master thesis: relativistic kinetic theory of particle ... · malism of relativistic kinetic theory...

Master thesis:

Relativistic Kinetic Theory

of Particle-Anti-Particle Asymmetries

and Dark Matter

by Dennis Janzen/ Master of Science: Physics([email protected])

Bielefeld University

Filing date: 31 May, 2019First Supervisor: Prof. Dr. Dominik J. SchwarzSecond Supervisor: Prof. Dr. Nicolas Borghini

Abstract

In this thesis the properties of Dark Matter particle species with a particle-anti-particle asym-metry are investigated assuming the theory of Cold Dark Matter (CDM). The observed densitycontribution of Dark Matter to the total matter density of the Universe restricts the modelling ofthe properties of Dark Matter. These restrictions have to be varied when introducing a particle-anti-particle asymmetry in the Dark Matter to accord with observations. In this presentation,Dark Matter is modeled as a Weakly Interacting Massive Particle (WIMP) that is assumed tobe heavy and have a very weak hypothetical interaction with visible matter (not to be confusedwith the Weak Nuclear Force of the Standard Model). This interaction is assumed to havemaintained thermodynamic equilibrium with visible matter at high temperatures in the earlyUniverse and also to have frozen out very early, explaining why Dark Matter does not interactwith visible matter anymore today.

The presented qualitative analysis of the freeze-out behaviour of the considered WIMPspecies is based on the formalism of relativistic kinetic theory and uses the Boltzmann equation.Regarding the WIMP paradigm and the CDM theory, these calculations lead to the constraintthat the thermally averaged cross-section of the considered species has to be on the weak scale inthe order of 10−37 cm2 in absence of a particle-anti-particle asymmetry. In the particle-symmetriccase the density contribution to the total density of the Universe is found to have no significantdependence on the particle’s mass. With an introduced asymmetry the interaction strengthof the WIMP is allowed to be higher to maintain the same magnitude of the WIMP’s densitycontribution. In an analytical approximation the abundance of the anti-particle is neglected anda relative WIMP density of ΩWIMP . 1 and a WIMP mass of ∼ 1 GeV is assumed. It is foundthat a particle-anti-particle asymmetry of D ∼ 10−9, corresponding to a ratio of the chemicalpotential to the temperature of µ/T ∼ 10−7, compensates an increase of the cross-section by afactor of 2 with respect to the symmetric case. In any case the cross-section is not allowed to bemuch weaker as the bound that is set in the symmetric case. In the presence of an asymmetrythe WIMP density contribution also becomes mass-dependent. For the same magnitude of theasymmetry D and a mass of ∼ 2 GeV the cross-section is demanded to be larger by a factor of4 with respect to the symmetric case. Hence, a presence of an asymmetry noticeably changesthe properties of a Cold Dark Matter WIMP species.

The first chapter presents observations and theories of Dark Matter and treats constraintsfrom the Friedmann-Lemaıtre-Robertson-Walker Universe and thermodynamics. Then, the for-malism of relativistic kinetic theory is presented in detail in the second chapter, starting formthe phase-space description and deducing the Boltzmann equation and its equilibrium solutions.The third chapter contains the calculation of freeze-out and decoupling applying the Boltzmannequation. The constraints on the properties of a symmetric WIMP species inferred from thefreeze-out calculations are discussed. For completeness, the evolution of an unstable particleand the associated increase of entropy due to the decays is shown for the symmetric case. In thelast chapter the formalism is extended by including possible particle-anti-particle asymmetries.Then, the asymmetry’s influence on the freeze-out behaviour and on the properties of the WIMPis presented.

Contents

1 Introduction 1

1.1 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The WIMP Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The Baryon Asymmetric Universe and Asymmetric Dark Matter Models . . . . . 5

1.4 The Friedmann-Lemaıtre-Robertson-Walker Universe . . . . . . . . . . . . . . . . 5

1.4.1 The FLRW Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Epochs in the Evolution of the Universe . . . . . . . . . . . . . . . . . . . 6

1.4.3 Density Contributions to the Energy Density of the Universe . . . . . . . 9

1.4.4 Evolution of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Relativistic Kinetic Theory 12

2.1 The Relativistic Phase-space Description and the Liouville Theorem . . . . . . . 12

2.2 The Phase Space Distribution and the Boltzmann Equation . . . . . . . . . . . . 16

2.3 The Collision Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 The Thermally Averaged Cross-section times Relative Velocity . . . . . . . . . . 20

2.5 Equilibrium Solutions of the Distribution Function . . . . . . . . . . . . . . . . . 23

2.5.1 Collisional Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2 Entropy evolution and the H-Theorem . . . . . . . . . . . . . . . . . . . . 26

2.5.3 The Equilibrium Distributions . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.4 The Limits of Radiation and Matter Domination . . . . . . . . . . . . . . 29

3 Evolution of a Symmetric Particle Species in the FLRW Universe 31

3.1 Equilibrium in the Early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The Boltzmann Equation in the FLRW Universe . . . . . . . . . . . . . . . . . . 32

3.3 Scaling out the Universe’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Calculating Relic Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.1 Hot Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 Example: Light Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 Cold Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.4 Example: Baryon-symmetric Universe . . . . . . . . . . . . . . . . . . . . 43

3.4.5 WIMPs as Cold Dark Matter and Heavy neutrinos . . . . . . . . . . . . . 44

3.5 Out of Equilibrium Decay and Entropy Production . . . . . . . . . . . . . . . . . 46

3.5.1 Calculating the Increase of Entropy during Re-heating . . . . . . . . . . . 47

4 Evolution of Asymmetries in the FLRW Universe 53

4.1 The Meaning of the Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 The Evolution Equation of an Asymmetric Species . . . . . . . . . . . . . . . . . 55

4.3 High Chemical Potentials and Quantum Effects . . . . . . . . . . . . . . . . . . . 57

4.4 Calculating Relic Abundances of an Asymmetric Species . . . . . . . . . . . . . . 59

4.4.1 Hot Asymmetric Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.2 Cold Asymmetric Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 Asymmetric WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Discussion and Conclusion 66

A Appendix: Conventions 68

B Appendix: Differential Geometry 69

C Appendix: Detailed Calculations 74

D Appendix: Python Code 76D.1 Freeze-out of a Symmetric Cold Relic . . . . . . . . . . . . . . . . . . . . . . . . 76D.2 Re-heating by the Decay of an Unstable Particle Species . . . . . . . . . . . . . . 78D.3 Freeze-out of an Asymmetric Cold Relic . . . . . . . . . . . . . . . . . . . . . . . 81D.4 Exclusion Plot of Allowed WIMP Cross-sections and Magnitudes of Asymmetry . 84

References 86

Register of illustration 88

Statutory declaration 89

I dedicate this thesis to all the people having patience with me while I was going throughrough times, to the great musicians of the twentieth century providing me with their marvelousmusic, keeping me from going insane (contrary to recently popular music) and to the electricitycompany that made their repayment ahead of time, such that I had enough money for this monthand could focus on this without worries.

1

1 Introduction

This thesis investigates the constraints on the properties of Dark Matter obtained from freeze-out calculations. The considered hypothetical particle species is a Weakly Interacting MassiveParticle species and a candidate for Cold Dark Matter.

First, an introduction to the observation of Dark Matter is given and the notions of theWIMP paradigm are sketched. Furthermore, the assumption of the existence of a particle-anti-particle asymmetry for Dark Matter is introduced in the context of Asymmetric Dark MatterModels. Then, the constraints of the Friedmann-Lemaıtre-Robertson-Walker model describingour Universe are presented. In the second chapter the formalism of relativistic kinetic theoryis constructed and discussed in detail. This formalism is the foundation for the freeze-outcalculations in chapters 3 and 4. The third chapter treats the evolution of a symmetric particlespecies regarding the WIMP paradigm and the forth chapter extends the discussion to the caseof a presence of a particle-anti-particle asymmetry in the Dark Matter sector. The main aimof this thesis is to show that the constraints on the properties of WIMP Cold Dark Mattersignificantly change with respect to the symmetric case when an asymmetry is introduced.

1.1 Dark Matter

Most of the Universe is dark. That does not refer to the emptiness of space, but to the cos-mological observations which indicate the existence of an additional kind of matter, labeled asDark Matter (DM). ‘Dark’ means that it does not interact with any electromagnetic radiationand is not the right attribute, ‘invisible’ would serve better, because this kind of matter doesnot only emit no light, it neither reflects nor absorbs light. Until now the only indication of itsexistence is given by its gravitational action.

The first speculations about the existence of Dark Matter have been made by Fritz Zwicky(1937) who investigated orbital velocities in the Coma Cluster. He pointed out that the massdensity in the Cluster has to be much higher than the luminous mass. Later, the Dutch scientistJan Hendrik Oort measured unexpectedly high orbital velocities around the Galactic Center forstars near our solar system. These velocities indicate a higher mass density of the Milky Waythan expected from mass-to-light ratios and Keplerian motion. The conclusion was the existenceof an additional part of matter and the first indirect observation of DM. The postulation ofmissing matter is based on the fact that rotational velocities fail to fit Kepler’s third law:

v(r) =

√GM(r)

r, (1.1)

where v(r) is the orbital velocity at a distance r from the Galactic Center, G Newton’s gravia-tional constant and M(r) the mass inside the radius r (Kolb and Turner, 1990, Sec. 1.7).

Later observations confirm that other galaxies also have higher rotational velocities in theirouter part (Oort, 1940). Rubin and Ford (1970) were the first astronomers carrying out aspectroscopic analysis to map the galaxy rotation curve of the Andromeda galaxy. This workestablished the foundation of modern surveys of galaxy rotation curves.

With the measured luminous mass distribution in the galaxy and the rotation curve it be-comes possible to model the velocity contribution of the DM and its density distribution in theshape of a halo (see Fig. 1).

After fitting, the observed data indicates that most of the galaxy’s mass is not located inthe galaxies buldge but in the DM halo. Regarding all measurements it was concluded that themass of a galaxy should in total be roughly five times the mass of visible matter. The density ofthe visible matter is approximately the density of baryons as leptons (electrons and neutrinos)

2 1 INTRODUCTION

Figure 1: The galaxy rotation curve of the galaxy NGC 3198 (van Albada et al., 1985): The totalrotation curve is composed of a contribution from the disk’s luminous mass and a DM halo component.

are much lighter compared to baryons (protons and neutrons). The observed ratio between theDM density parameter ΩDM and the density pareter of baryons ΩB is (Ade et al., 2016)

ΩDM ∼ 5 ΩB. (1.2)

The observation of galaxy rotation curves has not been the only indication of the existenceof DM. A further phenomenon in which the gravitational action of DM has been dicscovered isgravitational lensing. It describes the curving of spacetime by huge masses, such that nearbylight rays are bent towards the mass. The light rays from behind the heavy object seem to cometo the observer from another angle (see Fig. 2).

Figure 2: Model of gravitational lensing (Schneider et al. (1999), modified): A light ray comes to anobserver O from a source S by the deflection caused by a mass M, such that it is seen coming from anangle θ while in absence of deflection its actual angular seperation from the lensing object would beβ.

1.2 The WIMP Paradigm 3

The calculation of the deflection mass can be simply demonstrated in the limit of weaklensing. In this limit, deflection is happening far from the mass, such that the mass can beassumed to be point-like (Massey et al., 2010). The deflection angle α becomes

α =4GMDd

c2θ=

2RSDd

θ(1.3)

with the distance to the graviational lens (the massive object) Dd and the Schwarzschild radiusRS (Schneider et al., 1999). A further geometrical relation for the undeflected angle β is givenby

β = θ − Dds

Ddα, (1.4)

with the distance of the source to the lens Dds. By knowing Dd, the distance to the source andby measuring θ the deflection mass M can be calculated. Comparing this to mass calculationsfrom mass-to-light ratios there is also a difference found in mass which can be explained bypostulating the existence of DM. In the Sloan Digital Sky Survey over 350 000 gravitationallenses have been observed (Mandelbaum et al., 2006). For a galaxy with a luminous mass of theorder of 1010M (M, solar mass) the DM halo was found to be an order of magnitude higher.

Today the existence of dark matter is nearly undisputed, even if it is not the only way tointerpret the mentioned observations. Different other models have tried to modify the Newtoniangravity itself, but did not succeed in completely covering all phenomena. These kinds of modelswill not be covered here, instead the expected properties of DM are investigated.

Today the nature of DM is still unknown. Elementary particle physics are rather effectivelydecribed by the Standard Model (SM), including three of the four fundamental forces of physics.The SM still struggles with a quantum gravity theory and DM cannot be explained withoutassuming the existence of one or more additional particle species. The observation of DM givesrise to a completely new field of extensions to the SM trying to predict certain properties ofthe DM particles, but no theory could be confirmed by experiments so far. As mentioned, DMonly appears to us via its gravitational action. It has no electromagnetic interaction and italso does not interact via the strong nuclear force. Some DM models rely on the assumptionthat DM has some ‘weak interaction’ with visible matter, not necessarily but possibly the SM’sweak interaction. This assumption gives rise to the WIMP paradgim presented in the followingsection.

1.2 The WIMP Paradigm

Today’s collider experiments involve high particle energies upto ∼ 13TeV (Peilian, 2018). Untilnow Dark Matter has not been found, which leads to the assumption that a DM particle speciesmust be very heavy. Backtracking the expansion of the Universe it is expected that the Universewas hot and all particle species have been in thermodynamic equilibrium at some point in time(Kolb and Turner, 1990, Chap. 5). Due to the cooling of the Universe caused by the expansion,every interaction freezes out at some point as the interaction rates decrease with decreasingtemperature. At some point the interaction rates become smaller than the expansion rate andbecome impotent, i.e. they do not happen anymore (‘freeze out’ and ‘become impotent’ arefrom now on used interchangeably).

A further effect of the cooling is that annihilation reactions producing heavy particles becomeineffective, i.e. the temperature has fallen below the mass of the heavy particles and collisionalenergies do not suffice anymore to produce them. If the interaction rates of these annihilationsalso freeze out, the remaining heavy particles will not annihilate anymore and become over-abundant. Then, the species decouples and thermodynamic equilibrium with the rest of theUniverse is not maintained anymore. The consequence is that the heavy species will dominate

4 1 INTRODUCTION

the energy density of the Universe at some time. Following the evolution and the cooling of theUniverse this happens to every massive particle species at some point. After that the Universebecomes matter dominated. The time at which all species are in thermodynamic equilibriumis noted as the radiation domination epoch (Kolb and Turner, 1990, Chap. 3).

A rigorous assumption is that DM has some interaction with visible matter that has becomeimpotent at high temperatures, which is the reason DM does not interact with visible mattertoday. Due to the early freeze-out the considered interaction has to be weak (not to be confusedwith the Weak Nuclear Force). This weak interaction gives rise to the paradigm of WeaklyInteracting Massive Particles (WIMPs) (Kolb and Turner, 1990, Chap. 1). These particles haveto be massive to compensate the lack of matter observed in galaxy rotation curve surveys andgravitational lensing experiments.

Further constraints on the properties of the WIMPs are given by the investigation of structureformation in the Universe. The observed energy density in galaxies is about five orders ofmagnitude higher then the average density of the Universe (Kolb and Turner, 1990, Chap. 9).To allow density fluctuations in that order of magnitude it is not sufficient to only considerbaryonic matter in the calculation of structure formation. Observations of the Cosmic MicrowaveBackground (CMB) which marks the beginning of the end of the radiation domination epochalso also confirm that DM is needed to explain the magnitude of the density fluctuations (ibid.,Ch. 9). The growth of density fluctuations only becomes significant in the epoch of matterdomination. Observations of the CMB exhibit much higher density fluctuations than can beexplained by baryonic matter. The higher magnitude of these fluctuations can only be explainedby a very heavy particle species whose interactions froze out very early, i.e. by a WIMP species.Due to the early freeze-out, density fluctuations could grow to the observed order of magnitude.

The argument from structure formation indicates that the needed WIMP species has tobe a cold relic. That means that the WIMP species has to be sufficiently heavy, that itsdecoupling happens after the temperature of the Universe dropped below the WIMP mass. Sothe WIMP species already becomes non-relativistic before the freeze-out. These constraintsto the properties of DM give rise to the Cold Dark Matter (CDM) model, stating the WIMPparticle to be a cold relic (ibid.). There are a lot of possible candidates for CDM like heavyneutrinos, axions and supersymmetric particles (ibid.). In this thesis, the focus will not beon these different candidates which all are WIMPs. Instead the evolution of WIMPs will beinvestigated in general.

The arguments from structure formation exclude the model of Hot Dark Matter (HDM),i.e. excludes DM to be a hot relic. This means, that the observed density fluctuations in theUniverse cannot be explained by a WIMP decoupling when it is still relativistic. Possible HDMcandidates are light neutrinos, which will be briefly mentioned in Ch. 3.

Due to observational constraints the CDM model will be the model of interest in this thesis.Later chapters will explain the physical meaning of the epochs of matter domination and radi-ation domination and the associated terms of cold and hot relics in more detail. There, it willbecome clear how the evolution of a cold relic DM species will differ from the evolution of a hotrelic.

The observed density of visible matter, dominated by baryons, is expressed by the relativedensity: ΩB ∼ 0.042 (Perkins, 2009), where the total density parameter of the Universe isassumed to be Ω ≈ 1. Now, our Universe is known to be baryon-asymmetric. The observeddensity only that high due to this asymmetry. In Ch. 3 it will be shown, that most matterwould annihilate in the case of matter-anti-matter equality. As the contribution of DM to thetotal matter density in the Universe is dominant and five times higher than the contribution ofbaryonic matter (see Eq. (1.2)), the question arises, if DM also has some particle-anti-particleasymmetry. This idea gives rise to Asymmetric Dark Matter models. The following section will

1.3 The Baryon Asymmetric Universe and Asymmetric Dark Matter Models 5

give a brief sketch of the notions of these theories.

1.3 The Baryon Asymmetric Universe and Asymmetric Dark Matter Models

The fact that we exist and there is some structure as galaxies, stars and planets is alreadyevidence for the asymmetry of the Universe, because everything around us consists of matter andnot of anti-matter. In particle physics it was believed for a long time that all particle reactionsconserve some symmetry in the sense that exchanging all particles participating in a reaction bytheir corresponding anti-particle partner would not change any physics. This conservation law isstated as the CP-conservation, the conservation under exchanging the charge C and the parity Psimultaneously. But if this conservation held true in all physics, matter and anti-matter shouldbe equally abundant. Actually, a CP-violating interaction was found in the decay of the K0

meson which has two annihilation channels (Aaij et al., 2019). One annihilation channel has ahigher interaction rate than the corresponding CP-conjugated annihilation channel. Therefore,in net the products of the channel with shorter liftime become more abundant.

Any particle-anti-particle asymmetry has to be generated by such a CP-violating interaction(Kolb and Turner, 1990, Chap. 6). The focus will not be on the formalism of asymmetrygeneration in this thesis, but instead investigate the evolution of a particle species with a givenpre-existing asymmetry.

The previous section gave a sketch of the notion of the WIMP paradigm, where a WIMPDM particle species is assumed to have been in thermodynamic equilibrium with the rest ofthe Universe at some point. This thermodynamic equilibrium should have been maintained bya hypothetical interaction that has frozen out very early. Asymmetric Dark Matter (ADM)models assume that this interaction between the dark sector (summarizing all dark matterspecies) and the visible sector conserves some non-zero quantum number associated with theparticle-anti-particle asymmetry (Zurek, 2014). The consequence is that an particle-anti-particleasymmetry in the visible sector is directly connected to an asymmetry in the dark sector due tothe shared quantum number.

The aim of this thesis is to investigate in which way the properties of a Cold Dark MatterWIMP change, if one assumes an asymmetry in the dark sector. The intention is to determinethe dependecies of the DM density in the Uninverse, especially the influence of a DM asymmetryon the (indirectly) observed DM density today.

The evolution of a particle species is described by the formalism of relativistic kinetic the-ory. The formalism provides the tools for calculating the relic abundance of a particle speciesafter freeze-out and for the investigation of conserved quantities in thermodynamic equilibrium.Such a conserved quantity can be a quantum number associated with a particle-anti-particleasymmetry.

It is important to understand the conditions of this formalism in great detail because a lot ofinitial assumptions have to be made for the freeze-out calculation. Moreover, formalism needsinput from the Friedmann-Lemaıtre-Robertson-Walker (FLRW) model of our Universe and somethermodynamic relations. So, before constructing the framework of relativistic kinetic theorythe most importent features of this model and the required thermodynamics will be sketched.

1.4 The Friedmann-Lemaıtre-Robertson-Walker Universe

1.4.1 The FLRW Metric

This section is based on Kolb and Turner (1990, Chap. 2), so redundant citation will be omitted.

The fundamental quantity used in cosmological calculations is the metric tensor gµν . It

6 1 INTRODUCTION

describes the geometry of spacetime and is used to calculate apparent physical distances. Adistance can be associated with a line element defined by

ds2 = gµν dxµdxν , (1.5)

with two arbitrary displacement vectors dxµ.Our Universe is observed to be expanding and to be spatially homogenous and isotropic.

These properties are the basic assumptions in the FLRW model. In this model the line elementtakes the following form:

ds2 = gµν dxµdxν = dt2 −R2(t)

(dr2

1− κr2+ r2dθ2 + r2 sin2(θ)dφ2

). (1.6)

R(t) is the scale factor parametrizing the spacetime expansion as a function of time. Asadditionally, the Universe is observed to be flat the curvature parameter κ can be set to zero,such that Cartesian for the spatial part of the metric can be used:

ds2 = dt2 −R2(t)(dx2

1 + dx23 + dx2

3

). (1.7)

Hence the metric tensor elements are

g00 = 1, gi0 = 0 and gij = −R2(t)δij , (1.8)

and the metric’s determinant isg := det(gµν) = −R6. (1.9)

Inserting the metric tensor into the definition of the Christoffel symbols (see Appendix B, B.37),the only non-zero Christoffel symbols are

Γ0ij = RR δij and Γi0j = Γij0 =

R

Rδij = Hδij , (1.10)

with the Hubble expansion rate of the Universe H:

H =R

R. (1.11)

1.4.2 Epochs in the Evolution of the Universe

This section is based on Kolb and Turner (1990, Chap. 3).

The evolution of the Universe can be expressed by the evolution of spacetime, energy andtemperature. Together with the Hubble rate and the FLRW metric the description of spacetimeis given by the Friedmann equation,

R2

R2+

κ

R2=

8πG

3ρ+

Λ

3, (1.12)

with Newton’s gravtational constant G, the curvature parameter κ and the energy density ρ.The Friedmann equation is deduced by inserting the FLRW metric into the Einstein equation:

Gµν = 8πGTµν + Λgµν . (1.13)

Λ is the cosmological constant which is connected to the vacuum energy density and is so smallthat it can be neglected for now. Gµν is the Einstein tensor describing the properties of spacetime

1.4 The Friedmann-Lemaıtre-Robertson-Walker Universe 7

(see Eq. B.44) with respect to the energy distribution and Tµν is the total stress-energy-momentum tensor describing the energy distribution. Due to isotropy and homogeneity thereare no shear stresses in the FLRW Universe, so Tµν has to be diagonal:

Tµν = diag(ρ,−p,−p,−p), (1.14)

with p being the hydrostatic pressure.Further, the evolution of the energy density can be divided into several epochs, the epochs

of inflation (which will not be the focus in this thesis), matter domination, radiation dominationand vacuum energy domination. Due to the vanishing covariant derivative of the Einstein tensorand metric tensor (see Appendix B, Eq. B.43) the stress energy conservation holds1:

Tµν;ν = 0. (1.15)

The 1st law of thermodynamics for an adiabatically expanding Universe is deduced from the(µ = 0)-component:

T 0ν;ν =

∂T 00

∂t+ Γ0

ijTij + Γii0T

00 = ρ+ 3R

R(p + ρ) = 0

⇒ d(ρR3) = −pd(R3). (1.16)

Tµν takes into account all particle species. Regarding Eq. (1.16), the expansion is adiabaticbecause there are no additional terms on the right-hand side of Eq. (1.16), neither of heatexchange,

dQh = TdS, (1.17)

nor of creation and annihilation of particles i:

−∑i

µid(niR3), (1.18)

where dS is the change of entropy, T the temperature, ni the number density and µi the chemicalpotential of the particle i.

In general, now, an equation of state can be inserted:

ρ = wp (1.19)

Hence, we obtain different evolution of the energy density for different epochs of the Universeby integrating Eq. (1.16):

ρ = ρi

(R

Ri

)−3(1+w)

(1.20)

with initial values Ri = R(ti) and ρi. In the early Universe all particle species should have beenultra-relativistic at some point. This time is defined as the radiation domination epoch.Later most particles became non-relativistic and the Universe became matter-dominated.These two cases give rise to different equation of state parameters w resulting in different energydensity evolutions:

Radiation:

(w =

1

3

)ρ ∝ R−4, (1.21)

Matter : (w = 0) ρ ∝ R−3. (1.22)

1Note that with Eq. (B.28) and (1.8) Tµν is then given by Tµν = Tµνgµν = diag

(ρ, p

R2 ,pR2 ,

pR2

)

8 1 INTRODUCTION

Inserting these relations into the Friedmann equation (1.12), by integrating we get a relationbetween the scale factor and time (neglecting κ and Λ):

R

Ri=

√8πGρi

3

(t

ti

) 23(1+w)

, (1.23)

with an initial time ti. Setting R = Ri and t = ti Eq. (1.23) becomes

1 =

√8πGρi

3. (1.24)

Therefore, the relation (1.23) becomes

R

Ri=

(t

ti

) 23(1+w)

. (1.25)

The relations between R and t in the two epochs are then

Radiation:

(w =

1

3

)R ∝ t

12 , (1.26)

Matter : (w = 0) R ∝ t23 . (1.27)

The evolution of the Hubble rate (1.11) is in general given by the Friedmann equation (1.12),

H =R

R=

√8πG

3ρ, (1.28)

and in the respective limits using (1.25):

Radiation:

(w =

1

3

)H =

1

2

(1

t

), (1.29)

Matter : (w = 0) H =2

3

(1

t

). (1.30)

Inserting these relations into (1.28) gives:

Radiation:

(w =

1

3

)ρ =

1

4· 3

8πGt−2 (1.31)

Matter : (w = 0) ρ =4

9· 3

8πGt−2. (1.32)

We have obtained relations of the evolution of R, H and ρ during the different epochs, butin order to relate these quantities to the temperature T we need to model the evolution of ρwith the relativistic kinetic theory formalism in Ch. 2.

The final epoch to mention is the vacuum energy dominated epoch. At that time, theUniverse will have become so cold and diluted that the ρ-term in the Friedmann equation (1.12)can be neglected. As Λ and the associated vaccuum energy density are constant, i.e. unaffectedby expansion. Therefore, the following assumption can be made:

(Vacuum dominated epoch) : (w = −1) : ρ = −p. (1.33)

Inserting this into the first law of thermodynamics (1.16), ρ is found to be constant.


1.4.3 Density Contributions to the Energy Density of the Universe


Now, the decomposition of the Universe’s total energy density ρtot will be discussed.

