ϵH > 0 (smaller than one, though). Therefore, during inflation,
the comoving Hubble horizon is decreasing [92].
3.3 Primordial Perturbations
The subject of primordial perturbations is beyond the scope of the present work, however
we would like to briefly introduce some approximations in the study of these.
3.3.1 Scalar perturbations
During inflation, perturbations on the inflaton field are present due to quantum fluctua-
tions, at the linear level we may write
ϕ(x, t) = ϕcl(t) + δϕ(x, t), (3.29)
58 3. Inflation
where ϕcl(t) represents the classical homogeneous part of the field and δϕ(x, t) are small
(quantum) fluctuations of the field.
The equation of motion of the fluctuations, δϕ(x, t), is obtained after substituting ex-
pression (3.29) into the KG equation (3.14) (assuming both the classical field and the
fluctuations are stable), at the linear level in fluctuations, we get,
δ̈ϕ+ 3H ˙δϕ+
(
− 1
a2
∇2 + V ′′(ϕcl)
)
δϕ = 0. (3.30)
It is convenient to work in Fourier space rather than real space, the Fourier transform of
δϕ(x, t) reads
δϕ(k, t) =
∫
d3x
(2π)3
δϕ(x, t)eix·k, (3.31)
and the KG equation for δϕ in Fourier space reads as
δ̈ϕ+ 3H ˙δϕ− k2
a2
δϕ+ V ′′(ϕcl)δϕ = 0. (3.32)
Assuming that the scalar perturbations in the metric, in the Newtonian Gauge3, can be
written as
ds2 = (1 + 2Φ)dt2 − a2(t)(1− 2Ψ)δijdx
idxj, (3.33)
where Φ plays the role of the Newtonian potential4 and Ψ reflects spatial distortions.
Instead of solving for δϕ, we can characterized scalar fluctuations in terms of the gauge-
invariant Mukhanov-Sasaki variable, Q. In particular, in the Newtonian gauge is defined
in terms of the fluctuation δϕ and the scalar metric perturbation Ψ as [94]
Q = δϕ+
ϕ̇
H
Ψ =
∫
d3k
(2π)3
[
Qk(t)âk +Q∗
k(t)â
†
−k
]
, (3.34)
where the dot denotes a derivative with respect to cosmic time, and â†k and âk denote
creation and annihilation operators, respectively.
These operators satisfy the commutation relations
[
âk, â
†
k′
]
= δ(k − k′), and
[
âk, âk′
]
=
[
â†k, â
†
k′
]
= 0. (3.35)
3The Newtonian gauge is a specific choice of gauge in cosmological perturbation theory that simplifies
the equations. For more details on different cosmological gauges, referred to [93].
4That is were the name of the gauge comes from.
3. Inflation 59
Figure 3.1: Evolution of a fundamental mode, from subhorizon to the superhorizon mode,
where it freezes in. Illustration taken from reference [18].
We get that mode functions of Qk satisfy the equation of motion
Q̈k + 3HQ̇k +
[
k2
a2
+
3ϕ̇2
M2
p
− ϕ̇4
2H2M4
p
+ 2
ϕ̇Vϕ
HM2
p
+ Vϕϕ
]
Qk = 0, (3.36)
which is known as Mukhanov-Sasaki (MS or M-S) equation [95, 96]. We may solve this
equation during the slow-roll regime using Bunch-Davies [97] initial conditions,
Qk≫aH =
exp(−ikτ)
a
√
2k
, (3.37)
with adτ = dt being the conformal time5. During the slow-roll regime, Qk has solutions
given by
Qk =
√
π
2
ei(ν+
1
2)
π
2
√
−τ
a
H(1)
ν (−kτ), (3.38)
where we are using ν ≡ 3/2 + ϵH + ηH/2, defined in terms of the slow-roll parameters,
given in equation (3.27). The solution also involves a function H
(1)
ν which are the Henkel
functions of the first kind defined in terms of the Bessel functions, see reference [98].
The curvature perturbation, R, is determined by
Rk =
H
ϕ̇
Qk. (3.39)
5In this conformal time, the FLRW metric gµν is written conformal to a Minkowski metric, ηµν , i.e.,
gµν = a2ηµν
60 3. Inflation
Comoving Scales
Comoving
Horizon
Time  [ln(a)]
Inflation
Reheating
Radiation
horizon exit
density
fluctuation
horizon re-entry
Figure 3.2: Evolution of the comoving horizon as a function of the scale factor a.
Substituting our solution (3.38), we get
Rk(τ) =
H
ϕ̇
Qk(τ) =
√
π
2
ei(ν+
1
2)
π
2
√
−τH
aϕ̇
H(1)
ν (−kτ). (3.40)
In the real space the scalar curvature perturbation, R, has a correlation function (of two
points) denoted by ξR, that is,
ξR = ⟨R(x)R(x+ r)⟩. (3.41)
the correlation function tells us how the scalar perturbation at two points spatially sepa-
rated by a distance r is related. In Fourier space we may write
R(x) =
∫
d3k
(2π)3
R(k)eik·x. (3.42)
In Fourier space, the power spectrum of scalar perturbations, is given by
⟨R(k)R∗(k′)⟩ = 2π2
k3
PRδ(k − k′). (3.43)
where PR is the power spectrum of scalar perturbations6.
6The power spectrum, PR, is related to the correlation function, ξR(r), through a Fourier transform
ξR(r) =
∫
d3k
(2π)3
PR(k)eik·r.
3. Inflation 61
At the linear level, the power spectrum is given by
PR(k) =
k3
2π2
|Rk|2 , (3.44)
and using the solution (3.39) we get
PR(k) =
H4
4π2ϕ̇2
(
k
aH
)−2ϵH−ηH
≡ As
(
k
aH
)ns−1
, (3.45)
where we defined the amplitude of the scalar power spectrum, As, as As = H4/(4π2ϕ̇2)
and the tilt of the anysotropy spectrum, ns, as ns = 1 + (d lnPR)(d ln k) = 1− 2ϵH − ηH .
Therefore, we can parametrize the power spectrum as
PR = As
(
k
k∗
)ns−1
, (3.46)
where As and ns are evaluated at the end of inflation,
As∗ = PR(k∗)
∣
∣
a=aend
, and ns = 1 +
d lnPR
d ln k
∣
∣
∣
a=aend
. (3.47)
3.3.2 Tensor perturbations
In the case of tensor perturbations, we use standard transverse traceless perturbation
[99, 100], in this case we need to solve for the mode hk,γ
ḧk,γ + 3Hḣk,γ +
k2
a2
hk,γ = 0, (3.48)
where γ = +,× denotes the two polarization modes [100, 101], we get the corresponding
power spectrum [102]
∑
γ=+,×
〈
hk,γh
∗
k′,γ
〉
=
2π2
k3
PT δ(k − k′). (3.49)
Tensor perturbations correspond perturbation in the metric, these can be associated to
gravitational waves [103]. Finally, let us define the ratio of tensor to scalar perturbations,
r, as
r ≡ PT
PR
. (3.50)
62 3. Inflation
In an expanding universe the physical momentum k of a fluctuation δϕ redshifts as a−1.
At early times, when the scale factor a is extremely small, we got sub-horizon modes.
As the universe expands, the wavelength of the fluctuation stretches while the comoving
Hubble horizon reduces, at some point, their size are equal, see figure 3.1.
When the wavelength is larger than the comoving horizon, these flutuations (of physical
moment k) freeze-out, they remain constant in a super-horizon mode. After inflation, when
the comoving horizon grows (during reheating and radiation domination) these fluctuations
that were in the super-horizon limit re-enter the horizon, see figure 3.2. These fluctuations
could explain the anisotropies we measured in the CMB, that were the origin of large scale
structure formation [104], codified in the observational measurements abut ns and r.
For a detail treatment of primordial fluctuations we recommend the lectures by A. Riotto
[105].
3.4 Problems revisited
Now we will briefly discuss how this inflationary epoch solves the problems described in
section 3.1.
Horizon problem.
The comoving horizon7 dH , is obtained using the fact that light propagates in null
geodesics, ds2 = 0.
Using the FLRW metric (1.31), we get that dH is given by
dH =
∫
dt
a(t)
, (3.51)
During inflation a ∼ exp(Ht), therefore, dH ∼ exp(−Ht)/H < 1 it is exponentially
decreasing. Meaning that the entire observable universe was within a causal region
such that thermal equilibrium could be established.
Flatness problem.
Using expression (1.57), we have
Ωtot = 1 + ΩK → Ωtot − 1 =
K
a2H2
, (3.52)
During inflation, a(t) grows exponentially fast, making the term K/a2 negligible
compared to H−2 ∼ constant. This drives Ωtot to be exactly one. Any initial
curvature was flatten during this era. Inflation does not require the fine-tunning of
the initial conditions in the same way as the standard Big Bang models. Therefore,
inflation is the reason why our universe is spatially flat today.
7This comoving horizon measures the maximum distance light can travel up to a given time t.
3. Inflation 63
Monopole problem:
According to GUT theories, monopoles were created before inflation started. Once
inflation begins, the universe expands exponentially fast, diluting the number of any
pre-existing monopoles.
3.4.1 Alternatives to inflation
Although, as we will describe in the subsequent section, multiple inflationary models
(based on a single field, as the general treatment described before) can predict the primor-
dial tilt ns and the tensor to scalar ratio r compatible with CMB measurements. There
exist alternative scenarios to solve the shortcomings of moderns cosmology. Here, we
present some of these alternatives, although, let us be clear that we are not attempting
to describe any of them, we are limiting ourselves to make mention of a few generalities
about them.
f(R) Gravity.
As with the alternatives to Dark Matter, introducing a term f(R) into the Einstein-
Hilbert action (EH) [106], one can produce an epoch of accelerated expansion without
using an external field like the inflaton [107, 108]. This could also be applied to
explaining Dark Energy [109]. As we mention in chapter 2, the election of f(R) is,
somewhat, arbitrary [110] and they are complicated to be constrained [111].
Loop Quantum Cosmology.
Loop Quantum Cosmology (LQC) corresponds to applying the methodology of Loop
Quantum Gravity into Cosmology [112]. In LQC inflation is replaced by a quantum
bounce which is followed by a super-inflation phase. In these scenarios, the ex-
ponential expansion is caused by the quantum properties of the spacetime itself,
avoiding the universe to collapse into a singularity before bouncing [113]. The lack
of predictions experimentally verified about Loop Quantum Gravity is still an issue
[114, 115].
String Cosmology.
Models based on String Theory that are being study as possibles alternatives to the
inflationary paradigm [116, 117]. Or even more hypothetical alternatives based on a
concept known as branes [118, 119]. Needless to say, string theory is also a theory
lacking predictions experimentally verified [120].
The work being done on alternatives to inflation is very extensive [121, 122]. Regardless,
whatever the model we are studying, we must always keep in mind that the model should
(attempt) to be compatible with observations [123]. Linde, for example study if Ωtot ̸= 1
would disprove inflation [124], and while it does restrict some inflationary models it would
not be a definite no to inflation.
64 3. Inflation
0.94 0.96 0.98 1.00
Primordial tilt (ns)
0
.0
0
0
.0
5
0
.1
0
0
.1
5
0
.2
0
T
en
so
r-
to
-s
ca
la
r
ra
ti
o
(r
0
.0
0
2
)
Convex
Concave
TT,TE,EE+lowE+lensing
TT,TE,EE+lowE+lensing
+BK14
TT,TE,EE+lowE+lensing
+BK14+BAO
Natural inflation
Hilltop quartic model
α attractors
Power-law inflation
R2 inflation
V ∝ φ2
V ∝ φ4/3
V ∝ φ
V ∝ φ2/3
Low scale SB SUSY
N∗=50
N∗=60
Figure 3.3: Observation data by Planck satellite and theoretical predictions of selected
papers over the tilt of scalar anisotropies, ns (horizontal axis), and the tesor-to-scalar
ratio, r (vertical axis). Plot retrived by the Planck cosmological constraints 2018 [43].
3.5 Inflationary potential
We know that a real scalar field named the inflaton, ϕ, that slowly rolls down its own
potential, V (ϕ), exponentially expands the universe solving the shortcomings of standard
cosmology. Needless to say, there are multiple8 potentials V (ϕ) that satisfy the slow-
roll requirements. This naturally raises the question: how do we set on an inflationary
potential?.
