A Direct Measurement of the High-Mass End of the Velocity Dispersion Function at $z\sim0.55$ from SDSS-III/BOSS

We report the first direct spectroscopic measurement of the velocity dispersion function (VDF) for the high-mass red sequence (RS) galaxy population at redshift $z\sim0.55$. We achieve high precision by using a sample of 600,000 massive galaxies with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third Sloan Digital Sky Survey (SDSS-III), covering stellar masses $M_*\gtrsim10^{11}~M_{\odot}$. We determine the VDF by projecting the joint probability-density function (PDF) of luminosity $L$ and velocity dispersion $σ$, i.e. $p(L,σ)$, defined by our previous measurements of the RS luminosity function and $L-σ$ relation for this sample. These measurements were corrected from red--blue galaxy population confusion, photometric blurring, incompleteness and selection effects within a forward-modeling framework that furthermore correctly accommodates the low spectroscopic signal-to-noise ratio of individual BOSS spectra. Our $z\sim0.55$ RS VDF is in overall agreement with the $z\sim0$ early-type galaxy (ETG) VDF at $\log_{10}σ\gtrsim2.47$, however the number density of $z=0.55$ RS galaxies that we report is larger than that of $z=0$ ETG galaxies at $2.35\gtrsim\log_{10}σ\gtrsim 2.47$. The extrapolation of an intermediate-mass L-$σ$ relation towards the high-mass end in previous low-z works may be responsible for this disagreement. Evolutionary interpretation of this comparison is also subject to differences in the way the respective samples are selected; these differences can be mitigated in future work by analyzing $z=0$ SDSS data using the same framework presented in this paper. We also provide the sample PDF for the RS population (i.e. uncorrected for incompleteness), which is a key ingredient for gravitational lensing analyses using BOSS.