# Asymmetric dark matter annihilation as a test of non-standard cosmologies

###### Abstract

We show that the relic abundance of the minority component of asymmetric dark matter can be very sensitive to the expansion rate of the Universe and the temperature of transition between a non-standard pre-Big Bang Nucleosynthesis cosmological phase and the standard radiation dominated phase, if chemical decoupling happens before this transition. In particular, because the annihilation cross section of asymmetric dark matter is typically larger than that of symmetric dark matter in the standard cosmology, the decrease in relic density of the minority component in non-standard cosmologies with respect to the majority component may be compensated by the increase in annihilation cross section, so that the annihilation rate at present of asymmetric dark matter, contrary to general belief, could be larger than that of symmetric dark matter in the standard cosmology. Thus, if the annihilation cross section of the asymmetric dark matter candidate is known, the annihilation rate at present, if detectable, could be used to test the Universe before Big Bang Nucleosynthesis, an epoch from which we do not yet have any data.

a]Graciela B. Gelmini,
a]Ji-Haeng Huh
a]and Thomas Rehagen
\affiliation[a]Department of Physics and Astronomy, UCLA,

475 Portola Plaza, Los Angeles, CA 90095, USA
\emailAdd
\emailAdd
\emailAdd
\keywordsdark matter theory, dark matter experiments
\keywordsdark matter theory, dark matter experiments

## 1 Introduction

The nature of dark matter (DM) is one of the fundamental problems of physics and cosmology. Particles with weakly interacting cross sections and masses in the few GeV to 10 TeV range, WIMPs (Weakly Interacting Massive Particles), are among the best motivated DM candidates. DM particle candidates, such as WIMPs (but also sterile neutrinos and axions) are produced before Big Bang Nucleosynthesis (BBN), an epoch from which we have no data. BBN is the earliest episode (finishing 200 seconds after the Bang, when the temperature of the Universe is T MeV) from which we have a trace, the abundance of light elements D, He and Li. In order for BBN and all the subsequent history of the Universe to proceed as usual, it is enough that the earliest and highest temperature during the last radiation dominated period, the so called reheating temperature , is larger than 3.2 MeV [1].

The argument showing that WIMPs are good DM candidates, many times called the “WIMP miracle”, is more than 30 years old [2]. The density per comoving volume of non-relativistic particles in thermal equilibrium in the early Universe decreases exponentially with decreasing temperature, due to the Boltzmann factor, until the reactions which change the particle number become ineffective. At this point, when the annihilation rate becomes smaller than the Hubble expansion rate, the WIMP number per comoving volume becomes constant. This moment of chemical decoupling or freeze-out happens later for larger annihilation cross sections , which produces smaller WIMP densities. If the Universe is radiation dominated during decoupling and there is no subsequent change of entropy in matter plus radiation, the present relic density is

(1) |

which for weak order G gives the right order of magnitude of the DM density (and a temperature at freeze-out for a WIMP of mass ).

The “WIMP miracle” argument and the standard computation of relic densities rely on assuming that radiation domination began before the main epoch of production of the relics, that the entropy of matter and radiation has been conserved during and after this epoch, that WIMPs are produced thermally, i.e. via interactions with the particles in the plasma, and that there is no significant asymmetry between the WIMP particles and antiparticles. With these assumptions chemical decoupling happens at , thus all WIMPs with MeV decouple at temperatures higher than 4 MeV, when the content and expansion history of the Universe may differ from the standard assumptions.

The relic density, see e.g. Ref. [3] (and also the relic velocity distribution, see e.g. Ref. [4]) before structure formation of WIMPs (and other DM candidates, e.g. sterile neutrinos [5]) depends on the characteristics of the Universe (expansion rate, composition, etc.) before BBN. If these particles are ever found, they would be the first relics from the pre-BBN epoch that could be studied. Thus we will want to extract as much information about the Universe at the moment these particles decoupled as we can.

Here we present a new potentially detectable effect that non-standard cosmologies may have on asymmetric dark matter.

