Gravitational Wave Measurement of the $M_\mathrm{BH}$-$M_\mathrm{bulge}$ Intrinsic Scatter at High Redshift

The observed GWB spectrum is higher in amplitude than model predictions by a factor of 2-3. Using a semi-analytic model, we evaluate the effect of a high-scatter supermassive black hole (SMBH) scaling relation ($M_\mathrm{BH}$-$M_\mathrm{bulge}$) on models of the nanohertz gravitational wave background (GWB). By implementing an intrinsic scatter of the $M_\mathrm{BH}$-$M_\mathrm{bulge}$ relation, which is larger at higher redshift, but matches local observations, we find that the amplitude of GWB models increases to be consistent with the low-frequency end of the GWB spectrum. This amplitude increase is not uniform across frequencies, a strongly evolving scatter preferentially increases the number density of the most massive SMBHs which, in the GWB spectrum, minimizes the strength of the low-frequency turnover. Our models with positively evolving intrinsic scatter can reproduce the electromagnetically observed overmassive SMBHs at $4 < z < 6$ without changing the $M_\mathrm{BH}$-$M_\mathrm{bulge}$ normalization though we find that including moderate normalization evolution marginally improves fits to the GWB data. We conclude that the $M_\mathrm{BH}$-$M_\mathrm{bulge}$ relation which best describes the available GWB and electromagnetic data sets has intrinsic scatter that evolves as $\varepsilon(z) = \varepsilon_0 + (0.56 \pm 0.4) \log_{10}(1 + z)$ and normalization that evolves as $\alpha(z) = \alpha_0 (1 + z)^{0.84 \pm 0.35}$. The results of this work imply that the $M_\mathrm{BH}$-$M_\mathrm{bulge}$ relation we see today is not universal throughout cosmic time and that a diversity of seeding models and growth mechanisms may be at play in the early stages of SMBH-galaxy evolution.