1 Motivation and Introduction

Dark Matter Production Mechanisms with a Non-Thermal Cosmological History - A Classification


Dark Matter Production Mechanisms with a Non-Thermal

Cosmological History - A Classification

Gordon L. Kane, Piyush Kumar, Brent D. Nelson, Bob Zheng

Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109

Department of Physics, Yale University, New Haven, CT 06520

Department of Physics, Northeastern University, Boston, MA 02115, USA

1 Motivation and Introduction

Apart from its existence, the nature and non-gravitational interactions of dark matter (DM) are still very uncertain. The most popular class of dark matter models - Weakly Interacting Massive Particles (WIMPs) - rely on two key assumptions to reproduce the observed relic abundance. First, WIMPs are assumed to annihilate into Standard Model (SM) particles with an electroweak-scale cross section. Second, the universe is usually assumed to be radiation dominated between the end of inflation and matter-radiation equality. However, there are no clear indications that either of these assumptions are valid. With regards to the former, large regions of WIMP parameter space have been ruled out by various direct and indirect detection experiments. With regards to the latter, the earliest evidence for a radiation dominated universe arises during BBN, which occurs at temperatures of order an MeV. The energy budget of the Universe has not been probed for temperatures above that at the time of BBN. Of course, it is still possible that dark matter is a simple WIMP, but because of the above reasons it is well-motivated to go beyond the traditional WIMP paradigm, both in terms of DM candidates as well as the production mechanisms for DM.

A well-motivated alternative to the standard “thermal” cosmological history mentioned above is that of a non-thermal cosmological history, in which BBN is preceeded by a phase of pressureless matter domination. Such a situation is predicted in many top-down theories for new physics e.g. low-energy limits of supergravity and string/M-theory compactifications. These theories, under some very mild assumptions, contain gravitationally coupled scalars called moduli. When the Hubble parameter drops below moduli masses, moduli begin coherent oscillations and behave as pressure-less matter, dominating the energy density of the universe until the longest-lived one () decays to reheat the universe. In these cosmological histories, an electroweak-scale Wino provides a natural candidate for supersymmetric (SUSY) DM, provided that the modulus dominated phase ends at temperatures below a GeV or so [1, 2]. However, recent FERMI-LAT and HESS observations of Galactic Center photons have placed severe limits on Wino DM [3, 4]. If the Wino is stable, satisfying these constraints in the cosmological histories mentioned above requires a large hierarchy between the modulus and gravitino masses [3]. Such a hierarchy is quite unnatural for a broad class of models in which moduli stabilization sets the scale of supersymmetry breaking [5, 6, 7, 8]. This conclusion also holds if the lightest superpartner is some more general admixture of MSSM particles [9]. A simple way to avoid these constraints is to assume that the lightest visible sector superpartner, hereafter referred to as the LOSP, is unstable.

Motivated by the above statements, this work provides a comprehensive study of relic DM production in cosmological histories with a late phase of modulus domination. To perform as general an analysis as possible, we go beyond the standard WIMP picture by i). allowing for a wide range of DM masses and annihilation cross sections and ii). allowing for the possibility that DM is in kinetic equilibrium with some sector other than the visible sector. These two assumptions are well motivated in SUSY theories with an unstable LOSP, but can also be true in general. If the LOSP decays, DM is not a visible sector particle; a priori there is no reason to expect its DM mass or annihilation cross section to be near the electroweak scale. Moreover, if the DM resides in a sector that couples weakly to the visible sector, DM could be in kinetic equilibrium with a “dark sector” instead of the thermal bath of visible sector particles. As we will see, our results can be straightforwardly reduced to that of the single-sector case despite assumption ii).

The analysis of DM models in this framework can be effectively separated into three questions.

  • How does one classify production mechanisms for relic DM and accurately compute ?

  • What is the pattern of experimental and observational signals arising within such a framework?

  • What kind of DM candidates and interactions naturally arise in well-motivated theories?