To constraint the possible models, one must take into account observational data, par-
ticularly the values of the scalar spectral tilt, ns, and the tensor to scalar ratio, r. The
predictions of these parameters must be compatible with the Planck collaboration con-
straints on inflation 2018 results [43], as shown in figure 3.3. In this figure we have two
different sets of data. The first three shadow areas (gray, red and blue) correspond to the
value of ns and r, compatible with the correponding data sets. As measurements improve
over time and include more restrictions, the area is being reduced. The second type of
data correspond to the predictions of the selected inflationary models: Natural inflation
[125, 126], Hilltop quartic model [127], α-attractors [128], R2 inflation (Starobinsky) [129]
and single polynomial models as V ∼ ϕk with k a real positive number. From this plot, it
is clear that single polynomial potentials, are not suited to be the complete inflationary
potential. Given that, all of them predict values for r and ns that lie outside of the region
experimentally constrained.
8Maybe even infinite. although, these models should be theoretically motivated.
3. Inflation 65
Figure 3.4: Plot of the α-Attractors for k = 4 and different values of α, see equation
(3.54), models proposed by Kallos-Linde [128].
We can also see that potentials known as α-attractors (the yellow area in figure 3.4),
proposed by Kallosh and Linde [128] have theoretical predictions over ns and r that could
be fitted to observation. These α-attractors are given by
V (ϕ) = λM4
p
[√
6α tanh
(
ϕ√
6αMp
)]k
, (3.53)
for odd values9 of k and Mp being the Planck mass. The predictions over ns and r are
only sensitive to value of α, figure 3.4, where we plotted the potentials for different values
of α and k fixed to four.
3.5.1 T-models
The T -models, are a special case of the α-attractors, with α = 1. Thus, the corresponding
T -model inflationary potential is given by
V (ϕ) = λM4
p
[√
6 tanh
(
ϕ√
6Mp
)]k
. (3.54)
Near the minimum, these potentials take on the form
V (ϕ) ≈ λM4−k
p ϕk, (3.55)
9These potentials are derived within a framework of supergravity (and string theory), and by doing
so, we obtain that only odd-values are predicted, consult reference [130] for details.
66 3. Inflation
Figure 3.5: T-attractors proposed by Kallos-Linde [128], setting α = 1 and different values
of k, see expression (3.54).
For T-models, we know [130] that λ has to take on the value
λ =
18π2As
6
k
2N2
⋆
, (3.56)
where As = e3.044/1010 is the amplitude of the scalar power spectrum (equation (3.45))
and, for k = 4, N⋆ = 55.9. N⋆ are the number of e-folds10 between the horizon exit of the
CMB pivot scale (k⋆ = 0.05 Mpc−1) and the end of inflation.
T-attractors?.
All of them look the same for large field values, ϕ, ϕ≫Mp, regardless of the k value,
see figure 3.5. This means that there are no special initial conditions necessary for
inflation. Given that at large field values they look the same, the predictions over
r and ns are independent of k, we just need to fix λ to expression (3.56). Most
importantly, these predictions lie inside the observational allowed region constrained
by Planck, see the yellow area in figure 3.3.
As one might have suspected, any inflationary model that has a minimum of the
shape given in expression (3.55) could be considered. If this was the case, we would
differentiate the potentials based on their predictions over ns and r, because these
predictions are not sensible to the shape of the potential near its minimum (where
the potentials are similar).
We are using T-models because they are compatible with CMB and on top of that,
these models are theoretically inspired by supergravity, the details are discussed in
reference [128].
10The e-folds measure how much the scale factor grew in an exponential scale from an initial value,
ainitial, to the final value, afinal, N ≡ ln(afinal/ainitial).
3. Inflation 67
Different values of k?.
This is a fundamental part of the thesis: if all T -attractors are compatible with CMB
measurements, how can we determine which one describes our universe?. After the
slow-roll regime has ended, the shape of the potential near its minimum influences
the subsequent evolution of the universe. Studying this regime allows us to identify
possible differences among the different values of k.
When the inflaton ends slow roll, it starts oscillating about the minimum of its
potential. During these oscillations, the couplings of the inflaton to Standard Model
(SM) fields become relevant. These couplings could allow for inflaton decay into SM
particles. These stage of the universe is known as reheating [131, 132], and reheating
is sensible to the shape of the potential near the minimum (and the couplings of the
inflaton and SM fields) so this could help us solving this question.
Other physicists, like N. Bernal, have also studied the impact of the shape of the
minimum of the potential over the production of DM [133]. The effects of the shape
of the potential near the minimum, over the reheating epoch, will be studied in
chapter 5 and the effects over DM production will be studied in chapter 6.
k = 4?.
We are interested in the case k = 4, because, in this scenario the inhomogeneities on
the inflaton field (section 3.3) become predominant, this means that they grow ex-
ponentially fast (in the Fourier space) leading the inflaton to the non-linear regime,
process known as Fragmentation11 [134]. However, the fragmentation could have an
impact on reheating [135] and, in chapter 6, we will study the impact of fragmenta-
tion on WIMP production.
11The process of fragmentation will be discussed in chapter 5

69
Chapter 4
Reheating
4.1 Coherent oscillations regime . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Boltzmann equation during reheating . . . . . . . . . . . . . . . . . . . . . 78
After the period of cosmological inflation described in chapter 3, the universe has been
expanded about Ntot > N⋆ = 55.9 times its original size1. This caused any pre-existing
energy content in the universe to be diluted, leaving the universe in a nearly empty state.
In this chapter we will show a mechanism by which the universe undergoes a phase tran-
sition from a vacuum state to a radiation domination era. This process is known as
Reheating [136, 137]. Reheating is based on the idea that as the inflaton oscillates about
the minimum of its potential, it could decay into Standard Model (SM) particles. These
daughter particles would interact with themselves producing the rest of the particles creat-
ing a plasma with energy density, ρR. Eventually, the energy of these relativistic particles
equals the energy density of the inflaton field, ρϕ, that is
ρreh ≡ ρϕ(areh) = ρR(areh) =
π2
30
g∗(Treh)T
4
reh, (4.1)
where areh and Treh are the scale factor and the temperature of the thermal bath at the
end of reheating [138].
We will study scenarios where the inflaton couples to fermions and to bosons. At the end
of this chapter, we will solve the Boltzmann equation for Dark Matter (DM) during the
epoch of reheating. The difference with respect to the treatment shown in section 2.3, is
that here the entropy is not a conserved quantity.
1Remember that N⋆ is the amount of expansion (in exponential scale) between the horizon exit of the
CMB pivot scale (k⋆ = 0.05 Mpc−1.) and the end of inflation, for k = 4, N⋆ = 55.9, inflation could have
lasted longer.
70 4. Reheating
4.1 Coherent oscillations regime
As the inflaton begins a period of oscillations about the minimum of its potential, depend-
ing on the couplings of the inflaton to the SM fields, the inflaton could decay into SM
particles. If these couplings are small, the reheating process can be treated perturbatively.
In the perturbative regime, we treat the problem as if there were two perfect fluids; the
condensate, ρϕ, and the thermal bath, ρR, which are exchanging energy.
4.1.1 Reheating in a minimum V ∼ ϕk
Perturbative reheating can often be treated analytically as we will show next. The details
of the reheating process are sensitive to:
The specific potential:
The shape of the potential around the minimum. As we discussed previously, we are
using T -models given by equation (3.55),
V (ϕ) = λM4
p
[√
6 tanh
(
ϕ√
6Mp
)]k
≈ λM4−k
p ϕk,
(4.2)
where λ is a dimensionless constant that we normalized to equation (3.56),
λ =
18π2As
6
k
2N2
⋆
, (4.3)
with As = e3.044/1010 being the amplitude of the scalar power spectrum (equation
(3.45)) and, N⋆ = 55.9 (deduced for k = 4) is the number of e-folds between the exit
of the Planck’s pivot scale (k⋆ = 0.05 Mpc−1) and the end of inflation. Values that
are in good agreement with current data [130].
Couplings of the inflaton to SM fields:
The details of reheating will also depend on the couplings allowed between the in-
flaton and the SM particles.
As we discussed, inflation occurs at large field values (ϕ > Mp), and by definition it ends
when the acceleration of the universe comes to an end, ä = 0. We denote the scale factor
at end of inflation by aend, and while this scale factor at the end of inflation is simply
a reference point, the value of the energy density at moment2, ρend = 3V (ρend)/2 has
physical relevancy. This energy density at the end of inflation represents the the initial
conditions for reheating. We can see clearly that ρend depends on the potential.
2See equation (3.26)
4. Reheating 71
For the T-models, the inflaton field value marking the end of inflation is given by [139, 140]
ϕend =
√
3
8
Mp ln
[
1
2
+
k
3
(
k +
√
k2 + 3
)
]
, (4.4)
As the universe continues expanding the inflaton undergoes anharmonic oscillations dic-
tated by the Klein-Gordon (KE) equation, which is written as equation (3.11),
ϕ̈+ 3Hϕ̇+
1
a2
∇2ϕ+ V ′(ϕ) = −Γϕϕ̇, (4.5)
where the energy density, ρϕ, and the pressure, pϕ, of the inflaton field were given as
equations (3.9) and (3.10) in chapter 3. Using these quantities, for a spatially homogeneous
field, the equation of state of the inflaton field rewrites as
wϕ ≡
pϕ
ρϕ
=
1
2
ϕ̇2 − V (ϕ)
1
2
ϕ̇2 + V (ϕ)
, (4.6)
it is time dependent, nonetheless, we may compute a mean equation of state parameter.
First, let us multiply by ϕ the equation of motion (4.5), and then take the mean of it.
Noticing that over one period ⟨ϕϕ̇⟩ = 0, one can show that
⟨ϕ̇2⟩ ≈ ⟨ϕV ′(ϕ)⟩, (4.7)
substituting this relation into the equation (4.6) and using the potential given by (4.2),
we get that the mean of the equation of state parameter is simply [141, 142]
wϕ =
1
2
⟨ϕ̇2⟩ − ⟨V ⟩
1
2
⟨ϕ̇2⟩+ ⟨V ⟩
=
⟨ϕV ′⟩ − 2⟨V ⟩
⟨ϕV ′⟩+ 2⟨V ⟩ =
k − 2
k + 2
. (4.8)
Using this, the KG equation (4.5) (after averaging over oscillations), in terms of the energy
density ρϕ, reads
ρ̇ϕ = −
2k
2 + k
(3H + Γϕ) ρϕ, (4.9)
this equation admits an analytical solution when 3H ≫ Γϕ which is true at the beginning
of reheating. In this situation the KG equation may be integrated to obtain
ρϕ = ρend
(
a
aend
)− 6k
k+2
. (4.10)
72 4. Reheating
Moreover, during this epoch the Hubble expansion rate, H, is approximately
H(a) ≈
[
1
3M2
p
ρϕ(a)
] 1
2
= Hend
(
a
aend
)− 3k
2+k
. (4.11)
And we have defined
H2
end ≡
1
3M2
p
ρend. (4.12)
Therefore, the Hubble expansion rate is given by
H(a) =









Hend
(
a
aend
)− 3k
k+2
, for aend ⩽ a ⩽ areh,
Hend
(
a
aend
)−2
, for a ⩾ areh,
(4.13)
we defined the end of reheating a = areh as the moment when the inflaton and radiation
energy densities are equal, that is
ρϕ(areh) = ρR(areh) ≡ ρreh =
30
π
g⋆(Treh)T
4
reh. (4.14)
Notice that in order to not spoil Big Bang Nuclesynthesis (BBN) [143, 144] , the reheating
temperature must satisfy [145]
Treh ≥ TBBN = 4 MeV. (4.15)
During these anharmonic oscillations of the inflaton, the inflaton transfers its energy to
SM particles. This radiation has an energy density, ρR, which satisfies its own Boltzmann
equation (or a continuity equation) given by equation (1.60),
dρR
dt
+ 4HρR =
2k
2 + k
Γϕρϕ, (4.16)
the left hand side can be written as a−4[d/dt
(
ρRa
4
)
], and integration of both sides with
respect to cosmic time3, t, leads us to
ρR =
2
√
3k
2 + k
Mp
a4
∫ a
aend
Γϕ(a
′)
√
ρϕ(a′)a
′3
da′, (4.17)
3To do so, on the right-hand side we change variables from cosmic time, t, to the factor scale, a, making
use of the Friedmann equation (4.11).
