Raffelt in the course of the Elite Master Program

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(1)Master’s Thesis submitted for obtainment of the academic degree Master of Science (M.Sc.). keV Sterile Neutrino Dark Matter from Singlet Scalar Decays. by. Johannes König 16.09.2016 Advisor: Dr. habil. Georg G. Raffelt in the course of the Elite Master Program. Theoretical and Mathematical Physics.

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(3) Abstract Dark matter is one of the biggest mysteries in today’s particle- and astrophysics. A lot of observational evidence for it has been found so that there is hardly any doubt left on the existence of dark matter. Many different particle physics models, which for the most part can be put into a handful of basic categories, have been developed to explain this kind of matter. One of these classes is the set of models that try to use sterile neutrinos as the particles which constitute dark matter. There are several mechanisms that produce these sterile neutrinos in the early universe from the ubiquitous plasma, but some of them have issues to reproduce the measured properties of dark matter. The mechanism discussed in this work is the production of sterile neutrinos from the decay of a scalar particle. This process can generate distribution functions of the sterile neutrinos whose shapes are very different from thermal. Such distribution functions in turn open up new possibilities because they are not describable by one single number – the temperature T – but are more involved. Due to this fact, the whole distribution function has to be computed to adequately describe the situation and not just the number of the dark matter particles present today. This complication causes the need to solve the Boltzmann equation, the equation that is widely used in such settings, on the level of the distribution function. Many authors, however, employ it in an integrated form that does not give information on the distribution but only on its integral – the particle number density. We solve the Boltzmann equation for different parameter values of our model and finally compare our results to bounds mainly originating from observations regarding the formation of structure in the universe. Remark: Most of the material of this thesis is available as a preprint (ref. [1]) with Alexander Merle and Maximilian Totzauer as co-authors. Furthermore, it is also submitted to the Journal of Cosmology and Astroparticle Physics (JCAP).. 3.

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(5) Contents 1. Introduction. 7. 2. Theoretical background. 9. 2.1. Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.1.1. Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.1.2. Abundance . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.1.3. Structure formation aspects . . . . . . . . . . . . . . . . .. 11. 2.1.4. Particle candidates . . . . . . . . . . . . . . . . . . . . . .. 13. 2.1.5. Generic production mechanisms . . . . . . . . . . . . . . .. 14. 2.2. Sterile neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.2.1. Why sterile neutrinos? . . . . . . . . . . . . . . . . . . . .. 15. 2.2.2. Particle physics description of sterile neutrinos . . . . . . .. 17. 2.2.3. Production mechanisms for sterile neutrinos . . . . . . . .. 18. 2.2.4. Novelties in this work . . . . . . . . . . . . . . . . . . . . .. 19. 2.3. Structure formation . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 2.3.1. The free-streaming horizon λfs . . . . . . . . . . . . . . . .. 20. 2.3.2. The Lyman-α forest . . . . . . . . . . . . . . . . . . . . .. 22. 2.3.3. The linear matter power spectrum . . . . . . . . . . . . . .. 23. 2.4. More bounds from astrophysics and other sources . . . . . . . . .. 25. 3. Description of the setting and computational details. 29. 3.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 3.2. Summary of the production mechanism . . . . . . . . . . . . . . .. 30. 3.3. Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 3.3.1. Remarks on the numerics. . . . . . . . . . . . . . . . . . .. 4. Results. 40 43. 4.1. Stable scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 4.2. Full model – scalars decaying into sterile neutrinos . . . . . . . . .. 47. 4.2.1. Small scalar masses (mS ≤ mh /2) . . . . . . . . . . . . . .. 47. 4.2.2. Intermediate scalar masses (mh /2 < mS . mh ) . . . . . . .. 54. 4.2.3. Large scalar masses (mS & mh ) . . . . . . . . . . . . . . .. 59. 4.3. Comparison of the results to earlier work . . . . . . . . . . . . . .. 64. 5.

(6) Contents 5. Conclusion and Outlook. 69. A. Matrix elements and collision terms A.1. Matrix elements . . . . . . . . . . . . . . A.2. Collision terms . . . . . . . . . . . . . . A.2.1. 2-to-2 processes . . . . . . . . . . A.2.2. Higgs decay . . . . . . . . . . . . A.2.3. Scalar decay into sterile neutrinos. 71 71 72 72 74 75. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. B. Details on the interaction rate and derivation of rate equations. 79. C. Free-streaming horizon vs. linear power spectrum C.1. Comparison to free-streaming horizon . . . . . . . . . . . . . . . . C.2. Robustness of the power spectrum analysis . . . . . . . . . . . . .. 82 82 84. D. Details on anti-particle conjugation and Majorana mass. 85. E. List of abbreviations and some cosmological background E.1. Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2. Cosmological background . . . . . . . . . . . . . . . . . . . . . . .. 87 87 88. 6.

(7) 1. Introduction During the last decades our understanding of the universe has been greatly improved, yet there are a lot of mysteries that remain to be resolved. One of the most important among those is the nature of dark matter. Up to now the information we have about it is very limited. We know how much dark matter exists in the universe [2], its approximate distribution and we can say something about its typical velocities. We can also give bounds on its coupling strength to ordinary matter, because even with today’s precision we observe only gravitational interaction of dark matter. Probably the most interesting question about dark matter is what kind of particles it consists of, or if it consists of particles at all.1 As yet we do not know, but many scientists study models that include one or more particle(s) beyond the Standard Model (SM) which could be dark matter. Popular possibilities range from axions [4, 5] over supersymmetric particles [6, 7] to candidates arising from extra dimensions [8, 9]. Many of these are so-called WIMPs (weakly interacting massive particles) [10, 11], i.e. particles with a mass of typically around 100 GeV, which interact with the Standard Model particles and are usually stable. Another interesting option are sterile neutrinos. The Standard Model contains almost all fermions with left- and right-handed chirality. The only exception to this are the neutrinos, which are solely included as left-handed particles [12]. In order to fill this gap, it would be quite natural to discuss a theory that also includes right-handed neutrinos. These would be total singlets under the Standard Model gauge group, but they could mix with the left-handed neutrinos after electroweak symmetry breaking if an appropriate mass term is included. Furthermore, this could also solve the issue that in the Standard Model neutrinos are massless, while the observed neutrino oscillations strongly hint at a non-zero mass [13, 14]. Altogether, the special appeal of considering sterile neutrinos as dark matter candidates is that such a model could solve several problems of the Standard Model with comparatively little extra input. This thesis is structured as follows: chapter 2 gives an introduction to all the 1. An alternative approach is that the effect that is observed and usually associated with dark matter, is not to be explained by any new kind of matter but rather that the laws of gravity, which we use at the moment, are not correct but have to be modified. For a review see e.g. [3].. 7.

(8) 1. Introduction topics that will be touch upon in the rest of the thesis. This comprises the most important aspects of dark matter in general, then some details on sterile neutrinos and how they fit into the dark matter context, and finally some aspects of cosmic structure formation as well as other sources of bounds for our results. In chapter 3, the explicit form of the employed particle physics model will be introduced and general features of the Boltzmann equation and how it is treated in our framework will be addressed. Chapter 4 presents the results of the computations and discusses their viability w.r.t. various bounds. The last chapter gives a summary of the thesis and draws conclusions from the obtained results. Also an outlook on future projects that continue this work will be given.. 8.

(9) 2. Theoretical background This chapter is supposed to serve as an introduction to the various fields of particleand astrophysics that are touched in this thesis. Moreover, it makes connections among these fields and to the project itself wherever possible.. 2.1. Dark matter First, we give a short introduction to the facts about dark matter that are most important for the discussion to come.. 2.1.1. Evidence Why do we believe that dark matter exists? The answer to this question is that there is an overwhelming amount of evidence for something that is present in the universe but not included in the Standard Model or the theory of general relativity. We can see its effect by studying structures like galaxies or galaxy clusters: computing the gravitational effect of the matter we see and comparing this result to observations often reveals discrepancies. Some of the most striking examples are the following.. Galaxy rotation curves A rotation curve is the relation between the orbital velocity of constituents of a disk galaxy like stars or gas and the distance of these objects from the galaxy’s center. It is determined by the matter distribution within the galaxy. The peculiarity is that the mass profile inferred from the observed visible matter components does not match the distribution needed to produce the observed rotation curve. Figure 2.1 shows this effect in an example. That is one reason to believe that there is another type of matter in the galaxy which exactly provides for the missing mass – the dark matter [16]. It is different from ordinary matter because it apparently does not interact through any force other than gravity, which is why it is called dark.. 9.