## 2 Asymmetric dark matter

The idea of asymmetric DM is almost as old as the “WIMP miracle” argument: if DM particles and antiparticles have an asymmetry similar to the baryonic asymmetry this could explain why the baryonic and DM relic densities are similar, i.e. differing by a factor of a few and not by many orders of magnitude. In 1985, S. Nussinov [6] pointed out that if the asymmetry of technibaryons and usual baryons could be similar in the early Universe, their present number relic density would also be similar, which would mean that the ratio of their relic densities would be given by the ratio of the DM and baryon masses. It was then largely assumed then that the DM relic density was the critical density, thus (for ) would imply a ratio of the lightest neutral “Technibaryon (TB)” and nucleon mass to be GeV which was phenomenologically acceptable at the time. This argument does not exactly hold now, because the ratio of dark and visible matter relic densities is only about a factor of 5, but different versions of Technicolour models have been a fertile framework for 100 GeV-TeV mass asymmetric DM candidates (see e.g. [7]).

The first model unrelated to Technicolour for “light” asymmetric DM was proposed in 1986. Gelmini, Hall and Lin [9] produced several models for 5 to 10 GeV mass DM candidates, called “cosmions” (X), which could influence the physics of the Sun. In one of them the baryon minus “cosmion” number is assumed to be conserved and a common origin of an asymmetry in both and numbers, insures that the baryon and DM asymmetries are identical, so that GeV 5 to 10 (which was then considered enough just to account for the “galactic dark matter”). The same basic idea, i.e. the conservation of a linear combination of baryon and DM particle number and a shared origin of the baryon and dark matter asymmetry was subsequently realized in many models for heavier candidates, in the 10’s to 100’s GeV mass range (see e.g. Ref. [8]), until Kaplan, Luty and Zurek in 2009 [10] applied it again to candidates in the 5 to 15 GeV mass range, in a paper which generated renewed interest in asymmetric DM models (for a recent review see e.g. [11]). Light asymmetric DM particles not unlike “cosmions” were recently proposed as a means to solve the current discrepancy in the predicted composition of the Sun between helioseismological data and the revised Standard Solar Model [12].

The ideas we develop in the following apply to asymmetric DM WIMPs of any mass. We are not concerned here with the origin of the asymmetry. We assume only that one was generated before the chemical decoupling of the DM particles and antiparticles , so that

(2) |

where , , and are the respective relic number densities, is the entropy density, , dominated by the relativistic degrees of freedom and is a constant that characterizes the asymmetry. Here we take positive so and are respectively the majority and minority components of the DM at present. Moreover, in the following we will assume that and account for the whole of the DM, i.e.

(3) |

although the arguments can be easily changed if they account for only a fraction of the DM.

The evolution of the equilibrium values of , as function of are shown in Fig. 1.a. The equilibrium number densities and in the presence of an asymmetry differ by the chemical potential (in equilibrium )

(4) |

(5) |

Here, is the mass of and , and is the number of internal degrees of freedom of and of separately. For simplicity in the following we take (which assumes and are conjugate complex scalar fields). A different choice of (e.g. if and are two conjugate Dirac fermions) would not affect the results significantly. The results we find would change by factors of . The chemical potential can be written in terms of the asymmetry by substituting the equilibrium number densities into Eq. (2),

(6) |

and solving this quadratic equation for , to get

(7) |

Replacing Eq. (7) in Eqs. (4) and (5) we can find and as functions of , and . See Fig. 1.a for the evolution of and as a function of for and .

In the presence of the asymmetry , the annihilation of the majority component is considerably reduced after reaches the value at , . For the equilibrium number density per comoving volume of the majority component becomes almost constant while that of the minority component decreases faster than in the symmetric case as increases. It is easy to see why this is so by considering the annihilation rate per particle of and , and respectively,

(8) |

in equilibrium, whose evolution as function of is shown in Fig. 1.b. Here is the thermally averaged annihilation cross section and we are assuming that these annihilations are the only processes that can change the number of these particles.

For the and interaction rate in equilibrium become respectively larger and smaller than in the symmetric case because is much larger and is smaller than in the case.

The annihilation of each DM component and ceases when their respective interaction rates become smaller than the expansion rate of the Universe , which happens at their respective decoupling or freeze-out, for and for , defined by

(9) |

It is clear from Fig. 1.b that decouples earlier than , i.e. . At their respective freeze-out, each component acquires its relic density, fixed in comoving volume. However, little changes in the abundance of the majority component after it freezes-out, since already at the annihilations had become very suppressed.

In the standard cosmology the relic density of the minority component, which was computed for the first time in Refs. [14] and [15] and recently in Refs. [16] and [17], is exponentially small with respect to the majority component density. (Note that our asymmetry parameter is called in Ref. [14], in Ref. [15], in Ref. [16] and in Ref. [17].) This is why there is no asymmetric DM annihilation at present.

