Annu. Rev. Astron. Astrophys. 1993. 31:
689-716
Copyright © 1993 by . 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 *v*_{0}
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} < *v*_{0} 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 *H*_{0}*R* = 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.