(10) 2. Theoretical background. Figure 2.1.: Rotation curve of the spiral galaxy NGC 2841. The measured values are displayed as dots, whereas the solid lines are computed rotation curves determined from the visible matter content of the galaxy (vbary ), the dark matter part (vDM ) and the combined rotation curve from both contributions (vtot ). Figure taken from [15].. Bullet Cluster The Bullet Cluster (fig. 2.2) is the smaller of two galaxy clusters passing through each other. During this process the gas of each cluster is slowed down by electromagnetic interaction with the gas of the respective other cluster, whereas the stars and other constituents of galaxies basically pass through like collisionless particles [17], because their number in a given volume is very low. Due to this different behavior the gas clumps between the two clusters, but the stars are located farther out. By studying the gravitational lensing effect of the cluster, its mass distribution can be reconstructed. Most of the mass of ordinary matter comes from the gas, so we expect the mass density to peak close to the gas cloud in the middle, but that is not the case: it actually peaks away from the center on both sides. Using dark matter this can be easily explained because – like the stars – it passes through everything with almost no interaction during the collision of the clusters, so most of it is located where the stars are, i.e. farther out. Therefore, the total mass distribution is not concentrated in the middle but on the sides, just as observed. Apart from these two examples, there are many other sources of evidence for the. 10.

(11) 2.1. Dark matter. Figure 2.2.: The Bullet Cluster. Gas distribution from X-rays shown in red, mass distribution determined via gravitational lensing in blue. Image courtesy of NASA [18]. existence of dark matter, which is why most scientists have hardly any doubt about it.. 2.1.2. Abundance After accepting that dark matter exists, the next interesting question is: How much of it is out there? Regarding this there have been e.g. measurements of the cosmic microwave background (CMB) by the Planck collaboration [2], which determined that about 68% of the energy in the universe is dark energy and 32% is matter, of which baryonic matter only amounts to roughly 20%, the remainder being dark matter, i.e. there is four times as much dark matter in the universe as there is ordinary matter.. 2.1.3. Structure formation aspects Dark matter plays an important role in the formation of structures in the universe like galaxies or galaxy clusters. Also on larger scales dark matter dominates, i.e. the spatial distribution of galaxy clusters is determined by it. Observations reveal that, on large scales, galaxies and galaxy clusters form a filament-like structure as shown in fig. 2.4. The standard cosmological model, called the ΛCDM model, which includes a cosmological constant Λ and cold (i.e. very slow) dark matter reproduces. 11.

(12) Baryo. nic M atter. 2. Theoretical background. Dark Matter. Dark Energy. Figure 2.3.: Energy content of the universe: dark energy ≈ 68%, dark matter ≈ 26%, baryonic matter ≈ 6%.. Figure 2.4.: Some large scale structures in the universe as observed by the Sloan Digital Sky Survey (SDSS) compared to numerical results of the Millennium Simulation, an N -body simulation using the ΛCDM model and 1010 particles. Figure taken from [19].. 12.

(13) 2.1. Dark matter the experimental results very well on these scales. On the other hand, hot or warm dark matter, i.e. relativistic resp. rather fast dark matter, washes out structure because of its movement. A quantity that is widely used to quantify this effect is the so-called free-streaming horizon [20]. This is the average length scale that the dark matter particles would have freely moved since their production if they had not been subject to gravitational clustering or any other kind of restrictions. As already clear at this stage, this quantity uses the average velocity of the dark matter and neglects further information about its distribution function. This is why the free-streaming horizon can only be used as a rough estimate for bounds coming from structure formation. A more sophisticated quantity will be introduced in sec. 2.3. Despite the success of the ΛCDM model on large distances, it seems to encounter various problems on smaller scales, at least when comparing simulations to observations. One of them is that it predicts more satellite galaxies, i.e. small galaxies that orbit larger galaxies, than observed. But if this is really a problem of ΛCDM or if e.g. baryonic effects can explain these discrepancies is still unclear.. 2.1.4. Particle candidates We know only little about dark matter, and consequently there are many candidates for its particle nature. First of all, we do not know if there is only one type of dark matter, maybe there are more than one particle type that constitute the entirety of the dark matter in the universe. Apart from that question, there are many possibilities. Probably the most popular alternative is the so-called WIMP scenario. In this case dark matter is some kind of weakly interacting massive particle (WIMP), which still does not specify if this is a boson or a fermion etc. The basic idea is that this particle is in thermal equilibrium with all the Standard Model particles in the early universe. As the universe expands, the interaction rate between these particles and the Standard Model particles decreases until the particle eventually freezes out, which means that the interaction rate is low compared to the Hubble expansion, i.e. there is too little interaction with the thermal plasma to keep the particle in thermal equilibrium. From that point onwards (if the particle is stable) the distribution function will only change due to redshift of momentum. So, at the end of this process, the particle is present today with a certain abundance and a certain distribution of momenta. Choosing appropriate couplings, mass etc. of the particle, people try to fit the prediction to experimental data. One important thing to note here is that the particle has to interact only weakly because otherwise it could easily be produced in colliders like other particles of the Standard Model and, even more importantly, it would not be dark in the sense. 13.

(14) 2. Theoretical background that it would not have the property to virtually only interact gravitationally with everyhing else. Possible WIMP candidates are e.g. supersymmetric particles like a neutralino [21, 22], particles from other models like the inert Higgs model [23] or models including extra-dimensions [22]. Apart from WIMPs there is the possibility of axion dark matter. The axion is a postulated particle that was originally introduced to solve the strong CP problem of the Standard Model, but it can also serve as a dark matter candidate [5, 22]. Bar other, more exotic alternatives, the last large class of dark matter candidates are sterile neutrinos, which will be the scenario considered in this thesis.. 2.1.5. Generic production mechanisms Two of the most generic and most frequently considered production mechanisms are thermal freeze-out and freeze-in. The concept of thermal freeze-out is the following: at some high temperature in the early universe all particles are in thermal equilibrium and form a plasma. The interaction rate between the various particle species is high and therefore all species are kept in equilibrium because particles of a specific species are annihilated or decay but are also produced again from the plasma such that no species drops out of equilibrium. But the universe expands and this fact disturbs this equilibrium state in the sense that the interaction rate decreases with time because the mean free path of the particles grows. At some point the expansion causes the interaction rate to become too small to keep the species in thermal equilibrium. They cease to interact with each other and the abundance does not follow the equilibrium abundance anymore, but it stays constant in a comoving volume. The particles “stay as they are”, only their momentum gets redshifted and the number density decreases due to cosmic expansion. When that stage is reached, the species is said to have “frozen out”. Freeze-in is the somewhat opposite scenario [24]: we say that a particle freezes in if the coupling to the thermal bath is too weak to ever drag it into equilibrium. It gets produced by the plasma, but due to its low number density the backreaction is very inefficient. The small coupling causes the thermalization process to take a very long time, so long that it is not even completed when the temperature drops below the would-be freeze-out temperature. Thus, this species never actually reaches thermal equilibrium: first its abundance grows to reach equilibrium but this process is so slow that the particle species freezes in before completing, i.e. the interaction with the thermal plasma becomes small compared to the Hubble rate, and the particles virtually do not interact anymore. A particle species that freezes in is called a FIMP (f eebly interacting massive particle). Typical behaviors of the yield Y (i.e. the number density n divided by the entropy. 14.

