Quashing a suspected selection bias in galaxy samples having dynamically-measured supermassive black holes

Local early-type galaxies with directly-measured black hole masses, $M_{\rm bh}$, have been reported to represent a biased sample relative to the population at large. Such galaxies with Spitzer Space Telescope imaging have been purported to possess velocity dispersions, $\sigma$, at least $\sim$0.1 dex larger for a given galaxy stellar mass, $M_{\rm *,gal}$, than is typically observed among thousands of early-type galaxies imaged by the Sloan Digital Sky Survey. This apparent offset led Shankar et al. to reduce the normalisation of the observed $M_{\rm bh} \propto \sigma^5$ relation by at least $\sim$0.5 dex to give their "intrinsic relations", including $\sigma$-based modifications to the $M_{\rm bh}$-$M_{\rm *,gal}$ relation. These modifications were based on the untested assumption that the stellar masses had been derived consistently between the two samples. Here, we provide the necessary check using galaxies common to the Spitzer Survey of Stellar Structure in Galaxies (S$^4$G) and the Sloan Digital Sky Survey (SDSS). We find that the stellar masses of galaxies with and without directly measured black holes had appeared offset from each other due to the use of inconsistent stellar mass-to-light ratios, $\Upsilon_*$, for the optical and infrared data. We briefly discuss the "intrinsic relations" and why some of these will at times appear to have had partial success when applied to data based on similarly inconsistent values of $\Upsilon_*$. Finally, we reiterate the importance of the $\upsilon$ (lower-case $\Upsilon$) term, which we previously introduced into the $M_{\rm bh}$-$M_{\rm *}$ relations to help avoid $\Upsilon_*$-related mismatches.