Production of Sterile Neutrino Dark Matter at Large Lepton Asymmetry

Transcription

1 Production of Sterile Neutrino Dark Matter at Large Lepton Asymmetry Master Thesis submitted to the Faculty of Physics, Bielefeld University by Mandy Maria Wygas Supervisor and 1. assessor: Prof. Dr. Dietrich Bödeker 2. assessor: Prof. Dr. Dominik Schwarz Bielefeld, September 2015

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3 Abstract The constituents of Dark Matter (DM) are one of the puzzling questions in physics. In this thesis we are considering a sterile neutrino DM candidate with mass in the kev range. Thus, we discuss the cosmological constraints on the properties of sterile neutrinos and the mathematical classification of sterile neutrinos. The production rate for the in the presence of a large primordial lepton asymmetry resonantly produced sterile neutrinos will be derived in the framework of thermal field theory. The resonance and the impact of the non-thermal, i.e. colder, spectrum of the sterile neutrinos on it is studied. Furthermore, the impact of the large primordial lepton asymmetry on the lepton, baryon and charge chemical potentials µ around T QCD 200 MeV is investigated. For the analysis a grand canonical description up to O(µ 2 ) of the pressure is used. As a result, we obtain a linear dependence of the chemical potentials on the lepton asymmetry. Our numerical results for the dependence of the chemical potentials on the lepton asymmetry and the temperature, however, show a deviation to known results obtained by a different method. The image on the front page is property of symmetry magazine ( org/image/image-neutrinoscosmic). MW thanks symmetry magazine for the permission to use it in this thesis.

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5 CONTENTS Contents 1 Introduction 3 2 Cosmological Motivation The Universe Today Evolution of the Universe Dynamics of Cosmological Expansion Sample Cosmological Solutions Thermodynamics Entropy and Conserved Quantities Concept of Sterile Neutrinos Motivation Production in the Early Universe Mathematical Classification Cosmological Constraints on Sterile Neutrino Properties Impact of Sterile Neutrinos on the Evolution of the Universe Dark Matter Decay kev X-ray Line Basic Concepts of Thermal Field Theory Motivation Basic Equations for Fermions Two-point Correlation Functions Relations between Correlation Functions Resonant Production Rate of Sterile Neutrinos Derivation of the Production Rate Active Neutrino Spectral Function and Euclidean Propagator Relic Dark Matter Density Nonequilibrated Leptonic Flavours Point of the Resonance Order of Magnitude of the Resonant Production Comparison of R(T R ) with other Rates Hubble Parameter Active Neutrino Scattering Rate Impact of the Non-thermal Spectrum of Sterile Neutrinos Lepton Asymmetries and Chemical Potentials Grand Canonical Ensemble and Partition Function Equation of State and Net Particle Number Densities Derivation of the Susceptibilities Numerical Results for Large Lepton Asymmetries

6 CONTENTS 8 Conclusions 74 A Appendix 76 A.1 Results for the Non-thermal Spectrum A.2 Mathematica NB for Chemical Potentials A.3 Remarks B References 85 2

7 1 Introduction The basic pillars of modern physics are the Standard Model of elementary particle physics (SM) and the theory of General Relativity (GR). In the SM the electroweak and strong interactions of elementary particles are described by the underlying gauge group SU(3) C SU(2) L U(1) Y and are mediated by gauge bosons. The gravitational interactions are described in the framework of GR. Together the SM and GR can explain almost all phenomena observed in nature. Nevertheless, this cannot be a complete theory of nature. First, GR is based on a classical background, whereas the SM is a Quantum Field Theory (QFT). There is a need for an extended theory which combines both at a quantum background. Second, there are experimental observations, which cannot be explained in the framework of the SM and GR. Already in 1933, while observing the Coma galaxy cluster, the Swiss astronomer Fritz Zwicky claimed the existence of unseen matter, which he called dunkle Materie, dark matter (DM). He calculated that the gravitational mass of the galaxies has to be at least 400 times larger to receive a galaxy cluster which is bound together than one would expect due to their luminosity. Consequently most of the matter has to be dark [1]. Today, due to analyses and observations of the Cosmic Microwave Background (CMB) by the spacecrafts WMAP (Wilkinson Microwave Anisotropy Probe) and Planck, we know that the largest amount of the content of the universe is formed by 68 % dark energy (DE) and 27 % DM [2, 3]. Therefore, one of the most urgent questions in Cosmology and Particle Physics is the origin and content of DE and DM. As they cannot be described within the framework of the SM and GR, this requires physics beyond the SM. Several models have been developed to describe DE and DM. In this thesis we are concerned with the resonant thermal production of sterile neutrino DM. Sterile neutrinos are an extension to the SM particle content, which can solve several so far unsolved problems within the SM (cf. [4]). We are scrutinizing the possibility of a sterile neutrino DM candidate with mass in the kev-range, which is resonantly produced in the existence of a large primordial lepton asymmetry [5]. Therefore, we will discuss some general aspects of cosmological evolution and dynamics concerning sterile neutrinos in Chapter 2. In Chapter 3 we will introduce the concept of sterile neutrinos, i.e. their production, mathematical classification and cosmological constraints. The thermal field theory methods used for the calculation of the production rate of sterile neutrinos are explained in Chapter 4. In Chapter 5 the production rate of sterile neutrino DM is derived. The resonance of this production rate is examined in Chapter 6. The impact of a large primordial lepton asymmetry on the chemical potentials is investigated in Chapter 7. Throughout the whole thesis natural units and Einstein summation convention are used, i.e. c = = k B = 1 and repeated indices are summed over, unless otherwise stated. 3

8 2 COSMOLOGICAL MOTIVATION 2 Cosmological Motivation In this Chapter we outline some basic concepts of cosmology to motivate and pinpoint the concept of sterile neutrinos in this framework. Doing so, we want to allude to important cosmological epochs which become important while examining the production of sterile neutrinos. Therefore, we study the important aspects of temperature, expansion and composition of the universe today in Sec Furthermore, we give a brief overview of the evolution of the universe in Sec. 2.2 and the dynamics of cosmological expansion in Sec The Universe Today The present universe, more specific its observable part, is homogeneous and isotropic at large spatial scales. Homogeneity means that at scales larger than the largest structures in the universe (i.e. superclusters of galaxies, gigantic voids) all parts of the universe look the same. Isotropy means that there are no special directions in the universe (cf. [6]). Homogeneity and isotropy combined, form the cosmological principle, which guarantees that observations from Earth are representative for the whole universe and thus can be used to test cosmological models. However, homogeneity and isotropy do not imply that at each moment of time a 3-dimensional space is Euclidean (i.e. the universe has zero spatial curvature). Besides a 3-plane also a 3-sphere (positive spatial curvature) and a 3-hyperboloid (negative curvature) are homogeneous and isotropic spaces. Observational bounds, obtained by the temperature anisotropy of the CMB, can be interpreted as the spatial curvature of the universe is much greater than the size of the observable part of the universe. Therefore, the 3-dimensional Euclidean space is a very good approximation, especially locally [2]. The present CMB temperature is according to Ref. [7] T 0 = ± K. (2.1) Furthermore, the universe expands, so the distances between galaxies increase. One introduces the scale factor a(t), dependent on time, to describe this expansion. The distance between two far away objects is then proportional to a(t). The expansion rate is described by the Hubble parameter H = ȧ(t) a(t), (2.2) where ȧ(t) = da(t) dt denotes derivative with respect to time. The present value of the Hubble parameter is H 0 = H(t 0 ) = (67.8 ± 0.9) km s Mpc [3], where the subscript 0 of a quantity always indicates the present value. 4

9 2.1 The Universe Today The Hubble parameter has dimension [t 1 ], so the order of magnitude estimate for the age of the universe in natural units is given by H yrs. (2.3) A bold extrapolation of the cosmological evolution back in time leads to the Big Bang, the moment at which the classical evolution begins (cf. Sec. 2.3). The age of the universe is then the time passed since the Big Bang and the size of the observable part (i.e. the horizon size) is the distance travelled by signals emitted at the Big Bang and moving with the speed of light. The actual size of the universe however is larger than the horizon size, the spatial size may be infinite within the concept of GR (cf. [6]). The energy density in the universe in natural units can be obtained by dimensional analysis as ρ 0 H0 2G 1 = MP 2 l H2 0, where the gravitational constant G is related to the Planck mass via M P l = G 1/2 = GeV. The present energy density, also called critical energy density, in a spatially flat universe is given by which is numerically [3] ρ c = 3 8π H2 0 M 2 P l, (2.4) 6 GeV ρ c cm 3. (2.5) The energy density is composed of different constituents. This can be seen in the Friedmann equation [6] (ȧ(t) H 2 a(t) ) 2 = 8π 3 G (ρ M + ρ rad + ρ Λ + ρ curv ), (2.6) which relates the total energy density ρ tot = ρ M + ρ rad + ρ Λ and spatial curvature 8π 3 Gρ curv to the rate of the cosmological expansion (cf. Sec. 2.3). Here ρ M, ρ rad, ρ Λ are the energy densities of non-relativistic matter, relativistic matter (radiation) and dark energy, respectively. Typically the relative contributions of the different constituents to the energy density are parametrized by the density parameters Ω M = ρ M,0 ρ c, Ω rad = ρ rad,0 ρ c, Ω Λ = ρ Λ,0 ρ c, Ω curv = ρ curv,0 ρ c, (2.7) where ρ x,0 represents the present values of the energy densities, and ρ c according to Eq. (2.4). As a result of Eqs. (2.6) and (2.4) follows Ω i Ω M + Ω rad + Ω Λ + Ω curv = 1 (2.8) i at the present epoch. Due to observational results, the numerical values read [3, 6]: Ω M 0.32, (2.9) Ω Λ 0.68, (2.10) Ω rad = , (2.11) Ω curv < (2.12) 5

10 2 COSMOLOGICAL MOTIVATION One can see that the impact of relativistic particles (Ω rad ) and the spatial curvature (Ω curv ) is negligible today. One could also consider the density parameter of neutrinos [3] Ω ν < (2.13) Ω ν is small today and thus negligible, but it was important at former times (cf. Sec. 2.2). Furthermore it is hard to observe, because neutrinos are only weakly interacting. Our universe today consists of non-relativistic matter and mainly of dark energy. The present energy density of the non-relativistic matter can be subdivided into the sum of the mass densities of baryons, i.e. ordinary matter, and dark matter with the numerical values today[3] Ω M = Ω B + Ω DM, (2.14) Ω B = , Ω DM = (2.15) The components of dark energy and dark matter, which constitute the main part of the content of the universe today, are not known yet. The main difference between dark matter and dark energy is that dark matter is capable of clustering and most probably made of new stable particles which are non-relativistic. Whereas dark energy is homogeneously spread over space and is an unconventional form of energy of vacuum type (cf. [6]). There exist several different hypotheses for the constituents of dark matter and sterile neutrinos are one of them. 2.2 Evolution of the Universe The most convenient model which describes the evolution of the universe with dark energy and dark matter is called ΛCDM (Lambda Cold Dark Matter) model of cosmology. Therein the universe contains cold dark matter (CDM), i.e. particles which were non-relativistic at the time of decoupling, and a cosmological constant Λ which is associated with dark energy. Because the universe expands, it has been warmer and denser in the past. The Hot Big Bang Theory is the theory of such a universe. Back in time, or up in temperature, there existed important stages in the cosmological evolution (cf. Fig. 2.1), which we will briefly discuss here (cf. [6]). Recombination Recombination occurred at temperature T r = 0.26 ev 3000 K or equivalently at time t r 370, 000 yrs (cf. Sec. 2.3). This is an important epoch, since before recombination the photons scattered off free electrons in the baryon-electron-photon plasma and afterwards the ordinary matter was in the state of neutral gas, mainly hydrogen, 6

11 2.2 Evolution of the Universe Fig. 2.1: Stages of the evolution of the universe [6]. which was transparent to photons. The CMB, which we can observe today, comes from this recombination epoch and therefore carries information of the universe at this epoch. The CMB shows a very high degree of isotropy, which mirrors that the universe was highly homogeneous at recombination. Just small temperature fluctuations, which were caused by energy density perturbations (δρ/ρ ), are observed in the CMB. But these small density perturbations led to structure formation: first stars, then galaxies and then clusters of galaxies (cf. [8]). Before this recombination epoch there was the important transition from radiation to matter domination in the universe at T rm = 0.76 ev, which corresponds to the time t rm = 57, 000 yrs. Big Bang Nucleosynthesis (BBN) At much higher temperatures, whose order of magnitude is roughly determined by the scale of the binding energies in nuclei (1 10 MeV), protons and neutrons were free in the cosmic plasma. After the universe cooled down, neutrons were captured into nuclei. Resulting, light nuclei up to lithium-7 were formed in the primordial plasma. Heavier elements were not synthesized in the early universe, but during stellar evolution. The BBN epoch lasted from about 1 to 300 seconds after the Big Bang at temperatures T BBN 50 kev 1 MeV and it is the earliest epoch studied directly so far. 7

12 2 COSMOLOGICAL MOTIVATION Neutrino decoupling Neutrino interactions with the cosmic plasma terminated at T 2 3 MeV. Before, they were in thermal equilibrium with the rest of matter and afterwards they freely propagate through the universe. The role of neutrinos in the present universe is not very important, but the neutrino density in the early universe is an important parameter for the BBN, since the neutrino component affects the expansion rate through the Friedmann equation. Furthermore, relic neutrinos play some role in structure formation (cf. [6]). Cosmological phase transitions For even higher temperatures the epochs have not been directly probed, so one has to perform extrapolations. It is likely that the history of the universe goes back to temperatures of hundreds of GeV and even higher. At these temperatures there are some epochs of phase transitions. Starting from the latest, i.e. coldest, they are: QCD (Quantum chromodynamics) transition The QCD transition (or smooth crossover cf. e.g. [9]) is the transition from quark-gluon matter to hadronic matter. This temperature is determined by the energy scale of strong interactions and is about T QCD 200 MeV. For higher temperatures quarks and gluons behave as individual particles whereas for lower temperatures they are confined in colourless bound states, the hadrons. As a very important consequence the effective number of degrees of freedom g eff (T ) changes strongly during this transition (cf. [9]). As g eff (T ) enters the equation of state, it also influences the expansion rate through the thermodynamic pressure and thus the time of decoupling of DM and with that its relic density (cf. Sec. 2.3). At almost the same time, there was the transition associated with chiral symmetry breaking. Electroweak transition The electroweak transition at T ew 100 GeV results in the present phase with broken electroweak symmetry, Higgs condensate and massive W- and Z-bosons. Grand Unification transition What happened at even higher temperatures is very speculative. There are some hints towards Grand Unification at energies and temperatures T > GeV. If it was like this and if such temperatures existed in the early universe, then there was the corresponding phase transition at T GUT GeV. But it could also be possible that the maximum temperature of the universe was below T GUT and the Grand Unification phase did not exist. This is in fact the case in some models of inflation, where the reheating temperature is below T GUT (cf. [6]). 8

