In this paper we clarify the relation between the invariant relativistic relative velocity V _ { \texttt { r } } , the Møller velocity \bar { v } , and the non-relativistic relative velocity v _ { r } .
Adopting V _ { \texttt { 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 \langle \sigma v _ { \texttt { rel } } \rangle that appears in the density evolution equation for thermal relics is reformulated only in terms of V _ { \texttt { r } } and \mathcal { P } ( V _ { \texttt { r } } ) in a manifestly Lorentz invariant form ; ( ii ) the frame-dependent issues of the standard formulation in terms of the Møller 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 \langle \sigma V _ { \texttt { r } } \rangle 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 .