The power spectrum on small scales: Robust constraints and comparing PBH methodologies

We compare primordial black hole (PBH) constraints on the power spectrum and mass distributions using the traditional Press Schechter formalism, peaks theory, and a recently developed version of peaks theory relevant to PBHs. We show that, provided the PBH formation criteria and the power spectrum smoothing are treated consistently, the constraints only vary by $sim$10% between methods (a difference that will become increasingly important with better data). Our robust constraints from PBHs take into account the effects of critical collapse, the non-linear relation between $zeta$ and $delta$, and the shift from the PBH mass to the power spectrum peak scale. We show that these constraints are remarkably similar to the pulsar timing array (PTA) constraints impacting the black hole masses detected by the LIGO and Virgo, but that the $mu$-distortion constraints rule out supermassive black hole (SMBH) formation and potentially even the much lighter mass range of $sim$(1-100) $mathrm{M}_odot$ that LIGO/Virgo probes.