4. Reheating 73
where a general scale factor dependence of Γϕ has been assumed.
4.1.2 Fermionic Reheating
In the case of Yukawa coupling of the inflaton with a pair of fermions ψ and ψ̄,
Lint = −yϕψ̄ψ, (4.18)
with y being the Yukawa coupling, the decay rate of the inflaton is given by
Γϕ = α2 y
2
8π
mϕ, (4.19)
where α2 is numerically obtained after averaging over oscillations [135], the results show
that α = 1 for k = 2 and α = 0.711 for k = 4, therefore, we might define an effective
Yukawa coupling y2eff = α2y2. Additionally, mϕ is the effective mass for the inflaton field.
We associate the mass of the inflaton field, mϕ, to the second derivative of the potential
with respect to itself, m2
ϕ ≡ ∂2ϕV (ϕ) = Vϕϕ. In the case of the potential given by (4.2), we
get
m2
ϕ = ∂2ϕV (ϕ) = k(k − 1)λ
2
kM
2(4−k)
k
p ρ
k−2
k
ϕ (a), (4.20)
let us remark that for values of k ̸= 2, the inflaton mass mϕ features a field dependence.
Moreover, since Γϕ ∼ mϕ, if we consider k ̸= 2 this would lead to an inflaton decay rate
with a scale factor dependence.
Using this expression we can integrate the expression for ρR(a) given by equation (4.17),
and by doing this, we obtain
ρR(a) =
y2
8π
√
3k(k − 1)λ
1
kM
4
k
p
(
2k
14− 2k
)
ρ
k−1
k
end
(aend
a
)4
[
(
a
aend
)
14−2k
k+2
− 1
]
. (4.21)
Notice the dependence of ρR as function of the scale factor a, for a≫ aend,
ρR ∼ a−
6(k−1)
k+2 . (4.22)
The temperature of the relativistic species will be simply given by (equation (1.67b))
T (a) =
(
30
π2g⋆(T )
ρR(a)
) 1
4
→ T ∼ a−
3
2
(k−1)
(k+2) , (4.23)
74 4. Reheating
Expressions (4.8), (4.10), (4.22) and (4.23) provide us with the following scale factor
dependence for the (averaged) equation of state parameter, wϕ, the inflaton energy density,
ρϕ, the radiation energy density, ρR, (and therefore of the temperature associated to these
relativistic species, T ), those scale factor dependence are
⟨wϕ⟩ = wϕ =
k − 2
k + 2
,
ρϕ = ρend
(
a
aend
)− 6k
k+2
∼ a−
6k
k+2
,
ρR ∼ a−
6(k−1)
k+2
T ∼ ρ
1
4
R ∼ a−
3
2
(k−1)
(k+2)
(4.24)
all of them depend on the value of the integer k, which appears in the potential (4.2).
k = 2
We get an equation of state parameter wϕ = 0, like dust matter so the inflaton
energy density redshifts as matter, ρϕ ∼ a−3.
While the energy density refshifts as ρR ∼ a−3/2 so the temperature redshifts as
T ∼ ρ
1/4
R ∼ a−3/8, during dust-like reheating. Notice that for larger k values, the
temperature has a steeper dependence on the scale factor.
k = 4
In the scenario of interests to us, wϕ = 1/3, so the inflaton redshifts as if it was
radiation dominated, ρϕ ∼ a−3, while radiation ρR ∼ a−3, thereby, the temperature
T ∼ a−
3
4 .
The steeper dependence of the temperature verifies that, in this case, the energy
density of the inflaton redfshifts faster than a dust-like universe. Given that Tk=4 =
a−3/4 = a−0.75 ≫ a−3/8 = a−0.375 = Tk=2, this radiation-like will be further red-
shifted by expansion. The temperature in the bath is, in a sense, doubly redshifted
(production + expansion).
In the instantaneous thermalization approximation4, we must solve the continuity equa-
tions (4.9) and (4.16) in the case of fermionic reheating, with a coupling given by expression
(4.19). Using k = 4 our system of equation reads as
ρϕ + 4Hρϕ = −
4
3
Γϕρϕ,
ρR + 4HρR =
4
3
Γϕρϕ,
(4.25)
4This assumption means that we are considering that the particles that are being produced, for example
by decays of the inflaton, instantly reach thermal equilibrium. These particles form a distribution in
thermal equilibrium that will be described by either the Fermi-Dirac or the Bose-Einstein distribution
functions.
4. Reheating 75
0 5 10 15 20
10-47
10-37
10-27
10-17
Figure 4.1: Solutions to equations (4.25) and (4.26) in the case of a fermionic reheating
with a Yukawa coupling constant y = 6 × 10−4 as a function of the number of e-folds
N = ln a/aend.
where the Hubble parameter H is given by Friedmann equation (1.60),
H2 =
1
3M2
p
(ρϕ + ρR) , (4.26)
The numerical solutions are shown in figure 4.1, were we used a Yukawa coupling y =
6× 10−4. On the x axis we are plotting the number of e-folds N defined as
N = ln
a
aend
, (4.27)
that is, the horizontal axis measures the evolution of the scale factor from the end of
inflation. On the vertical axis we are plotting the energy densities of the inflaton and the
radiation. They are equal to each other at around Nreh = 15.7 e-folds, marking the end
of reheating.
We can estimate the evolution of the temperature because we know that T ∼ ρ
1
4 . The
temperature of the radiation plasma will also initially grow exponentially fast until it
reaches a maximum value that we will denote by Tmax. After the temperature reaches this
maximum, it decreases until it reaches the reheating temperature, Treh, and below.
76 4. Reheating
0 5 10 15 20
107
109
1011
1013
Figure 4.2: Temperature evolution of the thermal bath as a function of the e-Folds N =
ln(a/aend) number in the case of a fermionic reheating with a Yukawa constant y = 6×10−4
and in a potential of the form V ∼ ϕ4.
The evolution of the temperature, using numerical solutions, is shown in figure 4.2 we can
see that it does reaches a maximum, Tmax.
Maximum temperature.
In order to calculate the maximum temperature, one must first compute the scale
factor amax for which the temperature or the energy density is maximized. By doing
this, we obtain
amax = aend
(
2k + 4
3k − 3
)
k+2
14−2k
, (4.28)
this implies
T 4
max =
30
π2g⋆
ρR(a)
∣
∣
∣
a=amax
=
15y2
16g⋆π3
√
3k(k − 1)λ
1
kM
4
k
p ρ
k−1
k
end
(
3k − 3
2k + 4
)
3(k−1)
7−k
. (4.29)
Where we remove aend in favour of ρend. We can see that, in order to compute Tmax
we must specify the inflationary potential (to determine ρend).
4. Reheating 77
Reheating temperature:
We consider that reheating ends when the radiation density begins to dominate over
the inflaton density, i.e., whe ρϕ(areh) = ρR(areh), in figure 4.1 this condition is met
at Nreh = 15.7 e-folds. We may do some approximations in order to have a general
idea of when the reheating ends, assuming that a≫ aend
5
ρϕ(areh)
!
= ρR(areh) = ρend
(
a
aend
)− 6k
k+2 ∣
∣
∣
a=areh
≈ y2
8π
√
3k(k − 1)λ
1
kM
4
k
p
(
k + 2
14− 2k
)
ρ
k−1
k
end
(aend
a
)4 ∣
∣
∣
areh
,
simplifying our results, we get
(
areh
aend
) 3k
k+2
≈
(
16π(7− k)
(k + 2)
√
3k(k − 1)
) k
2 (
ρend
λM4
p
) 1
2
M4
p , (4.30)
using this, we determine the reheating temperature as
T 4
reh =
15 [3k(k − 1)]
k
2 y2kλ
24k−1π2+kg∗
(
k + 2
7− k
)k
M4
p , (4.31)
As one can see, the dependence on ρend has disapeared from Treh. For the T-attractor
model for k = (2, 3, 4) and y = 6 × 10−4, we find λ = (2, 0.9, 0.3) × 10−11 and
Treh = (4.1× 109, 1.0× 107, 3.2× 104) Gev, respectively, [130].
4.1.3 Bosonic Reheating
Alternatively, if the inflaton only decays into a pair of bosons S through the interactions
Lint = µϕSS, (4.32)
with µ being a coupling with mass dimension. Due to this coupling, the decay rate of the
inflaton (to bosons), is intensified
Γϕ =
µ2
eff
8πmϕ
, (4.33)
In the decay rate provided in equation (4.33), the effective coupling µeff ̸= µ (when k ̸= 2)
and it can be obtained after averaging over oscillations, for the details refer to [135].
5That is, assuming we are at the moment well after inflation has ended.
78 4. Reheating
As in the case of ferminic reheating, mϕ is the mass of the inflaton given by equation
(4.20). In this case, after integration of equation (4.16), the radiation energy density we
obtain is
ρR(a) ≈
√
3
8π
1
1 + 2k
√
k
k − 1
µ2
eff
λ
1
k
M
k−4
k
p ρϕ(areh)
1
k
(areh
a
) 6
2+k
[
1−
(aend
a
)
2(1+2k)
2+k
]
. (4.34)
Substituting into equation (4.23), leads to the following temperature relation for a≫ areh,
T (a) = Treh
(areh
a
) 3
2
1
2+k
, (4.35)
and using the Friedmann equation, we get that the Hubble parameter is given as
H(T ) ≈ Hreh
(
T
Treh
)2k
. (4.36)
4.2 Boltzmann equation during reheating
If we are interested in describing DM particle production during reheating, we need to
solve the Boltzmann equation under these circumstances. The problem with this, is that
entropy is being constantly supplied into the thermal bath, see equation (4.25), i.e.,
ρ̇R + 4HρR =
4
3
Γϕρϕ, (4.37)
therefore the entropy per comoving volume is not a conserved quantity6.
We deduced that, for k > 2, the radiation energy density, ρR, quickly reaches a maximum
and then redshifts more slowly than a4, see figure 4.1. Furthermore the decay products
are produced with a delta-function distribution and only through scatterings achieve a
thermal distribution.
Remember that for WIMPs the Boltzmann equation7 reads
ṅ+ 3Hn = −⟨σv⟩
(
n2 − n2
eq
)
. (4.38)
During radiation domination we introduced the yield Ỹ as equation (2.27).
6We derived the conservation of entropy when the continuity equation is equal to zero on the right
hand side, see section 1.6 for details.
7Deduced in section 2.3.
4. Reheating 79
30 32 34 36
10-10
10-7
10-4
Figure 4.3: Solution to the Boltzmann equation (4.38) for DM during reheating in the case
of a potential V ∼ ϕ4, with a fermionic reheating with a Yukawa coupling y = 6× 10−4 in
the s-wave approximation.
That is
Ỹ ≡ n
s
∼ a−3
a−3
, (4.39)
because after decoupling n ∼ a−3. Although, during reheating s ̸= a−3, instead, for this
reason, we define a different yield Y as
Y =
n
T 3
∼ a−3
a−3
. (4.40)
Instead of working with cosmological time t, let us introduce e-folds N , defined by
N ≡ ln
(
a
aend
)
. (4.41)
In these variables the Boltzmann equation takes the following form
dY
dN
= −
(
1
T
dT
dN
+ 1
)
3Y − ⟨σv⟩T
3
H
(
Y 2 − Y 2
eq
)
. (4.42)
80 4. Reheating
The Hubble parameter is given by expression (4.26)
H2 =
1
3M2
p
(ρϕ + ρR) , (4.43)
and the energy density of the inflaton is obtained as a solution to the KG equation,
expressed as equation (4.25), that is
ρ̇ϕ + 4Hρϕ = −
4
3
Γϕρϕ, (4.44)
In addition, in Boltzmann equation (4.42), we have the equilibrium yield, Yeq, which is
obtained directly from the definition of the yield, Y ,
Yeq =
neq
T 3
, (4.45)
where the density number in equilibrium, neq, is obtained, as indicated by equation (2.36),
by integrating the distribution function in equilibrium. Since, as we mention in chapter 2,
we are considering our WIMP candidate to be a Wino particle8. The distribution function
that we need to integrate is the Bose Einstein,
neq = gχ
∫
d3pχ
(2π)3
fχ,eq(E, t). (4.46)
Finally, under fermionic reheating, the width of the inflaton Γϕ is given by equation (4.19),
Γϕ→ψ̄ψ =
α2y2
8π
mϕ, (4.47)
where, as mention in section 4.1.2, for k = 4, α = 0.711 [135]. The final piece to solve the
Boltzmann equation (4.42) is the logarithmic derivative of the temperature.
