A Compaction Function Analysis of CMB $\mu$ distortion Constraints on Primordial Black Holes

TL;DR

This paper uses a compaction function-based Press-Schechter method to analyze how CMB $bc$ distortion constrains primordial black holes, suggesting previous limits may be underestimated and that spectral shape variations are negligible.

Contribution

It introduces a more precise method for calculating PBH abundance from $bc$ distortion constraints, highlighting potential underestimations in prior analyses.

Findings

PBH abundance could be higher than previous estimates.

Spectral shape variations have negligible impact on constraints.

The method improves reliability of PBH abundance calculations.

Abstract

Primordial black holes (PBHs) are considered viable candidates for dark matter and the seeds of supermassive black holes (SMBHs), with their fruitful physical influences providing significant insights into the conditions of the early Universe. Cosmic microwave background (CMB) $\mu$ distortion tightly constrain the abundance of PBHs in the mass range of $10^4 \sim 10^{11} M_{\odot}$ recently, limiting their potential to serve as seeds for the SMBHs observed. Given that $\mu$ distortion directly constrain the primordial power spectrum, it is crucial to employ more precise methods in computing PBH abundance to strengthen the reliability of these constraints. By a Press-Schechter (PS) type method utilizing the compaction function, we find that the abundance of PBHs could be higher than previously estimated constraints from $\mu$ distortion observations. Furthermore, our analysis shows that variations in the shape of the power spectrum have a negligible impact on our conclusions within the mass ranges under consideration. This conclusion provides us a perspective for further research on the constrain of PBH by $\mu$ distortion.