Black holes and WIMPs: all or nothing or something else

We consider constraints on primordial black holes (PBHs) in the mass range (10(-18)-10(15)) M-circle dot if the dark matter (DM) comprises weakly interacting massive particles (WIMPs) that form haloes around them and generate gamma-rays by annihilations. We first study the formation of the haloes 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(PBH) less than or similar to 2 x 10(-9) (m(x)/TeV)(1.1) in the mass range 2 x 10(-)(12) M-circle dot (m(x)/TeV)(-3.2) less than or similar to M <= 5 x 10(12) M-circle dot (m(x)/TeV)(1.1), where m(x) 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 less than or similar to 2 x 10(-)(12 )M(circle dot) (m(x)/TeV)(-3.2), the constraint on f(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 LIGONirgo 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(circle dot) and m(x) > 10 GeV. There is a region of parameter space, with M less than or similar to 10(-11) M-circle dot and m(x) less than or similar to 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.