Temperature term in Boltzmann equation.
During radiation domination, since entropy is conserved, we can prove9 that
1
T
dT
dN
= −1, (4.48)
and simplified the Boltzmann equation, though, during reheating this equation is
not valid so we must treated carefully.
8The s-partner (supersymmetric partner) of boson W.
9Directly from the continuity equation (1.60).
4. Reheating 81
The temperature we used to solve the Boltzmann equation was obtained using equa-
tion (4.23), i.e.,
T =
(
30
π2g⋆(T )
ρR
) 1
4
, (4.49)
where the energy density of radiation is the numerical solution to continuity equation
(4.37) and the solution can be seen in figure 3.2.
Finally, the solution to the Boltzmann equation equation (4.42), when using a Yukawa
coupling y = 6 × 10−3, in the s-wave approximation with ⟨σv⟩ = σs = 6 × 10−6 GeV−2,
is shown in figure 4.3. In this figure we are plotting the Yield of DM, Y = n/T 3, during
reheating near a minimum V ∼ ϕ4. On the x-axis we have the e-folds (measured from the
end of inflation). We can see, that after decoupling, this yield is constant in time (as we
said it would be in equation (4.39)). If we would have plotted Ỹ = n/s, this would not be
constant after decoupling. The computation of the relic density will be discuss later.

83
Chapter 5
Fragmentation
5.1 Inhomogeneities during oscillations . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Decay rate of the fragmentated inflaton . . . . . . . . . . . . . . . . . . . . 91
It is important to notice that, regardless of an homogeneous and isotropic universe at
large scale, we do see inhomogeneities such as stars, black holes, galaxies, clusters and
so on. We also measure perturbations on the Cosmic Microwave Background (CMB) as
deviations from the temperature TCMB,0 = 2.726 K. All of these can be tracked back to
primordial fluctuations in the inflaton field ϕ (and in the metric gµν). These fluctuations
grew to super-horizon modes freezing out during the inflationary epoch and they re-enter
the horizon at much later times [146], see chapter 3.
Nevertheless, during the oscillations of the inflaton field about the minimum of its po-
tential, a new form of fluctuations arise, these appear do self-interactions of the inflaton
field. These perturbations could for example produce Primordial Black Holes [147, 148]
or Gravitational Waves [149, 150]. Measuring these gravitational waves would give us
important information about the history of the universe [151, 152] and their production
is sensible to shape of the inflationary potential [135].
Analogous to primordial perturbations, at the linear level, when fluctuations are small, we
may write the inflaton field as
ϕ = ϕcl(t) + δϕ(x, t), (5.1)
where δϕ(x, t) are a result of self-interactions, and the time evolution of these inhomo-
geneities will be described by the Klein-Gordon (K-G or KG) equation (3.14),
ϕ̈+ 3Hϕ̇+
1
a2
∇2ϕ+ V ′(ϕ) = 0, (5.2)
in the present chapter we will study the evolution of inhomogeneities in a potential V ∼ ϕ4.
84 5. Fragmentation
5.1 Inhomogeneities during oscillations
To study the evolution of inhomogeneities during reheating, we start by substituting equa-
tion (5.1) into the KG equation (5.2), at the linear level in δϕ, the equation reads
δϕ̈+ 3Hδϕ̇+
(
− 1
a2
∇2 + V ′′(ϕ)
)
δϕ = 0, (5.3)
Assuming that, near the minimum, our potential is written as
V = λϕ4, (5.4)
that is k = 4.
Now, let us consider metric perturbations, remember that in our discussion in section 3.3
we introduce the gauge-invariant Mukhanov-Sasaki (MS or M-S) variable,
Q = δϕ+
ϕ̇
H
Ψ, (5.5)
where Ψ was the spatial scalar perturbation to the FLRWmetric. We deduced the equation
of motion for the mode Qk known as the Mukhanov-Sasaki equation (3.36), i.e.,
Q̈k + 3HQ̇k +
[
k2
a2
+
3ϕ̇2
M2
p
− ϕ̇4
2H2M4
p
+ 2
ϕ̇Vϕ
M2
p
+ Vϕϕ
]
Qk = 0, (5.6)
and we solved it during the slow-roll regime with Bunch-Davies initial conditions, and
using the solution, we computed the power spectrum of primordial scalar perturbations.
Alternatively, the Fourier transform the KG equation (5.3) for inflaton fluctuations reads
δ̈ϕ+ 3H ˙δϕ−
[
k2
a2
+ Vϕϕ(ϕcl)
]
δϕ = 0. (5.7)
These two equations are equivalent after the slow roll. Notice that during coherent oscil-
lations we have small field values, ϕ≪Mp, thus, we may disregard the factors
3ϕ̇2
M2
p
∼ ϕ̇4
2H2M4
p
∼ 2
ϕ̇Vϕ
M2
p
∼ ϕ, (5.8)
or alternatively, assuming Ψ = 0, such that the M-S variable, expression (5.5), reads as
Q = δϕ.
5. Fragmentation 85
That is, the M-S equation reduces to
Q̈k + 3HQ̇k +
[
k2
a2
+ Vϕϕ
]
Qk ≈ 0, (5.9)
which looks like the K-G equation of motion (5.7), thus, we may disregard the metric
perturbations during coherent oscillations.
Before we dive into looking for solutions to equation (5.7), we may consider the homoge-
neous component of the inflaton field, ϕcl(t) at early times. Let us parametrize this classical
oscillating component in terms of an envelope function ϕ0(t), encoding the redshift due to
expansion, and a quasi-periodic function P(t),
ϕcl(t) ≈ ϕ0(t)P(t). (5.10)
At the classical (homogeneous) level, the equation of state parameter was obtained to be
wϕ = (k − 2)/(k + 2), see equation (4.8). For k = 4 this result shows that wϕ = 1/3 as
if it was radiation dominated. In radiation domination, as we discussed in section 1.4,
ρϕ ∼ a−4. Using this condition, we determine that the decaying envelope of the oscillation
(for k = 4) redshifts as
ϕ0(t) = ϕend
(
aend
a(t)
)
, (5.11)
the time-dependence of the quasi-periodic function P(t) can be obtained using the K-G
equation (5.2). Using the fact that the envelope function is approximately constant over
one oscillation, we get [134],
ϕ̈+ 3Hϕ̇+ 4λϕ3 = 0 → Ṗ2 =
1
2
m2
ϕ
(
1− P4
)
, (5.12)
where the mass of the inflaton field is given by
m2(t) = Vϕϕ(ϕ0(t)) = 12λϕ2
end
(aend
a
)2
≡ m2
end
(aend
a
)2
. (5.13)
and we defined the effective mass of the inflaton at the end of inflation as
m2
end ≡ 12λϕ2
end. (5.14)
The equation (5.12) can be solved in terms of the Jacobi elliptic [153] functions
P(t) = sn
(
mϕt√
6
,−1
)
, (5.15)
86 5. Fragmentation
where we have the Jacobi elliptic function with argument mϕt/
√
6 and parameter −1.
Since we are in the scenario of radiation-like domination, we know that the scale factor
evolves as a ∼ t1/2, see equation (1.47). Hence, the conformal time τ =
∫
a−1dt ∼ t1/2 ∼ a.
We also know that the inflaton mass redshifts as mϕ ∼ a−1, therefore the argument of the
jacobi function may be expressed as mϕt = mend(τ − τend), where τend is the conformal
time at the end of inflation. Thereby, we can approximate the classical component as
ϕcl(τ) = ϕ0(τ)P(τ) ≈ ϕend
(
aend
a(τ)
)
sn
(
mend√
6
(τ − τend),−1
)
. (5.16)
5.1.1 Canonical Quantization
Now let us treat the fluctuations evolution dictated by equation (5.7), i.e.,
δ̈ϕ+ 3H ˙δϕ−
[
k2
a2
+ Vϕϕ(ϕcl)
]
δϕ = 0. (5.17)
Introducing the canonically normalized fluctuation
X ≡ aδϕ, (5.18)
and switching to conformal time1 τ = at, we verify that the kinetic term in the action of
the inflaton field (given by equation 3.6), transforms into
Sϕ =
∫
d4x
√−g
(
1
2
gµν∂µϕ∂νϕ− V (ϕ)
)
∼
∫
d3xdτ
(
1
2
∂τX∂τX + . . .
)
, (5.19)
which is a requirement to apply canonical quantization, although, the prize is that the
potential make take on a more complicated form.
We may quantize these canonical fluctuations, X, as
X(τ,x) =
∫
d3k
(2π)3/2
e−ik·x
[
Xk(τ)âk +X∗
k(τ)â
∗
−k
]
, (5.20)
where k denotes the comoving momentum, and âk and â
†
k are the annihilation and creation
operators, respectively. These operators satisfy the canonical commutation relations
[
âk, â
†
k′
]
= δ (k − k′) , and
[
âk, âk′
]
=
[
â†k, â
†
k′
]
= 0, (5.21)
1Conformal because in this conformal time, the FLRW metric gµν is expressed as a Minkowski metric
ηµν with a global factor a2, gµν = a2ηµν .
5. Fragmentation 87
we ensure that the corresponding commutation relations between the field, Xk, and its
momentum conjugate, X ′
k, imposing the Wronskian constraints XkX
∗′
k −X∗
kX
′
k = i.
Substituting expression (5.20) into the K-G equation (5.17), we get the equation of motion
for the k mode, Xk,
X
′′
k +
(
k2 − a′′
a
+ 12λϕ2
cla
2
)
Xk = 0, (5.22)
where ′ denotes derivative with respect to the conformal time, τ . Before solving this
equation, let us analyze a the terms inside the parenthesis.
Second term:
Observe that the second term decreases with time. To visualize this we use the
second Friedmann equation (1.41),
a′′
a
=
a2
6M2
p
(
4V − ϕ̇2
)
∼ λϕ4a2
M2
p
→ 0, (5.23)
because during reheating we have small field values ϕ≪Mp.
Third term:
Using the envelope function of the classical component given by equation (5.16), we
may estimate the evolution of the third term
a2ϕ2
cl ∼ a2
(
1
a
)2
∼ constant, (5.24)
thus, the third term is constant over time, and it is dependent on the condensate.
Based on these two observations, the equation of motion that we are solving, equation
(5.22), can be approximated as
X
′′
k +
(
k2 + 12λϕ2
cla
2
)
Xk = 0. (5.25)
If we introduce a new variable in the form of
z ≡ mend (τ − τend) , (5.26)
where mend is the effective mass of the inflaton at the end of inflation defined in equation
(5.14); using the solution of the classical field given by expression (5.16), we get
d2Xk
dz2
+
[
(
k
mend
)2
+ sn2
(
z√
6
,−1
)
]
Xk = 0, (5.27)
88 5. Fragmentation
10−4 10−3 10−2 10−1 1 10
k/mend
10−7
10−5
10−3
10−1
R
e
µ
k
Figure 5.1: Floquet chart of equation (5.27), a mode in the resonant band k/mend ∼ 0.73.
Illustration retrieved from reference [134].
where sn2(z/
√
6,−1) is the Jacobi elliptic function with argument z/
√
6 and parameter
−1 introduced in equation (5.15). In these new variables, the equation of evolution of the
canonically quantized fluctuation, Xk, takes the form of a Hills equation [154]. For this
equation, Floquet’s theorem guarantees that the solutions have the general form
Xk(z) = eµkzg1(z) + e−µkzg2(z), (5.28)
where g1(z) and g2(z) are periodic functions, and µk is a complex number, called the
Floquet exponents. If any Floquet exponent µk had a non-vanishing real part it would
inevitably lead to exponential solutions.
