© CAMBRIDGE UNIVERSITY PRESS 1999

1.4.3 Measurements on Scales of a Few Mpc

On smaller length scales, there are many measurements that are consistent with a smaller value of ₀ (e.g. Ch. 4; Ch. 11; Peebles 1993, Ch. 20). For example, the cosmic virial theorem gives (~ 1 h^-1 Mpc) 0.15 [(1 h^-1 Mpc) / (300 km s^-1)]², where (1 h^-1 Mpc) here represents the relative velocity dispersion of galaxy pairs at a separation of 1 h^-1 Mpc. Although the classic paper (Davis & Peebles 1983) which first measured (1 h^-1 Mpc) using a large redshift survey (CfA1) got a value of 340 km s^-1, this result is now known to be in error since the entire core of the Virgo cluster was inadvertently omitted (Somerville, Davis, & Primack 1996); if Virgo is included, the result is ~ 500-600 km s^-1 (cf. Mo et al. 1993, Zurek et al. 1994), corresponding to (~ 1 h^-1Mpc) 0.4-0.6. Various redshift surveys give a wide range of values for (1 h^-1 Mpc) ~ 300-750 km s^-1, with the most salient feature being the presence or absence of rich clusters of galaxies; for example, the IRAS galaxies, which are not found in clusters, have (1 h^-1 Mpc) 320 km s^-1 (Fisher et al. 1994), while the northern CfA2 sample, with several rich clusters, has much larger than the SSRS2 sample, with only a few relatively poor clusters (Marzke et al. 1995; Somerville, Primack, & Nolthenius 1996). It is evident that the (1 h^-1 Mpc) statistic is not a very robust one. Moreover, the finite sizes of the dark matter halos of galaxies and groups complicates the measurement of using the CVT, generally resulting in a significant underestimate of the actual value (Bartlett & Blanchard 1996, Suto & Jing 1996).

A standard method for estimating on scales of a few Mpc is based on applying virial estimates to groups and clusters of galaxies to try to deduce the total mass of the galaxies including their dark matter halos from the velocities and radii of the groups; roughly, GM ~ rv². (What one actually does is to pretend that all galaxies have the same mass-to-light ratio M/L, given by the median M/L of the groups, and integrate over the luminosity function to get the mass density (Kirschner, Oemler, & Schechter 1979; Huchra & Geller 1982; Ramella, Geller, & Huchra 1989). The typical result is that (~ 1 h^-1 Mpc) ~ 0.1-0.2. However, such estimates are at best lower limits, since they can only include the mass within the region where the galaxies in each group can act as test particles. It has been found in CHDM simulations (Nolthenius, Klypin, & Primack 1997) that the effective radius of the dark matter distribution associated with galaxy groups is typically 2-3 times larger than that of the galaxy distribution. Moreover, we find a velocity biasing (Carlberg & Couchman 1989) factor in CHDM groups b_v^grp v_{gal, rms} / v_{DM, rms} 0.75, whose inverse squared enters in the estimate. Finally, we find that groups and clusters are typically elongated, so only part of the mass is included in spherical estimators. These factors explain how it can be that our = 1 CHDM simulations produce group velocity dispersions that are fully consistent with those of observed groups, even with statistical tests such as the median rms internal group velocity vs. the fraction of galaxies grouped (Nolthenius, Klypin, & Primack 1994, 1997). This emphasizes the point that local estimates of are at best lower limits on its true value.

Another approach to estimating from information on relatively small scales has been pioneered by Peebles (1989, 1990, 1994). It is based on using the least action principle (LAP) to reconstruct the trajectories of the Local Group galaxies, and the assumption that the mass is concentrated around the galaxies. This is perhaps a reasonable assumption in a low- universe, but it is not at all what must occur in an = 1 universe where most of the mass must lie between the galaxies. Although comparison with = 1 N-body simulations showed that the LAP often succeeds in qualitatively reconstructing the trajectories, the mass is systematically underestimated by a large factor by the LAP method (Branchini & Carlberg 1994). Surprisingly, a different study (Dunn & Laflamme 1995) found that the LAP method underestimates by a factor of 4-5 even in an ₀ = 0.2 simulation; the authors say that this discrepancy is due to the LAP neglecting the effect of ``orphans'' - dark matter particles that are not members of any halo. Shaya, Peebles, and Tully (1995) have recently attempted to apply the LAP to galaxies in the local supercluster, again getting low ₀. The LAP approach should be more reliable on this larger scale, but the method still must be calibrated on N-body simulations of both high- and low-₀ models before its biases can be quantified.