#
Desy-Thesis-2012-039

Juli 2012

The Phase Transition

Implications for Cosmology and Neutrinos

###### Abstract

We investigate the possibility that the hot thermal phase of the early universe is ignited in consequence of the phase transition, which represents the cosmological realization of the spontaneous breaking of the Abelian gauge symmetry associated with , the difference between baryon number and lepton number . Prior to the phase transition, the universe experiences a stage of hybrid inflation. Towards the end of inflation, the false vacuum of unbroken symmetry decays, which entails tachyonic preheating as well as the production of cosmic strings. Observational constraints on this scenario require the phase transition to take place at the scale of grand unification. The dynamics of the breaking Higgs field and the gauge degrees of freedom, in combination with thermal processes, generate an abundance of heavy (s)neutrinos. These (s)neutrinos decay into radiation, thereby reheating the universe, generating the baryon asymmetry of the universe and setting the stage for the thermal production of gravitinos. The phase transition along with the (s)neutrino-driven reheating process hence represents an intriguing and testable mechanism to generate the initial conditions of the hot early universe. We study the phase transition in the full supersymmetric Abelian Higgs model, for which we derive and discuss the Lagrangian in arbitrary and unitary gauge. As for the subsequent reheating process, we formulate the complete set of Boltzmann equations, the solutions of which enable us to give a detailed and time-resolved description of the evolution of all particle abundances during reheating. Assuming the gravitino to be the lightest superparticle (LSP), the requirement of consistency between hybrid inflation, leptogenesis and gravitino dark matter implies relations between neutrino parameters and superparticle masses, in particular a lower bound on the gravitino mass of . As an alternative to gravitino dark matter, we consider the case of very heavy gravitinos, which are motivated by hints for the Higgs boson at the LHC. We find that the nonthermal production of pure wino or higgsino LSPs, i.e. weakly interacting massive particles (WIMPs), in heavy gravitino decays can account for the observed amount of dark matter, while simultaneously fulfilling the constraints imposed by primordial nucleosynthesis and leptogenesis, within a range of LSP, gravitino and neutrino masses. Besides its cosmological implications, the spontaneous breaking of also naturally explains the small observed neutrino masses via the seesaw mechanism. Upon the seesaw model we impose a flavour structure of the Froggatt-Nielson type which, together with the known neutrino data, allows us to strongly constrain yet undetermined neutrino observables.

###### Zusammenfassung

Wir untersuchen die Möglichkeit, dass die heiße thermische Phase des frühen Universums in Folge des Phasenübergangs, welcher die kosmologische Umsetzung der spontanen Brechung der mit , der Differenz von Baryonenzahl und Leptonenzahl , verknüpften Abelschen Eichsymmetrie darstellt, enzündet wird. Vor dem Phasenübergang durchlebt das Universum einen Abschnitt der Hybridinflation, gegen deren Ende das falsche Vakuum ungebrochener Symmetrie zerfällt, was tachyonisches Vorheizen sowie die Produktion kosmischer Strings nach sich zieht. Aus Beobachtungen gewonnene Einschränkungen dieses Szenarios erfordern es, dass der Phasenübergang bei der Skala der Großen Vereinheitlichung stattfindet. Die Dynamik des brechenden Higgsfeldes und der Eichfreiheitsgrade, zusammen mit thermischen Prozessen, generiert ein Vorkommen an schweren (S)neutrinos. Diese (S)neutrinos zerfallen in Strahlung, wodurch sie das Universum aufheizen, die Baryonenasymmetrie des Universums erzeugen und der thermischen Produktion von Gravitinos den Weg ebnen. Der Phasenübergang stellt folglich mitsamt dem (S)neutrino-getriebenen Aufheizprozess einen überzeugenden und testbaren Mechanismus zur Erzeugung der Anfangsbedingungen des heißen frühen Universums dar. Wir studieren den Phasenübergang im vollständigen supersymmetrischen Abelschen Higgsmodel, für welches wir die Lagrangedichte in beliebiger und unitärer Eichung herleiten und diskutieren. In Hinblick auf den anschließenden Aufheizprozess formulieren wir den kompletten Satz an Boltzmanngleichungen, deren Lösungen uns zu einer detaillierten und zeitaufgelösten Beschreibung aller Teilchenhäufigkeiten verhelfen. Angenommen, das Gravitino ist das leichteste Superteilchen (LSP), so impliziert die Forderung nach Konsistenz zwischen Hybridinflation, Leptogenese und Gravitino-Dunkler-Materie Beziehungen zwischen Neutrinoparametern und Superteilchenmassen, insbesondere eine untere Schranke an die Gravitinomasse von . Als Alternative zu Gravitino-Dunkler-Materie betrachten wir den Fall sehr schwerer Gravitinos, die durch Hinweise auf das Higgs-Boson am LHC motiviert sind. Wir stellen fest, dass die nichtthermische Produktion reiner Wino- oder Higgsino-LSPs, d.h. schwach wechselwirkender massereicher Teilchen (WIMPs), in den Zerfällen schwerer Gravitinos für die beobachtete Menge an Dunkler Materie innerhalb einer Bandbreite von LSP-, Gravitino- und Neutrinomassen aufkommen und zugleich den von primordialer Nukleosynthese und Leptogenese auferlegten Einschränkungen genügen kann. Abgesehen von ihren kosmologischen Auswirkungen, erklärt die spontane Brechung auch in natürlicher Weise die kleinen beobachteten Neutrinomassen vermöge des Seesaw-Mechanismus. Wir erlegen dem Seesaw-Model eine Flavour-Struktur vom Froggatt-Nielsen-Typ auf, welche es uns zusammen mit den bekannten Neutrinodaten erlaubt, bislang unbestimmte Neutrinoobservablen stark einzuschränken.

[0]Titlepagetitle

Gutachter der Dissertation: | Prof. Dr. Wilfried Buchmüller |

Prof. Dr. Mark Hindmarsh | |

Prof. Dr. Günter Sigl | |

Gutachter der Disputation: | Prof. Dr. Wilfried Buchmüller |

Prof. Dr. Jan Louis | |

Datum der Disputation: | 9. Juli 2011 |

Vorsitzender des Prüfungsausschusses: | Prof. Dr. Georg Steinbrück |

Vorsitzender des Promotionsausschusses: | Prof. Dr. Peter Hauschildt |

Dekan der Fakultät für Mathematik, | |

Informatik und Naturwissenschaften: | Prof. Dr. Heinrich Graener |

###### Contents

- ‣ DESY-THESIS-2012-039 Juli 2012 The Phase Transition Implications for Cosmology and Neutrinos
- 1 Introduction
- 2 Early Universe Cosmology
- 3 Framework for a Consistent Cosmology
- 4 Neutrino Phenomenology
- 5 Supersymmetric Abelian Higgs Model
- 6 Nonperturbative Dynamics
- 7 The Reheating Process
- 8 WIMP Dark Matter from Heavy Gravitino Decays
- 9 Conclusions and Outlook
- Acknowledgements
- A Statistical Thermodynamics
- B CP Violation in 2-to-2 Scattering Processes
- C Thermal Gravitino Production

###### List of Figures

- 2.1 Timeline of the hot thermal phase of the early universe
- 4.1 Neutrino mixing angles and .
- 4.2 Lightest neutrino mass and effective neutrino mass in tritium decay
- 4.3 Effective neutrino mass in decay and Majorana phase
- 4.4 Effective neutrino mass and violation parameter
- 5.1 Particle spectrum and interactions after spontaneous breaking
- 7.1 Comoving number and energy densities
- 7.2 Breakdown of the comoving number densities
- 7.3 Temperature of the thermal bath and lepton asymmetry
- 7.4 Comoving number densities after slight theoretical modifications
- 7.5 Contour plot of the reheating temperature
- 7.6 Contour plot of the baryon asymmetry
- 7.7 Relations between neutrino parameters and the gravitino mass
- 8.1 Lower bounds on and as functions of
- 8.2 Upper and lower bounds on as functions of and
- 8.3 Upper and lower bounds on and lower bounds on .
- 8.4 Upper bounds on and lower bound on as functions of
- C.1 and as functions of
- C.2 Rescaling prescription for the gravitino mass

## Chapter 1 Introduction

The cosmic microwave background (CMB) radiation [1] and the primordial abundances of the light nuclei [2] provide direct evidence for a hot thermal phase in the early universe. While the CMB represents a full-sky picture of the hot early universe close to its minimal temperature, primordial nucleosynthesis allows us to probe the history of the universe up to the first tenth of a second after the big bang. Going further back in time, beyond the generation of the light elements, the theoretical extrapolation becomes increasingly uncertain. Up to temperatures slightly above the electroweak scale, we are still able to make an educated guess about the evolution of the universe based on the established and well-tested physics of the standard model of particle physics. At temperatures around the scale of quantum chromodynamics (QCD), we thus expect the occurrence of a phase transition, in the course of which quarks and gluons become confined into hadrons. Similarly, one presumes a phase transition around the electroweak scale, which causes the Higgs boson, the electroweak gauge bosons as well as all fermions expect for neutrinos to acquire a mass via the Higgs mechanism. If indeed realized in the early universe, the electroweak phase transition would correspond to the cosmological implementation of electroweak symmetry breaking. Meanwhile, in anticipation of new insights from observations and experiments as to the physics beyond the standard model, we are at present merely able to speculate about the nature of the processes taking place at even higher energy scales. While the conclusive identification of a successor to the standard model is still pending, we know for sure that some processes must occur in the very early universe, which cannot be accounted for by the known laws of physics.

