European Research Universität Hamburg Council DER FORSCHUNG I DER LEHRE | DER BILDUNG

PRIMORDIAL NUCLEOSYNTHESIS IN THE PRESENCE OF MEV-SCALE DARK SECTORS

Dissertation

zur Erlangung des Doktorgrades an der Fakultät für Mathematik, Informatik und Naturwissenschaften

Fachbereich Physik der Universitat Hamburg

vorgelegt von

Marco Hufnagel

aus

Hamburg

2020

Gutachter der Dissertation: Dr. Kai Schmidt-Hoberg Prof. Dr. Geraldine Servant

Zusammensetzung der Prüfungskommission: Dr. Kai Schmidt-Hoberg Prof. Dr. Geraldine Servant Prof. Dr. Jochen Liske Prof. Dr. Gudrid Moortgat-Pick Dr. Torben Ferber Vorsitzender der Prüfungskommission: Prof. Dr. Jochen Liske

Datum der Disputation: 15.06.2020

Vorsitzender des Fach-Promotionsausschusses PHYSIK: Prof. Dr. Günter H. W. Sigl

Leiter des Fachbereichs PHYSIK: Prof. Dr. Wolfgang Hansen

Dekan der Fakultät MIN: Prof. Dr. Heinrich Graener

“Come on, Rory! It isn't rocket science, it's just quantum physics!” - The Doctor

“This is not the time for vanity. It's the time to show the universe how amazingly awesome I am!” - Captain Qwark

Dedicated to all the equations I have solved before. i

Abstract

In this thesis, we perform a comprehensive study of Big Bang nucleosynthesis constraints on different dark-sector models with MeV-scale particles which are neither fully relativistic nor fully non-relativistic during all relevant temperatures. To this end, we derive a generic set of equations that can be used to determine the light-element abundances for many different dark-sector scenarios. In particular, we take into account all relevant effects that might alter the creation of light elements in the early universe, including modifications to the Hubble rate and time-temperature relation, an adjusted best-fit value for the baryon-to-photon ratio due to an altered effective number of neutrinos, a modified neutrino-decoupling temperature as well as late-time modifications of the nuclear abundances due to photodisintegration. We then solve these equations for the case of dark sectors with MeV-scale particles that are created in the early universe and later decay into either dark or electromagnetic radiation. In the former case, the dark and the visible sectors are completely decoupled, and we show that even such scenarios can be severely constrained if the initial temperature ratio of both sectors is suffi­ ciently large. In the latter case, the particle decay can instead lead to a severe entropy injection into the SM heat bath, and we show that the final constraints can be very different from the naive order-of-magnitude estimates that are usually employed for such scenarios. We then turn to the case of MeV-scale dark matter that can annihilate into all kinematically available SM states, including electrons/positrons, photons, and neutrinos. In this context, we first up­ date the lower bound on the mass of thermal dark matter using improved determinations of the nuclear abundances. Afterwards, we calculate the corresponding constraints on the an­ nihilation cross-section of dark matter and show that, for p-wave suppressed annihilations, the bounds from nucleosynthesis are much stronger than the ones from the CMB and even competitive with the strongest bounds from other indirect searches. Finally, we apply our results to models with self-interacting dark matter as well as to models with axion-like par­ ticles. In the former case, we show that for scalar mediators, most parts of parameter space leading to sizable self-interactions are already excluded by a combination of direct-detection experiments and constraints from nucleosynthesis. In the latter case, we further evaluate the robustness of our constraints by allowing various additional effects that may weaken the bounds of the standard scenario. We find that, while the bounds can indeed be weakened, very relevant robust constraints remain. ii

Zusammenfassung

In dieser Arbeit betrachten wir Modelle mit verschiedenen dünklen Sektoren ünd üntersü- chen die Verwendbarkeit von Vorhersagen der primordialen Nukleosynthese zur Einschrän- küng des jeweiligen Parameterbereiches dieser Modelle. Dabei konzentrieren wir üns aüf Modelle mit Teilchen nahe der MeV-Skala, welche im Bereich der relevanten Temperatu­ ren weder ultra-relativistisch noch nicht-relativistisch sind. Zu diesem Zweck leiten wir ein generisches System von Gleichungen her, welches anschließend verwendet werden kann, um die Häufigkeit der leichten, im frühen Universum erzeugten, Elemente für verschiede­ ne Szenarien vorherzusagen. Dabei berücksichtigen wir alle Effekte, welche die Erzeugung dieser Elemente potenziell beeinflussen können, einschließlich einer möglichen Modifikati­ on der Hubble Rate und der Zeit-Temperatur-Beziehung, einer Änderung des Verhältnisses der Baryonen- und Photonenzahl aufgrund einer geänderten Anzahl effektiver Neutrinos, ei­ ner modifizierten Entkopplungstemperatur der Neutrinos und die nachträgliche Änderung der Änzahl leichter Elemente mittels Photodesintegration. In einem ersten Schritt wenden wir diese Gleichungen dann auf Modelle mit Teilchen nahe der MeV-Skala an, welche im frühen Universum zunächst erzeugt werden und anschließend entweder in dunkle oder elek­ tromagnetische Strahlung zerfallen. Im ersten Fall sind der dunkle und der sichtbare Sektor komplett entkoppelt und wir zeigen, dass auch der Parameterraum solcher Modelle deut­ lich eingeschrankt werden kann, solange das Verhältnis der Temperaturen beider Sektoren ausreichend groß ist. Im zweiten Fall führt der Teilchenzerfall zu einer deutlichen Entropie­ injektion in das Warmebad des SMs und wir zeigen, dass sich die Ergebnisse unserer umfas­ senden Analyse deutlich von den einfachen Abschätzungen unterscheiden konnen, welche normalerweise für solche Szenarien herangezogen werden. Anschließend betrachten wir Mo­ delle mit einem Dunkle-Materie-Kandidaten nahe der MeV-Skala, welcher zusätzlich in die kinematisch erlaubten SM-Teilchen, sprich Elektronen/Positronen, Photonen und Neutrinos, annihilieren kann. In diesem Zusammenhang aktualisieren wir zunächst die unter Schran­ ke an die Masse thermischer Dunkler Materie unter Verwendung aktuellster Messergebnisse. Nachfolgend berechnen wir, welche Einschränkungen sich für den Ännihilationswirkungs- querschnitt ergeben und zeigen, dass diese im Fall von p-Wellen unterdrückten Annihilatio­ nen, deutlich starker sind als jene vom CMB, und sogar vergleichbar mit den Ergebnissen anderer indirekter Suchen. Abschließend betrachten wir Modelle mit selbstwechselwirken­ der Dunkler Materie, sowie Modelle mit Axion-ahnlichen Teilchen. Im ersten Fall zeigen wir, dass für skalare Teilchen die meisten Bereiche mit phänomenologisch relevanten Selbstwech­ selwirkungen bereits durch eine Kombination indirekter und direkter Suchen ausgeschlossen sind. Im zweiten Fall untersuchen wir zusätzlich die Robustheit unserer Ergebnisse und zei­ gen, dass unterschiedliche Effekte die Resultate zwar abschwachen konnen, in jedem Fall aber deutliche Einschrankungen des Parameterraums verbleiben. iii

THIS THESIS IS BASED ON THE FOLLOWING PUBLICATIONS:

M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on MeV-scale dark sectors. Part I. Sterile decays, JCAP 1802 (2018) 044, [arXiv:1712.03972]

M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on MeV-scale dark sectors. Part II. Electromagnetic decays, JCAP 1811 (2018) 032, [arXiv:1808.09324]

P. F. Depta, M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on the anni­ hilation of MeV-scale dark matter, JCAP 1904 (2019) 029, [arXiv:1901.06944]

K. Bondarenko, A. Boyarsky, T. Bringmann, M. Hufnagel, K. Schmidt-Hoberg, and A. Sokolenko, Direct detection and complementary constraints for sub-GeV dark matter, JHEP 03 (2020) 118, [arXiv:1909.08632]

P. F. Depta, M. Hufnagel, and K. Schmidt-Hoberg, Robust cosmological constraints on axion-like particles, JCAP 05 (2020) 009, [arXiv:2002.08370]

OTHER PUBLICATIONS THAT ARE NOT PART OF THIS THESIS:

M. Garny, J. Heisig, M. Hufnagel, and B. Lülf, Top-philic dark matter within and beyond the WIMP paradigm, Phys. Rev. D97 (2018), no. 7 075002, [arXiv:1802.00814]

M. Garny, J. Heisig, M. Hufnagel, B. Lülf, and S. Vogl, Conversion-driven freeze-out: Dark matter genesis beyond the WIMP paradigm, PoS CORFU2018 (2019) 092, [arXiv:1904.00238] iv v Contents

Contents

1 Introduction 1

2 Particle dynamics in an expanding universe 5 2.1 The Friedmann equations ...... 5 2.2 The relativistic Boltzmann equation ...... 6 2.2.1 The Boltzmann equation in an expanding universe ...... 7 2.2.2 The collision operator in its general form ...... 8 2.2.3 Thermodynamical quantities ...... 11 2.2.4 Local thermal equilibrium ...... 13 2.2.5 A note on Maxwell-Boltzmann distributions ...... 17

3 Big Bang Nucleosynthesis 19 3.1 Particle evolution during nucleosynthesis ...... 19 3.1.1 Electromagnetic radiation ...... 20 3.1.2 Neutrinos ...... 22 3.1.3 Nuclei ...... 27 3.2 Comparison with observations ...... 34 3.2.1 Cosmological observations ...... 34 3.2.2 Theoretical predictions and their uncertainties ...... 35 3.3 Influence of additional dark-sector states ...... 37 3.3.1 Hubble rate ...... 38 3.3.2 Neutrino decoupling ...... 38 3.3.3 Time-temperature relation ...... 39 3.3.4 Baryon-to-photon ratio ...... 40 3.3.5 Photodisintegration ...... 40 3.3.6 Hadrodisintegration ...... 46 3.4 General solution strategy ...... 46 3.4.1 Cosmological evolution ...... 47 3.4.2 Light-element abundances ...... 48 3.4.3 Final constraints ...... 49

4 Bounds on sterile decays of light mediators 51 4.1 Setup and assumptions ...... 51 4.2 Evolution and influence of the dark sector ...... 53 4.2.1 Evolution of the mediator ...... 53 4.2.2 Evolution of the sterile neutrinos ...... 55 4.2.3 Hubble rate and time-temperature relation ...... 55 4.2.4 Combined evolution ...... 56 4.3 Results and discussion ...... 57 Contents vi

4.4 Impact of kinetic decoupling and inverse decays ...... 60 4.4.1 Impact of kinetic decoupling ...... 60 4.4.2 Impact of inverse decays and spin-statistics ...... 62 4.5 Summary ...... 64

5 Bounds on electromagnetic decays of light mediators 65 5.1 Setup and assumptions ...... 65 5.2 Evolution and influence of the dark sector ...... 67 5.2.1 Evolution ofthe mediator ...... 67 5.2.2 Hubble rate and time-temperature relation ...... 68 5.2.3 Photodisintegration ...... 70 5.3 Results and discussion ...... 72 5.3.1 Constraints on the usual freeze-out scenario ...... 72 5.3.2 Constraints on the pure freeze-in scenario ...... 78 5.4 Summary ...... 79

6 Bounds on residual dark-matter annihilations 81 6.1 Setup and assumption ...... 81 6.2 Cosmological evolution ...... 82 6.2.1 Evolution ofthe dark-matter candidate ...... 82 6.2.2 Neutrino decoupling in the presence of neutrino interactions ...... 86 6.2.3 Hubble rate and time-temperature relation ...... 87 6.3 Results and discussion ...... 88 6.3.1 Separation of constraints ...... 88 6.3.2 Constraints on the mass ofthermal dark matter ...... 88 6.3.3 Constraints on the annihilation cross-section from photodisintegration . 90 6.3.4 Comparison with other constraints ...... 93 6.4 Summary ...... 94

7 Bounds on scalar-portal dark-matter models 97 7.1 Definition and properties of the model ...... 98 7.2 Evolution and influence of the dark sector ...... 98 7.3 Results and discussion ...... 101 7.3.1 Constraints on different mass combinations ...... 101 7.3.2 Constraints in the context of dark-matter self-interactions ...... 103 7.4 Summary ...... 107

8 Bounds on axion-like particles 109 8.1 Definition and properties of the model ...... 109 8.2 Evolution and influence of the dark sector ...... 111 8.2.1 Evolution ofthe axion-like particle ...... 111 8.2.2 Freeze-out and re-equilibration ...... 113 vii Contents

8.3 Results and discussion ...... 115 8.3.1 Constraints on the vanilla ALP scenario ...... 115 8.3.2 Constraints beyond the vanilla case ...... 117 8.4 Summary ...... 122

9 Conclusions 123

A Reaction tables 127 A.1 Relevant reactions for Big Bang Nucleosynthesis ...... 127 A.2 Relevant reactions for Photodisintegation ...... 128

B Rates for the cascade processes 129

C Numerical solution techniques 135 C.1 Electromagnetic cascade ...... 135 C.2 Non-thermal nucleosynthesis ...... 136

D Additional plots regarding dark-matter annihilations 137

1

1 Introduction

In recent decades, modern physics has reached a point at which almost all known phenom­ ena of our world can be described by two powerful, yet partially disjoint theories. On the one hand, the Standard Model of Particle Physics (SM), has become the most important tool for describing our universe on small scales as it provides an excellent framework for predicting high-energy phenomena at a high level of accuracy. Not only does the SM allow for an ex­ ceptionally precise prediction of the anomalous magnetic moment of the electron [8] - one of the most precisely measured quantities in physics -, but it also led to the correct prediction of the Higgs boson, which has just recently been discovered by both ATLAS [9] and CMS [10.] On the other hand, the theory of General Relativity (GR) has become crucial for describing our universe on large scales as it provides an accurate description of all known gravitational interactions. GR has correctly predicted the existence of complex phenomena like black holes and gravitational waves, both of which have just recently been observed directly for the very first time [11, 12]. However, despite the overwhelming success of these theories, there re­ main some aspects of our universe that cannot be explained by either of them. One of these mysteries revolves around the nature and origin of dark matter (DM). More precisely, a vari­ ety of independent observations on different scales suggests that there exists a yet unknown type of matter, which is neither luminous nor baryonic but makes up more than one-quarter of the energy budget of our universe [13]. On the scale of galaxies, the existence of DM is backed up by measurements of galactic rotation curves [14], which suggest that the circu­ lar velocity of stars is constant beyond the visible disk. As it stands, this behavior conflicts with theoretical predictions, but can be resolved by embedding the galaxy into a halo of (un­ known) gravitationally-interacting particles. On the scale of galaxy clusters, additional evi­ dence comes from observations of the famous bullet cluster [15], which features an apparent displacement between the maximum of the gravitational potential and the region of highest baryonic matter density. Again, this behavior can be explained by assuming the presence of non-luminous matter that serves as an additional source for the gravitational field. Finally, on even larger scales, additional evidence for the existence of DM is also imprinted into the Cos­ mic Microwave Background (CMB). More precisely, the exact peak positions of the angular CMB power spectrum can only be explained by assuming that the early universe was populated by an additional (dark) type of matter, which was five times more abundant than the ordinary baryonic matter [13].1 However, despite the overwhelming evidence for the existence of DM, its precise nature remains shrouded in mystery. All of the previous observations only hint at gravitational interactions, and it is still unclear whether DM features any coupling to the SM besides gravity. From a model-building perspective, these observations therefore leave an infinite number of possibilities, ranging from models with a single weakly interacting massive particle (WIMP) (see e.g. [17] and references therein)to full-fledged (potentially completely de­ coupled) dark sectors that coexist alongside the one of the SM. Consequently, to narrow down

1See [16] for a review on the evidence for dark matter. Chapter 1. Introduction 2 the set of viable models, it is crucial to exploit all possible search strategies, including particle­ physics experiments like colliders as well as astrophysical and cosmological observations. In this thesis, we contribute to this endeavor and focus on the cosmological implications of DM, or more generally dark sectors, and aim at constraining different dark-matter scenarios by confronting them with the most recent observations from Big Bang Nucleosynthesis (BBN). In a nutshell, BBN describes an early phase in the expansion history of the universe, during which the first light elements such as helium and deuterium were created via nuclear fusion reactions of neutrons and protons, happening at temperatures in the MeV - keV range. Pri­ mordial nucleosynthesis plays a vital role in the cosmic chronology, since the amount of light elements that was created during this time (so far) constitute the earliest probe for our model of the universe.2 Strikingly, the observationally inferred values of the light-element abun­ dances agree exceptionally well with theoretical predictions within the SM, i.e. without the inclusion of DM, and any deviations are strongly constrained [18-23]. While this result makes a strong case for the validity of the SM, it also implies that the inclusion of DM - whatever it might be - must not distort primordial nucleosynthesis too much, as this would inevitably lead to conflicts with observations. This statement is especially important since the existence of an additional dark sector can alter primordial nucleosynthesis in many different ways: On the one hand, any dark sector introduces an additional energy density, which influences the expansion rate of the universe and thus the formation of light elements. This statement even remains true in the case of fully decoupled dark sectors. On the other hand, dark-sector states with electromagnetic decay channels can inject additional entropy into the SM heat bath and also cause late-time modifications of the nuclear abundances due to photodisintegration [24­ 26]. Using BBN, it is thus possible to constrain a variety of different dark-matter models, by first calculating the modified light-element abundances for the given scenario, and afterwards checking if the results are still compatible with the most recent observations. While this pro­ cedure has already been utilized many times in the past, most of these studies were limited to dark-sector states that are either non-relativistic (m » MeV) [22-29] or ultra-relativistic (m MeV) [20] for all temperatures relevant for BBN. As a result, all of these studies assume an explicit difference in scale between the masses of the dark-sector states and the temper­ atures that are relevant for BBN. While these assumptions can be used to correctly describe WIMP dark matter with m > O (GeV) or additional dark radiation with m ~ 0, they cannot be used for MeV-scale dark-sector states, which naturally appear in many models beyond the SM, e.g. in the form of dark photons or dark Higgs bosons. In this case, all relevant energy scales are essentially of the same order, including not only the two mentioned before but also the scale that is set by the binding energy of light nuclei, i.e. the scale that is relevant for pho­ todisintegration. This ultimately leads to a much more complicated phenomenology, since individual particles can transition from the relativistic to the non-relativistic regime during the time of BBN. Motivated by this, we complement the previous literature by performing a comprehensive study of BBN constraints on dark sectors that involve MeV-scale particles.

2In the future, it will, however, most likely be possible to probe even earlier times using gravitational waves. 3

To this end, in chapter 2 we first derive the relevant equations that govern the particle dynamics in an expanding universe. In addition to the Friedmann equations, which yield a global view on the cosmic expansion, we also inspect the relativistic Boltzmann equation in an expanding universe, which describes the evolution of individual particles in terms of a phase-space distribution function. In chapter 3 we then use these results to break down the relevant processes that occur during nucleosynthesis. After performing a detailed discussion in the context of the SM, we compose a list of all modifications that have to be performed in the presence of an additional dark sector. These considerations finally lead to a general for­ malism that allows to calculate the light-element abundances and thus BBN constraints for a variety of different dark-sector models. In the following chapters, we then apply the devel­ oped formalism to constraint different scenarios: In chapter 4 and chapter 5 we study two closely related scenarios with MeV-scale particles that are created in the early universe and later decay into either dark or electromagnetic radiation. In this context, we try to keep our assumptions simple and generic to derive rather model-independent constraints. In chapter 6 we instead consider MeV-scale dark-matter particles that can annihilate into all kinematically available SM states, i.e. electron/positrons, photons, and neutrinos. The resulting constraints are further compared with complementary bounds from the literature. In chapter 7 we apply our generic constraints from chapter 5 to models with dark-matter self-interactions that are mediated by scalar particles with Higgs-like couplings to the SM. Afterwards, we discuss if there remain any viable parts of parameter space with self-interaction cross-sections that are large enough to address the small-scale problems of the ACDM model. In chapter 8 we further apply our results from chapter 5 to models with axion-like particles that couple exclusively to photons. In addition to deriving general BBN constraints on this scenario, we also evaluate the robustness of these bounds by discussing different modifications to the model that might potentially weaken the constraints. Finally, we conclude in chapter 9. Chapter 1. Introduction 4 5

2 Particle dynamics in an expanding universe

To calculate the number of light elements that are created during BBN - possibly in the pres­ ence of an additional dark sector - it is crucial to first understand the equations governing the particle dynamics in the early universe. Naturally, all of the relevant processes take place within an expanding background metric, and consequently the evolution of each particle species is tightly coupled to the evolution of the universe itself. Therefore, in sec. 2.1 we first discuss the Friedmann equations, which relate the expansion of the universe to its total particle content. In sec. 2.2 we then utilize these results to derive general evolution equations for particle ensembles populating the expanding universe.

2.1 The Friedmann equations

According to the laws of general relativity, the evolution of the universe is governed by the Einstein field equation,

1 R^V 2 Rg^V = 8nGTHv • (2-1)

Here, R^v is the Ricci tensor, R the Ricci scalar, g^v the metric tensor, and T^v the energy­ momentum tensor. In this equation, we ignored the existence of a cosmological constant A, as we are only interested in the cosmology of the early universe after inflation when the contri­ bution of A was negligible. All cosmological observations suggest that our universe is homogeneous and isotropic on large scales. These observations put tight constraints on the allowed form of both the under­ lying metric and the energy-momentum tensor. More precisely, homogeneity demands that both quantities depend only on time, while isotropy implies that both tensors are diagonal and have spatial components proportional to unity. For the case of a flat universe, the most general metric that complies with these constraints is the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric,

(gHv)(t) = diag[1, -R(t),-R(t), -R(t)] , (2.2) which describes the large-scale evolution of a flat, homogeneous, and isotropic universe. In this metric, the volume element is given by dV = R(t)3d3x, meaning that the function R(t) can be interpreted as a scale factor which changes the physical distance over time. Regarding the energy-momentum tensor, the simplest possible realization is that of a perfect fluid with total energy density ptot(t) and total pressure Ptot(t),

(THv)(t) = diag [ptot(t), Ptot(t), Ptot (t), Ptot(t)] . (2.3)

So far the dynamics of the universe is only implicitly encoded in the time-dependence of the scale factor. To make this time-dependence explicit, we have to insert eqs. (2.2) and (2.3) into the Einstein field equation (2.1), which then yields two independent differential equations for Chapter 2. Particle dynamics in an expanding universe 6

R(t). These relations are commonly referred to as the first and second Friedmann equations and can be written in the form

1R(t)/R(t) = 3 ptot(t)=: H(t) , (2.4)

R(t)/R(t) = - 3 ptot(t) + 3Ptot(t)] = Hl(t) + H(t)2 . (2.5)

Here, the first equation can be used to define the Hubble rate H(t), which constitutes a useful measure for the expansion velocity of the universe. The second equation, in turn, can be brought into a more familiar form by first differentiating eq. (2.4) with respect to time and afterwards inserting eq. (2.5). This way, we arrive at the relation

ptot(t) + 3H(t) [ptot(t) + Ptot(t)] = 0 , (2.6)(2-6) which describes the conservation of energy and momentum inside an expanding universe. In fact, the same equation also emerges as the v = 0 component of the conservation law V T^v = 0, thus making this interpretation explicit. In the following, we shall refer to eq. (2.6) as the second Friedmann equation, as it is equiv­ alent to eq. (2.5). As it turns out, the second Friedmann equation is of particular importance as it provides a relation between the total energy density and pressure of the system and hence between the energy densities and pressures of the different particle species populating the universe.

2.2 The relativistic Boltzmann equation

After describing the expansion of the universe in terms of the Friedmann equations, we now turn to the evolution of individual particle species. For this purpose, we have to utilize the principles of statistical mechanics, as the large number of particles in the universe does not allow to track individual trajectories. Instead we have to work with particle ensembles, which are commonly characterized by a phase-space distribution function fx (t,x, p) - for a particle x - that depends on the time t, the position x, and the momentum p. This object encodes the number of particles x per phase-space volume and therefore can be related to the total particle number Nx via the relation

dNx = fx (t,x,P)d3 pd3x . (2.7)

Here, gx is the number of spin states of x.1 In statistical mechanics, the evolution of fx (t, x, p) is governed by the Boltzmann equation. However, we have to keep in mind that most particles in the early universe are ultra-relativistic and also that the universe itself is expanding. As a result, we cannot simply apply the Boltzmann equation in its classical form, but instead have to use a more general relativistic description.

1This definition implies that fx (t, x, p) is the same for all spin states. Also, particles and antiparticle both have their own distinct distribution function. 7 2.2. The relativistic Boltzmann equation

2.2.1 The Boltzmann equation in an expanding universe

Within a relativistically covariant formulation of statistical mechanics, the evolution of the phase-space distribution function fx (t, x, p) of a particle x is governed by the relativistic Boltz­ mann equation, which is given by [30, p. 5, eq. (20)]

PKdxca - j(t'x)pßPYdjfx(t'»,p) = C[fx,-](t'X'P) • (2-8)

= :L[fx ](t,X,p) Here,

= 1 gK3(dY + dSß* dSßY\ (2.9) ßY 2g dxß + dxY dxs is the Christoffel symbol of second kind and the two operators L[fx] and C[fx, ...] denote the Liouville and the collision operator, respectively. While L[f x] encodes the change of fx due to the effects of general relativity, C[fx, ...] encodes the change of fx due to particle-physics reactions as described by the underlying quantum-field theory. Unlike L[f x], which is only a functional of fx,C[fx,...] further depends on the distribution functions of all particles interact­ ing with x as indicated by the dots. In the context of this work, we are not interested in the general form of the Boltzmann equation for arbitrary space-time geometries but rather focus on the dynamics of particles in a homogeneous and isotropic universe as described by Friedmann, Lemaitre, Robertson, and Walker. In this case, the equivalence of all positions and directions in space implies that the phase-space distribution function can only depend on time and the magnitude of the mo­ mentum, i.e. fx(t,x,p) fx(t, p) with p = |p|. Consequently, all spatial derivatives in the Liouville operator vanish and eq. (2.8) simplifies to

(e»(P) d - j(t)pßPYdp) fx

j(t)PßPY = H(t)Ex(p)pj . (2.11)

After inserting this relation into eq. (2.10), we arrive at the adjusted Boltzmann equation d f.(p) H(t)pftp) = CLfKtP. (2.12) dt H (t)P dp Ex(p) which describes the evolution of the phase-space distribution function fx(t, p) in a homoge­ neous, isotropic, and expanding universe. In this form, the Boltzmann equation can be used to describe various aspects of early-universe cosmology, like the freeze-out/freeze-in of dark matter, the formation of light elements, and the decoupling of neutrinos. Such a description is possible by implementing the correct collision operator C[fx, ...](t, p) for the respective sce­ nario of interest. To fully utilize eq. (2.12), it is therefore crucial to understand the general form of C [fx, ...] (t, p), which we discuss in the following section. Chapter 2. Particle dynamics in an expanding universe 8

2.2.2 The collision operator in its general form

Let Rx be the set of all reactions involving x, Ir (Fr) the set of all particles in the initial (fi­ nal) state of the reaction r G Rx, and |Mr|2 the corresponding matrix element squared that is summed (but not averaged) over the spins of all initial- and final-state particles and that further includes the appropriate symmetry factors. Regarding the definition of Rx, reaction and inverse reaction are treated as two distinct elements and in case that x appears multiple times in a reaction, there is one contribution for each occurrence of x, i.e. xx X implies {xx' X, x'x X} C Rx. Using these definitions, the collision operator is then given by [31, pp. 119f., eq. (5.8)]

C ^...M Px ) = 1 £ £r g—1 |Mr|2 (2n)W £ Pk — £ Pl) 2 rGRx kGIr l GFr d3ps * n (i ± f) n f n (w>

In this expression, the ± sign in the factor (1 ± fm) evaluates to + (-) in case fm describes a boson (fermion) and the factor er is equal to +1 (-1) if x is a final-state (initial-state) par­ ticle of r. In order to understand the origin of this expression, it is helpful to decompose it into individual parts. According to the definition of the matrix element, the first term |Mr |2 (2n)4ö(£keir Pk — £iGFr Pl) is equal to the probability for the reaction r to happen with the given combination of momenta. However, this probability does not yet account for the abundance of initial-state particles, i.e. the number of available particles that can participate in the interaction. Hence, we further have to multiply the full expression by an additional factor nnGir fn, which is equal to the probability for finding initial-state particles with the re­ quired momenta. The remaining term nmGFr (1 ± fm) then ensures that the collision operator vanishes in local thermal equilibrium (cf. sec. 2.2.4). Physically, this product encodes quantum effects like Pauli blocking and Bose enhancement/stimulated emission [32]. Finally, to obtain the total probability for each reaction r, we have to integrate over the momenta of all partici­ pating particles other than x with the Lorentz-invariant integration measure d3Ps / (2n)32Es .2 With the collision operator defined above, the relativistic Boltzmann equation (2.12) isfully expressed in terms of the different phase-space distribution functions given the knowledge of all relevant matrix elements. For a given particle-physics model, i.e. Lagrangian, it is therefore possible to deduce the evolution of its particle content by (i) calculating the matrix elements and hence the collision operators for all relevant reactions, and (ii) using the resulting colli­ sion operators to formulate and solve the full system of Boltzmann equations for the various (anti-)particles and a given set of initial conditions. In the following, we illustrate this proce­ dure using a simple example: the (inverse) decay of a particle. It is worth noting, however,

2Similarly, we also have to sum over the spins of all particles other than x. However, this sum is already implicitly included in |Mr |2, which is defined as the matrix element squared, summed over the spins of all initial- and final-state particles. The factor gx-1 inC[fx,...] then compensates for the additional sum over the spin of x that is included in the definition of |Mr |2. 9 2.2. The relativistic Boltzmann equation that this example is not purely pedagogical, as the resulting Boltzmann equation is indeed relevant for most scenarios discussed in this thesis.

Example: (Inverse) Decay

Let x be some boson with mass mx that interacts with some other particle z with mass mz and its antiparticle z via the process x zz. In this example, the set of all relevant reactions is composed of the decay x — zz and the inverse decay zz — x,

Rx = {x zz, zz x} / (2.14) and the corresponding sets of initial- and final-state particles are given by

2-x^zz — Fzz—x — {x} and 2-zz—x — Fx—zz — {z, z} • (2.15)

Then, according to eq. (2.13) the collision operator for the particle x can be written as (here, we omit the function arguments for brevity)

d3 Pz d3 Pz C[fx,...] — - Iygx1 M x - |2 (2n)MPx - Pz - Pz)(1 ± fz)(1 ± fz )fx (2n)32Ez (2n)32Ez

+iy g-iiM zz -x |2(2n)4ö(pz + Pz - Px)(1 + fx)fzfz d3 pz d3 pz (2.16) (2n)32Ez (2n)32Ez

Invariance with respect to time reversal implies that the matrix elements for both reactions are the same, i.e. lMzz -x| — lMx -zz|. Moreover, the product g-1 lM.x .zz| is equal to the more familiar matrix element |Mx .zz |, which is averaged (not summed) over the spins of all initial-state particles and which can be expressed in terms of the total lifetime Tx of the particle x via the relation [33, p. 567, eq. (47.18)]

1 16nmx M |2 with ßz:— V1 - 4m2/m2. (2.17) Tx After inserting this expression into eq. (2.16), we find

mx 8n d3 pz d3 pz C[fx,...] — - (2n)4ö(px - pz - pz)Fxz (2.18) Tx ßz (2n)32Ez (2n)32Ez with Fxz :— (1 ± fz) (1 ± fz)fx - (1 + fx)fzfz. To further simplify this expression, we can use 3 momentum conservation to fix pz — px - pz upon integration over d pz. Then, by using the relation d3pz — 2np2dpzdcos(tfz) for the remaining integral and after combining all numerical factors, it follows that

1 w 2 C[fx,•••] — - — X - 5(Ex - Ez - Ez)FxzEp^ dpzdcos(tfz) . (2.19) Tx ßz J-1J0 z z E E pz —px-pz Chapter 2. Particle dynamics in an expanding universe 10

Energy conservation further demands Ex = [(px — pz)2 + m2]1/2 + Ez with the scalar product px • pz = pxpz cos(&z),3 which is equivalent to

cos($z) = ■2Ex2Ez—mx =: cz . (2.20) 2 px pz

This relation uniquely fixes the value of cos(tfz), which is mathematically described by means of the identity

Ez S(EX — Ez — Ez) = —-ö(cos($z) — cz) . (2.21) px pz

Since | cos($z)| < 1, the resulting integral over cos($z) is unequal to zero only if |cz | < 1, which directly translates into modified integration limits for pz of the form

pz G [p—, p+] with p± = ^E± — m2 and E± = Ex ±?xßz . (2.22)

Consequently, by integrating over cos(fiz) and after reinserting the full expression for Fxz, the collision operator may be written as

CLfx,...] = — m X — ff ±fxfz ±fxf] E dpz . (2.23) TX Tx z x J p— L J Ez P p P = px — p Inserting this expression into eq. (2.12), we obtain the relativistic Boltzmann equation for an unstable particle in an expanding universe (here, we implicitly assume energy-momentum conservation is used to reformulate pz in terms of pz),

1 p+ dfx (tz p) dfx (t p) mx mx pz dpz — H(t)p fx(t,p)+ X ßp fz (t, pz)fZ (t, pz) dt d p Ex Tx Ex Tx pzp Jp_ Ez

mx 1 pz d pz T fx(t' p) X ßzp fz(t, pz) + fz(t, pz) (2.24) Ex Tx Ez

The first term on the right-hand side of this equation encodes the usual relation for decaying particles Nx ~ — Nx / Tx with the inclusion of an additional gamma factor Yx = Ex /mx to account for (ultra-)relativistic particles with an effective lifetime Tx = Yx Tx > Tx in the lab frame. The remaining two terms describe the change of fx(t,p) due to quantum spin-statistics and inverse decays, and explicitly depend on the spectra of z and z. Thus, to determine the full evolution of x, we would also have to derive the two remaining Boltzmann equations for fz(t, p) and fz(t, p), which together with eq. (2.24) would form a set of coupled integro­ differential equations. However, since these equations are rather similar to the one for fx(t, p) we do not present them here. Let us note that it is possible to further simplify eq. (2.24) under certain assumptions, e.g. in case the particles z and z stay in thermal equilibrium throughout the evolution of x. We come back to this approximation at a later stage of this thesis, i.e. once we solve the Boltzmann equation for actual models.

