The Finite‐Size Effect of Galaxies on the Cosmic Virial Theorem and the Pairwise Peculiar Velocity Dispersions

Abstract

We discuss the effect of the finite size of galaxies on estimating small-scale relative pairwise peculiar velocity dispersions from the cosmic virial theorem (CVT). Specifically, we evaluate the effect by incorporating the finite core radius rc in the two-point correlation function of mass, i.e., ξρ(r) ∝ (r + rc)-γ, and the effective gravitational force softening rs on small scales. We analytically obtain the lowest order correction term for γ 2. Compared with the idealistic point-mass approximation (rs = rc = 0), the finite-size effect can significantly reduce the small-scale velocity dispersions of galaxies at scales much larger than rs and rc. Even without considering the finite size of galaxies, nonzero values for rc are generally expected, for instance, for cold dark matter (CDM) models with a scale-invariant primordial spectrum. For these CDM models, a reasonable force softening rs ≤ 100 h-1 kpc would have a rather tiny effect. We present the CVT predictions for the small-scale pairwise velocity dispersion in the CDM models normalized by the COBE observations. The implications of our results for confrontation of observations of galaxy pairwise velocity dispersions and theoretical predictions of the CVT are also discussed.