13 2.2 Evolution of the Universe Generation of baryon asymmetry The present universe contains baryons and practically no antibaryons. abundance is the ratio of baryon net and photon number density [10], The baryon η B = n B n B n γ = (2.16) The baryon number is conserved for sufficiently low energies which means that in the early universe the baryon asymmetry η B was of the same order of magnitude as now. That is why the baryon asymmetry is one of the most important parameters of cosmology. One big problem of cosmology however is to explain the very existence of such a baryon asymmetry as well as its value. It is not intuitive that the initial condition of the universe is not baryon-symmetric. The requirements for a baryon asymmetry are given by the three Sakharov conditions [11]: baryon number B violation, C-symmetry and CP -symmetry violation and interactions out of equilibrium. The problem of baryon number asymmetry cannot be solved within the SM. However there are many possible mechanisms of generation. One possible ansatz is baryogenesis via leptogenesis (see Refs. [12, 13]). Generation of dark matter It is not known of which particles non-baryonic clustered dark matter consists. It is assumed that there exist stable or almost stable particles which are not known in the SM. 1 Because of a lack of experimental information it is not possible to give a unique answer to the question of the mechanism of dark matter generation. One ansatz are Weakly Interacting Massive Particles (WIMPs). These are hypothetical stable particles with masses in the GeV-TeV range and annihilation cross sections comparable to weak cross sections, which did not totally annihilate in the cosmological evolution. Other possible dark matter particle candidates are axions, gravitinos, sterile neutrinos, etc. Depending on which model one considers, the generation of dark matter occurs at different points, i.e. temperatures, in the evolution of the universe (cf. [6]). In this thesis we follow the ansatz that all dark matter is made of a light sterile neutrino with mass in the kev range, which is produced most effectively at temperatures 1 There also exist alternative explanations to DE and DM such as Massive Compact Halo Objects (MACHOs) or Modified Newtonian Dynamics (MOND) which however seem to be disfavoured by observations of the Bullet Cluster (cf. [5] and references therein). 9

14 2 COSMOLOGICAL MOTIVATION T few GeV, around the region of the QCD phase transition. motivation, explanations and limits for this ansatz are given. In Chapter 3 the Inflationary epoch Hot Big Bang Theory has its problems. Some are due to the very special initial conditions which are needed. They are: why is the universe as large, warm, spatially flat, homogeneous and isotropic as it is and why does it have such a large entropy? One elegant solution is the inflationary scenario. The idea is that the hot cosmological epoch was proceeded by an epoch of exponential expansion (inflation). So, initially small regions inflated to very large size, which also may be orders of magnitude larger than the size of the observable part of the universe. This epoch would be followed by the epoch of post-inflationary reheating, where the inflaton energy is transferred to energy of conventional matter. Then the universe heats up and the Hot Big Bang epoch starts. For further information cf. Ref. [8]. 2.3 Dynamics of Cosmological Expansion The cosmological expansion is determined by the Einstein equations R µν 1 2 g µνr = 8πGT µν, (2.17) with R µν the Ricci tensor, g µν, µ, ν {0, 1, 2, 3}, the metric of 4-dimensional spacetime, R = g µν R µν the Ricci scalar and T µν = pg µν + (p + ρ)u µ u ν (2.18) the energy-momentum tensor with p the pressure and ρ the energy density in the universe and u µ the four-velocity of the perfect fluid in thermodynamic equilibrium, as which the universe can be approximated [14]. For a homogeneous and isotropic universe the energy-momentum tensor gets a diagonal structure [14] Thus, the 00-component of Eq. (2.17) has the form T µν = diag(ρ, p, p, p). (2.19) (ȧ a ) 2 = 8π 3 Gρ κ a 2, (2.20) with the curvature parameter κ = 1, 0, +1 for a 3-hyperboloid, 3-plane and a 3-sphere, respectively. This is the Friedmann equation (cf. Eq. (2.6)), which depends on two functions of time, a(t) and ρ(t), and therefore one needs additional equations to solve 10

15 2.3 Dynamics of Cosmological Expansion it. The covariant conservation of the energy-momentum tensor µ T µν = 0 yields (cf. [6]) ρ + 3ȧ (ρ + p) = 0. (2.21) a To obtain a closed system of equations determining the dynamics of a homogeneous and isotropic universe, one has to specify the equation of state, which is determined by the matter content in the universe. p = p(ρ) = wρ, (2.22) With Eqs. (2.20), (2.21) and (2.22) the dynamics of the cosmological expansion are completely determined. It has to be kept in mind that the universe consists of various types of matter, which have to fulfill Eqs. (2.21) and (2.22) separately and that the Friedmann equation (2.6) contains the sum over the different energy densities Sample Cosmological Solutions If the matter in the universe is in thermal equilibrium at zero chemical potential, then Eq. (2.21) can be written as dρ = 3d(ln a), (2.23) p + ρ where the left hand side is equal to ln s, with s the entropy density in the universe (cf. [6]). Hence, from Eq. (2.21) results sa 3 = const., (2.24) which means the entropy conservation in comoving volume. Or in other words, due to the expansion the entropy density decreases as s a 3. Let us consider for some solutions of Eq. (2.20) a spatially flat universe κ = 0, (2.25) which is a very good approximation of the real universe (cf. Sec. 2.1). This leads to the Friedmann equation (ȧ ) 2 = 8π Gρ. (2.26) a 3 Simple solutions of a universe filled with one type of matter can be obtained by determining the energy density as a function of the scale factor ρ = ρ(a) with the help of Eqs. (2.22), (2.23) and then computing the dependence of the scale factor of time with Eq. (2.26) (cf. [6]). 11

16 2 COSMOLOGICAL MOTIVATION The equation of state for non-relativistic matter is With Eq. (2.23) one obtains With Eq. (2.26) we obtain for the scale factor p = 0. (2.27) ρ = const. a 3. (2.28) a (t t s ) 2 3, (2.29) here t s is an arbitrary constant of integration, which henceforth will be set equal to zero, t s = 0. Relating this to the Hubble parameter in Eq. (2.2) one obtains H(t) = ȧ a = 2 3t. (2.30) During the radiation dominated (relativistic matter) period of the universe the equation of state is given by (cf. Sec ) which combined with Eq. (2.23) yields The scale factor is then and therefore p = 1 3 ρ, (2.31) ρ = const. a 4. (2.32) a t 1 2 (2.33) H(t) = ȧ a = 1 2t. (2.34) On the other hand the Hubble parameter can be expressed in terms of the temperature T, assuming thermal equilibrium between all types of particles and neglecting chemical potentials [6]: H(T ) = T 2 M P l = 1.66 g eff T 2 M P l, (2.35) where M P l = 90 8π 3 g eff M P l 0.6 geff M P l is the reduced Planck mass. Combining Eqs. (2.34) and (2.35) yields t = 1 2H = M P l geff (T )T, (2.36) 2 12

17 2.3 Dynamics of Cosmological Expansion where one has to consider that the effective degrees of freedom g eff (T ) themselves strongly depend on T at the QCD and electroweak transition regions (cf. [9]). For vacuum, i.e. dark energy dominated period, the equation of state reads The solution of the Friedmann Eq. (2.26) yields where the Hubble parameter is independent of time (cf. [6]). p = ρ = const. (2.37) a exp (H ds t), (2.38) H ds = 8π 3 Gρ vac (2.39) Thermodynamics In this thesis we want to study the influence of a large lepton asymmetry and thus a nonvanishing lepton chemical potential on the production of sterile neutrinos. Therefore, we need to state some aspects of a thermodynamic description for particles in the context of cosmology. The distribution function of a perfect fluid is defined as a density function in the phase space of the corresponding particle. Because of homogeneity and isotropy of the universe, the distribution function of a particle in the universe cannot be a function of the space r or the direction of the momentum p. Thus, it is of the form f = f( p, t). (2.40) Considering also particle masses m and helicity degrees of freedom g, one can derive the following quantities [14]: number density energy density n = g ρ = g d 3 p f( p, t), (2.41) (2π) 3 d 3 p (2π) 3 p 2 + m 2 f( p, t), (2.42) pressure p = 1 d 3 3 g p p 2 (2π) 3 f( p, t), (2.43) p 2 + m2 with the energy of a particle E = p 2 + m 2 according to special relativity. The internal degrees of freedom g are for example, g = 2 for massive leptons, g = 1 for neutrinos and g = 6 for quarks, counting particles and antiparticles separately. 13

18 2 COSMOLOGICAL MOTIVATION In kinetic equilibrium the distribution function f of a particle is given by f(e, µ, T ) = n F,B (E µ) = 1 e (E µ)/t ± 1, (2.44) where + corresponds to fermions (Fermi-Dirac distribution) and corresponds to bosons (Bose-Einstein distribution). These distribution functions depend on the energy of the particle E, the temperature T and the chemical potential of the particle µ. The chemical potential is the energy needed for adding a particle to the equilibrated system. If a particle number is not conserved, the corresponding chemical potential vanishes, µ = 0, in equilibrium. The chemical potentials of photons and gluons vanish, µ γ = µ g = 0, as their number is not conserved, since they can be produced and annihilated in any number. Thus, in kinetic equilibrium the thermodynamic variables in Eqs. (2.41)-(2.43) become with Eq. (2.44) and the substitution p = E 2 m 2 : n(e, µ, T ) = ρ(e, µ, T ) = g 2π 2 g 2π 2 p(e, µ, T ) = 1 g 3 2π 2 m m m de E E 2 m 2 e (E µ)/t ± 1, (2.45) de E2 E 2 m 2 e (E µ)/t ± 1, (2.46) ( E 2 m 2) 3 2 de e (E µ)/t ± 1. (2.47) Furthermore, for a chemically equilibrated system the chemical potentials of different particle species are related to another according to the reaction formulas, as a + b c + d (2.48) µ a + µ b = µ c + µ d. (2.49) In particular, if annihilation and pair production processes are in equilibrium, this means that the chemical potential µ ī of the antiparticle corresponding to the particle of species i with chemical potential µ i differs only by sign, i.e. µ ī = µ i, because of µ i + µ ī = 2µ γ = 0. All chemical potentials can then be expressed in terms of chemical potentials of conserved quantities and there are as many independent chemical potentials as independent conserved particle numbers. Thus, the net number density of a particle species i, which is the sum of the number density of a particle i minus its antiparticle ī, is given by n i = n i n ī = g i 2π 2 = g i 2π 2 m i m i dee E 2 m 2 i (f i f ī ) ( dee E 2 m 2 i 1 e (E µ i)/t ± 1 1 e (E+µ i)/t ± 1 ). (2.50) In general, the integrals in Eqs. (2.45)-(2.47) cannot be solved with elementary functions. But in the cases for T m (relativistic) and T m (non-relativistic) the 14

19 2.3 Dynamics of Cosmological Expansion formulas can be simplified. For the relativistic case, i.e. T m, µ, we can set the mass to zero and expand in µ/t [15]: n = { 1 ζ(3)gt π 2 6 gt 2 µ + O(T µ 2 ), bosons 3 ζ(3)gt π 2 12 gt 2 µ + O(T µ 2 ), fermions, (2.51) { π 2 ρ = 30 gt ζ(3)gt 3 µ + 1 π 2 8 gt 2 µ 2 + O(µ 3 ), bosons 7 π gt ζ(3)gt 3 µ + 1 4π 2 4 gt 2 µ 2 + O(µ 3 ), fermions, (2.52) p = ρ/3, (2.53) where ζ(3) = n=1 (1/n3 ) is the Riemann zeta function. In the case of vanishing chemical potential µ = 0 we get for the average particle energy in kinetic equilibrium { E = ρ π 4 n = 30ζ(3) T 2.701T, bosons. (2.54) T 3.151T, fermions 7π 4 180ζ(3) In the non-relativistic limit T m and T m µ, the second condition leads to a dilute system, i.e. occupation numbers 1, which is usually satisfied in cosmology when the first condition is satisfied, disregarding systems of high density like e.g. white dwarfs. Thus, one can approximate e E µ T ± 1 e E µ T, (2.55) which leads to the Maxwell-Boltzmann statistics for the distribution function f and thus is the same for bosons and fermions. One then obtains for the quantities n, ρ and p in the non-relativistic limit T m: ( mt n = g 2π ρ = n ( m + 3T 2 ) 3 2 e m µ T, (2.56) ), (2.57) p = nt ( ρ). (2.58) For the net particle number density we obtain to third order in the chemical potential for the relativistic and non-relativistic limit [14] n i (T, µ i ) (T m i,µ i ) = ( g i T 3 3π 2 ( g i T 3 6π 2 (T m i ) = 2g i ( mi T 2π ) 3 π 2 ( µ i ) ( T + µi ) ) 3 T + O(µ 5 i ), bosons π 2 ( µ i ) ( T + µi ) ) 3 T + O ( µ 5 ), (2.59) i, fermions 2 sinh ( µi T ) ( exp m i T ). (2.60) 15

20 2 COSMOLOGICAL MOTIVATION As mentioned at the beginning of this section this equations hold as long as all interactions are in chemical equilibrium, which is a very good approximation for particle interactions in the early universe. As long as the particle interactions are faster than the Hubble time t H = 1/H they stay in equilibrium. Comparing the typical interaction rates, this holds for SM particles until T 1 MeV for massless particles. Therefore, it is a good approximation to assume thermal and chemical equilibrium for the SM particles in the early universe (cf. [14]). If we compare the relativistic and non-relativistic particle number densities n, energy densities ρ and the pressure p, we see that they fall exponentially as the temperature drops below the mass of the particle. As the universe expands and cools down, the particles and antiparticles annihilate and cannot be produced in pairs again. The annihilation temperature is about T ann m i /3T and it is not an instantaneous process but takes several Hubble times (cf. [14]) Entropy and Conserved Quantities As already stated in Sec the early universe is in a local thermal equilibrium and thus we can describe it in the context of thermodynamics and derive an expression for the entropy in comoving volume with the help of the second law of thermodynamics, T ds = d(ρv ) + pdv µdn, (2.61) with T the temperature, S the entropy, ρ the energy density, V the volume, p the pressure, µ the chemical potential and N the particle number of the system. Hence, we get for the entropy density s = ρ + p µn T. (2.62) Thus, we can express the entropy density with the help of the formulas for n, ρ and p in Sec In the early universe T m i, µ i and thus for vanishing masses and chemical potential we get the well known result [14] s(t ) (m i,µ i =0) = 2π 2 45 T 3 i=bosons g i i=fermions g i (2.63) = 2π2 45 T 3 h eff, (2.64) where in the first line we sum over all relativistic bosonic and fermionic internal degrees of freedom, which in the second line is merged in the function h eff, which counts the effective relativistic degrees of freedom. The number of effective degrees of freedom itself depends on the temperature, since particle species annihilate with decreasing 16

