The peculiar Raychaudhuri equation
  Peculiar motions are commonplace in the universe. Our local group of
galaxies, for example, drifts relative to the Hubble flow at about 600 km/sec.
Such bulk flows are believed to fade away as we move on to progressively larger
scales. Recently, however, there have been reports of peculiar motions larger
and faster than typically expected. If these claims are correct, the role of
peculiar flows in shaping the kinematics of our universe has been probably
underestimated. Here, we use general relativistic techniques to analyse the
average kinematics of large-scale bulk motions and compute the nonlinear
Raychaudhuri equation of such flows. In doing so, we introduce two families of
observers. One at rest with the Hubble expansion and another following the
peculiar motion. We first derive the fully nonlinear expressions of the
"peculiar" Raychaudhuri equation in both frames and identify a range of
relative motion effects. Linearised around a Friedmann universe with
pressureless dust, the full equations reduce to a simple relation with no
explicit contribution from the matter component of the universe. Using
cosmological perturbation theory, we obtain a scale-dependent formula, which
also incorporates the effects of universe's inhomogeneity and anisotropy. When
the latter are secondary, the peculiar Raychaudhuri equation can be solved
analytically. Also, combining the resulting expression with data from recent
peculiar-velocity surveys, we estimate the relative deceleration/acceleration
rates of such bulk motions.
