Modeling the Phase-Space Distribution around Massive Halos
  The comparison between dynamical mass and lensing mass provides a targeted
test for a wide range of modified gravity models. In our previous paper we
showed, through numerical simulations, that the measurement of the
line-of-sight velocity dispersion around stacked massive clusters whose lensing
masses are known allows for stringent constraints on modified gravity on scales
of 2 - 15 Mpc/h. In this work we develop a semi-analytical approach based on
the halo model to describe the phase-space distribution and the line-of-sight
velocity dispersion for different tracers. The model distinguishes
contributions from the halo pairwise velocity and the virial velocity within
halos. We also discuss observational complications, in particular the
contribution from Hubble flow, and show how our model can incorporate these
complications. We then incorporate the effects of modified gravity
(specifically, f(R) and braneworld models), and show that the model predictions
are in excellent agreement with modified gravity simulations. More broadly, the
phase-space distribution provides a sensitive test of our understanding of
hierarchical structure formation when confronted with observations via this
model.