21 2.3 Dynamics of Cosmological Expansion temperature and thus h eff becomes smaller. Considering now non-vanishing fermionic chemical potentials leads to [15] s(t ) (m i=0) = 2π2 45 T 3 g i + 7 g i + 15 ( µi ) 2 8 8π 2 g i (2.65) T i=bosons i=fermions i=fermions = 2π2 45 T 3 (h eff + h eff ), (2.66) where h eff denotes a shift in the entropy density due to non-vanishing fermionic chemical potentials. Analogously we can derive the effective degrees of freedom from the energy density ρ i+ī = ρ i + ρ ī (2.67) i=b,f π2 15 g efft 4. (2.68) For relativistic fermions ρ can be solved exactly according to Eq. (2.46), which leads to [15] [ 7π 2 ρ i+ī = g i 120 T ( 4 T 4 µ ) i 2 1 ( + T 8π 2 T 4 µ ) ] i 4, (2.69) T i=fermions and thus to g eff = 15 π 2 i g i [ 7π ( µi ) 2 1 ( µi ) ] 4 + T 8π 2 T, (2.70) where the shift in the effective degrees of freedom due to the non-vanishing fermionic chemical potential is g eff = 15 ( µi ) 2 1 ( µi ) ] 4 8π 2 g i [2 + T π 2. (2.71) T i We note that g eff h eff and thus g eff (T, µ) h eff (T, µ). Non-vanishing chemical potentials would lead to an enhanced total energy and entropy density. For small chemical potential, e.g. µ/t O(0.1), this contribution is small and could be neglected [15]. Then g eff (T, µ) h eff (T, µ) for a wide range of temperatures (cf. [9]). The entropy s is important for describing quantities during the evolution of the universe, because it is conserved in comoving volume during the expansion of the universe. The entropy density scales as s a 3 as the net number densities n i corresponding to conserved quantities do, e.g. baryons, electric charge and leptons. If e.g. the baryon number B is conserved, the associated ratio n b /s is also conserved. After the electroweak phase transition T ew 100 GeV and before neutrino oscillations start at 17

22 2 COSMOLOGICAL MOTIVATION T 10 MeV, the baryon number B and lepton flavour numbers L α are conserved, since so far no process violating lepton or baryon number has been observed. However, sterile neutrino production will violate L α -conservation, this will be discussed in Chapter. 3. Moreover, the electric charge is conserved and we assume charge neutrality, i.e. Q = 0, which seems to be reasonable due to observations which point to neutrality and lack of currents on large scales. Additionally we assume that all globally conserved quantum numbers are also conserved locally. This is a good approximation for length scales larger than the largest length scales on which transport phenomena can show up, which is the mean free path of neutrinos. Taking the net particle number density n i, which is in thermal and chemical equilibrium given by Eq. (2.50), we can obtain five local conservation laws for the specific lepton flavour asymmetries l α, specific baryon asymmetry b and charge density q: l α = n α + n να s b = b i n i s i q = 0 = q i n i s i, α {e, µ, τ}, (2.72), b i = baryon number of species i, (2.73), q i = charge of species i, (2.74) with s the entropy density. For a specific temperature the free variables in the Eqs. (2.72)-(2.74) are the chemical potentials in the net particle densities n i (T, µ i ) and the specific lepton flavour asymmetries l α. The baryon number is fixed due to observational bounds to b O(10 10 ) [2]. After the onset of neutrino flavour oscillations only the total lepton number, i.e. specific lepton asymmetry, l α=e,µ,τ l α is conserved. For l there do not exist such stringent bounds, since the cosmic neutrino background cannot be measured directly, due to the weak interaction of neutrinos. The strongest constraints can be obtained by an analysis of BBN and CMB which yields l 0.02 (cf. e.g. [16]). In Chapter 7 relations between the chemical potentials and the asymmetries for the conserved quantum numbers B, L α and Q = 0 are derived for the case of large specific lepton asymmetries, l α b 0. 18

23 3 Concept of Sterile Neutrinos In this Chapter we give a brief introduction to the concepts of sterile neutrinos. After stressing the motivation to explain dark matter (DM) with sterile neutrinos in Sec. 3.1, we will give an overview of possible production mechanisms in Sec In Sec. 3.3 the required extensions of the SM to describe sterile neutrinos in the framework of QFT are introduced. In the end (Sec. 3.4), we will comment on the impact of sterile neutrinos on the evolution of the universe (Sec ) and on cosmological constraints on the sterile neutrino properties, focussing on the (radiative) decay. 3.1 Motivation The SM and GR can together describe almost all phenomena observed in nature. However, as already stated in Sec. 2.2, these concepts fail to explain DM. As most of the mass in the observable universe is not composed of baryonic matter, it is very important to find the origin and constituents of DM. In doing so, one can obtain a better comprehension of the content and evolution of the universe, e.g. explanation of clustering of matter and formation of structures. Even though the constituents of DM are unknown, it can be easily explained by new, massive, electrically neutral, long live and collisionless particles. Furthermore, the free streaming length in the matter dominated era of the universe has to be in congruence with the observed structure in the universe, i.e. the structure formation. Sterile neutrinos with certain properties could be such DM particles [4]. In general one distinguishes three types of DM candidates. Cold Dark Matter (CDM) are particles which were non-relativistic at the time of decoupling. Hot Dark Matter (HDM) are particles which were relativistic at the time of decoupling and remained so in the matter dominated epoch. Warm Dark Matter (WDM) particles were relativistic at freeze-out and became non-relativistic during the radiation dominated epoch. On large scale structures they behave like CDM and structures on smaller scales are suppresed due to the free streaming length. This behaviour is in good agreement with observations and numerical simulations (cf. [4]). Sterile neutrinos with mass in the kev-range, which we are considering in this thesis, are formally WDM. But their momentum distribution may be very different from Fermi-Dirac. In fact it is shifted to smaller momenta, cf. Sec. 6.3 and Ref. [17] for a more detailed study. Moreover, there is another motivation for introducing new (sterile) neutrinos. All known matter is composed of elementary fermions with spin 1 2. They can be described by Weyl spinors in an irreducible representation of the Poincaré group. Two representation of this group are known as left chiral and right chiral. All known elementary fermions, except neutrinos, have been observed left handed (chiral) as well as right handed. Only neutrinos have been observed left handed so far. There are two explanations for this: 19

24 3 CONCEPT OF STERILE NEUTRINOS right handed neutrinos do not exist in nature, or right handed neutrinos just have not been observed yet, because their interaction with other matter is so weak. As the left handed neutrino is electrically and colour neutral it only participates in the weak interaction of the SM. This interaction however does not couple to right handed fields. Therefore, the right handed counterpart would be a singlet under all gauge interactions of the SM (SU(3) C SU(2) L U(1) Y ) and thus is also called sterile neutrino. Nevertheless, it may have interactions within an extended gauge group beyond the SM. Sterile neutrinos could also explain several other phenomena, which cannot be explained within SM+GR: neutrino oscillations and masses, the cosmological origin of the baryon asymmetry. We just mention these aspects to give an overview of the applications of sterile neutrinos without surveying them. For a more detailed study on different aspects of sterile or right handed neutrinos see Refs. [4, 5]. 3.2 Production in the Early Universe It is not known when or how DM was produced in the early universe (cf. Sec. 2.2). If sterile neutrinos constitute DM - we are just considering the case that all DM is made of one type of particle - there are several possible production mechanisms which we will shortly review (cf. [4]). Non-resonant thermal production via mixing If the right handed, sterile neutrinos have non-zero Yukawa couplings F, i.e. nonvanishing Dirac-masses, they can be produced thermally from the primordial plasma via mixing to the left handed neutrinos, as proposed in the Dodelson-Widrow scenario [18]. If the production is efficient enough to bring them into thermal equilibrium, the density of the sterile neutrinos would be larger than the DM abundance Ω DM and thus result in overclosure of the universe, i.e. a younger universe with a larger Hubble rate, unless it got compensated by an extra radiation expected from sterile decay (cf. [19]). Avoiding overclosure yields an upper bound on the mixing angle. This production scenario would lead to the correct Ω DM if the mixing angle is small enough so that the sterile neutrinos never reach thermal equilibrium. This way they have a momentum distribution which is proportional to a Fermi-Dirac-distribution, but colder, and thus they are a WDM candidate. A production of sterile neutrinos by 20

25 3.3 Mathematical Classification this production mechanism alone is however excluded by x-ray bounds, since it cannot produce sufficient amount of DM [13, 20]. Resonant thermal production If a significant lepton asymmetry exists in the primordial plasma, the Mikheyev-Smirnov- Wolfenstein effect [21] can lead to a level crossing between the active and sterile neutrino dispersion relations. This would result in an enhanced production rate of the sterile neutrinos compared to the non-resonant production. A consequence of this is the depletion of the initial lepton asymmetry, due to the conversion of active neutrinos into sterile neutrinos. The momentum distribution of these sterile neutrinos is non-thermal and shifted to lower momenta compared to a thermal spectrum. This makes the sterile neutrino a better DM candidate due to avoiding problems associated with a too warm spectrum, i.e. the washout of large structures (cf. Sec. 3.4). This production mechanism was first proposed by Shi and Fuller [17]. It is necessarily accompanied by Dodelson-Widrow mechanism, i.e. non-resonant thermal production via mixing to active neutrinos. This is the production mechanism we will focus on in the following Chapters 5 and 6. We will not focus on the origin of the primordial lepton asymmetry here because this does not influence the calculations. For more general information about it see Refs. [12, 22] and for right handed neutrino specific information see Ref. [13]. Thermal production beyond the SM Sterile neutrinos form a singlet under the SM gauge and only interact via Yukawa couplings. But if there exists an extended hidden sector or extended Higgs sector there may be additional interactions beyond the SM interactions. The sterile neutrino may also be charged under some extended gauge group which is broken at high energies. This production mechanisms would also contribute to the thermal production of sterile neutrinos and could also lead to overclosure [5]. Non-thermal production Due to coupling to an inflaton, SM Higgs, other scalars or modified gravity, sterile neutrinos could also be produced non-thermally (cf. [5]). 3.3 Mathematical Classification To include sterile neutrinos in the framework of the SM with its interactions, we represent the right handed neutrinos as Majorana fermions with the spinors N I. They 21

26 3 CONCEPT OF STERILE NEUTRINOS form a singlet under all gauge interactions and only couple to left handed neutrinos via Yukawa coupling, just as the right handed charged leptons couple to the left handed charged leptons. Hence, the extended Minkowskian Lagrangian of the SM reads L = 1 2 N I i/ N I 1 2 M I N I N I F αi Lα φar N I F N αi φ I a L L α +L }{{}}{{} SM, (3.1) L L int S where M I are Majorana masses of the sterile neutrinos, which are chosen to be real in this basis, F αi are elements of a n n Yukawa matrix, L α are the weak interaction eigenstates of the active lepton doublets, φ = iτ2 φ is the conjugate Higgs doublet and a L (1 γ 5 )/2 and a R (1 + γ 5 )/1 are chiral projectors. Formally there is no restriction on the number of sterile neutrinos in Eq. (3.1). A more or less minimal setting is the so called νm SM (neutrino Minimal Standard Model), where the SM is extended by three right handed neutrino with specific mass pattern [23]. Therein the lightest sterile neutrino, i.e. the DM candidate, should correspond to I = 1, following normal hierarchy. For the calculation of the production rate of the DM sterile neutrino we will consider an extension of the SM to three additional sterile neutrino and scrutinizing I = 1. At temperatures where sterile neutrino production is effective (T few GeV)[24], the electroweak symmetry is broken, so one can use the vacuum expectation value of the Higgs-field φ = (v/ 2, 0), with v 246 GeV, in the Lagrangian in Eq. (3.1). Assuming that the mixing angles, defined as θ αi (M D) 2 αi M 2 I (3.2) with the Dirac mass matrix (M D ) αi vf αi 2, (3.3) are very small, θ 10 3, the sterile neutrino interaction eigenstate of type I is to a good approximation also its mass eigenstate [24]. Adding three sterile neutrinos in this way one introduce three new dimensional parameters M I, I {1, 2, 3}, in the Lagrangian, while in the SM there is just one dimensionful parameter: the Higgs mass m H (besides the Planck mass). Different sterile neutrino scenarios are classified by this parameters (cf. [4]). Sterile neutrinos with kev masses are a promising DM candidate we focus on. We will not consider the model building in detail. For a review on the model building of kev sterile neutrinos we refer to [5]. 22

27 3.4 Cosmological Constraints on Sterile Neutrino Properties Fig. 3.1: Different constraints on the sterile neutrino mass-mixing parameter space. The upper blue region is excluded by X-ray observations. The dark grey region M 1 < 1 kev is excluded by the Tremaine-Gunn bound. The upper solid black line corresponds to a non-resonant production of the observed DM abundance Ω DM in the absence of lepton asymmetry µ α = 0. In this plot (taken from [20]) µ α = 0 is the lepton asymmetry and not the chemical potential. The lower solid black line and the dashed line correspond to a resonant production, at µ α = and µ α = respectively, which yields the correct Ω DM. Bounds from structure formation have not been considered in this plot, since they are not independent of the lepton asymmetry, but depend on it in a complex manner [4]. 3.4 Cosmological Constraints on Sterile Neutrino Properties Impact of Sterile Neutrinos on the Evolution of the Universe One big issue in explaining DM with sterile neutrinos or in general in introducing new particles is, that their behaviour and properties should be in congruence with observations and experimental results. Since all astronomical observations to date are in good agreement with the ΛCDM model, the most important cosmological parameters regarding sterile neutrinos are, on one hand, the energy density parameters Ω B and Ω DM (cf. Sec. 2.1). Additional sterile neutrinos, which should explain DM, should lead to the correct observed total matter density today and in particular to the correct ratio of the baryonic to DM energy density. On the other hand, the effective number of neutrino species N eff in the radiation dominated epoch is also a very important cosmological parameter. It is a measure for the expansion history of the universe, since the Hubble parameter depends on the total energy density plus curvature as already seen in Eq. (2.6) and neutrinos also contribute to this energy density. Though the neutrino contribution to the energy density ρ is negligible today, this was not the case in the radiation dominated epoch. Conveniently, the contribution of neutrinos plus 23