(15) 2.2. Sterile neutrinos. frozen in. frozen out 0.001. 10-4. Y (T)≡n(T)/s(T). Y (T)≡n(T)/s(T). 0.001. Y (λFIMP). 10-5. Y (λFIMP/4) 10-6. 10-7. 10-4. 10-5. Y (λWIMP/4). in equilibrium. Y (λWIMP). 10-6. 10-7. Y therm. Y therm. not in equilibrium 10-8. 103. 102 T[GeV]. 101. (a) freeze-in. 1. 10-8. 103. 102 T[GeV]. 101. 1. (b) freeze-out. Figure 2.5.: Typical behavior of the yield Y in a freeze-in and a freeze-out case. The would-be equilibrium yield Y therm is shown for comparison. Both plots only differ by the coupling strength λ (λWIMP ∼ 400λFIMP ). density s to factor out the effect of the cosmic expansion) in the two cases is shown in fig. 2.5. In the FIMP case the final yield grows with the coupling strength, in the WIMP case it decreases with coupling strength. Both of these production mechanisms as well as the one that will be discussed in this thesis happen in the “early” universe. By “early” we mean here temperatures roughly in the range of 103 GeV to 1 GeV. In order to get a feeling for what this means in cosmic time please refer to fig. 2.6, where the most important events in the history of the universe together with the respective cosmic time and temperature are shown.. 2.2. Sterile neutrinos As already mentioned, the dark matter candidate in this thesis is a sterile neutrino. The purpose of this section is to motivate and explain the reasoning behind this particular choice and the differences to previous works by various research groups.. 2.2.1. Why sterile neutrinos? One might wonder why sterile neutrinos are considered here and not one of the other popular candidates for dark matter. There are several reasons for this. First of all, sterile neutrinos are believed to exist for reasons other than dark matter. For example, there is a seemingly odd gap in the Standard Model particle. 15.

(16) 2. Theoretical background. Figure 2.6.: History of the universe [25]. Typically a generic WIMP or FIMP production as well as the mechanism discussed in this thesis happens in the range of 103 GeV to 1 GeV, i.e. roughly from 10−11 s to 10−6 s after the Big Bang.. content, meaning that every fermion other than neutrinos exists in a left- and a right-handed version. So, it is deemed natural that also right-handed neutrinos exist. This is partially also motivated from various grand unification frameworks like SO(10), where right-handed neutrinos appear automatically [26]. Another. 16.

(17) 2.2. Sterile neutrinos point is that they could play an important role in the generation of mass for the active neutrinos. In the Standard Model the active neutrinos are massless, but as shown e.g. in the Super-Kamiokande experiment they oscillate [13,14]. This means that, if one starts by producing e.g. electron-neutrinos in a process and then detects them at some distance, the probability of finding mu- or tau-neutrinos is non-zero. This can be explained if the neutrinos have a mass, because that could lead to a mismatch between the interaction eigenstates (electron-, mu- and tau-neutrinos) and the mass (propagation) eigenstates, which are usually denoted by ν1 , ν2 , and ν3 . A similar effect is incorporated in the Standard Model in the quark sector (quark mixing [27]). But experimentally, e.g. from cosmological observations [28], we know that the neutrino masses must be way smaller than those of the other fermions. This is one reason for believing that the neutrino mass is not generated by a coupling to the Higgs doublet because the smallness could then only be explained by immense fine-tuning of the Yukawa couplings. There seems to be some mechanism that automatically achieves this strong suppression of the neutrino mass. Many candidates are considered in the literature, like radiative neutrino masses [29–31] or the probably most popular one: the seesaw mechanism [32], which uses sterile neutrinos to give mass to the active neutrinos.. 2.2.2. Particle physics description of sterile neutrinos In this segment some particle physics background about sterile neutrinos will be explained. Some more detailed computations and proofs can be found in appendix D. First, let us start with terminology: “sterile” neutrinos are the same as “right-handed” neutrinos. In this context, “right-handed” is to be understood in the sense of chirality not helicity. In the rest of the text the sterile neutrino will be denoted by N . Sterile neutrinos are called sterile because they are total singlets under the Standard Model gauge group SU (3)C ×SU (2)L ×U (1)Y : (1, 1, 0). The fact that they are right-chiral means in formulas PR N = N,. (2.1). with the right-chiral projection operator 1 + γ5 , 2 γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 .. PR ≡. (2.2) (2.3). 17.

(18) 2. Theoretical background The anti-particle N c is therefore purely left-chiral [33]: T. N c ≡ CN ,. (2.4). PR N c = 0.. (2.5). For a proof of this relation and more information on the matrix C please refer to appendix D. The fact that the sterile neutrino is a singlet under the SM gauge group and property (2.5) of the anti-particle makes it possible to include a term like 1 1 − mN c N + h.c. = − m N c N + N N c 2 2. (2.6). in the Lagrangian. It is gauge- and Lorentz-invariant. This term is called a Majorana mass term. The difference to a regular Dirac mass term is that only one Weyl spinor is needed. For example the electron mass term in the Standard Model has the form −meL eR + h.c. = −m (eL eR + eR eL ) ,. (2.7). where eL and eR are the left- resp. right-chiral electron, which are independent. On the other hand, in eq. (2.6) only one field enters: N . One very important requirement that makes it possible to write down a Majorana mass term is the fact that N is a singlet, which also implies that it is electrically neutral (singlet under U (1)em ).. 2.2.3. Production mechanisms for sterile neutrinos How can sterile neutrinos be produced in the early universe? One typically considered possibility is the so-called Dodelson-Widrow (DW) mechanism which was first proposed by Langacker [34] and then discussed in the dark matter context by Dodelson and Widrow in [35]. The idea is that the only interaction of the sterile neutrinos with other particles is by their mixing with the active neutrinos. This would be a regular FIMP scenario because due to the feeble interaction via the mixing the sterile neutrinos would never thermalize and eventually freeze-in. The drawback of this scenario is that it tends to produce too “hot” dark matter [36, 37]. This means that the velocities of the dark matter particles are too high to be consistent with structure formation because this movement washes out a lot of structure on length scales up to its free-streaming horizon and we would not observe as much structure as we do. The fact that this mechanism produces too hot dark matter is a result of the values of the parameters like the sterile neutrino mass or. 18.

(19) 2.2. Sterile neutrinos the temperature at which the sterile neutrinos are produced. These are restricted by bounds like the X-ray bound, which e.g. excludes too large sterile neutrino masses because the rate of processes N → γ + ν is proportional to the mass of the sterile neutrino in the fifth power [38]. This means that a very heavy sterile neutrino would produce so much radiation in these decays that it would have been observed already.1 However, this production mechanism is always there, it cannot be switched of. Nevertheless, we will not consider it in this thesis, since it was shown in [42] that the effect of Dodelson-Widrow production on the results we will obtain, is minor, at least for sterile neutrino masses above 3 keV. The next classic production mechanism for sterile neutrinos is the resonant version of the Dodelson-Widrow mechanism: the Shi-Fuller (SF) mechanism [43]. This scenario requires a primordial lepton number asymmetry. After its first discussion by Enqvist et al. in [44] it was proposed as a dark matter production mechanism by Shi and Fuller [43]. Since then a lot of research has been done on this field [45–51]. However, the work of different groups appear to yield different results and the compatibility with structure formation data is uncertain [52–54]. The third important mechanism is production via the decay of a parent particle. This particle can e.g. be a scalar [55, 56] (which is the scenario discussed in this thesis), a vector [57–59], a Dirac fermion [60], a pion [61, 62] etc. We want to discuss the case that this particle is a singlet scalar, which couples to the Standard Model via a Higgs portal. The value of this coupling determines if the scalar is a WIMP and freezes out (in the case of large coupling) [56, 63, 64] or if it is a FIMP and freezes in (small coupling) [65, 66]. Apart from this setting also a lot of alternatives have been discussed in the literature [67–76].. 2.2.4. Novelties in this work The model we will consider was studied before [56,63,65,66,70], but always different assumptions and simplifications were made. In [70] the authors limit their analysis to rate equations, which neglect the shape of the distribution function and only determine the evolution of yields with certain assumptions on the distributions.2 But also other simplifications have been made like assuming that the number of relativistic degrees of freedom is constant and that the scalar is very heavy (e.g. in [66]). The goal of this thesis is to provide a discussion in which these different 1. In 2014 two groups claimed to have found an X-ray signal at 3.55 keV [39, 40], which could be produced by 7.1 keV sterile neutrinos. However, the data is still ambiguous and this possible signal is actively discussed [41]. 2 Cf. appendix B.. 19.

