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 $ 10^4$h$^{-1}M_\odot$ are excluded from conforming all of the dark matter 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, due to the change of the mass density in PBHs with redshift, and therefore with scale, as they start affecting the gravitational potential at different times. 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 dark matter halo mass functions within these windows are similar} to those resulting from cold dark matter made of fundamental particles, but 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.