28 3 CONCEPT OF STERILE NEUTRINOS ν s θ αs Z 0 ν α ν α ν α Fig. 3.2: Principal decay mode for massive singlet neutrinos ν s, with mass less than twice the electron mass, mixing to an active neutrino of species α via θ αs. In the final state there are three light active neutrinos (where α, α {e, µ, τ}). unknown physics to ρ is parametrized as N eff ρ ν, where ρ ν is the contribution of one ultrarelativistic neutrino species. If there is no new physics and all neutrinos are effectively massless, N eff can be identified by the number of neutrino species. Each additional ultrarelativistic particle would in principal lead to a larger value of N eff, which influences the evolution of the universe. Therefore, sterile neutrinos cannot be relativistic during radiation dominance to be in congruence with the observed evolution of the universe (cf. [4]). The parameter space of sterile neutrino rest mass and vacuum mixing angle can also be constrained by dark matter stability considerations, X-ray observations, through kinematic arguments, e.g. phase space considerations (Tremaine-Gunn bound M 1 > 1 kev [25]), and structure formation simulations, in particular in the form of Lyman-α forest observations (cf. [26]). Fig. 3.1 shows that the mass-mixing parameter space is constrained in all directions by X-ray observations, the Tremaine-Gunn bound and the lepton asymmetry. Structure formation simulations have not been considered therein as they themselves depend on the lepton asymmetry (cf. [20]). In the following we will shortly discuss the constraints due to dark matter decay and focus on the radiative decays and the recently found 3.55 kev X-ray line. In doing so, we obtain some input values for the rest mass and vacuum mixing angle for the calculations in Chapter Dark Matter Decay To be an acceptable dark matter candidate a particle must have a lifetime τ at least of order of the age of the universe (τ > t uni yrs cf. Sec. 2.1). Therefore, significant constraints on the vacuum mixing angle and the rest mass can be made via the effects of the decay of sterile neutrinos. Sterile neutrinos ν s are unstable and decay via θ αs suppressed weak interaction, where θ αs is the posited mixing with the active neutrinos. The predominant decay channel is a tree-level decay mediated by a Z-boson into three light active neutrinos (ν s 3ν), as can be seen in Fig The corresponding 24

29 3.4 Cosmological Constraints on Sterile Neutrino Properties W ± γ l α γ ν s l α ν α ν s W ± ν α θ αs θ αs Fig. 3.3: Radiative decay modes for sterile neutrinos ν s mixing to an active neutrino via θ αs, with one active neutrino ν α and one photon in the final state (where α {e, µ, τ}). decay rate is given by [27] Γ νs 3ν = G2 F 768π 3 M 5 1 sin 2 2θ s 1 ( sin 2 2θ ) ( ) 5 M1, kev (3.4) with the Fermi constant G F = gw/4 2 2m 2 W GeV 2, M 1 the mass of the sterile neutrino and where we introduced the total vacuum mixing angle as (cf. [28]) sin 2 2θ 4θα1 2. (3.5) α=e,µ,τ Requiring that the lifetime should exceed the age of the universe (cf. Eq. (2.3)), i.e. τ 1/Γ νs 3ν s, leads to the combined constraint in the sterile neutrino mass-mixing parameter space ( sin 2 2θ ) ( ) 5 M (3.6) kev If the rest mass of the sterile neutrino is greater than twice the electron rest mass, i.e. M 1 1 MeV, then there would be an additional decay channel ν s νe + e with the rate Γ νs νe + e = 1/3 Γ ν s 3ν. Therefore, the overall decay rate would be enhanced by a factor of 4/3 [29]. The principal radiative decay modes for the decay ν s νγ are illustrated in Fig In the final state there is a light active neutrino and a photon, each of energy E γ = E ν = M 1 /2, which results of energy-momentum conservation in the decay of the non-relativistic, massive sterile neutrino into two (nearly) massless particles. The rate is given by [27] Γ νs νγ = 9G2 F α 2048π 4 M 1 5 sin 2 2θ (3.7) s 1 ( sin 2 2θ ) ( ) 5 M1, kev where α 1/137 is the electromagnetic fine structure constant. Another possible decay of a sterile neutrino would be via two-photon-emission ν s νγγ, which is strongly suppressed for the favoured mass region of M 1 O(keV) (cf. [27]). 25

30 3 CONCEPT OF STERILE NEUTRINOS Even though the one-photon radiative decay branch is smaller by a factor of 27α/8π compared to the 3ν channel in Eq. (3.4), the more important fact is that it leads to an electromagnetic imprint and is therefore much more viable for indirect detection. It should produce a signal in the diffuse X-ray background or a line in the X-ray spectrum in the direction where dark matter is accumulated, i.e. galaxy clusters, dwarf galaxies or galaxies. The position of the line would then fix the mass of the sterile neutrino, whereas the intensity would fix the mixing angle. Thus, this decay branch has been used to provide the most stringent constraints on many models for sterile neutrino dark matter [4] kev X-ray Line Recent analyses of the X-ray spectrum of galaxies and galaxy clusters have independently found a previously unidentified emission line at a photon energy of around 3.55 kev [30, 31]. This could possibly arise from the electromagnetic decay of a sterile neutrino with rest mass M kev into an active neutrino and a photon [32]. We will shortly review the results of the different groups. Ref. [30] detected an emission line at E γ = ( ) ± 0.03 kev in a stacked XMM-Newton spectrum of 73 galaxy clusters. Furthermore, the line was also detected in the Chandra ACIS-S and ACIS-I spectra of the Perseus cluster. Assuming that all dark matter is made of sterile neutrinos, according to their results (within a significance level of > 3σ) the mass and the respective mixing angle is M 1 = 7.1 ± 0.07 kev, sin 2 2θ (3.8) Ref. [31] identified a weak emission line at E γ 3.5 kev in the X-ray spectra of the Andromeda galaxy and the Perseus galaxy cluster, with the XMM-Newton. They obtain (within a significance level of 4σ) a mass for the sterile neutrino and a corresponding mixing angle of M 1 = 7.06 ± 0.05 kev, sin 2 2θ = (2.2 20) (3.9). Both references stated that the emission line is very weak and located within the vicinity of several known faint lines [30]. Furthermore, X-ray spectra of astrophysical objects are crowded with weak atomic and instrumental lines, of which not all may be known. Therefore, it is hard to exclude an astrophysical or instrumental origin of any weak line found in the spectrum of individual objects [31]. Summing up, the existence of the line, regarding the statistical significance σ, and its interpretation as having dark matter origin, are still up for debate (cf. e.g. [33] and references therein). Nevertheless the possibility of a dark matter origin of this emission line remains intriguing and several sterile neutrino dark matter models have 26

31 3.4 Cosmological Constraints on Sterile Neutrino Properties been adjusted to coincide with these values for the rest mass and mixing angle (cf. e.g. [32, 34]). In this thesis we also consider the possibility of a dark matter origin of this 3.55 kev X-ray emission line as input values for the model. Wherever a value for the rest mass or the mixing-angle is needed, the values are used in this thesis. M 1 = 7.1 kev, sin 2 2θ = (3.10) 27

32 4 BASIC CONCEPTS OF THERMAL FIELD THEORY Thermal Field Theory Quantum Field Theory Quantum Mechanics Special Relativity Electromagnetism Statistical Mechanics Classical Mechanics Fig. 4.1: Relation between different fields in physics leading to Thermal Field Theory. 2 4 Basic Concepts of Thermal Field Theory In this Chapter we are going to motivate the use of Thermal Field Theory in general and particularly in the context of the calculation of the production rate of sterile neutrinos. Furthermore, we will state the most important equations (for fermions) in this concept which we will need in the calculation of the production rate in Chapter Motivation Thermal (Quantum) Field Theory (TFT), or Quantum Field Theory at finite temperature, is the relativistic generalization of finite temperature non-relativistic Quantum Statistical Mechanics. It is a method to describe a large ensemble of multi-interacting particles in a thermodynamical environment. The main use is the description of canonical or grand-canonical ensembles, i.e. systems at fixed temperatures and chemical potential, and the calculation of expectation values of physical observables of a QFT at finite temperature in this formalism. In Fig. 4.1 one sees the relation between the different fields in physics and from what TFT emerged. TFT exhibits the following advantages, compared to other theories: it uses the path integral formalism, thus we just have to deal with commuting quantities, it is able to treat non-abelian gauge interactions such as QCD and it is Lorentz-covariant. 2 The diagram is inspired by a lecture of Alexei Bazavov given at Bielefeld University in

33 4.2 Basic Equations for Fermions In TFT there are mainly two different formalisms which can be distinguished: one based upon a complex-time contour and the other based on C algebra (Thermo-Field Dynamics, cf. [35]). In this thesis the first formalism is used and therefore described in the following. In the Matsubara formalism the TFT is built by incorporating a purely imaginary time variable into the evolution operator. This leads to the replacement of the continuous frequencies in momentum space by discrete imaginary ones, the Matsubara frequencies over which one has to sum in the Imaginary-Time Formalism (ITF). The real-time observables are then obtained by an analytic continuation. Alongside a Real- Time Formalism (RTF) which is based upon the choice of a contour in the complex plane, was developed by Keldysh, Mills and Schwinger. Main applications of TFT in high energy physics are: Heavy-Ion collisions: The formation of quark-gluon plasma (QGP) and the phase transition of QCD. Astrophysics: There are regions of extremely dense plasmas in the universe (cores of neutron stars, supernovae, white dwarfs,...) and high temperature (e.g. collapse of a supernova), where it is useful to use TFT to calculate the emission rates of neutrinos and axions [35]. Cosmology: The early universe before recombination is to very good approximation a perfectly thermalised plasma, so it can be used to study the behaviour of hot plasmas within TFT. Furthermore, one can study the reaction rates taking place in the hot plasma to determine abundances of some species e.g. of sterile neutrinos. This is what we are interested in in this thesis. For a detailed introduction to TFT see for example Refs. [35, 36]. 4.2 Basic Equations for Fermions In the concept of TFT one introduces a class of new variables which depend on a Minkowskian time t and a temperature T. Among these variables there are the production rates from thermal plasma, i.e. the production rate of sterile neutrinos which we are studying. Other examples for these variables are rates of oscillations and damping of waves in the plasma and transport coefficients (electric and thermal conductivity and bulk and shear viscosity). As we are dealing with the production rate of sterile neutrinos which are fermions, we will just consider the basic equations for fermions, i.e. for anticommutating operators. For bosonic operators they follow analogously, but obeying the commutation relation (cf. [37]). We use the conventions x = (t, x i ) for the Minkowskian space-time coordinates and Q = (q 0, q i ) for momenta. Euclidean coordinates are indicated by a tilde, x = (τ, x i ) and Q = ( q 0, q i ), obtained by a Wick rotation τ it, q 0 iq 0. Scalar products 29

34 4 BASIC CONCEPTS OF THERMAL FIELD THEORY will be defined as Q x = q 0 t + q i x i = q 0 t q x. The arguments of operators will denote implicitly, whether we are operating in the Minkoswkian or Euclidean spacetime. Heisenberg operators are defined as Ô(t, x) e iĥt Ô(0, x)e iĥt. (4.1) All observables of interest in Chapter 5 can be reduced to two-point correlation functions of elementary or composite operators due to Wick s theorem [36]. In the following subsections we will give some common definitions and relations between such correlation functions which we will need in Chapter Two-point Correlation Functions In this section fermionic operators, denoted by ˆν α (x), ˆ ν β (x) = ˆν β (x)γ0, are used. For further information see [36]. The operators could be elementary field operators, where the indices α, β label Dirac and/or flavour components, but they could also be composite operators of elementary field operators. We are considering a general density matrix of a grand-canonical ensemble with chemical potential µ L of the form ˆρ = Z 1 exp[ β(ĥ µ L ˆL)] (4.2) with ˆL d 3 x α=e,µ,τ ˆ ν α γ 0ˆν α the total lepton number operator and [ ] Z = Tr e β(ĥ µ L ˆL) (4.3) the quantum mechanical partition function of a grand canonical ensemble. Due to the canonical anticommutation relation one easily gets the relation This yields {ˆν α (x), ˆν β (x )} = δ (4) (x x )δ αβ, (4.4) [ˆν α (x), ˆL] = ˆν α (x). (4.5) e βµ L ˆLˆν α (0, x) = n=0 1 n! (βµ L) n ˆLnˆν α (0, x) = = ˆν α (0, x)e βµ L ˆLe βµ L. n=0 1 n! (βµ L) nˆν α (0, x)(ˆl 1) n (4.6) The two-point correlators are denoted by ˆν α (x)ˆ ν β (x ), where... = Tr[ˆρ... ] (4.7) 30

35 4.2 Basic Equations for Fermions denotes the expectation value with respect to the density matrix ˆρ. One can define classes of correlation functions (cf. [37]). The physical correlators are defined as Π > αβ (Q) dt d 3 xe iq x ˆν α (x)ˆ ν β (0), (4.8) Π < αβ (Q) dt d 3 xe iq x ˆ ν β (0)ˆν α (x), (4.9) ρ αβ (Q) dt d 3 xe iq x 1 { } ˆν α (x), ˆ ν β (0), (4.10) 2 αβ (Q) dt d 3 xe iq x 1 [ ] ˆν α (x), ˆ ν β (0), (4.11) 2 where ρ αβ is the spectral function. Retarded and advanced correlators can be defined as Π R αβ (Q) i dt d 3 xe iq x {ˆν } α (x), ˆ ν β (0) θ(t), (4.12) Π A αβ (Q) i dt d 3 xe iq x { } ˆν α (x), ˆ ν β (0) θ( t). (4.13) The time-ordered correlation function reads Π T αβ (Q) dt d 3 xe iq x ˆν α (x)ˆ ν β (0)θ(t) ˆ ν β (0)ˆν α (x)θ( t), (4.14) while the Euclidean correlator is β Π E αβ ( Q) dτ 0 ˆν d 3 xe (i q 0+µ L )τ iq x α (x)ˆ ν β (0), (4.15) and it is time-ordered by definition and can be computed with standard imaginary-time functional integrals in the Matsubara formalism (cf. [36]). The additional term in the exponential was added to compensate for the multiplicative factor in Eq. (4.16) Relations between Correlation Functions All correlation functions can be expressed in terms of the spectral function, which itself can be determined as a certain analytic continuation of the Euclidean correlator. Therefore, we first need to find a relation between the correlators defined in Eqs. (4.8) and (4.9). With the help of Eq. (4.6) one gets a relation between the two-point correlation functions: ˆν α (t iβ, x)ˆ ν β (0, 0) = 1 ] [e Z Tr β(ĥ µ L ˆL) e βĥ ˆν α (t, x)e βĥ ˆ ν β (0, 0) = 1 [ˆν Z Tr α (t, x)e βµ ˆLe ] L βµ L e βĥ ˆ ν β (0, 0) = 1 (4.16) [ Z e βµ L Tr e β(ĥ µ ˆL)ˆ ν ] L β (0, 0)ˆν α (t, x) = e βµ L ˆ ν β (0, 0)ˆν α (t, x). 31

