Annu. Rev. Astron. Astrophys. 1993. 31: 689-716 Copyright © 1993 by Annual Reviews. All rights reserved

3.3. The Velocity Distribution of Galaxies

The prime fact is that our Local Group has a velocity with respect to the CBR of about v₀ 600 ± 25 km s^-1, as directly measured by the dipole anisotropy of the CBR, with modest corrections for Local Group kinematics (Lubin & Villela 1986). If we interpret this in the conventional manner, it shows the effect of gravitational forces due to large-scale density irregularities, which provide the acceleration that, integrated over time, has caused our peculiar velocity. CDM models for structure with relatively low amplitude in the perturbation spectrum (₀ < 1) had difficulty producing such a large velocity but could always appeal to statistical fluctuations: We live, unluckily, in a galaxy which is on the tail of the peculiar velocity distribution. Utilizing the COBE normalization, this problem is diminished.

The second disturbing fact is that, in the frame of reference at rest with respect to the galaxies on average (the "Local Standard of Rest" frame in the notation of galactic stellar kinematics), the Local Group has a peculiar velocity v_p < v₀ and that, furthermore, so do most other galaxies. That is, the flow is quite cold with bulk motion exceeding random motion on all scales where measurements have been made.

This large coherence of the velocity field has two consequences. First, our velocity is not anomalous; most galaxies have a similar velocity with respect to the CBR. Second, this result is not likely to be in error, despite the well acknowledged difficulty of measuring proper velocities. The measurements indicate small proper velocities, and it is difficult to imagine systematic errors (or random ones) which would conspire to indicate small velocities with respect to us over a significant part of the sky if the truth were that there is a large directed velocity that cancels out our large directed velocity with respect to the CBR. The numerical basis for these remarks was presented in a paper by Groth et al (1989) in the framework of an analysis of the velocity correlation function (see also Gorski et al 1989).

Now let's address some of the quantitative details. In an exercise of precognition, Bertschinger et al (1990) compared the COBE normalized (i.e. unbiased) CDM model with observed large-scale velocities. The observed average peculiar velocities (with respect to the CBR) within spheres of 4000 and 6000 km/s centered on the Local Group were found to be 388 ± 67 km s^-1 and 327 ± 82 km s^-1, only modestly in excess of the predictions (287 km s^-1 and 224 km s^-1 on the same scales). However, this same model predicts a velocity dispersion on small scales which is now much too large. Simple theory allows one to estimate the one-dimensional pairwise velocity dispersion of galaxies separated by 1 h^-1 Mpc should be v_||, rms = 970 ± 160 km s^-1 in a standard CDM model (with COBE normalization ₈ = 1.08 ± 0.18), a result far in excess of the observed value 340 ± 40 by Davis & Peebles 1983). Numerical simulations by Cen & Ostriker (1993) and Ueda et al (1993) confirm that the CDM prediction normalized to COBE is far too high.

It is extremely unlikely that the small-scale velocity dispersion of galaxies is larger than the quoted value by a factor of 2.8 as required to bridge the differences (since unrecognized errors in observational procedures would have caused too large, not too small, a velocity dispersion to be "measured"). But, can the theoretical normalizations be in error? Is it possible that galaxies suffer a large enough "velocity bias" with respect to dark matter particles to lower the predicted dispersion greatly from that given by linear theory? Couchman & Carlberg (1992) argue on the basis of numerical hydro simulations that, on small scales, galaxies could have a considerably smaller velocity dispersion than dark matter particles. But, on the several megaparsec scale, Katz et al (1992) find little evidence for velocity bias. Cen & Ostriker (1992b), in a detailed hydrodynamic study, which allowed for galaxy formation, found a small effect. On the (2-20) h^-1 Mpc scales, the ratio of galaxy-to-dark matter proper velocities is expected to be close to 0.80; "velocity bias" is a real but quite small effect on appropriate scales. Thus, the standard CDM model, when normalized to the COBE-determined amplitude is in serious conflict with the observed pairwise velocity dispersion of galaxies.

Finally, let us examine the velocity field in a way that does not depend on the COBE result. Both the bulk flow <v> and the velocity dispersion , with respect to that bulk flow, are proportional to the amplitude of the perturbation spectrum, or, more physically, to gravitational potential fluctuations. But the ratio <v> / will be nearly independent of the still somewhat uncertain amplitude. We can imagine measuring both quantities on a patch of the universe of size R. Then <v>_R will measure the gravitational forces due to regions bigger than R, and _R measures the fluctuations on scales smaller than R. The Mach number, M(R) <v> / _R, thus measures the ratio of large-scale to small-scale power.

Ostriker & Suto (1990) showed how the Mach number could be calculated from standard linear theory and examined several of the popular cosmological scenarios. As expected, the standard CDM model predicts a "hot" flow M < 1 on scales hR > 10 Mpc, where the observations, as noted previously, strongly indicate a very cold flow. In a far more detailed study, Strauss et al (1993, references therein for earlier work) compared detailed numerical reconstructions of a standard CDM universe with the best currently available proper velocity data. The most conclusive results were from the AHM spiral galaxy data, using primarily Tully-Fisher distances. On the scale H₀R = 1500 km s^-1, they found a bulk flow of 460 km s^-1 with a 3-D velocity dispersion of 456 km s^-1, or a Mach number close to unity. Examining the simulated data in a fashion as close as possible to the real data, they found that fewer than 5% of the observers in a hypothetical CDM universe would find such a high Mach number. Thus, this test rejects the standard CDM scenario at the 95% confidence level; it is also independent of the COBE normalization.

To summarize this subsection, standard CDM models appear to predict significantly larger velocity dispersions for galaxies than are observed, whether one normalizes the theory to COBE or to the observed large-scale flows.