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 $\sigma$, i.e. $p(L,\sigma)$, defined by our previous measurements
of the RS luminosity function and $L-\sigma$ 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
$\log_{10}\sigma\gtrsim2.47$, however the number density of $z=0.55$ RS
galaxies that we report is larger than that of $z=0$ ETG galaxies at
$2.35\gtrsim\log_{10}\sigma\gtrsim 2.47$. The extrapolation of an
intermediate-mass L-$\sigma$ 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.