The numerical solutions to this equation are shown in figure 5.1. As we can see, there is a
resonant band at k/mend ≈ 0.73, which means that modes with this momentum will grow
exponentially (in Fourier space). When this happens, our quantization approach will not
sustain because it relies on quantizing small inhomogeneities which works well at early
times.
5.1.2 Alternative approach
Let us come back to equation the K-G equation for δϕ and use our knowledge that there
is a resonant band to our advantage. The K-G equation in question is given by expression
(5.22), which we may rewrite as
X ′′
k + ω2
kXk = 0. (5.29)
5. Fragmentation 89
10−27
10−23
10−19
10−15
10−11
ρ
φ
[M
4 P
]
ρϕ (condensate)
ρδϕ (particles)
ρϕ (total)
1 10 102 103
a/aend
0.32
0.34
⟨w
φ
⟩
1/3
Figure 5.2: Evolution of the energy density as a function of the scale factor. The energy
density of the classical oscillating field is in color green, while in orange we have the energy
density associated to inhomogeneities and finally in blue we have the total energy density
(spatially averaged). This plot corresponds to the evolution of the energy density of the
inflaton field when field is oscillating in a potential V ∼ ϕ4. At the bottom of the image
we have the oscillation-averaged equation of state parameter evolution. Figure retrieved
from [134].
Where we defined ω2
k as
ω2
k ≡ k2 − a′′
a
+ 12λϕ2
cla
2, (5.30)
where the classical part, the third term, is dominated by the quasiperiodic function P
defined in equation (5.16). As initial conditions, we choose the positive frequency Bunch-
Davies vacuum
Xk(τ0) =
1√
2ωk
, X
′
k(τ0) = −
iωk√
2ωk
. (5.31)
We are interest in the phase space distribution function of the fluctuations fδϕ, which is
given by [155],
fδϕ = nk =
1
2ωk
∣
∣
∣ωkXk − iX
′
k
∣
∣
∣
2
, (5.32)
90 5. Fragmentation
using this expression, the number density and energy density of the fluctuation will be
given by
nδϕ =
1
(2π)3a3
∫
d3knk,
ρδϕ =
1
(2π)3a4
∫
d3kωknk.
(5.33)
The resulting form of the phase space distribution function in the linear approximation
peakes at around k/mend = 0.73 [134], in full agreement with the results we got previously.
Therefore, we need to abandon the linear approximation and instead focus on solving the
complete K-G equation (5.2) in the nonlinear regime.
In the nonlinear regime the occupation numbers are large, nk ≫ 1, so the dynamics of the
system can be adequately described by assuming ϕ is a classical field, in that case, the
K-G equation reads
ϕ̈+ 3Hϕ̇− 1
a2
∇2ϕ+ Vϕ = 0, (5.34)
and it is solved over a configuration-space lattice [134]. The energy density of the inflaton
is computed from the spatial average2 of the energy-momentum tensor of ϕ, which is
denoted by a bar
ρϕ =
1
2
ϕ̇2 +
1
2a2
(∇ϕ)2 + V (ϕ), (5.35)
the results are shown in figure 5.2. We can see that the energy density associated to fluctu-
ations grow exponentially fast at around a/aend ∼ 102, and that they become predominant
over the classical oscillating field at around a/aend ∼ 2×102, when this occurs we say that
the inflaton has been fragmentated. It is fragmentated because now it is predominantly
composed of fluctuations. When the fluctuations start to grow, before fragmentation is
completed, we use the linear approximation. However, as occupation numbers of fluc-
tuations are larger (due to the anticipated Floquet resonant band) we use Cosmolattice
[156, 157] to solve the complete K-G equation over a configuration spatially averaged.
Whe also need to study the evolution of the equation of state parameter ωϕ = pϕ/ρϕ for
the energy density, ρϕ and the pressure pϕ of the complete inflaton given by expressions
(3.9) and (3.10), respectively, we get
ωϕ =
pϕ
ρϕ
=
(ϕ̇2)/2− (∇ϕ)2/6a2 − V (ϕ)
(ϕ̇2)/2 + (∇ϕ)2/2a2 + V (ϕ)
, (5.36)
2In lattice simulations the volume is finite.
5. Fragmentation 91
we can see that as the inflaton field is oscillating about the minimum so is the equation
of state parameter. That is why, as we did when dealing with reheating without fragmen-
tation3, we averaged over oscillations to create an (oscillation-averaged) equation of state
parameter, which without fragmentation led us to ⟨wϕ⟩ = (k− 2)/(k+2). With fragmen-
tation in an inflationary field V ∼ ϕk, we obtain the panel at the bottom of figure 5.2.
We can see that it starts at a value of 0.32 and quickly reaches 1/3, effectively constant
[158, 159].
5.2 Decay rate of the fragmentated inflaton
We used both canonical quantization and lattice codes to compute the energy density of
the inflaton field during reheating in a potential V ∼ ϕ4. In this section, we will study
the decay rates of the inflaton field, which at the linear level may be written as
ϕ = ϕcl(t) + δϕ(x, t), (5.37)
where ϕcl(t) corresponds to the classical oscillatory part of the condensate, the field that
is spatially homogeneous given by equation (5.16) and δϕ(xt) are the perturbations.
Assuming a Yukawa coupling of the inflaton into two fermions, as equation (4.18), we
would have
Lint = −yϕcψ̄ψ − yδϕψ̄ψ. (5.38)
If we are interested in studying the evolution of the phase space distribution of the decay
products, ψ, we must solve their Boltzmann equation, which after phase space integration
reads as4
ρ̇ψ + 4Hρψ =
∫
d3p
(2π)3
p0Ĉ[fψ] ≡ R(t), (5.39)
here we have introduced R(t), the radiation-energy production rate per unit volume. Ac-
cording to equation (5.38), this term will be written as
R(t) ≡ Rϕ(t) +Rδϕ(t), (5.40)
3Section 4.1.
4Corresponds to the Boltzmann equation (B.18).
92 5. Fragmentation
5.2.1 Decay of the coherent oscillations
Let us consider the condensate at early times, when it was composed in mainly oscillating
zero-mode [160, 161], in this case the continuity equation reads as equation (4.25),
ρ̇ϕ + 3H(1 + wϕ)ρϕ = −Rϕ(t), (5.41)
where we identified
R(t) = (1 + ωϕ)Γϕρϕ, (5.42)
and for a generic process ϕ→ A+B, it is known [162] that the decay rate may written as
Γϕ =
1
8π(1 + wϕ)ρϕ
∞
∑
n=1
|Mn|2En
√
[
1− 1
E2
n
(mA +mB)
2
] [
1− 1
E2
n
(mA −mB)
2
]
,
(5.43)
where the sum is taken over the harmonic modes of the classical harmonics-mode func-
tion P(t) (see equation (5.13)), with an energy En = nωϕ, additionally, Mn denotes the
transition amplitude in one oscillation for each mode.
In the case of an inflaton oscillating about the minimum of a quartic potential and assum-
ing that the masses mA and mB can be disregarded, that is, the decay products of ϕ are
relativistic, then, the decay rate of the condensate can be evaluated completely
Γϕcl→ψ̄ψ = α2 y
2
8π
m2
ϕ(t). (5.44)
where α is an efficiency factor that codifies the oscillations. And it is obtained numerically
by averaging over oscillations [135]. For k = 4 (k = 2) we get α = 0.71 (α = 1), using this
we may define an effective Yukawa coupling. y2eff = α2y2.
5.2.2 Decay of the fragmentated inflaton
Assuming a population of inflaton particles, δϕ, with a distribution function fδϕ(k), and
disregarding Pauli blocking and Bose enhancement, the collision term in the Boltzmann
equation can be written as
Ĉϕϕ[fψ] =
1
p0
∫
d3k
(2π)3
1
2k0
d3p′
(2π)32p0′
(2π)4δ(4) (k − p− p′) |M|2δϕ→ψ̄ψ fδϕ(k), (5.45)
where we have assumed that the process of two particles annihilating to create δϕ is
negligible.
5. Fragmentation 93
The particle production rate can be evaluated
Rδϕ =
∫
d3p(2π)3p0Ĉδϕ[fψ] = Γδϕ→ψ̄ψmϕnδϕ, (5.46)
where
Γδϕ→ψ̄ψ =
|M|2δϕ→ψ̄ψ
16πmϕ
√
1−
4m2
ψ
m2
ϕ
, (5.47)
if the decay products are relativistic, mψ ≪ E, their energies are negligible, leading to
Γδϕ→ψ̄ψ =
y2
8π
mϕ. (5.48)
5.2.3 Total decay rate
The total decay rate of the inflaton field, R(t), that accounts decays of the classical
oscilating field component, Rϕ, equation (5.42), and the fluctuations, Rδϕ, equation (5.46),
is given by
R(t) =
4
3
Γϕρ̄ϕ + Γδϕmϕnδϕ =
y2
8π
mϕ
(
4
3
α2ρ̄ϕ +mϕnδϕ
)
. (5.49)
where we defined the condensate component of the energy density5, ρ̄ϕ as
ρ̄ϕ =
1
2
¯̇ϕ+ V (ϕ̄). (5.50)
We can see that the contribution to the energy density of the inhomogeneities to the decay
rate, namely equation (5.49) has the following dependence
Rδϕ =
y2
8π
m2
ϕnδϕ ∼ m2
ϕ, (5.51)
In comparison to the classical condensate which has a dependence
Rϕ =
4
3
α2 y
2
8π
mϕρ̄ϕ ∼ mϕ. (5.52)
5The energy density of the spatially averaged inflaton field.
94 5. Fragmentation
This is important because we know that the mass of the inflaton, mϕ is associated to the
second derivative of the inflaton potential,
m2(ϕ) = Vϕϕ(ϕ0(t)). (5.53)
with fragmentation the inflaton mass redshifts faster than a−1 (which was the case without
fragmentation, see equation (5.13)). It is redshifting faster because the inflaton mass is
being converted into fluctuations.
It looks like the impact of fragmentation over the decay rate of the inflaton (to fermions)
is that the decay rate is smaller than the scenario without fragmentation, this means that
(keeping a fixed Yukawa coupling) reheating lasts longer. To conclude this statement, we
must compute the complete decay rate, equation (5.52), which means, we need to include
the the comoving density number of fluctuations nδϕ.
The comoving density number of fluctuations, nδϕ, is obtained by an integration over the
phase space of the distribution function of fluctuations, fδϕ, see equation (5.33),
nδϕ =
1
a3
∫
d3k
(2π)3
nk =
1
a3
∫
d3k
(2π)3
fδϕ, (5.54)
we do this integrals over spatially averaged configurations using lattice simulations, for
details refer to [134].
95
Chapter 6
Dark Matter relic abundance density
6.1 Reheating dynamics with fragmentation . . . . . . . . . . . . . . . . . . . 96
6.2 Wino Relic Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Finally, we would like to solve once again the Boltzmann equation in a FLRW universe
during reheating in an inflationary potential of the form V ∼ λϕ4. As we have seen this
induces a process known as fragmentation1. We will solve the Boltzmann equation and
compute ΩDM,0h
2 incorporating the effects of fragmentation into the process of reheating
and compare the results to those obtained at the background level2.
The Boltzmann equation during reheating was given by equation (4.42), and it reads
dY
dN
= −
(
1
N
dT
dN
+ 1
)
3Y − ⟨σv⟩T
3
H
(
Y 2 − Y 2
eq
)
, (6.1)
where the yield Y is defined in equation (4.40),
Y =
n
T 3
, (6.2)
and N ≡ ln (a/aend) represents the number of e-folds after the end of inflation, and was
defined in equation (4.41). Assuming a Yukawa coupling between the inflaton and two
fermions Lint = −yϕψ̄ψ, as equation (5.38). For this scenario we computed the decay rate
of the inflaton field (classical component plus inhomogeneities), given by equation (5.49),
R(t) = 4
3
Γϕρ̄ϕ + Γδϕmϕnδϕ. And finally, we will compute the DM relic density, thus, this
chapter may be considered the results chapter.