3For each value of px we can choose a coordinate system with px ~ e3. 11 2.2. The relativistic Boltzmann equation

2.2.3 Thermodynamical quantities

Up to this point, we have focused on how to determine the spectrum fx(t, P) for a given particle-physics model. However, in order to connect to the Friedmann equations (2.4) and (2.6,) we are also interested in quantities like the energy density px (t) and the pressure Px (t) for each particle x in the model. These quantities and others like the number density nx(t) and the entropy density sx(t) can be deduced from the full spectrum using a simple integration. Moreover, by using these quantities, it is possible to derive integrated Boltzmann equations, which are less general but easier to solve. Under certain assumptions, these equations can even be used to infer quantities like the number density or the energy density directly, without the need to derive them from the full spectrum. With this in mind, we now iterate over the various quantities and (i) discuss how to calculate them from the spectrum, and (ii) derive the corresponding integrated Boltzmann equation describing their evolution.

Number density

According to the definition of fx(t, P) in eq. (2.7), the number density nx(t) is obtained by integrating the phase-space distribution function over all possible momenta,

nx(t) = R fx(t' P) . <225)

Accordingly, the total number Nx(t) of particles x per comoving volume VR(t) := R(t)3 is given by

Nx(t) = nx(t)R(t)3 . (2.26)

By using the definition of the number density and after integrating both sides ofeq. (2.12) over j • gxd3p/ (2n)3, it is then possible to derive an integrated version of the Boltzmann equation, which takes the form

C[fx,...](t, p) gxd P n x (t) + 3 H(t)nx (t) (2.27) R3 p2 + m2 (2n)3 with nx(t) + 3H(t)nx(t) = Nx(t)R(t)-3. In this version of the evolution equation, the corre­ sponding left-hand side is fully expressed in terms of the number density and its derivatives. Let us note, however, that eq. (2.27) does not necessarily constitute a useful differential equa­ tion for nx(t). This is because such an interpretation is only possible if the integral over the collision operator canalso be expressedinterms ofthe number density, which only works un­ der certain assumptions, e.g. for non-relativistic and non-degenerate particle species. In fact, these assumptions are valid for the formation of light nuclei in the early universe, meaning that we can make use of this equation at a later stage of this thesis. Chapter 2. Particle dynamics in an expanding universe 12

Energy density and pressure

To determine the energy density px (t) from the spectrum, we have to multiply each state with its corresponding energy Ex (p) — a/p2 + m2 and afterwards integrate over all possible momenta,

Px (t) = VP2 + m2 fxp) ^2^ • (Z28)

Consequently, the total energy Ux(t) of the particle x per comoving volume VR(t) = R(t)3 is given by

Ux(t) = Px(t)R(t)3 . (2.29)

Similarly, the pressure Px(t) is obtained by integrating over all possible momenta after multi­ plying each state with the factor p2/3E [31], 2 Px (t) = 1 L^mp2 x fx (t, p) gg (2.30)

Regarding the relativistic Boltzmann equation, we may then integrate both sides of eq. (2.12) over J • Egxd3p/(2n)3 to obtain

Px (t) + 3H(t)[px (t) + Px (t)] = R g^C [fx,...](t, p) (2.31) with px(t) + 3H(t) [px(t) + Px(t)] = [Ux(t) + Px(t)VR(t)] R(t)— 3. Under certain assumptions, this expression can again be interpreted as a differential equation for Px(t), e.g. when describ­ ing non-relativistic (Px (t) ~ 0) or ultra-relativistic (Px (t) ~ px (t) /3) particles. However, even if such an interpretation is not possible, eq. (2.31) still directly connects to the second Fried­ mann equation (2.6) and therefore can be used to describe the heat flow between different sectors. To illustrate this statement, let us define the total heat Qx(t) of x via the relation

Qx(t) :— Ux(t) + Px(t)VR(t) , (2.32) which then further allows to define the volume heating rate qx (t),

qx(t) — Qx(t)R(t)-3 = px(t) + 3H(t) [px(t) + Px(t)] . (2.33)

From there, the second Friedman equation (2.6) can be transformed into the simple form

qtot( t) — 0 . (2.34)

According to this relation, the total heat of the universe is conserved and can only be trans­ ferred between different particle species. We can make use of this statement once we discuss additional dark-sector states that can decay into SM particles and thus effectively transfer heat from one sector to another. 13 2.2. The relativistic Boltzmann equation

Entropy density

The entropy density can be consistently defined as the integral over the Boltzmann entropy SB[fx](t, p) [34, p. 13, eq. (42)]

sx(t) — - R [fx ln(fx) T (1 ± fx) !n(1 ± fx)](t,p) (2n)3 • (2.35)

—SB[fx ](M

Here, the upper (lower) sign is used for bosons (fermions). Consequently, the total entropy Sx(t) of the particle x per comoving volume is given by (2.36) Sx(t) — sx(t)R(t)3 .

Using this definition of the entropy density, we may then derive an integrated Boltzmann equation describing the temporal evolution of sx(t). By making use of the relation

dSB fx](t p) — , / fx (fi p) \ dfx (t p) (2.37) d • 1 ± fx (t, p) d • which is true for both partial derivatives, and after integrating both sides of eq. (2.12) over f • ln[fx/ (1 ± fx)]gxd3p/(2n)3, we find

C Lfx,...](t, p) ln fx(t, p) gxd3p Sx (t) + 3H(t)sx (t) — - (2.38) R3 VF+m 1 ± fx (t, p) (2n)3 with sx(t) + 3H(t)sx(t) — Sx(t)R(t)-3. This equation describes the change of entropy due to particle-physics reactions in an expanding universe and is especially relevant for the discus­ sion of local thermal equilibrium in the next section. In combination with the second Fried­ mann equation it can further be used to describe the entropy transfer between different parti­ cle species, i.e. between the SM sector and a possible dark sector.

2.2.4 Local thermal equilibrium

All previously derived equations can be used to calculate fx(t, p) in full generality without the need for any further assumptions. Sometimes we are, however, interested in the special case of a particle species that stays in local thermal equilibrium (LTE) throughout most of its evolution. In this context, LTE implies (i) that a unique temperature Tx(t) can be assigned to the bath of x particles at any given time t, and (ii) that the corresponding spectrum is ap- proximately4 given by either a Fermi-Dirac distribution for fermionic x or by a Bose-Einstein distribution for bosonic x,

-1 Ex(p) - Hx(t) fx(t, p) — exp T1 (2.39) Tx (t)

4In fact, these distributions can only be considered exact solutions of the Boltzmann equation for a static uni­ verse. We discuss some of the subtitles this implies at the end of this section. Chapter 2. Particle dynamics in an expanding universe 14

Here, Hx (t) is the chemical potential of x and the — (+) sign is used for bosons (fermions). In this form, the distribution function complies with the second law of thermodynamics. By inserting eq. (2.39) into eq. (2.38), it follows that

S (t) = R(t)3 [ C '"M ) x ( ) — x ( ) x ( = ) x ( ) — x ( ) x ( ) f p E p H t g d3p 2 7 Q t H t N t (2.40) Sx(t) R(t) Jr Ex(p) Tx(t) (2n)3 (2.31) Tx(t) which is equivalent to

Tx (t)Sx (t) = Qx (t) — Hx (t)Nx (t) . (2.41)

Hence, the second law of thermodynamics separately holds true for each particle species in equilibrium. As a result, entropy is conserved if there is no heat transfer into other particle species (Qx (t) = 0) and if either the chemical potential is zero (hx (t) = 0) or the particle number is conserved (Nx (t) = 0). Even though it is possible for a particle bath to establish equilibrium by itself without the participation of any other particle,5 the more common sce­ nario features different particle species that maintain equilibrium among each other. In this case, all participating particles share the same temperature. Such a scenario usually implies a heat transfer between the different bath particles, meaning that only the total entropy can be conserved. Afterinsertingeq. (2.39) into eq. (2.35) the entropy density can further be written as

Tx (t)sx (t) — px (t) + Px (t) Hx (t)nx (t) , (2.42) which provides a direct relation between the various thermodynamical quantities in LTE. Another characterizing property of LTE is that the collision operator actually vanishes if all particles entering the calculation of C [fx, ...] (t, p) are in equilibrium with each other. To illustrate this statement, let r G Rx be some reaction involving x and r G Rx the corresponding inverse reaction. Invariance with respect to time reversal then implies that the respective matrix elements are equal, i.e. |Mr|2 = |Mr|2. Hence, the collision operator from eq. (2.13) involves terms of the form (cf. the example from sec. 2.2.2, especially eq. (2.18))

Cr ]( ) D / (2.43) Lfx,... t, p n (1 ± fm) n fn n (1 ± fm) n fn mGFr nGIr mGFr nGIr which is equal to zero if the particles participating in the reaction are all in equilibrium with each other, i.e. if all particles share the same temperature and have chemical potentials related via

Hn(t) = Hm(t) . (2.44) nGIr mGFr

The latter statement is commonly known as the principle of detailed balance. Consequently, in case of a static universe with H(t) = 0,eq. (2.39) wouldbe an exact solution of the Boltzmann equation for Tx(t) = const. However, for an expanding universe with H(t) = 0 this statement

5One notable example is the case of cannibal dark matter [35]. 15 2.2. The relativistic Boltzmann equation

is no longer true. Instead, the term proportional to C[fx, ...](t, p) in eq. (2.12) tries to bring the particles into equilibrium, while the term proportional to H(t) counteracts this attempt by red-shifting their momenta with a factor R(t)-1. Hence, eq. (2.39) can only be considered an (approximate) solution of the Boltzmann equation if the reaction rate

1 f Cr tfx, ...](t, px) gxd p rr(t) : — (2.45) nx (t) r3 p2 + m2 (2n)3 fulfills rr(t) » H(t) with x G Ir (cf. eq. (2.27)), i.e. if the scattering reactions are efficient enough to instantly re-establish equilibrium, irregardless of the expansion of the universe.6 This statement also implies that the reaction r can no longer ensure thermal equilibrium once rr (t) H(t). Afterwards, if there are no other reactions keeping x in equilibrium, the cor­ responding phase-space distribution naturally starts to deviate from the thermal spectrum given in eq. (2.39). Regarding the terminology, this result also forms the basis for discriminating between chemical and kinetic equilibrium. More precisely, the particle x is said to be in kinetic (chemical) equilibrium if there exists a reaction r G Rx with rr(t) » H(t) that fulfills Ir — Fr (Ir — Fr). Hence, kinetic (chemical) equilibrium of x breaks down once the last reaction decouples that conserves (changes) the particle number of x. Moreover, in chemical equilibrium, eq. (2.44) yields a useful relation between the different chemical potentials, while the same equation is trivial in kinetic equilibrium. At this point, it is worth noting that quantities in thermal equilibrium are usually better expressed in terms of temperature instead of time, as this allows to replace Tx(t) by simply Tx. However, whenever we express a quantity like the number density in terms of the tem­ perature, we have to introduce a new function, say nx, which fulfills nx(Tx(t)) — nx(t). For the sake of a simpler notation, though, we may drop the bar and instead discriminate the two functions solely by their argument. This notation does not introduce any ambiguities as long as we always specify the respective function argument. With this in mind, we may now turn to the calculation of quantities like the number den­ sity in LTE. In general, this procedure still requires a numerical treatment, since the occurring integrals cannot be evaluated analytically. However, it is possible to derive analytic expres­ sions in certain limits.

The non-relativistic limit

For Tx < mx and mx » px(Tx) we have Ex(p)/Tx ~ mx/Tx + p2/(2mxTx) » 1 and the spectrum is well approximated by

[mx — ^x(Tx)]/T —P2/(2mxTx) fx (Tx, p) e x x e (2.46)

6In this case C[ fx, ...](t, p) still oscillates between zero and some large value and therefore does not vanish when inserting the full solution of eq. (2.12). Chapter 2. Particle dynamics in an expanding universe 16

Inserting these expansions for Ex(p) and fx(Tx, p) into eqs. (2.25), (2.28), and (2.30) for the number density, energy density, and pressure, respectively, it follows that

3/2 g mxTx \ e-[mx-Mx(Tx)]/Tx x( x) n T H 2n ) 6 / (2.47)

Px(Tx) — mx ^2^ x nx(Tx) , (2.48)

Px (Tx ) — Txnx (Tx ) Px (Tx ) • (2.49)

For the pressure we therefore recover the ideal gas law, and the energy density is given by the number of particles per unit volume times their average energy (Ex) = mx + 3Tx /2. The latter relation complies with the classical result that each particle has three degrees of freedom, all of which contribute to the average energy with Tx/2. In the non-relativistic limit, it is also possible to directly relate the chemical potential to a given particle-antiparticle asymmetry. In general, particles and antiparticles in mutual chem­ ical equilibrium obey Tx = Tx and ^x(Tx) = —^x(Tx), implying that the net particle number can be written as

önx(Tx) := nx(Tx) - nx(Tx) (2—7 2gx ' e-m^x sinh[^(Tx)/Tx] • (2.50)

Rearranging eq. (2.50) with respect to (Tx) then yields

2n \3/2 ^nx (Tx ) emx/ Tx ^x (Tx) — Tx x arsinh (2.51) 2gx mxTx This equation illustrates that a known particle-antiparticle asymmetry can effectively be de­ scribed by means of a non-vanishing chemical potential.

The ultra-relativistic limit

For Tx » mx and Tx » (Tx), i.e. for non-degenerate particles, we have Ex(p)/Tx — p/Tx and the spectrum is well approximated by

fx(Tx, p) — [exp (p/Tx) T 1] 1 (2.52)

Inserting these expansions for Ex( p) and fx(Tx, p) into the respective definitions then yields

SVi x f1 for bosons nx ( Tx ) , (2.53) n [3/4 for fermions

n2 T4 f 1 for bosons Px(Tx) gxTx x [ , (2.54) 30 [7/8 for fermions

Px(Tx) Px(Tx) /3 . (2.55) 17 2.2. The relativistic Boltzmann equation

Within this limit, we thus recover the expressions for a photon gas with Py(T) « T4 and Py (T) = py (T)/3. In principle, it is also possible to derive analytic expressions in other limits. These are, however, less relevant for this work, which is why we do not discuss them here. A more detailed list of all approximations can, for example, be found in [31, pp. 62f.].

2.2.5 A note on Maxwell-Boltzmann distributions

For certain applications, it is common practice to approximate the equilibrium spectrum of certain particles by the classical Maxwell-Boltzmann distribution,

fx(Tx,p) — e [Ex(p) px(Tx^/Tx . (2.56)

This approximation is justified if quantum effects are negligible, i.e. in the absence of degener­ ate Fermi gases and Bose condensates. It is, however, important to note that in this approxima­ tion the collision operator no longer vanishes in LTE (cf. eq. (2.43)). In order to circumvent this problem, it is necessary to also drop the factors nmeFr (1 ± fm) describing Bose enhancement and Fermi blocking. Only then the collision operator takes the form

Cr [ x,-]( ) D / (2.57) f t, p n fn n fn neIr neIr which again vanishes if the spectra of all participating particles follow a Maxwell-Boltzmann distribution. Using this approximation for one particle therefore generally implies that (i) we also have to use the same approximation for all other particles participating in the interaction, and (ii) we must not consider any spin-statistical factors in the collision operator. However, it is worthnoting that the Maxwell-Boltzmann distributionnaturally approaches the full Fermi-Dirac/Bose-Einstein distribution in case ofnon-relativistic and non-degenerate particles. This is because for px(Tx) < Tx < mx, it is Ex(p)/Tx — mx/Tx + p2/(2mxTx) and the Maxwell-Boltzmann distribution is well approximated by

fx(Tx, p) — e [mx px(Tx)]/Tx x e p2/(2mxTx) , (2.58) which happens to be identical to the expansion in eq. (2.46). For non-degenerate particles, it then follows that fx(Tx, p)

3 Big Bang Nucleosynthesis

With the appropriate evolution equations at hand, we can now apply this formalism to make theoretical predictions for the light-element abundances - possibly in the presence of an ad­ ditional dark sector -, which can afterwards be compared to the most recent cosmological observations. To this end, we do not only have to derive Boltzmann equations for the number densities of nuclei, but also determine the evolution of the background plasma, which mainly drives the expansion of the universe. Therefore, in sec. 3.1 we first discuss the dynamics of the various background particles and afterwards determine the light-element abundances in the context of SM. In sec. 3.2 we then compare these results to the most recent cosmological observations and also compile a list of all relevant theoretical uncertainties. In sec. 3.3 we dis­ cuss how the previous calculations have to be modified inthe presence of additional dark-sector states, before finally breaking down our general solution strategy for the resulting equations in sec. 3.4.

3.1 Particle evolution during nucleosynthesis

The period of primordial nucleosynthesis plays a special role in the cosmic chronology, since the amount of light elements that was created during this time constitute the earliest probe of our universe. Consequently, all events happening at earlier times are inevitably subject to speculation.1 * The time span that is usually associated with BBN ranges from t ~ 1s to t ~ 104 s after the big bang, corresponding to temperatures between T ~ 1 MeV and T ~ 10 keV. During this time, the universe was mainly populated by photons, electrons, positrons, and neutrinos. All other particles had already become non-relativistic (T m) and hence irrelevant due to their suppressed number density (cf. eq. (2.47)). In the strong sec­ tor, gluons and light quarks had already coalesced into hadrons, thus forming neutrons and protons, which - for temperatures close to the binding energy of light nuclei - started to par­ ticipate in efficient fusion reactions. This finally led to the creation of the first light nuclei, including deuterium (D), helium-3 (3He), and helium-4 (4He) among others. However, since these elements featured a negligible abundance compared to other particles like photons, they hardly contributed to the total energy density of the universe and thus to the Hubble rate, ac­ cording to eq. (2.4). As a result, the cosmic expansion was dominantly governed by the back­ ground particles. Consequently, a careful treatment of primordial nucleosynthesis demands to understand the evolution of all particles in the heat bath, which is the main focus of this section.

1We know, however, that the universe most likely underwent (i) a phase of cosmic inflation to solve the homo­ geneity problem and (ii) a phase of baryogenesis to explain the apparent particle-antiparticle asymmetry. Chapter 3. Big Bang Nucleosynthesis 20

3.1.1 Electromagnetic radiation

Let us start our discussion with the collective evolution of the electromagnetic sector, which is composed of photons, electrons and positrons. During BBN, all of these particles were tightly coupled due to strong electromagnetic interactions like electron-positron annihilation e+e- o YY, Compton scattering e± y o e± y and double Compton scattering e± y o e± YY- In total, these processes were frequent enough to establish thermal equilibrium between all three particles, meaning that their spectra could be approximated by either a Bose-Einstein or a Fermi-Dirac distribution with a common temperature Te± = TY =: T. More precisely, electron-positron annihilation remains efficient until T « 16 keV [36] and even though chem­ ical equilibrium of electrons breaks down at lower temperatures, Compton scattering main­ tains kinetic equilibrium even longer until recombination at T « 0.23 meV [37]. Using the principle of detailed balance in the context of these reactions, we can further derive rela­ tions between the various chemical potentials. More precisely, applying eq. (2.44) to the cases of double Compton scattering and electron-positron annihilation yields (T) = 0 and

^e- (T) = -y.e+(T) =: ^e(T), respectively.2 *Consequently, the corresponding thermal spectra are given by

fY(T, p) = [exp(p/T) -1]-1 , (3.1)

-1 ((p2 + m2)1/2 ± ^e(T) fe±(T, p) = exp +1 (3.2)

With regard to the Friedmann equations, we are also interested in quantities like the energy density and the pressure of the electromagnetic sector. Regarding the photon bath, we can simply recycle the expressions for an ultra-relativistic, non-degenerate gas of bosons from sec. 2.2.4, which yields (gY = 2)

22 Py(T) = 15T , Py(T) = 45T4 - (3.3)

For electrons and positrons, these expressions are not appropriate, and instead, we have to in­ sert eq. (3.2) into the general integrals (2.28) and (2.30). Then, by substituting u = a/p2 + m2 /T for the integration variable, we find (ge+ = ge- = 2)

T4 r w p-± (t) = % f [u2 - (me/T)2]1/2u2 d 2 7 n2 t4 I ( /T - du =: 2 X T41+(me/T) , (3.4) me/T exp[u ± (T)] + 1 8 30

r w T4 [u2 - (me/T)2]3/2d _ 7 n2 t4 r , . P(:±(T) 3n2 Jme/T exp[u ± t(T)] + 1 du =:2 X 8 90t4 J+(me/T) (3.5)

2Once the process e+e o YY becomes inefficient, (T) = 0 remains valid, but ^e- (T) = -pe+ (T) might not be true anymore. However, it has been shown that this effect does not significantly change any observables [36]. 21 3.1. Particle evolution during nucleosynthesis with £e(T) := ^e(T)/T. Here, the integrals I+(me/T) and J+(me/T) can be evaluated nu­ merically for ^e(T) — 0 and are normalized in such a way that I+(0) = J+(0) = 1- The un­ known chemical potential ^e (T) that enters this expression can be calculated from the electron­ positron asymmetry according to eq. (2.51). Actually, the excess number of electrons over positrons is equal to the number of protons in the universe. With N being the set of all nuclei andZjtheatomicnumberofj e N, the difference between electrons and positrons is given by

öne(T) := n— (T) -ne+(T) = £ Zjnj(T) . (3.6) j e N

An explicit evaluation of this expression requires the number densities of the various nuclei, which we only calculate in sec. 3.1.3. However, we know that the universe features a very small baryon-to-photon ratio at the time trec of recombination, namely nb(trec)/ny(trec) = 6.138x 10 10[37]. Hence, by defining the number density of each nucleus with respect to the total amount of baryons, Yj(T) := nj(T)/nb(T), we find

öne(T) = nb(T) £ ZjYj(T) - 10 - 10n7(T) (3.7) j e N with ZjYj - O(1). For numerical calculations, the chemical potential is relevant only if ne(T) - - 10 10nY(T). Since ne±(T) — Vuk(T) » öne(T) for T » me, this con­ dition can only be fulfilled for T me when ne±(T)

n2 7 Pem(T) := Py(T) + Pe-(T) + Pe+(T) 2 + 4 x 8I+(me/T) T4 , (3.8) 30

n2 7 Pem(T) :=Py(T)+Pe-(T)+Pe+(T) 2 + 4 x 8J+(me/T) T4 , (3.9) 90 which then also yields the corresponding entropy density according to eq. (2.42),

Pem (T) + Pem (T) — (T }6ne (T) sem(T) := sy(T) + se-(T) + se+(T) = T

2n2 7 T3 2 + 4 x H+(me/T) (3.10) 15” 8 with H+(me/T) := 3/4I+(me/T) + 1/4J+(me/T) and ^e(T)öne(T) < pem(T) + Pem(T). Chapter 3. Big Bang Nucleosynthesis 22

-3

Figure 3.1: Left panel: The energy density of electrons and positrons (red), photons (blue) and neutrinos (orange) as a function of temperature. Right panel: The temperature of the electromagnetic sector (blue) and of the neutrino sector (orange) as a function of time (with inverted axes).

While the assumption of local thermal equilibrium allows to deduce the particle spectra without solving the full Boltzmann equation, we ultimately end up with quantities that are functions of temperature instead of time. To perform a translation between these quantities, we therefore have to determine the appropriate time-temperature relation T(t). This function can be derived from the second Friedmann equation (2.6), which states that the total heat of the system is conserved. Since the heat variation that is caused by the different nuclei is negligible due to their small abundance, it then follows that

qvv(t) = qem(t) = pem(t) + 3H(t) [pem(t) + Pem(t)] (3.11) with qvv (t) := Ej e{e^T} [qvj (t) + qv, (t)]. Then, by using the relation pem (t) = T (t) dpem(T )/dT and after rearranging the resulting expression with respect to T(t), we find

dT qvv (t) + 3H(T) [pem (T) + Pem (T)] (3.12) dt dpem (T)/dT which is an ordinary but non-linear differential equation for T(t). However, solving this equa­ tion still requires the knowledge of qvv(t), which naturally follows from the evolution of the neutrinos.

3.1.2 Neutrinos

For large temperatures, the three neutrino species are in thermal equilibrium with the electro­ magnetic plasma, since the weak interactions e+e- V[v and e±V[ e±V[ are still efficient 23 3.1. Particle evolution during nucleosynthesis enough to counteract the Hubble expansion. Yet, the corresponding rates are much smaller than the ones that thermalize the electromagnetic sector, and therefore already decouple much earlier at around T « 1.4 MeV [38, 39] - directly during BBN. A careful treatment of the un­ derlying decoupling process therefore requires to solve the full Boltzmann equation (2.12) for each neutrino species, which - assuming {fy. } : = {fVj, fVj | j E {e, t}} - is given by

dfvt (t p) dfVi(t p) Cweak [{fVj}, fe±](t, p) -H(t)p (3.13) dt d p P

Here, the corresponding collision operator Cweak [{fvj}, fe±](t, p) depends on the distribution functions of electrons, positrons and all three (anti-)neutrino flavors.3 4To solve this equation, a useful approximation is to assume that all neutrino species decouple instantaneously at a common temperature Tvd This approach allows to solve eq. (3.13) analytically, and there­ fore provides a much deeper understanding than a pure numerical treatment. We utilize this procedure in the following discussion but later also comment on the quality of this approx­ imation by comparing the result with numerical solutions from the literature (cf. [34, 41-43] and references therein). Before neutrino decoupling the weak interactions are still efficient enough to keep the neutrinos in thermal equilibrium with the electromagnetic plasma, meaning that each neutrino spec­ trum constitutes a Fermi-Dirac distribution with temperature Tv (T) = T,

fVi (T, p) = [exp (p/T) + 1]-1 for T > Tvd . (3.14)

In this expression, we explicitly assumed a vanishing chemical potential for neutrinos. This assumption is justified, because sphaleron processes - that are crucial for many models of baryogenesis - generally lead to a lepton asymmetry that is comparable in size to the one of baryons, which implies pVi (T)/T ~ p.e(T)/T 1. In fact, this statement is further backed up by constraints from BBN itself, which also favor a value of pVi (T)/T that is compatible with zero, i.e. tv (T)/T = 0.001(16) [ , p. 23, eq. (165)]. After neutrino decoupling the collision operator becomes small compared to the Hubble rate,

Cweak[{fVj}, fe±] (t, p) — 0, in which case the Boltzmann equation (3.13) is solved by

-1 fv. (T, p) = +1 for T < Tvd (3.15) with Rvd '■= R(Tvd). This function still resembles a Fermi-Dirac distribution with an effective temperature Tv(T) = TvdRd/R(T), which is a direct consequence of the momentum redshift after decoupling. Finally, the unknown scale-factor ratio Rvd/R(T) entering this expression can be determined by using eq. (2.31), which implies

qVi(t) = PVi(t) + 3H(t) [pVi(t) + PVi(t)] = 0 (3.16)

3A compilation of all terms entering this collision operator can be found in [40]. 4In fact, all functions of the form f (Tq, pR/Rj) solve eq. (3.13) for Cweak [{fvj}, fe± ](t, p) — 0. Eq. (3.15) then simply follows by imposing the correct initial condition at Tq = Tvd. Chapter 3. Big Bang Nucleosynthesis 24 in case of a vanishing collision operator. By combining this result with the second Friedmann equation (2.34), it then follows that the heat of the electromagnetic plasma is conserved,

qem(t) — qvv (t) — 0 (3.17) with qvv (t) — — Ej e {e,jiT} [qVj (t) + qVj (t)] like before. Consequently - according to the sec­ ond law of thermodynamics (2.41) - the variation of electromagnetic entropy after neutrino decoupling is given by

(jem(t) ^e(t) [Ne- (t) ^'Ne+(t)] Sem(t) — (3.18)

7 -te(t) [Ne— (t) - Nc+ (t)] - 0 (3.19) with Qem(t) — qem(t)R(t)3 — 0. In the last part of this expression, we used te(t) 1aswellas the fact that the rates of all reactions changing the relative number of electrons and positrons (ve + n p + e-, ve + p n + e+) are very small, due to the small amount of nucleons in the universe. As a result, the comoving entropy in the electromagnetic sector Sem(T) = sem(T) R(T) 3 is approximately conserved, which implies sem (T)R(T)3 — sem (Tvd)R(Tvd)3 or equivalently

Rvd/R(T) (3-0) (4y 1 + 4H+(me/T) T-) (3.20)

with H+(me/Tvd) - 1. By combining the results from eqs. (3.14) and (3.15), the full spectrum of (anti-)neutrinos can then be summarized as

f(v)i(T, p) = [exp (p/Tv(T)) + 1]-1 (3.21) with the neutrino temperature

f T for T > Tvd Tv (T) - I 1/3 1/3 (3.22) I (4/11)1/3 T [1 + 7/4H+(me/T)]1/3 for T < Tvd .

Regarding this expression, the most important result is that the neutrino temperature starts to deviate from the one of the electromagnetic sector as soon as the electrons and positrons become non-relativistic. More precisely, Tv (T) falls from Tv (T) — T for T » me to the smaller value Tv(T) — (4/11)1/3T - 0.71T for T me. This is because the entropy released by the electrons and positrons during their annihilation heats up the photon bath relative to the already decoupled neutrinos. In general, our approximate solution to the Boltzmann equation (3.13) states that the neu­ trinos retain a Fermi-Dirac distribution throughout their entire thermal evolution. However, we have to keep in mind that the employed approximation of instantaneous neutrino decou­ pling falls short when it comes to describing the actual transition period around T Tvd. In 25 3.1. Particle evolution during nucleosynthesis fact, a full numerical treatment reveals that the exact solution indeed starts to differ from a Fermi-Dirac distribution by < 1% at around T ~ 1 — 3 MeV [42, p. 7 fig. 1]. However, since these deviations are rather small, we still conclude that eq. (3.21) constitutes an appropriate approximation for the full solution within the scope of the SM. From there, the energy density and pressure of the full neutrino sector can be deduced from the ultra-relativistic limit discussed in sec. 2.2.4, which yields (gv = gVi = 1 since only left-handed neutrinos contribute)

7 n2 vv ( ) = [fr ( ) + v, ( )] (3.23) P T : £ T p T = 6 * 8 30T(T)‘ , i e {e,^,t}

7 n2 vv( ) = *( ) + v,( )] (3.24) P T : £ [P T P T = 6 * 8 90 T(T>4' i e {e,^,t}

By inserting Tv(T) from eq. (3.22), we then end up with the energy density that is presented in the left panel of fig. 3.1. Together, the electromagnetic sector and the neutrino sector dom­ inantly account for the total energy density psm(T) and pressure Psm(T) of the SM during BBN. The latter two quantities can therefore be written as (here, we explicitly neglect the en­ ergy density pb(t) and the pressure Pb (t) of baryons, as these quantities fulfill pb (t) Py(t) and Pb(t) Py(t) for all relevant temperatures due to the small baryon-to-photon ratio)

n2 + T4 Psm(T) ~ Pem(T) + pvv(T) 2 + 4 * 8I (me/T) + 6 * 8 () / (3.25) 30

n2 4 Psm(T) ~ Pem(T) + Pvv (T) 2+4 * 8 j+(me/t)+6 * 8 (T-^ T4 , (3.26) 90 which is a valid approximation for T m^ = 105.66 MeV, i.e. for all temperatures during and beyond nucleosynthesis5 - that is until matter-radiation equality. Nevertheless, at the end of this section, we shortly discuss how to extrapolate these quantities to higher temperatures. Having deciphered the evolution of the neutrinos, let us now come back to the time­ temperature relation. Solving the Boltzmann equation (3.13) numerically would allow to de­ duce the full function tjvi7 (t) entering the differential equation (3.12) for T(t). However, when approximating neutrino decoupling as instantaneous, we have to calculate tjvi7(t) separately for T > Tvd and T < Tvd. Before decoupling, tjvi7(t) D T(t), and after rearranging the resulting expression with respect to T(t), we find

dT = _3H(T) [psm(T) + Psm(T)] for T > Tvd . (3.27) dt dpsm (T)/dT

After neutrino decoupling, tjvi7(t) = 0, which yields upon insertion into eq. (3.12)

dT 3H(T) [pem(T)+ Pem(T)] f T < T (3.28) dt dpem (T )/dT r < T

5Remember that all other particles have already become non-relativistic and therefore Boltzmann-suppressed. Chapter 3. Big Bang Nucleosynthesis 26

Finally, the Hubble rate entering this expression can be deduced from the first Friedmann equation (2.4). When considering only the SM, the total energy density of the universe is approximately given by ptot(T) — psm(T) with psm(T) from eq. (3.25), which constitutes a valid approximation for all relevant temperatures. Hence, we obtain

H(T) — ^8nGPsm(T) . (3.29)

Inserting this expression into the equations for the time-temperature relation allows to deter­ mine the function T(t), which can then be used to re-express all particle spectra as a function of time. However, regarding the initial condition, the only sensible choice is limt_ ,0 T(t) — to, whose implementation requires some knowledge about the particle dynamics prior to nucle­ osynthesis, i.e. for T > 10 MeV. We discuss this topic in a short excursus at the end of this section and - for now - present the resulting functions T(t) and also Tv(t) — Tv(T(t)) in the right panel of fig. 3.1. Except during electron-positron annihilation, we have T(t) « 1/x/t and T — 1 MeV roughly corresponds to t — 1 s. In contrast, the neutrino temperature Tv(t) is not affected by the electron-positron annihilation and therefore stays « 1/ for all relevant times t > 10-2s. With this relation at hand, we have not only managed to determine the evolution of all background particles as a function of both time and temperature, but also to define the expan­ sion of the universe in terms of the Hubble rate (3.29). Therefore, we may now finally turn to the calculation of the light-element abundances that are created during BBN.

Excursus: The SM heat bath prior to nucleosynthesis

During nucleosynthesis, all massive particles except for electrons have already become non- relativistic and therefore irrelevant due to their suppressed number density. This is because all of these particles are kept in thermal equilibrium with the background photons and there­ fore can be described by either a Bose-Einstein or a Fermi-Dirac distribution, which gets sup­ pressed for T mx. However, this statement also allows to incorporate these particles by substituting 7 4 x 8I+(me/T) E gxKxI^ (mx /T) (3.30) x—Y,v in eq. (3.25). Here, the sum ranges over all particles in the thermal bath except for photons and neutrinos, (mx /T) is defined in accordance with eq. (3.4), i.e. (mx /T » 1) — 0, and Kx is 1 for bosons and 7/8 for fermions. Hence, prior to nucleosynthesis, the total energy density of the SM can be generalized to

4 2 n2 — —g*p( ) Psm(T) — 2 + E gxKJt (mx/T)+ 6 x 7f T4 * : T T4 . (3.31) x — Y,,v \ ' Here, we have further defined the effective number of relativistic energy degrees of freedom g*P (T), which constitutes a common way to parameterize psm(T). Analogously, it is also pos­ sible to derive generalizations for the total pressure of the SM (by replacing (mx /T) with 27 3.1. Particle evolution during nucleosynthesis

Jt (mx/T)) and the total entropy density of the SM (by replacing It (mx/T) with Ht (mx/T) and [Tv/T]4 with [Tv/T]3), which allows to define

n2 2n2 Psm(T) = ^g.P(T)T4 and Ssm(T) = —g*s (T)T3 (3.32) with the effective number of relativistic pressure (entropy) degrees of freedom g*p(T) (g*s (T)). Regarding the numerical evaluation of the various functions g.* (T), one non-trivial part is to incorporate the QCD phase transition, at which light quarks condense into hadrons, thus effectively changing the composition of the thermal plasma. However, this transition has al- readybeenextensivelystudiedinthe literature [31, 45, 46], and here we simply adopt the most recent results from [46, p. 2, fig. 1] whenever we have to perform calculations prior to nucle­ osynthesis. Especially, we can use eqs. (3.31) and (3.32) to determine the time-temperature relation for T > 10 MeV, which allows to incorporate the initial condition limt ,0 T(t) = to.