Here we will show that if the chemical decoupling of the minority component happens during a non-standard cosmological phase, the relic abundance of this component of asymmetric dark matter can be much larger than in the standard cosmology. Thus, if the annihilation cross section of the dark matter candidate is known, the annihilation rate at present, if detectable, could be used to test the Universe before Big Bang Nucleosynthesis, an epoch from which we do not yet have any data. In particular, because the annihilation cross section of asymmetric dark matter is typically larger than that of symmetric dark matter in the standard cosmology, the decrease in relic density of the minority component with respect to the majority component may be small enough that the annihilation rate at present of asymmetric dark matter, contrary to general belief, could be larger than that of symmetric dark matter in the standard cosmology.

## 3 Non-standard pre-BBN cosmologies

Many viable non-standard pre-BBN cosmological models have been proposed (see e.g. [18] for a review). Usually non-standard cosmological scenarios contain additional parameters that can be adjusted to modify the WIMP relic density. However these are due to physics that does not manifest itself in accelerator or DM detection experiments.

In the standard (STD) cosmology the Universe is radiation dominated, thus the expansion rate of the Universe depends on the temperature of the radiation bath as , but in non-standard models can decrease slower, , or faster, as decreases. This is what happens in “scalar-tensor” [19, 22] and “kination” [23] cosmological models, respectively. In both of these models the entropy of matter plus radiation is conserved (and it is dominated by the radiation component), thus the usual relation between and the scale factor of the Universe , namely , is the same as for a radiation dominated Universe.

In models in which the potential energy of a scalar field oscillating around its true minimum while decaying is the dominant component of the Universe just before BBN, the expansion rate decreases even faster, but there is entropy creation, thus . These models produce the largest departure of symmetric WIMP relic density from the standard density (see e.g. [3]) and we could expect their effect on asymmetric DM to be very large too. However here for simplicity we concentrate on models in which there is no entropy creation.

Scalar-tensor (ST) theories of gravity [19, 22] incorporate a scalar field coupled only through the metric tensor to the matter fields. In many of these models the expansion of the Universe drives the scalar field towards a state where the theory is indistinguishable from General Relativity at a transition temperature , but the effect of the scalar field changes the expansion rate of the Universe at earlier times, either increasing or decreasing it. Theories with a single matter sector typically predict an enhancement of before BBN. In Ref. [22], for large temperatures the is enhanced with respect to the standard expansion rate by a factor ( is the present temperature of the Universe), thus . At , drops sharply to values close to 1 at before BBN sets is.

Kination [23] is a period in which the kinetic energy of a scalar field (maybe a quintessence field) dominates over the potential energy so . Since as usual, . The contribution of the kinetic energy to the total density is usually quantified through the ratio of -to-photon energy density, at MeV so that at higher temperatures . Notice that at MeV, i.e. during BBN, the quintessence field cannot be dominant, thus .

## 4 Asymmetric DM relic density calculation

The relic densities of DM particles and anti-particle are calculated by solving their respective Boltzmann equation (for asymmetric DM, and are not self-conjugate, ). We assume that close to the moment of decoupling, the only reactions that change the number of DM particles and antiparticles are annihilations of pairs into Standard Model particles and the inverse process of pair creation. With this assumption we have

(10) |

Notice that in (10) we have disregarded the possibility of having and self-annihilation, which is instead considered in Ref. [24]. In terms of the dimensionless quantities , and , and treating as a constant, the Boltzmann equations become

(11) |

(12) |

In the following we take , a good approximate value for decoupling temperatures above the QCD phase transition. Eq. (11) implies , so the asymmetry defined in Eq. (2) is a constant. It is clear that is constant because we are only considering interactions of the form ’s where ’s are Standard Model particles, which leave the difference between the co-moving number density of the particle and anti-particle unchanged. We are assuming that the asymmetry has been produced prior to the epoch that we are considering.

