http://arxiv.org/abs/1311.4508
After clarifying the relation between the invariant relativistic relative velocity $v_{\texttt{r}}$, the M\o{}ller velocity $\bar{v}$, and the non-relativistic relative velocity $v_r$, the relativistic thermally averaged cross section times relative velocity $\langle \sigma v_\texttt{rel}\rangle$ that appears in the density evolution equation for thermal relics is reformulated only in terms the relativistic relative velocity $v_\texttt{r}$. Considering the annihilation of dark matter into a particle-antiparticle pair $f\bar{f}$, in the cases $m_f=0$, $m_f=m$ and $m_f \gg m$, we find that the coefficients of the expansion of $\langle \sigma v_{\texttt{r}}\rangle$ in powers of the relative velocity admit an exact analytical representation in terms of the Meijer $G$ functions that can be reduced to combinations of modified Bessel functions of the second kind.
Tue, 19 Nov 13
62/75