1Described in chapter 5.
2By background level we refer to the spatially homogeneous level, chapter 4.
96 6. Dark Matter relic abundance density
10 20 30 40
10-80
10-66
10-52
10-38
10-24
10-10
Figure 6.1: Evolution of the energy density of radiation, ρR, and the inflaton, ρϕ as function
of the e-Folds number, N = ln(a/aend). Numerical solutions to the system of equations
(6.4) where we fixed the Yukawa coupling constant to y = 6× 10−4 in both cases.
6.1 Reheating dynamics with fragmentation
In the perturbative image we need to solve the system of equations (4.25) modifying
the right hand side of them to incorporated the effects of the fragmentation, that means
writing
ρ̇ϕ + 3(1 + ⟨wϕ⟩)Hρϕ = −R(t) =
4
3
Γϕρ̄ϕ + Γδϕmϕnδϕ,
ρ̇R + 4HρR = R(t),
(6.3)
where the Hubble parameter is given by the Friedmann equation (1.60), which we may
approximated to
H2 ≈ 1
3M2
p
ρϕ. (6.4)
Now, let us explain the averaged-oscillated equation of state factor, ⟨wϕ⟩, that appears on
the continuity equation for the inflaton (6.3).
6. Dark Matter relic abundance density 97
10 20 30 40
10-80
10-66
10-52
10-38
10-24
10-10
Figure 6.2: Evolution of the energy density of radiation, ρR, and the inflaton, ρϕ as function
of the e-Folds number, N = ln(a/aend). Modifying the Yukawa coupling constant to get a
fixed reheating temperature of Treh = 0.1 GeV.
In section 4.1, when dealing with reheating without fragmentation, at the background
level, we deduced that the time-averaged equation of state parameter simplifies to3
⟨wBackground
ϕ ⟩ = k − 2
k + 2
, For k = 4 ⟨wϕ⟩ =
1
3
. (6.5)
We have discussed that for k > 2 (in particular k = 4), the fluctuations in the inflaton field,
δϕ(x, t), grow exponentially fast (in Fourier space) and at some point become preodomi-
nant over the classical oscillating field (in real space). When this happens, the inflaton is
no longer homogeneous, but rather, it has transition into a state were it is composed of
spatially localized fluctuations. We can associated these fluctuations with inflaton parti-
cles. We evaluated numerically the time evolution of ⟨wϕ⟩ including fragmentation, that
is, we time-average equation (5.36),
⟨ωϕ⟩ ≡ ⟨
pϕ
ρϕ
⟩ ≈ ⟨(ϕ̇
2)/2− (∇ϕ)2/6a2 − V (ϕ)⟩
⟨(ϕ̇2)/2 + (∇ϕ)2/2a2 + V (ϕ)⟩
. (6.6)
3See equation (4.8).
98 6. Dark Matter relic abundance density
The result of this evaluation is shown at the bottom panel of figure 5.2. From this figure,
we conclude that we in good approximation,
k = 4: ⟨wFrag
ϕ ⟩ ≈ 1
3
. (6.7)
The solutions to the system of equations (6.3) are shown in figure 6.1. In this figure we
have plotted the energy density of the inflaton field and radiaton; in two scenarios:
Solid lines: Assuming, an homogeneous inflaton and fixing the Yukawa coupling
to y = 6× 10−4.
Dashed lines: Including the effects of fragmentation into the dynamics, with the
same Yukawa constant y = 6× 10−4.
We can also see in figure 6.1 that at the background level the reheating process ends
before than the scenario with fragmentation (with a fixed Yukawa coupling to y = 6 ×
10−4). We get NBackground
reh ≈ 17.2 e-folds (as predicted in figure 4.1), meanwhile, including
fragmentation moves the end of reheating to NFrag
reh ≈ 37.5 e-fold. It lasted longer with
fragmentation as expected. Since the fluctuations include a factor Rδϕ ∼ m2
ϕ into the
decay rate of the inflaton, the total decay rate would be smaller, causing the reheating
process to last longer.
Since we do not know the precise value of the couplings, if they existed, between the
inflaton field and the SM fields. We can manipulate them (it, the Yukawa constant y) in
our favor.
If, at the background level, we lower the Yukawa coupling constant, to get a fixed reheating
temperature, Treh = 0.1 GeV. 4, we get the solutions shown in figure 6.2. At the back-
ground level we used a Yukawa coupling constant y = 1.6× 10−8 and with fragmentation
we used the previous coupling y = 6× 10−4.
One might consider that fragmentation can be ignored if at the end we can get the desired
reheating temperature in both cases however that is a fine tuning problem and secondly,
we will see shortly after this section that the fragmentation does have an impact in the
evolution of species outside of thermal equilibrium, such as DM.
6.1.1 Temperature evolution with Fragmentation
Finally on the discussion of reheating dynamics with fragmentation. Using our equations
deduced in chapter 1, in particular equation (1.69b), which dictates
ρR(T ) =
π2
30
g⋆(T )T
4 → T =
(
30
π2g⋆(T )
ρR(T )
) 1
4
. (6.8)
4We are fixing this value of the reheating number to compared our computations with the results in
reference [133].
6. Dark Matter relic abundance density 99
0 5 10 15 20 25 30 35
1
1000
106
109
1012
Figure 6.3: Evolution of the thermal bath temperature, T , as a function of the e-folds
N = ln(a/aend). The black dashed line corresponds to the scenario with fragmentation
and the solid blue line represents the evolution of the temperature at the background level.
In both cases, fixing the Reheating temperature to Treh = 0.1 GeV.
Using expression (6.8) we may obtain the temperature, of the thermal bath of relativistic
particles. Substituting the radiation energy density, ρR, obtained numerically by solving
the system of equations (6.3) (figures 6.1 and 6.2) and using the relativistic degrees of
freedom shown in figure 1.2. The resulting temperature is shown in figure 6.3. In this
figure we have plotted as a dashed black (solid line) line the evolution of the temperature
when we consider fragmentation (at the background level).
At the background level we obtained, equation (4.24),
TBackground ∼ ρ
1
4
R ∼ a−
3
2
(k−1)
k+2 , (6.9)
for k = 4 that means that
TBackground ∼ a−3/4 = a−0.75, (6.10)
in contrasts, when we include fragmentation we (numerically deduced) [134]
TFrag ∼ a−0.91. (6.11)
100 6. Dark Matter relic abundance density
28 30 32 34 36 38 40
10-10
10-8
10-6
10-4
0.01
Figure 6.4: Dark Matter yield, defined as Y = n/T 3 where n is the number density of DM
particles and T is the the temperature of the thermal bath, as a function of the e-folds
number N = ln(a/aend), where aend is the scale factor at the end of inflation. In color blue
are the results at the background level when we fixed the Treh = 0.1 GeV.
We can see that
TFrag ∼ a−0.9 > a−0.75 ∼ TBackground, (6.12)
the temperature is being faster when we include fragmentation.
6.2 Wino Relic Density
Finally, the Boltzmann equation for WIMP dark matter reads5
Ẏ = −
(
1
N
dT
dN
+ 1
)
Y − ⟨σv⟩T
3
H
(
Y 2 − Y 2
eq
)
, (6.13)
in the variables
Y =
n
T 3
, (6.14)
5Refer to section 4.2 for details.
6. Dark Matter relic abundance density 101
and using the e-folds numbers N defined as
N = ln
(
a
aend
)
, (6.15)
the resulting DM Yield Y is shown in figure 6.4. In this figure, we have plotted as a dashed
black line the yield when we consider fragmentation effects into our computations, and in
solid orange we have the yield of equilibrium given by equation (2.37) and in the solid blue
lines are the Boltzmann solution at the homogeneous level when fixing Treh = 0.1 GeV.
Switching to the Yield defined in equation (2.27),
Ỹ =
n
s
, (6.16)
and instead of e-folds we use the dimensionless variable x defined as equation (2.28),
x =
mχ
T
, (6.17)
where mχ = 100 GeV is the mass, that we are considering, for the Wino. Using these
variables, and fixing ⟨σv⟩ = σs ∼ 6× 10−10 GeV−2 and mχ = 0.1 GeV, the corresponding
yields DM, Ỹ , are shown in figure 6.5.
Equilibrium (solid orange): This line corresponds to
Ỹeq =
neq
s
, (6.18)
where the density number in equilibrium is numerically obtained using equation
(2.37).
Fragmentation (Dashed black): This line represents the DM yield when we
include the effects of fragmentation into the reheating dynamics. We are setting
the reheating temperature to Treh = 0.1 GeV. This temperature fixed our Yukawa
coupling to y = 6× 10−3.
Background level with fixed Reheating temperature (solid blue): Finally,
if, as we discussed in figure 6.2, in the background level we modify our Yukawa
coupling to y = 1.6× 10−8 to match the reheating temperature of Treh=0.1 GeV.
From this plots we compute the relic density abundance as we did in equation (2.45),
ΩDM,0h
2 ≡ ρDM,0
ρc,0
h2, (6.19)
102 6. Dark Matter relic abundance density
1 10 100 1000 104
10-11
10-9
10-7
10-5
0.001
Figure 6.5: Dark Matter (DM) yield defined as Y = n/s, where n is the number density
of DM and s is the comoving entropy of the thermal bath, given by equation (1.74). On
the horizontal axis we have the dimensionless variable x = mχ/T , where mχ = 100 GeV is
the mass of the WIMP (Wino), and we are setting thermally averaged annihilation cross
section to ⟨σv⟩ = 6× 10−10 GeV−2.
The orange solid line corresponds to the equilibrium yield, the dashed black line corre-
sponds to the scenario with fragmentation and a fixed reheating temperature Treh = 0.1
GeV. The blue solid line corresponds to the background level when fixing Treh = 0.1 GeV.
we get ρDM,0 = mχnχ,0 using our numerical solutions to the Boltzmann equation (figure
6.5), and we use the critical density of the universe today as it was provided in equation
(1.49) and little h as it was given in equation (1.30). The results are
Dashed black line: ΩDM,0h
2 ≈ 0.113,
Solid blue line: ΩBackground
DM,0 h2 ≈ 0.037.
(6.20)
Finally, at the background scenario where we fixed the Yukawa coupling, the freeze-out
process happened after the end of reheating, while there is radiation domination, see figure
4.1 and compare to figure 4.3.
If we compare all three scenarios in a single plot we get figure 6.6. As in figure 6.5,
the dashed black line represents the DM yield when there is fragmentation, the solid blue
represents the homogeneous level fixing the reheating temperature Treh = 0.1 GeV and the
green solid line represents the background level fixing the Yukawa coupling to Y = 6×10−4
6. Dark Matter relic abundance density 103
28 30 32 34 36 38 40
10-10
10-8
10-6
10-4
0.01
Figure 6.6: Dark Matter (DM) yield defined as Y = n/s, where n is the number density
of DM and s is the comoving entropy of the thermal bath, given by equation (1.74). On
the horizontal axis we have the dimensionless variable x = mχ/T , where mχ = 100 GeV is
the mass of the WIMP (Wino), and we are setting thermally averaged annihilation cross
section to ⟨σv⟩ = 6× 10−10 GeV−2.
The dashed black line corresponds to the scenario with fragmentation and a fixed reheating
temperature Treh = 0.1 GeV. The blue and green solid lines correspond to the background
level when fixing Treh = 0.1 GeV and Yukawa coupling y = 6× 10−4, respectively.
(as the value used with fragmentation). In this case the decoupling is taking place at the
standard scenario of radiation domination as described in chapter 2.
Using this we get the relic density
Solid green line: ΩBackground
DM,0 h2 ≈ 0.131. (6.21)
which is over production of DM.
104
Chapter 7
Conclusion
We revisited the WIMP paradigm in the standard scenario of radiation domination.
WIMPs are theoretical candidates to be Dark Matter (DM) see chapter 2. In this well
studied models, the freeze-out process occurs during radiation domination (when entropy
is conserved, equation (1.73)). From this analysis we showed that larger (smaller) values
of the thermally averaged annihilation cross section ⟨σv⟩ leads to smaller (larger) relic
densities, ΩDM,0h
2, see figure 2.8 and equation (2.46).