36 4 BASIC CONCEPTS OF THERMAL FIELD THEORY By inserting energy eigenstates in Eqs. (4.8) and (4.9) and making use of Eq. (4.16), one obtains Π > αβ (Q) = 1 [ ] dtd 3 xe iq x Tr e βĥ+iĥt βµ 1 Z }{{} e L ˆLˆν α (0, x)e iĥt }{{} 1 ˆ ν β (0, 0) m m m n n n = 1 dtd 3 xe iq x e β(em+µ L) e it(em En+q0) m ˆν α (0, x)e βµ L ˆL n n ˆ ν β (0, 0) m Z m,n = 1 d 3 xe iq x e β(em+µ L) δ(e m E n + q 0 ) m ˆν α (0, x)e βµ L ˆL n n ˆ ν β (0, 0) m, Z m,n (4.17) Π < αβ (Q) = 1 [ dtd 3 xe iq x Tr e βĥeβµ L ˆL 1 Z }{{} ˆ ν β (0, 0)e iĥt }{{} 1 ˆν α (0, x)e iĥt] m n n n m m = 1 dtd 3 xe iq x e βen e it(em En+q0) n ˆ ν β (0, 0) m m ˆν α (0, x)e βµ L ˆL n Z m,n = 1 d 3 xe iq x e βen δ(e m E n + q 0 βµ ) m ˆν α (0, x)e L ˆL n n ˆ ν β (0, 0) m Z }{{} m,n E n=e m+q 0 = e β(q0 µ L ) Π > αβ (Q), (4.18) which is the Kubo-Martin-Schwinger (KMS) relation in Fourier space: Consequently, ρ αβ (Q) = 1 2 Π < αβ (Q) = µ L ) e β(q0 Π > αβ (Q). (4.19) [Π > αβ (Q) Π< αβ (Q) ] = 1 2 Π> αβ (Q) (1 + e β(q0 µ L ) ) = 1 2 Π> αβ (Q)n F ( q 0 + µ L ) 1, (4.20) where n F (x) 1/[exp(βx) + 1], obeying n F ( x) = 1 n F (x), is the Fermi-Dirac distribution. Inversely, Π > αβ (Q) = 2 [ 1 n F (q 0 µ L ) ] ρ αβ (Q), (4.21) Π < αβ (Q) = 2n F (q 0 µ L )ρ αβ (Q). (4.22) Using the representation dω e iωt θ(t) = i 2π ω + i0 + (4.23) of the Theta-function, inserting it into the definition of the retarded propagator in Eq. (4.12) and rewriting the correlator as the Fourier transform of the spectral function in 32

37 4.2 Basic Equations for Fermions Eq.(4.10) yields Π R αβ (Q) = i = 2 dt dω = 2 = d 3 xe iq x 2θ(t) d 4 P (2π) 4 e ip x ρ αβ (P ) dω dp 0 e i(q0 p 0 ω)t dt 2π 2π ω + i0 + ρ αβ(p 0, q) dp 0 2πδ(q 0 p 0 ω) 2π 2π ω + i0 + ρ αβ (p 0, q) dp 0 π ρ αβ (p 0, q) p 0 q 0 i0 + and for the advanced propagator in Eq. (4.13) Knowing Π A αβ (Q) = and assuming ρ αβ is real, one can find dp 0 π (4.24) ρ αβ (p 0, q) p 0 q 0 + i0 +. (4.25) ( ) 1 1 ± i0 + = P iπδ( ), (4.26) Im Π R αβ (Q) = ρ αβ(q), Im Π A αβ (Q) = ρ αβ(q). (4.27) As the real parts of Π R αβ (Q) and ΠA αβ (Q) coincide, one obtains i[π R αβ (Q) ΠA αβ (Q)] = 2ρ αβ(q). (4.28) Analogously with the use of Eq. (4.26) it follows that Π T αβ (Q) = dp 0 π iρ αβ (p 0, q) q 0 p 0 + i0 + 2ρ αβ(q 0, q)n F (q 0 µ L ) (4.29) = iπ R αβ (Q) + Π< αβ (Q). Writing the correlator in Eq. (4.15) as a Wick rotation of the inverse Fourier transform of the left hand side of Eq. (4.8) and inserting Eq. (4.21) yields the spectral representation of the Euclidean correlator [36]: Π E αβ (Q) = dp 0 π ρ αβ (p 0, q) p 0 i[ q 0 iµ L ]. (4.30) This relation between the Euclidean propagator and the spectral function can be inverted by using Eq. (4.26) and thus the spectral function can be written as the discontinuity of the Euclidean propagator across the real axis: ρ αβ (q 0, q) = 1 2i Disc ΠE αβ ( q0 iµ L i[q 0 + i0 + ], q) = 1 2i Disc ΠE αβ ( iq0, q) (4.31) 1 2i [ Π E αβ ( i[q 0 + i0 + ], q) Π E αβ ( i[q0 i0 + ], q) ]. (4.32) 33

38 4 BASIC CONCEPTS OF THERMAL FIELD THEORY Combined with Eq. (4.27) this yields Im Π R αβ (q0, q) 1 2i Disc ΠE αβ ( iq0, q). (4.33) Comparing Eqs. (4.24) and Eq. (4.30) shows Π R αβ (Q) = ΠE αβ ( q0 iµ L i[q 0 + i0 + ], q). (4.34) Vice versa one can define (cf. [24]) Π E αβ ( i[q0 ± i0 + ], q) Re Π R αβ (q0, q) ± iim Π R αβ (q0, q). (4.35) The fermionic Matsubara sum over Eq. (4.30) can be carried out explicitly by making use of the relation (cf. [37]) T ω f 1 q 0 µ L iω f e iω f τ = β 0 dst ω f e iω f τ q 0 s+µ L s+iω f s = e (q0 µ L )τ n F (q 0 µ L ) (4.36) with the fermionic Matsubara frequencies ω f = 2πT (n ), n Z, and where we used the fact that we can write [37] T ω f e iω f τ = 2δ(τmod 2β) δ(τmod β) (4.37) and we assumed 0 < τ < β. Then it follows T ω f 1 q 0 µ L iω f e iω f τ = T ω f 1 q 0 µ L iω f e iω f (β τ) = e (β τ)(q0 µ L ) n F (q 0 µ L ). (4.38) As a consequence of these equations T e i(ω f +iµ L )τ (ω ω f + iµ L ) 2 + ω 2 = e µ Lτ T e iω f τ 1 (ω iω f ω f + µ L )(ω + iω f µ L ) f [ = e µ Lτ 2ω T ω f e iω f τ = 1 2ω A typical integral in this context yields 1 ω µ L + iω f + ω + µ L iω f [ n F (ω µ L )e (β τ)ω βµ L n F (ω + µ L )e τω]. 1 ] (4.39) β 0 dτe τ(iω f +α) = 1 + e βα iω f + α. (4.40) Further information, equations and relations can be found in [36, 37]. 34

39 5 Resonant Production Rate of Sterile Neutrinos In this Chapter we are going to derive an expression for the production rate of sterile neutrinos expressed through the active neutrino spectral function. In doing so we will use basic principles of thermodynamics and QFT as well as the basic principles of TFT introduced in Chapter 4. This section follows the lead of Refs. [24, 28, 34, 38]. 5.1 Derivation of the Production Rate For deriving the production rate we start with the Lagrangian in Eq. reminder (3.1), as a L = 1 2 N I i/ N I 1 2 M I N I N I F αi Lα φar N I F N αi φ I a L L α +L }{{}}{{} SM, (5.1) L L int S and the assumption, that the primordial lepton asymmetry is the same in all active species (ν e, e L, e R, ν µ, µ L, µ R, ν τ, τ L, τ R ). Strictly speaking this is not true, since flavour equilibrium through active neutrino oscillations is not expected to be reached before temperatures below 10 MeV [39, 40]. But assuming flavour equilibrium keeps the discussion simpler. Deviations because of nonequilibrated flavours will be discussed in Sec Furthermore, there is no lepton asymmetry reservoir, because the CPviolating reactions generating lepton asymmetry take place at temperatures above a few GeV (cf. [4]). Therefore, one also has to consider the depletion of the lepton asymmetry in time. We will not discuss this here further, see Ref. [34] for a detailed discussion. We assume as an initial condition that the concentration of sterile neutrinos at very high temperatures T 1 GeV was zero. This keeps the calculation simpler and notation clearer. Including a general ensemble of sterile neutrinos as initial state however enables to consider backreactions due to already produced sterile neutrinos during the production. These backreactions are completely neglected in assuming a zero concentration of the sterile neutrinos as initial condition. See Ref. [34] for results for an arbitrary ensemble of sterile neutrinos as initial condition. In addition the lightest sterile neutrino should never equilibrate due to the smallness of its Yukawa coupling (cf. Sec. 3.2). The production rate can be derived in analogy with fluctuation-dissipation relations, while disregarding the expansion of the universe first and inserting it later by hand. The Liouville-von Neumann equation reads i dˆρ(t) dt = [Ĥ, ˆρ(t)], (5.2) 35

40 5 RESONANT PRODUCTION RATE OF STERILE NEUTRINOS with ˆρ(t) the density matrix for the extended SM by sterile neutrinos and Ĥ its Hamilton operator. We split the Hamilton operator into Ĥ = ĤSM + ĤS + Ĥint, (5.3) i.e. the complete Hamiltonian of the SM, the free Hamiltonian of the sterile neutrinos Ĥ S and the interaction Hamiltonian Ĥint, describing the interactions between the sterile neutrinos and the particles of the SM. For solving (5.2) we assume the initial condition that the initial concentration of sterile neutrinos is zero, which is with the equilibrium density matrix of the SM ˆρ(0) = ˆρ SM 0 0, (5.4) ˆρ SM = Z 1 SM exp[ β(ĥsm µ B ˆB µl ˆLSM )], (5.5) where Z SM is the partition function of the SM β = 1/T, ˆB the baryon number operator, µ B the baryon chemical potential, µ L 0 the total lepton chemical potential and ˆL SM the total lepton number operator within the SM, ˆL SM d 3 x ] [ˆ lαl γ 0ˆlαL + ˆ lαr γ 0ˆlαR + ˆ ν αl γ 0ˆν αl, (5.6) α=e,µ,τ with ˆl α α and ˆν α, α {e, µ, τ}, the lepton and active neutrino operator respectively and ˆψ L a L ˆψ, ˆψR a R ˆψ. Thus, Eq. (5.4) describes a system with no sterile neutrino and chemically equilibrated lepton asymmetry between the SM particles. We will regard Ĥ 0 = ĤSM + ĤS of Eq. (5.3) as a free Hamiltonian and thus Ĥint as the interaction one. Then Eq. (5.2) for the density matrix in the interaction picture, i.e. ˆρ I exp(iĥ0t)ˆρ exp( iĥ0t), reads i dˆρ I(t) dt = [ĤI(t), ˆρ I (t)], (5.7) which can be verified easily. In Eq. (5.7) ĤI(t) = exp(iĥ0t)ĥint exp( iĥ0t) is the interaction Hamiltonian in the interaction picture. The time evolution of ˆρ I (t) can be computed in perturbation theory with respect to ĤI(t): ˆρ I (t) = ˆρ 0 i t 0 ˆρ I (t) = ˆρ 0 i t 0 dt [ĤI(t ), ˆρ I (t )] t t dt [ĤI(t ), ˆρ 0 ] + ( i) 2 dt dt [ĤI(t ), [ĤI(t ), ˆρ 0 ]] + O(Ĥ3 I ), 0 0 (5.8) where ˆρ 0 ˆρ I (0) = ˆρ(0). One has to consider that perturbation theory breaks down at time t t eq, where sterile neutrinos enter thermal equilibrium and their concentration 36

41 5.1 Derivation of the Production Rate needs to be computed differently. Here only times t t eq are of interest, because we consider sterile neutrino DM, which never thermally equilibrates (cf. Sec. 3.2, [24]). The distribution function of sterile neutrinos is associated with the operator d ˆN I (t, x, q) d 3 xd 3 q 1 V s=±1 â I;q,sâI;q,s, (5.9) where V is the volume of the system, â I;q,s is the creation and â I;q,s the annihilation operator of a sterile neutrino of type I, momentum q and spin state s, obeying the anticommutation relation {â I;p,s, â J;q,t } = δ(3) (p q)δ IJ δ st. (5.10) From now on just the production rate of the sterile neutrino of type I = 1, which correspond to the DM candidate sterile neutrino, will be considered. The phase space distribution function f 1 (t, q) is given by the expectation value of the operator in Eq. (5.9): f 1 (t, q) dn 1(x, q) d 3 xd 3 q = Tr [ d ˆN 1 d 3 xd 3 q ˆρ I(t) ] = d ˆN 1 (t, x, q) d 3 xd 3 q, (5.11) where the last expression denotes the expectation value with respect to ˆρ I (t). The production rate can be obtained by the derivative of Eq. (5.11) with respect to time. Keeping in mind that ĤI is linear in â I;q,s and â I;q,s (cf. Eqs. (5.13)-(5.15)), one gets no contribution for the term linear in ĤI by inserting Eq. (5.8) in Eq. (5.11). Furthermore, ˆρ 0 does not give rise to any contribution because it is time-independent. Therefore, the rate of the sterile neutrino production to second order is f 1 (t, q) = dn 1(x, q) d 4 xd 3 q = 1 V Tr { s=±1 â 1;q,sâ1;q,s t 0 dt [ĤI(t), [ĤI(t ), ˆρ 0 ]] }. (5.12) By only considering small temperatures T m W, with m W the mass of the W - boson, one can substitute the Higgs field in ĤI with its vacuum expectation value, so that Eqs. (5.1) and (3.3) combined yield ] Ĥ I = d 3 x [(M D ) α1ˆ ν α a R ˆN1 + (MD) α1 ˆ N1 a Lˆν α, (5.13) where ˆν α, α {e, µ, τ}, again are the active neutrino operators. ˆN1 is a Majorana spinor field operator of the sterile neutrino which evolves with the free Hamiltonian ĤS and therefore can be expressed as a free on-shell field operator d ˆN 3 p [ 1 (x) = â 1;p,s u 1;p,s e ip x + â (2π) 3 2p 0 1;p,s v 1;p,se ip x], (5.14) ˆ N 1 (x) = d 3 p (2π) 3 2p 0 s=±1 s=±1 [ â 1;p,sū1;p,se ip x + â 1;p,s v 1;p,s e ip x], (5.15) 37