A clear indication for physics beyond the standard model is the present composition of the universe [2]. First of all, it is astonishing that all matter in the universe which can be more or less well described by standard model physics seems to be almost exclusively made out of baryons and hardly out of antibaryons. This cosmic asymmetry between matter and antimatter calls for a nonequilibrium process in the hot early universe, in which a primordial baryon asymmetry is generated before baryons and antibaryons decouple from the hot plasma. Furthermore, as ordinary matter contributes with only to the energy budget of the universe, we are led to the conclusion that of the energy of the universe reside in unknown forms of matter and energy, viz. dark matter and dark energy. Dark matter, which encompasses of the total energy of the universe, is able to clump under the influence of the gravitational force and thus plays a crucial role in the theory of structure formation. Studies of the cosmic density perturbations, seen for instance in the CMB [1] or the distribution of galaxies in the neighbourhood of the Milky Way [3], imply that dark matter has to be present in the early universe long before the end of the hot thermal phase. Barring a modification of general relativity, the remaining of the energy of the universe have to be attributed to some form of dark energy, which is commonly identified as the energy of the vacuum and as such explained in terms of a cosmological constant . Further evidence for new physics derives from the properties of the CMB. The minute temperature anisotropies of the CMB exhibit correlations on scales exceeding the sound horizon at the time of photon decoupling and thus point to a mechanism in the very early universe capable of generating primordial metric fluctuations with super-horizon correlations. Finally, on the particle physics side, the flavour oscillations among the three standard model neutrino species [4, 5] represent the clearest evidence for physics beyond the standard model. These oscillations indicate that neutrinos have tiny, but nonzero masses, although the standard model stipulates them to be massless.

The most popular solution to the problem of the primordial density perturbations as well to other puzzles related to the initial conditions of big bang cosmology is inflation [6, 7, 8]—a stage of accelerated expansion in the very early universe driven by the energy of the vacuum. During inflation, the quantum fluctuations of a scalar field, the so-called inflaton field, are stretched to super-horizon scales, whereupon they freeze, remaining basically unchanged in shape until the onset of structure formation. The dynamics of the inflaton field correspond to those of an ensemble of inflaton particles in a coherent quantum state at zero temperature. Assuming inflation to be the source of the primordial perturbations, one arrives at the question as to the origin of the entropy of the hot plasma filling the universe during the thermal phase. In summary, we conclude that contemporary particle cosmology faces the task to explain the origin of the hot early universe as well as its initial conditions, i.e. the entropy of the thermal bath, the primordial baryon asymmetry and the abundance of dark matter.

In this thesis, we put forward the idea that the emergence of the hot thermal universe might be closely related to the decay of a false vacuum of unbroken symmetry, where denotes the difference between baryon number and lepton number . In such a scenario, the energy of the false vacuum drives a stage of hybrid inflation [9, 10], ending in a phase transition, in the course of which the Abelian gauge symmetry becomes spontaneously broken. Guided by the expectation that phase transitions might be in fact common phenomena in the early universe, we hence propose that also the very origin of the hot thermal phase has to be attributed to a phase transition, viz. the phase transition as we shall refer to it from now on. To be very clear about this point, we stress that the phase transition represents the cosmological realization of spontaneous breaking, similarly as the electroweak phase transition represents the cosmological realization of electroweak symmetry breaking.

Hybrid inflation ending in the spontaneous breaking of a local symmetry is an attractive scenario of inflation, as it establishes a connection between cosmology and particle physics. The symmetry breaking at the end of inflation may, in particular, be identified as an intermediate stage in the breaking of the gauge group of some theory of grand unification (GUT) down to the gauge group of the standard model. In this sense, the phase transition may be easily embedded into a grander scheme based on a GUT theory featuring as an additional gauge symmetry. A prime example in this context are GUT theories with gauge group [11]. We also note that incorporating into the gauge group of the theory is an almost trivial extension of the standard model. As it turns out, the global is already an anomaly-free symmetry of the standard model Lagrangian [12, 13]. Upon the introduction of three generations of right-handed neutrinos, it is then readily promoted to a local symmetry [14, 15]. Unfortunately, the statistical properties of the CMB temperature anisotropies rule out the simplest nonsupersymmetric version of hybrid inflation [1]. For this reason we will consider supersymmetric -term hybrid inflation [16, 17] in this thesis. Apart from its usual inner-theoretical and aesthetic virtues, including supersymmetry into our analysis also has an important phenomenological advantage. Invoking a discrete symmetry such as matter [18] or parity [19] renders the lightest superparticle (LSP) stable, turning it into an excellent particle candidate for dark matter [20, 21, 22]. Moreover, supersymmetry implies that each right-handed neutrino pairs up with a complex scalar to form a chiral multiplet. In the course of the phase transition, these neutrino multiplets acquire Majorana mass terms, such that after symmetry breaking the physical neutrino states consist of three heavy Majorana neutrinos and three heavy complex sneutrinos . The phase transition hence also sets the stage for the seesaw mechanism [23, 24, 25, 26, 27], which elegantly explains the tiny masses of the standard model neutrinos.

The decay of the false vacuum at the end of hybrid inflation is accompanied by tachyonic preheating [28, 29] and the production of topological defects in the form of cosmic strings [30, 31, 32]. Successful hybrid inflation in combination with the nonobservation of cosmic strings requires that the phase transition indeed has to take place at the GUT scale [33, 34]. Tachyonic preheating denotes the rapid transfer of the false vacuum energy into a gas of nonrelativistic Higgs bosons, entailing the nonadiabatic production of all particles coupled to the Higgs field [35]. After the phase transition the energy density of the universe is dominated by the abundance of Higgs bosons, which slowly decay into heavy neutrinos and sneutrinos. In combination with tachyonic preheating, the dynamics of the gauge degrees of freedom (DOFs) as well as thermal processes, the decay of the Higgs boson and its superpartners produces an abundance of heavy (s)neutrinos. These (s)neutrinos subsequently decay into radiation, thereby generating the entropy of the hot thermal phase, i.e. reheating the universe.

Hence, an important consequence of the phase transition is that the reheating process is driven by the decay of the heavy (s)neutrinos. This in turn automatically yields baryogenesis via a mixture of nonthermal and thermal leptogenesis [36]. Our work is thus closely related to previous studies on thermal leptogenesis [37, 38] as well as on nonthermal leptogenesis via inflaton decay [39, 40, 41, 42]. Furthermore, the fact that the reheating process is (s)neutrino-driven results in the temperature scale of reheating, i.e. the reheating temperature, being determined by the (s)neutrino lifetime and therefore directly related to (s)neutrino parameters. Of course, the final baryon asymmetry is also determined by (s)neutrino parameters and so we arrive at the remarkable conclusion that the initial conditions of the hot early universe cannot be freely chosen, but are fully controlled by the parameters of a Lagrangian, which could in principle be measured by particle physics experiments and astrophysical observations. The phase transition is hence not only a particularly simple mechanism for the generation of the initial conditions of the hot early universe, it is also testable in present-day and future experiments.

Assuming supersymmetry to be a local symmetry, the particle spectrum also features the gravitino—the spin- superpartner of the spin- graviton, which acts as the gauge field of local supersymmetry transformations. Due to the high reheating temperatures reached after the phase transition, thermal gravitino production during the reheating process is unavoidable [43, 44]. Depending on the superparticle mass spectrum, this may lead to various cosmological problems. As for a light stable gravitino, inelastic scatterings in the thermal bath may produce the gravitino so efficiently that it overcloses the universe. Meanwhile, the late-time decay of an unstable gravitino may alter the abundances of the light elements and thus spoil the successful theory of big bang nucleosynthesis (BBN) [45, 46, 47, 48, 49]. To avoid these problems, we will consider two particular superparticle mass spectra in this thesis. In the first case, we will assume the gravitino to be the LSP with a mass of , as it typically arises in scenarios of gravity- or gaugino-mediated supersymmetry breaking. Gravitino dark matter can then be thermally produced at a reheating temperature compatible with leptogenesis [50]. In the second case, we will take the gravitino to be the heaviest superparticle with a mass of . Such large gravitino masses are realized in anomaly mediation, which is a promising scenario of supersymmetry breaking, given the recent hints by the LHC experiments ATLAS and CMS that the Higgs boson may have a mass of about [51, 52]. A gravitino heavier than roughly can be consistent with primordial nucleosynthesis and leptogenesis [45, 53, 54], thus allowing us to circumvent all cosmological gravitino problems. In our second scenario, the nonthermal production of pure wino or higgsino LSPs, i.e. weakly interacting massive particles (WIMPs), in the decay of heavy, thermally produced gravitinos accounts for the relic density of dark matter.

In this thesis, we study the phase transition in the full supersymmetric Abelian Higgs model, for which derive the complete Lagrangian in arbitrary and unitary gauge. From this Lagrangian, we cannot only infer the decay rates of all particles under study, but also read off how the corresponding mass eigenvalues evolve with time in the course of spontaneous symmetry breaking. These time-dependent masses are an important input to the calculation of the particle abundances produced during tachyonic preheating. In order to describe the reheating process subsequent to the phase transition, we derive the Boltzmann equations for all particle species of interest. To facilitate our calculations, we treat the various contributions to the respective heavy (s)neutrino abundances separately, i.e. we formulate a separate Boltzmann equation for each contribution. Thanks to this novel technical procedure, we are able to solve a subset of Boltzmann equations analytically. Solving the remaining equations numerically, we obtain a detailed and time-resolved picture of the evolution of all particle abundances during reheating. An interesting result of our analysis is that the competition between cosmic expansion and entropy production leads to an intermediate period of constant reheating temperature, during which the baryon asymmetry as well as the thermal gravitino abundance are produced. The final results for these two quantities as well as the reheating temperature turn out to be rather insensitive to the influence of the extra superparticles not contained in the supersymmetric standard model. Likewise, the decay of the gauge DOFs shortly after preheating hardly affects the final outcomes of our calculations. Based on these observations, we conclude that the investigated scenario of reheating is quite robust against uncertainties in the underlying theoretical framework.