Let ρp be the energy density of all particles (the sum of the energy densities of matter andradiation). The Friedmann equation (1.12) can be rewritten by division by H2:

1 +κ

R2H2=

ρp3H2

8πG

+Λ

3H2, (1.34)

⇔ Ω ≡ Ω0 + Ωκ + ΩΛ = 1, (1.35)

with the density contributions,

Ω0 =ρpρc

=ρp

3H2

8πG

, (1.36)

Ωκ =ρκρc

= −3κ

8πGR2

3H2

8πG

= − κ

R2H2, (1.37)

ΩΛ =ρΛ

ρc=

Λ8πG3H2

8πG

=Λ

3H2, (1.38)

and the total density parameter Ω. Hence, the Friedmann equation becomes

H2 =8πG

3ρtot, (1.39)

with

ρtot = ρp + ρκ + ρΛ, (1.40)

and

ρκ = − 3κ

8πGR2, ρΛ =

Λ

8πG. (1.41)

Here, ρc is the critical density:

ρc =3H2

8πG. (1.42)

To understand what ‘critical’ means, we rearrange Eq. (1.34) and neglect Λ:

κ

R2H2=

ρp3H2

8πG

− 1 = Ω0 − 1. (1.43)

As H2R2 ≥ 0, three cases can be distinguished (Kolb and Turner, 1990, Chap. 3):

Ω0 > 1 ⇒ κ = +1 closed Universe, (1.44)

Ω0 = 1 ⇒ κ = 0 flat Universe, (1.45)

Ω0 < 1 ⇒ κ = −1 open Universe. (1.46)

In the case of a flat Universe, which is the observed state of the Universe, the density ρp is equalto the critical density. The closed case corresponds to a decelerated expansion of the Universe.Accelerated expansion only can be admitted for Ω0 ≤ 1. The flat case is the critical case toconstant expansion which is the reason why ρc is called critical.

10 1 INTRODUCTION

The density parameter is a useful quantity for estimating masses of particle species. Thecontribution Ω0 is composed of the contributions of all species in the Universe:

Ω0 =∑i

Ωi. (1.47)

For a species i the following relation has to hold (Kolb and Turner, 1990, Chap. 3):

ρi = Ωiρc = Ωih2 · 1.88× 10−29 g

cm3, (1.48)

where the value of the Hubble rate in (1.42) measured today (Kolb and Turner, 1990, Chap. 1)is

H0 = 100hkm

s ·Mpc, (1.49)

where the uncertainty is captured by

0.4 . h . 1.0, (1.50)

(ibid.). Given an observational constraint on Ωi, the particle’s mass mi can be estimated fromρi.

From observations we have the following density contributions (Perkins, 2009, Chap. 8):

all matter: Ωm ' 0.24± 0.03 (1.51)

baryonic matter: ΩB ' 0.042± 0.004 (1.52)

Dark Matter: ΩDM ' 0.20± 0.03 (1.53)

vacuum/dark energy: ΩΛ ' 0.76± 0.05 (1.54)

And as already mentioned it can be seen that DM dominates the matter density contribution.The total energy density observed to be ∼ 1 and is dominated by the vaccuum energy.

1.4.4 Evolution of Entropy


Before proceeding to the construction of the relativistic kinetic theory, it is useful to inves-tigate the evolution of the entropy. It is a useful quantity for reference when investigatingthermodynamic equilibrium, because the entropy per comoving volume is constant, if there areno annihilations or creations of particles.

The entropy density is defined by

s =ρ+ p−

∑i µini

T. (1.55)

In the case that there exists heat exchange and a change of the particle number, the first law ofthermodynamics (1.16) with (1.17) and (1.18) implies:

TdS = d(ρV ) + pdV +∑i

µid(niV ) = d[(ρ+ p−∑i

µini)V ]− V dp + V∑i

nidµi (1.56)

⇔ dS = d

[(ρ+ p−

∑i µini)V

T

]− V

Tdp +

V

T

∑i

nidµi +(ρ+ p−

∑i µini)V

T 2dT, (1.57)

with the comoving volume V = R3. By integrating we simply find Eq. (1.55) when multiplying(1.55) with R3.


Given the symmetry of the second derivatives,

∂2S

∂T∂V=

∂2S

∂V ∂T, (1.58)

differentiating Eq. (1.57) and sV with respect to T and V should yield the same:

∂2S

∂T∂V=∂2(sV )

∂T∂V︸︷︷︸(A)

=∂2

∂T∂V

[(ρ+ p−

∑i µini)V

T

]︸︷︷︸

(B)

− V

T

∂2p

∂T∂V+V

T

∑i

ni∂2µi∂T∂V

+(ρ+ p−

∑i µini)V

T 2

∂2T

∂T∂V. (1.59)

As the terms (A) and (B) above are equal we obtain the Gibbs-Duhem relation (Ehlers,1971):

−VTdp +

V

T

∑i

nidµi = −(ρ+ p−

∑i µini)V

T 2dT. (1.60)

So the general change of the entropy is given by

dS = d

[(ρ+ p−

∑i µini)V

T+ const.

], (1.61)

where some constant term has to be added as it is not possible to determine the absolute valueof entropy.

One could now repeat this calculation assuming that there are no particle reactions thatchange the particle number in a comoving volume. That would be expressed by omitting theterm containing dni in Eq. (1.56). Performing the same calculation and using the first law ofthermodynamics (1.16) for adiabatic expansion we obtain

d

[(ρ+ p)V

T

]= 0. (1.62)

Hence, there remains

dS = d

[−V

∑i µiniT

]. (1.63)

By the condition of symmetric second derivatives (1.58) a modified ’Gibbs-Duhem relation’ canbe deduced:

−V∑

i nidµiT

=V∑

i µiniT 2

dT. (1.64)

Thus, an entropy change in a comoving volume during adiabatic expansion is only due to achange of particle number,

dS = −∑i

µiTd(niV ), (1.65)

in absence of which the expansion is isentropic, i.e. the entropy in a comoving volume does notchange.

The necessary input of the FLRW model and several themodynamic relations for the con-struction of the formalism of relativistic kinetic theory have been presented. The next chapterwill treat the construction of this formalism in detail.

12 2 RELATIVISTIC KINETIC THEORY

2 Relativistic Kinetic Theory

The aim of this chapter is the construction of the theoretical framework for investigating theevolution of an arbitrary particle species, not focussing on DM explicitly for now. The relativistickinetic theory formalism provides a description of particle states in phase space and can be usedto deduce evolution equations of the thermodynamic ensemble. The most important equationis the Boltzmann equation which will be used in the freeze-out calculation of a symmetric andasymmetric WIMP species in Ch. 3 and Ch. 4. This section is mainly based on the treatmentsof Ehlers (1971) and Stewart (1971) that are much alike. Only other references will we markedto avoid redundant referencing.

2.1 The Relativistic Phase-space Description and the Liouville Theorem

The evolution of a gas of particles can be decribed in phase space. In classical kinetic theory itis possible to determine all particle’s states (~x, ~p) exactly and to compute the evolution of thesystem. Practically, we integrate over the occupied states of the particles and derive macroscopicquantities that describe the state of the gas. Before doing so, we have to consider the simultaneityproblem and the non-existence of inertial frames. Therefore, we look for a frame-independentdescription of phase space and then proceed to the macroscopic perspective (Ehlers, 1974).

The framework that to be constructed requires some assumptions:

1) The particles of the gas only have, 1a) long-range interactions expressed by a mean fieldand 1b) short-range interactions treated as point-like collisions. Between the collisionsthe particles move in the mean field like test particles. The model does not include a self-generated mean field by the particles, as would be expected e.g. for electrically chargedparticles.

2) The system is close to thermodynamic equilibrium equilibrium (explained in more detailthroughout this chapter and Ch. 3).

3) For 1) to hold, the gas is not too cold and dense and colliding particles have un-correlatedmomenta.

Keeping that in mind, let us continue with the construction of an 8-dimensional phase space asan extension of the classical case. Here, a point (xµ, pµ) is the union of a spacetime point xµ

with a 4-momentum pµ. Such a point describes the physical state of a particle. As we want todecribe a many-particle system we have to mind that in general relativity there exist no inertialframes. Since there is no preferred reference frame, we can pick an arbitrary one. Every classical3-dimensional reference frame is associated with a timelike hypersurface element σµ that servesas a frame-independent 3-dimensional volume element in relativistic spacetime. In any suchreference frame the time coordinate t is represented as a timelike 4-vector orthogonal to therespective hypersurface, hence there is no preferred time coordinate. The consequence is thatevery physical description in a chosen 3-dimensional reference frame has to be projected to thereference frame of an observer moving with a timelike 4-velocity uµ, where timelike means

uµuµ = 1. (2.1)

Let us define the oriented (Riemannian) volume element η on spacetime,

η =1

4!

√−g εµνρσ dxµ ∧ dxν ∧ dxρ ∧ dxσ, (2.2)

which is a differential form in the sense of differential geometry (see Appendix B, (B.15)).The ∧-symbol represents the totally anti-symmetric wedge product (B.16). g is the metric’s

2.1 The Relativistic Phase-space Description and the Liouville Theorem 13

determinant (1.9) and εµνρσ is the completely anti-symmetric Levi-Civita symbol. This volumeelement η is (Lorentz-)invariant, so the displacement (co-)vectors dxµ, dxν , dxρ, dxσ can bechosen arbitrarily, but linearly independent. The volume element of a 3-dimensional referenceframe representing a sub-manifold of spacetime is in general not Lorentz-invariant. We define itas

dV =1

3!

√−g εµνρσ uµdxν ∧ dxρ ∧ dxσ. (2.3)

In the observer’s rest frame, uµ = (1, 0, 0, 0), the 3-dimensional volume element becomes

dV =√−g dx1dx2dx3, (2.4)

and the wedges can be omitted when using orthonormal coordinates.We now can define the hypersurface element σµ as the boundary ∂η of η:

σµ = ∂η = dV uµ =1

3!

√−g εµνρσ dxν ∧ dxρ ∧ dxσ. (2.5)

For some spacetime region D and an arbitrary function (or tensor of arbitrary rank) Aµ therelation between η and σµ is then given by Gauss’ theorem:∫

DηAµ;µ =

∫∂D

σµAµ, (2.6)

with ∂D being the boundary of D and Aµ;µ the covariant derivative of Aµ (see Appendix A,(A.6)).

In a similar way we can define a Lorentz-invariant 4-dimensional volume element on momen-tum space with arbitrary dpµdpνdpρdpσ:

dP =1

4!

√−g εµνρσ dpµ ∧ dpν ∧ dpρ ∧ dpσ, (2.7)

or in orthonormal coordinates:

dP =√−g dp0dp1dp2dp3. (2.8)

In momentum space we have the advantage that the corresponding volume element will reduceautomatically to 3 dimensions if we only consider the physically accessible momenta:

i) p0 is identified with the particle’s relativistic energy E and is always non-negative2:

p0 ≡ E =(m2 − pipi

) 12 =

(m2 + (−gij)pipj

) 12 , p0 ≥ 0. (2.9)

ii) The spatial momentum components pi depend on p0 and they lie on thepositive mass shell (m ≥ 0):

m2 = gµν pµpν . (2.10)

With these two conditions we can define the 3-dimensional hypersurface element

Π = 2 θ(p0)δ(pµpµ −m2)dP. (2.11)

The factor 2 placed in front is due to normalization of the product of the Heaviside function

θ(p0) =

1, if p0 > 0,

0, if p0 ≤ 0,(2.12)

2Note that the spatial momentum components in (2.9) are still distinguished between co- and contra-variant,so pip

i = gijpipj (gii < 0).


with the Dirac delta-function δ(pµpµ −m2). By using the relation (Alt, 2006)

δ(f(x)) =

n∑i=1

δ(x− xi)f ′(xi)

, (2.13)

with the roots xi (if n is finite) of some differentiable function f(x), the hypersurface elementbecomes

Π = δ(p0 − E)dP

p0, (2.14)

and applying the delta-function,

Π =

√−gp0

δ(p0 − E)dp0dp1dp2dp3 =

√−gE

dp1dp2dp3, (2.15)

with the particle’s relativistic energy E given in (2.9). Having reduced the dimension of themomentum space the remaining phase space is now 7-dimensional with coordinates (xµ, pi) andin principle we could simply combine σµ with Π to obtain an integration measure. But thiswould not take into account the equations of motion of the considered gas particles. Theseequations have to be considered in the integration measure, because in general relativity thecurved spacetime itself causes the motion and functions that are defined on this spacetime needa covariant description of the spacetime.

To understand this we have to look at the worldlines xµ(λ) along which the particles movein spacetime, where λ is an affine parameter that has to be chosen coordinate-dependent. Thisnotion can be extended to worldlines in phase space (xµ(λ), pµ(λ)), so called phase orbits.These can be regarded as integral curves that are generated by a vector field L on phase space(see Appendix B):

L =d

dλ=dxµ(λ)

dλ

∂

∂xµ+dpµ(λ)

dλ

∂

∂pµ. (2.16)

Between collisions the particles follow geodesic motion, such that the eqations of motion are

dxµ(λ)

dλ= pµ and

dpµ(λ)

dλ= −Γµνρp

νpρ (Ehlers, 1974). (2.17)

The Γµνρ are the Christoffel symbols (see eq. (B.37)) that act as a mean field in which theparticles move as initially assumed. Thus the vector field that we define as the relativisticLiouville operator becomes

L = pµ∂

∂xµ− Γµνρp

νpρ∂

∂pµ. (2.18)

On a non-microscopic scale L defines a so-called phase flow, which is formed by all phaseorbits in the continuity limit3. This phase flow describes the motion of all particles. The aimis now to find a phase space volume element that is invariant under this phase flow, moreprecisely, an invariant 6-dimensional hypersurface element (as the observation of the system isalways relative to our reference frame). The invariant volume element will then be used as anintegration measure for computing macroscopic quantities to describe the change of the state ofthe gas with the phase flow.

To this end, let us consider the coordinate-invariant 7-dimensional volume element

Ω = η ∧Π. (2.19)

3In fact we have discrete particle phase orbits, but by choosing an adequate scale the vector field L will varysmoothly between two adjacent orbits in the continuum limit.

2.1 The Relativistic Phase-space Description and the Liouville Theorem 15

The phase space hypersurface element ω is obtained by contracting L with Ω, contracting viathe interior product of the vector L with the differential form Ω (see (B.21)):

ω : = L Ω

= pµσµ ∧Π︸︷︷︸(a)

− 1

2

√−gE

ε0ijk Γiνρpνpρ dpj ∧ dpk ∧ η︸︷︷︸

(b)

. (2.20)

An invariance with respect to the phase flow generated by L is expressed by a vanishing Liederivative. When the Lie derivative vanishes, L is called a Killing vector field (see AppendixB, B.48). ω arises naturally by taking the Lie derivative L L of Ω with respect to L (Ehlers,1971) defined by:

L LΩ = d(L Ω) + L dΩ, (2.21)

with d the exterior derivative operator (B.18). If r is equal to the dimension of the manifoldan r-form is defined on, the exterior derivative will vanish (B.20). As Ω is a 7-form on the7-dimensional phase space, consequently

dΩ = 0, (2.22)

and thus,L LΩ = d(L Ω) = dω. (2.23)

So Ω will be invariant under L, if dω vanishes. That this is the case can be shown locally inRiemannian coordinates, i.e.

uµ = (1, 0, 0, 0), (2.24)

∂p0

∂x0= 0, (2.25)

p0dp0 = pidpi, (2.26)√−g = 1, (2.27)

and Γµνρ = 0. (2.28)

Due to the vanishing Christoffel symbols the b)-term in (2.20) vanishes. For the a)-term weshow:

d(a) = d(pµuµdV ∧Π) = d(p0dV ∧Π) (2.29)

=1

4!

∂

∂xα(δ(p0 − E)

)dxα ∧ ε0νρσu

0dxν ∧ dxρ ∧ dxσ ∧ dp0 ∧ dp1 ∧ dp2 ∧ dp3

+1

4!

∂

∂pα(δ(p0 − E)

)dpα ∧ ε0νρσu

0dxν ∧ dxρ ∧ dxσ ∧ dp0 ∧ dp1 ∧ dp2 ∧ dp3. (2.30)

The second term in (2.30) is zero, because the wedge product vanishes for two identical co-vectors. Furthermore, choosing uµ as in (2.24) and with orthonormal coordinates,

d(a) =∂

∂x0

(δ(p0 − E)

)dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dp0 ∧ dp1 ∧ dp2 ∧ dp3, (2.31)

and due to (2.25) we obtain

d(a) =

(∂p0

∂x0

)︸︷︷︸

=0

(∂δ(p0 − E)

∂p0

)d4xd4p = 0. (2.32)


Hence, dω vanishes and Ω is invariant under the phase flow. The invariance of ω is given by thenilpotency of the interior derivative (B.22). In this case,

L · L = 0, (2.33)

such thatL Lω = d(L ω) + L dω︸︷︷︸

=0

= d(L · L Ω) = 0. (2.34)

With the obtained invariant measure ω the Liouville theorem can be formulated. Considera tube T of phase space orbits. We denote Σ as the tube’s intersection with the observer’shypersurface. Such a hypersurface represents a 6-dimensional classical phase space at some timet = const. and the measure ω on it reduces to

ω = pµσµ ∧Π. (2.35)

Further, the expression ∫Σω 6= 0 (2.36)

will hold if and only if Σ is not parallel4 to phase flow (Stewart, 1971). Assume that the tubeT is bounded by ∂T, where ∂T consists of the hypersurfaces Σ, Σ′ and Λ, where Λ is parallel tothe flow. With Stokes’ theorem ∫

Tdω =

∫∂Tω, (2.37)

we obtain

0 =

∫Tdω =

∫∂Tω =

∫Σω −

∫Σ′ω +

∫Λω. (2.38)

The last term vanishes due to the parallelism of Λ to the phase flow, so the resulting expressionis stated as the Liouville theorem: ∫

Σω =

∫Σ′ω. (2.39)

In words: the measure ω and the associated 6-dimensional phase space volume occupied by aset of gas particles remains constant following the phase flow of the particles.

2.2 The Phase Space Distribution and the Boltzmann Equation

For the further work it is important to mention that the validity of the deduced Liouvilletheorem is restricted to collisionless motion according to the equations of motion (2.17). Toinvolve collisions we have to define annihilations and creations in terms of the phase spacedescription. We denote a binary collision as

pA, pB −→ pC, pD (2.40)

with the respective momenta of the in-going particles pA and pB and the the out-going particlespC and pD. Such a binary collision gives rise to two endpoints of the phase orbits of the particlesA and B, called annihilations (x, pA) and (x, pB). On the other hand there are two initialpoints of the new worldlines of the particles C and D, called creations. In general, not allcollisions are binary, nor do collisions conserve the number of occupied states, so this change ofoccupied states has to be taken into account in the further description.

4A flow parallel to some hypersurface is geometrically orthogonal to the orientation of the surface, hence, thescalar product (inner product) between the flow-generating vector field and the hypersurface co-vector is zero.

2.2 The Phase Space Distribution and the Boltzmann Equation 17

Further, consider a 7-dimensional phase space region D. When counting the phase orbitsentering D positively and those exiting negatively, we obviously will count the difference betweencreations and annihilations in D. Let now D be tube-shaped region of orbits T as in the deductionof the Liouville theorem and let Σ be the intersection of the tube with the observer’s hypersurface(like the Σ in Eq. (2.39)). Further assume, that no states intersect with Λ. Using the foundinvariant measure ω, we can define the invariant 1-particle distribution function of theoccupied phase space states, denoted by f(xµ, pν). For now it is demanded that f is a functionalbeing non-negative for all pν . Physically, integrating f over the phase space volume occupied bya single particle state gives the occupation probability of this state. Assume that this probabilityW is always

W =

∫r−1

f(xµ, pν)ω ≤ g, (2.41)

where r−1 is the phase space volume of a single state and g is the number of allowed spin statesin such a phase space state, also called the spin degeneracy factor (Ehlers, 1969):

1

r=h3

g=

(2π~)3

g, (2.42)

where h is Planck constant the ~ the reduced Planck constant and

g =

2s+ 1 for mass m > 0 and spin s,

2 for mass m = 0 and spin s > 0

1 for mass m = 0 and spin s = 0.

(2.43)

Due to (2.41) the factor r has to be included in f to give it the right dimensions, such thatafter intgerating f over phase space the occupation probability is dimensionless. Doing so, theconstraint (2.41) on f becomes the following:

f ≤ r ∀pµ. (2.44)

This condition seems physically reasonable. For example, for fermions every state can be oc-cupied by maximally g particles due to the Pauli principle. However, this condition (2.44) isnot valid for bosons as will be discussed in Ch. 4. Nevertheless, we adapt the condition as itserves to keep the description more simple, initially. In Ch. 4 this constraint will be droppedand consequences of that shown.

Using the distribution function f(xµ, pν) the average number of worldlines intersecting withΣ and so the average number of occupied states in Σ is defined by

N [Σ] =

∫Σf(xµ, pν)ω =

∫Σf(xµ, pµ)δ(p0 − E)uµp

µ (−g)

p0d4p d3x. (2.45)

Now we can use the lemma from (B.24) in the following form:

df ∧ L · dΩ = L(f)Ω. (2.46)


The change of the number N of occupied states in the region T can now be defined by

N = N [Σ′]−N [Σ]−N [Λ]︸︷︷︸=0

= −N [∂T]

= −∫∂Tfω (from (2.45))

= −∫

Td(fω) (from Stokes′ theorem (2.37))

= −∫

Tdf ∧ ω (from dω = 0)

= −∫

Tdf ∧ L · Ω (from (2.20))

= −∫

TL(f)Ω (from (2.46)). (2.47)

The included minus-sign takes into account the orientation of phase space. For the same reasonin the deduction of the Liouville theorem (2.38) we counted the integral over Σ′ negatively. Theminus-sign is therefore necessary to count the states exiting T through Σ′ positively and thestates entering through Σ negatively.

If we define the phase space density of the change of occupied states as the collision termC(f), we obtain the Boltzmann equation:

L(f) = C(f). (2.48)

Having a model of the collision term, we have obtained an evolution equation of the system.In the special case of C(f) = 0 we obtain the Liouville equation:

L(f) = 0. (2.49)

This holds if there are no collisions or decays, or if the collisions cause not net variation of f .The second case is also labeled as detailed balance (Stewart, 1971, Sec. 2.6).

The Boltzmann equation can be recast into a different form by introducing the momentsof the distribution function f . Then the r-th moment of f is

Aµ1···µr :=

∫Πpµ1 · · · pµrfΠ. (2.50)

The two physically most interesting moments are

Nµ :=

∫ΠpµfΠ, (2.51)

and

Tµν :=

∫ΠpµpνfΠ. (2.52)

From (2.5) and (2.45) it is obvious that the flux of particles through σµ is

N =

∫Nµσµ =

∫fpµuµΠdV. (2.53)

When not integrating over dV , we obtain the particle number density measured in the localframe of an observer, who sees the system of particles in dV moving with a four-velocity uµ:

n =

∫fpµuµΠ. (2.54)

2.3 The Collision Term 19

Hence, Nµ is the particle 4-current. Tµν is the kinetic stress energy momentum tensor,where T 0

0 corresponds to the local energy density ρ and 13 T

ii to the hydrostatic pressure

p.Furthermore, for a region in spacetime η and the respective region in phase space D it can

be calculated: ∫ηηNµ

;µ =

∫∂ηNµσµ (from Gauss′ theorem (2.6))

=

∫∂η

∫Πfpµσµ ∧Π (from (2.51))

=

∫∂Dfω (from (2.35))

=

∫D

L(f)Ω = −N (from (2.47))

=

∫η

∫Π

L(f)η ∧Π (from (2.19)). (2.55)

Thus we can deduce the change of the particle 4-current from the Boltzmann equation:

Nµ;µ =

∫Π

L(f)Π. (2.56)

Similarly it can be proven that

Tµν;ν =

∫Πpµ L(f)Π, (2.57)

or for general moments of f :

Aµ1···µr;µr =

∫Πpµ1 · · · pµ(r−1) L(f)Π. (2.58)

2.3 The Collision Term

To obtain the evolution equation for the considered system a model for the collision term C(f)is required. For a binary collision with in-going particles A, B and out-going particles C, D wecan construct the collision bundle as the set of all collisions with the measure

η ∧ΠA ∧ΠB ∧ΠC ∧ΠC (Ehlers, 1971). (2.59)

In this description only 1-particle distribution functions are used, which from now on will bewritten as

fA = fA(xµ, pµA) and fC = rC ± fC(xµ, pµC), (2.60)

where the upper sign refers to bosons and the lower to fermions and rC is given in (2.42).Further, Boltzmann’s Stosszahlansatz can be introduced in a relativistic form. It states

that the number of collisions can be expressed as a functional of separable one-particle distribu-tion functions. In general, one would expect more complex higher order distribution functions,but to only work with the one-particle distribution function fits the initial assumption in section2.1 that our system is not too cold and dense, such that the momenta of colliding particles areun-correlated. According to the Stosszahlansatz for the considered binary collisions, the averagechange of the number of states is

−N =

∫ηΠAΠBΠCΠDfAfBfCfD|MpA,pB→pC,pD |

2(2π)4δ (pA + pB − pC − pD) , (2.61)


where the wedges are omitted from now on to make the description more simple. |MpA,pB→pC,pD |is the matrix element containing the particle physics of the respective reaction and the delta-function accounts for momentum conservation in the collision. Note that the functions f accountfor the effects of Pauli blocking for fermions and Bose enhancement for bosons (Blinovand Hook, 2017). For high phase space densities of the product particles C and D, Pauliblocking means that the considered binary collision is inhibited because the fermionic gas is ina degenerate state and the Pauli principle does not allow a higher occupation number of theparticle states. For a bosonic gas in the same case the reaction is enhanced as it is likely forbosons to occupy the same state. In Ch. 4.3 we will come back to the investigation of thesequantum effects.