3.1.3 Nuclei

The eponymous process ofprimordialnucleosynthesis is the creation of the first light elements due to nuclear fusion reactions of neutrons and protons, including pn Dy and Dn 3H7 among others (cf. app. A.1 for a compilation of all fusion reactions up to 6Li). During this whole process, the different nuclei undergo efficient Coulomb scattering with the background electrons, which effectively keeps them in kinetic equilibrium.6 In addition to this, protons and neutrons are also subject to weak interactions, which mainly determine the composition of nucleons at the start of nucleosynthesis. However, since both the nuclear fusion reactions and the weak interactions are conceptionally different, it is instructive to treat both of them separately at first, and only afterwards combine them into a single Boltzmann equation.

Nuclear fusion reactions

To derive a suitable Boltzmann equation describing the nuclear fusion reactions, it is useful to first discuss the evolution of the full baryon content in the universe. Within the scope of the SM, baryon number is conserved during nucleosynthesis, meaning that the total number of baryons N(t) = nb(t)R(t)3 per comoving volume element Vr(t) = R(t)3 fulfills N(t) = 0, or equivalently

nb(t) + 3H(t)nb(t) = 0 . (3.33)

At the time trec of recombination, nb(trec) is known in terms of the baryon-to-photon ratio n, which is measured to be y = nb(trec)/nY(trec) = 6.138 x 10-10 [37]. Using this initial

6Nevertheless, we cannot simply write down a thermal distribution for each nucleus. This is because we have no information on the various chemical potentials, and the principle of detailed balance only implies the trivial relation pe(T) + px(T) = P-e(T) + px(T). However, after solving the Boltzmann equation we could, in principle, determine px(T) by comparing the full solution with a thermal distribution. Chapter 3. Big Bang Nucleosynthesis 28 condition, the solution of eq. (3.33) becomes

nb(t) = yny(trec)R(trec)3/R(t)3^=) nnY(trec) exp — 3 H(t') dt' , (3.34) trec for which we have used to first Friedmann equation (2.4) to express the scale-factor ratio in terms of the Hubble rate. Regarding the evolution of the different nuclei, it is worth noting that we do not need to know their full spectrum. This is because we ultimately strive to compare our results with the most recent observations, which are only sensitive to relative number densities (see below). Hence it is sufficient to track the evolution of the various nuclei X e N := {n, p, D,3 He, ...} using the integrated Boltzmann equation (2.27), which - assuming {fN} := {fN|N e N}-takestheform

nX(t)+ 3H(t)nx(t)= [ Cfusion[{fN}fYgX^P . (3.35) ■'R3 y p2 + mX (2n)3

Here,thecollisionoperatorCfusion[{fN},fy](t,p)includesallnuclearfusionreactionsbetween the nuclei and therefore also depends on the photon spectrum. By redefining the number density of each particle relative to the total amount of baryons,

(3.36) this Boltzmann equation can then be simplified to

Yx (t) (3=3) 1 f Cfusion\{fN}/ fYKC P) gXd3p (3.37) nb (t) Jk3 p2 + mX (2n)3

As we have already discussed, this expression can only be interpreted as a differential equa- tionfor{YN(t)|N e N } if its left-hand side can also be expressed in terms of the various abundances. However, such a translation is indeed possible since allnuclei are non-relativistic during their creation at T ~ O(MeV) mX ~ O(GeV). To illustrate this statement, let us first consider fusion reactions of the form ij -H- kX with four distinct particles i, j, k, and X. In this case, the integrated collision operator for the forward reaction ij kX entering eq. (3.37) is given by (again omitting the function arguments for brevity)

J jkX f,...] (X^)3pX (2=3) /\Mij^kX|2(2n)4 5(pi + Pj—Pk— PX)

3 Pi d3 Pj d3 pk d3 pX * (1 ± fX)(1 ± fk)fifj (3.38) (2n)32Ei (2n)32Ej (2n)32Ek (2k)32Ex

To simplify this expression, let us emphasize that it is possible to neglect the spin-statistical factors. This is justified since all nuclei are non-relativistic and in kinetic equilibrium, which implies fw e— [mN—^/T 1 and consequently 1 ± fw — 1 for all nuclei N e N according to eq. (2.46). In case one of the final-state particles is a photon, energy conservation implies that 29 3.1. Particle evolution during nucleosynthesis

Ey is of the order of the binding energy of the created nucleus, Ey ~ O (MeV) ~ T, meaning that 1 + f7 — 1 is not necessarily true. However, this factor can instead by absorbed into the thermal rates, since the photon spectrum is known for all relevant temperatures. From there, the integrated collision operator can be simplified further by rewriting it in terms of the unpolarized cross-section (p, .kX, which is given by [33, p. 569, eq. (47.27)]

d3pk d3px 4iG -kX'Alol EiEj = Mij ,.X|2(2n)4J(p + pj - pk - px) (3.39) (2n)32Efc (2n)32Ex with Mij x |2 = (gigj) 1 Mij x |2 and the Moller velocity (E/EjVM0l)2 = (pi • pj)2 - m2m2. By inserting this expression into eq. (3.38), we find

gXd3pX gid3 pi gjd3 pj ,...] ^ij^kXvM0l fifj (3.40) (2n)3 Ex (2n)3 (2n)3

At this point it is worth recalling that all nuclei N G N stay in kinetic equilibrium due to efficient Coulomb interactions with the background electrons, meaning that each spectrum fN can be considered thermal with an unknown chemical potential pN, whose functional form is the only differentiating factor between kinetic and chemical equilibrium. Hence, by denoting all quantities that would describe a nucleon N in chemical equilibrium by a *, we find fN/fN = e(pN-pN)/T for non-relativistic species, which further implies7

fN (e p)=fN (t p) . (3.41) nN(t)

Consequently, by applying this relation to fi and fj in eq. (3.40), we obtain

gXd3px nz-nj gi d3 pi gjd3 pj Cij^kX L/x,...] Gij^fcXvM0lZj* (3.42) * j ( ) ( ) (2n)3 Ex nijn 2n 3 2n 3

From there, we define the thermal average of cross-section times Moller velocity via the rela­ tion

(Gij^kxvM0l) := / aij^fcxvM0b/i*zj* ^2np3 ^2J'3, (3.43) which simplifies the integrated collision operator to

j Cij^kX[/X,...] (2n)3E = 'kXvMol)ninj = nb(^'ij^kXvM0l) YiYj . (3.44)

In addition, we find that a similar calculation for the reverse reaction kX ij leads to the same result when substituting Y/Yy — YkYx. Consequently, after inserting this result into eq. (3.37), the variation of Yx due to fusion processes of the form z'j kX is given by

Yx D nb (Gij^kXvM0l)YiYj - (GkX^ijvM0l)YkYX . (3.45)

7This relation is also true for the participating photons, which stay in chemical equilibrium anyway. Chapter 3. Big Bang Nucleosynthesis 30

Finally, to obtain the fullevolutionofYx(t), we have to sum over all fusion reactions involving x, thus generalizing eq. (3.45) to processes with more particles in the initial or final state. In full generality, the evolution of the abundances then takes the form (adopting the notation from sec. 2.2.2)

Yx (t)= £ £r nb (t)1 |-1j [n Yi (t)^ , (3.46) r ERx iEI r in which (nrVM0l) already contains the relevant symmetry factors according to our definition of the matrix element. For 7 E Ir, we simply have Yy(t) = nY(t)/nb (t), which usually gets absorbed into the reaction rate. In total, we thus obtain one equation for each nucleon, mean­ ing that eq. (3.46) defines a set of ordinary but non-linear differential equations for a given set of measured reaction rates rr ex. {arVM0\}nYr 1 1. These rates are tabulated as functions of temperature (cf. references in app. A.1), which emphasizes the need to re-express eq. (3.46) in terms of T. This procedure naturally introduces the time-temperature relation via

YX = x 'd (3-47> and thus a dependence on the background particles that we have discussed before. However, we have to keep in mind that eq. (3.46) does not yet comprise all reactions that are relevant for nucleosynthesis, as we still have to discuss the additional weak interactions that influence the evolution of neutrons and protons.

Weak interactions

Before the first fusion reactions became efficient, the baryon content of the universe was com­ posed entirely of protons and neutrons. Back then, both nucleons participated in weak reac­ tions of the form nve pe-, ne+ pve and n pe-Ve, which kept them in thermal equi­ librium with the electromagnetic sector until decoupling at T « 0.77 MeV [44]. The spectra of neutrons and protons before this time are therefore well approximated by thermal distri­ butions with temperature Tn = Tp = T. Moreover, by applying the principle of detailed balance to the relevant weak reactions, we find pn (T) — pp (T) = pe(T) — pVe (T) T and consequently the ratio of protons and neutrons before decoupling can be written as

Yn (T) Q + pn (T) pp (T) e -Q / T = exp + (3.48) Yp (T) T T with Q := mn — mp ~ 1.293 MeV [19]. Since neutrons and protons were the only baryons at the time, we have Yn(T) + Yp(T) = 1, which together with eq. (3.48) implies

Yn(T) — [1 + eQ/T^ -1 and Yp(T) — [1 + e-Q/T^ -1. (3.49)

In combination with eq. (2.47), this relation can then be used to determine the chemical po­ tentials pn(T) — pp(T) and thus the spectra of neutrons and protons during chemical equi­ librium. However, to calculate the deviation from these distributions once the conversion re­ actions become inefficient, we instead have to solve the full Boltzmann equation for neutrons 31 3.1. Particle evolution during nucleosynthesis and protons, which is given by

1 Cconv \fn, fp,...] (t, p) gndp Yn (t) = (3.50) nb(t) r3 p2 + ml (2n)3

In the relevant collision operator Cconv\fn,fp, ...] (t, p), all reactions converting neutrons to pro­ tons involve terms of the form (cf. eq. (2.13))

C p [fn,...] D (1 — fp) fn — Y^bf* / (3.51) with 1 — fp — 1 and fn/nn = f* /nn* like before. As a result, we find Cn .p \fn,...] « Yn and a similar statement is true for reactions converting protons to neutrons, i.e. Cp n \fn,...] ex. Yp. In both cases, the remaining integrals no longer depend on fp or fn and therefore can be treated like simple prefactors. Hence eq. (3.50) can be brought into the general form

Yn (t) = (t)Yp (t) — \ p (t)Yn (t) = —Yp (t) . (3.52)

Here, the functions rn .p (T) and rp (T) can be interpreted as the total neutron-to-proton and proton-to-neutron conversion rate. Again, these terms are naturally known as a function of T (cf. [44,47] for a list of all analytic expressions), which again introduces a dependence on the background particles via the time-temperature relation. In general, eq. (3.52) influences the ratio of neutrons and protons even before the onset of nuclear fusion reactions and therefore needs to be combined with eq. (3.46) to precisely track the evolution of all nuclei in the early universe. However, while nuclear fusion only becomes efficient around T ~ 0.1 MeV [31], most weak interactions already decouple somewhat earlier at T « 0.77 MeV [44]. Based on this reasoning, both processes can partially be treated independently and we can first discuss the evolution of protons and neutrons prior to nucleosynthesis, before merging the results into a full numerical solution. To this end, we solve eq. (3.52) by using the procedure that is implemented in ALTERBBN [48, 49], which produces the numerical results that are presented in the left panel of fig. 3.2. As for the initial condition, we use eq. (3.48) to fix the neutron-to- proton ratio at T0 = 10 MeV. As a result, we find that for large temperatures, T > 1 MeV, the weak interactions are efficient enough to keep protons and neutrons in chemical equilibrium, meaning that their ratio obeys Yn(T)/Yp(T) = e-Q/T according to eq. (3.48) (dashed red). However, below T ~ 1 MeV, the weak reactions struggle to keep up with the expansion of the universe, which causes the neutron-to-proton ratio to deviate from its equilibrium value. Around T ~ 0.1 MeV - shortly before nuclear fusion reactions become efficient - all weak interactions except for the neutron decay have fully decoupled. In this region, the evolution of neutrons and protons can thus be approximated by a simple decay law of the form (dash- dotted green)

1 Yn (t) — n — .Yn (t) =-----Yn (t) = —Yp (t) (3.53) Tn Chapter 3. Big Bang Nucleosynthesis 32

-3

Figure 3.2: Left panel: The neutron-to-proton ratio Yn (T)/Yp (T) as a function of temperature (solid gray). For reference, we also show the equilibrium value of Yn (T)/Yp (T) according to eq. (3.48) (dashed red) as well as the solution that is obtained from the simple decay law (3.53) (dash-dotted green). Right panel: Evolution of the light-element abundances YX (T) relative to Yp (T) as a function of temperature. Here, we only show elements up to 7Be, since heavier ones would feature even smaller abundances. with the neutron lifetime Tn = 880.2 s [33]. As a result, we find that once the nuclear fusion reactions become efficient, corrections to the neutron and proton abundance from weak inter­ actions are mainly caused by the neutron decay. For even smaller temperatures T < 0.1 MeV, the amount ofnucleons drops sharply, because most of them get bound into heavier elements. However, to describe this process, we have to solve the full Boltzmann equation, including the nuclear fusion reactions from eq. (3.46).

Beta decays

Before we discuss the full solution, let us note that tritium (3H) is not stable after its creation but rather subject to beta decays of the form 3H 3He e-Ve with tsh = 3.89 x 108s. The lifetime of this transition exceeds the timescales that are relevant for nucleosynthesis. Hence it is justified to simply ignore this decay during BBN and afterwards - once the abundances are fixed - to treat the decay separately via the equation

1 Y3H(t) = -— Y3H(t) = -Y3He(t) , (3.54) T3h which essentially converts the full amount of 3H into 3He after nucleosynthesis. We therefore take this process into account by simply adding up the final values of Y3H and Y3He. 33 3.1. Particle evolution during nucleosynthesis

Full solution

To obtain a full numerical solution, we combine the weak interactions from eq. (3.52) with the nuclear fusion reactions from eq. (3.46) by adding up their corresponding right-hand sides. Regarding the initial condition, we utilize eq. (3.49) to fix Yn(T0) and Yp(T0) at T0 = 10MeV, while setting the abundances of all other nuclei to their chemical equilibrium value at this temperature, i.e. [31, p. 88, eq. (4.5)]8

/ 2n \3(Ax—1)/2 Yx(T)= gxnb(T)A—1 AX/22—Ax ----- Yp(To)ZxYn(T)AX—ZxeBX/T0 , (3.55) which follows from ^x(T0) = Zx^p(T0) + (Ax — Zx)^n(T0) according to the principle of detailed balance applied to the various fusion reactions. Here, ZX, AX, and BX are the atomic number, the mass number, and the binding energy of X, respectively. The resulting system of differential equations is then solved as a function of T by means of the linearization procedure thatisimplementedin ALTERBBN,where wealsoadoptthe includednetwork ofnuclear rates r1 rr « (^rVM0l)n[ r 1 . The results of this procedure are presented in the right panel of fig. 3.2. As already stated before, the bulk of all fusion reactions happens around T ~ 0.1 MeV, when the evolution of the neutrons is mainly governed by their decay. Nucleosynthesis cannot start any earlier as the universe first has to build up a suitable amount of deuterium before burning it into heavier elements. However, deuterium has a comparatively small binding energy of 2.22 MeV, meaning that its dissociation is still effective at larger temperatures. Only once a sufficient amount of deuterium has been synthesized, the heavier elements start to be rapidly created shortly after, whereby most nucleons are actually bound into helium-4, as this is the most tightly bound light nuclear species. While the temperature decreases, at some point the nuclear fusion reactions can no longer keep up with the expansion of the universe, which puts a natural end to nucleosynthesis at around T ~ 10 keV. Afterwards Yx (t) — 0 and the created abundances stay essentially constant over time. Using v1.4 of ALTERBBN with n = 6.138 * 10—10 [37] and Tn = 880.2 s [33], the final abundances of the lightest elements are given by (the following combinations of abundances can later be easily compared to observations)

4Y4He = (2.4745 ± 0.0005) * 10—1 , (3.56)

YD/Yp = (2.550±0.059) * 10—5 , (3.57)

Y3He/YD = (4.0±0.2)*10—1. (3.58)

Here, the errors are due to uncertainties in the nuclear-rate measurements, which we discuss in more detail below. Similar results are also obtained by other BBN codes, including PRI­ MAT [44] and PARTHENOPE [50]. To prove the predictive power of this calculation, the next step is to compare these results to the most recent cosmological observations.

8This is justified because, for T > 1 MeV, the fusion reactions keep the nuclei in chemical equilibrium [31]. Chapter 3. Big Bang Nucleosynthesis 34

3.2 Comparison with observations

3.2.1 Cosmological observations

Primordialnucleosynthesis predicts the creation of the first light elements in the early universe between 1 s and 104s after the big bang. All observations, however, are naturally performed at much later times, i.e. long after the formation of the first stars and hence after the onset of stellar nucleosynthesis. In the latter process, light elements are fused into heavier nuclei like iron, which inevitably changes the composition of elements in the universe - compared to its state shortly after BBN. From an observational point of view, it is therefore crucial to find astrophysical environments with a low metallicity, where the composition of light ele­ ments closely resembles the one shortly after primordial nucleosynthesis. In the last decades, significant progress has been made in finding such environments and thus in inferring the primordial abundances. In this context, the primordial abundance of helium-4 is best mea­ sured in blue compact galaxies by observing the recombination emission lines of helium and hydrogen. Here, the most recent measurements [51-53] suggest a combined value of[33]

Yp := 4Y4He = (2.45 ±0.03) x 10-1 . (3.59)

Regarding the determination of deuterium, several measurements exist from metal-poor dam­ ped Lyman-a systems [54-56], with a weighted average of [33]

D/1H := YD/Yp = (2.569 ±0.027) x 10-5 . (3.60)

There are also measurements ofhelium-3 fromthe hyperfine transition of3He+ [57]. However, these observations are argued to be subject to large astrophysical uncertainties, as the only available data comes from high-metallicity regions. The possibility of using them to constrain early-universe cosmology is therefore still debated in the literature [33, 58, 59]. It is widely accepted, though, that the ratio of helium-3 and deuterium can only be increased in these environments and therefore can be used as an upper bound when both species are observed simultaneously [59-61]. Based on this reasoning we also apply the results from [60], which suggest

3He/D := Y3He/YD = (8.3 ±1.5) x 10-1 . (3.61)

Finally, let us note that we do not consider any constraints from lithium, as the corresponding measurements are also believed to be subject to large systematic uncertainties, particularly due to stellar depletion [62]. In fact, theoretical calculations usually predict a lithium abun­ dance that is three times larger than the one that is inferred from observations. This discrep­ ancy is generally known as the cosmological lithium problem [63], and a viable solution to this issue has yet to be found. In principle, these observations can now be compared to the theoretical predictions that we calculated at the end of the last section. However, to perform a detailed analysis, we first have to take a closer look at the relevant theoretical uncertainties, which is the focus of the next section. 35 3.2. Comparison with observations

3.2.2 Theoretical predictions and their uncertainties

Regarding the theoretical prediction of the light-element abundances, we solve the set of equa­ tions that was derived in sec. 3.1.3, which - in case of the SM - leads to the results that were already presented in eqs. (3.56-3.58). However, as we have pointed out earlier, the underlying calculation is subject to some uncertainties, which we discuss in the following.

Uncertainties from the nuclear rates

First and foremost, the nuclear reaction rates entering eq. (3.46) are only known experimen­ tally and thus feature a measurement uncertainty. For each reaction, ALTERBBN therefore implements a mean, a high and a low value of the corresponding rate. To incorporate this uncertainty, we consequently compute three values YX, YX, high, and YX, low for each nuclear abundance at the end of nucleosynthesis, by using the mean, low, and high values of the nuclear rates, respectively. We then define the observable abundance ratios as

R1 := Yp , R2 := D/1H , R3 := 3He/D , (3.62) and afterwards approximate their theoretical uncertainty via

nRf = min ( | R - R,high |, | Ri - Ri,low | ) • (3.63)

As it turns out, this procedure leads to slightly larger errors compared to the formalism de­ scribed in [64], and therefore can be considered conservative.

Uncertainties from the baryon-to-photon ratio

Besides the nuclear fusion reactions, the final abundances are also affected by the uncertainty in the determination of the baryon-to-photon ratio. This is because quantities like D/1H show a strong dependence on n, which is further correlated with other cosmological parameters (cf. [37, p. 38, fig. 26]). Regarding the scenarios discussed in this thesis, we are mainly interested in the correlation between n and the effective number of neutrinos Neff, which - in case of the SM - is defined via (a generalization of this expression in the presence of BSM physics is discussed below)

pvv(trec) 7 n 4 4 (3.64) Neff = peff(trec) h Peff(trec)= 2 X 8 30 (trec) 11

In general, the correlation between both quantities can be expressed in terms ofthe probability distribution P for the pair (n, Neff), which is given by

2nan^Neä\/'1 - r2 x P(n, Neff) =

1 (n- n)2 (Neff - Neff)2 2r(n - 2)(Neff - Neff) exp (3.65) 2(1 - r2) n2 nN nnnNeff Neff Chapter 3. Big Bang Nucleosynthesis 36

Here, X denotes the mean and aX the uncertainty of X E {n, Neff} and r is the Pearson coeffi­ cient describing their correlation. By fitting P(n, Neff) to the 95% confidence-region ellipse in the Obh2 — Neff plane given in [37, p. 38, fig. 26] (TT/TE/EE+lowE+lensing+BAO)/ we find

fj = 6.128 * 10—10, an = 4.9 * 10—12, Neff = 2.991, aNef{ = 0.169, r = 0.677 . (3.66)

For a given value of Neff, the best-fit value of n is then determined by

n (Neff) = n + ran . (3.67) aNeff The most precise calculation in the SM - including QED effects [42, 65] and corrections from non-instantaneous neutrino decoupling - yields Neff = 3.043 [39] and consequently n = 6.138* 10—10, which matches the value that we used for our theoretical prediction in the pre­ vious section.9 From there, the variation of n can be translated into an uncertainty for the various abundance ratios by applying linear error propagation, which yields

btp aR>p = (3.68)

To calculate the derivatives dRi/dn entering this expression, we once again utilize v1.4 of ALTERBBN, which results in the following uncertainties for the observable abundance ratios

aYp = 0.0005 * 10—1 , aDtp1H = 0.024 * 10—5 , a3Hpe/D = 0.02 * 10—1 . (3.69)

While the corresponding errors on Yp and 3He/D arenegligible compared to the experimental uncertainties, we find that the error on D/1H instead leads to a rather significant contribution.

Combination of uncertainties

For the final comparison between theoretical prediction and observation, we combine both dominant error sources10 by defining

Ar, := (r, — R”'«)/]/(a£)2 + (aRtp/ + (off*) (3-70) as a proxy for the deviation of each observable Ri. By comparing our previous results from eqs. (3.59-3.61) with the observations from eqs. (3.56-3.58), we then find

AYp = 0.82 , AD/1H = —0.27 , A3He/D = —2.83 . (3.71)

Strikingly, the theoretical predictions for Yp and D/1H perfectly agree with cosmological ob­ servations within < 1a. The deviation for 3He/D is somewhat larger, but since the corre­ sponding measurement can only be used as an upper bound, no tension arises for A3He/D < 0.

9In the absence of BSM physics, it is actually not necessary to use the one-parameter extensions of PLANCK to determine the best-fit value of n. However, the one-parameter extension that we use here, becomes crucial once we consider the existence of additional dark sectors. 10We have checked that a variation of Tn within its uncertainty does not lead to a significant change in the abundance ratios. 37 3.3. Influence of additional dark-sector states

In summary, this result clearly illustrates the importance of primordial nucleosynthesis: Using the standard model of cosmology, we were able to reproduce the correct light-element abun­ dances, which substantiates the validity of this description for temperatures below 10 MeV. This statement then also leads to the main motivation of this thesis: All of the previous cal­ culations have been performed in the context of the SM, without the inclusion of any DM particles. Adding a dark sector, however, can influence primordial nucleosynthesis in many different ways and therefore lead to abundances that are no longer compatible with experi­ mental observations. By exploiting nucleosynthesis, it is thus possible to set stringent con­ straints on various dark-matter models by demanding that the additional particles do not distort the light-element abundances too much. In the next step, we therefore recapitulate the calculation from sec. 3.1 and break down the modifications that have to be performed in case additional dark-sector states are present.

3.3 Influence of additional dark-sector states

Let us now consider a universe that is not only composed of SM particles, but also comprises a dark sector, which involves a dark-matter candidate as well as other states such as mediators for dark-matter interactions. Then, for each particle x in the dark sector (DS), we have to solve a Boltzmann equation, which - assuming {f} := {fd | d G DS} - takes the form

dfx (t p) H(t) p dfx (t p) = Cdark[{fd. , p) (3 72) dt d p y/p2 + m2

Here, the collision operator Cdark[{fd},...](t, p) is different for each dark-sector particle and also depends on the underlying model, meaning that - in general - eq. (3.72) can only be made explicit for a given Lagrangian. However, in this section we strive for a more model­ independent discussion and therefore do not make any additional assumptions regarding the collision operator. Apart from adding eq. (3.72) to the set of all relevant Boltzmann equations, the presence of additional dark-sector states also has an indirect impact on the dynamics of other particles during nucleosynthesis. The aim of this section is to quantify these effects, for which we define the total energy density of the dark sector via pdark(t) := Ex G ds px(t) and also employ similar definitions for other quantities like the total pressure Pdark(t) and the total heating rate qdark(t). At this point, it is worth recalling that we are mainly interested in dark sectors that feature MeV-scale particles. In this case, the corresponding masses are of the same order as the temperatures that are relevant for BBN, mdark ~ Tbbn ~ O(MeV), which implies that some of the states might actually become non-relativistic directly during the creation of light nuclei. Consequently, such a setup has interesting implications for the evolution of the light-element abundances and also differs from most other scenarios that have previously been discussed in the literature [20, 22-29]. Chapter 3. Big Bang Nucleosynthesis 38

3.3.1 Hubble rate

First and foremost, the additional energy density of the dark sector influences the expan­ sion rate of the universe. More precisely, for pdark(t) ~ psm(t), the dark sector cannot be neglected when calculating the total energy density of the universe that enters the Hubble rate in eq. (2.4). Instead, we have to set ptot(t) = psm(t) + pdark(t), which implies

H(t) = y^~3 [psm(t) + pdark(t)] • (3.73)

An increased value of the Hubble rate leads to a faster expansion of the universe and con­ sequently to a different time-temperature relation. As a result, BBN happens on a shorter timescale, which can potentially distort the final light-element abundances via eq. (3.47). It is also worth noting that according to eq. (3.73), the spectra fx (t, p) with x G DS enter the Boltzmann equation (3.72) not only explicitly but also implicitly due to their contribution to the Hubble rate. We discuss how to solve the resulting system of equations in sec. 3.4.

3.3.2 Neutrino decoupling

The presence of additional dark-sector states also influences the decoupling of neutrinos. In this context, the most severe modifications occur if there are interactions between the dark sector and the neutrino sector. In this case, the collision operator in eq. (3.13) gets extended by additional terms,

Cweak [{fvj}zfe±] Cweak[{fvj}, fe±] + Cdark[{fj }, {fd}] , (3^74) which generally affect the decoupling process of neutrinos. For such a scenario, the approx­ imation of instantaneous neutrino decoupling is no longer valid, meaning that the various spectra fy. (t, p) can only be obtained by solving the full system of coupled Boltzmann equa­ tions for Vj and x G DS. A good example of such a scenario would be a dark mediator that decays into neutrinos at the time of their decoupling, which would naturally lead to a non­ thermal neutrino spectrum. Let us note, however, that we do not discuss any such scenarios in this thesis and therefore refrain from a further discussion of this possibility. If there are no interactions between the dark-sector particles and the neutrinos, their de­ coupling is still modified due to the increased Hubble rate. However, in this case it is possible to retain the approximation of instantaneous neutrino decoupling by implementing two ap­ propriate modifications. On the one hand, a larger Hubble rate starts to dominate the weak reaction rates at an earlier time and therefore leads to a different neutrino-decoupling temper­ ature. To quantify this effect, let us note that all weak reaction rates obey rweak(T) ex. GfT5 with the Fermi coupling constant GF [31, p. 91, eq. (4.18)]. Moreover, neutrino decoupling approximately happens when H(T) ~ rweak (T) ex GF T5 (cf. the discussion at the end of sec. 2.2.4), and hence we may approximate the modified neutrino-decoupling temperature by 39 3.3. Influence of additional dark-sector states means of the relation

Tv5d/H(Tvd) = T m Hsm T' -m) . (3.75)

Here, the additional index “sm” is used to denote quantities that have been determined in the absence of dark-sector states, especially Tvd,sm = 1.4 MeV. On the other hand, for qdark(t) = 0 after decoupling, we are no longer able to use entropy conservation in the electromagnetic sector to deduce the ratio Rvd/R(T) that enters eq. (3.15). In this case, we have to use a more general approach and determine the scale factor by means ofthe first Friedman equation (2.4), which yields upon integration (similar to what we did in eq. (3.34))

Rvd/R(T) = exp (- IT H(T) dTdT^ (3.76)

with Tvd from eq. (3.75). This approach introduces an implicit dependence on the various spectra fv via the Hubble rate, and therefore - upon insertion into eq. (3.15) - leads to an integral equation for the neutrino spectra after decoupling. We discuss how to (iteratively) solve the resulting equation in sec. 3.4.

3.3.3 Time-temperature relation

Regarding the time-temperature relation, the total heat variation qtot(t) that enters the second Friedmann equation (2.34) gets extended by an additional contribution from the dark sector, which implies

qvi7(t) qdark(t) = qem(t) = pem(t) + 3H(t) [pem(t) + Pem(t)] (3.77) with the total Hubble rate H (t) from eq. (3.73). By using the relation pem (t) = T (t) dpem(T )/dT and after rearranging the resulting expression with respect to T(t), we then find the modified equation11

dT qdark(t) + qvv(t) + 3H(T) [pem(T) + Pem(T)] 7„, dt _ dpem(T)/dT • (3.78)

This expression can be usedto deduce the time-temperature relation ofthe SM heatbathinthe presence of additional dark-sector states, assuming that the decay products rapidly thermalize with the background plasma, which is true for t < 108 s [66,67]. Especially, since this equation does not make any assumptions regarding the evolution of neutrinos, it even remains valid in the presence of dark-sector-neutrino interactions. From this general expression, the special case of instantaneous neutrino decoupling is obtained by repeating the steps discussed at the

11For T > pem(T) and Pem (T) need to be replaced by the energy density and pressure of the SM without neutrinos, i.e. by the expressions from eqs. (3.31) and (3.32) without the term a Tv(T)4. Chapter 3. Big Bang Nucleosynthesis 40 end of sec. 3.1.2, which yields

dT qdark(t) + 3H(T) [psm(T) + Psm(T)] f = — for T > Tv (3.79) dt dpsm(T)/ dT

dT qdark(t) + 3H(T) [pem(T) + Pem(T)] for T < Tvd . (3.80) dt dpem (T)/dT

As a result, a fully decoupled dark sector with tjdark (t) = 0 does not change the evolution of T(t) except for its contribution to the Hubble rate. However, for qdark(t) = 0, a heat exchange exists between the dark sector and the SM heat bath, which causes the photon temperature to drop slower or faster over time, depending on the transfer direction.

3.3.4 Baryon-to-photon ratio

The additional energy density from the dark sector also leads to a modification of the effec­ tive number of neutrinos, which gets additional contributions from the dark radiation that is still present at the time of recombination. Denoting the corresponding energy density by pdark-rad(t) C pdark(t), eq. (3.64) gets generalized to

pvv (t rec ) + p dark-rad (trec ) Neff = / (3.81) peff (trec ) and we have to use this expression whenever we calculate the best-fit value of the baryon-to- photon ratio n according to eq. (3.67). In addition to this, the additional energy density also alters the temporal evolution of the baryon-to-photon ratio prior to recombination. This is due to the modified Hubble rate and time-temperature relation that enter the calculation of nb(t) = n(t)nY(t) in eq. (3.34).

3.3.5 Photodisintegration

In the SM, there are no processes that alter the light-element abundances after nucleosynthesis has finished at T « 10 keV. Especially - for these temperatures - the thermal photons12 that populate the universe are no longer energetic enough to disintegrate the previously formed nuclei, meaning that reactions of the form Dy pn have already become inefficient. How­ ever, this situationmight change in the presence of additional dark-sector states that decay (or annihilate) into electromagnetic radiation after the end of nucleosynthesis. In such a scenario, the injected particles would initiate an electromagnetic cascade, thus creating non-thermal photons13 with energies that are potentially large enough to initiate disintegration reactions with the previously created nuclei. For this process to happen, the injected particles must have

12Here and in the following, a thermal particle is understood to originate from the thermal, i.e. equilibrated, background plasma. 13Eventually these photons would also thermalize with the background plasma, which, however, only happens on much larger timescales. 41 3.3. Influence of additional dark-sector states energies that are comparable to the binding energies of light nuclei, Eb ~ O(MeV), which is naturally expected for dark-sector models with MeV-scale particles. In the following, we break down the process of photodisintegration by (i) deriving the appropriate equations to calculate the non-thermal spectrum of photons originating from the electromagnetic cascade, andby (ii) quantifying the late-time modification of the light-element abundances due to photodisintegration reactions with these non-thermal photons. However, before tackling this problem, let us note that in the literature regarding photodisintegration, it is common to use a different normalization for the phase-space distribution function. More precisely, the redefined spectrum fx = /x is differential only in the energy and not in all three momentum components, i.e.14

fx(t, E) = nEpfx(t, p) with fx(t, E)dE = /(t, p)d3p . (3.82)

To stay consistent with the literature, we therefore also use this redefinition of the spectrum whenever appropriate.

Electromagnetic cascade

Once additional electromagnetic particles are injected into the plasma, they immediately ini­ tiate an electromagnetic cascade due to interactions with the thermal background species. By denoting the thermal particles with an index “th”, the most important reactions that drive this cascade are15

1. Double photon pair creation YYth e+e-

2. Photon-photon scattering 77th 77

3. Bethe-Heitler pair creation yN e+e-N with N G N

4. Compton scattering Ye- Ye-

5. Inverse Compton scattering e± Yth e± y .

Out of these processes, the most influential one is double photon pair creation, which is dom­ inant for all photons in the cascade that have energies above the electron-positron creation threshold of e- — m2 / (22T) [24]. As a result of this process, all photons with Ey > *E e- get rapidly converted into electrons and positrons, which depletes the abundance of non-thermal photons above this threshold - for a given temperature T. Regarding photodisintegration, the least energetic process Dy np has a threshold energy of ,np — 2.22MeV [28] and consequently this reaction is only efficient if *E e > EDY,. .n/, which equates to T < 5.35 keV.

