The velocity-density relation in the spherical model

Abstract

We study the cosmic velocity–density relation using the spherical collapse model (SCM) as a proxy to non-linear dynamics. Although the dependence of this relation on cosmological parameters is known to be weak, we retain the density parameter Ωm in SCM equations, in order to study the limit Ωm→ 0. We show that in this regime the considered relation is strictly linear, for arbitrary values of the density contrast, on the contrary to some claims in the literature. On the other hand, we confirm that for realistic values of Ωm the exact relation in the SCM is well approximated by the classic formula of Bernardeau, both for voids (δ < 0) and overdensities up to δ∼ 2–3. Inspired by this fact, we find further analytic approximations to the relation for the whole range δ∈[− 1, ∞). Our formula for voids accounts for the weak Ωm-dependence of their maximal rate of expansion, which for Ωm < 1 is slightly smaller that 3/2. For positive density contrasts, we find a simple relation that works very well up to the turnaround (i.e. up to δ≲ 13.5 for Ωm= 0.25 and neglected ΩΛ). Having the same second-order expansion as the formula of Bernardeau, it can be regarded as an extension of the latter for higher density contrasts. Moreover, it gives a better fit to the results of cosmological numerical simulations.