Successful hybrid inflation and leptogenesis constrain the viable range of neutrino mass parameters. Combining these constraints with the requirement that dark matter be made out of gravitinos, we find relations between neutrino parameters and superparticle masses, in particular a lower bound on the gravitino mass of . Similarly, we infer relations between the masses of the dark matter particle, the gravitino and the standard model neutrinos in the case of WIMP dark matter. Requiring consistency between hybrid inflation, leptogenesis, dark matter and BBN, we derive upper and lower bounds on the LSP mass as well as lower bounds on the gravitino mass, all of which depend on the lightest neutrino mass. For instance, given that the lightest neutrino has a mass of , a higgsino LSP would have to be lighter than , while the gravitino would need to have a mass of at least .

Our quantitative analysis of the reheating process by means of Boltzmann equations is based on a flavour model [55] of the Froggatt-Nielsen type [56]. Generally speaking, Froggatt-Nielsen flavour models are able to reconcile the large quark and charged-lepton mass hierarchies and the small quark mixing angles with the observed small neutrino mass hierarchies and the large neutrino mixing angles in a natural way. In this thesis, we point out that the Froggatt-Nielsen flavour structure, which we employ for our analysis, together with the known neutrino data, strongly constrains yet undetermined parameters of the neutrino sector. Treating unknown parameters as random variables, we obtain surprisingly sharp predictions for the smallest mixing angle, , the smallest neutrino mass, , and one Majorana phase, .

This thesis is organized as follows. In Ch. 2, we briefly review the basics of early universe cosmology, which we require as background material for our further discussion. We outline how the present composition of the universe calls for new physics beyond the standard model, discuss the main observational evidence for the hot thermal phase in the early universe, i.e. the CMB and BBN, and shortly touch on the other phase transitions, which we expect to take place in the early universe, i.e. the QCD and the electroweak phase transition. Finally, we review the electroweak sphaleron process, which is a crucial ingredient to leptogenesis. The reader acquainted with these rudiments of particle cosmology is invited to skip our introductory chapter and directly proceed with Ch. 3.

In Ch. 3, we develop a theoretical framework for a consistent cosmology, which addresses most of the problematic issues alluded to in Ch. 2. First, we motivate supersymmetric -term hybrid inflation as an attractive inflationary scenario and compile several useful formulae, which we need for our later analysis of the production of cosmic strings. Then we turn to the seesaw mechanism and the right-handed neutrinos. We introduce the superpotential for all quark and lepton superfields and subsequently use it to derive the mass and mixing matrices in the lepton sector. Next, we motivate leptogenesis as the most promising scenario of baryogenesis and elaborate on the two superparticle mass spectra, which we consider in this thesis. In the latter part, we particularly emphasize how the spectra under study circumvent the cosmological gravitino problems. Finally, we assemble all pieces of the puzzle and outline how the phase transition at the end of inflation gives rise to a consistent cosmology. We summarize all mechanisms for the production of particles during preheating and reheating and illustrate how the fact that the reheating process is driven by the decay of heavy (s)neutrinos directly implies relations between neutrino and superparticle masses. In conclusion, we present our Froggatt-Nielsen flavour structure and parametrize our entire model in terms of flavour charges.

In Ch. 4, we employ Monte-Carlo techniques to study the dependence of yet undetermined neutrino observables on the unknown factors contained in the Froggatt-Nielsen model. After a few technical remarks on our procedure, we list the surprisingly precise predictions for the various parameters in the neutrino sector and demonstrate that we are partly even able to reproduce them analytically.

In Ch. 5, we lay the theoretical foundation for our study of the phase transition and the subsequent reheating process. To be able to describe the dynamics of all physical particle species after spontaneous symmetry breaking, we require the Lagrangian of the supersymmetric Abelian Higgs model in unitary gauge. In a first step, we therefore derive the Lagrangian of a general supersymmetric Abelian gauge theory in arbitrary gauge. Then we evaluate this Lagrangian in unitary gauge for our specific field content, which readily provides us with all time-dependent mass eigenvalues and decay rates that we require for our further analysis.

In Ch. 6, we discuss the nonperturbative dynamics during the decay of the false vacuum of unbroken . First, we estimate the abundance of cosmic strings produced during the phase transition and restrict the parameters of hybrid inflation based on the requirement of successful inflation and the fact that no observational indications of effects related to cosmic strings have been found so far. In the second section of Ch. 6, we introduce the quench approximation for the waterfall transition at the end of hybrid inflation and generalize the common waterfall conditions [10], which only apply to the original, nonsupersymmetric variant of hybrid inflation, to the supersymmetric case. Furthermore, we compute the particle abundances generated during preheating.

In Ch. 7, we study the reheating process subsequent to the phase transition by means of Boltzmann equations. For a series of particle species, we first formulate template Boltzmann equations serving as proxies for their actual Boltzmann equations. After solving these template equations analytically and in full generality, we then apply our findings to our actual scenario. Moreover, we develop techniques to describe the evolution of the gravitational background analytically and to track the evolution of the temperature of the hot plasma by means of its own Boltzmann equation. In the next section, assuming the gravitino to be the LSP, we present the solutions of the Boltzmann equations for a representative choice of parameter values. Apart from a comprehensive discussion of the evolution of all particle abundances, we motivate a particular definition of the reheating temperature and check the robustness of the reheating process against small changes in the theoretical setup. In the third section of Ch. 7, we finally carry out a scan of the parameter space, from which we infer relations between neutrino and superparticle masses. To some extent, we are again able to reproduce our results analytically. For all important quantities we provide useful fit formulae.

In Ch. 8, we consider the production of WIMP dark matter in the decay of heavy, thermally produced gravitinos. After a short comment on the competition between our nonthermal WIMP production mechanism and thermal WIMP freeze-out, we present constraints on the neutralino, gravitino and neutrino masses and sketch the prospects for the experimental confirmation of our scenario.

In Ch. 9, we conclude and summarize our results. Furthermore, we give an outlook as to the possible directions into which the analysis presented in this thesis could be extended. The three appendices contain important supplementary material. In App. A, we summarize the formalism of Boltzmann equations and discuss the properties of particle species in kinetic and thermal equilibrium. In App. B, we provide the proof for an important relation, which is needed in the derivation of the Boltzmann equation for the lepton asymmetry and which is related to the violation in -to- scattering processes with heavy (s)neutrinos in the intermediate state. In App. C, we derive an analytical expression for the abundance of thermally produced gravitinos and illustrate how our quantitative discussion in Ch. 7 is easily generalized to gluino masses other than the one we employ in our analysis.

The discussion in Chs. 4, 7 and 8 is based on two projects in collaboration with Wilfried Buchmüller and Gilles Vertongen as well as on three projects in collaboration with Wilfried Buchmüller and Valerie Domcke, the results of which were respectively first published in Refs. [57, 58] and Refs. [59, 60, 61].

## Chapter 2 Early Universe Cosmology

The main intention of this thesis is to motivate and investigate the phase transition as the possible origin for the thermal phase of the hot early universe. Before we are ready to do so, we have to acquaint ourselves with the observational evidence for this phase and understand which physical processes have or may have taken place in it. For this reason we shall provide a brief review of early universe cosmology in this chapter, thereby compiling the background material for the further discussion. We will first discuss the present composition of the universe (cf. Sec. 2.1) and then some of the main events in the thermal history of the universe in reverse chronological order (cf. Sec. 2.2). We would like to emphasize that in this introductory chapter we will crudely restrict ourselves to aspects which are relevant for our purposes. More balanced and comprehensive presentations of the topic are for instance provided in standard textbooks [62, 63, 64] or dedicated review articles [65, 66, 2].

### 2.1 Composition of the Universe

Over the last years the observational progress has marked the advent of the era of precision cosmology. The combined data exhibits an impressive consistency and is in very good agreement with the currently accepted concordance model of big bang cosmology, the Lambda-Cold Dark Matter (CDM) model. Major evidence for this standard scenario of big bang cosmology derives from several cosmological observations, the most eminent being perhaps (i) the observed primordial abundances of the light elements, matching very well the theoretical prediction from BBN [2], (ii) the angular power spectrum of the temperature anisotropies in the CMB as measured by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite [1], (iii) the imprint of baryonic acoustic oscillations (BAOs) in the local distribution of matter as seen in galaxy surveys [3], (iv) direct measurements of the cosmic expansion rate, i.e. the Hubble parameter , by the Hubble Space Telescope [67], and (v) distance measurements based on type Ia supernovae (SNe) [68, 69].

All observed cosmological phenomena are consistent with the assumption that our universe is spatially flat [1, 70]. Indeed, combining the data on CMB anisotropies, BAOs and shows that presently, at , the total energy density of the universe does not deviate by more than from the critical energy density that is required for exact spatial flatness. In the following we shall hence neglect the possibility of a small spatial curvature and assume that , which is equivalent to saying that all density parameters sum to unity,

(2.1) |

This sum receives contributions from three different forms of energy or matter: radiation, matter and dark energy. In the present epoch the energy in radiation from beyond our galaxy is dominated by the photons of the CMB. Relic neutrinos which are presumed to be present in the current universe as a remnant of the hot early universe either belong to radiation or matter, depending on their absolute masses. The matter component splits into a small baryonic and a large dark nonbaryonic fraction. We shall now discuss in turn how photons, neutrinos, baryonic matter, dark matter and dark energy respectively contribute to .