In general, to obtain the total change of occupied states by the particle A we have to sumover all possible reactions in which A is involved. A reaction where a particle A is created isthen counted positively and a reaction where an A is annihilated is counted negatively. Thisis the reason there is a minus-sign on the left-hand side of Eq. (2.61) similar to Eq. (2.47).Furthermore we have to include a numerical factor γ for each reaction type, which preventscounting a reaction with two identical particles twice (Ehlers, 1971). For a binary collision it is

γABCD =

NA

NA!NB!NC!ND!, (2.62)

where Ni is the number of particles of type i involved in the collision. Considering all reactionsand their inverse reactions we have to sum over all particle species which interact with A. Then,the Boltzmann equation for the evolution of the particle species A, and hence, the collision term,becomes

L(fA) = C(f)

=1

2

∑b,c,d

γAbcd

∫ΠbΠcΠdfAfbfcfd|MpA,pb→pc,pd |

2(2π)4δ (pA + pb − pc − pd)

− 1

2

∑b,c,d

γAbcd

∫ΠbΠcΠdfAfbfcfd|Mpc,pd→pA,pb |

2(2π)4δ (pc + pd − pA − pb) . (2.63)

Since the terms do not change by exchanging c and d, the factor 12 is included to avoid counting

twice. If furthermore assuming CP-invariance in all reactions, i.e. invariance of the propertiesof the reaction under simultaneous charge conjugation C and parity transformation P (Ehlers,1971), then the matrix elements will be the same for the reactions and their inverses:

|MpA,pb→pc,pd |2 = |Mpc,pd→pA,pb |

2 = |MA,B↔C,D|2. (2.64)

Hence, with CP-invariance of all reactions the Boltzmann equation is

L[fA]

=1

2

∑b,c,d

γAbcd

∫ΠbΠcΠd(fAfbfcfd − fAfbfcfd)|MA,b↔c,d|2(2π)4δ (pA + pb − pc − pd) . (2.65)

2.4 The Thermally Averaged Cross-section times Relative Velocity

The last term in the Boltzmann equation above that we have to model is the matrix element|M|2. It is more usual to use the differential cross-section dQ instead or, in our case, thethermally averaged cross-section (times velocity). Consider now the case, that there is

2.4 The Thermally Averaged Cross-section times Relative Velocity 21

only one type of collision, the annihilation of particle A through: A+B→ C+D. The differentialcross-section is then

dQA,B→C,D =dRA,B→C,D

dF, (2.66)

(Cannoni, 2016). To keep the description frame-independent dQA,B→C,D has to be formulatedin a Lorentz-invariant way. Let dRA,B→C,D be the spacetime density of these collisions, thedifferential spacetime reaction rate. Its definition follows from (2.61):

dRA,B→C,D = −fAfBfCfD|MA,B→C,D|2(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD, (2.67)

(Cannoni, 2016). Physically, dRA,B→C,D gives the change of the number of particles A in somespacetime cell. The change takes particles A with a momentum in the range ΠA into accountand is due to collisions with particles B with a momentum in the range ΠB. Thereby, dRA,B→C,D

considers the likelihood of the product particles C and D to have momenta in ranges ΠC andΠD, respectively, regarding the effects of Pauli blocking and Bose enhancement. From (2.63),(2.61) and (2.67) it is clear that the Boltzmann equation for the considered collision is∫

ΠA

L(f)ΠA =

∫ΠA

∫ΠB

∫ΠC

∫ΠD

dRA,B→C,D +

∫ΠA

∫ΠB

∫ΠC

∫ΠD

dRC,D→A,B. (2.68)

Further, dF is the Lorentz-invariant relativistic differential flux, defined by

dF = dNµAdNBµvrel, (2.69)

where dNµA and dNBµ are the differential particle 4-currents and vrel is the Lorentz-invariant

relative velocity of the two colliding particles A and B (Cannoni, 2016). Consider a frame Swhere particle A moves with a velocity ~vA and another particle B with ~vB. The rest frame S1 ofthe particle B obviously moves with ~vB with respect to S. Then the magnitude of relative velocity|~v′A| between the particles as seen from the frame S1 is given by the Lorentz-transformation (seeAppendix C, Tsamparlis (2010) and Cannoni (2016)):

vrel = |~v′A| =

∣∣∣~vAγB− ~vB + (1− 1

γB)~vB

~vB·~vA

~v2B

∣∣∣1− ~vB · ~vA

. (2.70)

The magnitude of this relative velocity is Lorentz-invariant (see Appendix C). For co-linearvelocities of the particles the absolute value of the numerator above simplifies to∣∣∣∣~vA

γB− ~vB +

(1− 1

γB

)~vB

~vB · ~vA

~v2B

∣∣∣∣ = |~vA − ~vB| , (2.71)

so the relative velocity becomes:

vrel =|~vA − ~vB|1− ~vA~vB

≡ |v|1− ~vA~vB

(2.72)

Let us form now on assume the velocities ~vA and ~vB to be co-linear to keep the notation clear.To obtain a relativistic description we can infer

1− ~vA~vB =pµApBµ

p0Ap

0B

(2.73)

for the numerator in (2.72), and for the denominator:

|v| = |~vA − ~vB| =∣∣∣∣pµAp0

A

− pµBp0

B

∣∣∣∣ =

∣∣p0Bp

µA − p0

ApµB

∣∣p0

Ap0B

. (2.74)


Therefore the relative velocity is finally (Stewart, 1971)

vrel =|p0

BpµA − p0

ApµB|

pµApBµ. (2.75)

The dNµ from (2.69) are Lorentz-invariant, as can be seen from (2.51) (same definition, butwithout integration), and we can rewrite them as

dNµAdNBµ = fAΠA fBΠB p

µApBµ, (2.76)

such that the differential flux becomes (Cannoni, 2016)

dF = fAΠA fBΠB|p0Bp

µA − p

0Ap

µB|. (2.77)

Inserting (2.67), (2.75) and (2.77) into (2.66) we obtain the definition for the differential cross-section,

dQA,B→C,D =dRA,B→C,D

n0An

0B

|p0Bpµ

A−p0

Apµ

B|p0

Ap0

B

=−fAfBfCfD|MA,B→C,D|2(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD

fAΠA fBΠB |p0Bp

µA − p0

ApµB|

=−fCfD|MA,B→C,D|2(2π)4δ (pA + pB − pC − pD) ΠCΠD

|p0Bp

µA − p0

ApµB|

= −fCfD σA,B→C,D (2π)4δ (pA + pB − pC − pD) ΠCΠD, (2.78)

where σA,B→C,D is the scalar cross section given by (Stewart, 1971, Chap. 5):

σA,B→C,D =|MA,B→C,D|2

|p0Bp

µA − p0

ApµB|. (2.79)

With this we now can rewrite the spacetime reaction rate to

dRA,B→C,D = dQA,B→C,DdF

= −fAΠA fBΠB pµApBµ σA,B→C,DvrelfCfD(2π)4δ (pA + pB − pC − pD) ΠCΠD

= −fAd3pA fBd

3pB σA,B→C,DpµApBµ

p0Ap

0B

|p0Bp

µA − p0

ApµB|

pµApBµfCfD(2π)4δ (pA + pB − pC − pD) ΠCΠD

= −fAd3pA fBd

3pB σA,B→C,D|p0

BpµA − p0

ApµB|

p0Ap

0B

fCfD(2π)4δ (pA + pB − pC − pD) ΠCΠD

= −fAd3pA fBd

3pB σA,B→C,D |v| fCfD(2π)4δ (pA + pB − pC − pD) ΠCΠD. (2.80)

The term |v| in the last equation is given by (2.74). Note that this is not an invariant expression,however the frame dependence cancels with the dependence of the scalar cross section.

Now we expand the expression (2.80), such that

dRA,B→C,D = −fAd3pA fBd

3pBσA,B→C,D |v| f eq

A f eqB fCfD(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD

f eqA ΠA f

eqB ΠB

,

(2.81)

where the f eq are the distribution functions in thermodynamic equilibrium. By integrating weobtain the following shape for our reaction rate of the considered collision (Kolb and Turner,1990, Chap. 5), ∫

ΠA

∫ΠB

∫ΠC

∫ΠD

dRA,B→C,D = −nAnB〈σA,B→C,D|v|〉tot (2.82)

2.5 Equilibrium Solutions of the Distribution Function 23

with the local number densities nA and nB as given by (2.54) and the thermally averaged cross-section times velocity

〈σA,B→C,D|v|〉tot =

∫σA,B→C,D |v| f eq


neqA n

eqB

, (2.83)

(ibid.). The equilibrium number densities neqA and neq

B are defined analogously to nA and nB.Re-introducing all possible reactions from (2.68) the Boltzmann equation for the evolution ofthe particle species A finally becomes∫

ΠA

L(f)ΠA = −1

2

∑b,c,d

γAbcdnAnb〈σA,b→c,dvrel〉tot +

1

2

∑b,c,d

γAbcdncnd〈σc,d→A,bvrel〉tot, (2.84)

(Kolb and Turner, 1990).If we now want to apply simplifications from CP-invariance it is notthat simple to define a common thermally averaged cross-section for the reaction and the inversereaction. Even if we assume the matrix element |M| to be the same for both, the evolutionequation (2.84) for the species A is coupled to the evolution of the other paticles B, C andD, whose evolution is again coupled to other particles they interact with. Furthermore, in thedefinition (2.83) we have to take into account the quantum effects, whose magnitude highlydepends on the state of the gas mixture. Therefore a common 〈σ|v|〉 for both directions of thereaction can only be defined if we introduce further simplifications. With the notation (2.60)the distribution functions in (2.83) are

f eqA f eq

B fCfD = f eqA f eq

B (rC ± fC)(rD ± fD), (2.85)

such that (2.83) can be divided into

〈σA,B→C,D|v|〉tot = 〈σA,B→C,D|v|〉 ± 〈σA,B→C,D|v|〉q (2.86)

with

〈σA,B→C,D|v|〉 =


A f eqB rCrD(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD

neqA n

eqB

(2.87)

and

〈σA,B→C,D|v|〉q =



neqA n

eqB

, (2.88)

where the cross-section with index q accounts for the quantum effects. In the next chapterwe will see in which cases these quantum effects and the corresponding cross-section can beneglected.

In summary, the thermally averaged cross-section is a more useful quantity than the matrixelement to model the collision term. In this thesis only given cross-sections will be considered.Calculations of particular matrix elements from quantum field theory will not be shown. As wenow have a model for the collision term we can proceed to investigate solutions of the Boltzmannequation. A special case is the equilibrium solution.

2.5 Equilibrium Solutions of the Distribution Function

2.5.1 Collisional Invariants

The Boltzmann equation is a so-called non-linear Riccati equation (Kolb and Turner, 1990,Chap. 5), so analytic solutions do not exist in general. A special case that we already mentioned


is the Liouville equation (2.49) in the case of detailed balance, where collisions cause no netchange of the number of occupied states. We now want to find an analytic solution for localequilibrium. For detailed balance to hold the collision term has to satisfy

fAfBfCfD − fCfDfAfB = 0 (2.89)

for every such type of a binary collision and its inverse as seen in (2.63). With the definition off in (2.60) the above condition can be re-arranged to

fA

rA ± fA

fB

rB ± fB=

fC

rC ± fC

fD

rD ± fD. (2.90)

To find the local equilibrium solution f eqA we have to introduce collisional invariants (Ehlers,

1971). A collisional invariant φi is a non-negative function that is additively conserved incollisions:

φA + φB − φC − φD = 0. (2.91)

To ensure that the obtained conservation condition (2.90) is fulfilled, we define the collisionalinvariant for the species A as

φA = − ln

(fA

rA ± fA

). (2.92)

The minus sign is included to keep the constraint (2.44) on f satisfied, as we will see later. Nowwe solve for fA and find by recursively inserting fA

fA = exp(−φA)(rA ± fA) = exp(−φA)rA ± exp2(−φA)(rA ± fA) = · · ·

= −rA

( ∞∑k=0

(∓ exp(−φA))k

)− 1. (2.93)

And as φA ≥ 0, the geometric series above will converge (Forster, 2013):

∞∑k=0

xk =1

1− xfor |x| ≤ 1. (2.94)

Hence, we obtain the solution

f eqA =

rA

(exp(φA)∓ 1)(2.95)

As we defined φA > 0 and require f > 0 it can be seen that it was helpful to include the minussign in (2.92).

Since fA is a function of xµ and pµ, φA(xµ, pµ) also has to be. In his first formulationBoltzmann assumed φA to be twice differentiable with respect to momentum and obtained thefollowing Ansatz:

φA(xµ, pµA) = αA(xα) + βAµ(xα)pµA, (2.96)

where αA and βAµ are functions of spacetime5 xµ. To ensure that φA is positive and to preventdivergences of fA, βAµ has to be a timelike vector, such that the product βAµp

µA is non-negative.

Hence, it can be written

βAµ(xα) = βA(xα)uAµ(xα) with βA > 0. (2.97)

5While we denote the momenta with a particle index due to different masses, we could also assign that indexto the events xµ, but this is unnecessary as all particles physically move through the same spacetime.


and αA has to satisfyαA(xα) > −βA(xα)uAµ(xα)pµA. (2.98)

With the chosen Ansatz (2.95) the action of L on f in the Liouville equation becomes:

L(fA) = pµA∂fA

∂xµA− Γµνρp

νAp

ρA

∂fA

∂pµA= −

(pµAαA,µ + βAν,µp

νAp

µA − Γµνρp

νAp

ρAβAµ

)exp(φA)f2

Ar−1A

= − (pµAαA,µ + βAµ;νpµAp

νA) exp(φA)f2

Ar−1A = 0. (2.99)

Note that the Liouville equation also holds for φA, which can be found by dividing (2.99) by(− exp(−φA)f2

Ar−1A ):

L(φA) = 0. (2.100)

If the particle species A is massive with mA > 0, then αA and βµ naturally arise as conservedquantities. From (2.99) they have to satisfy

αA,µpµA = 0, (2.101)

βAµ;νpµAp

νA = 0. (2.102)

And to satisfy the Liouville equation for all pµA it must hold that

αA,µ = 0, (2.103)

βAµ;ν = 0 (mA > 0), (2.104)

βA,ν = 0. (2.105)

In the case of particle A being massless the Eq. (2.10) implies

gµνpµAp

νA = 0. (2.106)

Thus, to satisfy (2.102) by using (2.97) it must hold (Ehlers, 1974):

βAµ;ν = (βAuAµ);ν = βA,νuAµ︸︷︷︸!=0

+βAuAµ;ν ∼ βAgµν (mA = 0), (2.107)

where the highlighted term has to be zero to satisfy the Liouville equation for all pµA.Eq. (2.103) and (2.105) mean that in equilibrium (detailed balance) the dependence on

spacetime of αA and βA can be dropped while the spatial dependence of uAµ can only be droppedin the massive case using (2.104). Let us now for simplicity consider only the massless case.Investgating the αA and βAµ, it can be seen that αA depends on the particle type A and in abinary collision due to (2.91) it has to satisfy:

αA + αB = αC + αD. (2.108)

But, as we will see, βAµ is independent of the species A. Due to (2.91) the βAµ have to satisfy

βAµpµA + βBµp

µB = βCµp

µC + βDµp

µD. (2.109)

Also, as 4-momentum conservation holds, we can eliminate one of the momenta, i.e. pµA:

pµA = pµC + pµD − pµB. (2.110)

Hence,

βAµ(pµC + pµD − pµB) + βBµp

µB − βCµp

µC − βDµp

µD = 0

(βBµ − βAµ)pµB + (βAµ − βCµ)pµC + (βAµ − βDµ)pµD = 0 (2.111)


and the last equation implies that for arbitrary pµi 6= 0 (i = B, C, D) the following has to hold:

βAµ = βBµ = βCµ = βDµ = βuµ (2.112)

Therefore, the βµ are independent of the species. This is reasonable as we evaluate the dis-tribution functions in the collision term from the frame represented by the observer’s velocityuµ.

Furthermore, the chosen Ansatz (2.96) seems to be physically rigorous, because only scalarcollisional invariants as charge or baryon number and energy-momentum conservation are knownin physics, but Boltzmann (1896) found this solution by assuming φA to be twice differentiablewith respect to momentum. Still the question arises, why higher order terms of momentum arenot included in the Ansatz. We will now prove that there are no other collisional invariants thanour solution (2.96).

In general we can consider higher order terms of pµA, e.g. a second-order term

γAµνpµAp

νA, (2.113)

for which the γAµν need to be non-negative to keep φA non-negative. Analogously to Eq. (2.109)the γµν have to satify

γAµνpµAp

νA + γBµνp

µBp

νB − γCµνp

µCp

νC − γDµνp

µDp

νD = 0 (2.114)

and after eliminating pµA and pνA:

(γBµν − γAµν)pµBpνB + (γAµν − γCµν)pµCp

νC + (γAµν − γDµν)pµDp

νD

+γAµν(pµCpνD + pνCp

µD)− γAµν(pµCp

νB + pνCp

µB)− γAµν(pµBp

νD + pνBp

µD) = 0. (2.115)

Here it can be seen that due to the mixed terms all coefficients γµν have to be zero to satisfythe above equation. When we include higher order terms in φA these kind of mixed terms willalways occur und require the cofficients to be zero. Thus, we have proved that φA has to belinear and (2.96) is the most general solution.

In principle, the construction of the framework is now finished. It remains to find the physicalmeaning of αA and β, but for that it is useful to investigate the evolution of entropy.

2.5.2 Entropy evolution and the H-Theorem

With the obtained equilibrium solution of the distribution functions (2.95) it is now possible todeduce the corresponding expressions for thermodynamic quantities. By inserting (2.95) togetherwith the Ansatz (2.96) into the definitions (2.52) and (2.54) the particle number density nA, theenergy density ρA and the pressure pA become (Ehlers, 1971)

nA =gA

2π2~3

∫ ∞mA

√E2

A −m2AEAdEA

exp (αA + βEA)∓ 1, (2.116)

ρA =gA

2π2~3

∫ ∞mA

√E2

A −m2AE

2AdEA

exp (αA + βEA)∓ 1, (2.117)

pA =gA

6π2~3

∫ ∞mA

(E2

A −m2A

) 32 dEA

exp (αA + βEA)∓ 1, (2.118)

with mA the mass and EA the energy of the considered particle species and T the radiationtemperature. The entropy density sA can now be defined (and later it becomes clear that thishas to be true) by

sA = αAnA + ρAβ + pAβ. (2.119)


With the integral definitions of nA, ρA and pA in (2.116)-(2.118) we obtain

sA =gA

2π2~3

∫ ∞mA

[−αA − βEA

exp (αA + βEA)∓ 1∓ ln (1∓ exp (−αA − βEA))

]·√E2

A −m2AEAdEA (2.120)

We now want to give a brief proof of the H-theorem, which describes the evolution of theentropy. The definition in (2.120) can be easily generalized into a relativistic description. Theentropy 4-current for the particle species A is defined as

SµA(xµ) = −∫

Π(fA ln(r−1

A fA)∓ fA ln(r−1A fA))pµAΠ (2.121)

=

∫Π

(fA ln

(fA

fA

)± rA ln(r−1

A fA)

)pµAΠ, (2.122)

where S0A/γ = s is the local entropy density (with the Lorentz factor γ). For the total entropy

flux we have to sum over all particle species:

Sµ(xµ) =∑

A

SµA(xµ). (2.123)

For every particle species SµA(xµ) is proportional to a linear combination of the particle 4-current (2.51) with some integral over a term depending on pµA. Hence, we can apply (2.58) forthe evolution equation:

Sµ;µ =∑

A

∫ln

(fA

fA

)L(fA)ΠA, (2.124)

where the second term in (2.122) is proportional to φA and hence itself a collisional invariant.Therefore, the Liouville operator applied to this second term yields zero as in (2.100). Insertingthe collision term (2.65) becomes

Sµ;µ =1

2

∑A,B,C,D

γABCD

∫ [ln

(fA

fA

)(fAfBfCfD − fAfBfCfD)

· |MA,B↔C,D|2(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD

]. (2.125)

In the sum A can be exchanged with B and C with D without changing the terms, such thatthe expression can be rewritten to

Sµ;µ =1

8

∑A,B,C,D

γABCD

∫ [ln

(fAfBfCfD

fAfBfCfD

)

· (fAfBfCfD − fAfBfCfD)|MA,B↔C,D|2(2π)4δ (pA + pB − pC − pD) ΠAΠBΠCΠD

]. (2.126)

The terms in the intergrals above are of the shape

ln

(F

G

)(F −G) (2.127)

and are necessarily non-negative. With this the H-theorem is proven stating: the total entropyflux always increases or remains constant:

Sµ;µ ≥ 0. (2.128)


The constant case is then the case of detailed balance and isentropic expansion. When Sµ;µ = 0,the system is in thermodynamic equilibrium and the collision term in (2.126) has to vanish. Inother words:

ln

(fA

fA

)has to be a collisional invariant. (2.129)

Hence, the choice of the collisional invariant in (2.92) arises naturally.

2.5.3 The Equilibrium Distributions

To find the physical meaning of αA and β the argument of the entropy evolution in Sec. 1.4.4can be considered. It can be seen the differential of the entropy density in Eq. (2.119):

dsA = d(αAnA + βρA + βpA), (2.130)

(Ehlers, 1971). From the condition of symmetric second derivatives (1.58) the Gibbs-Duhemrelation can be found (see Eq. (1.60)):

dpA = −nA

βdαA −

ρA + pA

βdβ. (2.131)

Inserting this relation into Eq. (2.130) gives

dsA = αAdnA + βdρA (2.132)

and hence, the thermostatic potential:

dρA =dsA

β− αA

βdnA. (2.133)

Comparing this result with the first law of thermodynamics (1.16), the coefficients of the colli-sional invariant become

β(x) =1

T (x)and αA(x) = −µA(x)

T (x), (2.134)

with the temperature T and the chemical potential µA. With these quantities it is confirmedthat (2.119)) is really the entropy density as defined in Eq. (1.55).

As β is the inverse temperature it is independent of the particle species A. The dependence onspacetime of the collisional invariant is therefore only expressed by the dependence on spacetimeof β(x) and αA(x). These quantities are constants in thermodynamic equilibrium as they satisfy(2.103) and (2.105). Inserting these quantities into the equilibrium solution (2.95) gives

fA(xµ, pµA) =gA

h3

[exp

(−µA

T+uµp

µA

T

)∓ 1

]−1

, (2.135)

which is the Bose-Einstein distribution for bosons with the upper sign and the Fermi-Dirac distribution for fermions with the lower sign (Ehlers, 1971). These distributions takeinto account the quantum nature of the particles at high temperatures. Furthermore, in the restframe of the gas, where uµ = (1, 0, 0, 0), the minimum value of uµp

µA satisfies

uµpµA = p0

A ≥ mA. (2.136)

And due to (2.98) the chemical potential is restricted to

−αA =µA

T<uµp

µA

T. (2.137)


As this condition has to hold for all pµA, from Eq. (2.136) it can be shown that

−αA =µA

T<mA

T. (2.138)

This condition has to hold at any temperature T 6= 0. In further, it will be assumed that thetemperature has always been finite. As µ/T is constant, µ has to be neglibibly small or negative:

µA mA ∀ T <∞, T 6= 0. (2.139)

Remember that this relation is only valid when assuming (2.44), such that the distributionfunction is non-divergent and positive. In Chapter 4 the strictness of this constraint will beinvestigated in more detail.

At low temperatures (T mA, µA) the exponential function in (2.135) is much greater than1, so f simplifies to the Juttner distribution:

fA(xµ, pµA) =gA

h3exp

(µA

T− uµp

µA

T

). (2.140)

In the rest frame of the gas this becomes the Maxwell-Boltzmann distribution (Ehlers,1971)

fA(xµ, p0A) =

gA

h3exp

(µA

T− p0

A

T

). (2.141)

This solution can also be obtained by applying the low temperature limit in (2.90), such that

(r ± f) ' r. (2.142)

Then the conservation condition simplifies to

fA

rA

fB

rB=fC

rC

fD

rD. (2.143)

The solution to this eqution is given by (2.141).

2.5.4 The Limits of Radiation and Matter Domination

With the obtained equilibrium distribution it is now possible to numerically calculate the equi-librium values of thermodynamic quantities. The integrals in the definitions (2.116), (2.117),(2.118) and (2.120) have no general analytic solution, but some limiting cases approximationscan be made. In the relativistic limit it can be assumed that T mA. When also assumingT µA the thermodynamic quantities are given by (Kolb and Turner, 1990, Ch. 3)

nA =ζ(3)

π2geff

A T 3 with geffA = gA for bosons and geff

A =3

4gA for fermions, (2.144)

where ζ(3) = 1.20206... is the Riemann zeta function and

ρ =π2

30g∗T

4, (2.145)

p =ρ

3, (2.146)

s =π2

45g∗ST

3. (2.147)


Here, g∗ is the total number of degrees of freedom summing over all particles that are relativisticat the temperature T :

g∗ =∑

i=bosons,miT

gi

(TiT

)4

+7

8

∑i=fermions,miT

gi

(TiT

)4

, (2.148)

where Ti is the temperature of the respective species6. The total number of degrees of freedomin the calculation of the entropy density g∗S differs from Eq. (2.148) because of the differenttemperature dependence (Kolb and Turner, 1990):

g∗S =∑

i=bosons,miT

gi

(TiT

)3

+7

8

∑i=fermions,miT

gi

(TiT

)3

. (2.149)

The relativistic limit corresponds to the epoch of radiation domination (see Sec. 1.4.2).Here, it is convenient to express ρ, p and s as total quantities, because at high temperaturesall particles are assumed to interact with each other, such that the state of a particle speciesdepends on all other species. At low temperature interactions cease and the state of the speciesis then determined by the properties of the species itself. Therefore in the low temperature limit(T mA) we express the equilibrium values of quantities in terms of gA and not in terms of g∗and g∗S (Kolb and Turner, 1990, Ch. 3):

nA = gA

(mAT

2π

) 32

exp

(−(mA − µA)

T

), (2.150)

ρA = mAnA = gAmA

(mAT

2π

) 32

exp

(−(mA − µA)

T

), (2.151)

pA = nAT ρA, (2.152)

sA = nA

(mA − µA

T+

5

2

). (2.153)

The low temperature limit corresponds to the epoch of matter domination. In the transitionbetween these two epochs the evolution of the thermodynamic quantities has to be calculatednumerically.

The construction of the relativistic kinetic theory formalism is now finished. We startedwith the presentation of the phase space defining an invariant volume element and introducingthe distribution function of states. The Liouville operator provided the Boltzmann evolutionequation and we investigated its equilibrium solutions and obtained the Fermi-Dirac, Bose-Einstein and the Maxwell-Boltzmann (Juttner) distributions by defining collisional invariants.In the end we derived approximations for the equilibrium values of thermodynamic quantities.The Boltzmann equation and the equilibrium distribution functions will play an important rolein the next chapters. In chapter 4 the will see that the notion of collisional invariants is closelyrelated to the conservation of some quantum number which represents a particle-anti-particleasymmetry. In the following chapter we proceed to calculating relic abundance and investigatethe decoupling and freeze-out of a symmetric particle species.

6It is possible, that the temperature Ti of some species i will differ from the radiation temperature T , becauseits interactions become impotent at some temperature with the cooling of the Universe, such that it does notthermalize to equilibrium with the radiation and the other particle species (it still can thermalize with itself bythermal collisions).

31

3 Evolution of a Symmetric Particle Species in the FLRW Uni-verse

The formalism constructed in chapter 2 can now be applied to the FLRW Universe, wherethe constraints from Sec. 1.4 can be inserted. Before, the expectations should be discussed todistinguish the cases of the different epochs of radiation domination and matter domination. Theevolution of the various particle species in the Universe is very different for different masses andinteraction rates, which will be presented in the following. We will see that a WIMP Cold DarkMatter species decouples early due to its weak interactions. From the freeze-out calculation wecan set constraints on the properties of a WIMP CDM species. The presentation in this chapterfollows mainly the one of Kolb and Turner (1990, Chap. 5).

3.1 Equilibrium in the Early Universe

The relic abundances of the constituents of the Universe that are observed today (or not ob-served) can be modeled via the formalism of relativistic kinetic theory. In this modelling theterms of thermodynamic equilibrium and the interaction strength of the respective par-ticle species play an important role.

From our cosmological standard model there is a strong indication that all known constituentshave been in thermodynamic equilibrium at some point in time in the early Universe. ’Thermo-dynamic equilibrium’ summarizes three other terms: thermal equilibrium, kinetic equilibriumand chemical equilibrium.

Thermal equilibrium means when the gas and its constituents have the same temperature.If all interactions and thermal collisions happen rapidly, any local excess of energy by e.g. thedecay of a particle species will be distributed quickly and uniformly. This distribution of energydriving the system to thermal equilibrium is called thermalization.

Kinetic equilibrium prevails when the constituents of the gas follow the respective equi-librium phase space distribution function (2.135) (or (2.141) at low temperatures).

When chemical equilibrium is present we have detailed balance, i.e. all interactions chang-ing the particle number and their inverse reactions are in equilibrium (see Sec. 2.2). If all threecases are fulfilled then thermodynamic equilibrium is maintained.

There is not much known about the nature of DM, but a reasonable assumption is thatit was also in thermodynamic equilibrium with the visible matter initially. This equilibriumcould have been maintained by some as of yet undiscovered interaction, that could have beenpotent then. As we know that the Universe is expanding, we expect the temperature to decreasewith time. With the cooling also the particle interactions depending on temperature becomeimpotent at some point. If a particle species only interacts with the rest of the Universe7 viasuch an interaction becoming impotent, the species will drop out of equilibrium with the restof the Universe - or we say, it decouples. If an interaction becomes impotent we call it afreeze-out of the interaction. The relic abundance of a particle species primarily depends onwhether it decouples while it is still relativistic or already non-relativistic.

