† Corresponding author. E-mail:
Supported by the National Natural Science Foundation of China under Grant Nos. 11365022 and 11765021
We discuss the relic abundance of asymmetric Dark Matter particles in modified cosmological scenarios where the Hubble rate is changed with respect to the standard cosmological scenario. The modified Hubble rate leaves its imprint on the relic abundance of asymmetric Dark Matter particles if the asymmetric Dark Matter particles freeze–out in this era. For generality we parameterize the modification of the Hubble rate and then calculate the relic abundance of asymmetric Dark Matter particles and anti–particles. We find the abundances for the Dark Matter particles and anti–particles are enhanced in the modified cosmological models. The indirect detection signal is possible for the asymmetric Dark Matter particles due to the increased annihilation rate in the modified cosmological models. Applying Planck data, we find the constraints on the parameters of the modified cosmological models.
Recent Planck team’s full sky measurements of Cosmic Microwave Background (CMB) temperature anisotropies provide the precise values for the cosmological parameters, the amount of cold Dark Matter (DM) relic density is given as
Although the precise value of the Dark Matter relic density is given by the observations, the nature of Dark Matter is still a mysterious question for scientists. Experimental searches are ongoing to find the Dark Matter using direct detection or indirect detection. Parallel to the experiments, there are also many theoretical works which attempts to disclose the character of Dark Matter. One of the general belief is that the Weakly Interacting Massive Particles (WIMPs) are the good candidates for Dark Matter. Majorana particle neutralino is one of the favorite WIMP which is appeared in supersymmetry and its particle and anti-particle are the same. Because we have no evidence that shows that the Dark Matter should be Majorana particle, we still have other possibilities that the Dark Matter can be Dirac particles for which its particle and anti-particle are different.[2–3] The relic density of asymmetric Dark Matter is calculated in the standard cosmological scenario.[4–5] In the standard model of cosmology, it is assumed that Dark Matter particles were in thermal and chemical equilibirum in the radiation-dominated epoch after the period of last entropy production. Dark Matter particles decouple when they become nonrelativistic while the universe cools down. For asymmetric Dark Matter, the asymmetry is created well before the decoupling of the asymmetric Dark Matter particles according to the assumption, therefore there are more particles than the anti-particles in the beginning. Dark Matter anti-particles are annihilated away with the particles and the rest are the Dark Matter particles in the present universe. The direct ditection can be used to find the asymmetric Dark Matter particles in principle.
On the other hand, there is no observational evidence before Big Bang Nucleosynthesis (BBN). The alternative cosmological models like scalar-tensor gravity,[6] quintessence models with a kination phase,[7–10] brane world cosmology[11–14] or the late decaying of inflaton model[15–17] and anistropic expansion[18–19] predict that the expansion rate may have been different from the one expected by the standard Friedmann-Robertson-Walker model before BBN. If this occurs around the time when the asymmetric Dark Matter particles decouple from the thermal equilibrium, the relic abundance of asymmetric Dark Matter particles are affected by the modified expansion rate of the universe. The papers[20–25] already discussed the effects of the modified expansion rate on the relic density of asymmetric Dark Matter particles. In those papers, the authors concluded that the enhanced expansion rate leads to the earlier asymmetric Dark Matter particles freeze-out and enhanced relic density. In the modified cosmological models, the difference of the abundance between the particle and anti-particle is not large for reasonable annihilation cross sections. In order to match with the observed range of the Dark Matter relic density, the annihilation cross section should be large. Then the annihilation rate is increased and it leads to the possible indirect detection signal for asymmetric Dark Matter as well.[21,23,25] In this work, we present the generalized parametrization of the modification of the expansion rate of the early universe and derive the general equation to compute the relic density of asymmetric Dark Matter particles. Then we use the Planck data to find the constraints on the parameter space of the modified cosmological models.
The next section is devoted to the analysis of the relic density of asymmetric Dark Matter particles in modified cosmological scenarios both in numerical and analytical ways. In Sec.