14The second expression is only valid in combination with an integrand that does not depend on the angular components of the momentum. 15In this case, the reverse reactions do not exist, as it is not possible to create thermal particles from non-thermal particles. Chapter 3. Big Bang Nucleosynthesis 42

As a result, we find that photodisintegration is only relevant for temperatures past the end of nucleosynthesis, meaning that both mechanisms can be treated independently. To determine the non-thermal spectra that originate from this cascade process, we once again have to write down the appropriate Boltzmann equation, which has the general form [24] (omitting the time/temperature dependence for brevity)

dfx (E) ~ Sx (E) - rx (E)fx(E) + K (E, E'f (E') dE' (3.83) dt x with x G {e-,e+,7}. Here, rx(E) is the total interaction rate of particle x at energy E,

Kx'.x (E, E') is the differential interaction rate for scattering/conversion of particle x' with energy E' to particle x with energy E, and Sx (E) is the source term for the production of par­ ticle x with energy E. In this expression, we further ignored the term « H(t) on the left-hand side of the Boltzmann equation, which is justified since rx (t) » H(t) for all temperatures of interest [24, 29]. In this thesis, we do not perform a detailed derivation of all terms entering this equation, as this would involve the calculation of many collision integrals. However, to get a better understanding of its origin, we consider the exemplary reaction 77th — e+e- and discuss its contribution to the full cascade equation. In this case, the collision operator that enters the Boltzmann equation for fy « f7 is given by

C77th-eE f7 -] (2=3) - __L y g-1 |M?7th^e+e-|2(2^)46(p7 + Pyth - Pe+ - £e~ )

v (1 _ f )(1 _ f ) f f d3P7th d3Pe+ d3Pe- (3.84) ( fe+)( fe )'7'7th (_n)32E7th (2n)32Ee+(2n)32Ee- ’

Regarding this expression, the important realization is that we can again neglect the spin­ statistical factors and set 1 - fe± ~ 1. This is because the abundance of injected particles is generally constrained to be rather small, which leads to highly suppressed non-thermal spec­ tra with fe± 1.16 Afterwards, the only remaining unknown function that enters eq. (3.84) is f7, which can be moved outside of the integral. This allows to define

df7 C77ih -e' e [f7z"-] / r (3.85) dt D e7 ■ f177th-e+e- with some rate r77th .e ■ e- that can be expressed in terms of analytic functions. Since both sides of this equation are proportional to f7, and f7 « f7 according to eq. (3.82), we can then simply replace f7 by f7. With regard to eq. (3.83) we thus find r7(E) D r77th .e.e (E), which implies that rx (E) receives contributions from all reactions that features x in the initial state. An analog discussion for Kx/.x (E, E') then shows that there is one contribution to Kx/.x (E, E') for each reaction with x' in the initial and x in the final state. The corresponding analytic expressions have already been collated many times in the literature (cf. app. B for a list of references). However, given that a number of typos are present in these compilations, we provide an updated list of all rates and kernels in app. B for convenience.

16For example, this statement is verified by the results that are shown in fig. 5.2. 43 3.3. Influence of additional dark-sector states

Regarding the solution of eq. (3.83), the different interactions rapidly establish a quasi­ static equilibrium between the non-thermal particles.17 Hence, we seek for solutions with a vanishing temporal derivative, df(E)/dt = 0, which are determined by the equation

fx (E) = rJE) (sx (E) + /^Kx^x (E, E' f (E') dE') (3.86)

In this expression, the source term Sx(E) depends on the actual dark-matter model, but usu­ ally can be written in the form

Sx (E) = S^ ö(E — Eq ) + SxFSR)(E) (3.87) with Se±SR) (E) = 0 and sYFSR^ (E) a. S(± (cf. app. B). Here, the first term describes the creation of x from some dark-sector particle, i.e. DS e+e— / yy, and the second term accounts for the accompanying final-state radiation in case of injected electrons, i.e. DS e+e—7. Moreover, the presence of ö(E — E0) accounts for the fact that dark-sector particles with MeV masses or above are non-relativistic for T ~ O(keV) and hence inject monochromatic particles with some energy E0. We derive a general expression for S(x0) at the end of this section. In any case, the presence of ö(E — Eo) is difficult to handle numerically and it is beneficial to split this term from the remaining spectrum, which can be achieved by defining

sxo) 0(E — Eo ) Fx(E) := fx(E) (3.88) x (E) which then leads to the modified equation

K . (E, Eq')S() rx (E)Fx (e) = sxFSR)(E) + £ K . (E, E)Fx' (E') dE' (3.89) x fix' (Eo) Given the relevant rates and source terms, this equation determines the non-thermal spectra of all particles x E { y, e+, e— }, which can afterwards be used as an input for the disintegration reactions of light nuclei due to non-thermal nucleosynthesis. For future reference, let us finally note that for T > m2e/(22E0), the non-thermal distribution of photons can be approximated by the universal spectrum — ) 3/2 K0 for E < EX — X 2 fy,univ. K0 for Ex < E < Ec (3.90)

0 for E > EC withK0 = E0EX—2[2+ln(EC/EX)], EC = Eeth+e— = m2e /(22T), and EX = m2e /(80T) [24]. How­ ever, this approximation breaks down as soon as the energy of the injected particles is below the effective cutoff energy Eteh+e— for the production of electron-positron pairs [25], in which case it is crucial to solve eq. (3.89) explicitly. In this thesis, we always use the full solution of the cascade equation and only employ the universal spectrum as a validity check.

17Later on, these particles also thermalize due to kinetic interactions with the background plasma; however, this happens on much larger timescales. Chapter 3. Big Bang Nucleosynthesis 44

Non-thermal nucleosynthesis

The cascade equation (3.89) that we derived in the last section determines the non-thermal spectra originating from the electromagnetic cascade. The photons that are created in this process afterwards participate in photodisintegration reactions (Dy — np,3HeY — npp, ...) which cause a late-time modification of the light-element abundances after nucleosynthesis (cf. app A.2 for a list of all relevant reactions). Just like before, we can quantify this effect by deriving appropriate (supplementary) Boltzmann equations for the various abundances Yx (t). To this end, let us first consider reactions of the form Xy — ij with three distinct nuclei i, j, and X. In this case, the integrated collision operator that enters the equation for YX(t) is given by

J Cx; iL/x,...] (2=3) “/ |MXy—ij 22n)40(px + py -Pi -Pj)

d3 px d3 Py d3 pi d3 Pj x (1 ± fi)(1 ± fj )fXfY (3.91) (2n)32Ex (2n)32EY (2n)32Ei (2n)32E;-

Since the nuclei are non-relativistic for T ~ O (keV), we can once again neglect the spin­ statistical factors. Expressing eq. (3.91) in terms of the polarized cross-section then yields (cf. eq. (3.40)) 3 gx d3 px gxd px gYd Py cXy ■

The fact that the nuclei are non-relativistic also allows to set px = (mx,0) in (7xY -x and Vm0i, which especially implies Vm0i = 1, since the photons are hitting stationary targets in the comoving reference frame.18 As a result, J\Y .rj only depends on Ey and hence we can (i) independently perform the integral over d3px to obtain the number density nx, and (ii) rewrite fY in terms of fY by means of eq. (3.82), which yields 3 ] gxd px = nX Jo ^tXy -ijfY dEY = nbYX Jo ^tXy -iifY dEY . (3.93) ,."] (2n)3Ex

Consequently, after inserting this result into the full Boltzmann equation for Yx, we find that the variation of Yx due to photodisintegration reactions of the form xY - ij is given by

Yx D -Yx fY^XY-ij dEY . (3.94) 0

This relation can then be generalized to account for all relevant processes. More precisely, let j and j' range over all nuclei participating in photodisintegration reactions of the form jY — X

18Formally, the spectrum fx « ep2/(2mxT) can be interpreted as a Gauß function of px/T with mean value 0 and variance mx /T. Mathematically, such a function constitutes a delta sequence for mx /T 1, i.e. we have fN k ^mx/T(Px/T), which can be used to set px = 0 for non-relativistic particles. 45 3.3. Influence of additional dark-sector states and Xy j', respectively. Then the full Boltzmann equation for the temporal evolution of the abundance YX is given by

Yx (t) = £Yj (t) Jo f (t, Ej;(E) dE — Y;(t) £ f7(t, E)(Txy^f (E) dE . (3.95)

At first sight, the two eqs. (3.46) and (3.95) form a coupled system of equations. However, since nucleosynthesis and photodisintegration happen at different times - Tbbn ~ O(MeV) vs. Tpdi ~ O(keV) - both equations can be solved separately, thereby using the result of the first as an initial condition for the second. At this point, it is also worth noting that all disintegration reactions are subject to a certain threshold, which is essentially determined by the binding energy of the element in question. Depending on the energy of the injected particles, some reactions might therefore not be accessible at all. For example, photodisinte­ gration of helium-4 can only proceed if there are non-thermal photons with energies above ElHe-;, . 11 = 19.81 MeV (cf. app. A.2 for a list of all relevant threshold energies).

General expression for the source term

To calculate the appropriate source term that enters the cascade equation, let us first assume that the additional dark-sector states exclusively inject photons. From the previous discus­ sion we know that photodisintegration is only relevant if non-thermal particles are injected around T ~ O(keV). Consequently, all relevant particles have already got non-relativistic and therefore can only inject monochromatic photons with a fixed energy E0. As a result, the spectrum of injected photons must obey f^ (t, E) « ö(E — E0) and after employing the correct normalization from eq. (3.82), we find

j E) = PyE^ö(E — E0) = ö(E — E0) , (3.96) E E0 which yields upon differentiation with respect to time

df^t, E) pjt) z , x 7 = ~y[^ö(E — E0) . (3.97)

To determine the unknown function (t), let us note that the injection of non-thermal parti­ cles can only transfer heat from the dark sector into the final-state particles, which implies

— qdark (t) = qj) = pjt) + 3 H(t) [j + Pj)] , (3.98) and thus leads to the following equation upon insertion into eq. (3.97),

+ 3H (t)[pn(t) + Pjt)] Ö(E— E) = — qdaEko (t) Ö(E — E0) . (3.99)

In the previous discussion we have consistently neglected all dilution terms « H(t) according to [24, 29]; hence, after dropping the second term on the left-hand side of eq. (3.99), we can Chapter 3. Big Bang Nucleosynthesis 46 identify

S(0) (t) = ~qdark(t) , (3.100) E0 since df7n'(t,E)/dt = S7(t,E) = S^(t)3(E - E0) according to eqs. (3.83) and (3.87). If we instead consider an exclusive injection of electron-positron pairs, the derivation proceeds in a very similar way. However, in this case the heat that is released by the injection process gets equally distributed between electrons and positrons, which introduces an additional factor

1/2, q^t) — t/inje (t) = 2qe+ (t) = 2qinj (t). Finally, for the general case, we have to each term with the appropriate branching ratio, which yields

S70)(t) = BR77 x ~qdEk(t) and Se±) (t) = BRee x (t) . (3.101)

These expressions then allow to solve the cascade equations for all dark-sector scenarios that are discussed in this thesis.

3.3.6 Hadrodisintegration

There is also the possibility that the dark-sector particles do not inject electromagnetic but hadronic states. In this case, the resulting hadronic cascade would produce non-thermal pions and kaons, which would then influence the creation of light elements via additional contribu- tionsto eqs. (3.46) and(3.52). However, such a scenario is only relevant for heavier dark-sector particles with mass mdark ~ mn ~ O(100MeV), which are not the main focus of this thesis. We therefore refrain from a detailed discussion of this effect and instead refer to the many studies that have already been conducted in the existing literature [59, 68-70].

3.4 General solution strategy

In the last section, we derived all necessary equations to calculate the light-element abun­ dances in the presence of MeV-scale dark sectors. However, most of these equations are tightly coupled, which adds another level of complexity to solve them. In this section, we therefore outline the general strategy that we use to calculate the final light-element abun­ dances for a given dark-sector scenario. The important realization here is that we can break down the full calculation into individual chunks. First of all, the equations governing nucle­ osynthesis and photodisintegration are not coupled to the ones of the background particles, since the light-element abundances (at that time) are too small to alter the Hubble rate or the time-temperature relation significantly. Besides, nucleosynthesis and photodisintegration are well separated in time and can therefore also be covered individually. Hence, we can di­ vide the full calculation into three distinct parts, namely (i) the cosmological evolution of the background particles, which also includes the dark-sector states, (ii) the evolution of the light­ element abundances during nucleosynthesis, and (iii) the modification of the light-element abundances due to photodisintegration. 47 3.4. General solution strategy

3.4.1 Cosmological evolution

In the SM, Hubble rate and time-temperature relation can be calculated in a self-contained manner, meaning that the spectra of background particles can be determined without the a priori knowledge of the Hubble rate - as long as we assume instantaneous neutrino decou­ pling. However, this convenience cannot be maintained in the presence of additional dark­ sector states. More precisely, the Boltzmann equations (3.72) for the various spectra fd(t, p) explicitly depend on the Hubble rate, which cannot be fully determined without the knowl­ edge of fd(t, p) itself (cf. eq. (3.73)). Moreover, if entropy in the electromagnetic sector is no longer conserved, the Hubble rate also enters eq. (3.76), i.e. the determination of the neutrino spectra fy. (t, p), which again have a non-negligible contribution to the Hubble rate. All in all, these mutual dependencies result in a non-linear and highly coupled system of equations, which is hard to solve for most conventional ODE solvers. To overcome this problem, we instead utilize an iterative approach, which proceeds in the following steps:

0 - Setup Start the iteration with SM values for the Hubble rate H(0)(T), the time-temperature relation T(o) (t), and the energy density (T).

1 - Dark sector Solve the Boltzmann equations (3.72) for the dark-sector states with

H(t) = H(o)(T(o)(t)) (3.1o2) in order to determine the various spectra fd(o)(t, p). Usually, this step leads to a coupled set of integro-differential equations (cf. sec. 2.2.2); however, in this case the best solution tech­ nique depends on the actual collision operators, which is why it is best to postpone the cor­ responding discussion to a later stage of this thesis. Assuming that we managed to solve the corresponding system of equations, we afterwards calculate the thermodynamical quantities of the dark sector, including the total energy density pd(oa)rk(t) and pressure Pd(oark) (t) as well as the volume heating rate i/dark (t).

2 - Time-temperature relation Calculate the new time-temperature relation T(1)(t) by solving eq. (3.79) for T > TVJ) and eq. (3.80) for T < T|(d) with

H(T) = H(o)(T) and q/dark(t) = q/d(oa)rk(t) . (3.1o3)

This leads to an ordinary but non-linear differential equation, which can be solved numeri­ cally by using the vode integrator that is implemented in SCIPY [71], which applies a method based on backward differentiation formulas (BDF). However, since T(t) « 1/\/f for most times (cf. right panel of fig. 3.1), we substitute Z(t) = T(t)\/f in order to improve the numeri­ cal stability. Chapter 3. Big Bang Nucleosynthesis 48

3 - Hubble rate Calculate the new Hubble rate H(1)(t) according to eq. (3.73) with

Psm(t) = psm(T(1)(t)) and pdark(t) = Pdark(t) • (3d04)

4 - Neutrino decoupling Update the neutrino-decoupling temperature Td according to eq. (3.75) as well as the scale factor R(1)(T) according to eq. (3.76) with

H(T) = H(1)(t(1)(T)) and T(t) = T(1)(t) . (3.105)

This step leads to the new energy density of neutrinos, which can then be used to calculate the new energy density ps(m1)(T) ofthe SM.

5 - Repeat Replace (n) . (n+1) and repeat from step 1 until the quantities do no longer change signif- icantly between iterations.

This iterative procedure allows to consistently determine the Hubble rate H(t), the time­ temperature relation T(t), and the spectra of all dark-sector particles, which can then be used to determine the light-element abundances that are created during nucleosynthesis. In this context, we always check that the iterative procedure converges for all parameter points around the exclusion limit with a tolerance of 0.1%.

3.4.2 Light-element abundances

Regarding the evolution of the light-element abundances, we numerically solve the coupled set of Boltzmann equations (3.46) and (3.52) with the previously calculated time-temperature relation. To this end, we use the integration procedure that is implemented in ALTERBBN (more details can be found in the appendix of [48]) and also adopt the included network of nuclear rates. In this process, we set the baryon-to-photon ratio n to its best-fit value according to eq. (3.67) by calculating the respective value of Neff from eq. (3.81). In total, we perform this calculation three times by using the mean, low, and high values of the nuclear rates imple­ mented in ALTERBBN, which leads to the abundances YX, YX,high, and YX,low for all X G N . In the presence of dark-sector particles that inject electromagnetic radiation, we calculate the non-thermal spectra of photons, electrons, and positrons via eq. (3.89), which constitutes a Volterra integral equation of type 2 that can be solved numerically by using a discretization procedure similar to the one described in [24] (more details can be found in app. C.1). To this end, we include the appropriate source terms (which are specific to each dark-sector model) as well as the rates and kernels that are compiled in app. B. Regarding the late-time modification of the nuclear abundances due to photodisintegra­ tion, we solve the system of ordinary and linear differential equations (3.95), which can be performed semi-analytically (except for the integrals over the rates) by transforming it into a 49 3.4. General solution strategy single matrix equation (more details can be found in app. C.2). Here, we use the same baryon- to-photon ratio as before and implement the rates ofall reactions that are compiled in app. A.2 by adopting the analytical expressions for the rates 1 - 9 from [28]; however, we modify the prefactor of reaction 7 from 17.1 mb to 20.7mb as suggested by [29] in order to match the most recent EXFOR data. Again, this calculation is performed three times with initial conditions corresponding to Yx, Yx,high, and Yx,low, which then leads to the final abundances YX, YX high, and YX low for all X G N after photodisintegration.

3.4.3 Final constraints

Using these abundances, we then calculate the three observable abundance ratios R1, R2, and R3 (cf. eq. (3.62)) as well as their deviation from the observationally inferred values accord­ ing to eq. (3.70). From there, we consider a given model, or rather a given parameter point, excluded at 95% C. L. if

|Ar | > 1.96 (3.106) for at least one abundance ratio Ri. Repeating this process for all relevant parameter points then yields the full set of BBN constraints for the given model. Let us note that by using this procedure, we employ the envelope of the individual 95% C.L. constraints and do not consider their correlation to calculate a global 95% C.L. This approach is conservative as the latter procedure would lead to somewhat stronger constraints in regions where two different exclusion lines are close to each other.

Putting everything together, we have developed a general formalism that allows to calculate the light-element abundances and hence the BBN constraints for many different dark-sector scenarios. To apply this procedure, we only have to specify the Boltzmann equations for the various dark-sector states as well as the model-specific expressions for the Hubble rate, the time­ temperature relation, the effective number ofneutrinos and the source terms for photodisintegration. In the following, we apply this formalism to calculate constraints on the decay of MeV-scale mediators (cf. chs. 4 and 5), the annihilation of MeV-scale dark matter (cf. ch. 6) as well as constraints on specific models like scalar-portal dark-matter models (cf. ch. 7) and models with axion-like particles (cf. ch. 8). Chapter 3. Big Bang Nucleosynthesis 50 51

4 Bounds on sterile decays of light mediators

This chapter is based on the following publication:

M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on MeV-scale dark sectors. Part I. Sterile decays, JCAP 1802 (2018) 044, [arXiv:1712.03972]

In this thesis, we further implement the following improvements:

• Use the most recent values for the observed light-element abundances from [37]

• Include the correlation between n and Neff according to the discussion in sec. 3.2.2

In this chapter, we consider a fully decoupled dark sector with MeV-scale particles decaying into dark radiation, which are neither fully relativistic nor non-relativistic during all temper­ atures relevant for BBN. Since such scenarios do not feature any couplings to the SM, they are usually impossible to probe at terrestrial experiments such as colliders or dark-matter direct- detection experiments. However, as long as the temperature of the dark sector is not much smaller than the one of the SM, such a scenario would still significantly contribute to the total energy density of the universe and hence lead to an observable imprint on the light-element abundances that are created during BBN. By quantifying the influence of such scenarios onto nucleosynthesis, it is therefore possible to derive constraints for even completely decoupled dark sectors. Particularly, with this study we aim at complementing the existing literature, which already comprises studies for dark-sector particles that are purely ultra-relativistic [20] or non-relativistic [22, 23, 27] during BBN.

4.1 Setup and assumptions

Regarding the concrete setup, we extend the SM by a fully decoupled dark sector that contains three additional particles: (i) a dark-matter candidate x and its antiparticle x with mass mx ~ O(GeV), (ii) a self-conjugate, MeV-scale boson with mass m$ ~ O(MeV), and (iii) a very light fermion N and its antiparticle N with mass mN — 0, which essentially behaves like a sterile neutrino. Regarding the cosmological evolution of these particles, we further make the following assumptions:

• For large temperatures, the particle ty is thermalized within the dark sector, but even­ tually decouples (chemically and kinetically) 1 at some photon temperature Tcd, which

1After chemical decoupling, the dark sector could still stay in kinetic equilibrium until some temperature Tkd = Tcd, which could be described by additional elastic terms in the collision operator. However, in sec. 4.4.1 we show that our results are almost independent of the additional parameter Tkd, which is why we set Tkd = Tcd for simplicity. Chapter 4. Bounds on sterile decays of light mediators 52

Dark sector SM sector

Dark matter SM particles __ T j XX mK ~ (9 (GeV) 7, e±, y.±, t±, vb vif ... *d,cd ; ------n£ o £cd Mediator___ a "2- m

Figure 4.1: Sketch of the dark-sector model that is discussed in this chapter. The dark and the visible sector are completely decoupled (as indicated by the solid gray line), but there exist interactions between the mediator (p and the sterile neutrinos N (decay) as well as between the mediator and the dark-matter candidate x (decoupling).

corresponds to a different temperature Td cd in the dark sector. Such a setup can be re­ alized by introducing a coupling between (p and x, which leads to a freeze-out via the process XX W-

• The decoupling of (p happens before BBN (Tcd > 10 MeV) and consequently the dark­ matter particle x only enters the calculation implicitly by influencing the value of Td,cd' e.g. Tdcd ~ zn^/20 for a vanilla freeze-out scenario. After decoupling, x makes up the observed relic abundance and therefore only gives a negligible contribution to the en­ ergy density for the temperatures that are relevant for BBN.

• The particle (p can decay into sterile neutrinos via the reaction (p —> NN with some lifetime Tq.

• The dark sector and the SM sector are fully decoupled, meaning that any couplings between both sectors are considered to be sufficiently small.

• The particle N is only created via the decay of (p, meaning that its abundances is negligi­ ble at large temperatures. This assumption implies a small coupling between N and (p, corresponding to a large lifetime r^.

Given these assumptions, the full setup can be characterized by four different parameters, namely (i) the mass and (ii) the lifetime of the MeV-scale particle (p, (iii) the photon 53 4.2. Evolution andinfluence of the dark sector temperature Tcd at the time of decoupling, and (iv) the corresponding temperature ratio Zcd := Td,cd/Tcd.2 The full setup with all relevant parameters is also summarizedin fig. 4.1. In this context, the ultimate goal is to calculate the BBN constraints in the corresponding parameter space using the general solution strategy that we developed in sec. 3.4. To this end, we first employ the specific Boltzmann equations (3.72) for the dark-sector particles ty and N, and afterwards specify the influence of this particular dark-matter model on the process of nucleosynthesis.

4.2 Evolution and influence of the dark sector

4.2.1 Evolution of the mediator

Before decoupling, the particle ty is assumed to be in thermal equilibrium within the dark sector, and therefore can be described by a Bose-Einstein distribution with some temperature Td(T) = T. This statement also holds true at the time tcd of decoupling itself, which implies3

-1 (p2 + mty )1/2 fty (tcd, p) exp 1 (4.1) Ccd Tcd with Tcd = T(tcd) and a vanishing chemical potential p.ty (tcd) = 0. After decoupling, the evo­ lution of ty is dominated by its (inverse) decay ty NN and we have to solve the appropriate Boltzmann equation to determine the spectrum fty (t, p) for t > tcd. In this case, we can simply recycle the result that we derived at the end of sec. 2.2.2. More precisely - by substituting x = ty and z, z = N, N in eq. (2.24) - we find

dfty (t P) mty mty - H (t) P fty (t,p) + x 1 fN (fi PN )fN (t Pn) PNEd PN dt p p - EN

+ ET ft (t P) X p/ fN (t PN) + fN (t PN)] (4.2) with ßn = 1 for mN = 0 and by using the lower sign since N is a fermion. In general, this Boltzmann equation is coupled to the ones that govern the evolution of N and N, which leads to an interconnected system of integro-differential equations. However, in this thesis we try to reduce the complexity of this problem by making a conservative approximation:4 Since the sterile neutrinos are only produced via the decay of ty, their contribution is subdominant for t < Tty, which is why we set fN(t, Pn) = fN(t, Pn) = 0 in eq. (4.2). This way, we neglect the effects ofinverse decays and spin-statistics; however, we showinsec. 4.4.2that the inclusionof

2The parameter m^ is fixed by Td cj = Zcd Tcd, and mN is assumed to be negligible. 3Here, we assume that the decoupling does not significantly distort the thermal spectrum of ty. 4So far, we have not succeeded yet in implementing a stable numerical procedure that could solve the given set of equations for all relevant parameter points. For now, we therefore leave this endeavor for future work. Chapter 4. Bounds on sterile decays of light mediators 54 these effects does not substantially change the resulting constraints. Using this approximation, the Boltzmann equation (4.2) for ty greatly simplifies to

, p) H(t\p dftyp) mty f t p\ (43) — H(t}— Erf( p)' (3) This is a partial but linear - and no longer coupled - differential equation, which can be trans­ formed into an ordinary one by using the method ofcharacteristics. In this process, we can assume that H(t) isindependentof fty(t, p), which is in accordance with the iterative proce­ dure that we described in sec. 3.4, i.e. we use H(0) (t) to calculatefty(0) (t, p) and so on. For fixed H(t), the characteristic equation of eq. (4.3) is given by

ddp = — pH(t) pR(t) — w = 0 , (4.4) and consequently the integration constant w defines the corresponding family of characteristic curves. Physically, this result is a direct consequence of the momentum redshift that drives the evolution of ty after decoupling. Based on this relation, we can introduce a new function

fty(t, p*) := fty(t, p*/R(t)) with p* := w(t, p) = pR(t) , (4.5) which transforms eq. (4.3) into the ordinary form

dfty(t, p*) mty fty fa p*) (4.6) dt E

ftyp)= fty (tcd pR(t)/R(tcd)) exp |— - dt' | (4.7) Tty tcd p2R(t)2/R(t')2 + mty with the scale-factor ratio Rt/; w) , <«> according to the first Friedmann equation (2.4). The interpretation of this equation is rather intuitive: For stable particles (Tty TO), the evolution of ty is merely driven by the momentum redshiftaccordingtopR(t) = const. For finite lifetimes, we then also have to take into account the exponential decay term ex. exp[—t/(yTty)] with the appropriate time-averaged Lorentz fac­ tor y, which accounts for relativistic mediators with an increased lifetime in the lab-frame. In fact, since we did not assume anything about ty being non-relativistic or ultra-relativistic, this solution (4.7) is correct for all temperatures, including T ~ mty. Finally, the energy and number density of ty follow from a simple integration over the appropriate momentum phase-space. Regarding the number of spin-states gty that enter this calculation, we separately discuss scalar mediators with gty = 1, as well as vector mediators with gty = 3. In general, the decrease of fty(t, p) due to its decay is compensated by an increase in the abundance of sterile neutrinos, whose evolution is discussed in the next section. 55 4.2. Evolution and influence of the dark sector

4.2.2 Evolution of the sterile neutrinos

Regarding the evolution of the sterile neutrinos, we could again derive a Boltzmann equa­ tion from the general expression (2.12). However, in this particular case it is also possible to simplify the whole process by drawing on conservation of heat in the dark sector. More

precisely, since the dark sector is fully decoupled, we have qdark (t) = 0 or equivalently q$(t) + 4nn(t) = 0 with qnn(t) := qN(t) + qn(t). Moreover, the integrated Boltzmann equa­ tion (2.31) for $ is given by, C f$](t, p) = -m$f$(t, p)/(E$T$),

-qnn(t) = q$(t) - - m$n$(t) (4.9) L(p

In combination, these statements then lead to a complementary Boltzmann equation for the summed energy density of sterile neutrinos, Pnn(t) := Pn(t) + Pn(t),

P NN (t) + 4H (t)pN?N (t) = ~~ (4-10) T$

with Pnn (t) = Pnn (t)/3, since N is ultra-relativistic. This is a linear ordinary differential equation, which can be solved by using a separation of variables. By setting Pnn (tcd) — 0 in accordance with the assumption that N is only created via the decay of $, we thus find

Pnn(t) — m$n$(t') (RR(ty) dt' - (4-11)

With this result at hand, we have now fully solved the evolution equation for the dark-sector states, and we can now turn to the calculation of modified quantities like the Hubble rate and the time-temperature relation for this particular dark-matter model.

4.2.3 Hubble rate and time-temperature relation

First of all, the Hubble rate in this model receives additional contributions from the total energy

density of the dark sector, Pdark(t) = P$ (t) + Pnn(t), which implies

H(t) = 3 Psm(t)+ P$(t) + PNN(t)] , (4.12)

according to eq. (3.73). The additional energy density also leads to a modification of the ef­ fective number of neutrinos. Since the energy density of the mediator becomes negligible prior to recombination, the dark radiation receives contributions only from the sterile neutrinos.

Hence, by setting Pdark-rad(trec) = Pnn (trec) in eq- (3.81), we find

PVV (t rec ) + PNN (t rec) / (4.13) Peff(trec)

which implies a value of Neff that is increased by pvV (trec) /Peff (trec) compared to the SM value. According to eq. (3.67), we thus also end up with an increased best-fit value for the baryon- to-photon ratio. To determine the time-temperature relation, we can use heat conservation in the Chapter 4. Bounds on sterile decays of light mediators 56

dark sector, qdark(t) = 0, meaning that eqs. (3.79) and (3.80) can be written as

dT = _ 3H(T) [psm(T) + Psm(T)] for T > Tvd , (4.14) dt dpsm(T)/dT

dT = _ 'H T ', e m T . m T for T < Tvd (4.15) dt dpem (T)/dT with the full Hubble rate from eq. (4.12). As a result, we find that a fully decoupled dark sector can only indirectly influence the time-temperature relation via its contribution to the Hubble rate. The conservation of heat in the dark sector also implies vanishing source terms for photodisintegration,

S(y0)(t) = Se(±0)(t) = 0 , (4.16) according to eq. (3.101), which is the expected behavior for completely decoupled particles. Consequently, photodisintegration therefore does not need to be considered at all. Using these results, we can now apply our general solution strategy from sec. 3.4 to cal­ culate the BBN constraints for this particular dark-matter model. However, to gain a better understanding of the final results, let us first take a closer look at the evolution of pty (T) and Pnn(T) for some exemplary parameter points.

4.2.4 Combined evolution

In the left panel of fig. 4.2 we present the final solutions for pty (T) and pNN(T) for different choices of parameters. In general, the evolution of the dark sector features three distinct re­ gions. For t » Tty and T » mty, the particle ty is ultra-relativistic and therefore follows the expected scaling behavior pty (T) « R(T)— 4 for decoupled relativistic particles. However, once the boson becomes non-relativistic at around T ~ mty, the energy density of ty starts to fall off less steeply with pty(T) ex. R(T)— 3, which leads to a relative increase of its energy density com­ pared to the ultra-relativistic case. Finally, at t ~ Tty, the abundance of ty gets exponentially suppressed due to its decay. This decrease is compensated by the creation of sterile neutrinos, which re-establish the ultra-relativistic scaling pnr(T) « R(T)— 4. Depending on the choice of parameters, the decay of ty can also happen in the ultra-relativistic regime, in which case the region with pty(T) ex R(T)— 3 disappears, leading to a dark sector that is entirely composed of radiation (cf. the transition from the red curve in fig. 4.2 with Tty = 102 s to the blue one with Tty = 10 s). Qualitatively, the evolution of the dark sector therefore depends on the order of the mediator (i) becoming non-relativistic, and (ii) decaying into sterile neutrinos. More precisely, the longer the mediator can profit from the non-relativistic scaling before its decay, the larger the final energy density turns out to be in comparison with the one of the SM. Hence, we can already anticipate that the BBN bounds are strongest for sufficiently heavy and long-lived mediators. 57 4.3. Results and discussion

Figure 4.2: Left panel: Evolution of the energy densities P(p(T) (solid) and Pnn(T) (dashed) as a function of temperature for = 1 MeV, Tcd = 5 GeV, £cd = 1 and two different choices for the lifetime (red: = 102 s, blue: = 10 s). We also show the energy density pSm(T) of the SM for comparison (black). Right panel: The energy-density ratio pdark(T) /Psm(^) for the same two choices of parameters as well as the 95% C.L. for the naive scaling pdark(^) — ^Psm(T) for comparison. In both panels, the temperatures that are relevant for BBN are indicated in gray.

In the right panel of fig. 4.2 we further show the ratio pdark(^)/Psm(’T) for the same two choices of parameters. Utilizing our solution procedure to calculate constraints for the naive scaling pdark(^) = ^Psm(^) with some constant K, we find that the theoretically predicted abundances for a vector mediator are still compatible with observations at 95% C.L. if K < 4% (dashed black). In the full model, however, this ratio is not constant but instead varies by up to one order of magnitude. Hence, this simple scaling is not applicable and a more careful treatment is needed to infer the final exclusion limits. Especially, we find that the parameter point with = 10 s (blue) remains barely allowed, even though the final ratio is slightly larger them 4%.

