A Steep Slope and Small Scatter for the High-Mass End of the L-$\sigma$
  Relation at $z\sim0.55$
  We measure the intrinsic relation between velocity dispersion ($\sigma$) and
luminosity ($L$) for massive, luminous red galaxies (LRGs) at redshift $z \sim
0.55$. We achieve unprecedented precision by using a sample of 600,000 galaxies
with spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the
third Sloan Digital Sky Survey (SDSS-III), covering a range of stellar masses
$M_* \gtrsim 10^{11} M_{\odot}$. We deconvolve the effects of photometric
errors, limited spectroscopic signal-to-noise ratio, and red--blue galaxy
confusion using a novel hierarchical Bayesian formalism that is generally
applicable to any combination of photometric and spectroscopic observables. For
an L-$\sigma$ relation of the form $L \propto \sigma^{\beta}$, we find $\beta =
7.8 \pm 1.1$ for $\sigma$ corrected to the effective radius, and a very small
intrinsic scatter of $s = 0.047 \pm 0.004$ in $\log_{10} \sigma$ at fixed $L$.
No significant redshift evolution is found for these parameters. The evolution
of the zero-point within the redshift range considered is consistent with the
passive evolution of a galaxy population that formed at redshift $z=2-3$,
assuming single stellar populations. An analysis of previously reported results
seems to indicate that the passively-evolved high-mass L-$\sigma$ relation at
$z\sim0.55$ is consistent with the one measured at $z=0.1$. Our results, in
combination with those presented in Montero-Dorta et al. (2014), provide a
detailed description of the high-mass end of the red sequence (RS) at
$z\sim0.55$. This characterization, in the light of previous literature,
suggest that the high-mass RS distribution corresponds to the "core" elliptical
population.
