Dark matter freeze-out in modified cosmological scenarios

We study the effects of modifying the expansions history of the Universe on dark matter freeze-out. We derived a modified Boltzmann equation for freeze-out for an arbitrary energy density in the early Universe and provide an analytic approach using some approximations. We then look at the required thermally averaged cross sections needed to obtain the correct relic density for the specific case where the energy density consists of radiation plus one extra component which cools faster. We compare our analytic approximation to numerical solutions. We find that it gives reasonable results for most of the parameter space explored, being at most a factor of order 1 away from the measured value. We find that if the new contribution to the energy density is comparable to the radiation density, then a much smaller cross section for dark matter annihilation is required. This would lead to weak scale dark matter being much more difficult to detect and opens up the possibility that much heavier dark matter could undergo freeze-out without violating perturbative unitarity.

Figure 1

The required thermally averaged cross section to obtain the observed relic density ${\mathrm{\Omega }}_{\mathrm{CDM}}{h}^2=0.120$ as a function of mass. The black, red, orange, green, blue, and purple (from the top to the bottom) curves represent $\eta =0,1,10,{10}^2,{10}^3,{10}^4$, respectively. The solid colored curves represent $n_D=6$ while the dashed curves represent $n_D=8$. These coincide for the black (top) curve.

Figure 2

The required thermally averaged cross section as a function of $\eta$ to obtain the correct relic density ${\mathrm{\Omega }}_{\mathrm{CDM}}{h}^2=0.120$. The top figure is for $n_D=6$ while the bottom figure is for $n_D=8$. The various colors black, red, orange, brown, green, blue, and purple represent masses from ${10}^{-3}–{10}^3\mathrm{GeV}$ by increments of an order of magnitude, respectively.

Figure 3

The thermally averaged cross section required to obtain the correct relic density ${\mathrm{\Omega }}_{\mathrm{CDM}}{h}^2=0.120$ as a function of mass and $\eta$. The top figure represents $n_D=6$ while the bottom figure represents $n_D=8$. The scale represents ${\log }_{10}{\sigma }_0$.

Figure 4

The ratio of the thermally averaged cross section of the $n_D=6$ and $n_D=8$ cases required to obtain the correct relic density ${\mathrm{\Omega }}_{\mathrm{CDM}}{h}^2=0.120$ as a function of mass and $\eta$. ${\sigma }_0^{(6)}$ represents the cross section for $n_D=6$ while ${\sigma }_0^{(8)}$ represents the cross section for $n_D=8$.

Figure 5

The value of $c$ needed to obtain the correct relic density using the cross section values from Fig. 3. The top plot has $n_D=6$ and the bottom plot has $n_D=8$. Plotted is the value of ${\log }_{10}c$.

Figure 6

The difference in the obtained relic density using the values of $c$ from Eq. (30) and the observed relic density. The top plot has $n_D=6$ and the bottom plot has $n_D=8$.