Peculiar velocities in the $\Lambda$CDM universe

We present a unified analysis of the linear evolution of peculiar-velocity perturbations in the distribution of pressureless matter after recombination. Our study is carried out within the framework of full general relativity and encompasses both the earlier Einstein-de Sitter epoch and the subsequent $\Lambda$-dominated phase. Starting from a non-interacting, non-comoving mixture of radiation and dust, we derive the generalized differential equations governing the evolution of these velocity perturbations. We confirm that linear peculiar velocities ($v$) grow as $v\propto t$ throughout the Einstein-de Sitter era. In contrast, during a later phase of increasing $\Omega_{\Lambda}$-contribution, we find that the growth of the peculiar-velocity field is progressively suppressed, before reversing to decay. Nevertheless, part of the earlier velocity growth persists to the present. Hence, bulk-flow velocities at intermediate redshifts are expected to exceed those predicted by the $\Lambda$CDM model, followed by a decline in their value at lower redshifts. Interestingly, such peculiar-velocity profiles have already been reported in the literature.