42 5 RESONANT PRODUCTION RATE OF STERILE NEUTRINOS obeying the anticommutation relation in Eq. (5.10), with p 0 E p p 2 + M1 2 and P = (p 0, p). The spinors u, v satisfy the completeness relations u 1;p,s ū 1;p,s = /P + M 1, (5.16) s v 1;p,s v 1;p,s = /P M 1, (5.17) s and as they are describing Majorana fermions they behave under charge conjugation as u = C v T, v = Cū T, with C the charge conjugation matrix. Inserting the free field operators into Eq. (5.13) and sorting according to the exponentials yields Ĥ I = d 3 x s=±1 + d 3 p (2π) 3 2p 0 (5.18) { ) ((M D ) α1ˆ ν α a R â 1;p,s v 1;p,s + (MD) α1 â ip x 1;p,sū1;p,sa Lˆν α e ((M D ) α1ˆ ν α a R â 1;p,s u 1;p,s + (M D) α1 â 1;p,s v 1;p,s a Lˆν α ) e ip x}. Considering that the creation operator of the SM and of the sterile neutrino anticommute, as well as the respective annihilation operators, and by introducing Ĵ 1;p,s (x) (M D ) α1ˆ ν α (x)a R v 1;p,s + (M D) α1 ū 1;p,s a Lˆν α (x), (5.19) one can establish a shorter notation for ĤI Ĥ I = d 3 d 3 p [ x â 1;p,sĴ1;p,s(x)e ip x + (2π) 3 2p Ĵ 0 1;p,s (x)â 1;p,se ip x]. (5.20) s=±1 The next step is to insert Eq. (5.20) into the equation for the production rate in Eq. (5.12). We obtain an equation of the structure } Tr {Â[ ˆB, [ Ĉ, ρ SM 0 0 ]], (5.21) where  = ˆB = Ĉ = s=±1 â 1;q,sâ1;q,s, (5.22) d 3 x d 3 x d 3 p (2π) 3 2p 0 d 3 p (2π) 3 2p 0 m=±1 [ â 1;p,mĴ1;p,m(x)e ip x + Ĵ 1;p,m (x)â 1;p,me ip x], (5.23) [ â 1;p,nĴ1;p,n(x )e ip x + Ĵ 1;p,n (x )â 1;p,ne ip x ]. n=±1 (5.24) 38

43 5.1 Derivation of the Production Rate Considering the cyclicity of the trace and Tr(X Y ) = Tr(X)Tr(Y ) the structure of Eq. (5.21) can be simplified as } } Tr {Â[ ˆB, [ Ĉ, ρ SM 0 0 ]] = Tr(ρ SM )Tr {Â[ ˆB, [ Ĉ, 0 0 ]], } ( Tr {Â[ ˆB, [ Ĉ, 0 0 ]] = Tr {Â ˆBĈ 0 0 ˆB 0 0 Ĉ Ĉ 0 0 ˆB Ĉ ˆB )} = 0 {Â ˆBĈ ĈÂ ˆB } ˆBÂĈ + Ĉ ˆBÂ 0. (5.25) When inserting the expressions of Eqs. (5.22)-(5.24) only terms of the structure 0 ââ ââ 0 yield non-zero contributions, which correspond to the second and the third term in Eq. (5.25). The production rate Eq. (5.12) then becomes f 1 (t, q) = 1 t dt d 3 x d 3 x d 3 p d 3 p {ˆρ V (2π) 3 2p 0 (2π) 3 2p Tr SM 0 s,m,n=±1 0 [Ĵ 1;p,n (x )Ĵ1;p,m(x)e ip x ip x 0 â 1;p,nâ 1;q,sâ1;q,sâ 1;p,m 0 + Ĵ 1;p,m (x)ĵ1;p,n(x )e ip x+ip x ] } 0 â 1;p,m â 1;q,sâ1;q,sâ 1;p,n 0. (5.26) The expectation values evaluate due to the anticommutation relation of the operators given in Eq. (5.10) to 0 â 1;p,nâ 1;q,sâ1;q,sâ 1;p,m 0 = δ(3) (p q)δ ns δ (3) (q p)δ sm and we obtain for the production rate f 1 (t, q) = 1 1 t V (2π) 3 dt 2E q s=±1 0 = 0 â 1;p,m â 1;q,sâ1;q,sâ 1;p,n 0, (5.27) d 3 x d 3 x Ĵ 1;q,s (x )Ĵ1;q,s(x)e iq (x x ) + Ĵ 1;q,s (x)ĵ1;q,s(x )e iq (x x), (5.28) where the expectation value is meant to be with respect to ˆρ SM as usual. It must be pointed out that there is no annihilation or creation operator of the sterile neutrino left in this expression. Inserting the definitions for Ĵ and Ĵ, and generalizing α, β to not only count for the generation indices but also for the Dirac indices, one obtains correlators of the form ˆ ν β (x )ˆ ν α (x) and ˆν β (x )ˆν α (x), which must vanish, since the processes which generated the lepton asymmetry, ceased at temperatures above a few GeV [28] and lepton numbers are conserved within the SM. This then leads to f 1 (t, q) = 1 1 t V (2π) 3 dt d 3 x d 3 x (M 2E D) α1 (M D ) β1 q s=±1 0 [ v1;q,s a Lˆν α (x )ˆ ν β (x)a R v 1;q,s + ˆ ν β (x)a R u 1;q,s ū 1;q,s a Lˆν α (x) ] e iq (x x ) + ( x x ). (5.29) 39

44 5 RESONANT PRODUCTION RATE OF STERILE NEUTRINOS Now one can use the standard completeness relations for the spinors u, v (5.16), (5.17). The resulting mass terms M 1 get projected out by a L a R = 0 = a R a L. Hence, f 1 (t, q) = 1 1 t V (2π) 3 dt d 3 x d 3 x (M 2E D) α1 (M D ) β1 q 0 (a R /Qa L ) βα [ˆνα (x )ˆ ν β (x) + ˆ ν β (x )ˆν α (x) ] e iq (x x ) + ( x x ), (5.30) where α, β again can counted be for the generation indices or the Dirac indices. Using the translational invariance of the two-point correlation functions defined in Sec , ˆν α (x)ˆ ν β (x ) = f(x x ) = f( x ( x)) = ˆν α ( x )ˆ ν β ( x), (5.31) one can invert Eq. (4.8) ˆν α (x )ˆ ν β (x) = ˆν α ( x )ˆ ν β ( x) = P P = d 4 P (2π) 4 eip (x x ) Π > αβ (P ) d 4 P (2π) 4 e ip (x x ) Π > αβ ( P ), (5.32) and analogously Eq. (4.9). This yields for the total correlator of Eq. (5.30) d ˆν α (x )ˆ ν β (x) + ˆ ν β (x 4 P [ ] )ˆν α (x) = (2π) 4 e ip (x x ) Π > αβ ( P ) Π< αβ (P ), (5.33) and analogously with (x x ). Therefore, in total we obtain f 1 (t, q) = 1 1 t d V (2π) 3 dt d 3 x d 3 x 4 P 2E q 0 (2π) 4 (M D) α1 (M D ) β1 {(a [ ] R /Qa L ) βα e ip (x x ) Π > αβ ( P ) Π< αβ (P ) e iq (x x ) + ( x x ) }. (5.34) The next step is to carry out the integral over space and time coordinates and taking the limit t, so both terms of Eq. (5.34) yield t [ ] dt d 3 x d 3 x e i(q P )(x x ) + e i(p Q)(x x ) lim t 0 t [ = V (2π) 3 δ 3 (p q) lim dt e i(q0 p 0 )(t t ) ] + e i(q0 p 0 )(t t ) t 0 { 0 = V (2π) 3 δ 3 (p q) lim dt [ ] } e i(p0 q 0 )t + e i(p0 q 0 )t t }{{} t t t = V (2π) 3 δ 3 (p q) d te i(p0 q 0 ) t = V (2π) 4 δ 4 (P Q), (5.35) 40

45 5.2 Active Neutrino Spectral Function and Euclidean Propagator which removes the P -integration and cancels 1/V in Eq. (5.34). Returning to the convention that α, β label generations and expressing the Dirac part through a trace, one gets for Eq. (5.34) f 1 { ] } 1 (t, q) = (2π) 3 (M 2E D) α1 (M D ) β1 Tr /Q [Π > αβ ( Q) Π< αβ (Q) a R. (5.36) q Using Eqs. (4.21), (4.22) we can express the correlators through the active neutrino spectral function and employing that lepton generation conservation within the SM limits the indices α, β to be equal, one ends up with the master relation f 1 (t, q) = R(T, q) 1 = (2π) 3 q 0 α=e,µ,τ { M D 2 α1tr /Q [ n F (q 0 µ L )ρ αα (Q) + n F (q 0 + µ L )ρ αα ( Q) ] } a R, (5.37) where n F 1/[exp(βx)+1] is the Fermi-Dirac distribution, ρ αα is the spectral function related to the active neutrino of generation α (cf. Eq. (4.10)), and Q is the on-shell four-momentum of the right-handed neutrino, i.e. Q 2 = M 2 1 and E q = q 0. In an expanding universe, the physical momenta get redshifted, q(t) = q(t 0 )a(t 0 )/a(t), where a(t) is the scale factor (cf. Sec. 2.1). Therefore, the time derivative has to be replaced with d/dt = / t Hq i / q i. Thus, the phase space density of Eq. (5.11) obeys the equation ( ) t Hq i f 1 (t, q) = R(T, q). (5.38) q i 5.2 Active Neutrino Spectral Function and Euclidean Propagator Now one needs to derive an expression for the active neutrino spectral function ρ αα within the SM. The spectral function can be obtained in the imaginary-time formalism by a certain analytic continuation of the Euclidean active neutrino propagator, as has already been seen in Sec The Euclidean propagator is in general defined as Π E αα( Q) 1 = a L i /Q + i /Σ( Q) a i R = a /Q i /Σ( Q) L [ Q Σ( Q)] a 2 R, (5.39) without assuming any symmetry properties of the active neutrino self-energy Σ. With a Wick rotation one can transform it into a retarded Minkowskian propagator (cf. Eq. (4.34)): Π R αα(q 0, q) = Π E αα( iq 0 /Q + /Σ( Q), q) = a L Q 2 2Q Σ( Q) + Σ 2 ( Q) a R. (5.40) 41

46 5 RESONANT PRODUCTION RATE OF STERILE NEUTRINOS Writing the neutrino self-energy as Σ(q 0 ± i0 +, q) Re Σ(Q) ± iim Σ(Q) and hence Σ( [q 0 ± i0 + ], q) Re Σ( Q) iim Σ( Q), it follows from Eqs. (4.32) and (4.33) that [ Π R αα (q 0 + i0 +, q) Π R αα(q 0 i0 +, q) ] (5.41) ρ αα (Q) = 1 2i with Analogously = a L S I ( Q)[ /Q Re /Σ( Q)] S R ( Q)Im /Σ( Q) S R ( Q) 2 + S I ( Q) 2 a R, (5.42) S R ( Q) [Q Re Σ( Q)] 2 [Im Σ( Q)] 2, (5.43) S I ( Q) 2[Q Re Σ( Q)] Im Σ( Q). (5.44) ρ αα ( Q) = 1 2i [ Π R αα ( q 0 + i0 +, q) Π R αα( q 0 i0 +, q) ] (5.45) with = a L S I (Q)[ /Q + Re /Σ(Q)] S R (Q)Im /Σ(Q) S R (Q)2 + S I (Q)2 a R, (5.46) S R(Q) [Q + Re Σ(Q)] 2 [Im Σ(Q)] 2, (5.47) S I(Q) 2[Q + Re Σ(Q)] Im Σ(Q). (5.48) The next step is to perform some simplifications. The chemical potential changes the thermal distribution functions of the on-shell leptons that appear in the intermediate states, in the imaginary part Im Σ. Due to the fact that the case µ L /T 1 is of interest, these changes are not important and will be neglected. Therefore, one assumes that Im Σ( Q) = Im Σ(Q), as for µ L = 0 [38]. In Chapter 7 an analysis of the effect of a large lepton asymmetry on the chemical potential is performed. We will see there, that µ L /T 1 is satisfied for the required lepton asymmetry due to the values for the mass and mixing angle in Eq. (3.10), i.e. n νe /s O(10 6 ) with s the entropy density parametrised by Eq. (2.64) (cf. Sec. 5.3). The general structure of the real part Re /Σ αα (Q) can be written as [28] Re /Σ αα (Q) = /Qa αα (Q) + /ub αα (Q) + /uc αα (Q), (5.49) where u = (1, 0) is the plasma four-velocity. The first term with a αα (Q) can be neglected, because it is small compared to the tree-level term /Q. The other terms need to be kept at tree-level [24]. In doing so, the term proportional to /u has been split for symmetry reasons. The function b αα (Q) is defined to be odd in Q and the function c αα (Q) to be even in Q. b αα (Q) is independent of the chemical potential and for q m W reads [38] b αα (Q) = 16G2 F πα w q 0 [ 2φ(m α ) + cos 2 θ W φ(m να ) ], (5.50) 42

47 5.2 Active Neutrino Spectral Function and Euclidean Propagator with m α the mass of the charged lepton of generation α {e, µ, τ} and m να = 0 the mass of the SM active neutrino. Moreover the function [38] d 3 [ ] p n F (E) 4 φ(m) = (2π) 3 2E 3 p 2 + m 2 (5.51) E= p 2 +m 2 is finite and can be evaluated numerically, where φ(0) = 7π 2 T 4 /360. See Fig. 6.1 for a plot of b αα (Q). The function c αα (Q) must be proportional to the lepton chemical potential, i.e. the lepton asymmetry, and is therefore essential. The expression for c αα (Q) after a first order expansion in 1/m 2 W and under the assumption, that all active leptonic asymmetries are equal, i.e. n νe = n el = n er = n µl =..., can be found in [28]: c αα = 3 2G F (1 + 4 sin 2 θ W )n νe, (5.52) where G F is the Fermi constant and θ W is the electroweak mixing angle. The case for nonequilibrated leptonic flavours is considered in Sec With these assumptions one can rewrite /Q + Re /Σ(Q) /Q + /u(b + c), (5.53) [ 2 Q + Re Σ(Q)] M q 0 (b + c) + (b + c) 2, (5.54) /Q Re /Σ( Q) /Q + /u(b c), (5.55) [ 2 Q Re Σ( Q)] M q 0 (b c) + (b c) 2, (5.56) with the abbreviations b b αα (Q), c c αα (Q) here and below. In addition one can introduce [ ] I Q Tr /Qa L Im /Σ(Q)a R = 2Q Im Σ(Q), (5.57) ] I u Tr [/ua L Im /Σ(Q)a R = 2u Im Σ(Q). (5.58) Keeping in mind that Im /Σ(Q) has the same Lorentz structure as Re /Σ in Eq. (5.49) with Im /Σ /Qα(Q) + /uβ(q), (5.59) [Im Σ] 2 can be written with the help of Eqs. (5.57), (5.58), as [ ] 2 IQ 2 Im Σ(Q) + 2q0 I Q I u M1 2I2 u = 4q 2. (5.60) With these simplifications the production rate in Eq. (5.37) can finally be written as R(T, q) 1 (2π) 3 q 0 M D 2 α1 α=e,µ,τ { n F (q 0 + µ L ) 2S I(Q)[M1 2 + q0 (b + c)] S R (Q)I } Q SR 2 (Q) + S2 I (Q) + (c c, µ L µ L ) 43 (5.61),

