2013/11/30 by M. Cannoni · 53 citations
Mathematics · Physics and Astronomy · #Atomic physics #Dark Matter and Cosmic Phenomena #Degenerate energy levels #Distribution (mathematics) #Distribution function #Energy (signal processing) #High-Energy Particle Collisions Research #Limit (mathematics) #Mathematical analysis #Mathematics #Particle physics theoretical and experimental studies #Physics #Quantum mechanics #astro-ph.CO #hep-ph #math-ph #math.MP #nucl-th
paper · pdf · open access · doi:10.1103/physrevd.89.103533
published in Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields 89(10) (American Physical Society) · 9 pages, 2 figures. V2: revised text. The published version of this manuscript contains also most of the material of arXiv:1311.4494. In the arXiv the manuscripts are left separated and contain some details not present in the published merged version
openalex publication_date 2014/05/23 · arxiv created 2014/05/30 · arxiv updated 2014/06/02 · openalex created_date 2017/03/16 · openalex updated_date 2026/08/05
In this paper we clarify the relation between the invariant relativistic relative velocity Vr, the M\oller 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 ⟨\ensuremathσvrel⟩ 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 M\oller 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 ff, in the cases mf=0, mf=m, and mf\ensuremath≫m, we find that the coefficients of the low velocity expansion of ⟨\ensuremathσVr⟩ 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.