### 4.1 Analytical solutions

It is useful to obtain approximate analytical solutions to Eq. (14) and compare them with numerical solutions. Fig. 1.b makes it clear that in the period between , when becomes practically constant and the freeze-out of the minority component, decreases exponentially. Until freezes out, the densities of both DM components are those of equilibrium (for it is very close) and after, for , the production term in the Boltzmann equations is suppressed compared to the annihilation term (because ) and so we ignore it. Then, for , Eq. (14) becomes

(15) |

Integrating this equation from to (a good approximation for the value of at present) we get

(16) |

For asymmetric DM the ratio increases immediately after decouples, as decreases. The decoupling of is not instantaneous, as assumed when defining , and annihilations still continue for a while after departs from its equilibrium value. This can be seen e.g. in Figs. 9 and 10 where the solid black lines showing the numerical solutions for decrease after the freeze-out of the minority component at . So we keep only the term on the left hand side of Eq. (16) and obtain

(17) |

Using Eq. (2), i.e. we get for the majority component

(18) |

We have kept explicitly because we want to explore different possibilities for the Hubble expansion. Had we replaced by its functional form in the standard cosmology, these equations would be identical to those in Ref. [16], and the ratio of resulting from these equations is the same as in Ref. [17].

#### 4.1.1 Standard cosmology

In the standard cosmology, the expansion rate of the Universe is

(19) |

where is the reduced Planck mass, and is the effective number of relativistic degrees of freedom. Expanding the annihilation cross section in powers of the relative velocity, v, and then taking the thermal average, the annihilation cross section can be written in term of constants and as

(20) |

where the term is dominant for s-wave annihilation, and for p-wave annihilation.

#### 4.1.2 Kination

At the transition temperature at which the Universe becomes radiation dominated after the kination phase, the expansion rate of the Universe during kination, , coincides with the standard one, which allows us to fix the constant multiplying in and find

(23) |

This corresponds to having the kination parameter MeV, so for 1 MeV, during BBN.

#### 4.1.3 Scalar-tensor model

As mentioned above, in this model [22] the expansion rate of the Universe is given by

(25) |

where is a temperature dependent factor which transitions very fast to a value of 1 at the transition temperature , and is the present temperature of the Universe. In terms of , this factor is . Thus, for transition temperatures from 1 MeV to a few GeV, the Hubble parameter decreases suddenly by a few orders of magnitude at , when the change to the standard cosmology happens. Due to this sudden drop in the , and start re-annihilating and they freeze-out again shortly after. Usually and freeze-out in the re-annihilation phase before they can reach equilibrium and, in this case, we can still neglect the production term in the Boltzmann equations (again because ), and Eqs. (17) and (18) still hold. Using Eq. (25) in Eq. (17), we calculate the relic abundance of in the scalar-tensor cosmology,

(26) |

where , and is defined as before. We have taken for simplicity. As before, an expression for the relic abundance of can be found by using Eq. (2), i.e. .

#### 4.1.4 Estimation of the freeze-out temperature

To find we use the freeze-out condition Eq. (9) for , using in each case the corresponding written above and the reaction rate of , . Considering that until freeze-out and closely track their equilibrium values, and given in Eqs. (4) and (5), we can approximate by

(27) |

Using Eqs. (6) and (7) we get an expression for as function of the asymmetry . Using this form of , the freeze-out condition Eq. (9) becomes a transcendental equation for , which we can solve numerically. After solving for , we use the expressions calculated above to estimate the relic abundance of and .

### 4.2 Numerical solutions

The Boltzmann equations (13) and (14) can be solved numerically for each of the cosmologies we consider. Here we show examples of the evolution of and as a function of in each cosmology and later compare the numerically calculated relic density to the analytical results. In all of the following we present results for pure s-wave annihilation.

The final relic density of the minority component depends strongly on the relation between its freeze-out temperature or and the value at which the asymmetry becomes important and the minority and majority densities and become very different, given by (see in Fig. 2.a).

One can distinguish three cases. The first is when , i.e. the decoupling of , and also since in this case , happens before the split of and , then at freeze-out, and so the present relic abundance of and are practically the same. This case is very similar to that of symmetric DM.

The second case is when , i.e. when the decoupling of occurs well after the split between and , so that is exponentially smaller than , thus the present relic density is completely negligible when compared to the density.

The last case, in which the final is smaller but not much smaller than , happens if but , i.e. if the decoupling of occurs shortly after the split between and . Given a particular asymmetry , the range of cross sections that satisfies this relationship is narrow, approximately within a factor of three. We choose such that and constitute all of the DM, Eq. (3). In the standard cosmology the necessary annihilation cross section is very close to that of symmetric DM satisfying the same condition, and as soon as the cross section departs by a factor of a few from this value, the relic minority component density becomes exponentially small. In the kination and scalar-tensor cosmologies, for a fixed transition temperature, the required values of the annihilation cross section in this third case can be much larger than the standard symmetric cross section (although in each case the values must be in a reatively narrow range, within a factor of a few). This is illustrated in Fig. 3, which

presents the contours of (recall GeV) as a function of the annihilation cross section for which and make up all of the DM (Eq. (3) holds). As exemplified in Figs. 4 and 5, the contours are almost independent in the standard cosmology and if GeV in the other two. The contours of Fig. 3 do not apply if GeV and is close to 5 MeV (see Fig. 5). We use , with , , and where is the present entropy density of the Universe [26]. With these values of the various parameters, the contours satisfy the condition

