Stacked phase-space density of galaxies around massive clusters:
  Comparison of dynamical and lensing masses
  We present a measurement of average histograms of line-of-sight velocities
over pairs of galaxies and galaxy clusters. Since the histogram can be measured
at different galaxy-cluster separations, this observable is commonly referred
to as the stacked phase-space density. We formulate the stacked phase-space
density based on a halo-model approach so that the model can be applied to real
samples of galaxies and clusters. We examine our model by using an actual
sample of massive clusters with known weak-lensing masses and spectroscopic
observations of galaxies around the clusters. A likelihood analysis with our
model enables us to infer the spherical-symmetric velocity dispersion of
observed galaxies in massive clusters. We find the velocity dispersion of
galaxies surrounding clusters with their lensing masses of $1.1\times10^{15}\,
h^{-1}M_{\odot}$ to be $1180^{+83}_{-70}\, \mathrm{km/s}$ at the 68\%
confidence level. Our constraint confirms that the relation between the galaxy
velocity dispersion and the host cluster mass in our sample is consistent with
the prediction in dark-matter-only N-body simulations under General Relativity.
Assuming that the Poisson equation in clusters can be altered by an effective
gravitational constant of $G_\mathrm{eff}$, our measurement of the velocity
dispersion can place a tight constraint of $0.88 < G_\mathrm{eff}/G_\mathrm{N}
< 1.29\, (68\%)$ at length scales of a few Mpc about $2.5$ Giga years ago,
where $G_\mathrm{N}$ is the Newton's constant.