#### 2.1.1 CMB Photons

In the early 1990s the Cosmic Background Explorer (COBE) satellite experiment was the first precision measurement to confirm two key features of the CMB. Since COBE we know that the CMB has an almost perfect Planckian spectrum [71, 72] and that it is highly isotropic, with its temperature fluctuating across the sky only at the level of [73]. Together, these findings provide strong evidence for a hot thermal phase in the early universe preceded by an inflationary era (cf. Sec. 2.2.1). The mean CMB temperature is [74]. Given the thermal black-body distribution of the CMB photons, this temperature directly implies the following entropy, number and energy densities

(2.2) |

The present value of the critical energy density is determined by the current expansion rate. With the aid of the dimensionless Hubble parameter , which is defined through the relation , we are able to write as

(2.3) |

with denoting the Planck mass. The CDM fit to the combined CMB, BAO and data gives [1], such that , which results in a photon density parameter

(2.4) |

Barring some unknown form of dark radiation [75],
the only other significant contribution to the present-day entropy density
in radiation comes from neutrinos.^{1}^{1}1Note that in the recent cosmic past,
shortly after the onset of star formation, the entropy contained in black holes has come
to dominate over the entropy in radiation [76].
We thus conclude that photons are responsible for a large fraction of the
radiation entropy in the current universe,
but contribute only to a negligible extent to the total energy density.

#### 2.1.2 Relic Neutrinos

In the hot early universe neutrinos are produced and kept in thermal equilibrium via weak interactions. Around a temperature the rate of these interactions drops below the Hubble rate, causing the neutrinos to decouple from the thermal bath and evolve independently of all other species afterwards. The presence of a relic abundance of primordial neutrinos in the current universe is hence a fundamental prediction of the hot big bang scenario. It is doubtful whether this cosmic neutrino background (CNB) will ever be directly observed, as the low-energetic CNB neutrinos interact only extremely weakly [77]. By contrast, a series of physical processes in the early universe such as BBN, the evolution of the CMB temperature anisotropies or the formation of matter structures on large scales are fortunately sensitive to the influence of primordial neutrinos, which provides us with compelling indirect evidence for their existence [78, 79].

The observed oscillations between the three neutrino flavours
[4, 5] indicate that neutrinos have small
masses^{2}^{2}2In the following discussion we shall restrict ourselves to the relic
abundance of primordial neutrinos. If neutrinos are Dirac fermions, the abundance
of antineutrinos should at each time be approximately the same as the abundance of neutrinos..
This has a direct impact on their evolution after decoupling.
If neutrinos were massless, their temperature would decrease for the most
part in parallel to the photon temperature as the universe continues to expand.
Only at photon temperatures around the electron mass ,
and would behave slightly differently.
Around , the thermal production of electrons and positrons
begins to cease.
annihilations into photons then deposit the entire energy formerly contained
in electrons and positrons in the photon component, which slows down the decline of
for a short time, but not the decline of .
For massless neutrinos entropy conservation would imply
and neutrinos would presently have a density .
The energy density of massive neutrinos, however, experiences a slower
redshift due to the cosmic expansion than the energy density of massless
neutrinos.
While the energy of a massless neutrino goes to zero
as the universe expands, the energy of a neutrino mass eigenstate
with mass asymptotically approaches .
Once the energy of a massive neutrino is dominated by its mass rather
its momentum, it becomes nonrelativistic.
For sufficiently large neutrino masses, the energy contained in nonrelativistic
neutrinos thus outweighs by far the energy of neutrinos that are still relativistic,
such that the present neutrino density is well described by

(2.5) |

where the sum runs over all mass eigenstates that have turned nonrelativistic at some value of below , i.e., given the measured mass squared differences, over at least two out of three states. The lower bound on the sum of neutrino masses implied by the mass squared differences is roughly , so that . On the other hand, several cosmological observations constrain from above. Massive free-streaming neutrinos damp the growth of matter fluctuations and could thus leave an imprint in large-scale structure (LSS) observables [80, 81]. So far, no effects from neutrino masses have yet been observed. Instead, combining data from galaxy surveys, WMAP, BAO, and type Ia SNe, one is able to put an upper limit of on [82], which corresponds to .

After leaving thermal equilibrium, most neutrinos never again interact with other particles. The entropy and total number of neutrinos hence remain practically unchanged after decoupling, which is why we speak of the neutrinos as being frozen out. At the time neutrinos decouple, they are relativistic. Their entropy and number densities thus subsequently always evolve as the corresponding densities of massless neutrinos would do, independently of the fact that neutrinos are actually massive, turning nonrelativistic at lower temperatures. Because of this peculiar thermal history, neutrinos represent a prime example for what is often referred to as hot relics. With the aid of the would-be temperature of massless neutrinos, , we then obtain and .

In conclusion, we find that also neutrinos contribute only to a negligibly small extent to the total energy density of the universe,

(2.6) |

which follows from Eq. (2.5) and the bounds on the total neutrino mass,

(2.7) |

In return, their entropy density is almost as large as the one of the CMB photons. The present radiation entropy density , comprising the photon entropy density and the entropy densities of all hot relics, i.e. neutrinos in the standard hot big bang scenario, then turns out to be

(2.8) |

Note that, by definition, can also be written as the entropy of a thermal bath with an effective number of degrees of freedom at temperature ,

(2.9) |

The entropy associated with this density directly corresponds to the entropy inherent in the thermal bath during the hot phase of the early universe. A conclusive explanation for its origin is still lacking and it is a major task of modern particle cosmology to explore possible sources for this primordial entropy. A key motivation of this thesis is to demonstrate that the spontaneous breaking of at the end of inflation represents a viable scenario for its generation.

#### 2.1.3 Baryonic Matter

All forms of matter in the universe that can be more or less well described by standard
particle physics, such as gas clouds, stars, planets, black holes, etc., are baryonic, i.e. made out of ordinary atoms, whose nuclei are composed of protons and
neutrons.^{3}^{3}3In order to ensure that the universe as a whole is electrically charge neutral,
there has to be present one electron for each proton in the universe.
As a single proton is, however, roughly times heavier than an electron, the contribution
from electrons to the total energy presently stored in matter is negligibly small,
which is why we will not consider it any further.
The present abundance of these baryons, or more precisely nucleons, is conveniently
parametrized in terms of the baryon-to-photon ratio ,

(2.10) |

where is the mass of a single nucleon, denotes the present number density of baryons, and where we have used the value for stated in Eq. (2.2). In the standard BBN scenario with three generations of relativistic neutrinos, the primordial abundances of the light nuclei are solely controlled by the baryon-to-photon ratio (cf. Sec. 2.2.2). The measurement of these abundances hence provides us with an observational handle on . Matching the observed abundances with the theoretical BBN prediction, one finds at [2]

(2.11) |

One of the key predictions of standard cosmology is that between BBN and the decoupling of the CMB the number of baryons as well as the photon entropy are conserved such that the baryon-to-photon ratio remains unchanged between these two processes. This prediction can be observationally tested as the CMB power spectrum is fortunately very sensitive to the physical baryon density (cf. Sec. 2.2.1). Fitting the CDM model to the CMB data yields [1]

(2.12) |

which is consistent with the BBN result in Eq. (2.11) and hence serves as yet another endorsement of the standard picture. The agreement between the two determinations of is particular remarkable in so far as they probe completely different physical processes occurring in two widely separated epochs. Due to its high precision, we will from now on, after some additional rounding, use the CMB value as our estimate for the present baryon-to-photon ratio, , which corresponds to a baryon density parameter .

Depending on the perspective, we are led to the conclusion that the present abundance
of baryons in the universe is either exceptionally low or high.
First of all, it is surprising that BBN and the CMB concordantly imply that only
a fraction of roughly of the total energy of the universe resides in baryons.
In view of the fact that our universe appears to be spatially flat, one might rather
expect a baryon density parameter .
The low abundance of baryons is hence an indication for the presence of other
nonbaryonic forms of matter or energy, viz. dark matter and dark energy,
that account for of the energy budget of the universe.
On the other hand is remarkably large compared to the theoretical
expectation.^{4}^{4}4It
is also large compared to the observed abundance of luminous matter.
The density parameter of stars is smaller than by one order of magnitude,
[83].
Most baryons are thus optically dark, probably contained in some diffuse
intergalactic medium [84].
In the early universe the baryon-to-photon ratio freezes out when the baryons
decouple from the thermal bath at temperatures of .
Assuming that the universe is locally baryon-antibaryon symmetric down to
temperatures of this magnitude, the annihilation of baron-antibaryon pairs shortly before
decoupling would dramatically reduce the abundances of both baryons and antibaryons.
In consequence of this annihilation catastrophe the present baryon-to-photon would be
nine orders of magnitude smaller than the observed value,
[62, 85].
The most reasonable way out of the annihilation catastrophe is the possibility that
the universe possesses a baryon-antibaryon asymmetry at temperatures of
.
The excess of baryons over antibaryons at the time of annihilation would then explain
the large observed baryon abundance.

Further evidence for a primordial baryon asymmetry comes from the fact that
the observable universe seems to contain almost exclusively matter and
almost no antimatter.^{5}^{5}5Antiparticles of cosmic origin such as antiprotons
and positrons are seen in cosmic rays.
Their fluxes are, however, consistent with the assumption that they are merely
secondaries produced in energetic collisions of cosmic rays with the interstellar
medium rather than primordial relics.
If there were to exist large areas of antimatter in the universe, annihilation processes
along the boundaries between the matter and antimatter domains would result in characteristic
gamma ray signals.
As no such signals have yet been observed, the local abundance of antimatter can
be tightly constrained on a multitude of length scales, ranging from our solar
system, to galaxies and clusters of galaxies.
X- and gamma-ray observations of the Bullet Cluster, a system of two colliding galaxy clusters,
put for instance an upper bound of on the local antimatter fraction, thus
ruling out serious amounts of antimatter on scales of ,
which are the largest scales directly probed so far [86].
Furthermore, assuming that matter and antimatter are present in equal shares
on cosmological scales, one can show that the matter domain we inhabit
virtually has to cover the entire visible universe [87].