We extrapolated the WIMP mechanism to earlier times than radiation domination, in
particular during the epoch known as inflationary reheating. We do this because some
models with a mass and annihilation cross section that are in the allowed region of pa-
rameters to be detected directly, see figure 2.4; could not predict the relic density of DM
if studied during radiation domination. Perhaps, if studied at higher energy ranges, i.e.,
during reheating, they could predict the correct relic density and still be considered as
candidates to be DM.
We studied reheating scenarios in models based on T-models [128]. We considered these
models because they are compatible with Planck measurements about the primordial tilt
(of the scalar power spectrum), ns, and the tensor-to-scalar ratio, r, see equation (3.46)
and (3.50) respectively. The theoretical predictions by these T-models1 are shown by the
yellow area in figure 3.3.
An important property of these T-models (which depend on a real integer k see equation
(3.54)) is that the prediction about ns and r remained constant for different values of
k, see figure 3.5, although, they are different near the their minimum. To distinguish
them, we must analyze the evolution of the universe at the (coherent)-oscillations epoch
and determine if they predict indirect signals, for example, in the production of WIMPs
(Winos) that could help us distinguish them.
Near the minimum, these T-models can be written as a power series V ∼ λϕk, equation
(5.4). After the end of inflation, as the inflaton oscillates about this minimum, the cou-
plings of the inflaton to Standard Model (SM) fields become relevant, allowing for decays
1T-models correspond to α-attractors with α = 1, see equation (3.53).
7. Conclusion 105
from the inflaton field into SM Particles, eventually, reheating the universe, see chapter 4.
We assumed a Yukawa coupling of the inflaton field to two SM fermions, equation (4.18),
and assuming small couplings, we treated the reheating perturbatively. The solutions
to the system of equations (6.3), for y = 6 × 10−4 and k = 4 are shown in figure 4.1.
Alternatively, if we fixed the reheating temperature to Treh = 0.1 GeV, we get the solutions
shown in figure 4.2. In both of these figures, the dashed lines represent the scenario with
fragmentation, that we will be discussed next, and the solid lines represent the background
(homogeneous) level.
We studied the evolution of fluctuations, δϕ(x, t), present during the oscillations of the
inflaton field due to self interactions, in a potential with k = 4, chapter 5. We noticed
that, because of the shape of the potential, the fluctuations grow exponentially fast (in
the Fourier space), see figure 5.1 and become predominant over the classical condensate,
see the top panel in figure 5.2. When this happens, we say the the inflaton has been
fragmentated, because it is no longer a classical oscillating field but rather is composed of
spatial fluctuations we associate with inflaton particles.
When the inflaton has been fragmentated, the complete decay rate transforms into expres-
sion (5.49). This requires us to modify the dynamics of the reheating process described
by equation (4.25), to the expressions given in (6.3)2.
In these equations (6.3), we defined an oscillated-averaged equation of state parame-
ter, ⟨wϕ⟩. Without fragmentation (homogeneous level) we predicted ⟨wBackground:
ϕ ⟩ =
(k − 2)/(k + 2), equation (4.8). For k = 4 we get ⟨wBackground
ϕ ⟩ = 1/3. When we in-
clude fragmentation, we need to average expression (6.6). The numerical evaluation of
this corresponds to the bottom panel of figure 5.2. In this panel we see that, in the case
of k = 4, regardless of fragmentation, the equation of state parameter averages to
⟨wFrag
ϕ ⟩ ≈ 1
3
= ⟨wBackground
ϕ ⟩. (7.1)
Additionally, using the energy density of radiation computed in figure (6.2), we computed
the temperature of the thermal bath, using equation (6.8). The temperature evolution as
a function of the e-folds number, N = ln(a/aend) (where aend is the scale factor at the end
of inflation), is shown in figure 6.3. On this figure we can see that the temperature with
fragmentation no longer redshifts as TBackground: ∼ a−3/4 = a−0.75, but rather evolves as
TFrag ∼ a−0.93, see equation (6.12).
Finally, we are interested in the effects of fragmentation over the production of DM. To
study this, we must solve the Boltzmann equation during reheating with fragmentation,
given by equation (6.1). Fixing the reheating temperature to Treh = 0.1 GeV and the
mass of the Weakly Interacting Massive Particle (WIMP) to mχ = 100 GeV3. We then
solved the Boltzmann equation for a thermally averaged annihilation cross-section ⟨σv⟩ of
2The corresponding solutions are shown in figures 6.1 and 6.2.
3We set this values to compared our results with the work done, at the homogeneous level, in reference
[133].
106 7. Conclusion
the order of 10−10 GeV−2. Fixing the relic density to ΩDM,0h
2 = 0.1198± 0.0012, which is
the value reported by Planck 2018 results [9], fixed the cross section to ⟨σv⟩ ≈ 6× 10−10
GeV−2, and we kept it constant to compare with the predictions at the background level.
The results to the Boltzmann equation are shown in figure 6.4, we plotted Y = n/T 3,
where n is the density number of WIMPs and T the temperature given by expression
(6.8), as a function of the e-folds number N . The same solutions in the most common
variables Ỹ = n/s, where s is the entropy defined in equation (1.74), and x = mχ/T , are
plotted in figure 6.5. We can see that even though all of the scenarios predict a thermal
departure from equilibrium at around the same value x ∼ 10, the Yield’s freezes-out at
a different value. From this figure, we computed the DM relic density today and we got
equations (6.20) and (6.21),
Solid green line: ΩBackground
DM,0 h2 ≈ 0.131,
Dashed black line: ΩDM,0h
2 ≈ 0.113,
Solid blue line: ΩBackground
DM,0 h2 ≈ 0.037.
(7.2)
From this results, we conclude that:
If reheating takes place in potential V ∼ ϕ4, and the inflaton is coupled to a pair of fermions
through a Yukawa coupling and set Treh = 0.1 GeV, mχ = 100 GeV and ⟨σv⟩ = 6× 10−10,
we reproduce the relic density reported by Planck in the scenario with fragmentation. If,
at the background level, we fix the Reheating temperature (Yukawa coupling) we get less
(more) DM, see figures 6.5 and 6.6.
Freeze-out processes in scenarios with low reheating temperatures have been studied before
us [163]. However these did not include fragmentation effects, they were done at the
homogeneous level [133]. DM production in a the case k = 2, so that ⟨ωBackground
ϕ ⟩ = 0,
like an era matter-dominated have been considered as well, see reference [164].
The prospects of our work are extending the study to include particles with a p-wave
dominated annihilation cross section [165, 166]. Additionally, scenarios where the inflaton
couples to, for example, bosons or scalars [160, 167]. We briefly discussed the scenario of
bosonic reheating in subsection 4.1.3 at the homogeneous level. On top of that, we could
expand to consider different powers of k > 4 which also present fragmentation [134].
Finally, we can not ignored the lack of evidence for detection WIMPs [168, 169]. Hence,
alternative scenarios must be studied [170], one of the alternatives is known as non-thermal
DM or Feebly Interacting Massie Particles (FIMPs) [171, 172]. In this scenario the produc-
tion of particles never reaches thermal equilibrium [173, 174], thereby they are produced
in a freeze-in process [175]. For example, scalar DM [176], or in some MSSM, the Lowest
Supersymmetric Particle (LSP) is the gravitino (the s-partner of the graviton) which could
account for DM but never been in thermal equilibrium [177]. Or alternatively, gravitaion-
ally produced DM [178, 179] during reheating.
107
Appendix A
Friedmann metric detailed
A.1 Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2 Ricci’s tensor and scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3 Friedman equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.4 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
In this appendix we will derive the Friedmann equations and all the quantities associated
with this metric such as the Christoffel symbols which are relevant when solving the
geodesic and the Boltzmann equations. As we discussed in chapter 1, to obtain the
Friedmann equations, we need to substitute into the Einstein’s Field equations the metric
that satisfies the cosmological principle, that is, a metric that is spatially homogeneous
and isotropic and the energy momentum tensor given by equation (1.34).
The Einstein field equations are given by expression (1.27), that is
Rµν −
1
2
Rgµν = Gµν = 8πGNTµν + Λgµν , (A.1)
where on the left hand side we have the Einstein tensor, Gµν , defined in terms of the Ricci
tensor Rµν , the metric gµν and the Ricci scalar R; while on the right hand side has the
energy-momentum tensor Tµν and the cosmological constant Λ. The metric compatible
with our cosmological principle is the Friedmann-Lemâıtre-Robertson-Walker (FLRW)
metric given by equation (1.31), i.e.,
ds2 = gµνdx
µdxν = dt2 − a2(t)
(
dr2
1−Kr2 + r2dθ2 + r2 sin2 θdϕ2
)
, (A.2)
where a(t) is the scale factor, K = {+1, 0, 1} is the curvature parameter and {t, r, θ, ϕ}
108 A. Friedmann metric detailed
are comoving coordinates.
We know that the Ricci scalar is computed as the trace of the Ricci tensor Rµ
ν ,
R = Rµ
µ, (A.3)
the Ricci tensor is obtained as a contraction over two indices of the Riemann tensor Rρ
λµν ,
Rµν = Rρ
µρν , (A.4)
and the Riemann tensor Rρ
λµν is obtained as a function of the Christoffel symbols and its
derivatives, which in the mostly minus convention is given by,
Rρ
λµν = ∂µΓ
ρ
λν − ∂νΓ
ρ
λµ + ΓσλνΓ
ρ
σµ − ΓσλµΓ
ρ
σν . (A.5)
A.1 Christoffel symbols
In order to solve the the Einstein’s equations, we need to compute the Christoffel symbols
associated to our metric, these symbols are given by equation (1.16),
Γµαβ =
1
2
gµρ (∂αgρβ + ∂βgαρ − ∂ρgαβ) , (A.6)
and the inverse metric is obtained directly using the fact that we have a diagonal metric,
gµν =









1
− 1
a2(t)
(
1−Kr2
)
− 1
a2(t)r2
− 1
a2(t)r2 sin2 θ









. (A.7)
Time components: In this case we get the following Christoffel symbols
Γtrr =
aȧ
1−Kr2 ,
Γtθθ = aȧr2,
Γtϕϕ = aȧr2 sin2 θ.
(A.8)
The rest of the symbols that include a time index that are different from zero are
Γrtr = Γrrt = Γθtθ = Γθθt = Γϕϕt = Γϕtϕ =
ȧ
a
. (A.9)
A. Friedmann metric detailed 109
Radial components: For the symbols with radial components we get
Γrrr =
Kr
1−Kr2 ,
Γrθθ = −r
(
1−Kr2
)
,
Γrϕϕ = −r
(
1−Kr2
)
sin2 θ,
(A.10)
and
Γθrθ = Γθθr = Γϕrϕ = Γϕϕr =
1
r
. (A.11)
Angular components: The final non-zero symbols are given by
Γθϕϕ = −sin θ cos θ,
Γϕθϕ = Γϕϕθ = cot θ.
(A.12)
The rest of the symbols not listed here are all equal to zero.
A.2 Ricci’s tensor and scalar
Now we need to compute the Ricci tensor which is given by expression (A.4),
Rµν = Rρ
µρν = ∂ρΓ
ρ
µν − ∂νΓρµρ + ΓρρσΓ
σ
µν − ΓρµλΓ
λ
νσ, (A.13)
and the resulting non-zero component are listed next.
Time components: The only non-zero component in this case is
Rtt = −3
ä
a
. (A.14)
Radial components: Similarly, we get
Rrr =
aä+ 2ȧ2 + 2K
1−Kr2 . (A.15)
Angular components: The non trivial components are
Rθθ = r2
(
aä+ 2ȧ2K
)
, (A.16)
and
Rϕϕ = r2 sin2 θ
(
aä+ 2ȧ2 + 2K
)
. (A.17)
110 A. Friedmann metric detailed
Off-diagonal components: All of the mixed components of the Ricci tensor, such
as Rtr, Rtθ, and so on; are all zero as result of the FLRW spatial isotropy.
Finally, the Ricci scalar is given by equation (1.21), that is
R = Rµ
µ
= gµσRσµ
= gttRtt + grrRrr + gθθRθθ + gϕϕRϕϕ.
