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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.