The absence of antimatter in our universe thus allows for a different interpretation of the baryon-to-photon ratio . As the ratio of photons to antibaryons is practically zero, can also be regarded as a measure for the baryon asymmetry of the universe (BAU),

(2.13) |

To emphasize this different interpretation of the baryon-to-photon ratio we will write instead of in the following, where the subscript is supposed to refer to the total baryon number of the universe. Again, standard cosmology lacks an explanation for the origin of this primordial asymmetry. A second key motivation for this thesis is hence to identify a natural mechanism for the dynamical generation of the BAU that can be consistently embedded into an overall picture of the early universe. As we will demonstrate, leptogenesis after nonthermal neutrino production in the decay of Higgs bosons represents a viable and particularly attractive option.

#### 2.1.4 Dark Matter

A plethora of astrophysical and cosmological observations indicates that next to ordinary matter
some form of dark matter (DM), i.e. nonluminous and nonabsorbing matter which reveals
its existence only through its gravitational influence on visible matter, is ubiquitously
present in the universe.^{6}^{6}6For recent reviews on dark matter, cf. for instance
Refs. [88, 89, 90, 91].
Another ansatz to account for the various observed, but unexplained gravitational effects
is to modify the theory of general relativity.
While modifications of gravity (cf. in particular Refs. [92, 93])
are often able to explain isolated phenomena, they usually struggle to give a
consistent description of all observed phenomena, which is why we will not consider
them any further in this thesis.
Direct evidence for dark matter derives from all observable length scales.
The rotation curves of spiral galaxies as well as
the velocity dispersions of stars in elliptical galaxies
probe the abundance of dark matter on the scale of individual
galaxies.^{7}^{7}7Seminal works in this field have been the observations
by Vera Rubin and Kent Ford, who measured the rotation curve of the
Andromeda Nebula in 1970 [94],
as well as by Sandra Faber and Robert Jackson, who studied stellar velocities
in elliptical galaxies in 1976 [95].
This applies in particular to our own galaxy, whose rotation curve in combination
with other data allows to determine the fraction of dark matter in the neighborhood
of our solar system quite precisely [96].
On the scale of clusters of galaxies, peculiar galaxy velocities in viralized galaxy
clusters, X-ray observations of the hot intracluster gas and gravitational lensing
effects on background galaxies point to large amounts of dark
matter.^{8}^{8}8The first astronomer to stumble upon the problem of the
missing mass in galaxy clusters was Fritz Zwicky.
In 1933, observations of the Coma Cluster led him to conclude that the galaxies
in the cluster should actually fly apart, if there were not
large amounts of invisible matter present in it, holding them together [97].
Zwicky is hence usually credited as the discoverer of dark matter.
Especially compelling evidence for dark matter comes from detailed studies
of the Bullet Cluster, whose dynamics can only be understood if it is assumed
to be predominantly composed of very weakly self-interacting dark matter [98].
Finally, on cosmological scales the presence of dark matter is implied by the theory of
structure formation.
If the presently observed LSS of matter in the universe was to be traced back
only to the density fluctuations of ordinary baryonic matter at the time of photon decoupling,
the temperature anisotropies in the CMB would have to be at the level of .
However, the fact that they are actually two orders of magnitude smaller indicates
that baryonic density perturbations can, in fact,
not be the source of the required primordial wells of the gravitational potential.
Instead these potential wells have to be attributed to some form of nonbaryonic dark matter that,
unimpeded by photon pressure, is able to start clumping way before decoupling.
Furthermore, numerical simulations of structure formation show that most dark matter has to
be cold at the onset of structure formation, i.e. has to turn nonrelativistic long before
the energy in matter begins to dominate over the energy in radiation.^{9}^{9}9As light
neutrinos turn nonrelativistic only at very late times in the cosmological evolution, they
represent, in fact, a form of hot dark matter in the current universe.

By now the overwhelming observational evidence has firmly established the notion that nonbaryonic cold dark matter (CDM) is the prevailing form of matter in the universe. It is thus one of the key ingredients to the CDM model. Strong support for the CDM picture is again provided by the CMB power spectrum, which is next to the baryon density also sensitive to the total matter density (cf. Sec. 2.2.1). Assuming dark matter to be cold and nonbaryonic, the combined CMB, BAO and data allow for a precise determination of [1],

(2.14) |

which is roughly six times larger than the present baryon density as
inferred from the primordial abundances of the light elements or the CMB power spectrum.
With the aid of Eqs. (2.12) and (2.14),
the present density parameter of dark matter then turns out to
be^{10}^{10}10Later on we shall use a rounded version of the value
in Eq. (2.15), namely .

(2.15) |

We thus know quite certainly that dark matter accounts for roughly of the
energy budget of the universe.
The nature and the origin of dark matter have, however, remained mysterious puzzles so far.
At the present stage we are merely able to constrain to some extent its properties.
First of all, the mismatch between determinations of and ,
i.e. the present abundances of baryons in particular and of matter in general,
as well as arguments based on the theory of structure
formation indicate that dark matter has to be cold and nonbaryonic for the most
part.^{11}^{11}11Certain scenarios of warm dark matter or mixed dark matter which is composed of
a mixture of cold, warm and or hot components, are also admissible [99, 100].
Likewise, also small amounts of baryonic matter in the form of massive compact halo objects
(MACHOs) [101, 102] and or cold molecular gas
clouds [103] may well contribute to the dark matter in galaxy halos.
As it is dark, the particles constituting dark matter are usually assumed to
be electrically neutral.
Similarly, if these particles carried colour charge, they would strongly interact with
baryons, thus altering, for instance, the predictions of BBN and the appearance of the CMB.
Hence the dark matter particles are assumed to be colour-neutral.
Finally, they have to be perfectly stable or at least sufficiently long-lived in order to
explain the presence and influence of dark matter on cosmological time scales up to the current
epoch.
Interestingly, no known particle fulfills all these requirement
and thus the existence of dark matter is one of the strongest indications for
physics beyond the standard model.
Particle cosmology now faces the task to identify which hypothetical new elementary
particles could serve as dark matter particles, embed dark matter into a consistent
picture of the cosmological evolution, and explain in particular how its present
abundance is generated (cf. Eq. (2.15)).
Therefore, the third key motivation of this thesis is to demonstrate that several
well-motivated dark matter scenarios can actually be easily realized, if reheating
after inflation is triggered by the phase transition.
For the most part, we will consider a scenario in which thermally produced gravitinos
account for dark matter.
In Ch. 8, we will then turn to a setup in which either higgsinos
or winos represent the constituents of dark matter.

#### 2.1.5 Dark Energy

A crucial result of our discussion so far is that dark matter, baryonic matter,
neutrinos and photons together account for only roughly of the energy
budget of the universe.
The remaining have to be attributed to some form of dark energy
that, as opposed to dark matter, does not cluster under the influence of gravity.
At the present stage we almost do not know anything about the nature and the origin
of dark energy, whereby dark energy represents one of the greatest mysteries of modern physics.
At least some light on the properties of dark energy is shed by the fact that
the expansion of our universe is currently accelerating.^{12}^{12}12The accelerated
expansion of our universe became evident for the first time in measurements of the
distance-redshift relation of high-redshift type Ia SNe in
1998 [104, 105].
As matter and radiation on their own always lead to either a decelerating expansion
or an accelerated contraction, the dark energy has to be responsible for the
observed acceleration.
Assuming that dark energy can be described as a perfect fluid, just as all other forms of
matter and energy in the universe, the requirement that it be the source of the
accelerated expansion constrains its equation of state,
, where and
denote the pressure and the energy density of dark energy, respectively.
In other words: the accelerated expansion indicates that dark energy has a negative pressure.

There are several attempts to explain the presence of dark energy.
Many approaches assume, for instance, that dark energy corresponds to the energy of a
scalar field moving in some specific potential.
Depending on whether this field has a canonical kinetic term or not, dark energy is then
often referred to as quintessence [106, 107] or
essence [108].
An alternative possibility is that dark energy is entirely illusory, being in fact
an artifact of an incorrect treatment of gravity.
In this view, general relativity has to be modified in such a way
that the accelerated expansion can be accounted for without any recourse to dark
energy [109, 110].
The simplest solution, however, is provided by Einstein’s cosmological constant .
Including a term in the field equations of general relativity
corresponds to adding a constant vacuum energy density
with and equation of state to the energy budget
of the universe.
Although this ansatz is the least sophisticated one, it is consistent with all observations
and thus, along the lines of Occam’s razor, the explanation of choice for dark energy in the
CDM model.^{13}^{13}13Naively one might expect the energy density of the vacuum to be
related to the Planck scale, .
Interpreting dark energy as the energy of the vacuum, one then has to explain
why .
For a classic discussion of this so far unsolved problem cf. Ref. [111].
Our earlier results for the density parameters of all other forms of matter and energy
in the CDM model then allow us to calculate the density parameter of
dark energy [1],

(2.16) |

Finally, we remark that fitting the CMB, BAO and the SNe data from Ref. [68] to a relaxed version of the CDM model, in which and are allowed to differ from and , respectively, yields a dark matter equation of state [1], which is in excellent agreement with the assumption of a cosmological constant. For the moment being, as long as there is no commonly accepted explanation of dark energy in sight, we thus settle for a rather pragmatic approach and adopt the notion of a cosmological constant in this thesis, keeping in mind that it should be regarded as a placeholder for a future theory of dark energy that is still to come.