(20) 2. Theoretical background approaches are brought together. We will compute full distribution functions like in [66] – not only the yields – and also take into account that the number of relativistic degrees of freedom is not constant as well as consider scalar masses as low as 30 GeV, like in [70]. We will also compare our results to [66] and [70], wherever possible and useful. Especially the usage of the full Boltzmann equation on the level of distribution functions compared to the integrated equations (i.e. the level of yields/rate equations) is very important: the rate equations completely discard the shape of the distribution function. Instead they usually assume it to have a thermal, i.e. Boltzmann-like, shape. In cases where this assumption is not at all justified – examples will be given in chapter 4 – this can lead to significant errors also in the yields. Furthermore, if only the yields are available and the shape of the distribution function is guessed, the comparison of the results obtained to structure formation bounds can only be limited to the treatment of the free-streaming horizon. Solving the full Boltzmann equation and therefore determining the distribution function of the dark matter, both of these drawbacks are lifted as also more sophisticated quantities than the free-streaming length can be computed, which will be explained in the next section.. 2.3. Structure formation In this section more details about structure formation will be given. An experimental source of information will be discussed and a useful quantity to approximately compare structure formation data to a given distribution function will be introduced.. 2.3.1. The free-streaming horizon λfs The so-called free-streaming horizon λfs is a quantity that is typically discussed in the context of dark matter models and their relation to cosmic structure formation. It is defined as [66]:. λfs :=. ttoday Z tprod. 20. hv(t)i dt, a(t). (2.8).

(21) 2.3. Structure formation where t is cosmic time, a the scale factor, tprod the time of production of the dark matter, and the average velocity R∞. hv(t)i :=. p fN (p, t) m2N +p2. dp p2 √. 0. R∞. ,. (2.9). dp p2 fN (p, t). 0. with p the norm of the physical 3-momentum, fN the distribution function of the dark matter (here with index N for sterile neutrino) and mN the mass of a dark matter particle. λfs is the length which a dark matter particle in a given model would travel from the time of production until today on average, had it not been gravitationally trapped. The fact that this is an average is very important because, due to the integration, much information about the particles is lost. The distribution function is only used to the extent of computing the average velocity, but if the spectrum has a very non-thermal shape, the average velocity can be very misleading, since it can lie in between two characteristic velocities but can itself be very suppressed like we will see in examples in chapter 4. The usual procedure is now to compare the resulting free-streaming horizon with fixed values like 0.1 Mpc (about the size of a dwarf galaxy) and 0.01 Mpc (some value much smaller than the former). Then, if λfs is smaller than 0.01 Mpc the dark matter is called “cold” because the average length it has travelled is rather small and the temperature of a thermal distribution that would generate the corresponding average velocity would be low. This kind of dark matter is usually considered to be in agreement with structure formation data because due to the lack of movement only structure on very small scales is “washed out” by the dark matter. In the case when λfs is between 0.01 Mpc and 0.1 Mpc it is called “warm” and has travelled a mediocre distance, which causes the extinction of structures on small and very small scales. Finally, if the free-streaming horizon is greater than 0.1 Mpc we have “hot” dark matter because the movement is so large that too much structure is washed out to be in accordance with what we see in the universe. These critical values (0.01 Mpc and 0.1 Mpc) are only rough estimates and can easily be corrected by a factor of O(1) – resp. even more in the case of the boundary between cold and warm. Other references use slightly different values, so it is only to be seen as an order of magnitude estimate. The crudeness in employing the free-streaming horizon when investigating structure formation, which – as stated above – comes from the averaging process, is the reason why we go into more detail and compute the linear matter power spectrum,. 21.

(22) 2. Theoretical background which is determined by the distribution function of the dark matter and gives information about how strongly structure is suppressed on a given scale due to the dark matter momentum distribution. Its definition and some background information will be given in the following.. 2.3.2. The Lyman-α forest The so-called Lyman-α (Ly-α) forest is a set of absorption lines that appear in the spectrum of e.g. distant galaxies or quasars. A nice review on this topic is given in [77]. Fig. 2.7 shows as an example the spectrum of the quasar QSO1422+23 at redshift zem = 3.62.. Figure 2.7.: The Lyman-α forest: a set of absorption lines in the specturm of distant objects like quasars. Here the example of the zem = 3.62 quasar QSO1422+23. (Taken from [78] which uses data from [79].) The end of the forest at large wavelengths should be the line of first absorption. As we can clearly see, this is confirmed by computing the redshifted wavelength of the Ly-α light emitted by the quasar: λ = 1215.67 Å · (zem + 1) ≈ 5616 Å. These lines all come from the same atomic absorption line, namely the Lyman-α line of hydrogen. This line corresponds to the transition from one of the excited states with principal quantum number n = 2 to the ground state (n = 1). Its wavelength is λα = 1215.67 Å, but we not only see one absorption line but many of them. This can be explained by the redshift: the light that is emitted by the quasar hits a cloud of hydrogen. The absorption happens as the light passes through the cloud and produces a line at 1215.67 Å. Then the light travels through space before it encounters the next hydrogen cloud. By the time it arrives at the next cloud, the light has already redshifted and the absorption line that was produced by the first cloud has moved to a larger wavelength. Now the second cloud causes a second absorption line. This second line is again located at 1215.67 Å, so after the light has passed the second cloud, there are two lines in its spectrum, one at 1215.67 Å and another one at a larger wavelength. After N clouds there are. 22.

(23) 2.3. Structure formation N absorption lines in the spectrum, each at a different wavelength, but they all originate from the same process. From the positions and depths of the absorption lines, the matter distribution along the line of sight to the quasar can be inferred. As the gas follows the dark matter distribution, the dark matter model must be consistent with the amount of structure observed this way.. 2.3.3. The linear matter power spectrum As stated before, the free-streaming horizon is at best a rough estimator for the structure formation analysis of a dark matter model. To overcome the problem of averaging we take a look at a quantity called the linear matter power spectrum. It gives information on the correlations of the spatial distribution of matter and is defined via [80]: ∞. ξ(y) =. Z. Z 2 d3 k ik·y k dk sin(ky) e P (k) = P (k), 3 (2π) 2π 2 ky. (2.10). 0. where ξ is the correlation function of density fluctuations: ξ(y) := hδ(x)δ(x + y)i ,. (2.11). which due to isotropy only depends on y = |y|, and δ is the spatial density fluctuation: δ(x, t) :=. δρ(x, t) , ρ(t). (2.12). where ρ is the homogeneous “background” density and δρ is a small perturbation of it. The phase space distribution function of the dark matter particles can be used to compute P (k). In this work, the Cosmic Linear Anisotropy Solving System (CLASS)3 code has been used for this task. It “simulate[s] the evolution of linear perturbations in the universe and [...] compute[s] CMB and large scale structure observables” (from class-code.net). As input it uses the distribution function of the dark matter – which is the result of our Boltzmann equation – and parameters like the temperature at which this is the distribution function of the particles, and from that it computes e.g. the linear matter power spectrum corresponding to this distribution function today. After determining the linear power spectrum the question is: how do we use it? 3. See [81, 82] and class-code.net.. 23.

