Relativistic $<\sigma v_\text{rel}>$ in the calculation of relics abundances: a closer look
| Autor: | Cannoni, M. |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Phys. Rev. D 89, 103533 (2014) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1103/PhysRevD.89.103533 |
| Popis: | 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. Comment: 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 |
| Databáze: | arXiv |
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