3.1.5. Peculiar velocity - density relation

This is one of the most traditional methods to estimate the cosmic mass density. The principles are spelled out by Peebles (1980). There are two basic tools depending on the scale. For small scales (r < 1 Mpc) the perturbations developed into a non-linear regime, and the statistical equilibrium argument is invoked for ensemble averages that the peculiar acceleration induced by a pair of galaxies is balanced by relative motions (cosmic virial theorem). For a large scale (r > 10 Mpc), where perturbations are still in a linear regime, the basic equation is

(12)

with the density contrast. The contribution from a cosmological constant is negligible. The problem inherent in all arguments involving velocity is the uncertainty regarding the extent to which galaxies trace the mass distribution (biasing), or how much mass is present far away from galaxies.

Small-scale velocity fields:     The status is summarised in Peebles (1999), where he has concluded (10 kpc r 1 Mpc) = 0.15 ± 0.10 from the pair wise velocity dispersion (with samples excluding clusters) and the three point correlation function of galaxies via a statistical stability argument. Bartlett & Blanchard (1996) argued that it is possible to reconcile the observed velocity dispersion with ~ 1 if one assumes galactic halo extended beyond > 300 kpc. As Peebles (1999) argued, however, the halo is unlikely to be extended that much as indicated by the agreement of MW's mass at 100-200kpc and the mass estimate for MW + M31 in the Local Group.

Beyond a 10 Mpc scale, linear perturbation theory applies. An integral form of (12) for a spherical symmetric case (v / H₀ r = ^0.6 < >/3) applied to the Virgocentric flow gives 0.2 for v 200-400 km s^-1 and < > ~ 2, assuming no biasing (Davis and Peebles 1983). Recently, Tonry et al. (1999) argued that the peculiar velocity ascribed to Virgo cluster is only 140 km s^-1, while the rest of the peculiar velocity flow is attributed to the Hyd-Cen supercluster and the quadrupole field. For this case 0.06. We may have ~ 1 only when half the mass is well outside the galaxies.

Peebles (1995) argued that the configuration and kinematics of galaxies are grown following the least action principle from the nearly homogeneous primeval mass distribution. Applying this formalism to Local Group galaxies, he inferred = 0.15 ± 0.15. On the other hand, Branchini & Carlberg (1994) and Dunn & Laflamme (1995) argued that this conclusion is not tenable if mass is distributed smoothly outside galaxies as in = 1 CDM models. This seems, however, not very likely unless mass distribution is extended over 10 Mpc scale (Peebles 1999).

Large scale velocity fields:     There are a few methods to analyse the large-scale velocity fields based on (12). The direct use of (12) is a comparison of the density field derived from redshift surveys with measured peculiar velocities. Alternatively, one may use the density field reconstructed from observed velocity field for comparison with the actual density field, as in the POTENT programme (Dekel et al. 1990). A variant of the first method is to observe the anisotropy in redshift space (redshift distortion) (Kaiser 1987). As linear theory applies, always appears in the combination = ^0.6 / b where b is a linear biasing factor of galaxies against the mass distribution and can be inferred through non-linear effects. Much effort has been invested in such analyses (see e.g., Strauss & Willick 1995; Dekel et al. 1997; Hamilton 1998), but the results are still controversial. The value of ^0.6 / b derived from many analyses varies from 0.3 to 1.1, though we see a general trend to favour a high value. Notably, the most recent POTENT analysis using the Mark III compilation of velocities (Willick et al. 1997) indicates a high density universe = 0.5-0.7, and > 0.3 only at a 99% confidence level (Dekel et al. 1999).

The difficulty is that one needs accurate information for velocity fields, for which an accurate estimate of the distances is crucial. Random errors of the distance indicators introduce large noise in the velocity field. This seems particularly serious in the POTENT algorithm, in which the derivative . / ^0.6 and its square are numerically computed; this procedure enhances noise, especially for a small . The difficulty of inferring large scale velocity field may also be represented by the `great attractor problem'. Lynden-Bell et al. (1988) found a large-scale velocity field towards the Hyd-Cen supercluster, but also argued that this supercluster is also moving towards the same direction attracted by a `great (giant) attractor'. With Tonry et al.'s (1999) new estimate of the distance using SBF, this velocity field is modest, and Hyd-Cen itself serves as the great attractor that pulls the Virgo cluster, with the conclusion that is small.