(24) 2. Theoretical background The answer is that we compare it to bounds derived from Ly-α data. To be more specific we do not use the power spectrum directly but rather the ratio of this power spectrum to the power spectrum of perfectly cold dark matter. This quantity is called the squared transfer function T 2 (k): T 2 (k) ≡. P (k) . PCDM (k). (2.13). Ref. [83] gives an analytic fit formula for the squared transfer function of a thermal distribution with mass m that makes up all dark matter. Furthermore, limits on this mass are determined in [36]: a conservative analysis yields m & 2.0 keV (4σ C.L.), the result of a restrictive one is m & 3.3 keV (2σ). Although these bounds were obtained with an analysis using thermal distributions and for more precise bounds the real distribution function would have to be employed – which of course means a lot more effort – we will use the bounds stated above. This is justified by our procedure which is itself only approximate. The argument behind our following analysis is that the value of the squared transfer function for a certain wave number k indicates roughly speaking how much structure exists on the scale corresponding to k. If T 2 (k) is small, then there is little structure, if it is equal to one, however, the dark matter with the given distribution function allows for the same amount of structure on this scale as perfectly cold dark matter. We know that at large scales (for small k) CDM yields accurate results, so the squared transfer function should go to one for k → 0. On the other hand, on smaller scales we observe less structure than predicted by cold dark matter. This means that, for larger k, a value smaller than one is possible. The limits we use in some sense mark the “minimally allowed amount of structure”, so we exclude scenarios for which the squared transfer function takes smaller values than the bound for all wave numbers. Vice versa, we allow scenarios where it takes lager values everywhere. But it can also happen that there are regions where its value is larger and other regions where it is smaller. In order to deal with these situations, we use a method illustrated in fig. 2.8. The analysis goes as follows. First, the wave number k1/2 , which is defined as the wave number for which the squared transfer function has fallen off to 12 , is determined by: 1 T 2 (k1/2 ) = . 2. (2.14). With this, we say a bound allows the given sterile neutrino distribution function if 2 T 2 (k) ≥ Tlim (k) ∀k ≤ k1/2 ,. 24. (2.15).

(25) 2.4. More bounds from astrophysics and other sources. 1.0. T 2lim(k ). 0.8 0.6 0.5 0.4 0.2 0.0 0.5. 1. 5 10 k [h/Mpc]. 50. 100. Figure 2.8.: Illustration of the structure formation analysis. A limiting squared transfer function is displayed. If the squared transfer function from the sterile neutrino distribution is plotted into the same figure and it crosses into the red region, it is excluded, otherwise it is considered allowed by the bound. 2 where Tlim is the squared transfer function of the bound. Otherwise we say it is excluded by the bound. This is equivalent to saying it is excluded if the squared transfer function crosses into the red region in fig. 2.8. Choosing T 2 = 12 as a reference point is somewhat arbitrary. However, we will show in appendix C.2 that the final results do not change much if one chooses a smaller value. We will give more details on how far our results are compatible with structure formation and several examples for squared transfer functions in our setting in chapter 4.. 2.4. More bounds from astrophysics and other sources Apart from structure formation bounds coming from Ly-α data, several other bounds are considered: Tremaine-Gunn bound The Tremaine-Gunn (TG) bound [84] is a very important astrophysical bound that excludes large parts of the parameter space. It uses the fact that the phase space. 25.

(26) 2. Theoretical background density of fermions is restricted from above to restrict the mass m of fermionic dark matter particles: if the fermions are too light, one would have to pack more of them in the given phase space than possible, so they cannot explain the measured dark matter abundance. From astrophysical observations the most recent bound was found to be m & 0.5 keV [20]. Overclosure bound The overclosure bound marks the region in the parameter space which even for a sterile neutrino mass of 0.5 keV would overclose the universe. This means that the density from these neutrinos is larger than the critical density, i.e. the universe would be closed from that density alone. Since the sterile neutrino mass is determined by choosing it such that the density of the sterile neutrinos matches the observed dark matter density, the overclosure region is always contained in the area which is excluded by the TG bound. Model dependent collider bounds To be precise, for the case that is considered in this work, i.e. that the scalar does not develop a vacuum expectation value (vev), there are no relevant collider bounds available. If, however, we loosen up this restriction, we can find bounds from theoretical studies and collider data: in [85] the authors show that, in the regime of negligible mixing between S and the Higgs boson, perturbative unitarity sets a lower limit on the vev of S, s. hSi ≥. 3 mS . 16π. (2.16). If we furthermore assume a minimal setting, in which the sterile neutrino mass is solely produced by the vev of S in the Yukawa term of the Lagrangian, which means that mN = y hSi ,. (2.17). we can translate eq. (2.16) into a bound on the Yukawa coupling y: mN y≤ mS. s. 16π . 3. (2.18). Moreover, the mixing between S and the Higgs boson induces a correction to the W -boson mass. As a consequence of experimental agreement with the SM-value of the W -boson mass, the value of | sin(α)|, where α is the corresponding mixing. 26.

(27) 2.4. More bounds from astrophysics and other sources angle, is bound from above, i.e. a maximal mixing angle αmax is derived. Given the relations in eqs. (8) to (11) in [85], we can infer a limit on the Higgs portal coupling λ: |m2S − m2h | , λ ≤ y sin (2 αmax ) 2 v mN. (2.19). where v = 246 GeV [86] is the vev of the Higgs doublet. All of the above mentioned bounds will be shown in our results in chapter 4. Note however that the bounds from perturbative unitarity and the W -boson mass are model-dependent and only relevant for the minimal model in which mN = y hSi.. 27.

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(29) 3. Description of the setting and computational details This chapter discusses the exact form of the particle physics model, the setup and important equations like the Boltzmann equation and explains the steps that were taken to solve them.. 3.1. The model Our setup consists of the Standard Model of particle physics plus one real scalar S, which is a singlet under the Standard Model gauge group, and a sterile neutrino N . We postulate the following Z4 -symmetry: S → −S,. (3.1). N → iN,. (3.2). whilst not changing the Standard Model Lagrangian. The most general Lagrangian that respects this symmetry reads: L = LeSM + iN ∂/N + | {z } A. 1 µ y (∂ S) (∂µ S) − SN c N + h.c. − Vscalar + Lν . |2 {z } |2 {z } B. (3.3). C. The different parts of L denote the following: â LeSM is the Standard Model Lagrangian without the Higgs potential. â Term A is the kinetic term of the sterile neutrino. â Term B is the kinetic term of the scalar singlet. â Term C is a Yukawa-like coupling between the scalar and the sterile neutrino. â Vscalar = −µ2H Φ† Φ + 12 µ2S S 2 + λH (Φ† Φ)2 + λ4S S 4 + 2λΦ† ΦS 2 is the potential of the scalar fields, i.e. of the SM-singlet S and the Higgs SU (2)L -doublet Φ. â Lν is the part of the Lagrangian that gives mass to the active and/or the sterile neutrinos. We do not specify the exact form of this expression because the mass generation mechanism is irrelevant for the discussion in this work.. 29.

(30) 3. Description of the setting and computational details If µS is real, which leads to the scalar not developing a vacuum expectation value,1 the Z4 -symmetry is preserved. In order not to unnecessarily complicate computations, we assume that this is the case although the more general setup can be dealt with straightforwardly.2 This also means that the mass of the sterile neutrino has to be included in Lν . On the other hand, if the scalar developed a vacuum expectation value, term C would produce a Majorana mass for the sterile neutrino. Furthermore, if S does not get a vacuum expectation value and if we neglect the small correction from the coupling term to Φ, its mass is given by mS = µS .. (3.4). 3.2. Summary of the production mechanism The basic idea is that the sterile neutrino is the dark matter particle which is produced via the decay of the scalar S. This decay comes from the Yukawa term C in eq. (3.3), which provides the possibility of the decay of a scalar into two sterile neutrinos. The steps of the whole mechanism are the following: 1. S is produced in the early universe, i.e. as an initial condition we assume that it has no abundance. What happens here qualitatively depends on the value of the Higgs portal coupling λ: if it is large enough, the scalar has a high interaction rate with the Standard Model particles and is therefore quickly dragged into thermal equilibrium (WIMP case). It finally freezes out when the temperature drops too far. On the other hand, if λ is very small, this process of being attracted into equilibrium takes “too long”, meaning that, by the time when S would reach thermal equilibrium, the temperature is already below the freeze-out temperature of the scalar such that it never reaches equilibrium (FIMP case). 2. During all this time the scalar decays into sterile neutrinos such that they are also produced starting from zero abundance. We have to distinguish three main regimes here: a) FIMP case: Here the neutrinos are produced via the decay while the scalar abundance slowly grows because of the interaction with the Standard Model plasma and then decreases because this interaction 1 2. We assume here that all couplings are positive. If e.g. cubic terms appear in the potential, the results will of course change, but the effect should only be quantitative and not qualitative because such terms just open more channels, which could even approximately be absorbed in the Higgs portal coupling λ.. 30.