4.3 Results and discussion

In this section, we finally present the BBN constraints on sterile decays of MeV-scale medi­ ators (cf. fig. 4.1). As discussed above, there are four relevant parameters, the mass m

Tcd [GeV]

Figure 4.3: Constraints from BBN for different slices of parameter space for scalar (left) and vector mediators (right). In addition to the overall 95% C.L. limit (solid black), we also indicate the parts of parameter space that are excluded by individual elements (see text for details). 59 4.3. Results and discussion of helium-4 (pink/blue), over- and underproduction of deuterium (orange/gray), as well as overproduction of helium-3 relative to deuterium (green). In addition to this, we also consis­ tently indicate the regions of parameter space that are excluded due to an overly large/small value of Neff (dashed red/orange), which is calculated from eq. (3.81)andthencomparedto the most recent observations from PLANCK [37] with Neff = 2.99±0.21. Finally, we also show the resulting bounds in case y is not fixed by CMB observations (dash-dotted blue)5 as well as the two parameter points from fig. 4.2 for comparison (blue/red stars). Inthefirstrowof fig. 4.3, we show the resulting constraints in the mty — Tty parameter plane for Tcd = 5 GeV and Zcd = 1. For light mediators with a sufficiently small lifetime, ty decays when it is still relativis­ tic, in which case the dark sector simply behaves as extra radiation throughout its evolution. In this part of parameter space the ratio pdark(T)/psm(T) is approximately constant and by taking into account the relevant degrees-of-freedom we find

pdark (T) ~ f 058% for a scalar ty (4.17) T=IM„V~ fi74% for a vector ty , which is considerably smaller then the 4% limit that we deduced in the previous section. Con­ sequently, we end up with no BBN constraints in the bottom left corner of the mty — Tty param­ eter plane. This result also follows the general rule that the constraints are naturally stronger for a vector mediator as opposed to a scalar mediator. This is simply due to the higher number of spin degrees-of-freedom, which leads to a larger energy density according to the relation pty gty. Towards larger masses and lifetimes, the energy density of the dark sector increases compared to the one of the SM. This is because for T(t = Tty) < mty, the mediator starts to be­ come non-relativistic before its decay, in which case the overall energy density starts to profit from the pty(T) ex. R(T)—3 scaling of non-relativistic particles, compared to the pty(T)

In the second row of fig. 4.2, we show the resulting constraints in the Zcd — Tty parameter plane for Tcd = 5 GeV and mty = 1 MeV. In this case, an increase of the ratio Zcd = Tdcd/Tcd naturally leads to a hotter dark sector at the time of decoupling and therefore to a larger initial

5In this case, we consider a parameter point excluded, only if eq. (3.106) holds true for all n € [4.0,8.2] x 10—10. Chapter 4. Bounds on sterile decays of light mediators 60 energy density compared to the SM. Hence, we find stronger constraints for larger values of

Zcd, which become even more severe for sufficiently large lifetimes, i.e. when the mediator becomes non-relativistic prior to its decay. In the last row of fig. 4.3, we show the resulting constraints in the Tcd — Tty parameter plane with Zcd = 1 and mty = 1 MeV. In this context, we find that the bounds are largely insensitive to Tcd, except for decoupling temperatures close to the QCD phase transition at T ~ 200 MeV. In the latter case, the SM sector cools down much more quickly than the dark sector during the transition, which ultimately increases the energy density pdark(T) relative to psm(T) and consequently gives rise to stronger constraints. Finally, let us note that - for most parts of parameter space - the BBN constraints are subdominant to the corresponding ones from the CMB (Neff high). However, we might also be interested in scenarios with sterile neutrinos that decay (or otherwise disappear) before recombination or have eV-scale masses, in which case their contribution to Neff would be negligible. This would invalidate all constraints except for the one that is labeled BBN only, which is independent of all CMB measurement and therefore - albeit weaker - is significantly more robust.

4.4 Impact of kinetic decoupling and inverse decays

4.4.1 Impact of kinetic decoupling

So far, we have assumed that ty and x decouple both chemically and kinetically at a com­ mon temperature Tcd = Tkd. In general, the exact value of Tkd is model-dependent and its determination requires an elaborate calculation [72]; however, we show in this section that a prolonged phase of kinetic equilibrium only has a negligible influence on the energy density of the mediator and hence on the light-element abundances that are created during BBN. After chemical decoupling, but before kinetic decoupling, the mediator still retains a ther­ mal distribution, but ultimately develops a non-vanishing chemical potential pty (t) = 0. Hence, for these temperatures, the phase-space distribution function fty (t, p) is given by

P f (p2 + mty)1/2 — p,p(t) \ 1 1 fty(t, p)|tcd

For simplicity, let us assume that Tty to. In this case, we can use the relation px%(t) « pty (t) for t > tcd to deduce qty(t) ~ 0, which implies

pty(t) + 3H(t) [pty(t) + Pty(t)] ~ 0 . (4.19)

By using the relation pty (t) = T(t) x (dTd (T)/dT) x (dpty (Td)/dTd) and after rearranging the resulting expression with respect to dTd/dT, we thus obtain an equation that determines the dark-sector temperature relative to the one of photons,

dTd ~ pty (Td) + Pty (Td) dpsm(T)/dT (4.20) dT ~ dpty (Td)/dTd X psm(T)+ Psm(T) 61 4.4. Impact of kinetic decoupling and inverse decays

Figure 4.4: Left panel: The energy density p§(T) as a function of temperature for the extreme choices Tkd = Tcd (solid black) and Tkd 0 (dashed red). For comparison, we also show the corresponding curves in case p$(T) = m§n$(T)

n # (t) + 3H(t)n$ (t) = 0. (4.21)

By inserting the appropriate phase-space distribution function (4.18) into eqs. (4.19) and (4.21) we thus end up with a coupled set of integro-differential equations for Td (T) and (T), which we solve numerically using an iterative procedure. The results of this calculation are shown in the left panel of fig. 4.4 for the parameter point m$ = 100 MeV, Tcd = 10 GeV and Zcd = 0.5 as well as for the two extreme choices Tcd = Tkd (black solid) and Tkd 0 (red dashed). In the left panel of the same figure, we also show the ratio of both energy densities for comparison. Strikingly, this ratio is different from one only around T ~ m^; but even in this region the relative difference is always smaller than 1%. This result can be understood as follows: In the ultra-relativistic limit (T » m§) the tem­ perature of decoupled particles scales as Td(T) « R(T)-1, which is a direct consequence of the momentum-redshift pR(T) = const. In comparison, the energy density of kinetically cou­ pled mediators simply follows p$(T) « Td(T)4

1.10

1.08

P 1.06 ra

S 1.04

*13■ Q. 1.02

1.00 103 102 101 10° IO"1 IO"2 IO"3 [MeV] T [MeV]

Figure 4.5: Left panel: Part of the m

4.4.2 Impact of inverse decays and spin-statistics

In this section, we estimate how our BBN constraints are modified when also considering spin-statistics and inverse decays, both of which have been neglected for simplicity so far. When considering the full Boltzmann equations for (p and N (cf. eq. (4.2)), the collision oper­ ator vanishes in case both particles are brought into thermal equilibrium. As a result, such a thermalization would always occur once the decay rate overtakes the Hubble rate, which hap­ pens around t ~ Tq. For Td(f = r^) m^, this leads to a traditional decay scenario, since the approached equilibrium distribution of (p would already be Boltzmann-suppressed, in which case the thermalization causes an efficient decrease of the mediator abundance. However, for Td(t — T/>) m

t t )=( ) (4-22) with Td(Tp) = ZedTcdR(tcd)/R(vp)• Afterwards, p and N simply stay in thermal equilibrium and the corresponding dark-sector temperature can be deduced from the equation

dT; = (T)+ dT dpdark (Td W Psm (T) + Psm (T) in analogy to eq. (4.20). Here, pdar.(T;) = pp(T;) + pNn(T;) and both included energy densi­ ties are calculated from equilibrium distributions with temperature T;. Based on these equa­ tions, we then modify our calculation of the light-element abundances in the following way: For t < Tp, we calculate the energy density from eq. (4.7) by setting Tp to, correspond­ ing to a decoupled and stable mediator. For t > tp, we instead calculate the energy density by using equilibrium distributions for p and N with the modified temperature Td (T) from eq. (4.23). We then follow our general solution strategy to recalculate the BBN constraints for those parameter points with T; (t = Tp) » mp, corresponding to the region below the dashed black line in fig. 4.5. By employing this procedure, we end up with only very small modifica­ tions to the final light-element abundances. In fact, the corresponding differences are always within O(1%) and therefore do not suffice to put any (additional) constraints on this part of parameter space. However, we also find that the thermalization of the dark sector always leads to slightly larger energy densities, which can be explained as follows: As long as all par­ ticles in the dark sector are ultra-relativistic and obey P(T) ~ p. (T)/3, the second Friedmann equation (2.6) implies PdOrk(T)R(T)4 ™ T(;)(T)4R(T)4 = const and thus T(;) (T) « R(T)-1, which is independent of whether the mediator falsely decays or fully thermalizes (i.e. both with and without the star). Especially, this statement implies that both scenarios feature iden­ tical energy densities until T; ~ mp, e.g. until the mediator becomes non-relativistic and P(T) ~ p.(T)/3 is no longer fulfilled. During this transition, however, a fully thermalized dark sector cools down slower than T;(T) « R(T)-1 and therefore effectively increases its en­ ergy density relative to a scenario with mediators that have already decayed. To illustrate this effect, we show the relative difference of both energy densities in the right paneloffig. 4.5for a mediator with lifetime Tp = 10-3 s and mass mp = 10-1 MeV (red) or mp = 10-2 MeV (blue).6 We find that the energy density can generally be increased by up to 10% for a scalar mediator; however, this increase is still not large enough to exclude any additional points in this part of parameter space. Based on this result, we conclude that our constraints remain valid for

6Note that Trf* = mp does not correspond to T = mp , since T; < T for all parameter points. Chapter 4. Bounds on sterile decays of light mediators 64

Td (t = Tq) » m^; and even though a full calculation might still lead to changes in the tran­ sition regions TJ (t = Tq) ~ mQ and Td (t = Tq) ~ mQ, we find that the corresponding energy density generally becomes larger, which is why our results can be considered conservative.

4.5 Summary

In this chapter, we have calculated BBN constraints for MeV-scale mediators that decay into additional dark radiation. Such scenarios are notoriously difficult to constrain by laboratory experiments due to the absence of any non-gravitational interactions between the dark sector and the SM. However, using the predictive power of primordial nucleosynthesis, it is still possible to constrain such scenarios due to their additional contribution to the total energy budget of the universe. Consequently, such cosmological probes are especially important as they remain the only hope of constraining fully decoupled dark sectors in the foreseeable future. As a result, many studies of such scenarios have already been conducted in the past, which, however, only covered the limiting cases of non- or ultra-relativistic particles. In this chapter, we complemented the existing literature by considering MeV-scale mediators that are neither non-relativistic nor ultra-relativistic during the temperatures that are relevant for BBN. In this context, we have taken into account all relevant effects that alter the evolution of the light-element abundances, including a modification of the Hubble rate and the time­ temperature relation as well as corrections to the decoupling temperature of neutrinos (cf. sec. 3.3 for a detailed discussion ofall effects). We find that even a fully decoupled dark sector can be severely constrained if either the temperature ratio of the dark and the visible sector exceeds unity or the particle becomes non-relativistic before its decay and thus acquires a large energy density relative to the one of the SM. Finally, we have also discussed possible effects of inverse decays and spin-statistics but did not find any additional regions of parameter space that are excluded due to the generally larger energy densities. 65

5 Bounds on electromagnetic decays of light mediators

This chapter is based on the following publication:

M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on MeV-scale dark sectors. Part II. Electromagnetic decays, JCAP 1811 (2018) 032, [arXiv:1808.09324]

In this thesis, we further implement the following improvements:

• Use the most recent values for the observed light-element abundances from [37]

• Include the correlation between n and Neff according to the discussion in sec. 3.2.2

• Include the effects of spin-statistical factors and inverse decays

In this chapter, we consider a dark sector with MeV-scale particles decaying into electro­ magnetic radiation, which are neither fully relativistic nor non-relativistic during all tem­ peratures relevant for BBN. In such scenarios, the MeV-scale particles do not only become non-relativistic directly during BBN, but also have energies similar to the photodisintegration thresholds of most light nuclei. Such scenarios therefore naturally feature a variety of inter­ esting effects, including a non-trivial change of the Hubble rate, the injection of entropy into the electromagnetic plasma, and the late-time modification of light elements due to photo­ disintegration. While previous studies have already investigated similar scenarios, it has al­ ways been assumed that the decaying particles are non-relativistic during BBN [22-29, 73, 74] - except for some model-dependent cases [75]. In this chapter, we complement the previous studies and perform a more general discussion without any additional assumptions, which turns out to be relevant for many different phenomenological applications.

5.1 Setup and assumptions

Regarding the concrete setup, we extend the SM by an electromagnetically coupled dark sector that contains two additional particles: (i) a dark-matter candidate x and its antiparticle XX with mass mx ~ O(GeV), and (ii) a self-conjugate, MeV-scale boson Q with mass mQ ~ O (MeV), which couples to the electromagnetic sector of the SM. Regarding the cosmological evolution of these particles, we further make the following assumptions (here the first two points are simply copied from the previous chapter 4):

• For large temperatures, the particle Q is thermalized within the dark sector, but even­ tually decouples (chemically and kinetically) at some photon temperature Tcd, which corresponds to a different temperature Td,cd in the dark sector. Such a setup can be re­ alized by introducing a coupling between Q and x, which leads to a freeze-out via the process xX QQ. Chapter 5. Bounds on electromagnetic decays of light mediators 66

I ■ ■ ■ ■ ■ ■ SM sector Dark sector ■ ■ ■ I ■ ■ Dark matter Photons ■ br77 ■ XX mx ~ O(GeV) ■ 7 my = 0 ^,cd / ■ / Ccd \ Mediator Electrons, Positrons y i d nt# ~ O(MeV) !——► e~ e+ We — 0.511 MeV ■ Other particles I I I /A T±, vif Vi, . . I

Figure 5.1: Sketch of the dark-sector model that is discussed in this chapter. The dark sector is coupled to the electromagnetic part of the SM model and features interactions with the electrons and/or photons (decay). In addition to this, there are also interactions between the mediator and the dark-matter candidate x (decoupling).

• The decoupling of (p happens before BBN (Tcj > 10 MeV) and consequently the dark­ matter particle x only enters the calculation implicitly by influencing the value of T^cd, e.g. Tj/Cd ~ zn^/20 for a vanilla freeze-out scenario. After decoupling, x makes up the observed relic abundance and therefore only gives a negligible contribution to the en­ ergy density for the temperatures that are relevant for BBN.

• The particle (p can decay into electromagnetic radiation via the reactions (p yy and (p -4 e^e~ with a total lifetime and branching ratios BR77 and BRee, respectively. These decay channels mainly differentiate the given scenario from the one that we dis­ cussed in the previous chapter.

Given these assumptions, the full setup can be characterized by five different parameters, namely (i) the mass and (ii) the lifetime of the MeV-scale particle (p, (iii) one of the branching ratios with BR77 + BR^ = 1, (iv) the photon temperature Tcd at the time of decou­ pling, and (v) the corresponding temperature ratio £cd := Tjcd/^cd- The full setup with all relevant parameters is also summarized in fig. 5.1 (in a form that is directly comparable to the previous setup in fig. 4.1). Just like in the previous chapter, we now strive to calculate the BBN constraints in the cor­ responding parameter space using the general solution strategy that we developed in sec. 3.4. Consequently, the next step is to derive the specific Boltzmann equation (3.72) for the dark­ sector particle (p as well as modified expressions for the Hubble rate and time-temperature relation. 67 5.2. Evolution and influence of the dark sector

5.2 Evolution and influence of the dark sector

5.2.1 Evolution of the mediator

Compared to the previous chapter, we have not made any changes regarding the decoupling of ty, meaning that we can simply adopt eq. (4.1) for the mediator spectrum at the time tcd of decoupling, i.e.

-1 (p2 + mty )1/2 fty (tcd, p) exp 1 (5.1) Zcd Tcd with Tcd = T(tcd) and a vanishing chemical potential pty(tcd) = 0. To further derive the Boltzmann equation that describes the evolution of ty after decoupling, let us first assume that the mediator exclusively decays into either electron-positron pairs orphotons. Inthis case, we can simply recycle eq. (2.24) by substituting x = ty and (z, z) G {(e-, e+), (7,7)}, which yields

1 p+ dfty (t P) Pz d Pz - H(t) P C X ßp fz (t, pz )fz (t, pZ) dt ßzp Jp Ez

mty X ßßlp fz(t'pz)+ f(t'pz)! pz d pz T fty (t p) (5.2) Ez

Here, the uppersignis used forphotons and the lower sign is usedfor electron-positron pairs. To bring this equation into amore intuitive form, we can make use ofthe fact that the collision operator - or more precisely the right-hand side of eq. (2.43) - vanishes in thermal equilibrium. Hence, by applying this relation to the process ty zz, we find1 (here we explicitly indicate thermal spectra by an additional bar)

fz (fi pz )fz (t, pz ) = fty (t, p) ± fty (t, p) X fz (fi pz)+ fz(fi pz )] . (5.3)

Since electrons and photons both stay in thermal equilibrium for all relevant temperatures, we have fz (t, pz) = fz (t, pz) and fz (t, pz) = fz (t, pz), which allows to insert eq. (5.3) into the first integral onthe right-hand side ofeq. (5.2). From there, the Boltzmann equation simplifies to

- H(t) p dftyd^ = -d± (t' p) x [ft (t, p) - fty (t, p)] (5.4) with the modified decay term

mty pz d pz Dz±(t, p) = 1 ± ß p / fz (t, pz)+ f (t, pz/] Ez

m 1 + 2T j_/1 T exp[ (Ety + ßzp)/2T] \ (5.5) ßzp 1 T exp [-(Ety - ßzp)/2T] T=T(t)

1Remember that energy-momentum conservation is implicit in our notation. Also, since this relation is only valid in mutual equilibrium, all spectra that enter this equation have the same temperature T . Chapter 5. Bounds on electromagnetic decays of light mediators 68

In the last step, we have further evaluated the remaining integral analytically by inserting the respective thermal spectra for electrons, positrons, and photons. Finally, to obtain the full Boltzmann equation for ty, we have to sum over both decay chan­ nels and weight each term with the appropriate branching ratio, which yields

fdt P) - H(0pdf

This is a partial differential equation, which again can be transformed into an ordinary one by means of the method of characteristics. Following the same steps as in sec. 4.2.1, we find

(5.7) f dtP*) = -Dty(t*P /R(t)) x [fty(t,p* ) - fty(tp* /R(*))] with Dty (t, p) := BR77 D+(t, p) + BReeD- (t, p). As a result, we arrive at an inhomogeneous but ordinary and linear differential equation which can be solved semi-analytically (up to some integrals) by using both a separation of variables and a variation of constants. Consequently, by employing eq. (5.1) as an initial condition and after resubstituting p* = pR(t), we find

fty(t, p) = fty(tcd, pR(t)/R(tcd)) exp - D^t, pR(t)/R(f)) dt' tcd

tcd t with 0(t, p) := D^t, p} x fp(t, p). The first term of this expression resembles the usual decay law but with a temperature- and momentum-dependent effective lifetime Ttyf = D-1 (T, p) that does not only receive relativistic corrections in terms of an additional Lorentz factor, but also gets modified due to Pauli blocking/Bose enhancement and the inclusion of inverse de­ cays. The second term of this expression is independent of the initial condition and describes the creation of ty due to freeze-in from the electromagnetic bath. This term therefore consti- tutesanirreduciblecontributionto fty(t,p)whichdoesnotvanishforfty(tcd,p)=0. In this scenario, the decrease of fty(t, p) is not compensated by the appearance of sterile neutrinos, but rather by a change in the temperature evolution of the SM sector. To quantify this effect, we now turn to the calculation of the Hubble rate and the time-temperature relation for this particular dark-matter model.

5.2.2 Hubble rate and time-temperature relation

In this scenario, the energy density of the dark sector is fully determined by the evolution of the mediator, pdark(t) = Pty(t), and consequently the Hubble rate is given by

H(t) = 3 psm(t) + pty(t) , (5.9) 5.2. Evolution and influence ofthe dark sector 69 according to eq. (3.73). Moreover, since the energy density of the mediator becomes negligible prior to recombination, we have Pdark-rad(trec) = 0 and consequently the effective number of neutrinos in this model is given by

Pvv (trec) = 3( Tv (trec )\ 4 /11\4/3 (5.10) peff (trec) T(trec) 4 which directly follows from eq. (3.81). As a result, we find that Neff is only altered indirectly via modifications to the temperature ratio Tv(trec)/T(trec). More precisely, if the mediator decays after neutrino decoupling, the entropy that is released during the decay heats up the electromagnetic sector relative to the neutrino sector, which leads to a smaller value of Neff compared to the SM and consequently to a decreased best-fit value for the baryon-to-photon ratio. Moreover, the time-temperature relation receives additional contributions from the total heating rate of the dark sector. Since jdark(t) = fty (t), eqs. (3.79) and (3.80) can be written in the form

dT ity(t) + 3H(T) [psm(T) + Psm(T)] f T (5.11) dt dpsm (T)/dT r - 1

dT qty (t) + 3H(T) [pem (T) + Pem (T)] for T < Tvd (5.12) dt dpem(T)/dT with the total Hubble rate from eq. (5.9) and the volume heating rate 3 ‘fty(t) = - Ä3 Dtyp)f p) - fp2+mi gnp ' (5.13) which follows from integrating eq. (5.4) over J •Egtyd3p/(3n)3.2 Here, the important result is that the decay of the mediator has a direct impact onto the calculation of the time-temperature relation. More precisely, if the mediator injects additional heat into the electromagnetic bath (qty(t) < 0, ty e+e-/yy), the SM cools down more slowly. However, if the mediator is produced from the thermal bath (qty (t) > 0, e+e-/YY ty), the SM cools down faster in­ stead. Finally, the heat transfer of the dark sector also allows to calculate the source terms for photodisintegration and we find

sY0)(t)= BR77 x -2jty (t) and (t) = BRee x -q (t) , (5.14) mty mty according to eq. (3.101) with E0 = mty/2. Using these results, we can now apply our general solution strategy from sec. 3.4 to cal­ culate the BBN constraints for this particular dark-matter model. However, to gain a better understanding of the final results, let us first take a closer look at the influence of photodisin­ tegration on the creation of light elements for some exemplary parameter points.

2However, we find that for numerical calculations it is best to determine qfty (t) directly from its definition jty(t) = pty(t) + 3H(t) [pty(t) + Pty(i)], since the evaluation of fty(t, p) - fty(t, p) can easily lead to cancellation of significant digits. Chapter 5. Bounds on electromagnetic decays of light mediators 70

10-18 1 _ - [ ty YY Tty = 108s j 20 1 i 1 - i 1 - i------—— 1 10-22 —__ -—- X = Y 1 -24 | universal CM 10 - 1 > 1 l spectrum 1) -26 s 10 ■ 1 1 K ’ I Su 1 c2- ■ 1? I a. 10-28 ES li 1 s X 1° 1 X 10-30 ■ 1 i - i - | X = e- i 10-32 — i —— - I X = e+ i - 10-34 1 i 10-36 2 3 4 5 10 20 30 50 E [MeV ]

Figure 5.2: Comparison of the non-thermal spectra for a scalar ty decaying into ty e+e (left) and ty YY (right) with two different lifetimes Tty = 105 s (top) and Tty = 108 s (bottom). Here, we set mty = 100 MeV, Tcd = 10 GeV and fix Zed via the relation nty(Tcd)/ny(Tcd) = 1/2 x 10-5. All spectra are evaluated at T = T(t = Tty) and for comparison we further show the universal spectrum (dash-dotted green) as well as the lowest photodisintegration thresholds for deuterium and helium-4 (dashed black).

5.2.3 Photodisintegration

Unlike the source terms from eq. (5.14), all other contributions to the cascade equation are model-independent (cf. app. B), meaning that we can now solve the full equation for arbi­ trary parameter points. In fig. 5.2 we show the resulting photon (purple), electron (red), and positron (blue) spectra Fx(E) for a mediator with mass 2E0 = mty = 100MeV and lifetime Tty = 105 s (top) or Tty = 108s (bottom), decaying exclusively into e+e- (left) or YY (right). We 71 5.2. Evolution and influence of the dark sector

further fix np(Tcd)/ny (Tcd) = 1/2 x 10 5 at Tcd = 10 GeV, which can be realized by choosing Zcd 2 x 10-2, accordingly. Also note that the different spectra further depend on the tem­ perature and we set T = T(t = Tp) for definiteness, which corresponds to T = 3.63 keV and T = 0.11 keV for Tp = 105 s and Tp = 108 s, respectively. In all of these plots, we find that the electron and positron spectra are strongly suppressed with respect to the photon spectrum, which is true even in case the mediator decays exclusively into electron-positron pairs. This is because - below the threshold for double photon pair creation - non-thermal photons are frequently produced via inverse Compton scattering on the abundant background photons, while non-thermal electrons and positrons originate mainly from Compton or Bethe-Heitler scattering, both of which are strongly suppressed due to their low target densities (thermal electrons and nuclei). Consequently, we generally end up with a photon production rate that is much larger than the corresponding one for electrons and positrons. Besides, there also ex­ ists an intrinsic difference between the non-thermal spectra of electrons and positrons. This is because Compton scattering cannot efficiently produce non-thermal positrons due to the low amount of thermal counterparts, which ultimately translates parts of the electron-positron asymmetry to the non-thermal spectra (cf. lower panels of fig. 5.2). This disparity can only be resolved once double photon pair creation becomes accessible, which efficiently produces the same amount of electrons and positrons (cf. upper panels of fig. 5.2). For Tp = 105 s (cf. upper panels of fig. 5.2) we find that - below the threshold for double photon pair creation at Ee+ e- ~ m;e / (22T) « 3.27 MeV - our full solution of the cascade equa­ tion is in remarkable agreement with the universal spectrum from eq. (3.90), which constitutes an important consistency check for our calculations. However, while the universal spectrum vanishes by construction for E > Eeth+ e- , our calculation correctly includes the exponential suppression of the photon spectrum for larger energies due to the onset of efficient electron­ positron pair creation. This exponential suppression also implies that the photodisintegration of helium-4 cannot proceed efficiently, which is why we can already anticipate that mediators with such lifetimes mainly alter the abundance of deuterium.

For Tp = 108 s (cf. lower panels of fig. 5.2) we have Et+e- ~ m^ / (22T) « 107.9 MeV, which naturally leads to large deviations from the universal spectrum, instead. In fact, we find that the photon spectrum is never exponentially suppressed in this part of parameter space and therefore enables efficient disintegration of both deuterium and helium-4. With the non-thermal photon spectrum at hand, we may now solve eq. (3.95) to deter­ mine the evolution of the light-element abundances during photodisintegration. In fig. 5.3 we illustrate the results of this procedure for a comparable set of parameters with Tp = 105 s and BRee = 1. Since the initial amount of dark-sector particles is chosen to be rather small, np(Tcd)/n7(Tcd) = 1/2 x 10-5 1, we find that there are no modifications to standard nucle­ osynthesis, meaning that the abundances are still compatible with observations until t < 105 s. However, once the mediator starts to decay at Tp = 105 s, the resulting cascade of non-thermal Chapter 5. Bounds on electromagnetic decays of light mediators 72

8

Figure 5.3: Evolution of the nuclear abundances as a function of time during nucleosynthesis and photodisintegration for mp = 100 MeV, Tp = 105s, Tcd = 10 GeV, np (Tcd)/nY(Tcd) = 1/2 x 10-5 and BRee = 1. In addition to the theoretically predicted values (solid), we also show the ones that are inferred from observations (dashed). photons leads to efficient disintegration of deuterium, which significantly decreases its abun­ dance - much below the observationally inferred value. This ultimately leads to the exclu­ sion of this parameter point, which illustrates that photodisintegration is also sensitive to dark-sector abundances much below the one of photons. In the next section, we apply this procedure to the full parameter space in order to determine the full set of constraints from nucleosynthesis and photodisintegration.

5.3 Results and discussion

5.3.1 Constraints on the usual freeze-out scenario

In this section, we finally present the BBN constraints on electromagnetic decays of MeV-scale mediators (cf. fig. 5.1). As discussed above, there are five relevant parameters, the mass mp and the lifetime Tp of the mediator, one of the branching ratios with BRyy + BRee = 1, the photon temperature Tcd at the time of decoupling, and the corresponding temperature ratio Zcd = Td,cd/Tcd. In fig. 5.4 we first show the resulting constraints in the mp — Tp parame­ ter plane with Tcd = 10 GeV and Zcd = 1 for a scalar (left) and a vector mediator (right), 73 5.3. Results and discussion

Figure 5.4: Constraints from BBN in the m

0.1 MeV), which leads to the presence of an ad­ ditional thermal particle during nucleosynthesis. Hence, in order to retain the observed light­ element abundances, the mediator has to become Boltzmann-suppressed before - or at least shortly after - the onset of BBN, which effectively constrains masses below mp < 0.44 MeV and mp < 5.46 MeV in case of a scalar and vector mediator, respectively. If we instead con­ sider larger masses and lifetimes with T(t = Tp) mp, we face a more traditional decay scenario, and the constraints become stronger for larger masses and lifetimes. This is because a long-lived heavy mediator has more time to profit from the pp(T) « R(T)—3 scaling behav­ ior for non-relativistic decoupled particles, which generally leads to a larger relative energy density prior to its decay. However, as opposed to the results from the previous chapter, these constraints do not extend to arbitrarily small lifetimes but instead vanish below Tp ~ 0.1 s. This is because for smaller lifetimes, the mediator decays efficiently already before nucleosyn­ thesis, in which case the decay might still delay the temporal onset of BBN due to its influence on the time-temperature relation, but it can no longer alter its progression. In the transition region T(t = Tp) ~ mp both of these effects naturally blend together. Finally, let us note that the full parameter space faces stronger constraints in the case of vector mediators, which is simply due to their larger abundance. There also are some small differences between the two distinct decay channels, which is due to the difference between De—(t, p) and D+Y (t, p) enter­ ing eq. (5.8). However, if inverse decays and spin-statistics are not important, we simply have D— (t, p) ~ D+ (t, p) ~ mp/ (EpTp), which is why the results are identical for T(t = Tp) < mp (top right corner). In fig. 5.5 we show the same part of parameter space but for different values of Zcd, rang­ ing from Zcd = 1/4 to Zcd = 2. In general, by decreasing the value of Zcd, the constraints should become weaker due to the smaller initial abundance. However, we find that even upon reduction of the initial temperature ratio, the constraints for small masses and lifetimes remain largely unchanged. This is because - in this part of parameter space - the mediator always thermalizes with the SM heat bath prior to nucleosynthesis and thus acquires a tem­ perature Td(T) = T, which is independent ofits initial value Td(T) = T. In fact, this statement

3This would be the process that is usually associated with the decay of a particle. 75 5.3. Results and discussion

Figure 5.5: Overall constraints from BBN at 95% C.L. in the m

Figure 5.6: Constraints from BBN in the — £cd parameter plane for a scalar mediator that decays exclusively into electron-positron pairs. Here, we set ^cd = 10 GeV and show a de­ tailed view for = 100 MeV (left) as well as a compilation of the overall limits for different masses (right). large initial abundances, we also find that the minimally excluded value of partially drops below 0.1 s even if the mediator does not get fully thermalized (red, lower left panel). In this region of parameter space, the particle decays when it is semi-relativistic and consequently its lifetime is slightly prolonged due to the additional Lorentz boost. The limits for different temperature ratios can also be used to deduce the ones for different decoupling temperatures. This is because as long as the mediator is ultra-relativistic during decoupling, the resulting limit for a combination of parameters Tctj and £cd is equal to the one for

T'd = 10GeV and £d = \cd . (5.15)

Since the ratio *[g s(^cd)/^*s(T'cd)] 1>/3 Is usually rather small - except for decoupling tempera­ tures close to the QCD phase transition -, we find that the limits are largely insensitive to Tcj and consequently refrain from a more detailed discussion. Up to this point, the effects of photodisintegration were mainly negligible, since the acces­ sible region of parameter space was already excluded anyway. However, as we have pre­ viously pointed out, photodisintegration is indispensable for constraining mediators with very low abundances, for which the implied modifications of the Hubble rate and time­ temperature relation are already negligible (cf. fig. 5.3, which features H(Tcd)/n7(Tcd) = 1/2 x 10-5, red star in fig. 5.6). In the left panel of fig. 5.6 we therefore show the resulting con­ straints in the T

np (Tcd) = gpZcd (5.16) nY (Tcd) 2 and therefore can be directly calculated from the temperature ratio. This slice of parame­ ter space is especially intriguing as the resulting constraints are influenced by a variety of different effects, all of which become dominant for a distinct range of lifetimes: For small life­ times, Tp < 50 s, the limits dominantly arise from the increased Hubble rate. As expected, the bounds also become stronger for larger lifetimes, as long-lived particles have more time to profit from the pp(T) 6 x 103s, the limits dominantly arise from the photodisintegration of different elements. For example, a decaying mediator with Tp = 105 s leads to a non-thermal photon spectrum that is mainly capable of disintegrating deuterium (cf. upper panels of fig. 5.2). Instead, for Tp > 4 x 106 s, the disintegration of helium-4 into helium-3 also becomes accessible (cf. lower panels of fig. 5.2), meaning that in this region of parameter space the limit is mainly caused by an overproduction of helium-3 relative to deu­ terium. In this part of parameter space, the limits even extend to very small values of Zcd and thus to very small initial abundances. In the right panel offig. 5.6 we then show the same results for a variety of different masses, ranging from mp = 5 MeV to mp = 100MeV. In general, we find that the constraints become stronger for larger masses; however, there are two exceptions from this general behavior: For large lifetimes, Tp > 105 s, the constraints actually become partially weaker when in­ creasing the mass from mp = 50GeV (orange) to mp = 100MeV (red). This is because for mp = 100 MeV, the necessary relation for ultra-relativistic decoupling mp < Tdcd = Zcd Tcd implies Zcd > 10—2 for Tcd = 10 GeV; hence, for smaller temperature ratios, the mediator decouples while only being semi-relativistic, which leads to an additional suppression of the initial abundance. In addition to this, the bounds are also stronger for smaller masses when considering lifetimes below Tp < 10—1 s. In this part of parameter space, lighter particles simply profit from an additional Lorentz boost, which effectively delays their decay and thus extends the limit to smaller lifetimes. Last but not least, let us note that the mediator mass also determines the set of available photodisintegration reactions. This statement is especially relevant for mp = 5 MeV. In this case, the energy of injected particles E0 = mp/2 is only slightly larger than the minimal threshold energy of Et^^^np ~ 2.22 MeV, which is why this part of parameter space only features insignificant constraints from photodisintegration. Finally, let us note that for most parts of parameter space, the BBN bounds are subdomi­ nant compared to the ones from the CMB (Neff low). However, since the corresponding value Chapter 5. Bounds on electromagnetic decays of light mediators 78

tn* [MeV]

Scalar

Figure 5.7: Freeze-in contribution to the BBN constraints in the m

5.3.2 Constraints on the pure freeze-in scenario

Up to this point, we have considered a traditional freeze-out scenario by employing eq. (5.1) as an initial condition. However, we have also pointed out already that some parts of parameter 79 5.4. Summary

space are excluded independently of Zcd and Tcd due to the irreducible freeze-in contribution, which often suffices to thermalize the mediator independently of its initial condition. To quan­ tify this effect, we now study the resulting BBN constraints for a mediator with a vanishing abundance at early times, i.e. we set f (tcd, p) = 0 in eq. (5.8) with tcd 0. In fig. 5.7 we present the results of this calculation for scalar (left) and vector (right) mediators that decay exclusively into photons (top) or electron-positron pairs (bottom). First of all, we find that the limits in the left part of parameter space are very similar in shape to the ones from the freeze- out scenario for small Zcd, which is consistent with our previous discussion. In this context, the maximally excluded value of Tp is roughly determined by the condition T(t = Tp) ~ mp, above which the freeze-in is no longer effective since the thermal distribution of p is already Boltzmann-suppressed.4 However, below this line we find that even a post-BBN freeze-in can significantly alter the light-element abundances due to its indirect influence on the baryon-to- photon ratio. In addition to this, there also exist some islands towards larger masses, which are excluded due to photodisintegration. Hence, even this part of parameter space does not become completely unconstrained for Zcd 0. In general, the position of these islands can be explained by the threshold energies for the different photodisintegration reactions. For exam­ ple, the disintegration of deuterium cannot proceed unless mp/2 = E0 > E^^p ~ 2.22 MeV, which is why the first island only arises for mp > 4.44 MeV. The remaining island analogously corresponds to the threshold E4HeY^3Hp ~ 19.81 MeV with mp > 39.62 MeV.5 Finally, let us note that the constraints from photodisintegration are stronger in case the photons are injected directly (p YY vs. p e+e—), as this naturally results in a larger non-thermal photon spec­ trum.