(28) |

For the standard cosmology, there is only one contour that satisfies Eq. (28) (the left most contour, shown in blue). In kination and scalar tensor models the contour satisfying the same condition depends on the transition temperature , which in relevant models can take values between somewhat below 5 MeV and the standard cosmology freeze-out temperature. Thus, there is a range of contours which satisfy Eq. (28) for the kination and scalar-tensor models, as shown (dashed gray lines) in Fig. 3.a and Fig. 3.b respectively. Values of and above and to the left of the contours would lead to a relic density larger than the DM density. Given a particular contour, values of and below it would lead to a relic density of and smaller than the DM density.

For each DM particle mass there is a maximum value of , corresponding to the horizontal part of each contour, which satisfies Eq. (28) with , which (since ) means that and is negligible. Thus the horizontal part each contour, common to all of them, corresponds to the second case mentioned above, where . Only in the curved and vertical parts of the contours can be non-negligible with respect to . As the asymmetry becomes smaller, the contribution of to the relic density increases, and when =0 we recover the symmetric case. Eq. (28) and the definition of imply that for values of ranging from 0 to , the relic density of must be within a factor of 10 of the density of . The curved part of the contour corresponds to the third case mentioned above, where is larger but very similar to , for which the asymmetry is large but the relic density of is only a few orders of magnitude smaller than the density of .

As already mentioned the contours in Fig. 3 are only slightly affected by changes in in the standard cosmology, and if GeV also in kination and scalar-tensor models. Fig. 4 shows the almost overlapping contours for which and constitute the whole to the DM in the standard cosmology for of 10 GeV (red), 100 GeV (green), and 500 GeV (blue). The analytic solutions predict this behavior. For example, one can calculate the minimum annihilation cross section for a mass , which occurs when . For small , the exponential term in the analytic solutions (Eqs. (21) and (22)) can be expanded as . With this approximation, and substituting the analytic solutions into Eq. (28), we see that . Recalling that and using , as in the symmetric case, we calculate a minimum cross section , independent of .

Figs. 5.a and 5.b show the same as Fig. 4 but for the kination and scalar-tensor models, respectively. The solid contours correspond to models with a transition temperature of MeV, and the dashed contours to close to the standard cosmology freeze-out temperature. The figure shows that for GeV the contours change considerably with respect of those for GeV only if is low.

Notice that the value of in the horizontal part of the contours is unaffected by changes in in all three models, Figs. 4 and 5. Clearly, this must be so because there , and is negligible, thus the fixed DM density depends only on the product .

We mentioned three cases above in which either , or is larger but very close to , so that the minority relic number density is either equal to, negligible or just a few orders of magnitude smaller than the majority relic density , respectively. Figs. 6, 7 and 8 demonstrate these three cases in the standard cosmology, kination and scalar-tensor models respectively. In all three figures the bottom left panel shows the regime we are interested in, where the minority component relic density is only a few orders of magnitude smaller than the relic density of the majority DM component .

## 5 Comparison of analytic and numerical relic abundance solutions

Figs. 9 and 10 show that the analytic solutions we found above agree very well with the numerical solutions in all three cosmological models we examined. In the examples presented, both solutions agree within a factor of two. In both figures 100 GeV and but the cross section for each particular cosmological model is chosen so that is not much larger than , because we wanted to show examples where the relic density of the minority component is only a few orders of magnitude smaller than that of the majority component (see the figure captions).

We have ascertained that the analytical solutions for the relic abundances and after freeze-out reproduce the numerical solution to within a factor of four while (smaller ratios are completely irrelevant). As this ratio increases, the agreement is better, within a factor of two.