In this section, we introduce the parameterization of modification of the expansion rate of the universe and then derive the generalized equation to calculate the relic density for asymmetric Dark Matter. To describe the effects of the revised Hubble expansion rate in the early universe, we add the new dark density to the Friedmann equation as
The number densities of the asymmetric Dark Matter particles are resolved by the following Boltzmann equations,
New quantities Yχ = nχ/s,
From Eqs. (
In the standard frame of the Dark Matter particle evolution, it is assumed the particles χ and
Figure
First, we find the analytic solution for Eq. (
The Dark Matter relic density is given by Planck date as in Eq. (
Since the annihilation cross sections of the asymmetric Dark Matter particles in the modified cosmological scenarios are increased, then the annihilation rate is enhanced and the indirect detection is possible for the asymmetric Dark Matter particles, which is originally assumed to be detected only through the direct detection due to the suppressed abundance of the anti-particles. Therefore we can use the Fermi Large Area Telescope (Fermi-LAT)[27] data to find the limiting cross sections for the asymmetric Dark Matter in nonstandard cosmological scenarios. In the case of the symmetric Dark Matter, the annihilation rate is[21]
Then using Eq. (
Figure
In this work, we use the observed Dark Matter abundance to derive the constraints on the enhancement factor for the fixed asymmetry factor Λ. In the quintessence model (nD = 6), the constraint on the enhancement factor from BBN is η ˂ 30,[29] which matches partly with the limit η ˂ 9.1 × 103 that we obtained. In paper,[29] the bound on η was obtained at the temperature T = 10 MeV, here we take this bound as an example of a reference. As we mentioned earlier nD = 8 corresponds to the brane Randall-Sundrum II model,[11] which in our notations represents η = ρrad(T0)/(2λ) with λ being the tension of the brane which is related to the 5-dimensional Planck mass M5 by
We have studied the relic density of asymmetric Dark Matter particles in the modified cosmological scenarios. We have no observational evidence for the early pre-BBN universe. There are different cosmological models, which predicted the Hubble expansion rate before BBN may be faster or slower than the standard expansion law. If the Hubble expansion rate is changed in the modified cosmological scenarios, it affects the relic density of asymmetric Dark Matter particles. We have discussed in which extent the modification of the Hubble expansion rate affects the relic density of asymmetric Dark Matter in the modified cosmological models.
The Hubble expansion rate of the universe is increased in the modified cosmological scenarios like quintessence with kination phase, brane world cosmology or the late inflaton decay model and etc. For generality we found the parameterization of the enhancement of the Hubble expansion rate. Then we calculated the relic density of asymmetric Dark Matter particles in the modified cosmological scenarios where the expansion rate is faster than the standard cosmological scenarios. We found the resulting relic abundances are increased. Since the expansion rate of the universe is enhanced during the era of asymmetric Dark Matter decoupling, the asymmetric Dark Matter particles decouple from the thermal equilibrium earlier than the standard cosmological scenario and there are increased relic densities for particles and anti-particles. The increases depend on the enhancement factor η as well as the cross section and the factor nD. While using Planck date for the relic density of asymmetric Dark Matter, we found the cross sections are boosted for the modified cosmological models in order to satisfy the observed Dark Matter relic density. From the contour plot of (a, η), we found η is decreased (increased) for small (large) value of nD when the cross section is same for different nD.
The generalised formula obtained provides the relic density calculation of the asymmetric Dark Matter particles for the different modified cosmological scenarios where the expansion rate is enhanced. In the standard cosmological model, the abundance of minority component of asymmetric Dark Matter is suppressed. For the modified cosmological models, we obtained the enhanced relic densities both for particles and anti-particles, of course the increased amount of the relic abundance depends on the size of the annihilation cross section. The annihilation cross section should be boosted in order to provide the observed range of Dark Matter relic density (
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