Relativistic ⟨ σ v rel ⟩ in the calculation of relics abundances : a closer look

Abstract

In this paper we clarify the relation between the invariant relativistic relative velocity Vr, the Møller velocity ¯v, and the nonrelativistic relative velocity vr. Adopting Vr as the true physical relative velocity for pair collisions in a nondegenerate relativistic gas, we show that in the frame comoving with the gas (i) a probability density function P(Vr) for Vr exists, (ii) an exact formula for the mean value ⟨Vr⟩ exists, (iii) the thermally averaged cross section times relative velocity ⟨σvrel⟩ that appears in the density evolution equation for thermal relics is reformulated only in terms of Vr and P(Vr) in a manifestly Lorentz invariant form, and (iv) 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¯f, in the cases mf=0, mf=m, and mf≫m, we find that the coefficients of the low velocity expansion of ⟨σVr⟩ 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.