  We examine the effect of supermassive black hole (SMBH) mass scaling relation
choice on the inferred SMBH mass population since redshift $z \sim 3$. To make
robust predictions for the gravitational wave background (GWB) we must have a
solid understanding of the underlying SMBH demographics. Using the SDSS and 3D
HST+CANDELS surveys for $0 < z < 3$ we evaluate the inferred SMBH masses from
two SMBH-galaxy scaling relations: $\mathrm{M_{BH}}$-$\mathrm{M_{bulge}}$ and
$\mathrm{M_{BH}}$-$\sigma$. Our SMBH mass functions come directly from stellar
mass measurements for $\mathrm{M_{BH}}$-$\mathrm{M_{bulge}}$, and indirectly
from stellar mass and galaxy radius measurements along with the galaxy mass
fundamental plane for $\mathrm{M_{BH}}$-$\sigma$. We find that there is a
substantial difference in predictions especially for $z > 1$, and this
difference increases out to $z = 3$. In particular we find that using velocity
dispersion predicts a greater number of SMBHs with masses greater than $10^9
\mathrm{M}_\odot$. The GWB that pulsar timing arrays find evidence for is
higher in amplitude than expected from GWB predictions which rely on high
redshift extrapolations of local SMBH mass-galaxy scaling relations. The
difference in SMBH demographics resulting from different scaling relations may
be the origin for the mismatch between the signal amplitude and predictions.
Generally, our results suggest that a deeper understanding of the potential
redshift evolution of these relations is needed if we are to draw significant
insight from their predictions at $z > 1$

Astrophysics of Galaxies;High Energy Astrophysical Phenomena