Consider a species ψ (whitout loss of genarality, fermionic) which annihilates to some otherparticles X:

ψψ ←→ XX, (3.1)

where we assume thatmψ > mX . (3.2)

ψ and X are the corresponding anti-particles to ψ and X. We also assume the interactions ofX with the rest of the Universe to be still potent during the decoupling of the ψ-particles, such

7’The rest of the Universe’ will from now on mean: the other species and radiation.

32 3 EVOLUTION OF A SYMMETRIC PARTICLE SPECIES IN THE FLRW UNIVERSE

that X stays in thermodynamic equilibrium with the rest of the Universe. The process (3.1)above only happens in both directions while ψ is relativistic, i.e. as long as the temperature Tis much higher than the particle’s mass mψ, such that the average energy of a pair XX is highenough during an annihilation to recreate a ψψ-pair.

Generally the decoupling of the species depends on its interaction strength which is charac-terized by the interaction rate per particle

Γ ≡ neq〈σA|v|〉, (3.3)

where 〈σA|v|〉 is the thermally averaged product of the annihilation cross section (index A for’annihilation’) and relative velocity of the particles and neq is the equilibrium particle density.With the expansion of the Universe the temperature decreases. Let us assume that at early timesall interactions proceeded rapidly enough to keep all species in equilibrium and to hold againstthe departure from equilibrium caused by the expansion, so the number density n closely tracksthe equilibrium value nEQ that decreases with decreasing temperature. 〈σA|v|〉 also decreaseswith decreasing temperature or at least remains constant. Hence, the interaction rate Γ decreasesand at some point it becomes of the order of the Hubble rate H. Then the interactions becomeimpotent and the particle species decouples:

Γ & H (coupled), Γ . H (decoupled). (3.4)

Generally a species can decouple from a second species, but still can be coupled to a third one.It depends on the respective interaction rate compared to the Hubble rate. We further definea species to be decoupled if it is decoupled from all species and radiation it is defined to becompletely decoupled if the rate of thermal collisions ψψ ↔ ψψ drops below H, such that itcannot thermalize (maintain thermal equilibrium with the rest of the Universe) anymore. Thepoint of all particle species having decoupled from radiation is the beginning of the transitionbetween the radiation-dominated epoch to the matter-dominated epoch.

If ψ decouples (not necessarily completely) while it is still relativistic and the reaction in(3.1) can still proceed both ways, then it is called a hot relic. But if the species already becamenon-relativistic before decoupling it is called a cold relic. These terms are directly associatedwith Cold Dark Matter and Hot Dark Matter models. We will see that the evolution ofa cold relic species differs from that of a hot species and the consequences for the properties ofDM.

With the formalism that we constructed it is possible to compute relic abundances and ingeneral to describe cosmological milestones like inflation, big bang nucleosynthesis and recombi-nation of nuclei with electrons. First, we will assume a symmetric species, neither with an excessof particles nor anti-particles. Then, this species will be assumed the anticipated WIMP ColdDark Matter species and we will derive constraints on its properties. In the next chapter wewill extend the formalism to the evolution of particle-anti-particle asymmetries and a possibleasymmetries in the dark sector and consider the WIMP to be asymmetric.

3.2 The Boltzmann Equation in the FLRW Universe

The only input that the constructed formalism of kinetic theory requires is the FLRW metricwith its components given in Eq. (1.8). The presentation in Ch. 2 will now be revisited whileinserting the assumptions from the FLRW model.

With the determinant of the metric (1.9) we can adapt our integration measures. Theinvariant phase space volume element ω (2.35) then contains a factor R6 and the measures dVand Π a factor R3. Constructing the Boltzmann equation the action of the Liouville operator

3.2 The Boltzmann Equation in the FLRW Universe 33

on the distribution function becomes the following using Eq. (2.18) and (1.10):

L[f(xµ, pµ)] = p0∂f(xµ, pµ)

∂x0︸︷︷︸(I)

+ pi∂f(xµ, pµ)

∂xi︸︷︷︸(Ia):=0

− 2Γi0jp0pj

∂f(xµ, pµ)

∂pi︸︷︷︸(II)

−Γ0ijp

ipj∂f(xµ, pµ)

∂p0︸︷︷︸(III)

= p0∂f(t, pµ)

∂t︸︷︷︸(I)

− 2Hp0pi∂f(t, pµ)

∂pi︸︷︷︸(II)

−RRδijpipj∂f(t, pµ)

∂p0︸︷︷︸(III)

, (3.5)

where local coordinates, x0 = t, have been used. The Universe is assumed to be homogeneous andisotropic, so we can set the spatial derivative term (Ia) to zero. Together with our assumption ofuncorrelated momenta (see Sec. 2.1) the distribution function should not depend on the directionof the momentum, but only on the absolute value of the momentum or, due to the dependencebetween the components p0 and pi, only on p0: f(xµ, pµ) −→ f(E, |p|, t), with E given by Eq.(2.9) and |p| being the absolute value of the comoving momentum. Now, the expression (3.5)can be recast to a form that only depends on E.

At first, it is useful to perform the mass-shell intergral over the Boltzmann equation (3.5)and define

f(E, |p|, t) ≡∫f(pµ, t)δ(p0 − E)dp0, (3.6)

(Bernstein, 1988). Hence, we obtain

L[f(E, |p|, t)] = E∂f(E, |p|, t)

∂t︸︷︷︸(I)

− 2HE pi∂f(E, |p|, t)

∂pi︸︷︷︸(II)

−RR δij pipj∂f(E, |p|, t)

∂E︸︷︷︸(III)

. (3.7)

Consider the term (II). It is easily shown that

pi∂f(E, |p|, t)

∂pi= |p|∂f(E, |p|, t)

∂|p|, (3.8)

with the absolute value of the comoving momentum (Bernstein, 1988):

|p| =√−pipi =

√−gijpipj =

√R2|p|2 = R|p|, (3.9)

where |p| is the absolute value of the local momentum (Stewart, 1971). Moreover, we have

|p|∂f(E, |p|, t)∂|p|

= |p| ∂E∂|p|

∂f(E, |p|, t)∂E

=|p|2

E

∂f(E, |p|, t)∂E

, (3.10)

such that we obtain for the (II)-term:

(II) = −2H|p|2∂f(E, |p|, t)∂E

. (3.11)

Furthermore, we can rewrite the local momenta in the (III)-term in terms of comoving momentausing (3.9):

δij pipj = |p|2 = −|p|

2

R2, (3.12)

(Kolb and Turner, 1990, Chap. 5). Thus, we obtain the Liouville operator on f(E, |p|, t):

L[f(E, |p|, t)] = E∂f(E, |p|, t)

∂t−H|p|2∂f(E, |p|, t)

∂E(3.13)


We now can perform the rest of the integration of the Boltzmann equation (2.48) over

momentum space (we already integrated over dp0δ(p0−E), so it is left to integrate over R3 d3pE ),

and for that use the definition of the local number density from Eq. (2.54). We solve theintegral of the second term on the right-hand side in (3.13) by partial integration while usingf(|p| → ∞)→ 0, so we obtain the evolution equation for n:∫

L[f(E, |p|, t)]R3d3p

E= n+ 3Hn =

∫C[f ]Π. (3.14)

Actually, we could have obtained this equation more easily using Eq. (2.56), but we choseto reconstruct the formalism analogously to Chap. 2. In the rest frame of the gas we haveNµ = (n, 0, 0, 0) and thus,

Nµ;µ = n+ Γµµ0n = n+ 3Hn. (3.15)

It is now clear that in absence of interactions, i.e. when the right-hand side of Eq. (3.14) vanishes,the evolution of the number density goes as

n = ni

(RiR

)3

, (3.16)

with initial values Ri and ni.Further, it will be assumed that the only change in the particle number of some species ψ

proceeds via binary collisions like Eq. (3.1) (in general additional other reaction channels arepossible). Then, the collision term on the right-hand side of (3.14) (analogously to (2.65)) is∫

C(fψ)Πψ = −∫

ΠψΠψΠXΠX(2π)4δ4(pψ + pψ − pX − pX

)· |Mψψ↔XX |2

[fψfψfX fX − fXfX fψfψ

], (3.17)

where we already assumed the CP-invariance for the reaction.As we already mentioned in Sec. 2.4 it is not trivial to define a common thermally aver-

aged cross-section for the reaction and the inverse reaction because of the quantum effects ofPauli-blocking and Bose enhancement. In the low temperature limit (T m,µ) the quantumdistribution functions (the Fermi-Dirac and the Bose-Einstein distributions (2.135)) in the colli-sion term (3.17) simplify to Maxwell Boltzmann distributions (2.141). Furthermore the quantumeffects vanish as the functions f in the collision term (3.17) become

f = (r ± f) ≈ r. (3.18)

Thus the collision term becomes∫C[fψ]Πψ = −

∫ΠψΠψΠXΠX(2π)4δ

(pψ + pψ − pX − pX

)· |Mψψ↔XX |2

[fψfψrXrX − fXfXrψrψ

]. (3.19)

In Sec. 3.1 we assumed the interactions of the particles X and X with the rest of the Universeto happen rapidly enough to maintain equilibrium even when the interaction rates of the ψ-particles already dropped below H. Thus, in the comoving frame8 of the gas the distributionfunctions of X and X follow the Maxwell-Boltzmann equilibrium distribution:

fX ≡ f eqX = rX exp

(−p0X − µXT

), fX ≡ f

eqX

= rX exp

(−p0X− µXT

). (3.20)

8In the FLRW Universe the space is time-dependent due to the expansion, so the rest frame is always acomoving frame.

3.3 Scaling out the Universe’s Expansion 35

Now as the arguments of the exponential function above are collisional invariants, it holds that

p0ψ + p0

ψ = p0X + p0

X , (3.21)

µψ + µψ = µX + µX . (3.22)

Therefore we can write:

fXfXrXrX

= exp

(−

(p0X + p0

X− µX − µX)

T

)= exp

(−

(p0ψ + p0

ψ− µψ − µψ)

T

)=f eqψ f

eq

ψ

rψrψ. (3.23)

With this the factor in the collision integral becomes[fψfψrXrX − fXfXrψrψ

]=[fψfψrXrX − f

eqψ f

eq

ψrXrX

]. (3.24)

With this we can define a common thermally averaged annihilation cross-section. The termaccounting for the quantum effects (2.88) becomes zero in the low temperature limit. If weassume the particle species ψ to be symmetric, i.e. not to have an excess of particles over anti-particles, then the particle and anti-particle densities are equal: neq

ψ = neq

ψ. This is only possible,

if the chemical potentials in the definition (2.150) are equal: µψ = µψ. In Chapter 4 we will seethat this means that both chemical potentials have to be zero, wich will be assumed from nowon. Then, due to (3.22) also the chemical potentials µX and µX are zero.

With only zero chemical potentials the common thermally averaged annihilation cross-sectionas defined in (2.87) then becomes

〈σA|v|〉 ≡r2X

(neqψ )2

∫dΠψdΠψdΠXdΠX(2π)4δ


)|Mψψ↔XX |2f

eqψ f

eq

ψ, (3.25)

with rX = rX . And analogously to (2.84) the evolution equation for the density of the particlespecies ψ becomes ∫

L[fψ]Πψ = nψ + 3Hnψ = −〈σA|v|〉[n2ψ − (neq

ψ )2]. (3.26)

3.3 Scaling out the Universe’s Expansion

The state of a gas in the FLRW Universe is never static due to the expansion, not even in thestate of detailed balance. As we are especially interested in departures from equilibrium, it isworthwhile to consider the evolution of a quantity that is not affected by the expansion (Kolband Turner, 1990, Chap. 3). Let us define the variable

Yψ ≡nψs, (3.27)

that describes the species’ abundance with the entropy density s. In the case of detailed balancethe entropy in a comoving volume S = sR3 is constant as we have shown in Sec. 2.5.2, such that

Yψ =nψs

+ nψd1s

dt=nψs

+nψS

dR3

dt=nψs

+ 3nψSR2R =

nψs

+ 3nψsR3 R

R=nψs

+ 3Hnψs

(3.28)

⇔ nψ + 3Hnψ = sYψ. (3.29)

So the expansion effect cancels out and Yψ is proportional to the particle number in a comovingvolume (Kolb and Turner, 1990) as long as the process (3.1) is in equilibrium and we havedetailed balance.


In general, we get more specific information about the evolution of the Universe when focusingon temperature rather than time. Because of that we define an independent variable

x =mψ

T. (3.30)

Thus Eq. (3.26) can be rewritten to

−1

s〈σA|v|〉

[n2ψ − (neq

ψ )2]

=dYψdt

=dx

dt

dYψdx

=dx

dtY ′ψ. (3.31)

We now need a relation between x (or T ) and t. The evolution for the energy density differsin the epochs of radiation and matter domination as seen from (1.32), (2.145) and (2.151).Combining these equations we obtain in the radiation domination epoch

t = 0.301g− 1

2∗

mPl

T 2= 0.301g

− 12∗

mPl

m2ψ

x2, (3.32)

with the Planck mass

mPl =1√G. (3.33)

Note that the relation between the Hubble rate and the temperature is given with (1.29) by:

H = 1.66 g∗T 2

mPl= 1.66 g∗

m2ψ

mPl

1

x2=H(mψ)

x2, (3.34)

with H(mψ) the Hubble rate at temperature T = mψ (x = 1):

H(mψ) = 1.66 g∗m2ψ

mPl. (3.35)

The contribution of a non-relativistic particle species to the energy density from Eq. (2.151)is always suppressed by the exponential factor with decreasing temperature. As long as otherspecies are still relativistic, their contribution will follow (2.145) and dominate over the contri-bution of the non-relativistic species. And as we asssumed ψ to be more massive than X (seeEq. (3.2)) the relations (3.32) and (3.34) are good approximations in the regime T > mX .

Now, having an relation between x and t, the derivative in (3.31) is

dx

dt=

1

2

1√t

√1

0.301g

14∗

mψ√mPl

=1

2

1

t

√t

√1

0.301g

14∗

mψ√mPl

=1

2

x

t

Eq. (1.29)︷︸︸︷= Hx =

H(mψ)

x(3.36)

Hence, the evolution equation (3.31) becomes:

Y ′ψ = −〈σA|v|〉dxdt

1

s

[n2ψ − (neq

ψ )2]

= −sx〈σA|v|〉H(mψ)

1

s2

[n2ψ − (neq

ψ )2]

(3.37)

⇔ Y ′ψ = −sx〈σA|v|〉H(mψ)

[Y 2ψ − (Y eq

ψ )2]. (3.38)

UsingH(x) = H(mψ)x−2 (3.39)

and the definition of the interaction rate (3.3) Eq. (3.38) can be rearranged in a way to clearlysee the factor of the effectiveness of the interaction in (3.4):

Y ′ψ = − Γ

H(x)xY eqψ

( YψY eqψ

)2

− 1

. (3.40)

The obtained equation is an uncoupled single differential equation only depending on Yψ andx, which provides the equipment for calculating relic abundances of a particle species and forestimating its properties.

3.4 Calculating Relic Abundances 37

3.4 Calculating Relic Abundances

The relic abundance highly depends on the strength of the species’ interactions and on its mass.Furthermore, it is important whether the interactions drop below the Hubble rate while thespecies is still relativistic or already not. The evolution of the thermodynamic quantities inequilibrium are different in the relativistic (x 3) limit and in the non-relativistic (x 3)limit9 which we showed in Sec. 2.5.4. With the definition in (3.27) and the expressions for nψand s in (2.144), (2.147), (2.150) and (2.153) we write down the equilibrium values of Yψ in thetwo limits. Note that the number density in Yψ is the number density of the species ψ, but theentropy density is a total quantity dominated by the contributions of the relativistic species.And as we assume the rest of the Universe (species X) to be still relativistic when ψ is alreadynot, we will always write Yψ in terms of the relativistic limit of s in (2.147). Therefore theequilibrium values of Yψ are

Y eqψ (x) =

45ζ(3)

2π4

geffψ

g∗S= 0.278

geffψ

g∗S(x 3, ψ relativistic), (3.41)

Y eqψ (x) =

45

2π4

(π2

) 12 gψg∗S

x32 exp(−x) = 0.289

gψg∗S

x32 exp(−x) (x 3, ψ non-rel.), (3.42)

with geffψ = g for bosons and geff

ψ = 3g/4 for fermions as given in (2.144). Remember that we

assumed a zero chemical potential of all particle species ψ, ψ, X, X. In chapter 4 we willreintroduce it in our calculation and explain its physical meaning.

As Yψ follows the equilibrium value Y eqψ in thermodynamic equilibrium, according to (3.41)

it is constant when ψ is still relativistic and the number of degrees of freedom does not changedue to other particle species becoming non-relativistic. But when ψ has become non-relativistic,its abundance diminishes exponentially as seen in (3.42). Here, the question arises, why it isnot constant in that regime as Yψ is proportional to the particle number in a comoving volumewhich we expected to be constant. The answer is that it is only constant when all reactionsare in equilibrium, i.e. we have detailed balance. As ψ has become non-relativistic, the reaction(3.1) is not in chemical equilibrium anymore while ψ can still be in thermal equilibrium, i.e.have the same temperature as the rest of the Universe until the interactions that keep ψ in thatequilibrium freeze out. Y eq

ψ then gives the equilibrium value if the interactions would not freezeout.

It is now obvious that the abundance has a different evolution for hot or cold relics. Tounderstand this, consider the annihilation rate

ΓA = neq〈σA|v|〉 = sY eq〈σA|v|〉. (3.43)

In either case, relativistic or not, ΓA decreases with decreasing temperature and the interactionsfreeze out when ΓA ≈ H. As previously said, we expect the species to be in thermodynamicequilibrium before the decoupling, i.e.:

Y ≈ Y eq (x xf ), (3.44)

with xf being the value at the point of freeze-out. If the species becomes non-relativistic beforedecoupling, the departure from equilibrium is still small. After decoupling the abundance freezesin, so:

Y (x & xf ) = Y eq(xf ) (x & xf ). (3.45)

Now let us inspect the different cases.

9The average kinetic energy of a particle in a gas is EK = 32T (kB ≡ 1). The gas is relativistic as long as 2E

equals the particle’s mass (factor 2, because of two participants in a collision), thus x = 2 · 32kB ≡ 3 works as a

rule of thumb quite well.


3.4.1 Hot Relics

The hot relic case is simple as we do not necessarily need the evolution equation (3.38). Whena particle species is still relativistic (xf . 3) at the point of freeze-out of its interactions then itis defined to be a hot relic. From (3.45) we know that (asymptotic) final abundance is just theequilibrium value at freeze-out, so regarding (3.41) we obtain:

Yψ(x −→∞) ≡ Y∞ψ = Y eqψ (xf ) = 0.278

geffψ

g∗S(xf ). (3.46)

This shows that the final abundance of a hot relic only depends on g∗S(xf ). The interactionrate of the annihilation is proportional to the thermally averaged cross-section and to neq

ψ ∼ x−3

(see Eq. (3.3)). The higher the cross-section the later the annihilation rate will drop below Hand the bigger xf will be. g∗S decreases with higher xf and hence, the final abundance (3.46)grows with the strength of the interaction.

Futher on with the assumption of isentropic expansion after freeze-out the today’s particledensity is:

nψ0 = s0Y∞ψ , (3.47)

with s0 being the entropy density today. Let us estimate this value. We assume only the threeknown neutrino species and the photon to be relaticistic with both having a spin degeneracyfactor g = 2. Then the numbers of degrees of freedom today are given by (2.148) and (2.149)10:

g∗(T0) = 2 +7

8· 2 · 3 ·

(4

11

) 43

= 3.36, (3.48)

g∗S(T0) = 2 +7

8· 2 · 3 · 4

11= 3.91, (3.49)

assuming that today’s temperature (Kolb and Turner, 1990) is

T0 = 2.75K = 2.3× 10−13GeV = 11.675 cm−1. (3.50)

Thus with (2.147) the entropy density today is

s0 =2π2

45· 3.91 · T 3

0 = 2970

(T

2.75K

)3

cm−3. (3.51)

Hence, today’s number density would be

nψ0 = 2970Y∞ψ cm−3 = 825geff

g∗(xf )cm−3. (3.52)

With (2.151) we obtain the relic mass density just by multiplying with mψ:

ρψ0 = s0Y∞mψ = 2970Y∞

(mψ

eV

)eVcm−3. (3.53)

10After the decoupling of positrons and electrons the average radiation temperature admits a further contribu-tion by the annihilation: e+e− → 2γ. The contributed photon energy is always 2me per annihilation and whilethe background radiaton energy density decreases with expansion the additional energy delays this decrease. Thisprocess is called re-heating. The with the decoupling of the electrons the neutrinos lose their last indirect inter-action with radiation and do not thermalize with it anymore. The increase in temperature of the radiation dueto the annihilations is not transferres to the neutrinos, which is the reason that the neutrino temperature is lowerthan the radiation temperature.


With the critical mass density in Eq. (1.42) and with (1.48) the mass density contribution of ahot relic can be calculated:

Ωψh2 =

ρψ0h2

ρc= 7.83 · 10−2 geff

g∗S(xf )

(mψ

eV

). (3.54)

Applying the weakest contraint on Ω that the Universe is observed to be flat (see (1.46)), weobtain

Ωψh2 =

ρtotalh2

ρc. 1, (3.55)

if we assume the density contribution Ωψ of ψ to dominate Ω0 being the density parameter ofall particles. Because of

ρψ0 = mψnψ0, (3.56)

with (3.55) we can set a mass bound for a hot relic:

mψ . 12.77 eV ·g∗S(xf )

geff. (3.57)

We have seen that the evolution of a hot relic does not require to solve the Boltzmannequation. The final abundance does not depend on temperature, nor the density contributionof a hot relic does. Note that the density contribution depends on mass. Also, it becomes lessfor a weaker interaction strength. We will see that the behaviour of a cold relic is completelydifferent.

Even if Hot Dark Matter is excluded by observations, let us now take a look at an exampleof a light neutrino as candidate for Hot Dark Matter.

3.4.2 Example: Light Neutrinos

A light neutrino is a hot relic and a candidate for Hot Dark Matter. We know, that lightneutrinos decouple at (see Kolb and Turner (1990, Chap. 5))

T ∼ MeV, while g∗S = g∗ = 10.75 and geff = 1.5 (for massive Dirac-neutrinos). (3.58)

Inserting that into (3.54) and (3.57) we receive

Ωννh2 =

mν

91.52eVand mν = 91.52eV. (3.59)

This bound is called the Coswik-McCelland bound (Kolb and Turner, 1990). Since the densitycontribution goes as Ωψ ∼ g−1

∗S (xf ) and g∗S(xf ) is larger at higher temperatures, a speciesthat decouples earlier will have a smaller density contribution today. Assume that the neutrinodecouples at

T ∼ 300GeV, when g∗S ≈ g∗ ≈ 106.75. (3.60)

The density contribution is lower, roughly by a factor of 10:

Ωννh2 =

mν

910eV. (3.61)

So we see that the stronger the interaction, the more abundant a hot relic will be. Having seenthe freeze-out behaviour of a hot relic we now proceed to the investigation of cold relics.


3.4.3 Cold Relics

The evolution of a cold relic is more complicated. The particle becomes non-relativistic beforedecoupling (xf & 3). Hence, the inverse reaction in (3.1) becomes supressed. The equilibriumvalue now decreases exponentially with decreasing temperature as shown in (3.42). In this casewe have to use the Boltzmann equation, thus the relic abundance depends on the details offreeze-out contained in the annihilation cross-section. For the latter σA|v| ∝ v2n we distinguish

n = 0 for s-wave annihilation and n = 1 for p-wave annihilation. As 〈v〉 ∼ T12 we can parametrize

the averaged cross section as

〈σA|v|〉 ≡ σ0

(T

mψ

)n= σ0x

−n when non− rel. (x & 3). (3.62)

Note that this parametrization is only possible in the non-relativistic regime in which we canneglect quantum effects.

Inserting that parametrization into the evolution equation (3.38) we obtain

Y ′ψ = −λx−n−2(Y 2ψ − (Y eq

ψ )2), (3.63)

with

λ =

[sx〈σA|v|〉H(mψ)

]x=1

= 0.264

(g∗S(xf )√g∗(xf )

)mPlmψσ0, (3.64)

having used (2.147), (3.35) and (3.62). Note that g∗S and g∗ are not constant in general, but ifno other species becomes non-relativistic at the time ψ becomes non-relativistic, then they areconstants g∗S(xf ) and g∗(xf ).

In general the differential equation (3.63) can only be solved numerically. Fig. 3 shows ournumerical solution, for which we used a Runge-Kutta-Fehlberg method for solving differentialequations with an adaptive stepwidth (see Appendix D and Press et al. (2007)). The evolutionof the equlibrium law shows a maximum, which is unphysical. This maximum exists due to theapproximation (3.42). This will be tolerated because for large x, the approximation is sufficientlyclose to the result that is obtained using the exact expression of the number density (2.116).

Again, we can find analytic expressions in the relativistic and non-relativistic limits. Thedeparture from equilibrium can be expressed by

∆ ≡ Yψ − Y eqψ , (3.65)

and the evolution equation (3.63) can be written in terms of ∆:

∆′ = −Y EQψ

′ − λx−n−2∆(2Y eqψ + ∆). (3.66)

For early times (high T ), but before decoupling (1 < x < xf ) we assume both ∆ and ∆′

to be small as Yψ closely tracks its equilibrium value. By neglecting ∆′ in (3.66) we get

∆ ≈ −λ−1xn+2Y eqψ′

2Y eqψ + ∆

Y eqψ ∆︷︸︸︷≈ −x

n+2

λ·(

23x − 1

)Y eqψ

2Y eqψ

≈ −xn+2

2λ(

3

2x− 1)

x−11︷︸︸︷≈ xn+2

2λ, (3.67)

with

Y eq ′ =

(2

3x− 1

)Y eq, (3.68)

by differentiating (3.42) with respect to x.


1 10 100 1000

x=mT

−20

−15

−10

−5

0

ln(Y Y(

1))

⟨σA|v|⟩ = 10−⟩7cm2

⟨σA|v|⟩ = 10−⟩8cm2

⟨σA|v|⟩ = 10−⟩9cm2

ln(Yeq⟨Y(1))

Figure 3: Numerical solution of the Boltzmann equation for a symmetric particle species (based onKolb and Turner (1990, Chap. 5), produced by myself (see Appendix D.1)) having used the Runge-Kutta-Fehlberg method: The figure shows the freeze-out of a cold relic particle species, which ismassive and stable. We see the final abundance relative to the initial value at x = 1 for different cross-sections 〈σA|v|〉, where the abundance is higher for smaller cross-sections. Initially, the abundancefollows the equilibrium value Y eq until the point of freeze-out at which the interaction rate dropsbelow the Hubble rate. Then, the abundance lifts off to become approximately a constant. Thepoint of freeze-out happens earlier (smaller x) for small cross-sections. The equilibrium abundanceY eq shows an unphysical maximum due to an analytic approximation. This can be tolerated as thecalculated evolution fits well for large x, which is the regime of interest.