In this paper, we focus primarily on the first question by solving the Boltzmann equations for a two-sector system with a late phase of modulus domination. A brief overview of this framework, along with the Boltzmann equations describing its cosmological evolution, is presented in Section 2. We then classify all potential mechanisms for the production of relic DM, and compute for a large range of DM masses and annihilation cross sections. The entire parameter space of these DM models can be classified into four different parametric regimes, each with distinct production mechanisms for relic DM. We derive (semi) analytic approximations for in these different regimes, and confirm their validity by comparison with the numerical solution. This is the main new result of our work, and is discussed in detail in Section 3. Readers may go directly to Section 3.5, which contains a self-contained summary of the above results. In Section 4, we discuss the implications of our results for UV-motivated SUSY theories where the modulus mass is of order the gravitino mass, and show that this framework provides several viable alternatives to MSSM dark matter.

In the remainder of the paper, we briefly discuss the latter two questions listed above. In Section 5, we discuss potential experimental signatures of the DM models considered here. A significant portion of the parameter space predicts free-streaming lengths characteristic of warm dark matter. Furthermore, we identify a class of DM models in which the DM power spectrum is sensitive to the linear growth of subhorizon DM density perturbations during the modulus dominated era. This can lead to interesting astrophysical signatures, such as an abundance of earth-mass (or smaller) DM microhalos which are far denser than their counterparts in standard cosmologies [10]. Finally, Section 6 briefly addresses the third question, and describes work done in string theory models that have dark sectors. In a companion paper, we will elaborate further on some classes of these models. We present our conclusions in Section 7. The appendices contain technical results which will be referred to in the main text.

2 Overview of Two-Sectors - Models and Cosmology

The framework considered here consists of two sectors: a visible sector containing SM (and perhaps MSSM) particles and a dark sector containing the DM. Both the visible and dark sectors are assumed to have sufficient interactions such that thermal equilibrium is separately maintained within the two sectors, whose temperatures are and respectively. We assume that there exist very weak portal interactions between the two sectors, so that and may not be equal to each other. Finally, we assume that the Universe is dominated by the coherent oscillations of a modulus field at some time which is much earlier than when BBN occurs1. The cosmological framework described above is depicted schematically in Figure 1. The results of our work will be straightforward to reduce to the single sector case, see the discussion in Section 3.5.

As denoted in Figure 1, the visible sector contains radiation degrees of freedom , comprised of relativistic particles in equilibrium with the SM bath at temperature . We also track the abundance of an unstable WIMP-like particle which is in equilibrium with the visible sector. corresponds to the LOSP in the SUSY theories discussed in the introduction. The dark sector is assumed to contain a stable DM candidate , along with dark radiation . “Dark radiation” refers to dark sector particles which are in thermal equilibrium and are relativistic at a given dark sector temperature . Henceforth, visible (dark) sector quantities are denoted using unprimed (primed) variables. For simplicity and concreteness, we assume that no DM asymmetry is present, so DM particles and antiparticles need not be separately tracked in the Boltzmann equations. Relaxing this assumption is worth exploring in future studies, see for example [9]. Finally we make the assumption that , which is naturally expected for the supersymmetric theories discussed in the introduction and in Section 4.

Before moving on to study the cosmological evolution of this system, it is worth mentioning that there are constraints on hidden sector relativistic degrees of freedom during BBN and during recombination, through their contribution to the expansion rate of the Universe. These constraints are typically presented in terms of the number of effective extra neutrino species , which is related to the number of relativistic hidden sector degrees of freedom by:


where MeV, eV and . The current 95% CL bounds are  [12] and  [13]. We will discuss the implications of these constraints for the two sector models considered here in Section 3.1.2.

Figure 1: Schematic representation of the Two-sector Framework under consideration.