(A.18)
Using our previous results we get that the Ricci scalar in a FLRW universe is given by
R = −6
(
ä
a
+
ȧ2
a2
+
1
a2K
)
. (A.19)
A.3 Friedman equations
We know that the stress-energy tensor, Tµν , must be compatible with the cosmological
principle, so it must be diagonal, and we assume that Tµν corresponds to the stress-energy
tensor associated to an ideal fluid,
Tµν = (ρ+ p) uµvν − pgµν , (A.20)
where p is the pressure and ρ is the energy density, both of them isotropic, ρ = ρ(t) and
pressure p = p(t). Notice here a subtlety, this stress tensor has been defined in the mostly
minus convention, otherwise there would have been a +pgµν instead of −pgµν .
We choose uµ = (1, 0, 0, 0) to be the four-velocity vector in a comoving frame and using
our Friedmann metric we get
Tµν =






ρ
a2
1−Kr2p
a2r2p
a2r2 sin2 θp






. (A.21)
Time component (0− 0):
This component of the Einstein equation gives us the so-called first Friedmann equa-
tion
H2 ≡
(
ȧ
a
)2
=
8πGN
3
ρ− 1
a2
K +
1
3
Λ, (A.22)
A. Friedmann metric detailed 111
where we have defined the Hubble parameter as
H ≡ ȧ
a
, (A.23)
observations tell us that today ȧ > 0, that is, the universe is expanding.
Spatial components (i− i):
Beginning with the (1− 1) component, we get
2
ä
a
=
ȧ2
a2
+
1
a2
K + 8πGNp− Λ,
substituting the first Friedmann equation (1.47) into this relation leads us to the
second Friedmann equation
ä
a
= −4πG
3
(ρ+ 3p) +
1
3
Λ, (A.24)
while the (2 − 2) and (3 − 3) components lead to the same results (or a linear
combination of both previous equations).
Since both the metric gµν (and as a result of this, the Ricci tensor Rµν) and the
energy-momentum tensor Tµν are diagonal (homogeneous), the non-diagonal com-
ponents of the Einstein equations lead to a trivial relation of zero equals zero.
Therefore the evolution of our universe is described by the Friedmann equations which are
H2 =
(
ȧ
a
)2
=
1
3M2
p
ρ− 1
a2
K +
1
3
Λ,
ä
a
= − 1
6M2
p
(ρ+ 3p) +
1
3
Λ,
(A.25)
These two Friedmann equations conform a coupled system of differential equations for
three variables which are the energy density, ρ, the pressure, p, and the scale factor, a.
So it looks like we are missing one equation. Also, here we have introduced the reduced
Planck mass, Mp,
M2
p ≡
1
8π
M2
pl. (A.26)
defined in terms of the Planck Mass, Mpl,
M2
pl =
1
GN
, (A.27)
112 A. Friedmann metric detailed
A.4 The continuity equation
We know that the stress-energy tensor Tµν came with a conservation law given by (1.29),
∇µT
µν = ∂µT
µν + ΓνµσT
µσ + ΓµµσT
σν = 0, (A.28)
we already computed the Christoffel symbols for the FLRW metric and are given in equa-
tions (A.8)-(A.12), and we get T µν using our inverse metric given by equation (A.7) to
rise both indices of Tµν given by equation (A.21) and we get
T µν = gµαgνβTµν =








ρ
1
a2
(
1−Kr2
)
p
1
a2r2
p
1
a2r2 sin2 θ
p








, (A.29)
notice here the small subtlety, in these spherical coordinates, the tensor T µν is written as
T µν = gµαTαν =




ρ
−p
−p
−p




= Tµ
ν , (A.30)
that is, only with a mix of covariant and contravariant indices, the stress energy tensor
has no dependency on the scale factor, a.
Time component: When we compute for ν = t = 0,
∇µT
µ0 = ∂µT
µ0 + Γ0
µσT
µσ + ΓµµσT
σ0
= ∂0T
00 + Γ0
11T
11 + Γ0
22T
22 + Γ0
33T
33 + Γ1
10T
00 + Γ2
20T
00 + Γ3
30T
00
=
∂ρ
∂t
+
[
aȧ
1−Kr2
] [
1
a2
(1−Kr2)p
]
+
[
aȧr2
]
[
1
a2r2
p
]
+
[
aȧ
1−Kr2
] [
1
a2r2 sin2 θ
p
]
+
ȧ
a
ρ+
ȧ
a
ρ+
ȧ
a
ρ
= ρ̇+ 3
ȧ
a
(p+ ρ)
= 0.
Thereby, we get the energy conservation equation
ρ̇+ 3H (ρ+ p) = 0. (A.31)
or as it is sometimes called, the continuity equation.
A. Friedmann metric detailed 113
Spatial components: A simple computation reveals that the remaining three equa-
tions, for ν = i, vanish identically.
One can show, by direct computation, that the continuity equation (A.31) can be derived
by a linear combination of the Friedmann equations (given by expression (A.25)). That
means that we are still one equation short to fully solve Friedmann systmem of equations.
We are missing an equation will be an expression that relates p = p(ρ, a), the simplest
realization is to assume an equation of state
p = wρ, (A.32)
where w is called the equation of state parameter.
Therefore, to fully solve the Einstein equations for a metric compatible with our cosmo-
logical principle, we need to solve the following system of equations
H2 =
(
ȧ
a
)2
=
1
3M2
p
ρ− 1
a2
K +
1
3
Λ,
ρ̇ = −3 ȧ
a
(1 + w)ρ
p = wρ,
(A.33)
and sometimes might be useful to use the second Friedmann equation that was
ä
a
= − 1
6M2
p
(1 + 3w) ρ+
1
3
Λ, (A.34)
which as we said, it is a linear combination of the Friedmann and the continuity equations.

115
Appendix B
Boltzman Equation
B.1 Non relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B.2 Relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.3 In a FLRW universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
The Boltzmann equation, or as it is sometimes refer to, the Boltzmann transport equation,
describes the time evolution of the distribution function associated to a system, f , as a
function of the microscopic collisions.
Suppose that we have a thermodynamic system (which is in general not in thermal equi-
librium), but somehow, we know the actual distribution function f = f(t, r,v), the dis-
tribution function is defined as
f(t, r,v)d3rd3v ≡ the mean number of particles at a time t
that are located between r and r + dr
and have a velocity between v and v + dv.
(B.1)
The distribution function f provides a complete description of the macroscopic state of
the system and allows us to compute physical quantities of interest. Hence any problem
treating with non equilibrium thermodynamics can be solved by attempting to compute
the actual distribution function. To do this, we need to use the Boltzmann equation,
which reads as
L̂[f ] = Ĉ[f ], (B.2)
where L̂ and Ĉ are the Liouville and Collisions operators respectively.
116 B. Boltzman Equation
Let us define the mean number of particles as follows
n(r, t) ≡ the mean number of particles (irrespective of their velocity)
which at a time t are located between r and r + dr,
(B.3)
then, by this definition
n(r, t) =
∫
d3vf(r,v, t), (B.4)
where we integrate over all possible velocities.
Let us define χ = χ(t, r,v) as a general function that denotes a property of a particle
located at time t at the position r and with a velocity v, for example, χ might denote the
energy E. The mean value of χ at a time t is denoted by ⟨χ(r, t)⟩ and it is given as
⟨χ(r, t)⟩ ≡ 1
n(r, t)
∫
d3vf(r,v, t)χ(r,v, t). (B.5)
So computing the distribution function of the system is fundamental when trying to de-
scribe the macroscopic properties of a system. For example, the mean velocity ⟨v(r, t)⟩ ≡
u(r, t) of a particle at position r and time t is defined by
⟨v(r, t)⟩ ≡ u(r, t) =
1
n(r, t)
∫
d3vf(r,v, t)v. (B.6)
B.1 Non relativistic
The Boltzmann equation, in general, reads as equation (A.2), i.e.
L̂[f ] = Ĉ[f ].
in general, we have
df(t, r,v)
dt
=
∂f
∂t
+
∂f
∂ri
dri
dt
+
∂f
∂vi
dvi
dt
, (B.7)
so that, in the non relativistic regime we define the Liouville operator as
L̂ =
∂
∂t
+
dx
dt
·∇x +
dv
dt
·∇v =
d
dt
+ v ·∇x +
1
m
F ·∇v, (B.8)
where we use Newton’s relation F = m
dv
dt
.
B. Boltzman Equation 117
B.2 Relativistic
In GR (General Relativity), the four-position and four momentum are defined as
xµ = (t, r) ,
pµ = (E,p) ,
(B.9)
where E is the energy of the particle, r and p are the three-position and the three-
momentum vectors, respectively, in this scenario, the distribution function becomes a
function of these quantities, f → f (xµ, pµ). So that the Liouville operator now will be
written as
L̂[f ] =
df
dλ
(xµ, pµ) =
∂f
∂xµ
dxµ
dλ
+
∂f
∂pµ
dpµ
dλ
, (B.10)
as we discussed on chapter 1, the trajectory of a particle in GR obeys the geodesic equation
(1.15), which reads as follows
d2xµ
dλ2
+ Γµαβ
dxα
dλ
dxβ
dλ
= 0, (B.11)
where λ parametrizes the trajectory and Γµαβ are the Christoffel symbols which are given
by
Γµαβ =
1
2
gµν (∂αgβν + ∂βgαν − ∂νgαβ) . (B.12)
Using the relation
pµ = m
dxµ
dτ
=
dxµ
dλ
, (B.13)
we may expressed the geodesic equation in terms of the four-momentum as
dpµ
dλ
+ Γµαβp
αpβ = 0. (B.14)
Substituting both equations (B.13) and (B.14) into expression (B.10) we get that the
Liovulle operator in GR is written as
L̂[f ] =
∂f
∂xµ
pµ − ∂f
∂pµ
Γµαβp
αpβ = Ĉ[f ]. (B.15)
118 B. Boltzman Equation
B.3 In a FLRW universe
Finally, we will derive the form of the Boltzmann equation in a FLRW universe. First of
all, for an homogeneous and isotropic universe, we expect that the distribution function
shows this properties,
f(xµ, pµ) = f (|p| , t) = f(E, t), (B.16)
Now, we need to express the liouville operator in a FLRWmetric, to do this, we will use the
Christoffel symbols computed on appendix A which are given by equations (A.8)-(A.12),
we get
L̂[f ] = pµ
∂f
∂xα
− Γµαβp
αpβ
∂f
∂pµ
= p0
∂f
∂x0
− Γ0
αβp
αpβ
∂f
∂p0
= E
∂f
∂t
− Γ0
ijp
ipj
∂f
∂E
= E
∂f
∂t
− ȧ
a
gijp
ipj
∂f
∂E
= E
∂f
∂t
− ȧ
a
|p|2 ∂f
∂E
,
that is, in a FLRW metric, the Liouville operator readas as
L̂[f(E, t)] = E
∂f
∂t
− ȧ
a
|p|2 ∂f
∂E
. (B.17)
Now, let us integrate with respect to twice the phase space,
g
∫
d3p
E(2π)3
L̂[f ] = g
∫
d3p
(2π)3
∂f
∂t
− ȧ
a
g
∫
d3p
(2π)3
|p|2 ∂f
E∂E
=
d
dt
[
g
∫
d3p
(2π)3
f
]
− ȧ
a
g
∫
d3p
(2π)3
|p|2 ∂f
|p|∂|p|
=
dn
dt
− ȧ
a
g
∫
4π|p|3d|p|
(2π)3
∂f
∂|p|
=
dn
dt
+ 3
ȧ
a
g
∫
4π|p|2d|p|
(2π)3
f
=
dn
dt
+ 3
ȧ
a
[
g
∫
d3|p|
(2π)3
f
]
=
dn
dt
+ 3
ȧ
a
n.
B. Boltzman Equation 119
That is, after (twice the) phase space integration, the Boltzmann equation in a FLRW
metric is written as
ṅ+ 3Hn = g
∫
d3p
E(2π)3
Ĉ[f(E, t)]. (B.18)

121
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