Evolution of The Velocity-Dispersion Function of Luminous Red Galaxies:
  A Hierarchical Bayesian Measurement
  We present a hierarchical Bayesian determination of the velocity-dispersion
function of approximately 430,000 massive luminous red galaxies(LRGs) observed
at relatively low spectroscopic signal-to-noise ratio (SNR ~3-5 per 69 km
s^(-1)) by the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan
Digital Sky Survey III (SDSS-III). We marginalize over spectroscopic redshift
errors, and use the full velocity-dispersion likelihood function for each
galaxy to make a self-consistent determination of the velocity-dispersion
distribution parameters as a function of absolute magnitude and redshift,
correcting as well for the effects of broadband magnitude errors on our
binning. Parameterizing the distribution at each point in the
luminosity-redshift plane with a log-normal form, we detect significant
evolution in the width of the distribution toward higher intrinsic scatter at
higher redshifts. Using a subset of deep re-observations of BOSS galaxies, we
demonstrate that our distribution-parameter estimates are unbiased regardless
of spectroscopic SNR. We also show through simulation that our method
introduces no systematic parameter bias with redshift. We highlight the
advantage of the hierarchical Bayesian method over frequentist "stacking" of
spectra, and illustrate how our measured distribution parameters can be adopted
as informative priors for velocity-dispersion measurements from individual
noisy spectra.
