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 $\text{rhoann}$ 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 $3\mathrm{/}32\pi G^3{M}^2\lesssim \text{rhoann}$, or $M\gtrsim\Mann\equiv (3/32\pi G^3\rhoann)^{1/2}$\mathrm{.}Foraparticlemass$m$andfreeze-outtemperature$T_f=m/x_f$,theminimumblackholemassis$\Mann\approx 10^{10}\msun \times(x_f\sqrt{g_\star}/100\omcdm g_{\star S}m({\rm Gev}))$,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 $\sim G{M}_{BH}$ that could reach a total annihilation luminosity $\sim {10}^{59}\mathrm{e}\mathrm{r}\mathrm{g}{\mathrm{s}}^{-1}$. The absence of astronomical observations of such spectacular events suggests either: (i) the branching ratio for CDM annihilation to $e^{+}{e}^{-}$ pairs or quarks $\lesssim 10^{-10}$,whilethebranchingratioto$\nu{\bar{\nu}}$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.