(31) 3.2. Summary of the production mechanism becomes small compared to the Hubble function and S only decays into neutrinos. For late times the scalar has no abundance left and the sterile neutrino behaves like a frozen out species. b) WIMP case with large y: In this scenario the scalar is kept in equilibrium for a long time because the Higgs portal coupling is large. During this time in thermal equilibrium the scalar also decays into sterile neutrinos, but does not leave equilibrium as it is still produced by the plasma. When the scalar finally freezes out, the scalars (that are not interacting with the plasma anymore) decay into sterile neutrinos. The important fact here is that by far most of the sterile neutrinos are produced while the scalar is in equilibrium. c) WIMP case with small y: This is the opposite of case b). Due to the smallness of the Yukawa coupling the scalar decays only very slowly and so only comparatively few sterile neutrinos are produced while S is in equilibrium. Most of them come from the decay of frozen-out scalars. d) WIMP case with intermediate y: this is the transition region, in which both the in- and the out-of-equilibrium decays are important. This results in two distinct momentum scales in the distribution function of the sterile neutrinos.. In this thesis it is assumed that S has a mass of about 101 GeV to 103 GeV. The case of a very heavy scalar was already studied in [66], where some simplifications can be made, like assuming that the number of relativistic degrees of freedom was constant during the time span in which the scalar was produced in the early universe. This simplification, and most importantly the restriction of the mass of the scalar, will be lifted in the work at hand. This generalization has many consequences like different kinematically allowed/forbidden processes, most significantly for scalar masses smaller than half the Higgs mass the decay of a Higgs boson into two scalars is allowed, which turns out to immensely increase the coupling to the thermal bath. Apart from kinematically opening new possibilities the plain fact that the scalar mass is e.g. around 100 GeV makes temperatures below the electroweak phase transition (EWPT) relevant for the production of the scalar meaning that not virtually all of the scalars are already produced before the EWPT takes place.. 31.

(32) 3. Description of the setting and computational details. 3.3. Boltzmann equation The main tool to describe the evolution of the particles in the early universe is the Boltzmann equation. This classical equation3 is used to determine the evolution of the phase space distribution functions of the various particle species in time. In a very abstract form it reads L̂[f ] = C[f ].. (3.5). Here, L̂ is the Liouville operator, C are the collision terms describing particle interactions and f is the unknown distribution function for the species under consideration. From the distribution function, the number density n of the particles can easily be computed by integration: ∞. n(t) =. Z. Z g d3 p g dp 2 f (p, t) = p f (p, t), 3 (2π) 2π 2. (3.6). 0. where t is time, p is the absolute value of the physical particle momentum, g is the number of internal degrees of freedom of the particle and we used for the last equality the fact that the distribution function only depends on the absolute value of the momentum because we assume isotropy. In this work we use the input4 that all Standard Model particles which participate in the interaction with the scalar are in equilibrium until the scalar freezes out. Thus, only two coupled Boltzmann equations have to be solved: one for the scalar (fS ) and one for the sterile neutrino (fN ). The Liouville operator in a Friedmann-Robertson-Walker metric has the form [88] L̂ =. ∂ ∂ − Hp , ∂t ∂p. (3.7). with the time t, the absolute value of the 3-momentum p = |p| and the Hubble function H ≡ ȧ/a, where a is the cosmic scale factor and ˙ denotes the derivative w.r.t. time t. In order to find expressions for the collision terms, we need to compute matrix elements for the scattering resp. decay processes using quantum field theory. Then the collision term in the Boltzmann equation for the distribution. 3. As already apparent here, the fact that this equation is employed is an approximation in itself. For a full quantum description of the problem, Kadanoff-Baym equations have to be used, whose solution is computationally much more expensive if at all doable. The error caused by this simplification is hard to determine precisely, but is typically of the order of 10% of the final abundance [87] in a regular thermal freeze-out. 4 The validity of this statement was checked explicitly.. 32.

(33) 3.3. Boltzmann equation. regime. Table 3.1.: Different production channels for the scalar S of the regimes I–III. The t- and u-channel diagrams for hh ↔ SS are of higher order in λ and therefore not considered. production mechanism Φ. S. Φ h. S S. I. h. S. W+. S. II h h. h. S. W−. h. S. W+. S S. h h. S. S. W−. S. t. S. Z. S. Z. S. t̄. S. t. Z. S S. S. h. h S. S h. h. h. h. III. Z. h. h. S. t̄. h S. S. function of a particle species Φ for the occurrence of processes Φ + a1 + a2 + ... + an ↔ b1 + b2 + ... + bm ,. (3.8). has the form [88]: 1 Z C[fΦ ] = dPa1 dPa2 ...dPan dPb1 dPb2 ...dPbm 2EpΦ × (2π)4 δ (4) (pΦ + pa1 + pa2 + ... + pan − pb1 − pb2 − ... − pbm ) κΦ |M|2 × [fb1 ... fbm (1 ± fΦ ) (1 ± fa1 ) ... (1 ± fan ) − fΦ fa1 ... fan (1 ± fb1 ) ... (1 ± fbm )] . (3.9) The quantities in this equation are the following: Epi is the energy of the particle with momentum pi and mass mi , i.e. Epi =. q. m2i + p2i ,. (3.10). gi d3 pi , (2π)3 2Epi. (3.11). dPi is the phase space element dPi =. where gi is the number of internal degrees of freedom of the particle. κΦ is the number of Φ-particles produced or destroyed in the process. |M|2 is the squared. 33.

(34) 3. Description of the setting and computational details matrix element for the process, in which we average over both initial and final state spins and polarizations with appropriate factors to avoid double-counting indistinguishable particles in the initial resp. final states, and the fi ’s are the distribution functions of the particles participating in the process. The matrix elements for all relevant processes can be found in Appendix A. An overview over the processes of the interaction bewteen the scalar and the Standard Model is given in table 3.1, where regime I is valid for temperatures above the electroweak phase transition and all scalar masses5 , while regime II is for temperatures below the EWPT and masses larger than half the mass of the Higgs boson and regime III is for temperatures below the EWPT and scalar masses smaller than half the Higgs mass. For the sterile neutrino, the only channel is given by the decay of a scalar S into two sterile neutrinos via the Yukawa coupling. This is the only channel for all temperatures. Moreover, we use the “FIMP approximation” for the sterile neutrino, i.e. we neglect the inverse decay. Therefore, the two i.g. coupled Boltzmann equations decouple and we can solve the equation for the scalar on its own and afterwards just integrate the sterile neutrino equation. Now, we want to discuss which regime is relevant for which scenario: we always start our computation at high temperatures, i.e. in regime I. From here on, we continue to smaller temperatures until the there are virtually no more scalars around resp. all sterile neutrinos are produced. For a given scalar mass mS the production of S from the thermal plasma is most efficient if mS ∼ T . In the case of a small Higgs portal coupling λ (FIMP regime), the scalar is slowly produced and then freezes in soon after mS ∼ T . This means that for very high scalar masses (mS TEWPT ) basically only regime I is relevant. In the WIMP case, the scalar equilibrates quickly and freezes out at a scale related to its mass. An often used estimate is that the freeze-out temperature is given by Tfreeze-out ∼ mS /20. After freeze-out the scalar is not in contact with the plasma anymore and just decays into sterile neutrinos. So, for each mass we start in regime I and then proceed into regime II resp. III in case the scalar mass is larger resp. smaller than half the Higgs boson mass. Fig. 3.1 illustrates the temperature ranges in which the interaction between the scalar and the thermal plasma is relevant. For a scalar mass of 30, 60, 65, and 500 GeV each a FIMP (red arrow) and a WIMP case (blue arrow) are displayed. These results (for a stable scalar) were obtained with the full numerics used in this thesis. For a FIMP case the arrow is between the temperatures for which 10% and 90% of the final scalar yield are produced. An arrow for a WIMP 5. A remark on this regime: The Higgs doublet yields 4 degrees of freedom with a temperaturedependent mass mΦ (T ), which was taken from the Higgs potential given in ref. [89] (eq. (213) ff.).. 34.

