The reason peculiar velocities grow faster in general relativity than in Newtonian gravity

An increasing number of surveys has been reporting large-scale peculiar motions with sizes and speeds in excess of those allowed by the concordance cosmological model. These are the so called bulk flows, the presence of which has come to be treated as a problem for the $\Lambda$CDM paradigm. However, the limits of the $\Lambda$CDM model are based on Newtonian studies, which predict the mediocre $v\propto t^{1/3}$ growth-rate for the peculiar-velocity field ($v$). Recently, a few fully relativistic treatments have appeared in the literature, arguing for a much stronger velocity growth that could explain the reported fast and deep bulk flows. What separates the Newtonian from the relativistic studies is the gravitational input of the peculiar flux, namely of the kinetic energy triggered by the moving matter. The latter has no direct gravitational contribution in Newtonian theory, but it does so in general relativity. This drastically changes the driving agent of the peculiar-velocity field and boosts its linear growth. The aim of this work is to directly compare the two treatments, as well as identify and discuss the reasons for their different results. In the process, we also demonstrate how one could recover the relativistic growth-rate from a Newtonian setup by selectively including certain (typically ignored) source-free terms into the Poisson equation. This way, we provide a unified covariant comparison of the Newtonian, the quasi-Newtonian and the fully relativistic studies.