Résumé
Primordial black holes (PBHs) are very appealing dark matter (DM) candidates. It is highly plausible though, should they exist, that they would not make up all of the DM. Several studies showed that if the rest of DM is made of thermal particles, then these should accumulate around such PBHs, leading to the formation of very dense spikes in the radiation era. We contributed a detailed analytical study about this phenomenon, providing clear explanations as for the origin of scaling relations in the form of power-law density profiles with up to 3 different spectral indices, i.e. $3/4$, $3/2$, and $9/4$, and 4 asymptotic regimes. Here, we further derive an approximate analytical solution that enables fast numerical predictions for the density profiles of these spikes. We also address the specific case of self-annihilating DM species and derive new approximate analytical formulae. Our approximate density yields the correct annihilation rate within $\pm 15\%$ precision. We then focus on indirect detection in the cosmic microwave background and in extragalactic gamma-rays. We shed new and subtle light on how mutually exclusive PBHs and self-annihilating DM species can really be. In particular, the discovery of a population of sub-solar PBHs would set stringent constraints on the $s$-wave annihilation cross-section of these particles, a point so far missed in the literature.