For late times (low T ) after decoupling (x xf ), the abundance freezes in (Yψ = Y∞ψ )

while Y eqψ goes to zero exponentially. Conclusively, we can set ∆ ≈ Yψ ≈ Y∞ψ Y eq

ψ and ignore

terms depending Y eqψ and Y EQ

ψ′, such that Eq. (3.66) simplifies to

∆′ ≈ Y ′ψ = −λx−n−2∆2 ≈ −λx−n−2(Y∞ψ )2, (3.69)

∆′ ≈ −λx−n−2f (Y∞ψ )2. (3.70)

By integrating from xf to ∞ and neglecting ∆(xf ) we obtain

∆∞ = Y∞ψ =n+ 1

λxn+1f . (3.71)

For a cold relic we can determine the point of freeze-out xf in the following way: At the time(x = xf ) when the freeze-out occurs, Yψ is out of equilibrium and ∆ becomes of order Y eq

ψ ,which can be expressed by

∆(xf ) = c · Y eqψ = c a x

32f exp(−xf ), (3.72)

with a constant c of order unity and

a = 0.289(gψ/g∗S(xf )). (3.73)

The Ansatz (3.72) can be inserted into the early time solution (3.67), then

∆(xf ) ≈xn+2f

λ(2 + c). (3.74)


Again, setting (3.72) equal to (3.67) a recursive solution for xf is obtained:

xf ≈ ln[(2 + c)λac]−(n+

1

2

)ln(xf ) ≈ ln[(n+ 1)λa]−

(n+

1

2

)ln(xf ). (3.75)

From the numerically computed evolution we calculated the parameter c using the point offreeze-out, defined as

Γ(xf )

H(xf )xf= 1. (3.76)

Here, we included the additional factor xf in the denominator, because of the same shape it hasin Eq. (3.40). The numerical calculation results in a value of c ≈ 1.3.

The Boltzmann equation is a stiff differential eqaution and the fit of c highly depends onthe numerics of the method. Different authors state different fit values (see Cannoni (2015)).Furthermore the decoupling condition (3.76) is defined very vaguely. Moving on we use the fitobtained by Kolb and Turner (1990):

c(c+ 2) = n+ 1 c =√

2− 1 = 0.414 for n = 0. (3.77)

With this choice the relic abundance (3.71) takes the simple form:

Y∞ψ =3.79(n+ 1)xn+1

f

mPlmψσ0

√g∗(xf )

g∗S(xf )=

3.79(n+ 1)xfmPlmψ〈σA|v|〉

√g∗(xf )

g∗S(xf ). (3.78)

This leads to the following number density and density parameter today:

n∞ψ = 1.13 · 104

√g∗(xf )

g∗S(xf )

(n+ 1)xn+1f

mPlmψσ0cm−3, (3.79)

Ωψh2 = 1.07 · 109

√g∗(xf )

g∗S(xf )

(n+ 1)xn+1f

mPlσ0(GeV)−1. (3.80)

An interesting feature of (3.78) is that Y∞ψ depends on the inverse annihilation cross-section(albeit xf ∼ ln(σ0), the factor in the denominator is dominant). This mean that the weaker theinteraction, the higher the relic abundance. This is exactly the opposite case to the case of ahot relic, which has been shown in Sec. 3.4.1. For n = 0 (s-wave annihilation) the abundancedoes not depend on temperature anymore. Furthermore, if g∗ ≈ g∗S then√

g∗(xf )/g∗S(xf ) ≈ 1/√g∗(xf ). (3.81)

An earlier point of freeze-out xf due to a smaller cross-section will result in a higher numberof degrees of freedom. But in a regime in which no other species becomes non-relativistic thenumber of degrees of freedom is constant.

The abundance’s dependence on mass mψ is similar to the dependence on the cross-section.It has been shown that the mass has no effect on the abundance of a hot relic. For a cold relicthe abundance is higher the lighter the particle. Unfortunately, it is not easy to estimate a massfor a cold relic without knowing the mass-dependence of the cross-section because the expression(3.80) of the density parameter does not explicitly depend on the mass. Some examples withgiven cross-sections are presented in the following sections.

In principle an annihilation reaction can proceed via both s-wave and p-wave channels. Wecould now again present the similar calculation for cold relics with the following Ansatz for thethermally averaged cross-section analogously to (3.62) (Kolb and Turner, 1990, Ch. 5):

〈σA|v|〉 ≡ σ0(1 + c2x−2). (3.82)


But for simplicity we just give the results: The final abundance is then given by

Y∞ψ =3.79xf

mPlmψσ0

√g∗(xf )

g∗S(xf )·

(1

1 + 13c2x

−2f

), (3.83)

and the point of freeze-out by

xf = ln[(n+ 1)λa]−(n+

1

2

)ln(xf ) + ln

(1 + c2x

−2f

). (3.84)

The presentation of the calculations of freeze-out is now complete. We have seen, that thedifference in the evolution of hot relics differs significantly to the evolution of cold relics. Inthe case of a hot relic, the final abundance and density contribution to the total matter densityin Universe grow with higher cross-sections and both depend on the particle’s mass. For coldrelics the final abundance and the density contribution inversely depend on the cross-section.While the abundance can be smaller when the particle’ mass is higher, the density contributionis independent of mass, except of a factor of the number of effective degrees of freedom.

The next section gives an example of baryons considered as cold relics. After that, weinvestigate which properties a hypothetical WIMP species has to admit in the case it is a coldrelic.

3.4.4 Example: Baryon-symmetric Universe

Consider a baryon symmetric universe with equal nucleon and anti-nucleon densities and cal-culate their relic abundances. The nucleon-antinucleon annihilation cross-section (protons andneutrons are equal for simplicity) is defined by

〈σA|v|〉 = c1m−2π ≡ σ0, (3.85)

with the pion mass

mπ = 135MeV, (3.86)

and c1 a constant of order 1. This is the case of s-wave annihilation with n = 0. Inserting thisinto the formula for xf (3.75), in which we neglect the ln(xf )-term for simplicity, we obtain

xf ≈ ln [(2 + c)cλa] = ln [(n+ 1)λa] ≈ ln

[0.08

(gN√g∗(xf )

)mPlmBc1m

−2π

]≈ 42 + ln c1 ≈ 42,

(3.87)with the nucleon mass (Kolb and Turner, 1990)

mB ≈ mn ≈ mp ≈ 939 MeV, (3.88)

where mn is the neutron mass and mp the proton mass. Thereby we used g∗ ≈ g∗S ≈ 10 as in(3.58) and gB = 2.

Further, due to (3.87) the freeze-out temperature is about

Tf =mN

xf≈ 22.4 MeV (3.89)

and from (3.78) we calculate an abundance of

Y∞B = 7 · 10−20c−11 . (3.90)


Compared to observations this abundance is about 9 orders of magnitude lower than measured:

Y∞B =nBs≈ (6 - 10) · 10−11. (3.91)

Hence, the abundance is nine orders of magnitude higher in the baryon-asymmetric Universe.As mentioned in the introduction, in the baryon-symmetric case particles and particle’s wouldannihilate to a negligible abundance.

Now, we proceed to the investigation of the constraints, that are made on a WIMP speciesif it is considered to be Cold Dark Matter.

3.4.5 WIMPs as Cold Dark Matter and Heavy neutrinos

The density contribution in (3.80) only depends on the cross-section, if we neglect the logarithmin xf and choose a regime where g∗ = g∗S is constant. Let us now choose such a regime andassume a process of s-wave annihilation (n = 0). Using (3.80) the resulting density parameter is

Ωψh2 = 8.93 · 10−11 1√

g∗(xf )

xf〈σA|v|〉

(GeV)−2. (3.92)

With the maximally allowed density parameter for an accelerated expanding Universe Ωψh2 . 1

(see Eq. (1.45)) we obtain the condition:√g∗(xf )〈σA|v|〉

xf& 8.93 · 10−11 GeV−2 = 3.48 · 10−38cm2. (3.93)

Let us now take the following Ansatz for the cross-section (Kolb and Turner, 1990, Chap. 5):

〈σA|v|〉 = C · 10−37cm2 = C · 2.58 · 10−10GeV−2, (3.94)

with a constant C of order unity. With (3.75) the point of freeze-out will be

xf = 18 + ln(

C ·mψ

GeV

). (3.95)

The degeneracy factor can be chosen as gψ = 2 and the g∗-values in the range from 60 to106.75 and do not change the value 18 significantly. The upper value stands for the case that allStandard Modell particles are relativistic (Kolb and Turner, 1990, Chap. 3). The final abundanceof the particle is given by using (3.78):

Y∞ψ = 2.8 · 10−9 1

C

(GeV

mψ

). (3.96)

For comparison: In Fig. 3 we plotted the abundance for the considered cross-section (3.94) withC=1 and mψ = 2GeV and obtained a final abundance of Y∞ψ ≈ 6.43 · 10−9, which at least hasthe same order of magnitude so the analytical approximation fits relatively well.

Using (3.92) the density paramter becomes:

Ωψh2 =

0.045xfC

≈ 0.81

C. (3.97)

When considering Dark Matter to be a cold relic (Cold Dark Matter) the presented argumentleads to the conclusion that the cross-section of a DM particle has to be on the ’weak’ scale asseen in the Ansatz (3.94). The presented argument is the foundation of the WIMP paradigm.


The observed relative density of Dark matter is about

ΩDM = Ωψ + Ωψ ≈ 0.2, (3.98)

(see (1.53)). To obtain this value, only the constant C in the cross-section has to be matched.If we now assume that the WIMP species has a corresponding anti-particle and is symmetric,the obtained hypothetical cross-section of a Cold Dark Matter particle is

〈σA|v|〉 ≈ 8 · 10−37cm2. (3.99)

This cross-section is on the weak scale and it becomes clear as to why a heavy neutrino is ananticipated candidate for Cold Dark Matter.

Consider such a hypothetical heavy but stable neutrino species ν as a Cold Dark Mattercandidate with mν MeV that decouples after having become non-relativistic. The speciesannihilates through Z0-boson exchange to particle-antiparticle pairs i i, where i stands for νlight,e, µ, τ , u, d, s,. . . . The annihilation cross-section is different for Dirac and Majorana neutrinos.In the temperature regime T ≤ mν ≤MZ the approximation of the Fermi-coupling is valid withFermi’s constant GF . Then the cross-sections corresponding to the two cases are

〈σA|v|〉Dirac =G2Fm

2ν

2π

mi<mν∑i

(1− z2i )1/2 ·

[(C2

Vi + C2Ai)(1 +

1

2z2i )

], (3.100)

〈σA|v|〉Majorana =G2Fm

2ν

2π

mi<mν∑i

(1− z2i )1/2 ·

[(C2

Vi + C2Ai)8β

2i /3 + C2

Ai2z2i

], (3.101)

with zi = mi/mν , β = v/c, v the average velocity, CA = j3 and CV = j3 − 2q sin2 θW , wherej3 is the weak isospin, θW = mW /mZ the Weinberg angle and q the electric charge (Kolb andTurner, 1990, Ch. 5).

For Dirac neutrinos the sum in (3.100) is a constant ≡ c1, so we have s-wave annihilationand the cross section is

σ0 ≈ c1G2Fm

2ν

2π≈ 5

G2Fm

2ν

2π. (3.102)

Assuming gν = 2 and g∗ = 60 after the species becomes non-relativistic we can once more obtainvalues for xf , Y∞ν and Ωνh

2:

xf ≈ lnλa ≈ ln

(0.264 · 0.289 · 2√

60mPl

G2Fm

3ν

2πc2

)≈ 15 + 3 ln(mν/GeV) + ln(c1/5), (3.103)

Y∞ ≈ 6 · 10−9( m

GeV

)−3[1 +

3 ln(mν/GeV)

15+

ln(c1/5)

15

], (3.104)

Ωννh2 = 2Ωνh

2 = 3(mν/GeV)−2

[1 +

3 ln(mν/GeV)

15

]. (3.105)

Already included in this is the contribution of the anti-neutrinos ν. With the boundary ofΩννh

2 ≤ 1 or equivalently Ωνh2 ≤ 1

2 we obtain a mass limit of

m & 2GeV, (3.106)

which is also considered as the Lee-Weinberg bound (Kolb and Turner, 1990).In case of Majorana-neutrinos we have to consider both s- and p-wave annihilation as 〈σA|v|〉

contains a velocity-dependent term. As v depends on the temperature (or x) we can give thefollowing parametrization of the cross section analogously to (3.82):

〈σA|v|〉 ≡ c1G2Fm

2ν

2π(1 + c2x

−2) (3.107)


Then using (3.84) the point of freeze-out is

xf ≈ 15 + 3 ln(mν/GeV) + ln(c1/5) + ln

(1 +

c2

(15 + 3 ln(mν/GeV))2

). (3.108)

The last term can be neglected, so with (3.83) the final abundance is

Y∞ ≈ 6 · 10−9( m

GeV

)−3[1 +

3 ln(mν/GeV)

15

]·(

1 +c2

3[15 + 3 ln(mν/GeV)]−2

)−1, (3.109)

and the density contribution is then given by

Ωννh2 = 3(mν/GeV)−2

[1 +

3 ln(mν/GeV)

15

]·(

1 +c2

3[15 + 3 ln(mν/GeV)]−2

)−1. (3.110)

The last factor is slowly growing while the factor ∼ m−3ν is going to zero with increasing mass.

For c2 of order unity the last factor is ≈ 1 so the expression (3.110) again simplifies to (3.105)and we obtain the same mass bound (3.106).

We have now seen in detail how a symmetric Universe evolves. Until now we assumedthe abundance of a particle to be constant in a comoving volume after freeze-out. We alsoonly included binary collisions in our formalism and assumed ψ to be a stable particle. Beforeproceeding to the investigation of asymmetric particle species we want to treat the evolution ofan unstable particle and in which way the decaying affects the entropy.

3.5 Out of Equilibrium Decay and Entropy Production

For a decaying particle species ψ some interesting features occur. The decay becomes relevantafter the particle species becomes non-relativistic because the inverse reactions of the decayare still effective when it is still relativistic. Furthermore, the decay interaction rate obviouslycannot freeze out as it does not depend on a collision partner.

Assume that all other particles except of ψ are still relativistic and have interactions thatquickly (with a higher rate than H) establish thermal equilibrium with radiation. Then, duringthe decays, energy is transferred to radiation and entropy is produced which leads to a ‘re-heating’ of the Universe. Actually the Universe is not heated, but the temperature and so theradiation energy density decreases less steeply due to the additional energy received from thedecays.

Consider an already non-relativistic species that is relatively long-lived compared to the ageof the Universe (age at that time). At its decoupling is has an abundance of

Yψ(xf ) =nψ(xf )

s. (3.111)

It will be shown that after decoupling ρψ goes as

ρψ ∼ R−3. (3.112)

If there are no decays, the radiation energy density follows

ρR ∼ R−4, (3.113)

such that:ρψρR∼ R. (3.114)

3.5 Out of Equilibrium Decay and Entropy Production 47

Due to this relation ψ will dominate the Universe’s density at some time, if the species has arelatively long lifetime as assumed.

Further, the evolution of the species will follow a simple exponential decay law. We startfrom the rate equation: (Kolb and Turner, 1990)

d(ρψR3)

dt= −1

τ(ρψR

3). (3.115)

As ψ is already non-relativistic ρψ is a mass density. Thus, Eq. (3.115) tells us that the changeof the mass in a comoving volume only proceeds via decays and we obtain

ρψ + 3Hρψ = −ρψτ, (3.116)

whose solution is

ρψ(R) = ρψ(Ri)

(RiR

)3

exp

(− tτ

)≡ ρψi

(RiR

)3

exp

(− tτ

). (3.117)

3.5.1 Calculating the Increase of Entropy during Re-heating

The increase of entropy caused the decays will now be calculated. Taking the 2nd law owthermodynamics and inserting the rate equation (3.115) becomes

dS =dQ

T= −

d(R3ρψ)

T=R3ρψT

dt

τ(3.118)

⇔ dS

dt=R3ρψT

1

τ. (3.119)

From (2.147) it is known that

S = sR3 =π2

45g∗ST

3R3. (3.120)

Assuming that all other particle species, except ψ, are still relativistic and coupled to radiation,it is a good assumption that

g∗S ≈ g∗ = const. = g∗(x > 1) ≡ g∗. (3.121)

Hence, it can be written

S13 S =

(2π2g∗

45

) 13

R4 ρψτ. (3.122)

This differential equation has the solution(S

43f − S

43i

)=

4

3ρψiR

4i

∫ t

ti

1

τ

(2π2g∗

45

) 13(R(t′)

Ri

)exp(−t′/τ)dt′, (3.123)

with the approximately final value Sf and the initial values Si, ρψi and Ri. By rearranging someterms we obtain the relative increase of entropy:

SfSi

=

1 +4

3

ρψiR4i

S43i

· I

34

=

1 +4

3

mψnψi

s43i

· I

34

=

1 +4

3

mψYψi

s13i

· I

34

=

[1 +

4

3

(45

2π2g∗(Ti)

) 13 mψYψi

Ti· I

] 34

, (3.124)


where we used (2.151), (3.27) and (3.120). I is the integral:

I =

∫ t

ti

1

τ

(2π2g∗

45

) 13(R(t′)

Ri

)exp(−t′/τ)dt′. (3.125)

Now, the increase of entropy can be calculated. Nevertheless, let us first make an analyticalattempt and take a look at the evolution of the radiation energy density.

The first law of thermodynamics (1.16) with an additional term of heat exchange (1.17) dueto the decays of ψ takes the following form:

d(R3ρR) = −pRd(R3) + d(R3ρψ). (3.126)

If we use the equation of state for radiation (2.146) and Eq. (3.115) we obtain

d(R3ρR) = −ρR3d(R3) + (R3ρψ)

dt

τ, (3.127)

which can be recast as

ρR + 4HρR =1

τρψ =

ρψiτ

(RiR

)3

exp

(− tτ

). (3.128)

This equation has the following solution:

ρR =

(RiR

)4 [ρRi +

ρψiτ

∫ t

ti

R(t′)

Riexp(−t′/τ)dt′

], (3.129)

where the integral is of the same type as in (3.125). As already mentioned, in the absence ofdecays the radiation energy density goes as ρR ∼ R−4.

Together with the Friedmann equation (1.12) (with ρψ+ρR dominant) the equations (3.116),(3.122) and (3.128) provide a closed set of equations to calculate the evolution of R, S, ρψ andρR.

Let us figure out in which case a significant amount of entropy is produced. Until nowwe assumed a long-lived particle species. Consider a species ψshort with a very short meanlifetime and assume it immediately decays after becoming non-relativistic. As its equilibriumabundance Yψshort

is close to the equilibrium abundance Y eqψshort

when it is still relativistic, ithas the same abundance as the radiation and the rest of the Universe at that time. Then, atthe point of transition to the non-relativistic regime it is still very close to Y eq

ψshort. If ψshort

decays immediately, the energy per decay that is transferred to radiation is close to the presenttemperature. Therefore the amount of heating and produced entropy is very little and practicallynegligible.

On the other hand, as the equilibrium abundance of the radiation decreases with ∼ R−4, dueto (3.114) a sufficiently long-lived species ψ will dominate the energy density in the Universe atsome point. For simplicity we assume that the decay of all ψ-particles, that decay until the meanlifetime τ after having become non-relativistic, happens simultaneously at t = τ . Furthermore,ψ should become the dominant constituent of the Universe at some time tψ < τ . During thisperiod of time the Universe is matter dominated, such that R ∼ t2/3 (see Eq. 1.27). As we canneglect the exponential function in (3.117) for t < τ , we obtain

ρψ ≈ ρψi(RiR

)3

= ρψi

(tit

)2

, (3.130)


where we used the relation (1.25). Thus Eq. (3.128) becomes

ρR + 4HρR =1

τρψi

(tit

)2

, (3.131)

an inhomogeneous linear differential equation, whose solution is

ρR = ρRi

(RiR

)4

+5

3

ρψiτ

(t2it−t11/3i

t8/3

)≈ ρRi

(RiR

)4

+5

3

ρψiτ

t2it

(3.132)

as the second term ∼ t−8/3 in the bracket is much smaller than the first term ∼ t−1. From(3.132) we can see that for t < τ , after ψ having become non-relativistic, the radiation energydensity goes as ρR ∼ R−4 ∼ t−8/3 until the second term accounting for the decays becomesdominant: ρR ∼ t−1. In the latter period we have a growth of entropy. With (2.145) and(3.120) the relation between ρR and the entropy in a comoving colume S is

ρR =3

4

(45

2π2g∗

)1/3

S4/3R−4. (3.133)

Reordering this, we obtain for the growth of entropy:

S ∼ R3ρ3/4R ∼ R15/8 ∼ t5/4, (3.134)

using (1.27) and (3.132).Let us now calculate the relative entropy increase. To solve the integral in (3.125) we have to

make the assumption that ψ will dominate the energy density over the intervall of integration,so we can assume the Universe to be matter-dominated. Using the relation (1.25) and choosingw = 0 for the epoch of matter domination, the integral will become

I =1

τ

(2π2g∗

45

) 13∫ τ

0

(t

ti

) 23

exp(−t/τ)dt, (3.135)

choosing ti = 0 and integrating τ as we assume the decays to happen simultaneously at τ . Withthe relation (1.30) and the Friedmann equation (1.12) we obtain

ti =2

3

1

H(ti)=

2

3

(8πGρψi

3

)− 12

. (3.136)

Hence, the integral becomes

I =1

τ

(2π2g∗

45

) 13(

3

2

) 23(

8πGρψi3

) 13∫ τ

0t23 exp(−t/τ)dt, (3.137)

and if we now expand the exponential function to

exp

(− tτ

)≈ 1− t

τ, (3.138)

which is a valid approximation in the integration interval. The integration becomes trivial andwe obtain:

I ≈ 0.29

(8πGρψi

3

) 13

τ23

(2π2

45

) 13

g13∗ . (3.139)


100 101 102 103 104 105 106 107 108 109

RRi

−80

−70

−60

−50

−40

−30

−20

−10

0

ln(

ρρ ψ

(Ri))

0

1

2

3

4

5

ln(S S i

)

ρψ without decayρψ with decayρR without decayρR with decayln(S/Si) for τ ⋅Hi=108

ln(S/Si) for τ ⋅Hi=109

ln(S/Si) for τ ⋅Hi=1012

Figure 4: Re-heating caused by a decaying particle species (based on Kolb and Turner (1990, Chap. 5), producedby myself (see Appendix D.2)): The numerical solution of the relative increase of entropy produced by the decaysof an unstable particle species ψ after it has become non-relativistic is plotted for different life times relative tothe Hubble time (expressed as τHi). The relative increase grows with longer lifetimes. Furthermore the energydensities of radiation and the species ψ are plotted for τHi = 1012, where we see the delayed decrease of theradiation energy denstiy due to transferred energy of the decaying ψ-particles.

Inserting that into (3.124) and using ρψi = mψYψi we finally obtain the relative increase ofentropy:

SfSi

=

[1 + 0.60 g

13∗

(τ

mPl

) 23

m43ψY

43ψi

] 34

. (3.140)

For sufficiently large τ the first term 1 in (3.140) can be neglected, such that

SfSi≈ 0.68 · g1/4

∗mψYψiτ

1/2

m1/2Pl

. (3.141)

It is to note that the relative increase given in (3.140) goes to 1 while τ goes to zero which fitswith our argument that the amount of produced energy and entropy is negligible for short-livedparticle species because the radiation energy density is dominant before the decoupling of thespecies. The number of degrees of freedom g∗ is constant when no other species becomes non-relativistic. When this condition is not met, g∗ has to be thermally averaged over the integrationinterval. We assumed the decays until τ all to happen at τ , but in reality the decays of course willhappen continuously following a decay law. In this case decays at earlier times will contributeless to the relative increase of entropy than later decays.

Let us now calculate the increase of entropy numerically. Using the relation between the


scale factor and time (1.25) we can rewrite the integral in (3.125) to

I =2

3

tiτ

(2π2g∗

45

) 13∫ R

1

(R′

Ri

) 32

exp

(−(R′

Ri

) 32 tiτ

)dR′

=1

τHi

(2π2g∗

45

) 13∫ R

1

(R′

Ri

) 32

exp

(−2

3

(R′

Ri

) 32 1

τHi

)dR′, (3.142)

where we assumed that when ψ becomes non-relativistic, the age of the Universe ti is the Hubbletime. Thus, with (1.30) and (3.35) we have

ti ≡2

3

1

H(ti)≡ 2

3

1

Hi=

2

3

1

H(mψ). (3.143)

Inserting (3.142) into Eq. (3.140) the relative increase of entropy becomes

S

Si=

[1 +

4

3

mψYψiTi

1

τH(mψ)

∫ R

1

(R′

Ri

) 32

exp

(−2

3

(R′

Ri

) 32 1

τH(mψ)

)dR′

] 34

. (3.144)

Fig. 4 shows the numerical solution to (3.144) for different life times τ . A numerical Newton-Cotes integration method was used, the Simpson method (see Press et al. (2007)). Also, theevolution of the energy density of radiation and the species ψ has been plotted using (Eq. (3.117)and (3.129)). The following initial values have been assumed in the numerical computation:

mψ

Ti= 1, (3.145)

Yψi = 3.2 · 10−5, (3.146)ρψiρRi

= 10. (3.147)

In Fig. 4 we are able to see how the relative increase of entropy grows with longer life times τ .Furthermore, from the curve of the radiation energy density we see that the ’re-heating’ causedby the decays is no real heating, but the energy density decreases slightly slower when additionalradiation energy is released by the decays.

For the value τ ·Hi = 4 · 107 we numerically calculated an increase by a factor of 2. Let uscompare the numerical result (3.150) to the calculated analytic approximation. With (3.35) wehave

τHi = τH(mψ) = 4 · 107 ⇔ τ

mP l=

4 · 107

1.67

1

m2ψ

√g∗. (3.148)

Inserting this into the analytic approximation (3.140) we obtain:

SfSi

= 1.038. (3.149)

This value is very far off the numerical result, so the approximation is not very good for tooshort lifetimes τ .

Consider the numerically calculated increase for τ ·Hi = 1012, which we see in Fig. 1.4.4:

ln(Sf/Si) ≈ 5.5⇒ Sf/Si = exp(5.5) ≈ 244.7 . (3.150)

For the same initial values the analytic approximation (3.140) gives:

Sf/Si = 17.2. (3.151)


This value is also far off the numerical result. For higher life times the approximation becomesmore accurate.

The freeze-out calculations for a symmetric species is now complete. We used constraintsfrom the FLRW model and constructed an evolution equation for the abundance of a particlespecies by using the Boltzmann equation. We performed numerical calculations and analyticapproximations. The evolution of hot and cold relics is found to be a completely different one.While the abundance of a hot relic only depends on the number of effective degrees of freedomand its density contribution grows with higher cross-sections, the abundance of a cold relic is thehigher the weaker the interaction. Furthermore the density contribution of a hot relic is mass-dependent while the density contribution of a cold relic is not. We adapted these constraintsto a hypothetical Cold Dark Matter WIMP species and using the constraint of a flat Universeobtained a bound for the cross-section of a symmetric WIMP in the order of 〈σA|v|〉 ∼ 10−37 cm2.At last, for completeness we investigated the evolution of an unstable particle and the re-heatingit causes by the transfer of the decay energy to radiation.

In the following chapter we include the asymmetric case in our description and see how theobtained constraints from the symmetric case will change.

53

4 Evolution of Asymmetries in the FLRW Universe

In the case that the description seems to have a lack of references, note that the discussion wasmade by own motivation and own trains of thought, where no reference is given.

Until now a particle-anti-particle asymmetry between the ψ and ψ has not been consideredand it is clear that we have to introduce some extensions to our formalism. In chapter 2 wedid not make any assumptions on the symmetry. The first time the assumption that there isno asymmetry was made in Sec. 3.2, where the annihilation cross-section (3.25) for the reaction(3.1) has been defined for a symmetric species ψ. From that we constructed the evolution equa-tion (3.38) for a symmetric species. It is now straight-forward to see that, if we do not assumethe symmetry, nψ = nψ and Yψ = Yψ, the evolution equation will become

Y ′ψ = −sx〈σA|v|〉H(mψ)

[YψYψ − Y

eqψ Y eq

ψ

]. (4.1)

It is not trivial that s and 〈σA|v|〉 are the same for the asymmetric case, so our aim is nowto show that these quantities are independent of an asymmetry of the ψ-species and to find aquantity as a measure for the asymmetry. With this measure we want to investigate how theevolution equations of ψ and ψ will differ.