5.4 Summary

In this chapter, we have calculated BBN constraints for MeV-scale mediators that decay into electromagnetic radiation. Particles like this are predicted by various extensions of the SM, and their coupling strength is often subject to stringent upper limits from direct or indirect searches. Unless invoking additional, even lighter states, these bounds usually enforce macro­ scopic lifetimes, Tp ~ O(1s), which naturally conflict with predictions from nucleosynthesis. In this context, many studies have already been conducted in the past, but these only covered the limiting cases where the particle is either non-relativistic during BBN, or where it is ultra- relativistic and decays long after BBN. Hence, all of these studies have considered an explicit difference in scale between the mass of the particle and the temperatures relevant for BBN. However, when considering the decay of an MeV-scale particle, all relevant energy scales are essentially of the same order. This does not only include the two scales that were mentioned before, but also the scale that is set by the binding energy of the light nuclei, i.e. the scale that

4Naturally, the freeze-in already starts before t = Tp and consequently the resulting constraints also slightly extend past this line. 5Remember that 3H decays into 3He, meaning that an overproduction of tritium gets translated into an over­ production of helium-3. Chapter 5. Bounds on electromagnetic decays of light mediators 80 is relevant for photodisintegration. To complement the existing literature, we have therefore performed a more general study of BBN constraints for MeV-scale mediators that are neither fully relativistic nor non-relativistic during the temperatures relevant for BBN. To this end, we have solved the full Boltzmann equation, including effects from spin-statistics and inverse decays, without invoking any ultra- or non-relativistic approximations. Besides, we have also included all relevant effects regarding the prediction of the light-element abundances, includ­ ing a modification of the Hubble rate due to the additional energy density, a modification of the time-temperature relation due to an exchange of heat between the SM and the dark sector, a non-standard time-dependence of the baryon-to-photon ratio, a modified neutrino­ decoupling temperature, and the late-time modification of the nuclear abundances due to photodisintegration (cf. sec. 3.3 for a detailed discussion of all effects). We find that depend­ ing on the region of parameter space, all of these effects can dominate the final constraints, which consolidates the necessity of such a detailed analysis. Most importantly, however, we find that the final constraints are often very different from the naive order-of-magnitude esti­ mate Tp < 1s, which is often employed for such scenarios. In fact, for most parts of parameter space, we find that this bound is not conservative at all. Finally, let us note that the constraints derived in this chapter are mostly model-independent and only make minimal assumptions regarding the cosmological evolution of the dark sector. In ch. 7 and ch. 8, we have a second look at these limits and apply them to some actual models. 81

6 Bounds on residual dark-matter annihilations

This chapter is based on the following publication:

P. F. Depta, M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on the annihilation of MeV-scale dark matter, JCAP 1904 (2019) 029, [arXiv:1901.06944]

In this thesis, we further implement the following improvements:

• Use the most recent values for the observed light-element abundances from [37]

• Include the correlation between n and Neff according to the discussion in sec. 3.2.2

In this chapter, we consider MeV-scale dark matter that can annihilate into all kinematically viable SM states, i.e. electrons/positrons, photons, and neutrinos. While theoretical work and experimental searches so far have mainly focused on dark-matter particles with masses at the weak scale, the lack of signs for new physics at the LHC [76, 77] as well as stringent bounds on such scenarios from direct searches [78-80] have lately started to motivate the study of alternative scenarios. In the last few years, this has led to an increased amount of attention for models with sub-GeV dark matter, which are much less constrained by direct searches (see e.g. [81-85]). However, in this case, additional light states are typically required to obtain the correct relic abundance, which leads to additional experimental signatures that can be probed using astrophysical [86-88] and cosmological [1, 2, 18, 67] observations as well as collider searches [89-91] and beam-dump experiments [92-94]. In this chapter, we explore the cosmological implications of sub-GeV dark matter by performing a dedicated study of the BBN constraints originating from residual dark-matter annihilations. For GeV-scale particles such limits have already been derived in [95-97]; however, since these limits cannot be trivially extrapolated to smaller masses due to the presence of photodisintegration thresholds, our study constitutes a natural extension of the previous results from the literature. In this context, we also update the lower bound on the mass of thermal dark matter.

6.1 Setup and assumption

Regarding the concrete setup, we extend the SM by a dark-matter particle x with mass mx ~ O(MeV), which couples to both the electromagnetic and the neutrino sector of the SM. Unless explicitly stated otherwise, we do not make any additional assumptions about the nature of x, i.e. it can be a scalar, spinor or vector and either self-conjugate or not. Regarding the cosmological evolution of this particle, we further make the following assumptions:

• The particle x features interactions with the SM heat bath of the form xX XX with

X G {y,e-, Vi} and X G {7,e+, Vi}. These interactions keep x in chemical equilibrium until decoupling at some temperature Tcd. This temperature is usually set by the mass Chapter 6. Bounds on residual dark-matter annihilations 82

of the particles, which implies that the dark-matter candidate can still be in chemical equilibrium for temperatures that are relevant for BBN.

• After chemical decoupling, x might still be in kinetic equilibrium with the SM via inter­ actions of the form xX %X, which eventually decouple at a temperature Tkd < Tcd.

• At some point after chemical decoupling, the particle x acquires the correct relic abun­ dance. In case we consider annihilation cross-sections that are too large/small for this to happen, we assume the existence of additional processes that are capable of adjust­ ing the dark-matter abundance accordingly. This assumption is necessary to perform a meaningful comparison with other constraints that can be found in the literature (see below).

With these assumptions, the set of all relevant parameters is composed of (i) the mass mx of the dark-matter particle, (ii) the kinetic-decoupling temperature Tkd,1 as well as (iii) the to­ tal annihilation cross-section &xx^xx and the corresponding branching ratios BRxx into X G {y,e—,Vi}. Regarding the available interactions, we limit our discussion to two well motivated particle-physics scenarios: In scenario (S1) we assume that the dark-matter par­ ticle has negligible interactions with the neutrinos and therefore only annihilates into YY or e+e— with BRViVi = 0 and BRee + BRYY = 1. Such a scenario naturally arises in models with dark-matter interactions that are mediated by a Higgs-like scalar. In scenario (S2), we in­ stead consider equal branching ratios for dark-matter annihilations into e+e and veVe, i.e. BRVeVe = BRee = 1/2 and BRViVi = BRYY = 0 for i = e, which is naturally expected for models with dark-matter particles that dominantly couple to the first generation of lepton doublets. For mx < me, where the phase-space for annihilations into e+e— is closed, we assume equal branching ratio into photons and neutrinos, knowing that suchmasses are inanycase robustly excluded also for other choices of the branching ratios [98]. The next step is now to calculate the BBN constraints for these two scenarios using the general strategy that we developed in sec. 3.4, for which we still have to discuss the evolution and influence of the additional dark-matter particle.

6.2 Cosmological evolution

6.2.1 Evolution of the dark-matter candidate

In the previous chapters we have always been able to neglect the evolution of the dark­ matter particle due to its larger mass and consequently larger decoupling temperature Td,cd ~ O(GeV), which led to a negligible contribution of x during BBN. However, this convenience can no longer be maintained in this chapter as we are now interested in much lighter dark­ matter candidates with mx ~ O(MeV), which can very well decouple right during primordial

1The parameter Tcd can be derived from the dark-matter mass; however, in any case we find that this parameter does not enter our calculations at all. 83 6.2. Cosmological evolution nucleosynthesis. Regarding the derivation of the corresponding energy density, we can essen­ tially divide the calculation into two different parts. Before chemical decoupling at T = Tcd, x simply follows an equilibrium distribution, in which case the total energy density of dark­ matter particles is given by

2 px*(T) = AxKx x gx - t4 I± (mx/T) for T > Tcd , (6.1) in analogy to eq. (3.4). Here, X = 1 (A = 2) if x is self-conjugate (otherwise), and Kx = 1 (kx = 7/8) if x is a boson (fermion). After chemical decoupling, we assume that x eventually acquires the correct relic abundance (due to some additional processes), which implies [13, TT,TE,EE+lowP+lensing]

Qx = Px*(T0) = 0.120h-2 with pcrit = 3h Gef . (6.2) pcrit 8nG

Here, T0 = 2.725 K and Href = 100 km/ (s • Mpc), meaning that h is the value of the Hubble rate at T = T0 in units of 100 km/ (s • Mpc). Once this abundance has been reached, the comoving number density of x is approximately conserved. And even though residual annihilations are still taking place, their rate (£x. Jxx^XXVm0i(T) H(T) is much smaller than the Hubble rate and therefore can no longer significantly alter the x abundance. As a result, we find px* (T)R(T)3 = mxnx* (T)R(T)3 = const, which in combination with eq. (6.2) implies

3Hef X Qxh2 (-< h (t ’ dT0 px*(T) exp for T < Tcd . (6.3) 8n G

Finally, close to chemical decoupling, T ~ Tcd, the exact form of the energy density is irrelevant since (i) x is already Boltzmann-suppressed at this point and therefore does no longer influ­ ence the Hubble rate, and (ii) the temperatures Tcd ~ O (MeV) are still too large to allow for efficient photodisintegration, which only happens at T ~ O(keV). As a result, we find that eqs. (6.1) and (6.3) are indeed sufficient to determine the energy density of x for all compu­ tationally relevant temperatures, independently of the kinetic-decoupling temperature and annihilation cross-section. However, especially regarding the calculation of the thermally averaged annihilation cross­ section that is relevant for photodisintegration, we are also interested in the exact form of the dark-matter spectrum. Before chemical decoupling, x can actually be described by an equi­ librium distribution with temperature T and a vanishing chemical potential px(T) = 0. After chemical decoupling, the exact form of the spectrum (unlike the one of the energy density) critically depends on the temperature at which x kinetically decouples from the SM heat bath. More precisely, before kinetic decoupling, x can still be described by a thermal distribution with temperature T and a chemical potential px(T) = 0, which adjusts in such a way that eq. (6.3) is fulfilled. However, after kinetic decoupling, the evolution ofx is determined solely by the mo­ mentum redshift p pR (T) / R (Tkd), meaning that the corresponding distribution can still be considered thermal but now has an effective temperature TkdR(Tkd)2/R(T)2 and a modified Chapter 6. Bounds on residual dark-matter annihilations 84

chemical potential. Consequently, for T Tcd, x can still be described by the non-relativistic limit of the Bose-Einstein/Fermi-Dirac distribution

fx(T, p) — e-[mx-^x(T)]/Tx(T) x e-P2/[2mxTx(T)] for T < Tcd (6.4) with an effective temperature

(T for T > Tkd Tx(T) = (6.5) Tkd R(Tkd)2/R(T)2 for T < Tkd and a chemical potential (T) that adjusts in such a way that eq. (6.3) is fulfilled.2 While the annihilation rate is much smaller than the Hubble rate after decoupling and therefore does no longer influence the energy density, residual annihilations might still be sufficient enough to cause a late-time modification of the nuclear abundances [99]. To quan­ tify this effect, we still have to calculate the volume heating rate qdark(t) « (Pxx^xx}nx (t) that enters the corresponding source terms (3.101). To this end, we first consider the case of Majo­ rana dark matter with px* (T) = px(T) and x = x, but later comment on how the results are modified in case of other scenarios. For T ~ O (keV) mx, the reverse reaction XX xx is already strongly suppressed and consequently the integrated Boltzmann equation (2.31) for x can be written as (for T Tcd)3

Px + 3H[Px + Px] (— -/ lMxx^xx|2(2n)MPx + Px' -PX -PX)(Ex + Ex')

d3 Px d3 Px' d3 PX d3 Px x (1 ± fX)(1 ± fX }fxfx' (6.6) (2n)32Ex (2n)32E^ (2n)32Ex (2n)32Ex

Here, the sum Ex + Ex' follows from the fact that x appears twice in the initial state of the reaction xx XX (cf. sec. 2.2.2). Moreover, for T Tcd ~ O(MeV), the particle x is already non-relativistic, meaning that its energy is roughly given by Ex ~ mx T. Consequently, by employing energy conservation we find Ex ~ Ex T and thus 1 ± fx — 1, which again allows to drop the spin-statistical factors in the collision operator (cf. eq. (2.39) with E/T » 1). From there, we can rewrite eq. (6.6) in terms of the polarized cross-section (cf. eq. (3.39)), which yields

33 gxd Px gxd Px' px + 3H[px + px] — 2 y Exaxx^XXvM0lfxfx (6.7) (2n )3 (2n )3

= -2(Exaxx^XX vM0l }nnx (6.8)

2The chemical potential drops out of all relevant calculations, which is why we do not present the full expres­ sion here. 3Remember that in our notation Mxx .\x already contains all relevant symmetry factors for identical particles, i.e. in this case a factor 1/2. 85 6.2. Cosmological evolution

Figure 6.1: Thermally averaged dark-matter velocity squared as a function of temper­ ature for = 10 MeV and different values of T^j, ranging from = 100 eV (purple) to ^kd = 1 MeV (red). The gray regions indicate the temperatures and velocities that are relevant for photodisintegration, CMB observables, and dark-matter annihilations in typical present- day halos. with a thermal average (•) that is defined in analogy to eq. (3.43). Since x Is non-relativistic for all temperatures relevant for photodisintegration, we can set E% ~ m% and afterwards perform the usual non-relativistic expansion of [45,100],

(^XX^Mol) 2 + 2^ = 2 + 2 X ~ln----- (6’9) with the relative velocity = 6T^(T)/zn^, the s-wave contribution a and the p-wave con­ tribution b. Here, we have factored out the symmetry factor 1/2 for identical particles, meaning that a and b are calculated from the cross-section without any additional factors that compen­ sate for double counting, which makes for an easier comparison with the existing literature. Hence, by putting everything together we find

taW =4x(f) =Px(Ö+3H(t)[px(T) + Px(t)] ~ -[amx+6bTx(t)]nx(t)2 (6.10)

1 a + 6b^l px(f)2 . (6.11) Chapter 6. Bounds on residual dark-matter annihilations 86

For the general case, including dark-matter particles that are not self-conjugate, we have to replace Px(t) X-1px* (t), while using the same definitions for a and b. While the s-wave contribution to this expression remains constant over time, the corresponding p-wave contri­ bution is suppressed by the temperature of the dark-matter particle. We illustrate this behav­ ior in fig. 6.1 for mx = 10 MeV and different values of Tkd. Clearly, we see that the suppression of the annihilation cross-section is much less severe during photodisintegration than it is at recombination, which is why we can already anticipate that BBN bounds on p-wave annihi­ lating dark matter are much stronger than the corresponding ones from the CMB. Using these expressions, we now gathered all of the necessary ingredients to calculate the Hubble rate and the time-temperature relation for this particular dark-matter scenario. However, since this model features an explicit coupling to the neutrino sector, we still have to discuss a subtle change to the calculation of the neutrino-decoupling temperature.

6.2.2 Neutrino decoupling in the presence of neutrino interactions

Insec. 3.3.2 we concluded thateq. (3.75) is an appropriate way of approximating the neutrino­ decoupling temperature ifthere are no additional interactions between the dark sector and the neutrinos. However, in scenario (S2), we explicitly assume a non-vanishing coupling between the dark-matter candidate and the electron neutrino. This additional interaction can serve as a “bridge” for keeping the neutrinos in thermal contact with the photon heat bath at tem­ peratures beyond the one determined by eq. (3.75), i.e. beyond the temperature at which the interactions between electrons and neutrinos become negligible.4 To incorporate this effect, we denote the latter temperature by TV^,e± and additionally calculate the temperature at which the DM-induced equilibration of neutrinos becomes inefficient by the relation

1Ve (Tvd,x) = \?xx -VeVeVM0l) A- nx* (Tvd,x) = H(Tvd,x) , (6.12) according to the discussion at the end of sec. 2.2.4. Here, rx% -V.Ve (T) is the annihilation rate of dark-matter particles into electron neutrinos, which equals the annihilation rate into electron­ positron pairs according to the assumptions for scenario (S2). From there, we then approxi­ mate the temperature of neutrino decoupling via

TVd = min(TVd,e±, TVd,x ) . (6.13)

Clearly, this resultis only an approximation for the solution ofthe full Boltzmann equation for neutrinos; nevertheless, this description is still sufficient to gain a qualitative understanding of the impact on the creation of light elements in the early universe. A more detailed discussion of this topic can be found in [98], which uses an approach based on energy-transfer rates, but only presents the results in the two limiting cases \jxx evm0i) •VeV.vM0l) and lJxx.lXlJ vm0i ) \Jxx -Vei/eVM0l )• Finally, let us note that we do not have to make any changes

4In this context, the flavor of the final state neutrinos is not important due to effective oscillations (see e.g. the discussion in [98]). 87 6.2. Cosmological evolution to the calculation of the neutrino-decoupling temperature when considering scenario (S1). In this case we simply have Tvd = Tvd,e±, meaning that Tvd is determined by eq. (3.75) as usual.

6.2.3 Hubble rate and time-temperature relation

In this scenario, the Hubble rate receives additional contributions from the dark-matter candi­ date, which implies

H(t) = ^8n3G [Psm(t) + Px* (t)] , (6.14)

according to eq. (3.73) with pdark(t) = px*(t). Also, since x has already become non-relativistic at the time of recombination, we have pdark-rad(trec) = 0 and consequently the effective number of neutrinos is given by

N = pvi7 (trec) = 3 ( Tv (trec) \ fUA ' (6.15) Neff Peff (trec) 3 T(trec) 4 according to eq. (3.81). Hence, the exact value of Neff depends on the ratio Tv(trec)/T(trec), which can be increased or decreased compared to the SM case, depending on the temper­ ature at which the dark-matter candidate decouples from the electromagnetic and neutrino sector. However, for both scenarios discussed in this work, the presence of x can only de­ crease the temperature ratio and thus the value of Neff. This is because for pJ** VeVeVM0l)

^axx+ 'e- VMol) (which is true for both scenarios), the dark-matter candidate is never in equi­ librium with only the neutrinos and therefore can never heat up the neutrino sector relative to the electromagnetic sector. Regarding the time-temPerature relation, significant modifications only arise if the dark-matter particle has not yet become non-relativistic, meaning that we can use eq. (6.1) as an approximation for the relevant energy density. Consequently, by inserting qx(t) 3 px* (t) T(t) into eqs. (3.79) and (3.80) we find

dT 3H(T) [psm+x(T) + Psm+x(T)] for T > Tvd , (6.16) dt dpsm+x(T)/dT

dT 3H(T) [pem+x(T) + Pem+x(T)] for T < Tvd (6.17) dt dpem+x (T)/ dT with p.+K(T) := p.(T) + px*(T) and P.+x(T) := P(T) + Px*(T). In the case of scenario (S2), we thereby explicitly have to use the neutrino temperature that is determined by eq. (6.13). Finally, the source terms for Photodisintegration are determined by the volume heating rate qx (t) from eq. (6.11), which yields

-qx(t) (± )( ) = eex -qx(t) S^0) (t) = BR77 x and Se 0 t BR / (6.18) mx 2mx according to eq. (3.101) with E0 = mx. At this point, it is worth noting that annihilations into neutrinos do not lead to an electromagnetic cascade and therefore do not further influence Chapter 6. Bounds on residual dark-matter annihilations 88 the light-element abundances due to photodisintegration. However, the possible existence of this annihilation channel generally implies BR77 + BRee = 1 and therefore still has an indirect impact on the results.

6.3 Results and discussion

6.3.1 Separation of constraints

Instead of performing a combined analysis of all effects that are relevant for this model, we instead divide the discussion into two separate parts, as this leads to partially mode­ independent results, which can also be used outside the scope of this scenario. In sec. 6.3.2, we first neglect the effects of photodisintegration and only5 calculate the constraints that arise from the additional thermal particle that is present during BBN. In the case of scenario (S1), these constraints are independent of the annihilation cross-section, as long as we assume that its value is large enough to keep the dark-matter particle in equilibrium until T ~ mx, i.e. un­ til it gets Boltzmann-suppressed. In scenario (S2), however, the annihilation cross-section ex­ plicitly enters the calculation of the neutrino-decoupling temperature according to eq. (6.13). In this case, we limit our discussion to s-wave annihilations with a thermal cross-section of -VeVeVM0l) — 4 x 10-26 cm3/s (8 x 10-26 cm3/s) for self-conjugate (not self-conjugate) dark matter according to [101, 102]. These considerations ultimately lead to constraints onthe mass mx of the dark-matter particle, which can be used to update the existing bounds on thermal darkmatter [103, 104]. In sec. 6.3.3, we instead consider constraints that solely6 originate from photodisintegration due to residual dark-matter annihilations. This part of the calculation ex­ plicitly depends on the cross-section, and we separately consider s-wave as well as p-wave annihilations for different combinations of parameters. When comparing our results with other constraints in sec. 6.3.4, both of these regions are then finally combined into an overall constraint.

6.3.2 Constraints on the mass of thermal dark matter

In this section, we present the BBN constraints that arise from the additional dark-matter particle that is present during nucleosynthesis. In fig. 6.2 we show the corresponding results for different dark-matter types (top to bottom) and both previously defined scenarios (S1) (left,

BR77 + BRee = 1) and (S2) (right, BRee = BRVeVe = 1/2). Here, we utilize the same color scheme as before and separately highlight an exclusion due to over- and underproduction of helium- 4 (pink/blue), over- and underproduction of deuterium (orange/gray), as well as values of Neff that are too large/small when compared to the most recent observations from PLANCK

5Here, we use the light-element abundances that are obtained after nucleosynthesis in the presence of x and do not perform any further modifications. 6Here, we start with SM values for the light-element abundances and only consider modifications due to pho­ todisintegration. 89 6.3. Results and discussion

10

Figure 6.2: Constraints on the mass of thermal dark matter for different dark-matter types (top to bottom) and both previously defined scenarios (S1) (left) and (S2) (right). Here, we sepa­ rately highlight the masses that are excluded due to an over- or underproduction of individual elements (see text for details).

(dashed red/orange). In general, we find that the constraints from BBN and CMB are almost identical. Depending on the dark-matter type and annihilation channel, both observations rule out thermal dark matter with masses slightly below 7 - 10 MeV. Here, the only exception is a real scalar annihilating into electromagnetic radiation, which features BBN bounds that are considerably weaker than the corresponding ones from the CMB. Let us note, however, that the bounds from BBN are significantly more robust, since the CMB constraints arise from a value of Neff that is too small compared to observations and therefore can easily be evaded by invoking additional dark radiation. For the scalar case, we also find a very good agreement with the constraints from fig. 5.4 for small lifetimes Tp < 10-1 s. Since the latter bounds also originate from a thermal particle that is present during BBN, this result constitutes an important consistency check. For a dark-matter particle that annihilates into electromagnetic radiation (cf. left panel of fig 6.2), we obtain CMB limits that are in good agreement with the results from [98] and slightly stronger than the ones from [103, 104], which is simply due to the usage of more recent data from PLANCK. Similarly, we also find that our BBN limits are significantly stronger than the ones presented in [103, 104], which is due to much improved values for the measured light-element abundances (especially deuterium). For a dark-matter particle that annihilates into electrons and neutrinos with equal branching ratios (cf. left panel of fig 6.2), the limits generally turn out to be slightly stronger. This is because the dark-matter­ neutrino interactions can delay the decoupling ofneutrinos by up to two orders ofmagnitude, which leads to a larger energy density of neutrinos during BBN and consequently to stronger Chapter 6. Bounds on residual dark-matter annihilations 90 constraints. However, these results are not completely independent of the annihilation cross­ section, and we find that decreasing (increasing) the annihilation cross-section also makes these limits slightly weaker (stronger).

6.3.3 Constraints on the annihilation cross-section from photodisintegration

To complement the previous results, this section is devoted to constraints from photodisinte­ gration due to residual dark-matter annihilations. In fig. 6.3 we show the corresponding re­ sults for s-wave (top) and p-wave annihilations (bottom) of Majorana dark matter with branch­ ing ratios according to the scenarios (S1)7 (left, BRee = 1) and (S2) (right, BRee = BRVeVe = 1/2), which in the latter case leads to constraints that are less stringent by a factor of two. Regard­ ing the p-wave constraints, we fix Tkd = 100eV for concreteness8, whichis at the lower end of values that are still consistent with Lyman-a measurements [105]. Other choices of Tkd are dis­ cussed below. Beside the overall 95% C.L. limit (solid black), we also indicate the parts of pa- rameterspacethatareexcludeddueto an over- and underproduction of helium-4 (pink/blue), an over- and underproduction of deuterium (orange/gray) as well as an overproduction of helium-3 relative to deuterium (green). We find that for small masses, the bounds from pho­ todisintegration are mainly determined by the disintegration thresholds of deuterium and helium-4, which cause the corresponding bounds to vanish below mx — E^^np = 2.22 MeV and mx — E4He7 ,p3| i = 19.81 MeV, respectively. For larger masses, mx > 25 MeV, we find that photodisintegration can both leadto anunder- or overproduction of deuterium depending on the exact value of the annihilation cross-section. This implies the existence of a very narrow region of parameter space (between the solid gray and orange lines) that is still compatible with the observed value of deuterium; however, this region is always robustly excluded by an overproduction of helium-3 relative to deuterium or an underproduction of helium-4. For mx > 100 MeV the constraints eventually transition into a simple power law, but with dif­ ferent exponents for s-wave and p-wave annihilating dark matter. In case of s-wave annihi­ lations we have aexcl(mx) ~ mx, since the energy that is injected into the plasma scales as

Einj« |tjx| = x a/mx a/mx. However, for p-wave annihilations, the thermally averaged cross-section introduces an additional dependence on mx via the term Tx(T)/mx in eq. (6.9), which implies Einj « b/mp and consequently bexcl (mx) « mp; hence, the stronger mass de­ pendence of the limits in case of p-wave annihilations. In general, we find that the strongest bounds are obtained for mx ~ 150 MeV and mx ~ 10 MeV for s-wave and p-wave annihilat­ ing dark matter, respectively. At this point, the former scenario almost allows to probe the value a — 4 x 10-26cm3/s [102], which is expected for a thermal relic (dash-dotted black). In the latter case, however, the corresponding value b — 2a/(6/20) ~ 3 x 10-25 cm3/s with 6Tcd/mx — 6/20 is missed by almost two orders of magnitude. In fig. 6.4 we show the corresponding results for different values ofthe kinetic-decoupling

7For scenario (S1) the results explicitly depend on BRee and BR77 . In this section, we set BRee = 1, but we also present the corresponding results for BR77 = 1 in app. D. 8In a specific particle-physics model, Tkd could be calculated unambiguously. 91 6.3. Results and discussion

Figure 6.3: Constraints from photodisintegration for s-wave (top) and p-wave (bottom) anni­ hilations of Majorana dark matter and both previously defined scenarios (SI) (left) and (S2) (right). In addition to the overall 95% C.L. limit, we also indicate the parts of parameter space that are excluded by individual elements (see text for details). temperature, ranging from = 10 eV to T^a = 1 MeV. For small decoupling temperatures, Tkd < 100 eV, the particle stays in kinetic equilibrium for all temperatures that are relevant for photodisintegration, meaning that the bounds become completely independent of T^a- For larger temperatures, T^a > 100 eV, the bounds becomes weaker for increased values of T^d, since the thermally averaged cross-section — 6bT^(T)/zn^ decreases faster beyond decoupling (cf. fig. 6.1). Finally, for T^a > 10 keV, decoupling happens even before the onset of photodisintegration, in which case the constraints simply scales as TkaR(Tka)2 according to eq. (6.5). To complement these results, in fig. 6.5 we further show the constraints as a function of Chapter 6. Bounds on residual dark-matter annihilations 92

Figure 6.4: Constraints from photodisintegration for p-wave annihilations of Majorana dark matter into electron-positron pairs in the m^ — b parameter plane and different values of de­ coupling temperatures The meaning of the differently colored regions is identical to the one used in fig. 6.3. 93 6.3. Results and discussion

Figure 6.5: Constraints from photodisintegration for p-wave annihilations of Majorana dark matter in the — b parameter plane and two different choices of the mass mx. The meaning of the differently colored regions is identical to the one used in fig. 6.3.

Tkd for mx = 10 MeV (left) and m% = 100 MeV (right). These masses roughly correspond to the global (local) minimum of the exclusion limits that are presented in fig. 6.3. As a re­ sult, we find that the bounds indeed become independent of at ~ 0(1 keV) and T^h ~ 0(100 eV) for = 10 MeV and m% = 100 MeV, respectively. For larger decoupling temperatures, the bounds simply scale with T^R(T^)2f which becomes especially apparent by the small bump in the constraints at T^a < 1 MeV, corresponding to the onset of electron­ positron annihilations.

6.3.4 Comparison with other constraints

In fig. 6.6 we finally compare our combined constraints with complementary cosmological and astrophysical searches. Here, we show a comparison for both s-wave (left) and p-wave (right) annihilations while setting T^a = 100 eV for definiteness. In case of s-wave annihilations we find that the CMB bounds on exotic energy injections [13, 106, 107] (red) are most stringent for all parts of parameter space and allow to probe values of a that are at least two orders of magnitude smaller then the ones that are constrained by BBN (purple). Depending on the mass of the dark-matter particle, bounds from the diffuse extragalactic background [108,109] (orange) or from gamma-ray observations of Milky way dwarf galaxies [110] (green) are com­ petitive or more stringent than the bounds that have been derived in this work. However, in case of p-wave annihilations we find that - especially for small masses - the BBN con­ straints are highly competitive with all other searches, including additional constraints from the high-redshift intergalactic-medium temperature [107] (light blue) and from observations of charged cosmic rays [111] (dark blue). More precisely, we find that the bound from BBN Chapter 6. Bounds on residual dark-matter annihilations 94

Figure 6.6: Comparison of the BBN constraints derived in this work (dark red) with com­ plementary bounds on the cross-section for s-wave (left) and p-wave (right) annihilations of dark matter. Here, we show bounds from the CMB [13, 106, 107] (purple, updated with the results from fig. 6.2 for small masses), from the high-redshift intergalactic-medium temper­ ature [107] (light blue), from the diffuse extragalactic background [108, 109] (orange), from gamma-ray observations of MW-satellite dwarf galaxies [110] (green), and from cosmic-ray observations [111] (dark blue). overtakes the corresponding one from the CMB by at least one order of magnitude, which is mainly due to increased dark-matter velocities (ü^) oc during photodisintegration with T ~ O(keV) as compared to recombination with T ~ <9(eV) (cf. fig. 6.1). Recent results also suggest that p-wave annihilations of MeV-scale dark matter can be constrained by ob­ servations of charged cosmic rays [111] (dark blue). In general, these constraints are slightly stronger than the bounds from BBN, but they are also subject to some systematic uncertainties (mainly due to the model that is used for the cosmic-ray propagation). We therefore conclude that the resulting bounds from BBN clearly serve as an important complementary constraint for p-wave annihilations of MeV-scale dark matter.