#### 2.1.6 Stages in the Expansion History

The identification of the key items in the cosmic energy inventory as well as the determination of their respective contributions to the total energy density mark milestones of modern cosmology. Together with the current expansion rate , the density parameters fully determine the present state of the universe on all scales on which the cosmological principle holds. On top of that, they also allow to trace the evolution of the universe back in time up to temperatures of , i.e. until weak interactions begin to bring about interchanges between the abundances of the different species. Below the threshold for pair production, , the energy densities of photons, matter and dark energy can, for instance, be written as functions of the cosmological redshift in the following way,

(2.17) |

with denoting the coefficient in the equation of state of species . We respectively have , and . The energy density of a nonrelativistic neutrino species with typical momentum and mass evolves similarly to the matter energy density ,

(2.18) |

Once the typical neutrino momenta begin to exceed , the respective
neutrino species becomes relativistic,^{14}^{14}14Given the allowed range of the total neutrino mass
(cf. Eq. (2.7)), matching the two expressions for
in Eqs. (2.18) and (2.19) and solving for
shows that the heaviest neutrino, which eventually contributes most
to , turns nonrelativistic at a redshift of .
so that its energy density henceforth runs in parallel to ,

(2.19) |

The density of the total radiation energy is given as usual, , with counting the effective number of relativistic degrees of freedom.

In the present epoch dark energy dominates the total energy of the universe, . However, as the energy densities of radiation, matter and dark energy scale differently with redshift , this changes as we go back in time. First, at the energy contained in matter catches up with dark energy, . Then, at radiation takes eventually over as the dominant form of energy in the universe, . The above scaling relations for the energy densities imply

(2.20) |

where we have used that and . These two redshifts correspond to the following photon temperatures,

(2.21) |

as well as to the following values of the cosmic time ,

(2.22) |

which are to be compared to the age of the universe, [1].

In summary, we conclude that the universe experiences at least three
dynamically different stages in its expansion history.
(i) In the very recent cosmic past, , the energy of the universe
is dominated by the vacuum contribution, which, due to its negative pressure, causes
the expansion to accelerate.
(ii) Between and most energy is contained in pressureless matter.
Note that it is in this epoch that matter structures are able to form in the
universe.^{15}^{15}15Curiously enough, the matter-dominated era lasts sufficiently long
to allow for the formation of such complex structures as galaxies, solar systems and human
beings, which, from the perspective of mankind, appears to be a
fortunate cosmic coincidence.
The question of why dark energy becomes relevant exactly at the present time, i.e. why presently rather than
or , is one of the greatest puzzles of modern cosmology.
Cf. e.g. Ref. [112].
(iii) For radiation is the most abundant form of energy in the universe.
When speaking of the hot thermal phase of the early universe or the
hot early universe, we actually refer to this phase of radiation domination.
During the radiation-dominated era the universe is filled by a hot plasma in thermal equilibrium
that becomes increasingly hotter and denser as one goes further back in time.
In the approximation of a constant number of relativistic degrees of freedom
, the temperature of the thermal bath scales inversely
proportional to ,

(2.23) |

where we have normalized to its value at the time of neutrino decoupling. As the temperature continues to rise, more and more particle species reach thermal equilibrium with the bath, causing to increase. Turning this picture around, we may equivalently say that in the hot early universe various species decouple one after another from the thermal bath in consequence of the declining temperature. These departures from thermal equilibrium shape the present state of the universe. Up to now we have already discussed the decoupling of neutrinos at and the decoupling of baryons at . As we will see later on, similar nonequilibrium processes at even higher temperatures may be responsible for the relic density of dark matter and the baryon asymmetry of the universe. In fact, the very aim of this thesis is to describe a possible origin for the hot thermal phase of the early universe, namely the spontaneous breaking of at the end of inflation, that naturally entails the simultaneous generation of entropy, baryon asymmetry and dark matter.

### 2.2 The Hot Thermal Phase

The hot early universe represents the stage for a great variety of physical processes taking place over an enormous range of energy scales (cf. Fig. 2.1 for an overview of the main events in its thermal history). As a final preparation before turning to our own scenario, we shall now discuss in more detail the decoupling of the CMB, primordial nucleosynthesis, the QCD and the electroweak phase transition as well as electroweak sphalerons.

#### 2.2.1 The Cosmic Microwave Background

Towards the end of the radiation-dominated phase, at temperatures of
, protons, i.e. hydrogen nuclei, are kept
in thermal equilibrium via the steady interplay of radiative recombination
and photoionization processes.
However, as the plasma cools in the course of the expansion, photoionization becomes
less efficient, the hydrogen nuclei begin to bind free electrons into neutral
atoms and the ionization fraction of hydrogen freezes out at a vanishingly small value.
This process is usually referred to as hydrogen recombination.^{16}^{16}16Prior to
hydrogen recombination, at , helium decouples in a
similar way.
As hydrogen is still fully ionized at this time, the universe remains opaque after
helium recombination.
Due to the high abundance of thermal photons in the plasma it takes place at a temperature
significantly below the binding energy of hydrogen, .
In fact, the temperature has to drop to
until the fractional ionization reaches a value of .
As the abundance of free electrons continues to decrease even further, the rate
of Thomson scatterings between thermal photons and plasma electrons falls below
the Hubble rate .
At the mean free photon path
equals the Hubble radius , or equivalently ,
and most photons scatter for the last time.
This moment of last scattering marks the time when the
photons decouple and the universe becomes transparent to radiation.
After decoupling the photons freely propagate until they eventually reach us
in the form of CMB radiation.
In this sense the CMB represents a full-sky picture of the early universe at a temperature
of , i.e. at a redshift
and a cosmic time .

To be precise, the decoupling of the CMB actually occurs during the matter-dominated era (cf. in Eq. (2.21)). But as its origin is inextricably linked with the thermal history of the universe, it represents nonetheless one of the main physical phenomena associated with the hot big bang [113]. In particular, the fact that the CMB has an almost perfect Planckian spectrum may be regarded as key evidence for an early stage during which the universe was filled by a hot plasma in thermal equilibrium. Alternative attempts to explain the origin of the CMB, such as the idea put forward by the proponents of the steady state theory proposing that the CMB may in fact be starlight thermalized by dust grains, typically end up with a superposition of blackbody spectra corresponding to different temperatures.

The CMB not only provides striking evidence for the hot thermal phase,
as we have seen in Sec. 2.1, it also allows to precisely
determine a multitude of cosmological parameters that enter into the theoretical description
of the early universe.^{17}^{17}17For reviews on the physics of the CMB
and its potential to constrain cosmological models, cf. for instance
Refs. [114, 115].
The primary CMB observable encoding cosmological information is the variation
of the CMB temperature across the sky, which is conveniently characterized by
the angular power spectrum of the relative temperature
fluctuations,

(2.24) |

Except for the dipole anisotropy, which is interpreted as being due to the
motion of the earth relative to the absolute CMB rest frame, the CMB temperature
anisotropies directly correspond to the density perturbations inherent in the
baryon-photon fluid at the time of last scattering.
Several physical processes leave their imprint in the observed power spectrum.
(i) The tight coupling between photons and baryons leads to higher temperatures
in regions of high baryon density.
(ii) Photons that have to climb out of potential wells after decoupling are
gravitationally redshifted.
This translates into a shift of the observed
with respect to the intrinsic temperature fluctuation,
which is usually referred to as the Sachs-Wolfe effect [116].
Similarly, decaying gravitational potentials traversed by the CMB photons
on their way from the surface of last scattering to the observer induce small
boots in the observed CMB temperature.
This is known as the integrated Sachs-Wolfe effect.
(iii) The non-zero velocity of the plasma at decoupling results in a Doppler
shift in the frequency of the CMB photons.
(iv) Perturbations in the gravitational potential, induced by the
growing density fluctuations of dark matter, as well as photon pressure
drive acoustic oscillations in the photon-baryon fluid, which gives rise to a
series of acoustic peaks in the CMB power spectrum.^{18}^{18}18Perturbations
in the photon-baryon fluid can only evolve causally as long as they extend over
scales smaller than the sound horizon.
This explains the position of the first acoustic peak in the CMB power spectrum.
It is located at an angular scale of roughly or equivalently
at , which corresponds to the angular diameter of the sound
horizon at last scattering.
These four effects, but in particular the acoustic peaks,
are very sensitive to the parameters of the underlying cosmology.
Barring a few degeneracies, the CMB power spectrum encodes
information about at least ten basic cosmological parameters.