2.1 Cosmological Evolution

The cosmology of the framework can be studied by writing down the Boltzmann equations for the time evolution of the relevant quantities which comprise the total energy density of the Universe. This includes the modulus energy density , the energy density arising from and with number densities and respectively, and the energy densities of radiation in the visible and dark sector, denoted by and respectively. The relevant parameters in the Boltzmann equations turn out to be:


Here is the decay width of the unstable particle, and denote the masses of and respectively, while and denote the thermally averaged annihilation cross-section of and respectively. and are the relativistic degrees of freedom in the visible and dark sectors at a given temperature . The quantities and denote the branching fractions of the modulus to and respectively2. Given the assumption , approximately denotes the fraction of the energy density from the modulus going to dark radiation, with the remaining fraction going to visible radiation. Finally, following established convention we define in terms of the decay width of the modulus as follows:


where GeV is the Planck scale, and is the number of relativistic degrees of freedom in the visible sector at . We will discuss the physical intepretation of in Section 3.1.2.

A priori, the nine parameters in (2) can vary over a wide range of values, and could affect the computation of the DM relic abundance in a variety of ways. However, we will show that for , the DM production mechanisms only depend on a subset of the parameters in (2), in particular:


where if decays to and if does not decay to . Note that there is no dependence on parameters measuring the attributes of the LOSP ! Furthermore, as will be discussed in Section 4, the parameters and are completely determined by the masses and couplings of the modulus . Thus these parameters are insensitive to the details of the dark sector. In the forthcoming analysis, we find it useful to choose benchmark values for the following parameters:


The theoretical motivation for these benchmark values will be clear from the discussion in Section 4. With these parameters fixed, the DM abundance will depend only on and , and we will see that these can take a wide range of values for viable DM production mechanisms. As mentioned above, for most of the paper we take since this is naturally obtained if is not Planck suppressed. In Appendix 9, however, we will briefly discuss the case .

The Boltzmann equations which describe this system are a natural generalization of those which are applicable to a single sector framework within a modulus dominated Universe, as studied in [14, 15]. As pointed out in these papers, it is more convenient to define dimensionless variables corresponding to the energy and number densities and also to convert derivatives with respect to time to those with respect to the (dimensionless) scale factor , with . Thus, following [14, 15] we define:


and are the thermally averaged , energies assuming that and are in kinetic equilibrium. The Boltzmann equations in terms of these comoving dimensionless variables are:


with , and


and are related to the and equilibrium number densities via:


where counts the degrees of freedom of and for bosonic (fermionic) . is given by (9) with primed variables replacing unprimed variables.

Note that we have assumed in (2.1) that decays to ; we neglect inverse decays, as the dynamics which fix occur when (see Section 3.2) at which point inverse decays are exponentially suppressed. The thermally averaged decay rate is given by:


where is the decay rate in the rest frame, and and are modified Bessel functions of the second kind. The quantities and are respectively the thermally averaged partial widths for and . In the remainder of this work, we focus on the case where all decay channels yield such that (2.1) is valid; this corresponds to and both being charged under the DM stabilization symmetry. It is also possible for to instead decay directly to visible radiation, as is the case for R-parity violating SUSY models. In this case and are essentially decoupled in the Boltzmann equations, which significantly simplifies the analysis. In Section 3 we focus on the more complicated case where decays to , and discuss how relaxing this assumption affects our results.

The above differential equations are solved subject to the following initial conditions:


These initial conditions are somewhat unphysical as they imply at . However, at early times the visible and dark radiation energy densities are subdominant, so this approximation is justified. is the initial value of the Hubble parameter which fixes the initial energy density of the modulus field, parameterized by . As we will see, in most cases the DM relic abundance is largely insensitive to the initial condition .

3 Solution of the Boltzmann Equations and the Dark Matter Abundance

Given the system of equations (2.1), it is possible to numerically solve it for various choices of the parameters in (2). However, in order to get a good physical intuition of the qualitatively different mechanisms at play, it is advisable to study various approximate (semi) analytic solutions which are applicable in different regions of the parameter space. We carry out such an exercise in this section. In Appendix 10, we compare our approximations to the full numerical analysis and find very good agreement.

3.1 Useful Approximations

We now derive useful approximations which allow us to obtain semi-analytic expressions for in Section 3.2. To start with, it is worth noting that remains constant until to a very good approximation. Thus in the following analysis we set throughout the period of modulus domination, considerably simplifying the Boltzmann equations. Our strategy will be to use physically well-motivated approximations to first solve for , , and , and then use these solutions to study the equation for .

