Relativistic $<\sigma v_{\text{rel}}>$ in the calculation of relics abundances: a closer look

In this paper we clarify the relation between the invariant relativistic relative velocity $V_{\text{r}}$, the M\o{}ller velocity $\bar{v}$, and the non-relativistic relative velocity $v_r$. Adopting $V_{\text{r}}$ as the true physical relative velocity for pair-collisions in a non-degenerate relativistic gas, we show that in the frame co-moving with the gas (i) the thermally averaged cross section times relative velocity $<\sigma v_{\text{rel}}>$ that appears in the density evolution equation for thermal relics is reformulated only in terms of $V_{\text{r}}$ and $\mathcal{P}(V_{\text{r}})$ in a manifestly Lorentz invariant form; (ii) the frame-dependent issues of the standard formulation in terms of the M\o{}ller velocity, as well as “superluminal” relative velocities, are not present in this formulation. Furthermore, 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 low velocity expansion of $<\sigma V_{\text{r}}>$ 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.