Diagnosing the massive-seed pathway to high-redshift black holes: statistics of the evolving black hole to host galaxy mass ratio

Supermassive black holes (SMBHs) with masses of $\sim {10}^9{\mathrm{M}}_{\odot }$ within the first billion year of the universe challenge our conventional understanding of black hole formation and growth. One pathway to these SMBHs proposes that supermassive stars (SMSs) born in pristine atomic cooling haloes (ACHs) yield massive seed BHs evolving to these early SMBHs. This scenario leads to an overly massive BH galaxy (OMBG), in which the BH to stellar mass ratio is initially $M_{\mathrm{b}\mathrm{h}}\mathrm{/}{M}_{\ast }\ge 1$, well in excess of the typical values of $\sim {10}^{-3}$ at low redshifts. Previously, we have investigated two massive seed BH candidates from the $\mathtt{\text{Renaissance}}$ simulation and found that they remain outliers on the $M_{\mathrm{b}\mathrm{h}}-M_{\ast }$ relation until the OMBG merges with a much more massive halo at $z=8$. In this work, we use Monte-Carlo merger trees to investigate the evolution of the $M_{\mathrm{b}\mathrm{h}}-M_{\ast }$ relation for $50,000$ protogalaxies hosting massive BH seeds, across $10,000$ trees that merge into a ${10}^{12}{\mathrm{M}}_{\odot }$ halo at $z=6$. We find that up to $60\mathrm{\%}$ (depending on growth parameters) of these OMBGs remain strong outliers for several 100 Myr, down to redshifts detectable with {\it JWST} and with sensitive X-ray telescopes. This represents a way to diagnose the massive-seed formation pathway for early SMBHs. We expect to find $\sim 0.1-1$ of these objects per {\it JWST} NIRCam field per unit redshift at $z\gtrsim 6$. Recently detected SMBHs with masses of $\sim {10}^7{\mathrm{M}}_{\odot }$ and low inferred stellar-mass hosts may be examples of this population.