4.1 The Meaning of the Chemical Potential

In previous chapters and the description of symmetric particle species we deduced the constraint(2.139) allowing only negligibly small µψ compared to mψ or negative muψ. To find the evolutionequations for ψ and ψ it is important to understand how the asymmetry is directly connectedto the chemical potential.

Let us now make the assumption that the annihilation products X and X of the reaction(3.1) have negligible chemical potentials compared to the chemical potentials of ψ and ψ:

µX = µX ≡ 0. (4.2)

Due to the conservation of the chemical potential in chemical equilibrium (see (3.22)) it has tohold:

µψ = −µψ ≡ µ. (4.3)

Here, we defined the absolute value of the chemical potential of ψ and ψ by µ. It is also possibleto find the relation (4.3) without assuming (4.2). Instead it can be assumed that the endproducts of the annihilation cascade of ψ (and X) are photons. Consider such an annihilationcascade:

X + X ←→ · · · ←→ i · γ. (4.4)

From observations we know that the photon number is not conserved in reactions in generaland it is possible that i photons can be produced in a reaction (Baierlein, 2001) with a notfixed integer i. Consider now that X directly annihilates to a different number of photons, forexample via the two channels:

A) : X + X ←→ γ + γ, (4.5)

B) : X + X ←→ γ + γ + γ. (4.6)

Further, we know that µ/T is a collisional invariant, so due to Eq. (3.22) the following has tohold for the respective channels in chemical equilibrium:

A) : µX + µX = 2µγ , (4.7)

B) : µX + µX = 3µγ . (4.8)

54 4 EVOLUTION OF ASYMMETRIES IN THE FLRW UNIVERSE

And this implies that

2µγ = 3µγ ⇔ µγ = 0. (4.9)

This gives the first hint that the chemical potential or, more precisely, the collisional invariantµ/T is connected to the conservation of particle number or at least the conservation of somequantum number that a particle carries.

That the chemical potential of photons is zero has the consequence that

µX + µX = 0⇔ µX = −µX , (4.10)

(Kolb and Turner, 1990). Consequently, infinite negative potentials that have been allowed bythe constraint (2.139) are now excluded. So (2.139) is tightened in the following way: In everyannihilation cascade having an unfixed number of photons as final products, it has to hold:

µj = −µj , (4.11)

|µj | = |µj | mj ∀ T <∞, T 6= 0, (4.12)

for all annihilating particles j. These relations also apply to ψ and X. Thus, we found (4.3)without assuming zero chemical potential of X and X. Regardless, we assume µX = µX = 0anyway and explain the motivation of this assumption at the end of the next section.

The connection between the chemical potential and the the conservation of some quantumnumber can now be found by considering the difference of the number densities of ψ and ψ. Withthe definition of the equilibrium number density in (2.116) and with the distribution function(2.135) we obtain (Kolb and Turner, 1990, Chap. 3):

nψ − nψ =gψ

(2π)3

∫ ∞ψ

dEψEψ(E2ψ −m2

ψ)12

1

exp(Eψ−µT

)± 1− 1

exp(Eψ+µT

)± 1

, (4.13)

where we already used the absolute value of the chemical potential of ψ in (4.3) and assumedthat the chemical potential of ψ is positive and for ψ negative.

The integral above has no general analytic solution. Only in the relativistic and non-relativistic cases we get approximations (Kolb and Turner, 1990, Chap. 3):

nψ − nψ =gψT

3

6π2

(π2(µT

)+(µT

)3), (T mψ), (4.14)

nψ − nψ = 2gψ

(mψT

2π

) 32

exp(−mψ

T

)sinh

(µT

), (T mψ). (4.15)

In both cases the difference can only be zero if the collisional invariant µ/T is zero. This is onlypossible if µ = 0 for all temperatures. So a symmetric species in thermodynamic equilibriumwill always have zero chemical potential as already mentioned in Sec. 3.2. The magnitude ofan asymmetry of ψ today directly depends on the collisional invariant µ/T as a once fixed µ/Tremains constant. Thus we have found a measure for the particle-anti-particle asymmetry.

Let us now reconsider the cross-section in the evolution equation (4.1). Analogously to itsdefinition in Eq. (3.25) we can define it for the asymmetric case (we already assume a low-temperature regime in which quantum effects can be neglected):

〈σA|v|〉 ≡r2X

neqψ n

eq

ψ

∫dΠψdΠψdΠXdΠX(2π)4δ


)|Mψψ↔XX |2f

eqψ f

eq

ψ. (4.16)

4.2 The Evolution Equation of an Asymmetric Species 55

Further, when the temperature is much lower than mψ, the product of the distribution functionsin (4.16) are Maxwell-Boltzmann distributions (2.141) (in the local frame of the gas), such that

f eqψ f

eq

ψ= rψ exp

(−

(Eψ − µ)

T

)rψ exp

(−

(Eψ + µ)

T

)= r2

ψ exp

(−

(Eψ + Eψ)

T

)exp

(µ− µT

)= r2

ψ exp

(−

(Eψ + Eψ)

T

). (4.17)

Similarly for the product of the number densities in the denominator in (4.16) with the definition(2.54) we have

neqψ n

eq

ψ=

(∫fψdp

3ψ

)·(∫

fψdp3ψ

)= r2

ψ exp

(µ− µT

)(∫exp

(−EψT

)dp3

ψ

)·(∫

exp

(−EψT

)dp3

ψ

). (4.18)

Thus, we see that the defined cross-section is independent of the chemical potential and thereforeindependent of a particle-anti-particle asymmetry.

As the species X is still relativistic when ψ becomes non-relativstic, the contribution of ψto the entropy density is negligible, such that the total entropy density is given by Eq. (2.147).Note that to find the expression (2.147) we assumed T µ for all chemical potentials of thelighter relativistic particles. For now we investigate the evolution of ψ with non-zero chemicalpotential and neglect all other chemical potentials of particles that are lighter than ψ. Hence,s is also independent of µ and the evolution equation (4.1) is valid for an asymmetric particlespecies after it has become non-relativistic.

4.2 The Evolution Equation of an Asymmetric Species

The next step is to show in which way the evolution of the abundance of the particle ψ will differto the evolution of the abundance of the anti-particle ψ. The evolution equation (4.1) containsthe independent variables Yψ and Yψ. A connecting relation between these variables is required.This relation is clearly given by the asymmetry. Let us define the quantum number D servingas measure of the asymmetry:

D ≡ Yψ − Yψ =nψ − nψ

s. (4.19)

D is conserved in a comoving volume, as the only considered interaction of ψ is (3.1), whichdoes not change the net particle number nψ−nψ. D still could be conserved if some interactionwould violate the net particle number and the products of such a reaction would carry on thequantum number, but these kinds of interactions are not discussed in this thesis.

Further, consider the equilibrium values for the two limits in Eq. (3.41) and (3.42). In theasymmetric case with µ 6= 0 they become

Y eqψ (x) = Y eq

ψ(x) = 0.278

geffψ

g∗S(x 3, ψ relativistic) (4.20)

Y ψ eq

ψ(x) = 0.298

gψg∗S

x32 exp (−x± α) (x 3, ψ non-rel.) (4.21)

with α = µ/T being constant. The upper sign in the exponential function stands for ψ andthe lower for ψ. Note that the relativistic expression (4.20) is given for T µ while (4.14) isdeduced for arbitrary µ. So for T µ, Eq. (4.20) is independent of µ.


In the non-relativistic limit the product Y eqψ Y eq

ψis also independent of µ:

Y eqψ Y eq

ψ=

(0.145

gψg∗S

)x3 exp (−2x) ≡ (Y eq

sym)2. (4.22)

Y eqsym is given by (3.42) and is the equilibrium abundance in absence of an asymmetry.

Including the relation (4.19) for the asymmetry in the evolution equation (4.1) we can rewritethe product YψYψ by expressing Yψ in terms of D and Yψ:

YψYψ = Yψ(Yψ −

(Yψ − Yψ

))= Y 2

ψ −DYψ. (4.23)

Similarly we can express Yψ in terms of Yψ and D, such that:

YψYψ =(Yψ +

(Yψ − Yψ

))Yψ = Y 2

ψ +DYψ. (4.24)

Hence, we obtain two evolution equations for ψ and ψ:

Y ′ψ = −λx−n−2[Y 2ψ −DYψ − (Y eq

sym)2]

(4.25)

Y ′ψ = −λx−n−2[Y 2ψ +DYψ − (Y eq

sym)2]. (4.26)

Here we used λ given by (3.64), which is independent of the asymmetry. Before we can solvethese equations similarly to the way we did in Ch. 3 we want to have an expression for D interms of µ.

Using the Eq. (4.14) and (4.15) in the respective limits we have:

D = 0.242gψg∗S

(π2(µT

)+(µT

)3)

(T mψ) (4.27)

D = 0.58gψg∗S

x32 exp(−x) sinh

(µT

)(T mψ). (4.28)

D has to be constant, but the expression (4.28) is not constant due to the exponential factor,which is a consequence of the approximation as D is only given exactly by solving (4.13). As µ/Tis conserved, Eq. (4.27) is at least constant when g∗S is constant. Consequently the expression(4.27) for D is constant when fixing

g∗S ≡ g∗S(x 1) ≡ g∗0. (4.29)

Thus, we can use (4.27) as the relation between the D and µ. Even if the expression (4.27) isderived for T mψ we can use it also in the non-relativistic regime as D remains constant.

Having found that a non-zero chemical potential is directly connected to a particle-anti-particle asymmetry we can now explain the motivation of the assumption (4.2). We assumedthat ψ carries the quantum number D which is defined to be conserved. If the species ψ hasa non-zero chemical potential a further interaction must have been present at some time. Thisinteraction must have been responsible for the production of the asymmetry (Kolb and Turner,1990, Ch. 6) as already mentioned in the Sec. 1.3. In this thesis the formalism of asymmetryproduction will not be discussed. We want to keep this lack of knowledge concentrated in theproperties of ψ and not distribute it to the species X. So we assume that the species X is onlyproduced by the reaction (3.1) and all other species are part of the annihilation cascade of X.Hence, the chemical potentials of all particles except of ψ can be set to zero and do not have aneffect on the total entropy density.

The two obtained evolution equations (4.25) and (4.26) for ψ and ψ take into account theparticle-anti-particle asymmetry. Before we can proceed to calculating relic abundances of anasymmetric species, it is necessary to mention the case of very high chemical potentials whichare connected to the phenomena of Bose-Einstein condensation and degenerate fermions. Wewill see that the magnitude of the chemical potential has an influence on the freeze-out scenario.

4.3 High Chemical Potentials and Quantum Effects 57

4.3 High Chemical Potentials and Quantum Effects

Until now our complete decription of freeze-out was based on the constraint |µ| mψ. Thisconstraint was obtained in Sec. 2.5.3 by the assumption (2.44), such that f < r and f does notdiverge. In fact the strictness of this assumption has not been checked. Let us now drop theconstraint (2.44) and allow chemical potentials,

|µ| > mψ, (4.30)

and investigate the form of the distribution function f for different cases of bosonic and fermionicspecies. Remember that we do not drop the constraint (4.3) by dropping (4.12) because (4.12)was concluded by (4.3) and not the other way round.

Consider the Bose-Einstein distribution and a bosonic species:

f =r

exp(ET −

µT

)− 1

. (4.31)

The lowest energy state a particle in general can occupy is E = m. If the bosonic species ismassive and µ < m, then f will not diverge. But if we have µ ≥ m we always have divergence atE = µ, i.e. E = µ is the ground state and the states E < µ cannot be occupied. The divergencedoes not have to be present at all temperatures. Suppose that µ = m at a temperature T = m.As the temperature varies and µ/T is always constant µ has to vary synchronously to T , whichmeans:

µ ≥ m for T ≥ m −→ f divergent, (4.32)

µ < m for T < m −→ f non-divergent. (4.33)

Even if we make µ arbitrarily small, f would diverge for a sufficiently high temperature. Thisexplains, why we assumed the µ to be negligible compared to the mass while T is not too high.

The divergence marks the transition to a Bose-Einstein condensate (Kolb and Turner, 1990,Chap. 3), which simply means that it is likely that a limitless amount of bosons coherentlyoccupies the same state. We will not focus on the description of this phenomenon in this thesis.

Furthermore for negative chemical potentials we also do not have a divergence above theground state. The higher the magnitude |µ| of the negative chemical potential will be, the lowerthe occupation number will be for the ground state:

f(E = m) =r

exp(m−(−µ)

T

)− 1

µ→−∞−−−−−→ 0. (4.34)

This behaviour is also shown by a massless bosonic species with negative chemical potential.A fermionic species in kinetic equilbrium follows the Fermi-Dirac distribution:

f =r

exp(ET −

µT

)+ 1

. (4.35)

This function has no divergences for any combination of µ and m. The ground states for differentpositive chemical potentials are

f(E = m) .r

2(µ < m), (4.36)

f(E = m) ≈ r

2(µ = m), (4.37)

f(E = m)µ→∞−−−→ r (µ > m). (4.38)


For the last case of chemical potentials higher than the mass we have a degenerate Fermi-gas(Kolb and Turner, 1990, Chap. 3). Depending on the magnitude of the chemical potential thegroundstate is occupied with a probability of asymptotically g. Due to Pauli-blocking otherparticles are forced into higher energy states. Reconsider the case of µ = m at T = m. Withµ/T being constant we obtain:

µ ≥ m for T ≥ m −→ f degenerate, (4.39)

µ < m for T < m −→ f non-degenerate, (4.40)

with degenerate meaning, that the lowest energy states are occupied with a probability ofapproximately g. As µ/T is always constant µ decreases with decresing T . At some point therewill always be a temperature at which the chemical potential will be equal to the particle’s mass.When µ then drops below m, the gas transits from a degenerate state to a non-degenerate state.It is now important to distinguish at which temperature the equality µ = m will be reached.Connected to this, three cases of degeneracy can be distinguished:

1 :|µ|T

< 1 gas becomes non-degenerate when still relativistic: T > m, (4.41)

2 :|µ|T

= 1 gas becomes non-degenerate and non-relativistic simultaneously: T = m, (4.42)

3 :|µ|T

> 1 gas becomes non-degenerate when already non-realtivistic: T < m. (4.43)

In the next section we will see how the freeze-out scenarios in these three cases differ from thesymmetric case.

That way we see the different shapes of the distribution function and the correspondingphenomena when dropping the constraint (4.12).

Further, the expressions of the thermodynamic quantities in the relativistic limit (2.144)-(2.147) were defined for high temperatures compared to mass and negligible chemical potentials(corresponding to case 1 in (4.41)) while the non-relativistic expressions have been deducedfor arbitrary µ. We can now find different approximations of the integrals (2.116)-(2.118) and(2.120) assuming |µ| > T > mψ (case 3 in (4.43)).

For a fermionic species ψ the relativistic limits of thermodynamic quantities are given in thefollowing for a case-3-chemical potential (4.43) (using (2.116), (2.117), (2.118) and (1.55)). If ψhas µψ > 0, then (Kolb and Turner, 1990, Chap. 3):

nψ =1

6π2gψµ

3ψ (µψ T mψ), (4.44)

ρψ =1

8π2gψµ

3ψ, (4.45)

pψ =1

8π2gψµ

3ψ, (4.46)

sψ =ρψ + pψ − µψnψ

T= 0. (4.47)

The contribution to the entropy density sψ is zero for a highly degenerate species.As a bosonic species would form a Bose-Einstein condensate for µ > T > mψ and f would

diverge, the description of the bosonic system is obviously not possible as the integral is notdefined. Therefore a Bose-Einstein condensate has to be treated seperately, which will not befurther investigated.

Eq. (4.3) showed that the absolute value of the chemical potentials of ψ and ψ must beequal. Therefore, the thermodynamic quantities of ψ are given by the non-relativistic limits

4.4 Calculating Relic Abundances of an Asymmetric Species 59

of the integrals (2.116), (2.117), (2.118) and (1.55) with high magnitudes of negative chemicalpotentials (still case 3 in (4.43)) (Kolb and Turner, 1990, Chap. 3):

nψ = exp(µψT

) gψπ2T 3 = exp

(−µT

) g

π2T 3, (4.48)

ρψ = exp(−µT

) 3g

π2T 4, (4.49)

pψ = exp(−µT

) g

π2T 4, (4.50)

sψ =ρψ + pψ + µnψ

T=(

4 +µ

T

) gψπ2T 3 exp

(−µT

)µT−−−→ 0, (4.51)

with µ = µψ = −µψ. Here, the contribution to the entropy density sψ is suppressed by theexponential factor also negligible compared to the total entropy density. This also holds true forthe other quantities, especially for the number density. For higher magnitudes of the chemicalpotential (case 3) the number density of anti-particles is suppressed. Note that for bosons withµ < m including negative µ the thermodynamic quantities are also given by (4.48)-(4.51).

We now have seen the three cases in which the magnitude of the chemical potential can beclassified. The next step will be the calculation of relic abundances of an asymmetric fermionicspecies and we will see the influence of the chemical potential on the freeze-out scenario and themagnitude of the particle-anti-particle asymmetry.

4.4 Calculating Relic Abundances of an Asymmetric Species

The calculation of relic abundances in the asymmetric case will now be presented analogously tothe presentation in Sec. 3.4. Remember that we assume all interactions to be rapid compared tothe Hubble rate in the early Universe, such that all species are in thermodynamic equilibrium.At that time the displacement from equilibrium is assumed to be small and all abundancesfollow the equilibrium distributions in kinetic equilibrium. The abundances will follow theseequilibrium laws until the interactions of the particles freeze-out and finally will become

Y∞ψ ≈ Yeqψ (xf ), (4.52)

with the point of freeze-out xf .

In Eq. (3.3) the annihilation interaction rate per particle Γ is defined including neqψ in the

symmetric case. While the cross-section is independent of the asymmetry, neqψ clearly is not. In

fact, in the asymmetric case the interaction rates per particle differ between ψ and ψ becauseneqψ > neq

ψ. As expected, the interaction rate per particle Γψ for ψ is suppressed for higher

magnitudes of chemical potentials. Thus, it earlier drops below H, such that freeze-out happensearlier. On the other hand Γψ is enhanced and the freeze-out of ψ is delayed.

In the following it will be shown how different magnitudes of the chemical potential of ψ willchange the abundance in the two different freeze-out scenarios of hot and cold relics.

4.4.1 Hot Asymmetric Relics

So far we have obtained three different equilibrium laws the abundances of ψ and ψ follow whenthey are still relativistic. These equilibrium laws correspond to the cases 1 in (4.41) and 3in (4.43) and are given by Eq. (4.20), (4.44), (4.48). We still can use the entropy density inEq. (2.147) as the contributions (4.47) and (4.51) to the total entropy density are zero in the


relativistic case and negligible in the non-relativistc case:

Case 1: Y eqψ (x) = Y eq

ψ(x) = 0.278

geffψ

g∗S, (4.53)

Case 3: Y eqψ (x) =

15

2π4

gψg∗S

(µT

)3, (4.54)

Case 3: Y eq

ψ(x) =

45

π4

gψg∗S

exp(−µT

). (4.55)

These three expressions are constant, if g∗S is constant, i.e. if no other species becomes non-relativistic. Assuming g∗S = g∗S(xf ) the three expressions (4.53)-(4.55) state the final abun-dances Y∞ψ and Y∞

ψfor the cases 1 and 3. Note that case 1 corresponds to negligible chemical

potentials and is so a case close to particle symmetry. For the case 2 with |µ|/T ∼ 1 there is noanalytic approximation and the integral (2.116) has to be solved numerically.

Further, we can again use the found expressions for Y∞ψ and Y∞ψ

to find n∞ψ , n∞ψ

, ρ∞ψ , ρ∞ψ

, Ωψ

and Ωψ as we did in Sec. 3.4.1. But as we want to focus on Cold Dark Matter in further, we willskip the more detailed calculations for hot relics and focus on the evolution of cold asymmetricrelics.

4.4.2 Cold Asymmetric Relics

We remember, that the freeze-out of cold relics happens when the species has already becomenon-relativistic. Before decoupling the abundance follows the equilibrium value given in (4.21)and then levels off to remaining constant. Note that Eq. (4.21) has been derived for arbitraryµ/T . The calculation for an asymmetric cold relic is very similar to the symmetric case thatwe calculated in Sec. 3.4.3. There we derived approximated analytic solutions for early and latetimes. In this section we will shortly recapitulate this calculation including the asymmetry.

Let us start from the evolution equations (4.25) and (4.26) and assume s-wave channelannihilation (n = 0). We can express these equations in terms of the departure of equilibriumfor µ = 0:

∆+ = Yψ − Y eqsym, (4.56)

∆− = Yψ − Y eqsym, (4.57)

where Y eqsym is given by (3.42). With these quantities the evolution equations become:

∆′+ = −Y eqψ′ − λx−2

[∆+(2Y eq

sym + ∆+)− YψD]

(4.58)

∆′− = −Y eq

ψ′ − λx−2

[∆−(2Y eq

sym + ∆−) + YψD]. (4.59)

These equations are similar to the equation (3.66) for the symmetric case, only with an additionalterm ∓DY±. From here, the calcualtion is similar to the presentation in Sec. 3.4.3. We willonly briefly sketch the calculation and give the formulae with the additional term and use thenotation:

Yψ = Y+, and Yψ = Y−. (4.60)

At early times when the abundances of ψ and ψ closely follow the equilibrium value wecan neglect ∆± and ∆′±, such that we can approximate:

∆± ≈x2

2λ± Y±D

2Y eqsym + ∆±

. (4.61)


This expression is similar to (3.67). If we now neglect ∆± in the additional term and use

Y± = Y eqsym exp

(±µT

), (4.62)

then the early time solution for the asymmetric case becomes:

∆± ≈x2

2λ± 1

2exp

(±µT

)D (1 < x xf ). (4.63)

At late times the equilibrium value has decreased exponentially, so Y eqsym and Y eq

sym′ are

negligible and ∆± ≈ Y± ≈ Y∞± . Thus the evolution equations simplify to:

∆′± ≈ −λx−2(∆2± ∓ Y±D

)≈ −λx−2

f

((Y∞± )2 ∓ Y∞± D

). (4.64)

This approximation is similar to (3.70). Now integrating from xf to ∞ the late time solutionbecomes:

∆∞± ≈ Y∞± ≈xfλ±D. (4.65)

One would expect an additional term of D/2 to the symmetric case, but instead have an ad-ditional term D. The factor 1

2 must have been lost in the definition of the departures fromequlibrium ∆± (4.56) and (4.57) defined as departures from the symmetric equilibrium law(3.42).

To obtain an expression for the point of freeze-out xf we now infer the Ansatz,

∆±(xf ) = ±c Y eqsym, (4.66)

similar to the Ansatz (3.72). Note that the departure form equlibrium for the anti-particlespecies is negative. Inserting the Ansatz into the calculation for early times, the early timesolution becomes:

∆±(xf ) ≈x2f

(2± c)λ± 1

2exp

(±µT

)D. (4.67)

Now setting this expression equal to the Ansatz (4.66) and using (3.42) we obtain an expressionfor xf :

xf (±) = ln(λa)− ln

(1± λD

2cexp

(±µT

))− 1

2ln(xf ), (4.68)

where for simplicity it has been assumed that (2 ± c)(±c) = 1. Note that the additional termcontaining D is negative for the anti-particle species ψ and positive for ψ. That means, inpresence of an asymmetry the point of freeze-out of ψ is delayed and for ψ it happens earlier aswe expected.

Finally we can give the formulae for the final abundances, number densities and today’sdensity contributions:

Y∞± = 3.79

√g∗

g∗S

xfmψmPlσ0

±D, (4.69)

n∞± = s0Y∞± = 1.13 · 104

√g∗

g∗S

xfmψmPlσ0

cm−3 ± 2970D cm−3, (4.70)

Ω±h2 = 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1 ± 2.82 · 108Dmψ

GeV. (4.71)

The expressions for the anti-particle ψ become negative for a sufficiently large D, which is anerror due to the approximation. For such magnitudes of D we simply expect the final abundanceof ψ to be zero.


1 10 100

x=mT

−20

−15

−10

−5

0

ln(YY(1))

⟨σA|v|⟩ = 10−37cm2⟨ μ=0μT =1μT =1⟩5ln(Yeq

sym/Yeqsym(1))

Figure 5: Numerical solution of the Boltzmann equation for an asymmetric particle species (based onGraesser et al. (2011), produced by myself (see Appendix D.3)): The figure shows the freeze-out of anasymmetric cold relic particle species ψ, which is massive and stable. We see the final abundances of ψand the anti-particle ψ relative to the initial value of the symmetric equilibrium law Y eq

sym at x = 1 fordifferent ratios of the chemical potential to the temperature µ/T . In all cases the same annihilationcross-section was used. The dark blue line shows the evolution of the abundance in absence of anasymmetry (µ/T = 0). The other filled lines show the abundances of the anti-particles ψ and thedashed lines the abundances of ψ. The higher the ratio µ/T the smaller the abundance of the anti-particle will be while the abundance of ψ increases with increasing µ/T . The point of freeze-out,when the abundances levels off from Y eq

sym, happens earlier for the particles while the freeze-out of theanti-particles is delayed in the presence of an asymmetry.

Having obtained the analytic approximations, let us consider the numerical solution to theevolution equations of ψ and ψ (4.25) and (4.26). Fig. 5 shows the evolution for differentmagnitudes of µ/T . A significant difference to the symmetric case emerges for µ/T > 0.5, wherethe final abundances of ψ and ψ differ in one order of magnitude. Furthermore, we clearly seethat the freeze-out of ψ happens the earlier the higher the asymmetry. Note, that there is againan unphysical maximum in the evolution equilibrium abundance which will be tolerated.

4.4.3 Asymmetric WIMPs

We will now investigate, which combinations of D and 〈σA|v|〉 can influence the density contri-bution of relic WIMPs, which we consider as a candidate for Cold Dark Matter.