Approximate solutions for , and

Consider first the Boltzmann equation for . With , it is straightforward to solve for :


Thus, as expected, remains approximately constant at , and only begins to decay vigorously when the dimensionless scale factor satisfies , with


Now consider the equations for and . As can be seen from (2.1), in addition to the modulus decay term these equations contain the and annihilation terms as well as the decay term. However, it turns out that for all these terms are quite sub-dominant compared to the modulus decay term. This is because if , the energy densities of and are subdominant to during the modulus dominated era; a more detailed argument for this is presented in Appendix 8. Given this approximation, the solutions to (2.1) do not depend on the branching fractions of . Thus the approximate solutions for and can be found readily by integrating the modulus decay term:


In the second line of (14), represents the late time solution for , i.e when the scale factor . Note that during the radiation dominated era. We have approximated as


where and approximately correspond to the temperatures at which the integrand (14) peaks. These temperatures characterize the transition between modulus and radiation domination, and are defined more precisely in Section 3.1.2. To obtain the result above for , we have expanded the function obtained after the integration as a series expansion in with and kept the leading term. This can be justified by taking as given by (11), where is the Hubble parameter when the modulus starts dominating the energy density of the Universe. Thus, for with ,3 one finds .

Temperature-scale factor relation and the “maximum” temperature

The temperature of a system is measured by the radiation energy density, and the relation between the two is given in general by:


In a radiation dominated Universe, it is well known that remains constant with time, giving . However, the situation is different within a modulus dominated Universe since does not remain constant with time. It can be shown that at early times when , and the temperatures and scale factor are related approximately by [14]:


where , the maximum temperature attained during modulus domination, is given by:


Thus, we see that the temperature has a more complicated dependence on the scale factor compared to that in radiation domination. Using the fact that with , one finds that . From (14) it is straightforward to relate the visible and dark sector temperatures:


Combining (18) and (19) gives for the dark sector. As mentioned in Section 2.1, bounds on at both MeV and eV constrain and , which through (19) can be mapped into a constraint on . Comparing (19) with the bound (1), we see that the resulting constraint on is insensitive to assuming . Taking and , the constraints (1) imply (BBN) and (CMB).

In the presence of dark radiation, as defined in (3) no longer corresponds to the visible sector temperature when , assuming the modulus has completely decayed (). Instead, we define the temperatures ,  as the visible and dark sector temperatures when :


The bounds from discussed above imply . Hence, for simplicity we will set . It is also useful to compute the scale factor which corresponds to the temperature . We compute by substituting and in (16):


From the definition of in (13), we see that .

We caution the reader that the definitions of , and established above are limited in the following sense. The above expressions for , and were derived from assuming that the universe has reached radiation domination, i.e. and . However, modulus decay is a continuous process which occurs when , but does not have a well-defined start or end point. Upon solving the Boltzmann equations, one finds that when , the modulus has not finished decaying and the radiation dominated phase has not yet been reached . In the next subsection we will verify this fact graphically, utilizing the full numerical solutions for and (see Figure 2 below). Despite this ambiguity, we find , and to be useful qualitative proxies for the temperature and scale factor at which the universe transitions from the modulus dominated to radiation dominated era.

Approximate solution for

Now consider the Boltzmann equation for . Motivated by earlier statements, we are interested in the case where is a LOSP with weak scale mass and annihilation cross section; thus will be exponentially suppressed for temperatures of a few GeV. In our analysis, we will mostly consider the situation that the LOSP decays before the modulus (typically much before), i.e. . Such a condition can be naturally achieved since the modulus decays by Planck-suppressed operators. In Appendix 9, we will briefly consider the case where .