## 6 Present asymmetric DM annihilation rates

Given a particular value of the annihilation cross section and almost identical values of the asymmetry, very different minority over majority asymmetric DM density ratios may result in the three pre-BBN cosmologies we explored, if the chemical decoupling happens during these different cosmological phases. Thus, if the annihilation cross section of the DM candidate is known, the annihilation rate at present, if detectable, could be used to test the Universe before BBN, an epoch from which we do not yet have any data. In particular, because the annihilation cross section of asymmetric DM can be larger than that of symmetric DM in the standard cosmology, the decrease in the minority component relic density may be compensated or even overcompensated by the increase in annihilation cross section so that the annihilation rate at present of asymmetric DM, contrary to general belief (see e.g. Ref. [12]), could be non-negligible and even larger than that of symmetric DM in the standard cosmology. There are experimental upper bounds on how large the annihilation cross section of asymmetric DM could be, but they are model dependent (see e.g. Ref. [13]). Assuming a “neutralino like” DM candidate in the standard cosmology (for which particle and anti-particle coincide) the standard annihilation rate at present can be written as

(29) |

where the factor corresponds to identical particles annihilating, is the DM energy density, is the velocity averaged self-annihilation cross section, which is similar to the early Universe thermally averaged s-wave dominated cross section . For s-wave annihilation with , and assuming that the candidate accounts for the whole of the DM, this is approximately . Note that in the early Universe is equal to half of the annihilation cross section needed for non-identical particles with no asymmetry () to make up all of the DM. Numerically, we find this cross section to be .

For asymmetric DM, the annihilation rate is given by , and we have assumed all along that and , thus

(30) |

The ratio of the asymmetric DM annihilation rate to the standard symmetric annihilation rate is then

(31) |

In the standard cosmology, this ratio never exceeds one, but in the kination or scalar-tensor pre-BBN cosmological models, it may exceed one for a range of annihilation cross sections, as shown in Figs. 11, 12 and 13. This range of cross sections depends on the transition temperature at which the cosmology becomes standard.

In Figs. 11, 12 and 13 we provide examples in which the asymmetric DM annihilation rate in a non-standard pre-BBN cosmological model is greater than the standard symmetric annihilation rate for . We have chosen particular annihilation cross sections and transition temperatures in each case. In Fig. 11 the transition temperature is low, =5 MeV and for , i.e. , for the scalar-tensor model we get , so the annihilation rate is times greater than the standard symmetric DM annihilation rate. For the kination and standard models the present annihilation rate is negligible in this example, or smaller respectively than the standard symmetric rate. For kination, , and this ratio is even smaller in the standard cosmology. Thus, in this example, the scalar-tensor model is easily distinguishable from the other two.

For smaller annihilation cross sections we can select kination and scalar-tensor models which would produce almost identical effects in the annihilation rate, as demonstrated in Fig. 12. In this figure, the transition temperatures, =5 MeV for kination and for the scalar-tensor model, are chosen so that the contours of for which and make up all the DM in both models are very similar. For , both models give a present asymmetric DM annihilation rate times larger than the standard symmetric DM annihilation rate. If such a rate were detected, we could see that the cosmology is non-standard, but could not distinguish between the kination and scalar-tensor models chosen here.

For asymmetric DM annihilation cross sections very close to the standard symmetric DM annihilation cross section it is difficult to distinguish between any of the three pre-BBN cosmological models. In order to prevent the Universe from being overdense, the transition temperature of the non-standard cosmological models must be set very close to the standard cosmology freeze-out temperature. This means that and only evolve in the non-standard cosmological phase for a very short time after freeze-out. In Fig. 13, and the transition temperature is high, for both kination and scalar-tensor models. All three models produce similar relic densities, and the present asymmetric DM annihilation rate is times smaller than the standard symmetric DM annihilation rate for all of them. Thus the annihilation rate could not help us distinguish between the different pre-BBN cosmological models.

These examples show that if the mass, approximate particle-antiparticle asymmetry, and annihilation cross section of a DM candidate is known, then if the present annihilation rate is detectable, one could possibly distinguish between these different pre-BBN cosmological models. A higher annihilation rate than predicted by the model of symmetric DM in the standard cosmology would imply the existence of a non-standard cosmology during pre-BBN epoch.

### 6.1 Fermi Space Telescope bounds on non-standard asymmetric DM models

In non-standard pre-BBN cosmological models the annihilation rate of asymmetric DM can be so large as to already be rejected by present bounds. The Fermi-LAT collaboration has published upper bounds [25] on the annihilation cross section of a symmetric DM candidate as a function of its mass, from a combined analysis of Milky Way satellites, assuming that it constitutes all of the DM and annihilates only into particular final states, which correspond to limits on the annihilation rate in Eq. (29), . Here we use the and modes, which impose the limit of equal to