(35) 3.3. Boltzmann equation. Figure 3.1.: Temperature range of the plasma temperature T at which the scalar singlet S is efficiently produced in the freeze-in scenario (red) and in the freeze-out scenario (light blue). We display four different masses (30, 60, 65, and 500 GeV) of the scalar that span the variety of possible production times. TEWPT indicates the temperature when the electroweak phase transition happens, while TPT is supposed to give a rough idea of when the Higgs and the gauge bosons become strongly Boltzmann suppressed, even if still equilibrated with most of the remaining SM degrees of freedom.. has no real starting point because the scalar is in equilibrium, but the end point is defined by the temperature at which the yield of the scalar is 10 times as large as it would be if the scalar were still in equilibrium. As one can easily see, for very large masses really only regime I is relevant. This approximation was also used in [66], whereas for smaller scalar masses also the lower temperature regimes are relevant resp. even most important (e.g. in the 30 GeV FIMP case displayed). To give one concrete example, the collision term for the scalar with a mass larger than half the Higgs mass at a temperature of, e.g., 50 GeV has the form S S S S S CII [fS ] = Chh↔SS [fS ] + CW + W − ↔SS [fS ] + CZZ↔SS [fS ] + Ctt̄↔SS [fS ].. (3.12). Here, hh ↔ SS of course includes both contributing diagrams. Moreover, the only fermion with a relevant impact in all of our cases is the top quark.. 35.

(36) 3. Description of the setting and computational details. r = mh/T 10. -1. 101. 1. 102. YSeq(T). 10-3. mS in GeV: 30. 10-5. 60 65 100. 10-7. 500 1000. 10-9 103. 102. 101. 1. T [GeV] Figure 3.2.: Yield of the scalar S if it were in equilibrium for all temperatures. Shown for the different masses of S discussed in this work. The “time” variable r = mh /T , where mh is the mass of the Higgs boson, will be used later.. Coming back to the explicit form of the collision terms, the expression given in eq. (3.9) is generally simplified by the following line of thought: First of all, we assume that fi 1 such that we can neglect all the 1 ± fi terms. This is a very good approximation in all the relevant cases we consider, but it does need not hold in general. Moreover, it is assumed that the equilibrium distributions of fermions and bosons can be well approximated by a Maxwell-Boltzmann distribution: √ 2 2 fieq (p, T ) = e− mi +p /T .. (3.13). Assuming this, the equilibrium number density of a species is given by neq i (T ). ∞ gi Z gi mi 2 eq 2 = 2 dp p fi (p, T ) = 2 mi T K2 , 2π 2π T. (3.14). 0. where K2 is a modified Bessel function of the second kind. Figure 3.2 shows the equilibrium yield of the scalar for the different scalar masses that are considered in this work. Using the Boltzmann approximation for the equilibrium distribution functions and the already mentioned fact that all the Standard Model particles that participate in interactions with the scalar are in thermal equilibrium, we can infer from energy conservation, i.e. from the corresponding δ-distribution in (3.9),. 36.

(37) 3.3. Boltzmann equation that e.g. for a ii ↔ SS process we have fieq (q)fieq (q 0 ) =e−. √. m2i +q 2 /T −. e. √. m2i +q 02 /T. 0. 0. = e−Ei (q)/T e−Ei (q )/T = e−(Ei (q)+Ei (q ))/T =. 0. =e−(ES (p)+ES (p ))/T = fSeq (p)fSeq (p0 ).. (3.15). In total, the whole bracket . 0. . . fi (q)fi (q ) 1 + fS (p). 0. . 0. . . 1 + fS (p ) − fS (p)fS (p ) 1 ± fi (q). 0. 1 ± fi (q ). . (3.16) simplifies to [fSeq (p)fSeq (p0 ) − fS (p)fS (p0 )] .. (3.17). The integration over q and q 0 can be performed without being touched by the distribution functions, which means that the collision term looks like Cii↔SS [fS ] =. 1 Z gi p02 q dp0 [fSeq (p)fSeq (p0 ) − fS (p)fS (p0 )] 2 3 02 2Ep (2π) 2 mS + p ×. Z |. d cos θdφdQdQ0 (2π)4 δ (4) (p + p0 − q − q 0 )|Mii↔SS |2 . {z. gi I(p,p0 )/8. (3.18). }. I(p, p0 ) was computed numerically on a grid before dealing with the Boltzmann equation,6 which saves a lot of time during the actual process of finding a solution. In order to make our lives easier, we rewrite the Boltzmann equation by introducing suitable variables. Up to now, we work with the non-linear partial-integro differential equation !. ∂ ∂ − Hp f = C[f ]. ∂t ∂p. (3.19). Solving it is also numerically very challenging. To simplify it to some extent, new variables are introduced. They are chosen such that the left-hand side of the target equation consists only of one derivative and the multiplication by a function. In. 6. In order not to overload the notation, we suppressed the possible temperature-dependence of I, which appears in the process φφ ↔ SS because the mass of φ changes with T . See eq. (A.14) for more details.. 37.

(38) 3. Description of the setting and computational details order to find variables that achieve this effect, start with general variables ξ and r: ξ = ξ(p, t),. (3.20). r = r(p, t).. (3.21). Plug these into the Liouville operator: !. ∂ξ ∂ ∂r ∂ ∂ξ ∂ ∂r ∂ + − Hp(r, ξ) + . L̂ = ∂t ∂r ∂t ∂ξ ∂p ∂r ∂p ∂ξ. (3.22). We only want the first term to remain so, before anything else, we demand that r does not depend on p and therefore eliminate the corresponding derivative. Doing this makes r a pure “time” variable, which is very useful. We obtain: L̂ =. ∂r ∂ ∂ξ ∂ ∂ξ ∂ + − Hp(r, ξ) . ∂t ∂r ∂t ∂ξ ∂p ∂ξ. (3.23). Next, we require ∂ξ ∂ξ = Hp(r, ξ) . ∂t ∂p. (3.24). This is a partial differential equation. Fixing the initial condition at t = t0 ξ(p, t0 ) = ξ0 (p) ∀p,. (3.25). where ξ0 is some differentiable function, it has the solution !. ξ(p, t) = ξ0. a(t) p . a(t0 ). (3.26). This means if we fulfill the requirements that r only depends on t and the dependence of ξ on p and t is given by (3.26), the Liouville operator has the simple form L̂ =. ∂r ∂ . ∂t ∂r. (3.27). One possible choice for these variables is7 m0 , T 1 a(t) ξ= p= T0 a(t(T0 )) r=. 7. gs (T0 ) gs (T ). !1/3. p , T. (3.28). Here, we use that there is a 1-to-1 correspondence between time t and temperature T .. 38.

(39) 3.3. Boltzmann equation where gS is the effective number of entropy degrees of freedom (cf. eq. (3.30) and appendix E.2). The quantities m0 and T0 are a reference mass and reference temperature respectively, which we choose to equal the Higgs mass m0 = T0 = mh .. (3.29). For the last equality in (3.28) we used the fact that the entropy density s is covariantly constant [88]: 2π 2 s(T )a (T ) = gs (T ) T 3 a3 (T ) = const. 45 3. (3.30). Relation (3.30) can also be used to derive the time-temperature relation8 dT = −HT dt. T gs0 +1 3gs. !−1. (3.31). by taking its derivative w.r.t. time t. Plugging this into the Liouville operator yields L̂ = rH. m0 gs0 +1 3rgs. !−1. ∂ . ∂r. (3.32). By this transformation we have achieved that the equation does not anymore contain derivatives w.r.t. both variables p and t and an integral over p, but we only have one derivative w.r.t. r and the integral over ξ, which makes processing easier. These new variables not only simplify our equation in the desired way but they also have other nice properties: r can be thought of as the “time” variable. It increases if and only if t increases. The fact that the exact relation is more complicated is not important because we only want to be able to chronologically order things. ξ on the other hand is the “momentum” variable, but obviously is not just the a(T ) momentum. Through the factor a(T it already includes the cosmological redshift 0) of momentum and the factor 1/T gives a useful normalization to the temperature because this sets the scale of typical momenta in a thermal distribution. With these properties we have that the distribution as a function of ξ and r does not change if it is only subject to cosmological redshift, e.g. after freeze-out. In formulas this means that the collision terms are all vanishing. Thus, the Boltzmann equation. 8. A prime 0 denotes the derivative w.r.t. the temperature T .. 39.