48 5 RESONANT PRODUCTION RATE OF STERILE NEUTRINOS with S R (Q) = M q 0 (b + c) + (b + c) 2 + I2 Q 2q0 I Q I u + M1 2I2 u 4q 2 (5.62) S I (Q) = I Q + (b + c)i u. (5.63) Eq. (5.61) is the master formula which will be the starting point for the following discussions. The given expressions are valid beyond perturbation theory, but are in practice used with approximate perturbative expressions for the functions b, c and I Q (cf. Refs. [28, 34, 38]). 5.3 Relic Dark Matter Density After obtaining a final distribution function f 1 for the sterile neutrinos by integrating Eq. (5.38) (cf. [34] for details) one has to relate it to the present DM energy density. As today sterile neutrinos are non-relativistic, their energy density is given by (cf. Eqs. (2.7) and (2.42)) d 3 q ρ 1 = Ω 1 ρ c = 2 0 (2π) 3 M 1f 1,0, (5.64) where f 1,0 is their distribution function today at temperature T 0. The dark matter density can be written according to Eq. (2.7) as with ρ c the critical energy density of Eq. (2.4). Thus ρ DM,0 = Ω DM ρ c, (5.65) ρ 1 ρ DM,0 = Ω 1 Ω DM = 2M 1 Ω DM ρ c d 3 q 0 (2π) 3 f 1,0. (5.66) As we want sterile neutrinos to account for all DM this ratio has to be equal to 1. Thus for a fixed total mixing angle sin 2 2θ we can get a corresponding value for the required asymmetry n νe /s according to Eq. (5.66) to satisfy that sterile neutrinos can account for all DM. With the values for the mass and mixing angle according to Eq. (3.10) and equilibrated leptonic flavours one obtains [34] n νe s = for M 1 = 7.1 kev, sin 2 2θ = , (5.67) where only non-vanishing Yukawa couplings to the electron flavour were assumed. For nonequilibrated leptonic flavours a higher lepton asymmetry for a fixed mixing angle is needed to obtain the correct amount of DM. For a detailed analysis for different scenarios, i.e. different mixing angles and equilibrated and nonequilibrated leptonic flavours, we refer to [34]. 44

49 5.4 Nonequilibrated Leptonic Flavours 5.4 Nonequilibrated Leptonic Flavours We restricted our analysis of the production rate to the case of equilibrated leptonic flavours and thus equilibrated lepton asymmetries and the same chemical potential µ L for all flavours. We did this for simplicity reasons and in concordance with Refs. [24, 28, 38]. In fact, flavour equilibrium through active neutrino oscillations is not reached at the temperatures of interest around T 200 MeV, where the production rate of sterile neutrinos obtains a resonance (cf. Chapter 6). Flavour equilibrium will only be reached at about temperatures below 10 MeV [39], as well for large lepton asymmetries [40]. Hence, in general one has to assign different lepton asymmetries and subsequently different chemical potentials to the different flavours, i.e. µ L µ Lα, α {e, µ, τ} (cf. Chapter 7). This will lead to a change in the density matrix of the SM in Eq. (5.5): ˆρ SM = Z 1 SM exp[ β(ĥsm µ B ˆB α=e,µ,τ µ Lα ˆLα )], (5.68) where ˆL α is the lepton number operator of the SM associated with the flavour α (cf. Eq. (5.6)), ] ˆL α d 3 x [ˆ lαl γ 0ˆlαL + ˆ lαr γ 0ˆlαR + ˆ ν αl γ 0ˆν αl. (5.69) This modification will at the end lead to a change in the master relation in Eq. (5.37): f 1 1 (t, q) = (2π) 3 q 0 α=e,µ,τ { M D 2 α1tr /Q [ n F (q 0 µ Lα )ρ αα (Q) + n F (q 0 + µ Lα )ρ αα ( Q) ] } a R. (5.70) Furthermore, in Sec. 5.2 we also assumed equilibrated leptonic flavours in the derivation of the function c αα (Q) in Eq. (5.52). For nonequilibrated leptonic flavours the function reads [28] c αα = 2G F n βl + 2s W n βr + 2n να + n νβ, β α (1 + 2s W ) n αl (1 2s W ) β α β (5.71) where s W sin 2 θ W. Especially this function complicates the (analytic) calculation for nonequilibrated leptonic flavours, because it yields a lot more unknown parameters, i.e. the different number asymmetries, than in the case of equilibrated leptonic flavours. For an analysis of the production rate with non-equilibrated leptonic flavours see [34]. 45

50 6 POINT OF THE RESONANCE 6 Point of the Resonance In this Chapter we will first simplify the production rate in Eq. (5.61). Therefore we will discuss some assumptions and estimates to obtain a simpler equation to estimate the order of magnitude of the resonant production in Sec Then we will cross-check if the assumptions made during the calculation of the production rate are really valid at the point of the resonance. Therefore, we compare the order of magnitude of the resonant production rate at the resonance with the other important rates during the production, i.e. the Hubble parameter (Sec ) and the active neutrino scattering rate (Sec ). We will conclude with the analysis of the impact of the non-thermal spectrum of the sterile neutrinos in Sec In doing so, we will consider equilibrated leptonic flavours throughout the whole section for simplicity. Even though this is not true at temperatures T 10 MeV, as discussed in Sec. 5.4, this is sufficient for obtaining an estimate of the order of magnitude for the resonant production rate R(T, q). 6.1 Order of Magnitude of the Resonant Production We go on examining Eq. (5.61) to simplify it. As already stated, the production of the dark matter sterile neutrinos, with masses in the kev range, takes place at temperatures below a few GeV. Therefore, it is a good approximation to set the term I u, defined in Eq. (5.58), in Eqs. (5.62), (5.63) to zero, thus they simplify to S R (Q) = M q 0 (b + c) + (b + c) 2 + I2 Q 4q 2, (6.1) S I (Q) = I Q. (6.2) Eq. (6.1) can be simplified further by recognizing that the last term is much smaller than the other terms, as I Q is small and q 2 = Q 2 + q 2 0 = M q2 0 q2 0 T 2 is large (cf. Eqs. (6.6), (6.12), (6.13)). Thus, the simplified production rate reads R(T, q) 1 (2π) 3 q 0 α=e,µ,τ M D 2 α1 { n F (q 0 I Q [M1 2 + µ L ) (b + c)2 ] [M q0 (b + c) + (b + c) 2 ] 2 + IQ 2 }{{} R + n F (q 0 I Q [M1 2 µ L ) (b c)2 ] } [M q0 (b c) + (b c) 2 ] 2 + IQ 2. }{{} R + (6.3) 46

51 6.1 Order of Magnitude of the Resonant Production b αα /G F 2 T 4 q α = e α = μ α = τ T [ev] Fig. 6.1: Function b αα (Q) of Eq. (5.50) in units of G F T 4 q 0 as a function of temperature and active neutrino flavour α {e, µ, τ}. This production rate has a resonance where the denominator is as small as possible, thus the function F(T ) M q 0 (b ± c) + (b ± c) 2 ) (6.4) = M q 0 (b c ) + (b c ) 2 ) (6.5) vanishes. Which sign one has to account for in Eq. (6.4) depends on whether the function c is positive or negative. In Eq. (6.4) + corresponds to a negative c and to a positive c which leads to Eq. (6.5). As the function c is either positive or negative, a resonance can only occur in one of the terms R or R + of Eq. (6.3) at a time. The next question is how the function F(T ) behaves. Thus, one has to analyse the functions b and c. The function b αα of Eq. (5.50) is plotted in Fig. 6.1 for α {e, µ, τ}. Analysing the plots and considering a resonance temperature of T R O(10 2 ) MeV, one gets an approximated function for b of b 80G 2 F T 4 q 0. (6.6) The exact numerical pre-factor of b is not very important here, since just the order of magnitude of the resonance will be determined. The function c in Eq. (5.52) can also be expressed in terms of the specific lepton asymmetry nνe s η νe. Knowing that s is parametrized by the effective number of relativistic degrees of freedom h eff (cf. Eq. (2.64)), one ends up with c αα (T ) = 2 2π 2 G F (1 + 4 sin 2 θ W )h eff η νe T 3. (6.7) 15 47

52 6 POINT OF THE RESONANCE F(T) T [ev] Fig. 6.2: Function F(T ) of Eq. (6.4) as a function of temperature for M 1 = 7.1 kev, η νe = and q 0 T (cf. Sec. 6.3). In the zoom-in one can identify the first root of F(T ) at T ev. So for very low temperatures T M 1, F(T M 1 ) M1 2 and therefore is positive. At very large temperatures T M 1, the functions can be approximated as b T 4 and c T 3 and thus F(T M 1 ) q 0 b is positive as well. At intermediate temperatures the function c overtakes the others for sufficiently large η νe and sufficiently small q 0 /T, which is satisfied as sterile neutrinos are produced non-thermally, which leads to two roots of F(T ). Fig. 6.2 demonstrates this behaviour of the function F(T ). One can also find these roots by a simple calculation (cf. [34]). First, take F(T ) at fixed temperature and consider F(T ) as a function of q 0. The function b is linear in q 0 and c is constant, therefore one can introduce b = bq 0. Thus, Eq. (6.4) becomes F = M bq 2 0(2 + b) 2q 0 c (1 + b) + c 2! = 0. (6.8) Which yields the solutions q± 0 = c (1 + b) ± c 2 M1 b(2 2 + b). (6.9) b(2 + b) Thus, there are roots of Eq. (6.4) for c 2 > M1 b(2 2 + b). The Jacobian at q 0 = q±, 0 F(T ) q 0 c = 2 2 M1 b(2 2 + b), (6.10) is non-zero and therefore the solution in Eq. (6.9) can be inverted and taken for fixed q 0 and variable temperature T, which yields two real roots, as can be seen in the plot 48

53 6.2 Comparison of R(T R ) with other Rates of F(T ) in Fig By an analysis of Eq. (6.3) one can see that the root at the lower temperature gives the dominant contribution to the production of sterile neutrinos. Considering the order of magnitude of the rate R(T, q) at the resonance in Eq. (6.3) one has to make several estimates: As mentioned before M 1 T and µ L /T 1. In general the imaginary part of the neutrino self-energy can be expressed as a function Im Σ = G 2 F T 4 q 0 f(q, T ), where f is a non-trivial dimensionless function, which is numerically of order unity [38]. Recalling the Lorentz structure of Im /Σ in Eq. (5.59) then yields QIm Σ α(q)q 2 + β(q)t = α(q)m β(q)t M 1 T β(q)t, (6.11) where it was used that Q 2 = M1 2 is the on-shell four-momentum of the sterile neutrino and α(q) is small compared with the tree-level term /Q, and finally I Q = 2QIm Σ 2G 2 F T 4 q 2 0. (6.12) In the function c of Eq. (6.7) one can approximated (1 + sin 2 θ W ) 2, where θ W is the Weinberg angle. Thus, c 4 2π 2 G F h eff η νe T 3 3.7G F h eff η νe T 3. (6.13) Comparison of R(T R ) with other Rates For an analytic comparison of the order of magnitude of the production rate R(T R ) at the temperature of resonance T R with other rates one can simplify Eq. (6.3) further by neglecting terms of O(b 2, c 2, bc). The production rate then peaks to a good approximation where the function F (T ) M q 0 (b c ) (6.14) vanishes. We are considering the case of a positive function c > 0, the case c < 0 results analogously. Therefore, the term R in the production rate in Eq. (6.3) is small compared to the term R + and thus the production rate can be approximated as R(T, q) n F (q 0 µ L ) (2π) 3 q 0 n F (q 0 µ L ) (2π) 2 2q 0 α=e,µ,τ α=e,µ,τ I Q M D 2 α1m1 2 F 2 (T ) + IQ 2 (6.15) M D 2 α1m 2 1 δ(f (T )). (6.16) 49

54 6 POINT OF THE RESONANCE The second line is due to the fact that I Q is very small and consequently one can identify a representation of the Dirac delta-function (Cauchy-Lorentz-distribution). One has to remark that strictly speaking I Q is not infinitely small and therefore the resonance is not infinitely narrow like a delta distribution. Nevertheless, compared to the temperature range we are considering this is a good approximation, which will be discussed in Sec The temperature of resonance T R can now be calculated by setting F (T ) of Eq. (6.14) to zero. We then obtain M 2 1 = 2q 0 (c b). (6.17) Now one can make additional assumptions to simplify the calculation further: For a typically assumed lepton asymmetry of η νe , which yields the correct abundance of DM for sin 2 2θ = (cf. [34]), and a temperaturedependent effective number of relativistic degrees of freedom h eff (T ) (cf. [9]), the order of magnitude of the functions b and c of Eqs. (6.6), (6.13) can be approximated as b T 5 MeV 4, (6.18) T 3 c h eff (T ) MeV 2. (6.19) The dominance of c over b for temperatures up to T O(10 3 ) MeV becomes visible. For a higher lepton asymmetry c b is valid also for higher temperatures. For this reason it is possible to neglect the contribution of b in calculating the production rate at resonance, which corresponds to temperatures T R O(10 2 ) MeV. Even though sterile neutrinos are not produced thermally the deviation of the thermal spectrum is very small, so that it is a good approximation in calculating the order of magnitude of the resonance to assume q 0 T and q T. In Sec. 6.3 the impact of the non-thermal spectrum is analysed. We thus obtain the temperature of resonance T R with the help of Eqs. (6.17) and (6.13): M π 2 G F h eff η νe TR 4, 15 ( 8 ) 2π T R G F 15 h 4 eff η 4 ν e M 1 2 1, T R h eff η 4 ν e ( ) 1 M1 2. (6.20) kev Even though the effective degrees of freedom h eff (T ) strongly depend on T at the relevant temperatures due to the QCD phase transition at T QCD 200 MeV and 50