In Sec. 3.4.5 we estimated an annihilation cross-section of a symmetric WIMP species (see(3.94)). Let us consider the other extreme case for a very high asymmetry where Ωψ becomeszero in our approximation (4.71) and again estimate the WIMP cross-section by assuming thelimit of an accelerated expansion of the Universe,

ΩDMh2 = (Ωψ + Ωψ)h2 . 1, (4.72)

to be satisfied:

Ωψh2 = 0 = 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1 − 2.82 · 108Dmψ

GeV(4.73)

⇔ 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1 = 2.82 · 108Dmψ

GeV, (4.74)


⇒ Ωψh2 = 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1 + 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1 (4.75)

= 2 · 1.07 · 109

√g∗

g∗S

xfmPlσ0

GeV−1. (4.76)

Here, we have assumed xf,ψ ≈ xf,ψ ≡ xf . Compared to the symmetric expression (3.92) Eq.(4.76) differs by a factor of 2, so a first expectation is that 〈σA|v|〉 has to be twice as high as in(3.94) to satisfy (4.72):

〈σA|v|〉 = 2 · C · 10−37cm2 = C · 5.16 · 10−10GeV−2. (4.77)

If we still assume xf ≈ 18 as in (3.95), then the density contribution would become (similarlyto (3.97)):

Ωψh2 =

0.023xfC

≈ 0.40

C. (4.78)

To satisfy (4.72) we obtain:C ≥ 0.4. (4.79)

Let us now assume the extreme case C= 0.4. The asymmetry D is given with (4.74) by:

D = 3.79

√g∗

g∗S

xfmψmPlσ0

= Y∞sym = 2.8 · 10−9 1

C

(GeV

mψ

)= 7.0 · 10−9

(GeV

mψ

)(4.80)

with Y∞sym given in (3.96). With (4.27) this magnitude of D corresponds to:

7.0 · 10−9

(GeV

mψ

)= 0.242

gψg∗S

(π2(µT

)+(µT

)3)

(4.81)

⇔(π2(µT

)+(µT

)3)

= 2.89 · 10−8 g∗Sgψ

(GeV

mψ

). (4.82)

Assuming only very small µ/T , we can approximate:(π2(µT

)+(µT

)3)≈ π2

(µT

)(4.83)

⇒ µ

T≈ 2.92 · 10−9 g∗S

gψ

(GeV

mψ

). (4.84)

Assuming a mass of mψ ∼ 1 GeV and a number of degrees of freedom g∗S = 106.75, Eq. (4.84)becomes:

µ

T≈ 1.56 · 10−7. (4.85)

Inserting this into (4.27) the asymmetry D is:

D = 7.45 · 10−9. (4.86)

This is a rather small value to compensate a doubled cross-section. It is also interesting thatthe density contribution only depends on the mass in presence of an asymmetry (4.71). For thesame magnitude of the asymmetry (4.86), inserting a mass of 2 GeV into (4.71) the cross-sectionhas to be higher by a factor 4.

By that example we see that our calculation is only valid for very small µ/T correspondingto the case 1 (4.41). We took the highest possible value for D, where the expression for theabundance of ψ (4.69) becomes zero and obtained D ∼ 10−9. For higher D the expression(4.69) would become negative and our approximation will not be valid anymore. In principle


one could set Y∞ψ

= 0 for higher values of D, which seems physically reasonable but as we see,

the approximation shows linear behaviour, where solving the Boltzmann equation is a highlynon-linear problem, so for higher D the results would be incorrect.

The above calculated example is a special simplified case, where we could set Ωψh2 = 0

and find 〈σA|v|〉 and D to satisfy (4.72). For lower values of D, the abundance Y∞ψ

has to be

included in the density contribution ΩDMh2. Let us now find a relation between 〈σA|v|〉 and

µ/T at constant ΩDMh2 = 1.

Due to (4.72) we haveΩψh

2 + Ωψh2 = 1. (4.87)

Together with (4.71) we have a set of three equations for four variables: Ωψh2, Ωψh

2, D and〈σA|v|〉. Hence, to find a relation between D and 〈σA|v|〉 we have to fix one of the variables.Let us describe our desired relation in the vicinity of particle-anti-particle symmetry and set

Ωψh2 =

1

2. (4.88)

Then with (4.87) and (4.71) we obtain:

a

〈σA|v|〉+ bD +

1

2= 1, (4.89)

⇔ D =1

2b− a

b〈σA|v|〉, (4.90)

where

a = 1.07 · 109

√g∗

g∗S

xfmPl

GeV−1, (4.91)

b = 2.82 · 108 GeV

mψ. (4.92)

Using (4.27) and the approximation (4.83) the relation between 〈σA|v|〉 and µ/T in the case ofvery small asymmetries becomes:

µ

T= 0.42

g∗Sgψ

(1

2b− a

b〈σA|v|〉

). (4.93)

In Fig. 6 we have plotted this relation. There, we clearly see, that higher cross-sections haveto be compensated by a higher asymmetry and a higher ratio µ/T to obtain the same densitycontribution ΩDMh

2. Cross-sections below the value (3.94) with C = 0.4 are not allowed. Thisis the value found in absence of an asymmetry. Note that the density contribution only dependson the mass in the asymmetric case (4.71), where a higher cross-section can be compensatedif the particles mass is higher. In that case D has not to be that large. Nevertheless, in thesymmetric case the cross-section in (3.94) is the lowest possible cross-section to satisfy (4.72).

Finally, we have seen in which way the properties of a Cold Dark Matter WIMP species haveto change with respect to the symmtric case if we introduce an asymmetry. In this chapter weinvestigated the physical meaning of the chemical potential and saw that the ratio µ/T is directlyconnected to a conserved scalar quantum number according to the notion of collisional invariants.Such a non-zero quantum number can define a particle-anti-particle asymmetry. Furthermore,we investigated the behaviour of the Fermi-Dirac and the Bose-Einstein distribution functionsfor high ratios µ/T . Thereby we defined three cases of the magnitude of the degeneracy for afermionic species. In the case 3 of high ratios µ/T > 1 the final abundance of a hot relic could beincreased by higher ratios. For cold relics the behaviour of freeze-out does not differ for high or


10−10 2×10−10

[⟨σA|v|⟩] =Gev−20⟨00

0⟨2⟩

0⟨⟩0

0⟨7⟩

1⟨00

1⟨2⟩

1⟨⟩0

1⟨7⟩

2⟨00

μ μ

1e−8

ΩDM>1

ΩDM<1

ΩDM=1

Figure 6: Exclusion plot of allowed ratios of the chemical potential to temperature µ/T and allowed annihilationcross-sections 〈σA|v|〉 (produced by myself (see Appendix D.4)): The black line marks the upper limit of allowedµ/T for corresponding annihilation cross-sections, such that the density contribution of the WIMP Dark Matteris ΩDM ≤ 1, where Ω = 1 is the upper bound when demanding accelerated expansion Universe. All combinationsof µ/T and 〈σA|v|〉 above and to the left of the black line are excluded.

low ratios µ/T as the analytic approximations of the thermodynamic quantities have been madefor arbitrary ratios. Moreover, we adapted the approximations from Sec. 3.4.3 to the asymmetriccase and obtained different behaviour of the abundance and the density contribution with respectto the symmetric case. A higher asymmetry demanded the cross-section to be higher to satisfythe constraint of a flat Universe. In the presence of an asymmetry the density contribution ofa cold WIMP relic became mass-dependent and a higher mass then also allowed only highercross-sections.

Hence, the aim of this thesis is achieved and it has been shown, that the constraints on anasymmetric Cold Dark Matter WIMP species noticeably change when introducing a particle-anti-particle asymmetry for the WIMP species.

66 5 DISCUSSION AND CONCLUSION

5 Discussion and Conclusion

The focus of this thesis is the investigation of constraints on an Weakly Interacting MassiveParticle species as a candidate for Cold Dark Matter. These constraints are deduced fromcalculations of the freeze-out behaviour of the species and change with respect to the particle-symmetric case if a particle-anti-particle asymmetry is introduced. First, the relativistic kinetictheory formalism is constructed inferring the Boltzmann equation and its equilibrium solutions.Using the Friedmann-Lemaıtre-Robertson-Walker for a homogeneous and isotropic expandingUniverse the constructed formalism provides an evolution equation to describe the freeze-outof the WIMP’s interactions and to calculate its relic abundance and the density contributionto the total matter density in the Universe. The evolution and the calculated abundance differsignificantly for different particle masses and different annihilation cross-sections of the particlespecies. The freeze-out calculation is first presented for a symmetric WIMP species and thena second time considering a particle-anti-particle asymmetry. For the symmetric case also theevolution of an unstable particle and the corresponding increase of entropy due to the decayhave been discussed for completeness. Comparing the obtained constraints and assuming theWIMP to be asymmetric, the species’ cross-section and/or mass are required to be higher tostay consistent with the observed density of Dark Matter in the Universe.

Requiring the density contribution of the WIMP species to be ΩWIMP . 1 in the symmetriccase the lower bound of a WIMP cross-section is found to be of the order of〈σWIMP|v|〉 & 10−37 cm2 according to the literature value. Moreover the density contributionof the WIMP does not significantly depend on the WIMP mass in absence of an asymmtry.Contrary to that, an analytic approximation for an asymmetric WIMP with mass m ∼ 1 GeVyields the insight that an asymmetry in the order of D ∼ 10−9 corresponding to a ratio of thechemical potential to the temperature of µ/T ∼ 10−7 demands the cross-section to be twice ashigh when demanding the density contribution not to change. In the asymmetric case the densitycontribution is also dependent on the WIMP mass. For the same magnitude of asymmetry anda mass of m ∼ 2 GeV the cross-section has to be higher by a factor of 4 with respect to thebound that has been set in the symmtric case. In any case the cross-section can not be lowerthan this bound of 10−37 cm2.

The presentation of this thesis begins with discussion of relativistic kinetic theory whichis the foundation of freeze-out calculations. The formalism is treated in great detail. Themathematical accuracy of its description in chapter 2 is much higher than in the other chapters.This is absolutely required to fully understand the assumptions made in the calculation offreeze-out and the dependencies on thermodynamic equilibrium and the interaction strength.Furthermore, kinetic theory provides the notion of collisional invariants and is used as thefoundation of the argument of conserved quantities as the considered asymmetry represented bythe conserved quantum number D. Moreover the formalism shows the connection between theasymmetry and a non-zero chemical potential.

The performed freeze-out calculations require solving the Boltzmann equation which hasno general analytic solution. Hence, the analysis in chapters 3 and 4 is more qualitative andmakes use of analytic approximations. These approximations limit the discussion to specialcases and carry an error that has to be regarded in all calculations. The numerical solutionsof the Boltzmann equation show large errors as the Boltzmann equation is very stiff, such thatslightly different choices of the initial value result in a different outcome.

The most important insights of this thesis are that the presence of an asymmetry lifts thelower bound of the WIMP cross-section set in the symmetric case and that the density contri-bution of the asymmetric Cold Dark Matter species becomes mass-dependent. This result wasfound by an analytic approximation that is only valid for low asymmetries below D ∼ 109. In

67

general the dependence between the asymmetry, the cross-section and the relative densities ofthe WIMPs and anti-WIMPs are only given by three equations. One of these four variables isalways undetermined. By a special case of the used approximation the density contribution ofthe anti-particle of the WIMP becomes zero and the mentioned results are found. A furtherworthwhile investigation would be to find a fourth relation to obtain a set of equations of fullrank. Another possible approach could be calculating slight changes of the used approximationand extrapolating the behaviour of the asymmetry with respect to the cross-section and viceversa.

This thesis required putting a lot of effort into understanding the foundation of relativistickinetic theory and the notion of freeze-out. Nevertheless it was worthwhile and intriguing inwhich way a so sophisticated formalism could yield a concise and meaningful result.

68 A APPENDIX: CONVENTIONS

A Appendix: Conventions

Natural Units

The used values in this section are taken from Kolb and Turner (1990).The definition of natural units requires

[Energy] = [Mass] = [Temperature] = [Length]−1 = [Time]−1. (A.1)

Instead it holds

E = m[c2] = T [kB] = k[~] = ω

[~c

], (A.2)

where E is energy, c speed of light, m mass, T temperature, kB the Boltzmann constant, kthe wave numeber, ~ the reduced Planck constant and ω the circular frequency. We can set allconstants in brackets to 1 and then we have the following conversion factors (Kolb and Turner,1990):

1GeV = 1.7827× 10−24g = 1.1605× 1013K (A.3)

1GeV−1 = 1.9733× 10−14cm = 6.5822× 10−25s (A.4)

Notation

The signature of the metric tensor gµν is

sign(gµν) = (+,−,−,−).

The Einstein summation convention is used. For (contra-variant) vectors Aµ partial derivativesare abbreviated by

∂Aµ

∂xν=: Aµ,ν (A.5)

and covariant derivatives by∂Aµ

∂xν+ ΓµνρA

ρ =: Aµ;ν . (A.6)

The covariant derivative for a co-variant vector is

∂Aµ∂xν

− ΓρνµAρ =: Aµ;ν . (A.7)

69

B Appendix: Differential Geometry

In this thesis I assume the reader’s knowledge of the basics of differential geometry, i.e. theknowledge of vector spaces and topological spaces, manifolds and coordinate transformations.The further terminology that we use will be shortly summarized in the following. The presenta-tion of this Appendix Chapter follows the work of Nakahara (2003), where all formula are takenfrom there.

Vectors, Co-vectors and Tensors

In differential geometry a tangent vector on an m-dimensional manifold M is defined by acurve c(t) in M with some scalar parameter t. For some function f on M the vector X can beapplied as a differential operator to f at any point p = c(t) with coordinates xµ:

X[f ] =df(c(t))

dt

∣∣∣∣t

=dxµ(c(t))

dt

∣∣∣∣t

∂f

∂xµ= Xµ ∂f

∂xµ. (B.1)

So the tangent vector is given by

X = Xµ ∂

∂xµwith Xµ =

dxµ(c(t))

dt

∣∣∣∣t

. (B.2)

All tangent vectors at a point p in M form a tangent space TpM with the basis vectors ∂/∂xµ.The union of all tangent spaces of M ,

TM =⋃p∈M

TpM, (B.3)

is called the tangent bundle over M .As TpM is a vector space, there exists an associated dual vector space, the co-tangent space

T ∗pM . We call the elements of T ∗pM a co-tangent vector or a differential one-form:

ω = ωµdxµ. (B.4)

With the basis co-vectors dxµ we can define the inner product 〈 , 〉 : T ∗pM × TpM → R ofa vector V = V µ∂/∂xµ and a co-vector ω = ωµdx

µ by

〈ω, V 〉 = ωµVν

⟨dxµ,

∂

∂xν

⟩= ωµV

νδµν = ωµVµ. (B.5)

With this we can identify the action of a vector on a function V [f ] with the inner product ofthe vector with the function’s differential, a one-form:

df =∂f

∂xµdxµ. (B.6)

So we obtain

〈df, V 〉 = V µ ∂f

∂xν

⟨dxν ,

∂

∂xµ

⟩= V µ ∂f

∂xµ= V [f ]. (B.7)

The concept of vectors and co-vectors can be generalized to tensors of type (q, r). Such atensor T is an element of the tensor space T q

r,pM at a point p and can be written as

T = Tµ1···µq

ν1···νr∂

∂xµ1⊗ · · · ⊗ ∂

∂xµq⊗ dxν1 ⊗ · · · ⊗ dxνr (B.8)

With an associated multi-linear product a (q, r)-tensor maps q co-vectors and r vectors to R.

70 B APPENDIX: DIFFERENTIAL GEOMETRY

Vector Fields, Flows and Lie Derivatives

Tangent vectors are defined by given integral curves c(t). Alternatively, we can define a vectorfield V over M by smoothly assigning a vector to each point p in M . This vector field generatessmooth integral curves in M , where the integral curves c(t)|xµ0 = σ(t, xµ0 ) at some point xµ0 aresolutions of the differential equation:

V [σ(t, xµ0 )] =d

dtσ(t, xµ0 ) (B.9)

where the initial condition isσ(0, xµ0 ) = xµ0 . (B.10)

The collection of all generated integral curves with initial points xµ defines a flow σ(t, xµ) inM, parametrized by a map σ : R×M →M satisfying

σ(t, σ(s, xµ0 )) = σ(t+ s, xµ0 ) (B.11)

with t, s ∈ R.We now consider the phase flows σ(t, x) and τ(s, x) generated by the vector fields X and Y ,

respectively. With these we can define the Lie derivative of Y with respect toX, more precisely,the change of the vector field Y along the phase flow σ(t, x) generated by X. This is simplythe difference between the vector Y |x′=σ(ε,x) transported back to x by (σ−ε)∗ : Tσ(ε,x)M → TxMand the vector Y |x:

LXY = limε→0

1

ε

((σ−ε)∗[Y |x′=σ(ε,x)]− Y |x

). (B.12)

After some rearrangement the Lie derivative becomes

LXY = (Xµ∂µYν − Y µ∂µX

ν)∂ν (B.13)

with ∂µ = ∂/∂xµ. For the Lie derivative of a one-form ω along X we obtain

LXω = (Xµ∂µων + ∂νXµωµ)dxν . (B.14)

Differential Forms

A differential r-form is a totally anti-symmetric (0, r)-tensor defined by

ω =1

r!ωµ1···µrdx

µ1 ∧ · · · ∧ dxµr , (B.15)

where the ωµ1···µr are totally anti-symmetric. The ∧ is the totally anti-symmetric wedgeproduct of r one-forms defined by

dxµ1 ∧ · · · ∧ dxµr =∑P∈Sr

sgn(P )dxµP (1) ⊗ · · · ⊗ dxµP (r) . (B.16)

The function P is the permutation function and Sr the set of all possible permutations of theindices µ1 to µr. The sgn-operator assigns a sign to evergy permutation, where an even numberof single permutations of two indices yields a positive sign and an odd number of permutations anegative sign. Because of the total anti-symmetry the wedge-product vanishes in the case of twoidentical co-vectors in the product. An r-form ω and a q-form ξ are combined to a (q+ r)-formby the exterior product:

ω ∧ ξ =1

r!q!

∑P∈Sr

sgn(P )ωP (1)···P (r)ξP (r+1)···P (q)dxµP (1) ⊗ · · · ⊗ dxµP (r+q) . (B.17)

71

Analogously to (B.6) we can define the exterior derivative dr, mapping an r-form to anr + 1-form, by

drω =1

r!

(∂

∂xνωµ1···µr

)dxν ∧ dxµ1 ∧ · · · ∧ dxµr . (B.18)

The exterior derivative is nilpotent,dr+1dr = 0. (B.19)

If ω is an m-form with m, the dimension of M , then the exterior derivative on dω will containa wedge product of to identical co-vectors. Because of the anti-symmetry of the wedge productit becomes:

dω = 0, (B.20)

as the wedge product of two identical co-vectors vanishes.A useful extension of the inner product of a one-form and a vector is the interior product

iX , which maps an r-form to an (r − 1)-form with respect to a vector X:

iXω =1

(r − 1)!Xνωνµ2···µrdx

µ2 ∧ · · · ∧ dxµr . (B.21)

The interior product is also nilpotent (Nakahara, 2003, Sec. 5.4)

i2X = 0. (B.22)

Similar to the identity (B.7) for the inner product we can deduce

df ∧ (iXω) = df ∧ 1

(r − 1)!Xνωνµ2···µrdx

µ2 ∧ · · · ∧ dxµr

= df ∧ 1

(r − 1)!Xνωµ1µ2···µr

⟨dxµ1 ,

∂

∂xν

⟩dxµ2 ∧ · · · ∧ dxµr

=

⟨X, df ∧ 1

(r − 1)!ωµ1µ2···µrdx

µ1 ∧ · · · ∧ dxµr⟩

= 〈X, df ∧ ω〉 = 〈X, df〉ω(B.23)

and so we finally obtain the Lemma:

df ∧ (iXω) = X[f ]ω. (B.24)

Reconsidering the Lie derivative of a one-form we can rewrite expression (B.14) to the Liederivative of a general r-form ω:

LXω = (diX + iXd)ω. (B.25)

The Lie derivate commutes with the interior product:

LXiX = iXLX . (B.26)

Riemannian Geometry and General Relativity

A manifold M is called a Riemannian manifold if provided with a Riemannian metric g.g is a symmetric positive-definite tensor field of type (0,2):

g = gµνdxµdxν . (B.27)

72 B APPENDIX: DIFFERENTIAL GEOMETRY

Its trace gives the dimension of the manifold m:

gµνgµν = m. (B.28)

Let us now define the connection as the directional derivative with respect to some vectorV (i.e. acting on another vector W ),

∇VW = V µ∇∂µW ν∂ν = V µ

(∂W ν

∂xµ+ ΓνµκW

κ

)∂ν . (B.29)

The directional derivative with respect to a (contra-variant) basis vector ∂µ will then be thecovariant derivative:

∇∂µW ν = ∇µW ν = W ν;µ =

∂W ν

∂xµ+ ΓνµκW

κ. (B.30)

For a (co-variant) co-vector ων it will be

∇µων = ων;µ =∂ων∂xµ

+ Γκµνωκ. (B.31)

The covariant derivative of the metric tensor is

∇νgλµ = ∂νgλµ − Γκνλgκµ − Γκνµgλκ. (B.32)

The connection ∇κ will be called a metric connection if it leaves the metric invariant:

∇κgµν = 0. (B.33)

Now the connection coefficients Γκµν can be divided into a symmetric part Γκ(µν) and an anti-

symmetric part Γκ[µν]:

Γκµν = Γκ(µν) + Γκ[µν] (B.34)

with

Γκ(µν) =1

2

(Γκµν + Γκνµ

)and Γκ[µν] =

1

2

(Γκµν − Γκνµ

). (B.35)

Further, we have

Γκ(µν) = Γκµν +1

2(T κν µ + T κ

µ ν + T κµν), (B.36)

where T κµν is the torsion tensor. If the spacetime is torsion free than the connection coefficientsare equal to the Christoffel symbols:

Γκµν =1

2

(∂gνλ∂xµ

+∂gµλ∂xν

− ∂gµν∂xλ

)gκλ. (B.37)

From that we can construct the Riemann curvature tensor:

Rκλµν =1

2

(∂2gκµ∂xλ∂xν

−∂2gλµ∂xκ∂xν

− ∂2gκν∂xλ∂xµ

+∂2gλν∂xκ∂xµ

)+ gζη(Γ

ζκµΓηλν − ΓζκνΓηλµ). (B.38)

By contracting two indices we obtain the Ricci tensor,

Rµν = Rλµλν , (B.39)

and by further contraction the Ricci curvature scalar:

R = Rµµ. (B.40)

73

Now the Riemann tensor satisfies the Bianchi identities:

Rκλµν +Rκµνλ +Rκνλµ = 0 (first Bianchi identity) (B.41)

∇κRξλµν +∇µRξλνκ +∇νRξλκµ = 0 (second Bianchi identity). (B.42)

By contracting the second Bianchi identity twice, ξ with µ and λ with ν, after some rearrange-ment we obtain,

∇µGµν = 0 (B.43)

with the Einstein tensor Gµν , which Einstein used in his formulation of general relativity:

Gµν = Rµν − 1

2gµνR. (B.44)

Killing Vector Fields

A vector field ξ is called a Killing vector field, if the flow generated by it is an isometry, i.e.M is invariant under the transformation

xµ 7→ xµ + εξµ (B.45)

with ε being infinitesimal. For the metric tensor gµν(xµ) this means

∂(xκ + εξκ)

∂xµ∂(xλ + εξλ)

∂xνgκλ(xµ + εξµ) = gµν(xµ). (B.46)

A Killing vector field will satisfy the Killing equation,

ξρ∂ρgµν + ∂µξκgκν + ∂νξ

λgµλ = 0, (B.47)

which can be shorter expressed in a vanishing Lie derivative

(Lξg)µν = 0. (B.48)

So every vanishing Lie derivative with respect to a vector field indicates a Killing vector field.

Volume Elements and Orientation

A volume element is used as a measure for intergration on M . Integration is only possible ifM is orientable. Orientability is given if for any two sets of coordinates xµ, yν describing asub-manifold U ⊂M the Jacobian determinant is positive:

J = det

(∂xµ

∂yν

)> 0. (B.49)

If M is orientable and has a metric gµν , then there arises a natural invariant volume elementΩ, invariant under coordinate transformation

Ω =1

m!

√|g| εµ1µ2···µmdxµ1 ∧ dxµ2 ∧ · · · ∧ dxµm (B.50)

with g the metric’s determinant (Nakahara, 2003, Sec. 7.9) and the totally anti-symmetric Levi-Civita symbol εµ1µ2···µm . Thereby the invariance is given for any choice of linearly independentco-vectors dxµ. In a more general perspective Ω is a differential form. The wedge product hasthe advantage to provide orientation when constructing a volume element. In physics we oftenwork in orthonormal coordinates. In that case the volume element simplifies by the completecontraction with the Levi-Civita symbol to a product of scalars:

Ω =√|g| dx1dx2 · · · dxm (B.51)

In general relativity it is more convenient to work with the general notation of differential formsas there is no Lorentz-invariance of the three-dimensional volume.

74 C APPENDIX: DETAILED CALCULATIONS

C Appendix: Detailed Calculations

C.1 Lorentz-invariance of the Relative Velocity

(The used properties of Lorentz-transformations in this section are based on the work of Tsam-parlis (2010). The treatment of the invariant relative velocity is based on (Cannoni, 2016).We will now proof that the magnitude of the relative velocity vrel is Lorentz-invariant. For thiswe consider the Lorentz-transformation of a worldline dxµ of particle A moving with a velocity~vA in a frame S to dx′µ in a frame S1, which moves with a velocity ~vB with respect to S:

dx′µ =

(dt′

dx′i

)= Λµνdx

ν

=

(γvB −vBjγvB

−vBiγvB δij + (γvB − 1)vBivBj

~v2B

)(dtdxi

)

=

(γvBdt− γvBvBjdxj

dxi − γvBvBidt+ (γvB − 1)vBivBjdxj~v2B

), (C.1)

where γvB is the Lorentz-factor

γvB =1√

1− ~v2B

. (C.2)

Now assume that S1 is the rest frame of a particle B which moves with the velocity ~vB in S.Then the relative velocity between the particles as seen from S1 is

v′Ai =dx′idt′

=dxi − γvBvBidt+ (γvB − 1)vBi

vBjdxj~v2B

γvBdt− γvBvBjdxj(C.3)

=

dxidt − γvBvBi

dtdt + (γvB − 1)vBi

vBjdxjdt

~v2B

γvBdtdt − γvBvBj

dxjdt

(C.4)

=

vAiγvB− vBi + (1− 1

γvB)vBi

vBjvAj

~v2B

1− vBjvAj(C.5)

with

vAi =dxidt. (C.6)

In vectorial notation (C.5) becomes

~v′A =

~vAγvB− ~vB + (1− 1

γvB)~vB

~vB·~vA

~v2B

1− ~vB · ~vA. (C.7)

Otherwise the relative velocity from the rest frame of the particle A, let it be S2, is analogously:

~v′B =

~vBγvA− ~vA + (1− 1

γvA)~vA

~vA·~vB

~v2A

1− ~vA · ~vB. (C.8)

75

Now the absolute value of ~v′B is given by

(1− ~vA · ~vB)2|~v′B|2 =

(~vB

γvA

− ~vA +

(1− 1

γvA

)~vA

~vA · ~vB

~v2A

)2

=~v2

B

γ2vA

+ ~v2A +

(1− 2

γvA

+1

γ2vA

)~v2

A

(~vA · ~vB)2

~v4A

− 2(~vA · ~vB)

γvA

+ 2

(1− 1

γvA

)1

γvA

(~vA · ~vB)2

~v2A

− 2

(1− 1

γvA

)~v2

A

~vA · ~vB

~v2A

=~v2

B

γ2vA

+ ~v2A +

(1 +

1

γ2vA

)(~vA · ~vB)2

~v2A

− 2(~vA · ~vB)2

~v2A

1

γ2vA

− 2(~vA · ~vB)

=~v2

B

γ2vA

+ ~v2A +

(1− 1

γ2vA

)(~vA · ~vB)2

~v2A

− 2(~vA · ~vB)

= ~v2B(1− ~v2

A) + ~v2A + (~vA · ~vB)2 − 2(~vA · ~vB)

= (~vB − ~vA)2 + (~vA · ~vB)2 − ~v2A~v

2B. (C.9)

The last expression stays invariant when exchanging A and B, thus

vrel = |~v′A| = |~v′A|, (C.10)

which proofs that the magnitude of the relative velocity is Lorentz-invariant.

