Power spectrum of density fluctuations, halo abundances and clustering with primordial black holes

We study the effect of dark matter (DM) being encapsulated in primordial black holes (PBHs) on the power spectrum of density fluctuations $P(k)$; we also look at its effect on the abundance of haloes and their clustering. We allow the growth of Poisson fluctuations since matter and radiation equality and study both monochromatic and extended PBH mass distributions. We present updated monochromatic black hole mass constraints by demanding $<10%$ deviations from the $Lambda$ cold dark matter power spectrum at a scale of $k=1$hMpc$^{-1}$. Our results show that PBHs with masses $>10^4$h$^{-1}M_odot$ are excluded from conforming all of the DM in the Universe. We also apply this condition to our extended Press-Schechter (PS) mass functions, and find that the Poisson power is scale dependent even before applying evolution. We find that characteristic masses $M^*leq10^2 $h$^{-1}M_odot$ are allowed, {leaving only two characteristic PBH mass windows of PS mass functions when combining with previous constraints, at $M^*sim10^2$h$^{-1}M_odot$ and $sim10^{-8}$h$^{-1}M_odot$ where all of the DM can be in PBHs. The resulting DM halo mass functions within these windows are similar} to those resulting from cold dark matter made of fundamental particles. However, as soon as the parameters produce unrealistic $P(k)$, the resulting halo mass functions and their bias as a function of halo mass deviate strongly from the behaviour measured in the real Universe.