Thermal Relic Abundances of Particles with Velocity-Dependent Interactions

Abstract

We reexamine the evolution of thermal relic particle abundances for the case where the interaction rate depends on the particle velocities. For the case of Sommerfeld enhancement, we show that the standard analytic approximation, modified in a straightforward way, provides an estimate of the relic particle abundance that is accurate to within 10% (in comparison to less than 1% error for the non-Sommerfeld-enhanced case). We examine the effect of kinetic decoupling on relic particle abundances when the interaction rate depends on the velocity. For the case of pure p-wave annihilation, the effect of kinetic decoupling is an increase in the relic abundance, but the effect is negligible when the kinetic decoupling temperature is much less than the chemical decoupling temperature. For the case of Sommerfeld-enhanced s-wave annihilations, after kinetic decoupling occurs, annihilations continue to change the particle abundance down to arbitrarily low temperatures, until either matter domination begins or the Sommerfeld effect cuts off. We derive analytic approximations to give the final relic particle abundances for both of these cases.

Figures (5)

• Figure 1: The evolution of relic particle densities for the case of Sommerfeld-enhanced $s$-wave annihilations of a 500 GeV mass particle with $\sigma_0 = 3 \times 10^{-26}$ cm$^3$ s$^{-1}$ as a function of the indicated value of the coupling $\alpha$. Top and bottom curves correspond to the limiting cases of no Sommerfeld enhancement, and $1/v$ enhancement. Horizontal lines are the analytic estimates for the final relic abundances in these two cases (i.e. $n=0$ and $n=-1/2$, respectively, in equations \ref{['xf']} and \ref{['Yinf']}).
• Figure 2: A comparison between the analytic approximation for the ratio in relic abundance for $s$-wave annihilation with and without a $1/v$ Sommerfeld enhancement to the corresponding numerical results over several orders of magnitude of the parameter $\lambda$.
• Figure 3: The effect of kinetic decoupling on the evolution of the relic particle abundance for the case of $p$-wave annihilation of a 500 GeV mass particle with $\sigma_0 = 3 \times 10^{-26}$ cm$^3$ s$^{-1}$. Horizontal line gives our analytic estimate of the final relic abundance.
• Figure 4: The effect of kinetic decoupling on the evolution of the relic particle abundance for the case of $s$-wave annihilation for a 500 GeV mass particle with $\sigma_0 = 3 \times 10^{-26}$ cm$^3$ s$^{-1}$, in the limit where the Sommerfeld enhancement scales as $1/v$. Horizontal lines give our analytic estimates of the final relic abundances.
• Figure 5: As Fig. 4, for Sommerfeld-enhancement coupling of $\alpha = 0.01$, a value for which the Sommerfeld effect by itself is negligible without kinetic decoupling. Note the strong effect of kinetic decoupling upon the relic particle abundances.