6.4 Summary

In this chapter, we have calculated BBN constraints on MeV-scale dark-matter particles that can annihilate into the kinematically available SM states, namely electrons/positrons, pho­ tons, and neutrinos. Such scenarios are not only testable by direct and astrophysical searches but also leave their imprint on cosmological observables, including the light-element abun­ dances. In this context, we have derived the corresponding constraints for two different scenarios that are most natural from a particle-physics point of view, i.e. annihilations into electromagnetic radiation as well as annihilations into electron-positron pairs and/or electron 95 6.4. Summary neutrinos with equal branching ratios. To this end, we included all relevant effects that can alter the light-element abundances, with a particular emphasis on their late-time modifica­ tion due to residual dark-matter annihilations. As a result, we found that constraints from BBN and CMB independently constrain masses below 7 - 10MeV, depending on the dark­ matter type, which serves as an important target value regarding future searches for low-mass dark matter. Additionally, nucleosynthesis also leads to strong constraints on the annihilation cross-section of dark matter with masses above 10 MeV. While s-wave annihilations are dom­ inantly constrained by bounds on exotic energy injection at the time of recombination, these CMB limits become much weaker in case of p-wave annihilations due to the strong velocity suppression. However, since this suppression is less severe during photodisintegration, we find that the CMB limits, in this case, are easily overtaken by the ones from BBN, which are even competitive with the most recent constraints from charged cosmic rays. Chapter 6. Bounds on residual dark-matter annihilations 96 97

7 Bounds on scalar-portal dark-matter models

This chapter is based on the following publications:

M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBN constraints on MeV-scale dark sectors. Part II. Electromagnetic decays, JCAP 1811 (2018) 032, [arXiv:1808.09324]

K. Bondarenko, A. Boyarsky, T. Bringmann, M. Hufnagel, K. Schmidt-Hoberg, and A. Sokolenko, Direct detection and complementary constraints for sub-GeV dark matter, JHEP 03 (2020) 118, [arXiv:1909.08632]

In this thesis, we further implement the following improvements:

• Use the most recent values for the observed light-element abundances from [37]

• Include the correlation between n and Neff according to the discussion in sec. 3.2.2

• Include the effects of spin-statistical factors and inverse decays

In this chapter, we apply our results from ch. 5 to the case of models with self-interacting dark matter and scalar mediators that couple to the SM via Higgs-like couplings, a scenario that has already been extensively studies in the literature [112-116]. In this setup, the cou­ pling constant between dark-matter particles and SM states is usually severely constrained, meaning that the corresponding annihilation channel is strongly suppressed and no longer allows to obtain the correct relic abundance. However, this problem can be circumvented by employing hidden-sector freeze-out, which becomes viable in case the dark-matter particle x is much heavier than the mediator namely mx m§. Interestingly, the latter condition also implies the presence of sizable dark-matter self-interactions, which can be used to address cer­ tain small-scale problems of the ACDM model. Additionally, the velocity dependence of the self-scattering cross-section ensures that strong limits from large-velocity systems like galaxy clusters are rendered harmless [117-123]. Nevertheless, there remain strong bounds from direct-detection experiments and the CMB. Especially, for s-wave annihilations of dark mat­ ter into subsequently decaying vector mediators, most parts of parameter space are already excluded due to strong re-ionization bounds from the CMB [124, 125]. Inthecaseofscalarme- diators, however, only velocity-suppressed p-wave annihilations are possible, meaning that the overwhelming bounds from the CMB do not apply. On the downside, scalar mediators usually feature long lifetimes due to their small Yukawa-like couplings, thus bringing them into potential conflict with BBN observations. To further narrow down the parameter space for scalar particles that mediate dark-matter self-interactions, we thus perform a dedicated study of BBN constraints on this particular scenario and afterwards discuss if there remain any parts of parameter space with self-interaction cross-sections that are large enough to ad­ dress the small-scale problems of the ACDM paradigm. Chapter 7. Bounds on scalar-portal dark-matter models 98

7.1 Definition and properties of the model

In this chapter, we specifically focus on models that extend the SM by a Dirac dark-matter particle x and a scalar mediator ty, whose interactions can be parameterized by a Lagrangian of the form

1 1 m f Lxty = 2drfd

Here, v= 246 GeV is the electroweak vacuum expectation value and mf is the mass of the

SM fermion f. Moreover, the coupling constant gx determines the strength of dark-matter self-interactions and 9 can be interpreted as the mixing angle between the scalar mediator ty and the Higgs boson h. In addition to dark-matter self-interactions of the form xx XX, this Lagrangian also introduces a variety ofnumber-changing interactions, including the (inverse) decay ty ff as well as the annihilation of dark-matter particles into either SM states xX ff or scalar mediators xX tyty. The corresponding Feynman diagrams of these processes are collectively presented in fig. 7.1. Depending on the mediator mass mty, the mediator can not only decay into fermions (2me > mty > 2mn), but also into hadrons (mty > 2mn) or photons (mty < 2me), whereby the latter process is loop-induced. A calculation of the total lifetime Tty is rather involved and suffers from sizable uncertainties due to the hadronic contributions close to the QCD phase transition. In this work, we therefore simply utilize the results from [126, p. 4, fig. 1] and [127, p. 89, eq. (2.45)] for mty > 2me and mty < 2me, respectively. At this point, it is worth noting that we are mainly interested in MeV-scale mediators, which either decay into electron-positron pairs or photons. Nevertheless, hadronic decay channels still become relevant when calculating the cross-section for the process xx tyty (see below). In general,alloftheinteractionsthatare showninfig. 7.1 significantly influence the cosmological evolution of ty and x- At large temperatures, T » mx, the reactions xx tyty and xx ff keep ty and x in thermal equilibrium with each other and with the SM sector, respectively.1

However, both interactions eventually decouple (independently) at T ~ mx ~ O(GeV), and afterwards ty remains fully decoupled until its decay at t ~ Tty. Consequently, we find that the evolution of this model can easily be mapped onto the general scenario that we discussed in ch. 5 by calculating the appropriate values of Tcd and Zcd for each combination of mx and mty. To perform this mapping, we now take a closer look at the cosmological evolution of the dark sector and derive a procedure to approximate Tcd and Zcd for each relevant parameter point.

7.2 Evolution and influence of the dark sector

In order for x to be a suitable dark-matter candidate, the coupling constant gx has to be fixed in such a way that the freeze-out of x via the process xx tyty leads to the correct relic abun­ dance. When fixing gx by this requirement, the evolution of the dark sector usually proceeds

1In a full model, there might also be interactions between ty and the W and Z bosons, which we do not discuss in this thesis. 99 7.2. Evolution and influence of the dark sector

Figure 7.1: Feynman diagrams for the most important interactions of the model that is defined via the Lagrangian (7.1). In addition to (inverse) decays of the mediator (top left), the dark­ matter particle also features annihilations into (p(p (top right) and ff (bottom). in a way that is best visualized by the left panel of fig. 7.2, which shows the rates Tr(T) (cf. eq. (2.45)) for the most relevant reactions r from fig. 7.1 and the exemplary parameter combina­ tion m% = lOGeV, = 100MeV, and sin(0) = 10’4. At large temperatures, both r^^,yy(T) (blue) and r^_^(T) (orange) exceed the Hubble rate H(T) (green), meaning that^; and (p are kept in thermal equilibrium with the SM heat bath and therefore adopt a temperature that is equal to the one of photons, Tj(T) = T. However, the reaction XX ~> ff eventually decouples at a temperature Tvd, and even though x ant^ afterwards remain in mutual equilibrium with each other, they start to develop a temperature that differs from the one of photons, Td(T) f T. Subsequently, also the process XX W decouples at a photon temperature Tc^ < Tvd, corre­ sponding to a temperature ratio £cd = Td(Tcd)/Tca 1. Afterwards, x approaches the correct relic abundances and (p remains decoupled until its (inverse) decay at t ~ via the processes (p ff (purple) and ff—>(p (red). Based on this setup, we calculate the values of Tcd and £cd for a given set of parameters m%, m

r^^yj(Tvd) = [(t7x^yjüM0i)nA:](Tvd) = H(Tvd) . (7.2)

In this context, we take into account all possible annihilation channels by using the cross­ section [93,128]

- 2 (S - Chapter 7. Bounds on scalar-portal dark-matter models 100

10-10

10-151-

> u 10-20 k tZ) OJ +->

1 2

10-30 103 102 101 100 10-1 10-2 10-3 T [GeV ]

Figure 7.2: Rates rr (T) of the reactions r that are shown in fig. 7.1in comparison to the Hubble rate H(T) for mx = 10GeV, mty = 100MeVandeithersin(0) = 10-4 (left) orsin(0) = 7 x 10-6 (right). which includes the full decay rate r$ = t- 1 « sin(0)2 from [126,127] and therefore accounts for all electromagnetic and hadronic decay channels. After decoupling from the SM sector,2 the dark sector remains in thermal equilibrium but eventually develops a different tempera­ ture Td(T) = T. To calculate Td(T) we may employ separate heat conservation in both the visible and the dark sector, which yields

dTd I = Pdark(Td) + Pdark(Td) dPsm(T)/dT (7 4) dT Tvd 'T>Tcd dpdark(Td)/dTd Psm(T) + Psm(T) inanalogyto eqs. (4.20) and (4.23). Bysolving this equation with the initial condition Td(Tvd) = Tvd, we obtain the function Td(T) that determines the evolution of the dark-sector tempera­ ture until x and ty decouple at Tcq. The latter quantity, in turn, is approximated via the relation rxx^tyty(T) ~ H(T), which yields

rXx 'tyty( Td(Tcd) ) = I'^XX 'tytyvM0l)nJ ( Td(Tcd) ) = H(Tcd) . (7.5)

In this expression, the modified cross-section &Xx^tyty explicitly takes into account Sommerfeld enhancements due to multiple mediator exchanges in the initial state, which can be important for a sufficiently large hierarchy mx mty. Following [129-131], this cross-section can be approximated by3

^Xx XtyVM0l — (Sp 1) x \^XXty’VvM0l] p + ^XXty’VvM0l . (7.6)

2Elastic scattering of the form tyf O tyf might lead to a prolonged phase of kinetic equilibrium; however, this only has a small effect since both scattering partners are relativistic around Tvd. 3Since ty is a scalar, the s-wave contribution to the cross-section vanishes. 101 7.3. Results and discussion

Here, ^xx -tytyVMal is the full cross-section of the process xx . tyty, \axx tytyVMal]p is the cor­ responding p-wave contribution and Sp is the relevant Sommerfeld factor that is given in [132, p. 18, eq. (B.2)]. In combination, eqs. (7.4) and (7.5) determine the temperature Tcd as well as the temperature ratio Zcd = Td (Tcd )/Tcd. However, we have to keep in mind that the calculation critically depends on the exact value of gx, which cannot be fixed with­ out the a priori knowledge of Tcd and Zcd, as the latter two quantities influence the final relic abundance. To overcome this problem, we start with a rough estimate of the form gx ~ 0.01 y/mx/GeV as motivated by the results from [133] and afterwards apply a recursive procedure until h2 = 0.120 [13] is satisfied. This way, we obtain a consistent set of values

(Tvd, Tcd, Zcd, gx) foragivenset of input parameters (mx, mty,sin(9)). We further checked that this approximate way of calculating Tcd and Zcd via the assumption of instantaneous decou­ pling is in O(1%) agreement with the full solution of the Boltzmann equation from DARK- SUSY [101], which is sufficient for our purposes. Let us note, however, that the previous calculation partially breaks down for very small values of sin(9). More precisely, at some point the rate (T)

7.3 Results and discussion

7.3.1 Constraints on different mass combinations

In this section, we finally present the corresponding BBN constraints in fig. 7.3 for fixed mass ratios mx/mty (left) as well as fixed mediator masses mty (right) in the mty - sin(9) and mx - sin(9) parameter planes, respectively. Beside the overall 95% C.L. (solid black), we also separately highlight over- and underproduction of helium-4 (pink/blue), over- and under­ production of deuterium (orange/gray), overproduction of helium-3 relative to deuterium (green) as well as values of Neff that are too large/small when compared to observations from PLANCK (dashed red/orange). As already mentioned before, we also indicate the parts of

4Lifetime and branching ratios are calculated from the full decay rates that are presented in [126, p. 4, fig. 1] for mty > 2me and in [127, p. 89, eq. (2.45)] for mty < 2me . Chapter 7. Bounds on scalar-portal dark-matter models 102

[MeV]

Figure 7.3: Constraints from BBN for fixed mass ratios m%/ m

(right). In addition to the overall 95% C.L. constraints (solid black), we also indicate the parts of parameter space that are excluded due to individual elements (see text for details). In addition, we explicitly highlight all parameter combinations that never lead to a thermaliza­ tion of the dark and the visible sector (shaded white). 103 7.3. Results and discussion parameter space that do not ensure thermal equilibrium between the dark and the visible sector without the inclusion of additional processes (shaded white). Finally, let us note that the standard freeze-out scenario breaks down for almost degenerate dark-sector states with mx < 10m$ (hatched) [134], meaning that our calculation is no longer reliable in this region. Overall, we find that the results significantly depend on the mediator mass while the de­ pendence on mx is rather subtle (but still not negligible). This is because the latter quantity only modifies the parameters Tcd and Zcd, while the former one critically determines the avail­ able decay channels of and therefore its lifetime. More precisely, for m§ < 2me, the scalar can decay only into photons, which leads to large lifetimes and consequently severe constraints (cf. fig.5.4). In this part of parameter space, the mediator is essentially stable and therefore acts as extra dark radiation during all temperatures that are relevant for nucleosynthesis, which ultimately leads to conflicts with observation. For 2me < m§ < 2m^, however, the mediator decays mainly into electron-positron pairs. In this part of parameter space, the constraints are very similar to the ones in the lower left panel of fig. 5.4, but slightly distorted since Tcd and Zcd are different for each parameter point. In fact, we find 10 MeV < Tcd < 1 GeV for 2me < m^ < 2mM and similarly 0.8 < Zcd < 1.5. Let us note that this region only features small effects from inverse decays and spin-statistics with O(5%) corrections, since the media­ tor predominantly decays at T(t = tm§, which corresponds to a more traditional decay scenario. However, including these effects still leads to slightly stronger constraints, which is why they cannot simply be neglected. Instead, for m§ > 2m^, the limits are rather weak due to small lifetimes above the muon threshold. In this part of parameter space, the mediator essentially decays long before the onset of BBN and therefore does not leave any observable imprints for the values of sin(0) that we consider. Finally, for m§ > 2mn, hadronic decay channels would become relevant for small values of sin(0), see e.g. [126], which are, however, beyond the scope of this work. At this point, it is important to note that - for large parts of parameter space - the dark and the visible sector have never been in thermal equilibrium (shaded region). When employing conservative bounds that do not make any additional assumption about the early-universe cosmology, the limits would therefore naturally end at the dash-dotted line. For smaller mix­ ing angles, the limits critically depend on the initial temperature ratio, and while our choice

Zcd = 1 for Tcd still leads to stringent constraints, the other extreme choice Zcd = 0 implies slightly weaker results, since only freeze-in contributions would remain [135].

7.3.2 Constraints in the context of dark-matter self-interactions

In this section, we interpret the previous results in the context of dark-matter self-interactions and discuss if there remain any parts of parameter space with self-interaction cross-sections that are still large enough to address the small-scale problems of the ACDM paradigm. To this end, we also compare our results with complementary constraints from SN1987a supernova Chapter 7. Bounds on scalar-portal dark-matter models 104

Figure 7.4: Constraints from BBN in the context of self-interacting dark matter in the mty — mx parameter plane for sin(0) — 5 x 10—5. In addition to the overall constraints from BBN (purple), we also indicate the parts of parameter space that lead to a transfer cross-section aT of dark matter at a typical small-scale velocity of v — 30 km/s between 0.1 cm2/g < aT/mx < 1 cm2/g (light blue) and 1 cm2/g < aT/mx < 10 cm2/g (dark blue). For comparison, we also present complementary constraints from direct-detection experiments [78-80, 136, 137] (brown; solid — current, dashed — future) as well as from cross-sections that are too large on the scale of galaxy clusters with v ~ 1Mm/s [117-123] (green). neutrinos [87],5 rare kaon decays [116], and direct-detection experiments. In the latter case the strongest constraints are set by XENON1T [78] for large dark-matter masses and by either CRESST-II [79, 136] or CDMSLITE [80] for small dark-matter masses. Additionally, we also consider the future final stage of the CRESST-III experiment [137], which allows to study the prospects of future direct-detection experiments.6 In fig. 7.4 we first show the resulting con­ straints in the mty — mx parameter plane for a fixed value of sin(0) — 5 x 10—5. We find that most regions of parameter space that lead to a transfer cross-section aT of dark matter at a typ­ ical small-scale velocity of 30 km/s between 0.1 cm2/g < aT/mx < 1 cm2/g (light blue) and

5Constraints from horizontal branch stars [138] are only relevant for mty < 30 keV. 6All bounds from direct-detection experiments are evaluated by employing the code DDCALC 2.0.0 [139, 140], which is modified in such a way that the non-standard dependence of the recoil rate on the momentum trans­ fer due to the presence of the light mediators is properly taken into account. Details on our implementation of CRESST-III can be found in [141]. Status of the limits: August, 2018. 105 7.3. Results and discussion

Figure 7.5: Constraints from BBN in the sin(0) — mx parameter plane for fixed self-interaction cross-section = 3cm2/g (left) and (?T/mx = 10cm2/g (right). Beside the overall BBN constraints (purple), we also show complementary constraints from direct-detection exper­ iments [78-80, 136, 137] (brown; solid = current, dashed = future), rare kaon decays [116] (red), SN1897a supernova [87] (gray), as well as cross-sections that are too large on the scale of galaxy clusters with v ~ IMm/s [117-123] (green). lcm2/g < (Tt/Wx < 10cm2/g (dark blue) [142-144] are already excluded by current direct- detection experiments (solid brown). Most of the remaining parameter space - which corre­ sponds to mx < 0.5 GeV - can then be excluded by BBN constraints (purple); however, this exclusion is not necessarily robust for all masses, since the dark and the visible sector never thermalize for mx < 0.2 GeV (purple hatched), in which case our results are subject to addi­ tional, non-minimal assumptions (cf. sec. 7.2). Other constraints from large-velocity systems such as galaxy clusters with v ~ 1 Mm/s [117-123] (green) do not impose any additional re­ strictions on the viable parameter space. However, it is worth noting that CRESST-III (dashed brown) will be able to probe all remaining regions with 1 cm2/g < or I mx < cm2/g in the near future. To further investigate which values of sin(0) still lead to viable regions of parameter space with sizable self-interactions, in fig. 7.5 we show the resulting constraints in the sin(0) — mx parameter plane while fixing the mediator mass in such a way that (Tt/tHx = 3cm2/g (left) or ar/™x = 10cm2/g (right) for v ~ 30km/s. The latter identification is indeed possible for mx < 10 GeV, in which case each value of CT/mx results from a unique choice of mx. Overall, fig. 7.5 illustrates the strong complementary between constraints from direct detec­ tion (brown) and rare kaon decays (red) on the one hand, and constraints from BBN (purple) and supernova neutrinos (hatched gray) on the other hand.7 Taken together, all of these con­ straints only leave a very narrow window of viable parameters for pt/mx = 3 cm2/g that is

7The supernova bounds are hatched due to their intrinsic uncertainty [145]. Chapter 7. Bounds on scalar-portal dark-matter models 106

in# [MeV]

Figure 7.6: Left panel: Largest value of the mixing angle that is still allowed by direct-detection experiments and rare kaon decays. Right panel: Constraints from BBN in the m

0.5 GeV, they become insensitive to sin(0) for m% < 0.5 GeV, in which case the mixing angle is fixed to the largest value that is still com­ patible with rare kaon decays, i.e. sin(0) = 1.9 x 10“4 (cf. fig. 7.5, brown+red). In general, we find that large parts of the parameter space with 0.1 cm2/g < or< 10 cm2/g are already excluded. Mediator masses with < 2me are ruled out due to the large lifetimes Tq that are implied by the decay channel (p —> 77, which lead to long-lived mediators and consequently to severe constraints. For larger masses, 2me < < 2mtlf the BBN bounds become less 107 7.4. Summary stringent when increasing the value of mx, since direct-detection constraints only allow for small values of sin(0), corresponding to smaller lifetimes and weaker constraints. Finally, for mty > 2m^ the mediator can quickly decay into muons, with lifetimes Tty < 0.1 s well below the onset of BBN, which leaves this part of parameter space completely unconstrained. Con­ sequently, except for a small window between mty ~ 1 — 3 MeV and mx ~ 0.2 — 0.6 GeV (also cf. fig. 7.5), only combinations of mty and mx along the first and second resonance peak are still consistent with observations. However, a detailed discussion of this finely tuned part of parameter space would require a careful treatment of the thermal history of the dark sector taking into account effects associated to the resonant enhancement of the annihilation cross­ section [146, 147], which is beyond the scope of this thesis.

7.4 Summary

In this chapter, we have calculated BBN constraints on models with self-interacting dark mat­ ter and scalar mediators that couple to the SM via a Higgs-like coupling. Such models are especially compelling since they lead to dark-matter self-interactions that allow for address­ ing the small-scale problems of the ACDM model, while also evading the upper bound on the cross-section that is set at the scale of galaxy clusters. However, the coupling constant in such models is also subject to severe constraints from direct-detection experiments, rare kaon decays, and SN1987a supernova neutrinos. To complement these results, we addition­ ally determined the BBN constraints on this scenario by carefully taking into account the cosmological evolution of the dark-sector states. Moreover, we have included all relevant effects that can potentially alter the light-element abundances by mapping the full model onto the general scenario that we discussed in ch. 5. Overall, we find that most parts of parameter space with sizable dark-matter self-interactions are ruled out by a combination of direct-detection experiments and constraints from nucleosynthesis. In fact, only a very small region of parameter space remains viable, which is located between mty ~ 1 — 3 MeV and mx ~ 0.2 — 0.6 GeV. Additionally, all of the remaining parameter combinations with 1cm2/g < aT/mx < 10cm2/g can be fully tested with upcoming low-threshold direct- detection experiments like CRESST-III. Chapter 7. Bounds on scalar-portal dark-matter models 108 109

8 Bounds on axion-like particles

This chapter is based on the following publication:

P. F. Depta, M. Hufnagel, and K. Schmidt-Hoberg, Robust cosmological constraints on axion-like particles, JCAP 05 (2020) 009, [arXiv:2002.08370]

In this chapter, we consider models with axion-like particles that have masses in the keV — GeV range. Axions or axion-like particles (ALP) arise in many different models beyond the SM as the Nambu-Goldstone bosons of some broken symmetry. In addition to the usual QCD axion [148-152], ALPs also arise in string compactifications [153, 154], models with a broken continuous R-symmetry [155], or in the context of the relaxation mechanism [156, 157]. Suf­ ficiently light ALPs are very long-lived and consequently constitute a promising dark-matter candidate [158-161]. In addition, they can also be held responsible for astrophysical anoma­ lies, such as the unexpected transparency of the universe to high-energy 7-rays [162] or the presence of a mono-energetic X-ray line around 3.5keV [163, 164] (see however [165]). Heav­ ier ALPs with smaller lifetimes - while not being a suitable dark-matter candidate - have instead been proposed as a possible explanation for the difference between the measured and the theoretically predicted anomalous magnetic moment of the muon [166, 167] as well as for the apparent resonances that are observed in nuclear transitions of beryllium [168] and helium [169]. Since ALPs can also serve as dark-matter mediators that allow to evade the strong constraints from direct-detection experiments [92, 170-172], it is not surprising that such models, as well as their experimental prospects, have already been extensively stud­ ied in the literature [90, 92, 94, 173-181]. In this work, we update and extend the previous research by discussing the impact of such models on the light-element abundances that are created during nucleosynthesis (also see [75, 182-185]). As it turns out, the resulting con­ straints are complementary to the ones from searches in the laboratory and therefore provide important additional information regarding the validity of certain regions in parameter space. To evaluate the robustness of these results, we further analyze the impact of additional effects that might potentially weaken the constraints, including extra dark radiation, non-vanishing neutrino chemical potentials, and potentially low reheating temperatures.

8.1 Definition and properties of the model

In this work, we mainly focus on ALPs that couple exclusively to photons; however, we later comment on the possible modifications that might occur in models with more general cou­ pling structures. In this case, the Lagrangian for the axion-like (pseudoscalar) particle ty can be written as

Lty — 1 d.tyd^ty — 1 mtyty2 — gtyYtyFHvF*v . (8.1) Chapter 8. Bounds on axion-like particles 110

7

7

Figure 8.1: Feynman diagrams for the most important interactions of the model that is defined via the Lagrangian (8.1). In addition to (inverse) decays (left), the ALP can also undergo efficient Primakoff interactions (right).

Here, g^ is the coupling constant between ALPs and photons, F^v = — dyAp is the electromagnetic field strength-tensor that includes the photon field A^, and FHV is its dual. This Lagrangian introduces a three-point vertex between two photons and the ALP, thus enabling the ALP (inverse) decay (p 77 as well as Primakoff interactions q±(p q^y with any charged particle q. The corresponding Feynman diagrams of these interaction are presented in fig. 8.1. In principle, (p also undergoes Compton scattering of the form (py (py; however, this process is suppressed with the fourth power of which is already constrained to have rather small values (see below). Given this Lagrangian, the lifetime 77 of the ALP takes the form 647T / (8-2) which allows to use 77 and g^ interchangeably for a given ALP mass. Both of the reactions that are depicted in fig. 8.1 also dominantly determine the cosmological evolution of (p. Since the interaction term in the Lagrangian is UV-divergent, we find that the cross-section for Pri­ makoff interactions increases with the energy. Consequently, for large temperatures, this pro­ cess ensures chemical equilibrium between (p and the SM; however, it eventually decouples at a temperature Tfo once its rate can no longer keep up with the expansion of the universe. Finally, after decoupling, the evolution of (p is mainly determined by its (inverse) decay. As a result, we find that the previously defined model of axion-like particles shares many sim­ ilarities with the general scenario that we discussed in ch. 5, and many parts of parameter space can indeed be translated into each other by setting £cd = 1 and Tc^ = Tfo. To make this statement a little more explicit, we now take a closer look at the cosmological evolution of (p as determined by the Lagrangian in eq. (8.1). 111 8.2. Evolution and influence of the dark sector

8.2 Evolution and influence of the dark sector

8.2.1 Evolution of the axion-like particle

Except for the inclusion of additional Primakoff interactions, the ALP evolves in a very sim­ ilar way as the mediator from ch. 5. Consequently, we can simply recycle the corresponding Boltzmann equation (5.6),

- H(t) p = -[d, (t, p) + K (t, p)] x f (t, p) - f, (t, p)] (8.3) with D,(t, p) = D+ (t, p) from eq. (5.5) (cf. ch. 5) and an additional term K,(t, p) that describes the change of , due to Primakoff interactions. An elaborate calculation reveals that the latter term is approximately given by [184, 185]1

K (t \ V ,Ya i/\ , [ E,(mi + T)]2 q (T\ g 4 3 2 (8.4) Kf (f'p) “ L ln X + mY[m2 + (m, + 3T)2]) Qn(T) T=T(t)

In this expression, the sum goes over all charged fermions in Qf = {e±, p±,... }, n (T) is the number density of particle species i, Qi is the corresponding electric charge in units of e, a = e2/(4n) is the fine-structure constant, and m7 = eTg*q(T)1/2/6 [185] is the thermal (plasmon) mass of the photons with g*q (T) being defined via

E Q2n(T) =: ^g^q(t)T3 . (8.5) ieQf n

At this point it is worth noting that a non-vanishing value of mY can also inhibit ALP (inverse) decays and therefore lead to a modification of D,(t,p). However, we find that the corre­ sponding effect is negligible for all interesting parts of parameter space and consequently we set mY = 0 outside of eq. (8.4). As we have already pointed out earlier, the Primakoff inter­ actions keep , in chemical equilibrium with the SM at temperatures T Tfo. Regarding the solutionofeq. (8.3), we therefore set2

-1 (p2 + m2,)1/2 f,(t0,p) = 1 (8.6) exp ---- T0------as an initial condition at T0 := T(t0) = max(20Tfo, 10 MeV), where Tfo is approximated via the relation H(Tfo) = rt± , ,t±7 (Tfo) (also cf. eq. (8.9) below), and T0 > 10 MeV ensures that our calculation starts before the onset of primordial nucleosynthesis. Afterwards, we can apply

1This expression is only valid for charged fermions and therefore does not account for W bosons or light charged hadrons. However, in the parameter region of interest, neglecting these particles still leads to conser­ vative constraints, since additional contributions to K, (t, p) would merely delay the chemical decoupling of , and therefore generally lead to larger energy densities and hence stronger constraints. 2In this case, we explicitly track the decoupling of ,, which is slightly more specific than the general procedure that we used in ch. 5. Chapter 8. Bounds on axion-like particles 112

Figure 8.2: Approximate values for Tfo (Primakoff decoupling) and Tre (re-equilibration) as a function of the mass m

D(t,p) +K to, which yields

f(^o,p^W/R(fo)) exp a(t',pR(t)/R(t'))dt'^

+ ^^(t',pR(t)/R(t'))exp (-j\(t",pR^/R(t"^ dt") dt' (8.7)

with a(t, p) := D^ft, p) + K^t, p) and ß(t, p) := ot(t, p) x fp(t, p). Similarly, we find that the expressions for the Hubble rate, the effective number of neutrinos, the time-temperature relation, and the source terms for photodisintegration are identical to the ones from sec. 5.2.2, when using the modified volume heating rate

= - f3 [^(t p) + K^t, p)] x [/^(t, p) - f^t, p)] ^p2 + m* . (8.8)

Consequently, we do not intend to go over the corresponding expressions again. In the context of our general solution strategy from sec. 3.4, we solve the full Boltzmann equation without using any further approximations. However, in order to gain a better understanding of the parameter space, it is instructive to derive approximate formulae for the temperature Tfo at which the Primakoff processes decouple as well as for the temperature Tre at which the ALP is re-equilibrated due to the (inverse) decay. This is the focus of the next section. 113 8.2. Evolution and influence of the dark sector

8.2.2 Freeze-out and re-equilibration

Within 10% accuracy, the freeze-out temperature Tfo can be approximated by the recursive relation [185, 186]

g*P (Tfo) 10-9 GeV-1 Tfo — 123 GeV . (8.9) g*q f gpY

Analogously, the re-equilibration temperature Tre is approximately given by [75]

_ 1 /2 2.20 (|} g,p(Tre)-1/4MeV for mp > Tre T re — (8.10) 1.69 (?)-1'3g.f (Tre)-1/6 (MV)1/3 MeV for mp < Tre •

Given these quantities, the qualitative evolution of p is determined by the order of the ALP (i) freezing out from the SM heat bath, (ii) becoming non-relativistic, and (iii) re-equilibrating via the (inverse) decay. In fig. 8.2 we show the dependence of Tfo and Tre on the mass mp and either the lifetime Tp (left) or the axion-photon coupling (right). In this figure, the dark (light) gray region indicates the combinations of parameters, for which the ALP is always (only for temperatures relevant to BBN) in thermal equilibrium with the SM heat bath. The distinct kink that is present in each Tre-contour happens at Tre = mp. This is because - for smaller masses - the additional Lorentz boost increases the effective lifetime of the particle, meaning that a smaller “real” value is needed to end up with the same re-equilibration temperature. To complement this discussion, in fig. 8.3 we further show the full cosmological evolution of the thermal bath for different (phenomenologically distinct) combinations of Tfo and Tre. In this figure, we do not only show the energy densities of the various bath particles (left), but also the collision terms Dp(T, p = T) and Kp(T, p = T) in comparison to the Hubble rate H(T) (right).

• mp = 10-2 MeV and Tp = 10-2 s, corresponding to Tfo Tre (cf. top panels of fig. 8.3): If the Primakoff process does not decouple before the (inverse) decay of p becomes ef­ ficient, the ALP always stays in thermal equilibrium with the SM heat bath, since one of the collision terms always exceeds the Hubble rate. Consequently, we find that the ALP features a sizable abundance until T ~ mp, i.e. until it becomes non-relativistic. In general, this behavior is typical for parameter points in the dark gray region of fig. 8.2. In this case, the energy density of p (green) simply tracks its equilibrium value (orange) for all relevant temperatures.

• mp = 10-2 MeV and Tp = 104 s, corresponding to Tfo ~ Tre (cf. middle panels of fig. 8.3): In case Tfo and Tre are very similar, the phenomenology is rather non-trivial and different scenarios are possible. Regarding the example shown in fig. 8.3, the ALP first decouples from the thermal bath, afterwards gets partially re-equilibrated and eventually becomes non-relativistic. Chapter 8. Bounds on axion-like particles 114 R(T)

x

(T)/po x p

4 R(T)

x

px(T)/po

4 R(T)

x

px(T)/po

Figure 8.3: Energy densities (left) and rates (right) as a function of temperature for different choices of parameters. The energy densities are normalized in such a way that the neutrinos obey Petree)/po x R(frec)4 = at the time of recombination. 115 8.3. Results and discussion

• mp = 10 MeV and Tp = 104 s, corresponding to Tfo » Tre (cf. lower panels of fig. 8.3): If the Primakoff process decouples before the (inverse) decay of p becomes efficient, the

ALP fully decouples at Tfo, afterwards potentially becomes non-relativistic at T ~ mp and eventually re-equilibrates at Tre. Moreover, the latter process can be interpreted as a traditional decay scenario in case p has already gotten non-relativistic, i.e. for Tre mp? This behavior is typical for parameter points in the white region of fig 8.3. In this case, the model can also easily be mapped to the scenario that we extensively studied in ch. 5

by setting Zcd = 1 and Tcd = Tfo.

After roughly mapping out the corresponding parameter space, we now apply our general solution strategy to determine the full set of BBN constraints for the given model of axion-like particles.

8.3 Results and discussion

8.3.1 Constraints on the vanilla ALP scenario

In this section, we present the BBN constraints for models with axion-like particles as a func­ tion of the relevant parameters mp and Tp. To this end, in fig. 8.4 we first show the corre­ sponding contours of constant Yp (top left), D/1H (top right) and Neff (bottom). These results are further supplemented by the corresponding observationally inferred values (solid gray) as well as their variation at +95% C.L. (upper limit, dash-dotted gray) and -95% C.L. (lower limit, dashed gray). As a result, this figure already provides a first estimate for the region of parameter space that is still allowed when being confronted with most recent cosmological observations (filled gray). Let us note, however, that these results do not yet comprise any the­ oretical uncertainties (e.g. for the nuclear rates, cf. sec. 3.2.2), which implies that the final limits will be slightly weaker. Nevertheless, this plot still allows to performa direct comparison with previous results from the literature, especially the ones that are presented in [75, p. 8, fig. 5]. While the latter results perfectly agree for Yp and Neff, we find larger values for D/1H, which ultimately leads to less stringent constraints when compared to [75]. Consequently, both find­ ings are partially incompatible. Concerning this matter, we carefully checked our results with the ones presented in [103] (which should be identical for Tp < 10-2 s, see below) and found good agreement. We therefore believe that the constraints from [75] are too stringent due to an overestimation of the D/1H underproduction, corresponding to an underestimation of deuterium itself. In fig. 8.5 we finally show the full set of BBN constraints in the mp - Tp parameter plane (including all relevant uncertainties). In addition to the overall 95% C.L. (solid black), we

3In the previous chapters, we have always used the condition T(t = Tp) ~ mp to determine if the mediator follows a traditional decay. In fact, we find that within O(1%) accuracy T(t = Tp) « Tre for Tre > mp, which also includes T(t = Tp) ~ Tre ~ mp. Both expressions only differ for Tre < mp, when the Lorentz boost becomes important. Chapter 8. Bounds on axion-like particles 116

Figure 8.4: Resulting values for yp (green), D/1 H (blue) and Neff (orange) in the parameter plane. For comparison, we also show the observationally inferred values (solid gray) as well as their variation at ±95% C.L. (dashed/dash-dotted gray). also separately highlight over- and underproduction of helium-4 (pink/blue), over- and un­ derproduction of deuterium (orange/gray), overproduction of helium-3 relative to deuterium (green) as well as values of Neff that are too large/small when compared to observations from PLANCK (dashed red/orange). In general, we find that these results closely resemble the ones from fig. 5.4 for electromagnetic decays of light mediators, which can be understood as fol­ lows: In the top right comer of parameter space (corresponding to the white region in fig. 8.2), the Primakoff interactions decouple before the ALP decay, which can be mapped onto the sce­ nario from ch. 5 by setting £cd = 1 and Tcj = Tfo. We have to keep in mind, however, that the results in fig. 5.4 are only shown for a fixed decoupling temperature, while Tfo is differ­ ent for each parameter point. Then again, we have already discussed that the bounds from 117 8.3. Results and discussion

Figure 8.5: Constraints from BBN on models with axion-like particles. In addition to the overall 95% C.L. constraints (solid black), we also indicate the parts of parameter space that are excluded by individual elements (see text for details). fig. 5.4 are largely insensitive to the exact value of Tcd = T(o (cf. eq. (5.15)), which implies that the constraints are still largely comparable in this part of parameter space. However, such an identification is no longer possible in the lower left corner of parameter space (corresponding to the dark gray region in fig. 8.2), in which case the Primakoff interactions only decouple after the decay of (p has become efficient. Nevertheless, for these masses and lifetimes, the mediator from ch. 5 is always thermalized with the SM heat bath prior to the onset of nucleosynthesis. Consequently, in both cases the constraints originate from an additional thermal particle that is present during nucleosynthesis and therefore are - once again - identical. Small differences only arise in the transition region when Tfo ~ Tre. For such choices of parameters, the Pri­ makoff interactions additionally decouple below the QCD phase transition, which leads to slightly stronger constraints compared to fig. 5.4 that manifest themselves in form of the small island around ^3 MeV and « 1 s, corresponding to Tfo ~ 200 MeV.

