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

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 $\rhoann$ 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 $3/32\pi G^3M^2\lesssim\rhoann$, or $M\gtrsim\Mann\equiv (3/32\pi G^3\rhoann)^{1/2}$. For a particle mass $m$ and freeze-out temperature $T_f=m/x_f$, the minimum black hole mass is $\Mann\approx 10^{10}\msun \times(x_f\sqrt{g_\star}/100\omcdm g_{\star S}m({\rm Gev}))$, 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 $\sim GM_{BH}$ that could reach a total annihilation luminosity $\sim 10^{59} {\rm erg s^{-1}}$. 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.