(40) 3. Description of the setting and computational details reads ∂ f (ξ, r) = 0, ∂r. (3.33). which is exactly the statement discussed. This property will be particularly useful when displaying the evolution of the distribution functions in time, as the graph just stays the same after freeze-out resp. after the production of sterile neutrinos is completed in case of the sterile neutrino distribution function.. 3.3.1. Remarks on the numerics Solving the resulting equation ∂ f= ∂r. m0 gs0 3rgs. +1. rH. C[f ]. (3.34). is still tough. To make it easier, we do not regard this as a partial integro-differential equation, but rather as a system of coupled ordinary differential equations by discretizing the ξ-space, i.e. we replace the unknown function of two variables by N unknown functions of one variable, f (ξ, r) → f (r) ≡ (fξ1 (r), fξ2 (r), . . . , fξN (r)),. (3.35). and approximate the integral over ξ by a sum over these fixed ξ-values.9 To illustrate this, consider the collision term for a 2 → 2 process and start with its general form given in eq. (A.14) – the others are treated analogously: S Cii↔SS [fS ](p, T ). Z∞ 2 02 0 1 gi p dp eq = [fS (p)fSeq (p0 ) − fS (p)fS (p0 )] I(p, p0 , T ). 3 S 32(2π) Ep EpS0 0. (3.36) I(p, p0 , T ) is computed numerically on a grid before solving the equation and is then interpolated, so it is callable like a normal analytically given function. First,. 9. We usually have N = 100 or N = 200 of these fixed ξ-values. It turns out that, in order to get a good approximation of the function and the integral, it is useful to choose them such that they are equidistant in the log(ξ)-space.. 40.

(41) 3.3. Boltzmann equation we write this term in the new variables ξ and r: S [fS ](ξ, r) = Cii↔SS. . . . 3 m0 Z∞ 2 02 0 gi ξ dξ r S S 3 32(2π) Ep(ξ,r) Ep(ξ 0 ,r) 0 gS (r) gS (r0 ). . . . . . . . × fSeq p(ξ, r) fSeq p(ξ 0 , r) − fS p(ξ, r) fS p(ξ 0 , r). m0 . r (3.37). . . I p(ξ, r), p(ξ 0 , r),. And now we discretize it, such that we get a collision term for each ξj : S Cii↔SS, ξj [fS ](r) =. . . . 3 m0 N X r S 32(2π)3 Ep(ξ j ,r) k=1 gS (r) gS (r0 ). . . . gi2 ξk2 ∆ξk S Ep(ξ k ,r). . . . . × fSeq p(ξj , r) fSeq p(ξk , r) − fS p(ξj , r) fS p(ξk , r). . m0 . r (3.38) . I p(ξj , r), p(ξk , r),. The system of ordinary differential equations ∂ fξ (r) = ∂r 1 ∂ fξ (r) = ∂r 2 .. . ∂ fξ (r) = ∂r N. m0 gs0 3rgs. +1. rH m0 gs0 +1 3rgs rH m0 gs0 3rgs. +1. rH. Cξ1 [f ](r), Cξ2 [f ](r),. CξN [f ](r),. (3.39). is still a stiff problem. This means that many solving algorithms need extremely small stepsizes (in r) to remain stable. Due to this fact and for general speed aspects, we implemented this system of ODEs in Matlab and used the builtin solver ode15s [90,91], which is especially useful when dealing with stiff equations. Our initial conditions are that both abundances are negligible: fS (ξ, rstart ) = 0 ∀ξ,. (3.40). fN (ξ, rstart ) = 0 ∀ξ,. (3.41). mh for rstart m . If the scalar is a FIMP this makes sense right away, and for the S WIMP case the initial condition does not matter because the distribution will be dragged into equilibrium extremely quickly.. 41.

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(43) 4. Results In this chapter the outcome of the calculations is presented and the viability of the model under various viewpoints is discussed. The Boltzmann equation was solved for different values of λ, y and mS in the following approximate ranges:1 10−10 . λ . 10−4 ,. (4.1). 10−9 . y . 10−6 ,. (4.2). 30 GeV ≤ mS ≤ 1000 GeV.. (4.3). For discrete values of the scalar mass mS the λ − y parameter space is scanned in regions that are not a priori excluded by experiments. The most important of these easily checkable bounds is the Tremaine-Gunn bound (cf. section 2.4). It excludes all parameter space which leads to a sterile neutrino mass of less than 0.5 keV. As the overclosure region is always contained in the area that is excluded by the Tremaine-Gunn bound, it does not have to be checked. Only regions that are not excluded by these bounds are considered. The scalar masses chosen in this work cover the different regimes as shown in table 4.1. The biggest qualitative difference will turn out to appear between the Table 4.1.: Scalar mass values considered and their properties. mS = 30 GeV mS = 60 GeV mS = 65 GeV mS = 100 GeV mS = 500 GeV mS = 1000 GeV 1. mS ≤ mh /2: Higgs boson can decay into two scalars; relevant production temperatures below EWPT mS ≤ mh /2: Higgs boson can decay into two scalars; relevant production temperatures mainly below EWPT mS > mh /2: Higgs boson cannot decay into two scalars; relevant production temperatures mainly below EWPT mS > mh /2: Higgs boson cannot decay into two scalars; relevant production temperatures above and below EWPT mS > mh /2: Higgs boson cannot decay into two scalars; relevant production temperatures above EWPT mS > mh /2: Higgs boson cannot decay into two scalars; relevant production temperatures above EWPT. The exact ranges of λ and y depend on the scalar mass mS because the excluded regimes themselves depend on it. Furthermore, the values for mS are discrete.. 43.

(44) 4. Results mass values in gray and the ones in white, because the masses for the gray cases are smaller than half the Higgs boson mass, and the decay of a Higgs boson into two scalars will prove to be extremely strong in keeping the scalar in thermal equilibrium, or, in the FIMP case, in producing many S particles. For each triple (mS , λ, y) the time evolution of the distribution functions of the scalar and the sterile neutrino is computed. Then the neutrino mass is determined by matching the resulting abundance to the observed dark matter abundance, and structure formation bounds will be applied which will exclude further regions in the parameter space. The structure formation bound used here derives from the Lyman-α forest data. We use the final sterile neutrino distribution function to compute the linear matter power spectrum, from which we in turn derive the squared transfer function, i.e. the ratio of this power spectrum and the power spectrum of perfectly cold dark matter (CDM), to compare it to the bounds given in [36], as discussed in section 2.3.. 4.1. Stable scalar To understand the different effects that happen for different parameters it is very instructive to first have a look at the case when y = 0, i.e. considering a stable scalar. Of course this does not make sense if we want to describe dark matter consisting of sterile neutrinos because if y vanishes, the scalar does not decay and no sterile neutrinos are produced at all. But for better understanding it is useful to ignore the decay into neutrinos for the moment. We start by examining the interaction rate Γint of the scalar with the thermal Standard Model bath.2 Fig. 4.1 shows the interaction rate in comparison with the Hubble function H. A typical estimate is that a particle that is initially in equilibrium freezes out when Γint ≈ H, because after that the interaction rate is too low to keep the species in equilibrium. In fig. 4.1(a) one can see the cases with very small coupling parameter λ. Starting with the initial condition of vanishing scalar abundance, i.e. virtually no scalars are present, the interaction rate is too low ( H) to ever fully drag them into equilibrium, as can be seen by noticing that Γint always stays below the Hubble rate. This is a standard FIMP case. Fig. 4.1(b) shows the case of an intermediate coupling strength λ. Here, it depends on the mass of the scalar if it reaches equilibrium or if it is a pure FIMP: for the masses 30 GeV and 60 GeV (the faint lines), at around r ≈ 3 the interaction rate becomes high enough for the scalars to reach equilibrium and freeze out afterwards. 2. See Appendix B for more details.. 44.