First of all, four parameters characterize the power spectra of primordial density fluctuations as well as primordial gravitational waves. These primordial scalar and tensor perturbations, as they are also referred to, eventually evolve into the CMB temperature fluctuations. The parameters characterizing their power spectra, and , hence determine the initial conditions for the evolution of the CMB anisotropies. Usually, and are taken to be power-laws,

(2.25) |

where is the comoving momentum scale and stands for an arbitrary reference scale. Technically, denotes the power spectrum of the curvature perturbation , which measures the spatial curvature of a comoving slicing of spacetime. represents in fact the sum of two power spectra, and , which respectively account for the two physical polarization modes and of the general traceless and transverse spatial metric perturbation. Note that due to rotational invariance . The great virtue of the three perturbations , and is that they are time-independent at early times, i.e. as long as they extend over scales larger than the Hubble radius . So far, the CMB data has revealed no sign of tensor modes. Thus, only the curvature perturbation amplitude as well as the scalar spectral index have been measured up to now. Neglecting potential tensor contributions and using a reference scale , the combined WMAP, BAO and data yields [1],

(2.26) |

For comparison, the COBE data implies an amplitude at roughly the same scale . This result is usually referred to as the COBE normalization of the scalar power spectrum [117]. WMAP, BAO and the SNe data from Ref. [68] together yield a tight upper bound on the tensor-to-scalar ratio, at . A measurement of the tensor spectral index is beyond the scope of any experiment in the near future. In single-field slow-roll models of inflation (cf. Sec. 3.1) does not represent an independent parameter in any case. It is rather directly related to the tensor-to-scalar ratio via the consistency relation, , which reduces the number of free parameters fixing the initial conditions of the CMB anisotropies to three. The background cosmology setting the stage for the evolution of the CMB anisotropies is described by at least five parameters: the expansion rate , the energy densities of matter and baryons, or equivalently and , the density parameter of dark energy , and the coefficient in the equation of state for dark energy. In Sec. 2.1, we discussed in detail the numerical values of these parameters according to the CMB data in combination with other cosmological observations. Finally, one astrophysical parameter influences the CMB power spectrum: the integrated optical depth , which characterizes the amount of CMB photons that undergo Thomson scattering owing to the reionization of the universe in the recent cosmic past. completes the set of standard parameters usually included in analyses of the CMB power spectrum. Beyond this set further parameters, such as the density of massive neutrinos or the running of the scalar spectral index , may be taken into account as well.

While the CMB stands out as one of the main pillars of the picture of the big bang,
it also shows very plainly some of the severe problems big bang cosmology is facing
with regard to its initial conditions.
First of all, the observation that presently does not
deviate by more than from unity gives rise to the flatness problem.
In a decelerating universe the deviation from exact flatness
always grows as some power of the cosmic
time.^{19}^{19}19Given a scale factor ,
scales like .
During the phases of radiation and matter domination we respectively have
and .
The total density parameter of a universe
exhibiting a small, but non-zero curvature in the present epoch
must hence approach unity to arbitrary precision as one goes back in time.
In other words, the initial value of must be unnaturally
fine-tuned.
Second, at the time of last scattering the past or particle horizon,
i.e. the distance scale characterizing the radial extent of causally connected domains,
is of corresponding to an angular diameter of
in the sky.
By contrast, the CMB is highly isotropic across the entire sky, which is to say
that at the time of decoupling the photon temperature is almost perfectly homogeneous
over a huge number of causally disconnected regions.
Again, this high degree of homogeneity can only be achieved by an unnatural
fine-tuning of the initial conditions, a puzzle which is known as the horizon problem.
Furthermore, the minute deviations from an exactly isotropic temperature, that
we do observe in the CMB, finally lead to the third and perhaps most severe problem.
The mechanism responsible for the high degree of homogeneity over a multitude
of causally disconnected regions also has to explain why the temperature fluctuations
around the homogeneous background are precisely at the level of and, in particular,
why they are correlated over scales exceeding the causal horizon at decoupling.
This problem may be translated into the following two fundamental questions:
(i) what is the origin of the primordial scalar and tensor perturbations and
(ii) which statistical properties do they have?
As we will see in Sec. 3.1, all these three problems concerning the initial
conditions of the hot big bang can be successfully solved in inflationary cosmology.

#### 2.2.2 Primordial Nucleosynthesis

Primordial or big bang nucleosynthesis (BBN), i.e. the generation of the
light elements during the first of the radiation-dominated era,
represents the earliest testable nonequilibrium process in the history
of the universe which can be accounted for by well-understood standard model physics
only (cf. Fig. 2.1).^{20}^{20}20For reviews on BBN, cf. for
instance Refs. [118, 119].
At present it hence provides the deepest reliable probe of the early universe.
The overall agreement of the observed primordial abundances of the light elements
with the predictions of BBN serves as a strong corroboration of hot big bang cosmology,
underpinning our picture of the early universe to a similar extent as the anisotropies
in the CMB.

Before the onset of BBN, at temperatures or correspondingly at times , the weak interactions , , and keep the neutron-to-proton ratio in thermal equilibrium, with denoting the neutron-proton mass difference. Around a temperature of the rate of neutron-proton interconversion processes eventually drops below the Hubble rate and the neutron-to-proton ratio freezes out at . Subsequent to freeze-out, still continues to decrease due to neutrons undergoing decay, . At the time the neutrons decouple from the thermal bath, the temperature has already fallen below the binding energy of deuterium, . The synthesis of deuterium, however, does not yet commence because of the large abundance of highly energetic photons that immediately dissociate each newly formed deuterium nucleus. This delay in the production of the light elements is referred to as the deuterium bottleneck. It is overcome once the number of photons per baryon above the deuterium photodissociation threshold has decreased below unity, which happens at a temperature or roughly at the end of the first three minutes. The breaking of the deuterium bottleneck marks the onset of BBN. At last deuterium can be efficiently produced and further processed into heavier elements such as helium-3, helium-4 and lithium-7.

Independently of the nuclear reaction rates, virtually all free neutrons end up bound in helium-4, which is the most stable one among the light elements. At the neutron-to-proton ratio has decreased to and the primordial mass fraction of helium-4 can be estimated as

(2.27) |

which corresponds to a ratio by number of helium-4 to hydrogen of . Deuterium, helium-3 and lithium-7 are produced in much smaller numbers. At the end of BBN around , when the temperature has dropped to and most nuclear reactions have become inefficient, and are of , while is of . The complicated network of nuclear reactions that lead to these primordial abundances is described by a coupled system of kinetic equations that needs to be solved numerically [120, 121]. Besides the temperature or equivalently the cosmic time , the Hubble rate and the nuclear reaction rates that enter into these equations are functions of only one cosmological parameter: the number density of baryons during BBN. As is directly related to the present value of the BAU, , this explains why the observed primordial abundances of the light elements give us a handle on (cf. Sec. 2.1.3).

The abundance of primordial deuterium is inferred from spectra of high-redshift
quasar absorption systems, while primordial helium-4 is observed
in low-metallicity regions of ionized hydrogen.
The spectra of old metal-poor, i.e. population II stars in the spheroid
of our galaxy allow to determine the primordial abundance of lithium-7.
All in all, the theoretical BBN predictions match the observed abundances of deuterium,
helium-4 and lithium-7 quite well within the range stated in
Eq. (2.11).^{21}^{21}21Data on helium-3 solely derives from the solar system and
high-metallicity regions of ionized hydrogen in our galaxy, which makes it difficult to infer
its primordial abundance.
On top of that, the theory of stellar helium-3 synthesis is in conflict with
observations.
For these two reasons, helium-3 is usually not used as a cosmological probe.
An obvious curiosity, however, is that the lithium-7 abundance points to a value of
that is smaller by at least than the value jointly
favoured by the abundances of deuterium and helium-4.
This discrepancy is known as the lithium problem [122]
and potentially indicates effects of new physics.

Leaving aside the lithium problem, we conclude that BBN is able to correctly predict the primordial abundances of the light elements over a range of nine orders magnitude. This success is a milestone of big bang cosmology, encouraging us to believe that the laws of physics which we are able to test in laboratory experiments also apply to the very first moments of the universe. We are thus confident that modern particle physics allows us to speculate about the history of the universe at still earlier times, , although as of now we have no means of observationally accessing them. Furthermore, the success of BBN provides us with a powerful tool to constrain deviations from the standard cosmology.

The helium-4 abundance, for instance, is very sensitive to the value of and thus the presence of additional relativistic species during BBN [123]. Increasing above its standard value entails a faster Hubble expansion, which results in the neutrons decoupling at earlier times. The neutron-to-proton ratio then freezes out a correspondingly higher temperature, leading to a larger abundance of primordial helium-4 (cf. Eq. (2.27)). Deviations from the standard value of are usually parametrized in terms of an effective number of neutrino species . Before annihilation, is given as [124] and is related to through . In turns out that the primordial helium-4 mass fraction scales with as [125], which allows to place limits on by means of the measured abundance of primordial helium-4. In combination with the seven-year WMAP data, one finds at [126].

Likewise, the late-time decay of a massive nonrelativistic particle which is not included in the standard BBN scenario may as well alter the primordial abundances of the light elements. Similarly to additional relativistic species, the presence of such a particle modifies the expansion rate prior to its decay. On top of that, if the new particle dominates the energy density of the universe at the time of its decay, a significant amount of entropy is produced while its decay products thermalize. This changes the time-temperature relationship and results in a diluted baryon-to-photon ratio. Based on these effects, one can derive an upper bound on the lifetime of the decaying particle or equivalently a lower bound on the temperature of the thermal bath at the time the entropy production is completed [127]. If the process of entropy production shortly before BBN is identified with the reheating of the universe after inflation, this lower bound on the temperature corresponds to the lowest possible value of the reheating temperature (cf. Sec. 3.1.1). Combining the observed primordial abundances of deuterium and helium-4 with CMB and LSS data, one obtains at [128].

Independently of whether a long-lived massive particle dominates the energy density of the universe or not, it may after all spoil the success of standard BBN through the cascade processes induced by its decay. Charged particles or photons emitted in radiative decays of the long-lived particle entail electromagnetic showers [129]. Sufficiently energetic photons produced in these showers are then able to photodisintegrate previously formed light nuclei. Moreover, given appropriate couplings and on condition that they are kinematically allowed, decays into colour-charged particles trigger hadronic cascade processes [47]. These involve energetic pions, kaons, neutrons, protons as well as the corresponding antiparticles, all of which are able to react with the light nuclei in various ways. The hadrons emitted in the decays of the long-lived particle induce, for instance, extraordinary interconversion processes between the background nucleons. This leads to an enhancement of the neutron-to-proton ratio after neutron decoupling and thus to a larger abundance of helium-4. At the same time, the energetic hadrons are also able to dissociate background helium-4 nuclei and to produce the other light elements nonthermally. If the decaying particle is electrically charged, it can form bound states with background nuclei, which again changes the nuclear reaction rates. Especially, the production of lithium-6 may be catalyzed in this way [130].

