  The formation of black holes inevitably affects the distribution of dark and
baryonic matter in their vicinity, leading to an enhancement of the dark matter
density, called spike, and if dark matter is made of WIMPs, to a strong
enhancement of the dark matter annihilation rate. Spikes at the center of
galaxies like the Milky Way are efficiently disrupted by baryonic processes,
but mini-spikes can form and survive undisturbed at the center of dwarf
spheroidal galaxies. We show that Fermi LAT satellite data allow to set very
stringent limits on the existence of mini-spikes in dwarf galaxies: for thermal
WIMPs with mass between 100 GeV and 1 TeV, we obtain a maximum black hole mass
between 100 and 1000 $M_\odot$, ruling out black holes masses extrapolated from
the M-{\sigma} relationship in a large region of the parameter space. We also
performed Monte Carlo simulations of merger histories of black holes in dwarf
spheroidals in a scenario where black holes form from the direct collapse of
primordial gas in early halos, and found that this specific formation scenario
is incompatible at the 84% CL with dark matter being in the form of thermal
WIMPs.