8.3.2 Constraints beyond the vanilla case

So far, we have derived constraints in case of a cosmological history that closely resembles the usual ALP cosmology with a high inflationary scale. To evaluate the robustness of these Chapter 8. Bounds on axion-like particles 118 constraints with respect to changes in the particle-physics model, we now discuss some (non­ minimal) modifications to the previous scenario, all of which have the potential to weaken the resulting constraints. In particular, we discuss the effects of (i) additional dark radiation and non-vanishing neutrino chemical potentials, as well as (ii) low reheating temperatures.

Effects of additional dark radiation and non-vanishing neutrino chemical potentials

A rather simple modification of the cosmological scenario is the addition of extra dark radiation, which can even be completely independent of the already discussed ALP sector. To quantify this effect, we introduce an additional energy density of the form

7 n~ Pa(T)= 2ANeff x 830Tv(T)4 , (8.11) which we incorporate into our general procedure by (i) changing the prefactor of pvv(T) in eq. (3.23) from 6 to (6 + 2ANeff), and by (ii) adopting the following expressions for the Hub­ ble rate and the time-temperature relation, accordingly. Another, arguably more contrived (cf. sec. 3.1.2) modification is the addition of a non-vanishing neutrino chemical potential pv (T), which leads to modified neutrino spectra of the form

fV{(T, p) = [exp (p/Tv(T) T tv) + 1]-1 with tv := Pv(T)/Tv(T) = const . (8.12)

Here, the - (+) sign is used for neutrinos (antineutrinos). Such a modification can potentially influence standard nucleosynthesis in two different ways: On the one hand, for tv = 0, the energy density of neutrinos from eq. (3.23) gets multiplied by an additional factor

ANefftv + 1 = 7^ x |PolyLog ^4, —e + PolyLog ^4, — etv^ | , (8.13) which, however, can be absorbed into the value of ANeff from eq. (8.11). On the other hand, the chemical potential tv of electron neutrinos also influences the neutron-to-proton ratio from eq. (3.48), which gets modified to

Yn (T) _ e—Q/T—tv (8.14) Yp (T) since pn (T) — pp (T) ~ pve (T) = 0. Consequently, we find that the neutron-to-proton ratio at freeze-out is suppressed by an additional factor exp(—tv) compared to the vanilla case. Based on the previous two modifications, we then calculate robust BBN constraints by adjusting ANeff and tv in such a way that the constraints are maximally weakened for each individual point (m$, t$) in parameter space. The results of this procedure are presented in fig. 8.6, where we separately indicate (i) the standard ALP scenario (dashed, cf. fig. 8.5), (ii) an ALP scenario with an additional contribution to ANeff (dash-dotted), and (iii) an ALP scenario with an additional contribution to ANeff and a non-vanishing chemical potential tv (solid). For comparison, we also follow [94] and show complementary constraints from electron [187-190] and proton beam-dump experiments [92,191-193], SN1987a supernova cooling [194], visible 119 8.3. Results and discussion

Figure 8.6: Combined constraints from BBN and CMB in comparison with complementary constraints from beam-dump experiments and observations of horizontal branch stars and supemovae. In addition to the constraints that result from standard assumptions (dashed blue), we also show the remaining constraints when optimizing only NGff (dash-dotted blue) or both NGff and £,v (solid blue). decays of ALPs produced in SN1987a [195] (the latter two are dashed due to their recently debated reliability [145]), and horizontal branch star cooling [185]. In general, we find that case (ii) significantly weakens the constraints compared to case (i), which is mainly due to an increase of NGff. In fact, an inclusion of the additional parameter ANGff always allows to adjust NGff in such a way that it is compatible with CMB observations. However, such an adjustment does not necessarily lead to the weakest constraints, since a larger value of NGff also increases the predicted abundances of deuterium and helium-4, thus strengthening the BBN constraints. In fact, for 3 MeV < m

200 MeV, the constraints are mainly due to an underproduction of deuterium, while NGff remains slightly below its central value. Around the exclusion line, the required value of ANGff ranges from 0.16 < ANGff < 0.67 and 0.67 < ANGff < 22 for m

10 MeV, respectively. For large masses - especially for > 100 MeV with NGff > 10 - these values suggest a severe cancellation of different effects, which corresponds to a significant tuning of different parameters against each other. Moreover, we find that the constraints are further weakened when transitioning from case (ii) to case (Hi), i.e. when allowing for an additional variation of £v. However, we have to keep in mind that these results require a simultaneous optimization of a priori indepen­ dent variables, which leads to an even more severe tuning of unrelated parameters. In gen­ eral, we find that the required values of and ANGff vary between 0 < < 0.12 and Chapter 8. Bounds on axion-like particles 120

[MeV]

Figure 8.7: Constraints from BBN (solid) and PLANCK (dashed) on axion-like particles for different reheating temperatures between Tr = 10 MeV (blue) and Tr = 109 MeV (red).

0.54 < ANGff < 3.8 for < 10 MeV and between 0.12 < < 0.35 and 3.8 < ANGff < 29 for m

10 MeV.

Effects of a low reheating temperature

Up to this point, we have assumed that the reheating temperature Tr after cosmic inflation reaches values above the Primakoff freeze-out temperature Tfo, thus ensuring that (p always starts out in thermal equilibrium. However, in case of a low reheating temperature, the ALP production merely proceeds via "freeze-in"4 and therefore is suppressed at large tempera­ tures, which potentially leads to weaker constraints. To quantify this effect, we consider dif­ ferent reheating temperatures in the viable range 10 MeV < Tr < 1019MeV [196, 197] and discuss their implications for the final constraints. To this end, we extend our model by an additional free parameter Tr and replace the initial condition from eq. (8.6) by

o,p)*/( = O (8.15) with To = Tr, i.e. we conservatively assume that there was no initial ALP abundance at higher temperatures (e.g. from inflaton decays). In fig. 8.7 we present the results of this procedure for different values of Tr, ranging from Tr = 10 MeV (blue) to Tr = 109 MeV (red). At this

technically, this is a UV"freeze-in", which explicitly depends on Tr. 121 8.3. Results and discussion point, it is important to note that we can explicitly compare these results with the ones from electromagnetic decays of light mediators in fig. 5.7; however, there are some differences be­ tween both figures due to the additional freeze-in contribution from Primakoff interactions. For Tr = 10 MeV (blue) and small masses, m$ < 20 MeV, this additional contribution leads to slightly stronger limits compared to fig. 5.7 with an exclusion line that roughly follows Tfo ~ Tr (cf. fig. 8.2). Above this line, the Primakoff interactions are no longer efficient after reheating, since the total freeze-in abundance of ty is roughly proportional to TR/Tfo. In contrast, for mty > 20 MeV, the constraints originating from photodisintegration become slightly weaker. This is because for Tfo Tr, the freeze-in contribution from Primakoff interactions is irrele­ vant and the one from inverse decays is suppressed compared to fig. 5.7 due to the finite value of Tr to (in ch. 5 we essentially assumed Tr to). Increasing the reheating temperature from Tr to TR > Tr then influences these constraints in two different ways: (i) The exclu­ sion line following Tr ~ Tfo for smaller masses rescales to Tr < TR Tfo (cf. the transition from blue to green), and (ii) the Primakoff freeze-in becomes more relevant for the photodis­ integration constraints, which leads to growing islands for mty > 20 MeV. Even though the corresponding freeze-in abundance « TR/Tfo is rather small in this region, i.e. for Tr » Tfo, photodisintegration still leads to strong constraints as it is also sensitive to very small abun­ dances (cf. orange/red). For very large values of TR, both regions eventually merge (cf. the transition from green to orange) and the previous results from fig. 8.5 are finally recovered for TR = 109 MeV. All in all, we therefore conclude that the constraints are indeed weakened for low reheating temperatures. However, it is worth noting that values as low as TR = 10MeV can certainly be considered extreme as it might be challenging to even develop a consistent cosmological picture that allows such small reheating temperatures.

Effects of more general coupling structures

Let us finally comment on axion models with more general coupling structures. In particular,

ALPs might also couple to the gluon field (Lty D gtyGGpvGpvty/4) or have derivative couplings to axial-vector fermion currents (Lty D gtyf (dpty)fY5YPf mfgtyftyfysf). The impact of such couplings onto cosmological observables has already been discussed in [185], and the limits are generally expected to be similar when interpreting Tty as the total lifetime of the axion-like particle. However, while the limits in the gray region of fig. 8.2 directly apply, the remaining parts of parameter space might face slightly stronger or weaker limits, depending on which effects are present. Specifically, additional couplings might lead to the following modifica­ tions:

• Larger production rates due to additional production channels, albeit possible interfer­ ences between Primakoff and Compton processes could occur.

• Smaller freeze-out temperatures of the ALP. If the freeze-out is shifted to temperatures for which ty has already gotten non-relativistic, the constraints become weaker as the total energy density of ty is smaller compared to the one of a decoupled particle. Chapter 8. Bounds on axion-like particles 122

• Other final states that might lead to additional modifications of the light-element abun­ dances. For m§ < 2m^, the only additional viable final-state particles beyond photons are electrons and neutrinos, both of which are suppressed due to their small Yukawa couplings. For m§ > 2m n, the ALP might also decay into hadrons, in which case the constraints become significantly stronger due to the onset of hadrodisintegration.

8.4 Summary

In this chapter, we have calculated BBN constraints for models with axion-like particles that feature masses in the keV - GeV range. Such particles are part of many models beyond the SM and can potentially explain the difference between the measured and the theoretically predicted anomalous magnetic moment of the muon as well as the unexpected resonances that are observed in certain nuclear transitions. In general, axion-like particles can not only be constrained by collider experiments and astrophysical observations, but also leave an im­ print on certain cosmological observables, in particular the light-element abundances that are created during nucleosynthesis. In this work, we have performed a detailed study of the latter constraints by including all effects that might potentially modify the primordial light­ element abundances, namely a modification of the Hubble rate due to the additional energy density, a modification of the time-temperature relation due to an exchange of heat between the SM and the dark sector, a non-standard time-dependence of the baryon-to-photon ratio, a modified neutrino-decoupling temperature, and the late-time modification of the nuclear abundances due to photodisintegration (cf. sec. 3.3 for a detailed discussion of all effects). In general, we find that the constraints closely resemble the ones from ch. 5 and therefore are strongest for sufficiently heavy and long-lived ALPs, but also generally exclude masses below m§ < 0.45 MeV, independently of the lifetime. To evaluate the robustness of these constraints, we have further discussed some modifications to the vanilla ALP scenario, all of which have the potential to weaken the final results severely. Specifically, we have concentrated on mod­ ifications that can be treated independently of the ALP sector, including the addition of extra dark radiation, a non-vanishing neutrino chemical potential, and a potentially low reheating temperature. While those effects can indeed weaken the final constraints, we also find that significant and robust limits remain, which nicely complement the results from laboratory experiments and astrophysical observations. 123

9 Conclusions

The remarkable agreement between the observationally inferred light-element abundances and their predictions within the SM puts severe constraints on any dark sector that is present at the time of nucleosynthesis (T ~ MeV — keV). In this thesis, we have calculated these constraints for different scenarios with MeV-scale particles, which are neither fully relativistic nor non-relativistic during all temperatures relevant for nucleosynthesis, a scenario that has not yet been extensively studied in the literature. To this end, we have derived a generic set of equations that can be used to calculate the light-element abundances and thus constraints for many different dark-sector scenarios. In this process, we have explicitly taken into account all relevant effects that might alter the creation of light elements in the early universe. Specifically, we have considered

1. a modified Hubble rate due to the additional dark energy density,

2. a modified time-temperature relation due to an increased Hubble rate and a potential in­ jection of entropy into the SM heat bath,

3. a modified best-fit value of the baryon-to-photon ratio due to the adjusted effective number ofneutrinos,

4. a modified neutrino-decoupling temperature due to an increased Hubble rate (even though this effect was only incorporated via the approximation of instantaneous neutrino de­ coupling),

5. late-time modifications of the nuclear abundances due to photodisintegration using the full solution of the cascade equation instead of the universal-spectrum approximation.

In ch. 4 we then first applied this formalism to the case of fully decoupled dark sectors with MeV-scale particles that are created in the early universe and later decay into dark radiation. We found that even fully decoupled dark sectors can be severely constrained if either (i) the temperature-ratio of the dark and the visible sector is of order unity, or (ii) the particle be­ comes non-relativistic prior to its decay and therefore acquires a large energy density relative to the one of the SM (cf. fig. 4.3). As a result, we found that the bounds are strongest for suf­ ficiently heavy and long-lived mediators and that it is even possible to constrain arbitrarily small lifetimes when considering sufficiently heavy mediators.

In ch. 5 we then considered the case of dark sectors with MeV-scale particles that are created in the early universe and later decay into electromagnetic radiation. We found that the final constraints can be very different from the naive order-of-magnitude estimate Tty < 1s, which is usually employed for such scenarios (cf. fig. 5.4). In fact, for very small mediator masses mty < 1 — 10 MeV, the inverse decay YY ty inevitably thermalizes the mediator with the SM heat bath, which allows to exclude arbitrarily small lifetimes. On an even more general note, Chapter 9. Conclusions 124 the inverse decay constitutes an irreducible freeze-in contribution that can be used to con­ strain sufficiently light and short-lived mediators, independently of their initial abundance (cf. fig. 5.7). However, even for larger mediator masses - for which the inverse decay is less relevant - we were still able to constrain lifetimes down to t$ ~ 0.1 s, which is still one order of magnitude smaller than the naive estimate.

In ch. 6 we calculated the corresponding constraints for MeV-scale dark-matter particles that can annihilate into all kinematically available SM states, including electrons/positrons, pho­ tons, and neutrinos. In this context, we first updated the bound on the minimal mass for thermal dark matter and found that BBN and CMB independently exclude masses below mx < 7 — 10 MeV (cf. fig. 6.2). These results were then combined with constraints on the annihilation cross-section of dark matter, which we calculated for both s-wave and p-wave annihilations. In the former case, we found that the CMB constraints from exotic energy injec­ tions are much stronger than the ones from nucleosynthesis. However, in the latter case, the constraints from BBN turned out to be significantly stronger than the ones from the CMB and even competitive with the most recent bounds from cosmic-ray observations (cf. fig. 6.6).

In ch. 7 we then applied our results from ch. 5 to models with self-interacting dark matter and scalar mediators that couple to the SM fermions via Higgs-like couplings. As a result, we found that most parts of parameter space with sizable dark-matter self-interactions are already excluded by a combination of direct-detection experiments and constraints from nu­ cleosynthesis (cf. fig. 7.4). More precisely, we found that only a very small region of parameter space remains valid, which is located between m$ ~ 1 — 3 MeV and mx ~ 0.2 — 0.6 GeV (cf. fig. 7.6). However, most of this small region will be fully testable in the future with upcoming low-threshold direct-detection experiments like CRESST-III.

Finally, in ch. 8 we applied our results to models with axion-like particles that couple exclu­ sively to photons. In general, we found that the resulting bounds are very similar to the ones from ch. 5 and consequently are strongest for sufficiently heavy and long-lived ALPs, but also constrain masses below m$ < 0.45 MeV, independently of the lifetime (cf. fig. 8.5). To evalu­ ate the robustness of these constraints, we further allowed various additional effects that may weaken the results of the standard scenario. Specifically, we have concentrated on modifica­ tions that factorize from the ALP sector, including extra dark radiation, non-vanishing neu­ trino chemical potentials (cf. fig. 8.6), and a potentially low reheating temperature (cf. fig. 8.7). While we found that those effects can indeed weaken the final bounds, very relevant robust constraints remained, which nicely complement the results from laboratory experiments and astrophysical observations (cf. fig. 8.6).

Overall, the studies that have been performed in this thesis can be used further to constrain the overwhelming set of possible dark-matter models and therefore help to unravel the mys­ tery of DM itself. In particular, we hope that these results facilitate future experimental and theoretical studies, thus assisting the search for an unambiguous sign of dark matter. Appendix

127

A Reaction tables

A.1 Relevant reactions for Big Bang Nucleosynthesis

Ref.

[ 98 ] P + n D + Y [ 99 ] D + n 3H + Y [ 98 ] D + p 3He + Y

[ 98 ] D + D 3H + P [ 98 ] D + D 3He + n

[ 00 ] D + 4He 6Li + Y [ 98 ] 3H + p 4He + Y [ 98 ] 3h + D 4He + n

[ 98 ] 3h + 4He 7Li + Y

[ 98 ] 3 He + n 3H + P [ 9S ] 3 He + n 4He + Y 4He [ 0( ] 3 He + D + P [ 98 ] 3 He + 4He 7Be + Y [ 0 1 ] 6Li + n 7Li + Y [ 0( ] 6Li + n 4He + 3H

[ 00 ] 6Li + p 7Be + Y [ 00 ] 6Li + p 4He + 3 He [ 00 ] 6Li + 4He 10B + Y

A complete list of all reactions that are used in ALT ER B B N can be found in [49, p. 11, tab. 1]. Appendix A. Reaction tables 128

A.2 Relevant reactions for Photodisintegation

Ref. Eth [MeV]

[ ] D + Y - p + n 2.22 [ ] 3H + Y - D + n 6.26 [ ] 3h + Y- p + n+n 8.48 [ ] 3He + Y- D + p 5.49 [ ] 3He + Y- n + p+p 7.12 [ ] 4He + Y- 3H + p 19.81 [ ] 4He + Y- 3He + n 20.58 [ ] 4He + Y- D + D 23.84 [ ] 4He + Y- D + n+p 26.07 [ ] 6Li + Y- 4He + n+p 3.70 [ ] 6Li + Y- X + 3A 15.79

In the following fig. A.1 we further show the cross-sections for some of these reactions as a function of energy.

Figure A.1: Cross-section of different photodisintegration reactions as a function of energy. 129

B Rates for the cascade processes

In this appendix, we collect for reference all relevant total and differential interaction rates rx (E) and Kx-x (E, Ef) for the cascade processes of high-energetic photons, electrons, and positrons on the background photons, electrons, and nuclei (see eqs. (3.86) and (3.89)).

Target densities

The thermal photon spectrum f7 (e) is given by

e2 1 / (B.1) fY (

nb(T) = n X nY(T) = y X ’ n3^T3 . (B.2)

For the number density of background electrons ne(T) we thus obtain

ne(T) = £ZNnN [Yp(T) + 2Y4h(T)] x nb(T), Yn(T) = nNT) . (B.3)

At the times relevant to photodisintegration (t > 104 s), BBN has already terminated and the nuclear abundances YN(T) are approximately constant. Hence, in the following, we neglect the temperature dependence ofYN(T), and fix them to their values directly after BBN.

Final-state radiation: DS e+e 7

Following [202], the source term for final-state radiation can directly be calculated from the source term of electrons or positrons via the expression [203, 204]

£ 1 + (1 — x)2 j/ 4E0 (1 — x) x 0 (* —M—x) S(YFSR)(E) = (B.4) n x \ m’ with x = E/ E0.

Double photon pair creation: YYth e+e

The rate for double photon pair creation is given by [24]1

r!DP) E = J. X £/E d 3 X C ds s • '■ß 1 — 4me /s (B.5)

1Correcting a typo in eq. (27) of [24]. Appendix B. Rates for the cascade processes 130 with the total cross-section

This process is only relevant above the threshold of production of electron-positron pairs E > m%/ (22T), allowing us to set r7DP)(E) = 0 for E < m2 / (22T). The differential rate for double photon pair creation entering the calculation of the electron and positron spectrum2 was originally calculated in [205] and is given by3 22 1 , ~ j-f?(e) (E, E' ) = x 3 dt 2 , E', e) , (B.7) E'3 m2JE' e2 with 4(E' + e)2 l (4eE(E' + e - E)\ G(E, E', e) = E(E' + e - E) ln m2(E' + e)

+ m_____ . (E' + e)4 + e(E' + e) E2(E' + e - E)2

2 [2e(E' + e) - m2] (E' + e)2 8e(E' + e) + m2E(E' + e - E) m2 (B.8) for me < E-m < E < EUm,

2E±m = E' + e ± (E'-e)^J1 - ml f (B.9) and G(E, E', e) = 0 otherwise. As explained above, we further set K^D^i (E, E') = 0 for

E' < m2 /(22T).

Photon-photon scattering: YYth YY

The total and differential interaction rate for photon-photon scattering have been originally calculated in [206], and are given by4

8n4 4 r(PP) (E) = 1946 63 e 17 (E) = 50625n a4 me x / (B.10) and

21 2 2 ' a4 8n4 T6 1 E (E) KK((PP) (E eE ) = 1112 x 8 x x E'2 (B.11) E, 10125n m8e 63

In principle, these expressions are only valid for E < m2/T [24]. However, for energies larger than this, photon-photon scattering is in any case negligible compared to double photon pair creation, making it unnecessary to impose this additional constraint.

2Here, the notation 7 e± in the index of Kx- ,x indicates that the corresponding expression is valid for x' x G {7 e+, 7 e- } and consequently enters eq. (3.89) twice. 3Correcting a typo in eq. (28) of [24]. 4Correcting a typo in eq. (31) of [24] and in eq. (5) of [25]. 131

Bethe-Heitler pair creation: yN Ne+e

The total rate for Bethe-Heitler pair creation at energies E > 4me and up to order m^ / E2 can be written as [24, 207]5

Oe — m * (E Z:-. m) [y^a) - ™

|ln(2k)3 - ln(2k)2 + 6 - ln(2k) + 2Z(3) + - 7 (B..12)

4 D e k=E/m Here, we only take into account scattering off 1H and 4He, which implies

E ZNnN(T) - E Z^n(T) = [Yp (T) + 4Y4^ (T)] * n (T) , (B.13) N N e {p,4He} since the abundances of all other nuclei are strongly suppressed. Furthermore, for energies in the range 2me < E < 4 MeV, the interaction rate is essentially constant [29]: rYBH) (E) — (E = 4 MeV). The differential rate for Bethe-Heitler pair creation is given by [24, 208]

KY2( ± (E, E') = (EZN«n(T)J * B E E) * 0(EZ- E - me) , (B.14) with the differential cross-section

d^BH (Ez E>) = a\ / p+ p- \ 4 - 2E+ E-HO dE m2 * E'3 * 3 p2+ p2

6-+ l+ 1+1- + m 2e p3 p3+ p+ p-

8E+E E'2 +L 3------+ E2+ E2- + p2+p2- - me2E+E- p3+ p3” 3p+p-

E E-3 p2+ + -E- 3- p2 \ 2m|E + - 1 e+ (B.15) 2p+p- p3+ p3- where we have defined

E- := E , E+ := E'- E , P± := E± - m2 (B.16)

(B.17) E+ E- + p+ p- + m2 E± + p± L := ln / 1± := ln E+ E- - p+ p- + m2 E± - p±

5We checked that higher order terms do not change the final results. Appendix B. Rates for the cascade processes 132

The ©-function appearing in eq. (B.14) ensures that we fulfill energy conservation in the inte­ gration of Ef over the range [E, to] in eq. (3.89).

Compton scattering: Yeth Ye

The total rate for Compton scattering can be found in [24, 25] and is given by

(CS) 2na2 1 1 8 1 rY (E) — 2 x ne(T) x (1 -4 - 42)ln(1+x) + m2e x x 2 + x 2(1 + x)2 x—2E/me (B.18)

Furthermore, the differential rate for the energy of the scattered photon reads [24, 25]6 2 KCSY (E, E') — ©(E - E' /(1 + 2E'/me)) x x ne(T)x me 1 \E' E / me me\2 (1 1 \ E'2 E + E' + l E eJ me\E E') / (B.19) with the ©-function corresponding to the vanishing of the rate above the Compton edge. On the other hand, following [24], the differential rate relevant for the spectrum of elec­ trons can be deduced from eq. (B.19), namely

(E, E') — kYY7 (E' + me - E, E') . (B.20)

Inverse Compton scattering: e± Yth e±y

The differential rate for production of photons from inverse Compton scattering was origi­ nally calculated in [209] and can be written as

Y(E, E')— 2na2 x 1. £ de YY F(E, E',e) x 0(E' - E - me) . (B.21)

For e < E < 4eE'2 / (m2 + 4eE'), the function F(E, E', e) is given by7

r2 q2 F(E, E', e) — 2q ln(q) + (1 + 2q)(1 - q) + (1 - q) , (B.22) with

— 4e E' — E r (B.23) e m2e ' q re (E'- E) ' and F(E, E', e) — 0 otherwise.8 Again, the ©-function in eq. (B.21) ensures energy conserva­ tion upon integration of E' over the range [E, to].

6Correcting a typo in eq. (10) of [25]. 7Correcting a typo in eq. (49) of [24]. 8 According to [209], the function F (E, E', e) takes a different form for E < e. However, this part of parameter space is practically irrelevant for our considerations. 133

The total rate for inverse Compton scattering entering the calculation of the electron and positron spectrum is given by [24, 209]9

(E) = 2na2 * -1 dE7 de F(E7, E, e) . (B.24) 1 E2 Jo 1 Jo e '

Finally, the differential rate for the production of electrons and positrons can be written as [24, 2o9]

k|±c^e± (E, E') = 2na2 * £ de fYee) f(E' + e - E, E', e) . (B.25)

Additional processes not considered in our calculation

Other processes such as

• Coulomb scattering e±e- e±e- and Ne- Ne-,

• Thompson scattering N7th Ny,

• Magnetic moment scattering Ne- Ne- or

• Electron-positron annihilation e+e- 77 are suppressed by the small density of background electrons or nuclei ne, nN n7 and can therefore be neglected.

9Correcting a typo in eq. (48) of [24]. Appendix B. Rates for the cascade processes 134 135

C Numerical solution techniques

In this appendix, we give some more details on the numerical procedures that we use in the context of our general solution strategy described in sec. 3.4.

C.1 Electromagnetic cascade

To solve eq. (3.89) numerically, we can exploit the fact that all spectra vanish for E > E0 . Moreover, we are only interested in solving the equation up to some minimal energy Emin , which is set by the lowest threshold energy of the photodisintegration reactions and in this work we specifically set Emin = 1.5 MeV. Given the relevant energy range [ Emin, E0 ], we then define a grid of energies e := eyi with i G {0,..., N}, (e0, £n) = (Emin, E0), and an equidistant spacing in y, i.e. Ay = [ln(Eo) — ln(Emin)] /N. Here, we explicitly set N = 200, which is sufficient to ensure convergence for all relevant parameter points. By evaluating eq. (3.89) at each individual grid point we then find

K(€i, Eo)sX0) /• ln(E0) rx (ei )Fx (ei) = SX ) (ei) + £ + dyeyKx'^x (ei, ey )F*/ (ey) ' ( ) x' rx (E0) ./ln ei

"K x(ci, E0)SX0) sXFSR) (c) + £ + ~2 2 £ ejKx'—x (ei, ej)Fx' (ej) x _ rx' (E0 ) 2 j=i+1

+ eiKx'—x (ci, ci )Fx' (ci ) + E0 Kx' -x (cu E0 )Fx' (E0 ) (C.1)

In the last step, we have used the trapezoidal integration rule and the sum £jN=i+1 is under­ stood to vanishes for i = N — 1. This expression is valid for i < N, while for i = N we simply have

SxFSR)(E0) + £ Kx'-x(E0,E0)S(0 Fx(E0) = (C.2) rx(E0) + £ rx(E0)rx'(e)

Assuming that Fx(Ej) has already been calculated for j > i, eq. (C.1) can be interpreted as a linear system ofthree equations for the unknown variables Fx(Ei). Consequently, by defining

F(ei) := [F7(ei),Fe— (ei),Fe+ (ei)]T,wehave

F (ei) = a(ei) + B(ei )F (ei) (C.3) with 1 r Kx-x (ei, E0 )sx) Ay N—1 la(ei)] x + ~2 2 £ ejKx -x (ei, ej)Fx' (ej) rx ( i) £ _ rx' (E0) e x' 2 j=i+1 S(FSR) x (ei) + E0 Kx'—x (ei, E0)Fx' (E0) + / (C.4) rx (ei)

Ay eiKx—x (ei, ei) )] (C.5) [B(et xx' 2 rx (ei) Appendix C. Numerical solution techniques 136

Given the knowledge of [a(ez-)]x and [B(q)] ,, the linear eq. (C.3) can then be solved by using standard techniques to calculate the values of F(ez-). However, [a(ez-)]x and [B(q)] , explicitly depend on F(ej) for j > i. Hence, to determine F(E) for all energies, we start with eq. (C.2) for i = N and use it to calculate F(cn-1 ), which is then used together with F(eN) to determine F (cn-2 ) and so on.

C.2 Non-thermal nucleosynthesis

To solve eq. (3.95) numerically, we first define

rr(t) := Jo fy(t, E)ar(E) dE , (C.6) which transforms eq. (3.95) into the form

Yx(t) = E Yj(t)rjY^x(t) - Yx (t) E1 X (t) . (C.7) j j'

Consequently, by defining Y(t) := [Y„(t), Yp(t), Yd(t),... and after substituting t T by means of the time-temperature relation, we find

= r(t)y (t) (c.8) with

[R(T)]XX := dT * rx'7^x(T) - X'xx E' xy - (t) . (C-9)

Here, the matrix [R(T)] xx' can be obtained via numeric integration, and eq. (C.8) is an ordi­ nary system of differential equations, which is solved by

Y(T) = exp R(T') dT') Yo (C.10)

with the initial condition Y(To) = Yo at To = 102T(t$), i-e- sufficiently long before the decay of the particle. Here, the function exp (•) is the matrix exponential, which can be evaluated nu­ merically by (i) diagonalizing the matrix Jto R(T')dT' with the corresponding unitary trans­ formation UR(T), (ii) taking the exponential of the diagonal matrix by exponentiating each eigenvalue individually, and (iii) transforming the resulting matrix back using UR-1(T). 137

D Additional plots regarding dark-matter annihilations

mx [MeV] mx [MeV]

Figure D.l: Constraints from photodisintegration due to p-wave annihilations of Majorana dark matter into photons in the m% — b parameter plane for different values of decoupling temperatures T^j. The meaning of the differently colored regions is identical to the one used in fig. 6.3. Appendix D. Additional plots regarding dark-matter annihilations 138

Figure D.2: Constraints from photodisintegration due to s-wave (top) and p-wave (bottom) annihilations of Majorana dark matter into photons. The meaning of the differently colored regions is identical to the one used in fig. 6.3.

In this appendix, we complement the discussion of scenario (SI) from sec. 6.3.3 by showing the results for Majorana dark matter that decays exclusively into photons instead of electron­ positron pairs. In fig. D.2, we compare the results for s-wave and p-wave annihilations, while fixing 7^ — 100 eV in the latter case (in analogy to fig. 6.3), and in fig. D.l we show the p-wave results for different kinetic-decoupling temperatures ranging from Tkd = 10 eV to T^ — 1 MeV (in analogy to fig. 6.4). 139 Bibliography

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Acknowledgments

“There's no point in being grown up ifyou can't be childish sometimes.” - The Doctor

First of all, I am very grateful to Kai Schmidt-Hoberg for his excellent supervision, many inter­ esting discussions, and for giving me the opportunity to graduate at DESY in the first place. Not only did you always provide me with excellent research topics, but you also gave me the necessary freedom to evolve into an independent researcher. I sincerely appreciate that you never treated me like a less-experienced physicist but always made me feel like a valuable member of your group. You did an awesome (prima...koff) job, and you can rest assured that I will never forget the meaning of the word dedication. :P

I am also thankful to Geraldine Servant for agreeing to co-referee my thesis, to Jochen Liske, Gu- drid Moortgat-Pick and Torben Ferber for joining my disputation committee as well as to Andreas Ringwald for agreeing to join, even though it did not work out in the end.

I would also like to thank all of my collaborators. First and foremost, a special shout out to Sebastian Wild for essentially being an additional supervisor and for always taking the time to answer my questions. Your support really made me grow as a physicist. A special “Thank You” also goes to Torsten Bringmann for coming up with exciting new ideas and for expanding my knowledge via many interesting discussions. I also want to thank Camilo Garcia Cely for giving me helpful advice on countless occasions.

Thanks to all of my friends and office mates at DESY for giving me such a fantastic time. A special “Thank You” goes to Frederik Depta for being an awesome office mate and a valuable collaborator. Thanks for enduring my (mostly pretty amazing) jokes and countless wire at­ tacks. I am well aware of your inner desire to throw stuff at me, and for that, I forgive you. Thanks to Jonas Wittbrodt for being my personal programming tutor, Pascal Stienemeier for hosting amazing board-game evenings, Felix Giese for enduring (and most likely also enjoy­ ing) several Marvel movies, Ilija Buric for never succeeding in convincing me that Dr. Zhivago is better than The Doctor (but nice try anyway), Thibaud Vantalon for reminding me that it is best to always lock my computer and Janis Kummer for giving me a warm welcome on my very first day at DESY.

Thanks to Fabian Prolingheuer for proofreading my thesis. I really appreciate the effort you put into correcting my (repeated) grammar mistakes. :P Also, I am grateful to Margitta Kircher for helping me to get through some difficult times.

A very special “Thank You” to Tabea Armbrust, Yannick Baumer, Dennis Willsch, Madita Willsch and Bernhard Klemt for being the best friends I could have ever wished for. You helped me become the best version of myself, and I would not stand where I stand today without the support, the love, and the laughter that you brought into my life. I am grateful for every sec­ ond that we have spent together, and I am very much looking forward to all the ones that are yet to come.

Lots of love and thanks to my family. Thanks to my mum, my dad, my sister Kathrin, and my brother Patrick for their relentless support and for encouraging me to always pursue my dreams. This work is as much your achievement as it is mine. You were with me every step of the way, giving me love, courage, and a hand to hold, without ever asking for anything in return. I am also grateful for all the joy that my niece Malia and my nephew Linus brought into my life. You might still be too young to understand, but you already made such a difference in my life. Your smile was there when I needed it the most. I love you, all of you!

Eidesstattliche Versicherung

Hiermit versichere ich an Eides statt, die vorliegende Dissertationsschrift selbst verfasst und keine anderen als die angegebenen Hilfsmittel und Quellen benutzt zu haben.

Die eingereichte schriftliche Fassung entspricht der auf dem elektronischen Speichermedium.

Die Dissertation wurde in der vorgelegten oder einer ähnlichen Form nicht schon einmal in einem früheren Promotionsverfahren angenommen oder als ungenügend beurteilt.

Hamburg, den 30.04.2020

(Marco Hufnagel)