55 6.2 Comparison of R(T R ) with other Rates hence the right hand side in Eq. (6.20) is not independent of T, the variation of h eff is not so important as it only contributes as h 1/4 eff. Therefore, h eff will be considered to be of the order h eff 50 (cf. [9]) in the analytic calculation of the order of magnitude of the production rate. Now it is possible to calculate the order of magnitude of the production rate Eq. (6.15) at T R. For F (T ) = 0 we get With n F (q 0 µ L ) 1 M D 2 α1 = θ2 α1 M 2 1 R(T, q) n F (q 0 µ L ) (2π) 3 q 0 (cf. Eq. (3.2)) and the already stated assumptions follows 1 R(T R ) 8(2π) 3 G 2 F α=e,µ,τ M 4 1 T 7 R M D 2 M1 2 α1. (6.21) I Q α=e,µ,τ With the total mixing angle in Eq. (3.5) one obtains 4θ 2 α1. (6.22) R(T R ) 1 8(2π) 3 G 2 F M 4 1 T 7 R sin 2 2θ. (6.23) Inserting T R of Eq. (6.20) and G F yields ( ) R(T R ) ev sin 2 2θh 7 4 eff η M1 2 νe. (6.24) kev To receive a value only depending on the variables η νe, sin 2 2θ and M 1 kev, h eff 50 is inserted. Fixing the order of magnitude of the total mixing angle to the cosmological constraints discussed in Sec. 3.4, sin 2 2θ 10 11, one finally obtains: ( R(T R ) ev h 7 4 eff η 7 4 sin 2 ) ( 2θ M1 νe kev ( ev η 7 4 sin 2 ) ( 2θ M1 νe kev ) 1 2 ) 1 (6.25) Hubble Parameter An important parameter which one has to consider while dealing with interaction or production rates is the Hubble parameter. As long as the interaction or production rate 51

56 6 POINT OF THE RESONANCE is larger than the Hubble parameter the interaction or production occurs. If however the Hubble parameter is larger than the interaction or production rate, the interaction or production freezes out due to the fact that the expansion of the universe is faster than the interaction. The Hubble parameter at the early radiation, i.e. epoch is given by Eq. (2.35), as a reminder H(T ) = T 2 M P l 1.66 g eff T 2 relativistic matter, dominated M P l. (6.26) With Eq. (6.20) the Hubble parameter at the temperature of resonance is ( ) H(T R ) ev η 1 2 M1 ν e, (6.27) kev where it was used that g eff h eff (cf. Sec and Ref. [9]). Comparing the production rate R(T R ) to H(T R ), one obtains R(T R ) H(T R ) h 7 4 eff η 9 4 νe ( sin 2 ) ( ) 2θ 1 M kev 1 (6.28) for typical values for sin 2 2θ O(10 11 ), η νe O(10 6 ), M 1 O(keV) and h eff 50. Thus, the production rate of the sterile neutrinos is much bigger than the Hubble parameter R(T R ) H(T R ) and in fact the mixing of the active to sterile neutrinos occurs at the resonance temperature, for the here made assumptions and simplifications as q 0 T (cf. Sec. 6.3) Active Neutrino Scattering Rate We assumed that the SM particles, in particular the active neutrinos, stay in thermal equilibrium during the calculation of the production rate of the sterile neutrinos (cf. Sec. 5.1, Eq. (5.5)). Thus, the interaction between the active neutrinos has to be much larger than the production rate. The active neutrino scattering rate results from the scattering cross-section σ a G 2 F E2, where E is the energy of the active neutrino, which in this case can be roughly approximated as E T (cf. Eq. (2.54)). With the cross-section one can calculate the mean free time τ = 1/ σnv, where n T 3 is the number density for relativistic particles and v 1 is the velocity in units of the speed of light [6]. In general one obtains the scattering rate from Γ = 1/τ. Therefore, the active neutrino scattering rate is given by At the resonance temperature T R of Eq. (6.20) it yields Γ a (T ) G 2 F T 5. (6.29) Γ a (T R ) ev h eff η 4 ν e ( ) 5 M1 2. (6.30) kev 52

57 6.3 Impact of the Non-thermal Spectrum of Sterile Neutrinos Comparing the active neutrino scattering rate to the Hubble rate one receives Γ a (T R ) H(T R ) h eff η 4 ν e ( ) 3 M1 2 kev 1. (6.31) So the active neutrinos are in thermal equilibrium compared to the Hubble parameter. Comparing the production rate of the sterile neutrinos R(T R ) to the active neutrino scattering rate Γ a (T R ) yields R(T R ) Γ a (T R ) h 3 eff η3 ν e After inserting h eff 50 and η νe O(10 6 ), one obtains R(T R ) Γ a (T R ) ( η νe 10 6 ) 3 ( sin 2 ) ( ) 2θ 2 M (6.32) kev ( sin 2 ) ( ) 2θ 2 M (6.33) kev Thus the interaction of the active neutrinos stays in thermal equilibrium during the production of sterile neutrinos for typical values of sin 2 2θ O(10 11 ), η νe O(10 6 ) and M 1 O(keV). As long as the asymmetry η νe is small enough, i.e. η νe < O(10 3 ) at constant sin 2 2θ and M 1, the active neutrinos stay in thermal equilibrium. A very large lepton asymmetry would lead to a even more rapid conversion of the lepton asymmetry due to the production of the sterile neutrinos (cf. [?]). Moreover a much larger asymmetry is contradictory to a largest possible lepton asymmetry leading to successful Big Bang Nucleosynthesis (cf. Sec. 2.2) of η νe and the maximal asymmetry in the νmsm, which is η νe [28]. 6.3 Impact of the Non-thermal Spectrum of Sterile Neutrinos In the last two Sections 6.1 and 6.2 it was assumed that the energy of the produced sterile neutrinos is of order q 0 T. It was stated that the sterile neutrinos are produced non-thermally as first proposed by Shi and Fuller [17]. Nevertheless like in Ref. [28] it was assumed that the deviation of the thermal spectrum, E 3.151T (cf. Sec ), is very small and therefore the calculations in the previous sections have been performed with q 0 = T for determining the order of magnitude of the resonance of the production rate. However, one should cross-check the impact of the non-thermal, cold energy-spectrum of the sterile neutrinos. Therefore, an analysis of the impact of q 0 T on the resonance of the production rate in Eq. (6.3) is performed. Thereby only the assumptions and simplifications of Sec. 6.1 are used. Additionally, we used a simplified relation between the chemical potential µ L and the lepton asymmetry n α /s according to Eq. (2.59) up to first order in µ L : η νe = n ν e s 15 µ L 4π 2 h heff (T ) T, (6.34) 53

58 6 POINT OF THE RESONANCE which holds for T µ L, m. A more detailed analysis of the relation between the chemical potential and the lepton asymmetry is performed in Chapter 7. Furthermore, we used the values sin 2 2θ = , η νe = and M 1 = 7.1 kev (cf. [34]) and sterile neutrino energies q T = q 0 /T = E /T, where E 3.151T is the energy expectation value in thermal equilibrium (cf. Eq. (2.54)). As can be seen in the upper diagram in Fig. 6.3, a smaller energy q 0 of the sterile neutrinos shifts the point of the resonance towards higher temperatures and results in a higher production rate. It becomes visible that the height of the production rate is growing exponentially with smaller energy q 0. This reflects that resonantly produced sterile neutrinos obey a cold momentum-distribution. Furthermore, there is little variation of the location of the resonance temperature T R for q T. It is between T R ( ) MeV and thus always in the regime of the QCD phase transition. This stresses the importance of the QCD phase transition for the sterile neutrino production rate, again. Another aspect which is reflected in Fig. 6.3 is the relative smallness of the width of the resonance compared to the temperature regime it is located in. This aspect has been used in Sec. 6.2 to identify a representation of the Dirac delta-function in the production rate in Eq. (6.16). To examine whether active neutrino scattering does or does not occur during the resonant production of the sterile neutrinos, the width of the resonance, i.e. the period of the resonant production of sterile neutrinos t res, is calculated and compared to the mean free time τ = 1/Γ of the active neutrino scattering rate. If the mean free time of the active neutrinos is much shorter than t res, the active neutrinos are scattering and thus stay in thermal equilibrium during the production. Regarding the width of the resonance, the FWHM (Full Width at Half Maximum) has been calculated. As the production rate R(T, q) in Eq. (6.3) is given in dependence of temperature T and not time t, one can get the period of the resonance according to Eq. (2.36) as t res = M P l geff (T 1 )T1 2 1 geff (T 2 )T 2 2, (6.35) where T 1,2 are the temperatures at which the resonance declines to half its value. Using Eq. (2.36) for the determination of t res we assumed that we have a radiation dominated universe and neglected matter contributions to the equation of state of matter in Eq. (2.22). This is a relatively good approximation, since matter-radiation equality occurred at a colder temperature, T rm 0.76 ev (cf. Sec. 2.3). For simplicity in the calculation of Eq. (6.35) it was assumed again, that g eff (T ) h eff (T ) (cf. Sec and Ref. [9]). A study of the exact time-temperature relation in the presence of sterile neutrinos has been performed in [41]. The results for the temperature of the resonance T R, the height of the production rate R(T R ), the period of the resonance T res = T 2 T 1 and t res and the mean free time of the active neutrinos τ a (T R ) for energies between q 0 = ( )T = q T T in steps of 0.1T are listed in Tab. A.1 in Appendix A.1. First, we would like to stress again the fact that the period of the resonance T res is orders of magnitude smaller compared to 54

59 6.3 Impact of the Non-thermal Spectrum of Sterile Neutrinos R(T) [ev] T [ev] R(T) [ev] T [ev] q T =q 0 /T Fig. 6.3: Impact of different sterile neutrino energies q 0 on the resonance of the production rate (6.3) as a function of temperature for M 1 = 7.1 kev, η νe = and sin 2 2θ = [34]. Here q T = q 0 /T is changing from 0.01(dark red) (dark blue) in steps of 0.1. A general overview of the impact on the height and the location of the resonance is given in the upper diagram and a zoom-in is given in the lower diagram. 55

60 6 POINT OF THE RESONANCE T[eV] q T =q 0 /T Fig. 6.4: Behaviour of the temperature of resonance T R and the period of the resonance T res = T 2 T 1 for different sterile neutrino energies q T = q 0 /T for the simplified production rate in Eq. (6.3). T R T res [1/eV] t res τ a (T R ) 1/R(T R ) 1/H(T R ) q T =q 0 /T Fig. 6.5: Behaviour of the period of the resonance t res, the active neutrino scattering rate τ a (T R ), the inverse Hubble parameter 1/H(T R ) and the inverse of the production rate 1/R(T R ) for different sterile neutrino energies q T = q 0 /T for the simplified production rate in Eq. (6.3). 56

61 6.3 Impact of the Non-thermal Spectrum of Sterile Neutrinos the temperature of resonance T R. This is illustrated in Fig. 6.4 and could also already be seen in the lower diagram in Fig Second, comparing the period of the resonance t res to the active neutrino scattering rate τ a (T R ) in Fig. 6.5 we see that τ a (T R ) is orders of magnitude smaller than t res for all considered energies q 0 and therefore the active neutrinos stay in thermal equilibrium during the production of sterile neutrinos. We also calculated the Hubble parameter H(T R ) for the different energies to compare it to production rate at the resonance temperature, to cross-check if the expansion of the universe is slower than the production of the sterile neutrinos, i.e. R(T R ) > H(T R ) and 1/R(T R ) < 1/H(T R ) respectively. The results are also listed in Tab. A.1 in Appendix A.1. In Fig. 6.5 the inverse of the Hubble parameter 1/H(T R ) and the production rate 1/R(T R ) are plotted for different momenta of the sterile neutrinos q T = q 0 /T. We can see that for relatively small momenta, i.e. q T 1.5, the expansion of the universe is much smaller than the production of the sterile neutrinos, i.e. 1/R(T R ) < 1/H(T R ). However the production rate approaches the Hubble parameter and finally there is a crossing of the production rate and the Hubble parameter. Thus, in the approximation of the production rate in Eq. (6.3) no sterile neutrinos of momenta greater than q T (cf. Tab. A.1) are produced, because the expansion of the universe is faster than the production at the corresponding temperature T R. Together with the height of the resonant production rate R(T R ) (cf. Fig. 6.3 and Tab. A.1) this reflects the fact that lower momentum states are occupied more than higher momentum states. Consequently, we find that resonantly produced sterile neutrinos are produced with a cold momentum-distribution as stated by Shi and Fuller [17]. 57

62 7 LEPTON ASYMMETRIES AND CHEMICAL POTENTIALS Fig. 7.1: Sketch of a QCD phase diagram in the n B T plane for a fixed specific lepton asymmetry l. The exact phase diagram, whether it has a critical end point (T e, µ e ) is still under debate, as is the critical temperature T c (µ B = 0). Plot taken from [46]. 7 Lepton Asymmetries and Chemical Potentials In Sec. 6.3 we used a simplified relation between the lepton chemical potential and the lepton asymmetry for small chemical potentials µ L /T 1 and thus small lepton asymmetries according to Eq. (2.59) (cf. Sec ). However, it would be interesting to know which impact a large lepton asymmetry compared to the baryon asymmetry, l = α l α b, α {e, µ, τ}, has on the evolution of the lepton chemical potentials µ Lα and on the relation between the chemical potential and the asymmetry. Related studies has been performed by [34, 41]. In addition, the impact of a large lepton asymmetry on the baryon chemical potential µ B is worth studying, since it is well established that the order of the QCD phase transition depends on the baryon density, or equivalently on the baryon chemical potential µ B (cf. [42, 43, 44]). As discussed in Sec. 2.2 the SM predicts a spontaneous breaking of the chiral symmetry of QCD and confinement of quarks into hadrons at a (pseudo-)critical temperature T c. Thus, this is a very interesting point or epoch in the evolution of the universe. But only very little is known experimentally about this QCD transition at T c 165 MeV [45]. Starting with the critical temperature T c, going on with it is not known whether it is a real phase transition or a smooth crossover, even though there are hints for the smooth albeit rapid crossover due to lattice QCD (cf. e.g. [45]). Conveniently one draws a QCD phase diagram in the µ B T plane, where the cosmic QCD phase transition is assumed to take place at T = T c and µ B 0. In Fig. 7.1 the most current expected shape of the QCD phase diagram in µ B T plane is shown. Conveniently one assumes the lepton asymmetry to be of the order of the baryon asymmetry, l O(b), but as already stated in Sec the lepton asymme- 58