The velocity coherence scale: a novel probe of cosmic homogeneity and a potential standard ruler

We introduce the velocity coherence scale \(R_v\), the scale at which the spherical volume average of the trace of the velocity correlation tensor transitions from scaling faster than the sphere radius to scaling more slowly. This corresponds to the radius at which the average motion of galaxies along their separation vectors transitions from correlated to anti-correlated. We derive a theoretical estimator for \(R_v\) by defining the bulk in spheres \(B_R\), a velocity-field analogue of the mean scale counts used in density-field correlation analyses. We show that, for a statistically homogeneous matter distribution, the logarithmic derivative of \(B_R\) and the correlation dimension \(D_2\) share the same asymptotic behaviour and therefore can be used to estimate the scale of transition to statistical homogeneity. Furthermore, we show that in standard \(Λ\)CDM cosmologies the velocity coherence scale is tightly connected to the matter-radiation equality scale \(k_{eq}\), and that its value in comoving coordinates is redshift-independent. These results highlight the potential of \(R_v\) both as a standard ruler and as a physically motivated scale characterising the onset of cosmic homogeneity. We present a proof of concept using measurements of the PV correlation functions from SDSS. We show that the main challenge in determining \(R_v\) is the limited precision of PV measurements compared to density ones, as they typically rely on smaller samples with larger uncertainties that scale roughly linearly with survey depth. Fitting our theoretical estimators for \(R_v\), we obtain \(R_v \approx 138^{+23}_{-54}\,\mathrm{Mpc}/h\). Finally, we show that more precise determinations should be achievable with current and upcoming peculiar velocity surveys.