76 D APPENDIX: PYTHON CODE

D Appendix: Python Code

D.1 Freeze-out of a Symmetric Cold Relic

from math import ∗import numpy as npimport pandas as pdimport matp lo t l i b . pyplot as p l timport matp lo t l i b . t i c k e r as mticker

mass=2.0 #GeV Lee−Weinberg Boundm planck = 1.221∗10∗∗19 #GeVG fermi =1.16637∗10∗∗(−5) #1/GeV∗∗2c r o s s = 2.56811∗10∗∗(−10) #1e−37 cm∗∗2 WIMPdo f ent ropy =60.0dof =2.0power=0def absorbed parameter ( Cross ) :

Para= 0 .264∗ ( dof ∗∗0 .5 )∗m planck∗Cross∗massreturn Para

parameter = 1

#Equi l i b r ium e v o l u t i o ndef Y eq (x , i n i t , Cross ) :

Yeq in i t = 0 .298∗ ( dof / do f ent ropy )∗ i n i t ∗∗ ( 1 . 5 )∗exp(− i n i t )∗ absorbed parameter ( Cross )

e q u i l = 0 .298∗ ( dof / do f ent ropy )∗x ∗∗ ( 1 . 5 )∗ exp(−x )∗ absorbed parameter ( Cross )/ Yeq in i t

return e q u i l

#D i f f e r e n t i a l equa t iondef dYdx(x ,Y, i n i t , Cross ) :


f=−parameter∗x∗∗(−power−2)∗(Y∗∗2−(Y eq (x , i n i t , Cross ) )∗∗2)∗ Yeqin i t

return f

def n( x ) :N = dof ∗mass ∗∗3∗(1/(2∗ pi ) )∗∗1 . 5∗ x∗∗(−1.5)∗ exp(−x )return N

def Gamma(x , Cross ) :gamma = Cross∗n( x )return gamma

def HubbleRate ( x ) :H = 1.67∗ do f ent ropy ∗∗0 .5∗mass ∗∗2/( m planck∗x∗∗2)return H

D.1 Freeze-out of a Symmetric Cold Relic 77

def E f f e c t i v e n e s s (x , Cross ) :E = Gamma(x , Cross )/ ( HubbleRate ( x )∗x )return E

def RungeKuttaFehlberg ( In i tPo int , EndPoint , StepWidth , Epsi lon , Cross ) :x i n t e r v a l l =[ ]Y=[ ]Yequi = [ ]E = [ ]h=StepWidthi n i t = I n i t P o i n tmessage = True

Y. append (1)Yequi . append (1)x i n t e r v a l l . append ( I n i t P o i n t )E. append ( E f f e c t i v e n e s s ( x i n t e r v a l l [ 0 ] , Cross ) )s t a t e = x i n t e r v a l l [ 0 ]i = 1while ( s t a t e <= EndPoint ) :

x i n t e r v a l l . append ( x i n t e r v a l l [ i−1]+h)Y. append (0)E. append ( E f f e c t i v e n e s s ( x i n t e r v a l l [ i ] , Cross ) )Yequi . append ( Y eq ( x i n t e r v a l l [ i ] , i n i t , Cross ) )K 0=dYdx( x i n t e r v a l l [ i −1] , Y[ i −1] , i n i t , Cross )K 1=dYdx( x i n t e r v a l l [ i−1]+h/2 , Y[ i −1]+(h/2)∗K 0 , i n i t , Cross )K 2=dYdx( x i n t e r v a l l [ i−1]+h/2 , Y[ i −1]+(h/2)∗K 1 , i n i t , Cross )K 3=dYdx( x i n t e r v a l l [ i−1]+h/2 , Y[ i −1]+(h/2)∗K 2 , i n i t , Cross )K 4=dYdx( x i n t e r v a l l [ i−1]+h , Y[ i−1]+h∗K 3 , i n i t , Cross )i t e r a t i o n 1 = Y[ i −1]+(h /6)∗ ( K 0+2∗K 1+2∗K 2+K 3 )i t e r a t i o n 2 = Y[ i −1]+(h /24)∗ ( K 0+2∗K 1+2∗K 2+2∗K 3+K 4 )e r r o r=abs ( i t e r a t i o n 2−i t e r a t i o n 1 )h=h∗( Eps i lon / e r r o r )∗∗ (1/4)Y[ i ]= i t e r a t i o n 1s t a t e = x i n t e r v a l l [ i ]i f message == True :

i f E[ i ] < 1 :print ( ” po int o f f r e e z e−out : ”+ str ( x i n t e r v a l l [ i −1])+ ” , abundance Y: ” + str (Y[ i −1]) + ” , Y eq : ”+ str ( Yequi [ i −1]) + ” , f i t t i n g constant c : ”+ str ( (Y[ i −1]/ Yequi [ i −1])−1))message = False

i += 1

Y=np . l og (Y)Yequi=np . l og ( Yequi )YData= pd . DataFrame ( ’ x−Values ’ : x i n t e r v a l l , ’Y−Values ’ :Y,


’Yeq−Values ’ : Yequi )

return YData

Y RKF1=RungeKuttaFehlberg (1 , 1000 , 0 .0000001 , 0 .000001 , c r o s s )Y RKF2=RungeKuttaFehlberg (1 , 1000 , 0 .000001 , 0 .000001 , 0 .1∗ c r o s s )Y RKF3=RungeKuttaFehlberg (1 , 1000 , 0 .000001 , 0 .000001 , 0 .01∗ c r o s s )p l t . f i g u r e ( f i g s i z e =(11 ,7))p l t . x l a b e l ( r ’ $x=\ f r a c mT$ ’ , f o n t s i z e =20)p l t . y l a b e l ( r ’ $\ ln \ l e f t (\ f r a c YY(1)\ r i g h t ) $ ’ , f o n t s i z e =20)p l t . yl im ((−20 ,1))p l t . xl im ( (1 , 1 0 0 0 ) )p l t . x s c a l e ( ’ l og ’ )ax=p l t . gca ( )ax . xax i s . s e t m a j o r f o r ma t t e r ( mticker . Sca larFormatter ( ) )ax . xax i s . g e t ma jo r f o rmat t e r ( ) . s e t s c i e n t i f i c ( Fa l se )ax . xax i s . g e t ma jo r f o rmat t e r ( ) . s e t u s e O f f s e t ( Fa l se )p l t . y t i c k s (np . arange (−20 , 5 , s tep =5))

l i n e 1=p l t . p l o t (Y RKF1 [ ’x−Values ’ ] , Y RKF1 [ ’Y−Values ’ ] , ’ r ’ )l i n e 2=p l t . p l o t (Y RKF2 [ ’x−Values ’ ] , Y RKF2 [ ’Y−Values ’ ] , ’ c ’ )l i n e 3=p l t . p l o t (Y RKF3 [ ’x−Values ’ ] , Y RKF3 [ ’Y−Values ’ ] , ’ b ’ )l i n e 4=p l t . p l o t (Y RKF1 [ ’x−Values ’ ] , Y RKF1 [ ’Yeq−Values ’ ] , ’ k ’ )ax . l egend (

( r ’ $\ l a n g l e \ sigma A\ ver t \mathrmv\ ver t \ rang l e =10ˆ−37\mathrmcmˆ2$ ’ ,

r ’ $\ l a n g l e \ sigma A\ ver t \mathrmv\ ver t \ rang l e =10ˆ−38\mathrmcmˆ2$ ’ ,

r ’ $\ l a n g l e \ sigma A\ ver t \mathrmv\ ver t \ rang l e =10ˆ−39\mathrmcmˆ2$ ’ ,

r ’ $\ ln (Y \mathrmeq /Y( 1 ) ) $ ’ ) , prop= ’ s i z e ’ : 16)p l t . s a v e f i g ( ’ f r eezeoutsymmetr i c . eps ’ , bbox inches=’ t i g h t ’ )p l t . show ( )

D.2 Re-heating by the Decay of an Unstable Particle Species


parameter = 4.27∗10∗∗(−5)

de f integrand (x ,TauH ) :f = x ∗∗ ( 1 . 5 )∗ exp (−(3/2)∗(1/TauH)∗x ∗ ∗ ( 1 . 5 ) )re turn f

de f r h o p s i ( x ) :

D.2 Re-heating by the Decay of an Unstable Particle Species 79

Psi = x∗∗(−3)re turn Psi

de f rho psiwD (x ,TauH ) :Ps i = r h o p s i ( x )∗ exp (−(3/2)∗(1/TauH)∗x ∗ ∗ ( 1 . 5 ) )re turn Psi

de f rho R (x , i n i tRPs i ) :R = (1/ in i tRPs i )∗x∗∗(−4)re turn R

def Simpson ( In i tPo int , EndPoint , StepWidth , TauH, in i tRPs i ) :x i n t e r v a l l =[ ]I =[ ]IR =[ ]S =[ ]R=[ ]RwD=[]Ps i =[ ]PsiwD =[]i n i t = I n i t P o i n th = StepWidth/2N=(EndPoint−I n i t P o i n t )/ StepWidthn=i n t (N)f o r j in range (0 , n ) :

x i n t e r v a l l . append ( I n i t P o i n t+j ∗StepWidth )

f o r i in range ( l en ( x i n t e r v a l l ) ) :i f i ==0:

S . append (1)I . append (0 )IR . append (0 )Psi . append (1 )PsiwD . append (1)R. append (1/ in i tRPs i )RwD. append (1/ in i tRPs i )

e l s e :S . append (0)I . append (0 )IR . append (0 )Psi . append ( r h o p s i ( x i n t e r v a l l [ i ] ) )R. append ( rho R ( x i n t e r v a l l [ i ] , i n i tRPs i ) )PsiwD . append ( rho psiwD ( x i n t e r v a l l [ i ] ,TauH) )RwD. append (0)N = (h /3)∗ ( integrand ( x i n t e r v a l l [ i −1] , TauH)

+4∗ in tegrand ( x i n t e r v a l l [ i−1]+h , TauH)+integrand ( x i n t e r v a l l [ i ] , TauH) )

IR [ i ] = IR [ i −1]+(1/TauH)∗N


I [ i ]= I [ i−1]+NS [ i ]=(1+ parameter ∗(1/TauH)∗ I [ i ] )∗∗ ( 3 / 4 )RwD[ i ] = R[ i ]+( x i n t e r v a l l [ i ] )∗∗(−4)∗ IR [ i ]

S=np . l og (S)Psi=np . l og ( Psi )PsiwD=np . l og (PsiwD)R=np . l og (R)RwD=np . l og (RwD)SData= pd . DataFrame ( ’R−Values ’ : x i n t e r v a l l ,

’S−Values ’ : S , ’ Psi−EDensity ’ : Psi ,’ Psi−withDecay ’ : PsiwD , ’ Radiation ’ : R,’ RadwithDecay ’ :RwD)

re turn SData

Homer1=Simpson (1 , 1000000000 , 1000 , 10∗∗8 , 10)Homer2=Simpson (1 , 1000000000 , 1000 , 10∗∗9 ,10)Homer3=Simpson (1 , 1000000000 , 1000 , 10∗∗12 , 10)

p l t . f i g u r e ( f i g s i z e =(11 ,7))

p l t . x l a b e l ( r ’ $\ f r a c RR i $ ’ , f o n t s i z e =20)p l t . y l a b e l ( r ’ $\ ln \ l e f t (\ f r a c \ rho \ rho \ p s i ( R i )\ r i g h t ) $ ’ ,

f o n t s i z e =20)

p l t . yl im ((−80 ,0))p l t . xl im ( (1 , 10∗∗9 ) )p l t . x s c a l e ( ’ log ’ )

l i n e 1=p l t . p l o t (Homer3 [ ’R−Values ’ ] , Homer3 [ ’ Psi−EDensity ’ ] , ’ b−−’,l a b e l = r ’ $\ rho \ p s i $ without decay ’ )

l i n e 2=p l t . p l o t (Homer3 [ ’R−Values ’ ] , Homer3 [ ’ Psi−withDecay ’ ] , ’ b− ’ ,l a b e l = r ’ $\ rho \ p s i $ with decay ’ )

l i n e 3=p l t . p l o t (Homer3 [ ’R−Values ’ ] , Homer3 [ ’ Radiation ’ ] , ’ r−−’,l a b e l = r ’ $\ rho R$ without decay ’ )

l i n e 4=p l t . p l o t (Homer3 [ ’R−Values ’ ] , Homer3 [ ’ RadwithDecay ’ ] , ’ r− ’ ,l a b e l = r ’ $\ rho R$ with decay ’ )

ax2 = p l t . gca ( ) . twinx ( )p l t . yl im ( ( 0 , 5 . 7 ) )p l t . y l a b e l ( r ’ $\ ln \ l e f t (\ f r a c S S i \ r i g h t ) $ ’ , f o n t s i z e =20)

l i n e 5=p l t . p l o t (Homer1 [ ’R−Values ’ ] , Homer1 [ ’ S−Values ’ ] , ’ k− . ’ ,l a b e l = r ’ $\ ln (S/ S i ) $ f o r $\ tau\ cdot H i = 10ˆ8$ ’ )

l i n e 6=p l t . p l o t (Homer2 [ ’R−Values ’ ] , Homer2 [ ’ S−Values ’ ] , ’ k−−’,l a b e l = r ’ $\ ln (S/ S i ) $ f o r $\ tau\ cdot H i = 10ˆ9$ ’ )

l i n e 7=p l t . p l o t (Homer3 [ ’R−Values ’ ] , Homer3 [ ’ S−Values ’ ] , ’ k− ’ ,l a b e l = r ’ $\ ln (S/ S i ) $ f o r $\ tau\ cdot H i = 10ˆ12$ ’ )

D.3 Freeze-out of an Asymmetric Cold Relic 81

l n s = l i n e 1+l i n e 2+l i n e 3+l i n e 4+l i n e 5+l i n e 6+l i n e 7l ab s = [ l . g e t l a b e l ( ) f o r l in l n s ]ax2 . l egend ( lns , labs , l o c =3, bbox to anchor =(0 ,0 .1 ) ,

prop= ’ s i z e ’ : 12)p l t . s a v e f i g ( ’ en t ropy inc r ea s e . eps ’ , bbox inches =’ t ight ’ )p l t . show ( )

D.3 Freeze-out of an Asymmetric Cold Relic


mass=2.0 #GeV Lee−Weinberg Boundm planck = 1.221∗10∗∗19 #GeVG fermi =1.16637∗10∗∗(−5) #1/GeV∗∗2c r o s s = 2.56811∗10∗∗(−10) #1e−37 cm∗∗2 WIMPdof ent ropy =60.0dof =2.0power=0de f absorbed parameter ( Cross ) :

Para= 0 .264∗ ( dof ∗∗0 .5 )∗m planck∗Cross∗massre turn Para

parameter = 1

#Actua l ly D i s always p o s i t i v e , but f o r negat ive alpha#i t i s negat ive and f i t s the term in the d i f f e r e n t i a l#equat ion to be p o s i t i v e , so t h i s i s f i n ede f D(x , alpha ) :

d = 0 .242∗ ( dof / do f ent ropy )∗ ( p i ∗∗2∗ alpha+alpha ∗∗3)re turn d

#Equi l ibr ium evo lu t i onde f Y eqsym (x , i n i t , Cross ) :


e q u i l = 0 .298∗ ( dof / do f ent ropy )∗x ∗∗ ( 1 . 5 )∗exp(−x )∗ absorbed parameter ( Cross )/ Yeq in i t

r e turn e q u i l

de f Y eqplus (x , i n i t , Cross , alpha ) :Yeq in i t = 0 .298∗ ( dof / do f ent ropy )∗ i n i t ∗∗ ( 1 . 5 )

∗exp(− i n i t+alpha )∗ absorbed parameter ( Cross )e q u i l = 0 .298∗ ( dof / do f ent ropy )∗x ∗∗ ( 1 . 5 )

∗exp(−x+alpha )∗ absorbed parameter ( Cross )/ Yeq in i tr e turn e q u i l

#D i f f e r e n t i a l equat ion


de f dYdxPlus (x ,Y, i n i t , Cross , alpha ) :Yeq in i t = 0 .298∗ ( dof / do f ent ropy )∗ i n i t ∗∗ ( 1 . 5 )

∗exp(− i n i t+alpha )∗ absorbed parameter ( Cross )f=−x∗∗(−power−2)∗((Y∗∗2−(Y eqsym (x , i n i t , Cross ) )∗∗2)

∗Yeqin i t − D(x , alpha )∗Y)return f

de f n(x , alpha ) :N = dof ∗mass ∗∗3∗(1/(2∗ pi ) )∗∗1 . 5∗ x∗∗(−1.5)∗ exp(−x−alpha )re turn N

def Gamma(x , Cross , alpha ) :gamma = Cross∗n(x , alpha )re turn gamma

def HubbleRate ( x ) :H = 1.67∗ do f ent ropy ∗∗0 .5∗mass ∗∗2/( m planck∗x∗∗2)re turn H

def E f f e c t i v e n e s s (x , Cross , alpha ) :E = Gamma(x , Cross , alpha )/ ( HubbleRate ( x )∗x )re turn E

de f RungeKuttaFehlberg ( In i tPo int , EndPoint , StepWidth , Epsi lon ,Cross , alpha ) :

x i n t e r v a l l =[ ]Yplus =[ ]Yequisym = [ ]Yequiplus = [ ]E = [ ]h=StepWidthi n i t = I n i t P o i n tmessage = True

Yplus . append ( exp ( alpha ) )Yequisym . append (1 )Yequiplus . append (1 )x i n t e r v a l l . append ( I n i t P o i n t )E. append ( E f f e c t i v e n e s s ( x i n t e r v a l l [ 0 ] , Cross , alpha ) )s t a t e = x i n t e r v a l l [ 0 ]i = 1whi l e ( s t a t e <= EndPoint ) :

x i n t e r v a l l . append ( x i n t e r v a l l [ i−1]+h)Yplus . append (0 )E. append ( E f f e c t i v e n e s s ( x i n t e r v a l l [ i ] , Cross , alpha ) )Yequisym . append ( Y eqsym ( x i n t e r v a l l [ i ] , i n i t , Cross ) )Yequiplus . append ( Y eqplus ( x i n t e r v a l l [ i ] , i n i t , Cross ,

alpha ) )K 0=dYdxPlus ( x i n t e r v a l l [ i −1] , Yplus [ i −1] , i n i t , Cross ,

D.3 Freeze-out of an Asymmetric Cold Relic 83

alpha )K 1=dYdxPlus ( x i n t e r v a l l [ i−1]+h/2 , Yplus [ i −1]+(h/2)∗K 0 ,

i n i t , Cross , alpha )K 2=dYdxPlus ( x i n t e r v a l l [ i−1]+h/2 , Yplus [ i −1]+(h/2)∗K 1 ,

i n i t , Cross , alpha )K 3=dYdxPlus ( x i n t e r v a l l [ i−1]+h/2 , Yplus [ i −1]+(h/2)∗K 2 ,

i n i t , Cross , alpha )K 4=dYdxPlus ( x i n t e r v a l l [ i−1]+h , Yplus [ i−1]+h∗K 3 ,

i n i t , Cross , alpha )i t e r a t i o n 1 = Yplus [ i −1]+(h /6)∗ ( K 0+2∗K 1+2∗K 2+K 3 )i t e r a t i o n 2 = Yplus [ i −1]

+(h /24)∗ ( K 0+2∗K 1+2∗K 2+2∗K 3+K 4 )e r r o r=abs ( i t e r a t i o n 2−i t e r a t i o n 1 )h=h∗( Eps i lon / e r r o r )∗∗ (1/4)Yplus [ i ]= i t e r a t i o n 1s t a t e = x i n t e r v a l l [ i ]i f message == True :

i f E [ i ] < 1 :p r i n t (” po int o f f r e e z e−out : ”+ s t r ( x i n t e r v a l l [ i −1])+ ” , abundance Y: ”+ s t r ( Yplus [ i −1]) + ” , Y eq : ”+ s t r ( Yequiplus [ i −1])+ ” , f i t t i n g constant c : ”+ s t r ( ( Yplus [ i −1]/ Yequiplus [ i −1])−1))message = False

i += 1

Yplus=np . l og ( Yplus )Yequisym=np . l og ( Yequisym )Yequiplus = np . l og ( Yequiplus )YData= pd . DataFrame ( ’ x−Values ’ : x i n t e r v a l l ,

’ Yplus−Values ’ : Yplus , ’Yeqsym−Values ’ : Yequisym )

re turn YData

Y RKF=RungeKuttaFehlberg (1 , 500 , 0 .000001 ,0 .000001 , c ros s , 0)

Y RKFPlus1=RungeKuttaFehlberg (1 , 500 , 0 .00000001 ,0 .000001 , c ros s , 1)

Y RKFMinus1=RungeKuttaFehlberg (1 , 500 , 0 .00000001 ,0 .000001 , c ros s , −1)

Y RKFPlus2=RungeKuttaFehlberg (1 , 500 , 0 .00000001 ,0 .000001 , c ros s , 1 . 5 )

Y RKFMinus2=RungeKuttaFehlberg (1 , 500 , 0 .00000001 ,0 .000001 , c ros s , −1.5)

p l t . f i g u r e ( f i g s i z e =(11 ,7))p l t . x l a b e l ( r ’ $x=\ f r a c mT$ ’ , f o n t s i z e =20)


p l t . y l a b e l ( r ’ $\ ln \ l e f t (\ f r a c YY(1)\ r i g h t ) $ ’ , f o n t s i z e =20)p l t . yl im ((−20 ,1))p l t . xl im ( ( 1 , 5 0 0 ) )p l t . x s c a l e ( ’ log ’ )ax=p l t . gca ( )ax . xax i s . s e t m a j o r f o r ma t t e r ( mticker . Sca larFormatter ( ) )ax . xax i s . g e t ma jo r f o rmat t e r ( ) . s e t s c i e n t i f i c ( Fa l se )ax . xax i s . g e t ma jo r f o rmat t e r ( ) . s e t u s e O f f s e t ( Fa l se )p l t . y t i c k s (np . arange (−20 , 5 , s tep =5))

l i n e , = p l t . p l o t (Y RKF[ ’ x−Values ’ ] , Y RKF[ ’ Yplus−Values ’ ] , ’ b− ’)l i n e1 ,= p l t . p l o t (Y RKFPlus1 [ ’ x−Values ’ ] ,

Y RKFPlus1 [ ’ Yplus−Values ’ ] , ’ c ’ )l i n e2 ,= p l t . p l o t (Y RKFMinus1 [ ’ x−Values ’ ] ,

Y RKFMinus1 [ ’ Yplus−Values ’ ] , ’ c−−’)l i n e3 ,= p l t . p l o t (Y RKFPlus2 [ ’ x−Values ’ ] ,

Y RKFPlus2 [ ’ Yplus−Values ’ ] , ’ r ’ )l i n e4 ,= p l t . p l o t (Y RKFMinus2 [ ’ x−Values ’ ] ,

Y RKFMinus2 [ ’ Yplus−Values ’ ] , ’ r−−’)#l i n e 5=p l t . p l o t (Y RKFPlus3 [ ’ x−Values ’ ] ,

Y RKFPlus3 [ ’ Yplus−Values ’ ] , ’ r ’ )#l i n e 6=p l t . p l o t (Y RKFMinus3 [ ’ x−Values ’ ] ,

Y RKFMinus3 [ ’ Yplus−Values ’ ] , ’ r−−’)l i n e7 ,= p l t . p l o t (Y RKFMinus1 [ ’ x−Values ’ ] ,

Y RKFMinus1 [ ’ Yeqsym−Values ’ ] , ’ k ’ )

#l ine5 , , r ’ $\ f r a c \muT = 2$ ’p l t . l egend ( [ l i n e , l i n e1 , l i n e3 , l i n e 7 ] ,

[ r ’ $\ l a n g l e \ sigma A\ ver t \mathrmv\ ver t \ rang l e= 10ˆ−37\mathrmcmˆ2 ,\mu = 0$ ’ , r ’ $\ f r a c \muT = 1$ ’ ,r ’ $\ f r a c \muT = 1.5 $ ’ ,

r ’ $\ ln (Y \mathrmsymˆ\mathrmeq /Y \mathrmsymˆ\mathrmeq (1 ) ) $ ’ ] ,prop= ’ s i z e ’ : 16)

p l t . s a v e f i g ( ’ asymmetr i c f reezeout . eps ’ , bbox inches =’ t ight ’ )p l t . show ( )

D.4 Exclusion Plot of Allowed WIMP Cross-sections and Magnitudes ofAsymmetry


do f ent ropy =106.75dof =2.0mass=1.0 #GeV Lee−Weinberg Boundm planck = 1.221∗10∗∗19 #GeV

D.4 Exclusion Plot of Allowed WIMP Cross-sections and Magnitudes of Asymmetry 85

x f = 18sigma 0 = 1.032∗10∗∗(−10)

a = 0.5∗ s igma 0b = 2.82∗10∗∗8

de f alpha ( sigma ) :A = 0 .42∗ ( do f ent ropy / dof )∗ (1/ b)∗(0.5−( a/ sigma ) )re turn A

Sigma = np . arange ( sigma 0 , (2)∗ sigma 0 , 0 .0001∗ s igma 0 )Alpha = alpha ( Sigma )

p l t . f i g u r e ( f i g s i z e =(11 ,7))p l t . x l a b e l ( r ’ $\ l e f t [\ l a n g l e \ sigma A\ ver t \mathrmv\ ver t \ rang l e \ r i g h t ]

= \mathrmGevˆ−2$ ’ , f o n t s i z e =20)p l t . y l a b e l ( r ’ $\ f r a c \muT$ ’ , f o n t s i z e =20)p l t . yl im ( ( 0 , 0 . 00 0000 02 ) )p l t . xl im ( ( 0 . 9∗ sigma 0 ,2∗ s igma 0 ) )p l t . x s c a l e ( ’ log ’ )ax=p l t . gca ( )ax . yax i s . s e t m a j o r f o rm a t t e r ( mticker . Sca larFormatter ( ) )ax . yax i s . g e t ma jo r f o rmat t e r ( ) . s e t s c i e n t i f i c ( True )ax . yax i s . g e t ma jo r f o rmat t e r ( ) . s e t u s e O f f s e t ( True )

l i n e , = p l t . p l o t ( Sigma , Alpha , ’ k ’ )

p l t . l egend ( [ l i n e ] , [ ’ $\Omega \mathrmDM = 1 $ ’ ] , l o c =6,prop= ’ s i z e ’ : 16)

ax . t ex t ( 0 . 3 , 0 . 7 , r ’ $\Omega \mathrmDM>1$ ’ ,t rans form=ax . transAxes , f o n t s i z e =16,v e r t i c a l a l i g n m e n t =’top ’ )

ax . t ex t ( 0 . 7 , 0 . 3 , r ’ $\Omega \mathrmDM<1$ ’ ,t rans form=ax . transAxes , f o n t s i z e =16,v e r t i c a l a l i g n m e n t =’top ’ )

p l t . s a v e f i g ( ’ e x c l u s i o n p l o t . eps ’ , bbox inches =’ t ight ’ )p l t . show ( )

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Register of Illustration

Fig. 1: van Albada et al. (1985)

Fig. 2: Schneider et al. (1999) (modified)

Fig. 3: produced by myself (see Appendix D.1), based on Kolb and Turner (1990, Chap. 5)

Fig. 4: produced by myself (see Appendix D.2), based on Kolb and Turner (1990, Chap. 5)

Fig. 5: produced by myself (see Appendix D.3), based on Graesser et al. (2011)

Fig. 6: produced by myself (see Appendix D.4)

Statutory Declaration

I declare that this Master Thesis is completely written by myself. All contents and thoughts,that originate from others or literal quotation are marked. Furthermore this Thesis was not partof any other publication nor used to achieve any other academic graduation.

Bielefeld, 31 May, 2019

master thesis: relativistic kinetic theory of particle ... · malism of relativistic kinetic theory...

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