Black Holes and WIMPs: All or Nothing or Something Else
We consider constraints on primordial black holes (PBHs) in the mass range $( 10^{-18}\text{-}10^{15} )\,M_{\odot}$ if the dark matter (DM) comprises weakly interacting massive particles (WIMPs) which form halos around them and generate $\gamma$-rays by annihilations. We first study the formation of the halos and find that their density profile prior to WIMP annihilations evolves to a characteristic power-law form. Because of the wide range of PBH masses considered, our analysis forges an interesting link between previous approaches to this problem. We then consider the effect of the WIMP annihilations on the halo profile and the associated generation of $\gamma$-rays. The observed extragalactic $\gamma$-ray background implies that the PBH DM fraction is $f^{}_{\rm PBH} \lesssim 2 \times 10^{-9}\,( m_{\chi} / {\rm TeV} )^{1.1}$ in the mass range $2 \times 10^{-12}\,M_{\odot}\,( m_{\chi} / {\rm TeV} )^{-3.2} \lesssim M \lesssim 5 \times 10^{12}\,M_{\odot}\,( m_{\chi} / {\rm TeV} )^{1.1}$, where $m_{\chi}$ and $M$ are the WIMP and PBH masses, respectively. This limit is independent of $M$ and therefore applies for any PBH mass function. For $M \lesssim 2\times 10^{-12}\,M_{\odot}\,( m_{\chi}/ {\rm TeV} )^{-3.2}$, the constraint on $f^{}_{\rm PBH}$ is a decreasing function of $M$ and PBHs could still make a significant DM contribution at very low masses. We also consider constraints on WIMPs if the DM is mostly PBHs. If the merging black holes recently discovered by LIGO/Virgo are of primordial origin, this would rule out the standard WIMP DM scenario. More generally, the WIMP DM fraction cannot exceed $10^{-4}$ for $M > 10^{-9}\,M_{\odot}$ and $m_{\chi} > 10\,$GeV. There is a region of parameter space, with $M \lesssim 10^{-11}\,M_{\odot}$ and $m_{\chi} \lesssim 100\,$GeV, in which WIMPs and PBHs can both provide some but not all of the DM, so that one requires a third DM candidate.