The Hosts and Mass Distribution Function of the Most Massive Black Holes: Implications for the nHz Gravitational Wave Background

Abstract

We use the core-radius, $r_γ,$ of a galaxy's central core along with a volume-limited sample of brightest cluster galaxies (BCG) to determine the $z=0$ mass distribution function of the most massive super-massive black holes (SMBH). We recover a $M_{BH}-r_γ$ relationship between the SMBH mass, $M_{BH},$ and $r_γ$ of form $M_{BH} \approx1.3\times10^{10}~(r_γ/{\mathrm{kpc}})~M_\odot,$ with $σ= 0.25\pm0.03$ dex scatter in $M_{BH}.$ We derive a core distribution function (CDF), based on HST and DLS images of the BCGs. The CDF falls steeply as $N(r_γ)\sim r_γ^{-3}$ for $r_γ>0.50$ kpc. Only $\sim5\%$ of BCGs have $r_γ>1$ kpc. Combining the CDF and $M_{BH} - r_γ$ relationship yields the BH mass distribution function (MDF). For $M_{BH} \geq 7\times10^{9}~\mathrm{M_\odot}$ (the mean $M_{BH}$ at $r_γ=0.50$ kpc) we derive: $\log\left(\frac{dN(M_{BH})}{d{M_{BH}/10^{10} M_\odot}}~[\rm Mpc^{-3}]\right)=(-3.05\pm0.42)~\log\left(\frac{M_{BH}}{10^{10}M_\odot}\right)~-~5.11\ (+0.41, -0.36)$. Cores with $r_γ\geq 0.50$ kpc are found almost exclusively in BCGs, implying that SMBHs with $M_{BH} \geq7\times10^{9}~\mathrm{M_\odot}$ may also be exclusive to BCGs. Core-radius correlates with $α,$ a parameter that characterizes the growth of BCG envelopes by mergers, suggesting that the most massive SMBHs grew in tandem with their BCG hosts. The BH MDF embodies a significant population of SMBH with $M_{BH} >7\times10^{9} M_\odot.$ When combined with the Liepold & Ma (2024) BH MDF to account for SMBH with $M_{BH} < 7\times10^{9} M_\odot,$ it can comfortably explain the nHz gravitational wave background.