In the Boltzmann equation for , the decay term grows like ; thus we are interested in the solution for in the low temperature regimes where can be neglected (this approximation is justified in Appendix 9). With this approximation, the Boltzmann equation for can be written as:


where is the critical value required for annihilations to be efficient for a given value of the Hubble parameter. More precisely, is given by:


Now, if the processes for depletion of (the first and second terms on the right hand side of (22)) and the production of (the third term in the right hand side of (22)) are larger than itself, then these are each faster than the Hubble rate and one rapidly reaches a situation where the two processes cancel each other, giving rise to what is known as quasi-static equilibrium (QSE) [16]. The QSE solution is found by equating the right hand side of (22) to zero:


Given the criteria described above (24), QSE occurs when:


Upon inspection, one finds that the QSE condition (25) is equivalent to the familiar condition . Thus, we see that as long as , the QSE condition will be satisfied during the modulus dominated era such that for .

We can gain further insight into the QSE solution for by rewriting (24) as:


Physically, the QSE solution for occurs when moduli decay into , and decay into , balance one another; this explains the dependence of on . In the above expression, is defined as:


Note that for the benchmark choice of parameters in (5), and not extremely small, is quite large (compared to a WIMP cross-section ). We expect the same qualitative conclusion as long as the portal coupling is not extremely tiny. Thus for supersymmetric models where is the LOSP, we expect , and hence in the QSE solution for in the second line of (26).

Figure 2 shows a plot of the solutions for the values of and (normalized to their maximum values) as functions of the scale factor for the choice of benchmark parameters as in (5). As can be seen from (12), (14) and (26), respectively, the solutions for and do not depend on , or to a good approximation. Moreover the solution for depends does not depend on for most models of interest in which as we have just discussed above.

Figure 2: Plots of the exact solutions for and (normalized to their maximum values) as functions of the scale factor . We have taken , , GeV and GeV. All other parameters taken to their benchmark values (5). The dashed vertical line represents the scale factor defined in (21), which characterizes the transition between a modulus dominated and a radiation dominated universe.

3.2 Classifying Production Mechanisms for Relic Dark Matter

We now move on to studying the main quantity of interest – the Boltzmann equation for , whose solution will give us the expression for the relic abundance of dark matter in terms of a subset of the parameters (2) appearing in the Boltzmann equations. More precisely, the relic abundance is given by:


In the above expression, is the temperature at any very late time in which the universe has become radiation dominated () and the comoving abundance has become constant. The parameters GeV and are the present day temperature and radiation relic density. Taking and using (16) to relate and , (28) can be written as:


In order to derive semi-analytic approximations for and , we will solve the Boltzmann equation for given the approximations stated in the previous sections. In the following we will show that , so is insensitive to as mentioned above.

Using the approximate solutions for and in (12), (14) and (26), respectively, we can reduce the system of Boltzmann equations in (2.1) to a single ordinary differential equation for the evolution of :


where is defined in (26). Note that if does not decay to , the term in (30) is absent. Using a similar definition for the critical annihilation for as was used for in (23), one can rewrite (30):


where if decays to 4, and if does not decay to . Just as for the case of , if the processes of depletion of (first term on the right hand side of (31)) and production of (second, third and fourth terms on the right hand side of (31)) are each greater than itself, will rapidly reach a quasi-static equilibrium (QSE) attractor solution such that terms on the right hand side of (31) cancel among themselves:


Comparing (31) and (32), and using (26) for the QSE solution for , we see that the QSE conditions hold when:


Note that in contrast to for , does not necessarily enter QSE during the modulus dominated phase. One reason for this is that in contrast to , which is assumed to be a WIMP, we are exploring a much more general set of possibilities for the mass and interactions of the DM particle .

In order to understand better the broad possibilities that could arise for , it is important to find the conditions necessary for QSE to hold at . If the QSE conditions hold at , then the positive contribution to from modulus decay is annihilated away such that maintains its QSE value. In this case, the final abundance is insensitive5 to modulus decay parameters such as and . Conversely if QSE does not hold at , will be sensitive to contributions from modulus decay, along with other sources for production during the modulus dominated era. Comparing (31) and (32), we see that requiring places a lower bound on . Keeping the above statements in mind, it is useful to define a critical annihilation cross section such that