Spherical Gravitational Collapse of Annihilating Dark Matter and the Minimum Mass of CDM Black Holes

Abstract

Spherical gravitational collapse of a cold gas of annihilating particles involves a competition between the free-fall rate $\propto \sqrt{\rho }$ and the (s-wave) annihilation rate $\propto \rho$. Thus, there is a critical density $\text{rhoann}$ above which annihilation proceeds faster than free fall. Gravitational collapse of a cloud of (initial) mass $M$ to a black hole is only possible if $3/32\pi G^3{M}^2\lesssim \text{rhoann}$, or $M\gtrsim \text{Mann}\equiv (3/32\pi G^3\text{rhoann}{)}^{1/2}$. For a particle mass $m$ and freeze-out temperature $T_f=m/x_f$, the minimum black hole mass is $\text{Mann}\approx 10^{10}\text{msun}\times (x_f\sqrt{g_{\star }}/100\text{omcdm}{g}_{\star S}m(\mathrm{Gev}))$, where $g_{\star S}$ and $g_{\star }$ are degeneracy factors. The formation of a black hole of initial mass $M_{BH}$ is accompanied by the annihilation of about $M_{ann}$ released in a burst lasting a time $\sim G{M}_{BH}$ that could reach a total annihilation luminosity $\sim 10^{59}\mathrm{erg}{\mathrm{s}}^{-1}$. The absence of astronomical observations of such spectacular events suggests either: (i) the branching ratio for CDM annihilation to $e^{+}{e}^{-}$ pairs or quarks $\lesssim 10^{-10}$, while the branching ratio to $\nu \bar{\nu }$ is $\lesssim 10^{-5}$; or (ii) CDM is not made of annihilating particles, but may be in some non-annihilating form, such as axions; or (iii) CDM black holes never form.