Establishing the Dark Matter Relic Density in an Era of Particle Decays

TL;DR

This work tackles the question of how an early non-standard cosmological epoch, where a dominant φ component enforces $H \propto T^{2+n/2}$ before decays drive a transition to $H \propto T^4$, alters dark matter relic-density calculations. It develops a Boltzmann framework without entropy conservation using dimensionless variables and derives the coupled evolution equations, alongside the maximum bath temperature $T_{Max}$ and the onset of radiation domination at $A_{\times}$. The authors compute DM relic densities for freeze-in, freeze-out during reheating, and non-thermal production, revealing that the exponent $n$ leaves a lasting imprint on the relics for freeze-in and non-thermal production, while freeze-out during reheating is comparatively less sensitive. By generalizing prior results on early matter and kination domination, the paper provides analytic expressions and guidance for predicting DM abundance in broad non-standard cosmologies, with implications for interpreting experimental constraints and guiding future cosmological probes.

Abstract

If the early universe is dominated by an energy density which evolves other than radiation-like the normal Hubble-temperature relation $H\propto T^2$ is broken and dark matter relic density calculations in this era can be significantly different. We first highlight that for a population of states $φ$ sourcing an initial expansion rate of the form $H\propto T^{2+n/2}$ for $n\geq-4$, during the period of appreciable $φ$ decays the evolution transitions to $H\propto T^4$. The decays of $φ$ imply a source of entropy production in the thermal bath which alters the Boltzmann equations and impacts the dark matter relic abundance. We show that the form of the initial expansion rate leaves a lasting imprint on relic densities established while $H\propto T^4$ since the value of the exponent $n$ changes the temperature evolution of the thermal bath. In particular, a dark matter relic density set via freeze-in or non-thermal production is highly sensitive to the temperature dependance of the initial expansion rate. This work generalises earlier studies which assumed initial expansion rates due to matter or kination domination.

Figures (4)

• Figure 1: The figure shows the bath temperature $T$ as function of $A$ for different values of $n$, assuming that initially the expansion rate is $H\propto T^{n/2+2}$ and that there is negligible energy in the radiation bath or dark matter $R(a_I)=X(a_I)=0$. We fix $\Phi_I=\Phi(a_I)=10^{10}$ GeV, or equivalently (see eq. (\ref{['IC']})) this corresponds to, for example, $H_I=1$ eV and $T_{\rm RH}=100$ GeV. The curves follow eq. (\ref{['Tapprox']}). Of the cases shown $n=-1$ corresponds to a matter-like $\phi$ (blue, solid), $n=0$ is radiation-like $\phi$ (dashed), and for $n=1$ then $\phi$ redshifts faster than radiation (dotted). Note that the maximum temperature drops (and occurs earlier) for increasing $n$ following eq. (\ref{['amax']}).
• Figure 2: (Left). The point $A_{\times}$ at which $R(a_\times)=\Phi(a_\times)$ as $\Phi_I$ is varied and for different $n$, found by solving the coupled differential eqns. (\ref{['phie']}) & (\ref{['rade']}) with the initial conditions of eq. (\ref{['IC']}). The line styles match Figure \ref{['fig:1']}, with $n=0$ is dashed and $n=1$ dotted. The point $A_{\times}$ signifies the breakdown of eq. (\ref{['phiearly']}), which underlies Figure \ref{['fig:1']}. (Center). For a given value of $T_{\rm RH}$ (the reheating temperature after $\phi$ decay) $A_{\times}$ is associated to a specific temperature $T_{\times}$ via eq. (\ref{['TTT']}). Here we show $T_{\times}$ as a function of $\Phi_I$ for $T_{\rm RH}=1$ TeV. The black dashed curve indicates the maximum temperature $T_{\rm Max}\sim0.3\times (M_{\rm Pl}H_I T_{\rm RH}^2)^{1/4}$. Changes in $T_{\rm RH}$ simply scales the y-axis and the relative orientations of the lines are unchanged. (Right). We illustrate the temperature evolution for two cases with $n=-2$ and $n=-3$, taking $T_{\rm RH}=1$ TeV and $\Phi_I=10^{10}$ GeV, and we highlight where $T_{\rm Max}$ and $T_{\times}$ occur.
• Figure 3: (Left). Shaded areas indicate regions of the $T_{\rm RH}$-$M_X$ plane for which $A_{\rm Max}<A_*<A_{\times}$ for $n=-2$ and different values of the initial expansion rate $H_I$ and we also require $\Phi_I(H_I,T_{\rm RH})<M_{\rm Pl}$. (Right). As an example we fix $n=-2$ and $H_I=10^{-4}$ GeV and plot the $T_{\rm RH}$ which gives the observed relic density $\Omega_{\rm Obs}$ by freeze-in as $M_X$ is varied assuming $\alpha_s=10^{-18}$ and $\alpha_p=0$, as described in eq. (\ref{['relicFI']}). We highlight the region $A_{\rm Max}<A_*<A_{\times}$ and $\Phi_I<M_{\rm Pl}$ for $H_I=10^{-4}$ GeV (matching the left panel), outside of this region the relic density curve is unreliable and we indicate this by dashing the line. Note that $T_{\rm RH}\sim1$ TeV and $H_I\sim10^{-4}$ GeV corresponds to $\Phi_I\sim10^{17}$ GeV.
• Figure 4: Plot shows lines in the $T_{\rm RH}$-$M_X$ plane for which the dark matter relic density is reproduced via non-thermal production, following eq. (\ref{['nontrd']}). Taking three different exponents of the initial expansion rate $n=1$ (dotted), $n=0$ (dashed) and $n=-1$ (solid) and pameterising the $\phi$-dark matter branching fraction in terms of $\eta\equiv b\cdot{\rm GeV}/m_\phi$ we show three different values $\eta=0.5,~10^{-4},~10^{-7}$. The plot fixes the initial Hubble rate to be $H_I=$eV. The right panel shows an enlargement of the dashed rectangle of the left panel and illustrates the difference between $n=0,1$, and $-1$.