The effect of stellar encounters on the dark matter annihilation signal from prompt cusps

Prompt cusps are the densest quasi-equilibrium dark matter objects; one forms at the instant of collapse within every isolated peak of the initial cosmological density field. They have power-law density profiles, $\rho \propto r^{-1.5}$ with central phase-space density set by the primordial velocity dispersion of the dark matter. At late times they account for $\sim 1\%$ of the dark matter mass but for $>90\%$ of its annihilation luminosity in all but the densest regions, where they are tidally disrupted. Here we demonstrate that individual stellar encounters, rather than the mean galactic tide, are the dominant disruptors of prompt cusps within galaxies. Their cumulative effect is fully (though stochastically) characterised by an impulsive shock strength $B_* = 2\pi G\int\rho_*({\bf x}(t))\, \mathrm{d}t$ where $\rho_*$, the total mass density in stars, is integrated over a cusp's entire post-formation trajectory. Stellar encounters and mean tides have only a small effect on the halo annihilation luminosity seen by distant observers, but this is not true for the Galactic halo because of the Sun's position. For a 100 GeV WIMP, Earth-mass prompt cusps are predicted, and stellar encounters suppress their mean annihilation luminosity by a factor of two already at 20 kpc, so that their annihilation emission is predicted to appear almost uniform over the sky. The Galactic Center $\gamma$-ray Excess is thus unaffected by cusps. If it is indeed dark matter annihilation radiation, then prompt cusps in the outer Galactic halo and beyond must account for 20-80% of the observed isotropic $\gamma$-ray background in the 1 to 10 GeV range.