In order to determine the net effect of a long-lived massive particle on the primordial abundances of the light elements, it is necessary to compute the distributions of the various decay products of the decaying particle as functions of time. These spectra then allow to calculate the rates of the photo- and hadrodissociation, neutron-proton interconversion, and nonstandard production processes induced by the decay of the unstable particle. Requiring the impact of the decaying particle to remain small, such that the consistency between the theoretical BBN predictions and the astrophysical observations is maintained, one can derive constraints on the mass, lifetime and abundance of the unstable particle prior to its decay [48, 49]. In Ch. 8 we will in particular consider bounds on the properties of a very heavy gravitino decaying shortly before BBN [131].

#### 2.2.3 Phase and Topological Transitions

The synthesis of the light elements marks the earliest process in the hot early
universe that is firmly established on the basis of observations.^{22}^{22}22Recall that
BBN enables us to trace the evolution of the hot thermal phase up to temperatures as high as
or equivalently cosmic times as early as (cf. Sec. 2.2.2).
The exact nature of all nonequilibrium processes occurring prior to BBN, such as the
generation of the BAU or the primordial metric perturbations, are currently still subject to
speculations.
On the other hand, the standard model of particle physics describes
the interactions of elementary particles with great precision all the way up to the
TeV scale.
Based on standard model physics one is thus able to make an educated guess about the
history of the universe up to or equivalently
.
In the following we shall in particular elaborate on the phase and topological transitions
which presumably take place in the very early universe.

##### QCD Phase Transition

At temperatures well above the scale of quantum chromodynamics
(QCD),^{23}^{23}23The QCD scale corresponds to the energy scale
at which, according to its renormalization group running in perturbative QCD,
the strong coupling constant formally diverges.
, most quarks, antiquarks and
gluons interact only very weakly with each other.
Instead of being bound in baryons or mesons, they freely propagate through the thermal bath
as independent degrees of freedom, forming what is referred to as a quark-gluon plasma.
However, as the temperature decreases, the strong force becomes increasingly stronger,
until at a temperature all colour-charged particles get
confined in hadrons, i.e. pions for the most part.
This transition from the quark-gluon plasma to hadronic matter is known as
the QCD or quark-gluon phase transition (cf. Fig. 2.1).
Its order parameter, , keeping track of the progress of the QCD
transition as it unfolds, is given by the vacuum expectation value (VEV) of the quark condensate
operator, .
While initially vanishes, it is of
at the end of the QCD phase transition.
As the quark condensate operator transforms nontrivially under chiral transformations,
we conclude that the QCD phase transition entails the spontaneous breaking of
the global chiral symmetry in the quark sector.
We also note that, according to numerical lattice calculations, the QCD phase
transition is most likely a smooth crossover rather than a first or second
order phase transition.
One thus expects that it does not leave any observationally detectable
imprint in the cosmic evolution.

##### Electroweak Phase Transition

The QCD phase transition is believed to be preceded by the electroweak (EW) phase transition, occurring close to the Fermi or electroweak scale (cf. Fig. 2.1). At temperatures all standard model particles are massless and the universe is said to be in the symmetric phase. The order parameter of the electroweak phase transition, , is identified with the VEV of the Higgs product operator , with denoting the standard model Higgs doublet, . By definition, vanishes in the symmetric phase. Once the temperature drops below a critical value , the Higgs boson , the electroweak gauge bosons and all fermions except for neutrinos acquire masses through the Higgs mechanism. This is reflected in the order parameter obtaining a nonzero value that approaches in the zero-temperature limit. Both the explicit value of as well as the order of the electroweak phase transition depend on the Higgs boson mass . The LHC experiments ATLAS and CMS recently presented hints that the Higgs boson may have a relatively large mass, [51, 52]. Based on this value for , one finds [64]. Furthermore, given , the electroweak phase transition turns out to be a smooth crossover without any dramatic cosmological consequences.

After the phase transition the universe is in the Higgs phase and the electroweak symmetry is said to be spontaneously broken to the electromagnetic symmetry,

(2.28) |

This terminology is, however, not quite correct as transforms as
a singlet under all gauge transformations, so that the electroweak symmetry remains
intact even after the electroweak phase transition.
What happens instead is a rearrangement of the physical degrees of freedom,
proceeding in such a way that after the phase transition the electroweak symmetry
is realized in a nonlinear fashion.
It would hence be more appropriate to speak of the electroweak symmetry as being
hidden subsequent to the phase transition.
However, as it is more common to refer to it as being broken,
we will adopt this terminology in the following.^{24}^{24}24Likewise, when referring
to some Higgs product operator acquiring a VEV ,
we will also sometimes write , although we actually mean
.
We note that this discussion applies in particular also to the phase transition, during which the gauge symmetry actually becomes hidden
rather than broken.

##### Electroweak Instanton and Sphaleron Transitions

As the temperature approaches the electroweak scale, also nonperturbative processes which simultaneously violate baryon number and lepton number gain in importance. Their emergence is a direct consequence of the fact that the electroweak dynamics are governed by a chiral and non-Abelian gauge theory. First of all, we note that both global and transformations represent accidental symmetries of the standard model Lagrangian. Hence, both and are conserved in the standard model at the classical level. Due to the chiral nature of the electroweak interactions, they are, however, violated at the quantum level through the triangle anomaly, which results in the divergences of the baryon and lepton number currents, and , being nonzero [12, 13],

(2.29) |

Here, counts the number of fermion families, represents
the Levi-Civita symbol in four dimensions, and
are the field strength tensors of the weak and hypercharge gauge fields, and
and denote the corresponding gauge couplings.
The second ingredient to the nonconservation of and is
the complicated structure of the vacuum of the gauge theory.
As for any non-Abelian gauge theory, the vacuum
manifests itself in infinitely many, homotopically distinct,^{25}^{25}25Gauge
configurations belonging to different homotopy classes are transformed into
each other via large gauge transformations. pure gauge configurations, each
of which is characterized by a specific integer topological charge or Chern-Simons
number .
An important observation is that distinct realizations of the vacuum differing
by are connected to each other via a non-contractible
loop in field configuration space [132].
The field configuration of highest energy along this path is known as the
sphaleron [133].
Corresponding to a saddle-point of the energy functional of the gauge-Higgs system,
the sphaleron represents a classical, spatially localized and static, but unstable
solution of the electroweak field equations.
Its energy determines the height of the potential barrier by
which two adjacent realizations of the vacuum are separated,

(2.30) |

Now combining the nontrivial topology of the vacuum with the fact that the currents and have nonzero divergences (cf. Eq. (2.29)), one can show that both and are violated in topological vacuum transitions,

(2.31) |

In the standard model, in which we have , the smallest jump in and is hence . The difference between and is, by contrast, always conserved in topological transitions. This is also evident from the vanishing divergence of the current, (cf. Eq. (2.29)).

Topological transitions between different realizations of the vacuum come in two different varieties. One possibility is tunneling through the potential barrier via instantons. The instanton rate is, however, proportional to and thus severely suppressed. This is to say that in the standard model - and -violating processes are completely negligible at low temperature. On the other hand, in the hot plasma filling the universe during the radiation-dominated era, thermal fluctuations can lead to sphaleron transitions over the potential barrier [134]. In the Higgs phase, the sphaleron rate is proportional to [135] and hence becomes unsuppressed as soon as . Although the barrier is large at zero temperature, , it rapidly melts away as the temperature approaches the critical value from below (cf. Eq. (2.30)). That is why sphalerons already reach thermal equilibrium at a temperature slightly below the critical temperature, [136], rather than at temperatures as high as . Conversely, we may say that at the sphaleron processes freeze-out, so that for both and are conserved. This means in particular that at the latest around the baryon asymmetry is fixed to its present value.

During the restoration of the electroweak symmetry the potential barrier vanishes completely. Hence, the actual sphaleron configuration in the sense of a saddle-point of the energy functional no longer exists in the symmetric phase. Instead, at , topological transitions occur due to thermal fluctuations in the electroweak gauge fields. In the following we shall, however, refer to these transitions as sphaleron processes nonetheless. In the symmetric phase, sphaleron transitions occur at rate per unit volume [137] where . This result can be used to show that sphalerons are in thermal equilibrium up to a temperature . At higher temperatures the sphaleron rate is again outweighed by the expansion rate.

Above the electroweak scale all standard model gauge and Yukawa interactions as well as the electroweak sphaleron and QCD instanton processes are in thermal equilibrium. This implies relations between the chemical potentials of all fermions and Higgs particles, which, together with the requirement that the total hypercharge of the thermal bath be zero, can be used to derive the sphaleron-driven equilibrium values of and [138],

(2.32) |

with denoting the number of Higgs doublets. The standard model (SM) only contains one Higgs doublet , while in its minimal supersymmetric extension, the minimal supersymmetric standard model (MSSM), two Higgs doublets, and , are required in order to ensure anomaly freedom,

(2.33) |

From Eq. (2.32) we conclude that if , sphaleron processes always completely wash out any baryon asymmetry, which is generated in some nonequilibrium process at . By contrast, as is conserved in topological transitions, any primordial asymmetry is guaranteed to survive until sphaleron freeze-out. From this perspective, the baryon asymmetry, which we presently observe in the universe, points to a nonequilibrium process above the electroweak scale that is responsible for the generation of a primordial asymmetry. As we will see in Sec.