Uncertainties in the supermassive black hole abundance and implications for the GW background

The present-day mass function of supermassive black holes is the most important observable quantity for the prediction and theoretical interpretation of the gravitational wave background (GWB) measured by pulsar timing arrays (PTAs). Due to the limited sample size of galaxies with dynamically inferred SMBH masses, more readily measurable galaxy properties $X$ that correlate with the black hole mass are used as labels (via scaling relations $M_{\bullet}-X$), which can then be counted in a larger galaxy catalog to produce a measurement of the mass function. Estimating the amplitude of the GWB from the local mass function is therefore simpler than general measurements of scaling relations and galaxy mass/luminosity functions for two reasons: the contribution to the characteristic strain is dominated by a narrow range of masses, and the mass proxy $X$ is always marginalized over. While consistent errors in $X$ in both catalogs are irrelevant, relatively small biases between them can produce significant shifts in the predicted SMBH abundance. In this work, we explore measurements of the SMBH mass function using different mass proxies through a set of catalogs with a number of redundant measurements between them. This enables us to investigate internal inconsistencies that lead to discrepancies in the final black hole abundance, while minimizing observational systematic biases induced by combining disparate sets of measurements. We focus on 3 proxies: the velocity dispersion $\sigma$, K-band luminosity $L$, and a combination of $L$ and radius $R$ defined by the fundamental plane. We show that all three can be reconciled to some degree, but highlight the remaining dependence on poorly-quantified systematic corrections between the scaling relation catalogs and the mass function catalogs, as well as the potential impact of selection effects.