Measuring the homogeneity scale using the peculiar velocity field

Abstract: We propose an innovative definition of the scale at which the Universe becomes homogeneous based on measurements of velocities rather than densities. When using the matter density field, one has to choose an arbitrary scale (e.g. within 1\% of the average density) to define the transition to homogeneity. Furthermore, the resulting homogeneity scale is strongly degenerate with the galaxy bias. By contrast, peculiar velocities (PV) allow us to define an unambiguous scale of homogeneity, namely the distance at which the velocities between pairs of galaxies change from being on-average correlated to anti-correlated. Physically, this relates to when the motion of pairs of galaxies is influenced by the matter density between them, rather than beyond. The disadvantage is that peculiar velocities are more difficult to measure than positions, resulting in smaller samples with larger uncertainties. Nevertheless, we illustrate the potential of this approach using the peculiar velocity correlation functions obtained from the Sloan Digital Sky Survey PV catalog, finding an homogeneity scale of $R_H\approx 133\substack{+28 \ -52}\, \rm{Mpc/h}$. Finally, we show that more precise measurements are within reach of upcoming peculiar velocity surveys, and highlight this homogeneity scale's potential use as a standard ruler within the standard cosmological model.