Annu. Rev. Astron. Astrophys. 1998. 36:
599-654
Copyright © 1998 by . All rights reserved |

**3.3. Velocity Statistics**

The spatial statistics listed above make use of only half the phase space coordinates of particles, neglecting velocity information (or, in redshift space, using radial velocity in place of distance). Although it is difficult to measure sufficiently accurate extragalactic distances to obtain reliable peculiar velocities (Strauss & Willick 1995), different cosmological models vary substantially in their predictions (especially as a function of the density parameter ), making investigation of peculiar velocities well worthwhile.

The simplest velocity statistic is the velocity distribution of single
particles, *f*(*v*)4
*v*^{2} *dv* where *v* is the magnitude of peculiar
velocity. This statistic is poorly studied despite the fact that it is
the lowest-order distribution function appearing in the BBGKY kinetic
theory of gravitational clustering
(Peebles 1980).
Either the radial
component or one Cartesian component may be used instead of the magnitude
of the three-dimensional velocity; because of isotropy, all three
distributions are simply related.
Inagaki et al (1992),
Raychaudhury & Saslaw
(1996) compared the predictions of the thermodynamic theory of
Saslaw & Hamilton
(1984) with simulations starting from
Poisson initial conditions, and they found good agreement.
Cen & Ostriker
(1993c) examined *f*(*v*) for galaxies formed in their CDM
simulations and noted that an exponential distribution fits better
than a Maxwellian.

Davis et al (1997)
have proposed using the small-scale velocity dispersion
_{1}, the second
moment of *f*(*v*) after filtering out long-wavelength
velocity contributions, to provide an estimate of
through the
Layzer-Irvine cosmic energy equation. In practice, they measure this
single-particle dispersion using a single-particle weighting of a pairwise
velocity. This statistic has the advantage of being less sensitive to
sampling fluctuations from clusters of galaxies than the pair-weighted
pairwise velocity dispersion (e.g.
Zurek et al 1994,
Somerville et al 1997).
Alternatively, a redshift dispersion may be measured as a function of
density, as suggested by
Kepner et al (1997b).

Besides examining the velocities of individual particles or galaxies,
study is possible of the "fluid" or "bulk flow" velocity obtained by
averaging the velocity over a window of radius *R*,
(*R*).
Kofman et al (1994)
showed analytically and with *N*-body simulations that in the
mildly nonlinear regime, the distribution of
remains close to Maxwellian for Gaussian initial conditions.
The steep decline of the Maxwellian for bulk flows that are much larger than
the rms leads to a sensitive test of power on large scales (e.g.
White et al 1987b,
Bertschinger &
Juszkiewicz 1988). Other
velocity field statistics include the cosmic Mach number
= *V(R)* /
(*R*),
where (*R*) is the
small-scale velocity dispersion within the same window
(Ostriker & Suto
1990). Bulk flows are sensitive
to large-scale power, while small-scale dispersions are sensitive to smaller
scales, so
is sensitive to
the shape of the power spectrum.

Another statistic applied to the large-scale velocity field is the
distribution of velocity divergence, *f*
()
*d* where
=
^{.}
(Bernardeau 1994).
The velocity divergence is
attractive theoretically because in the linear regime, it is proportional
to the density fluctuation (see Equations 6 and 7);
in the mildly nonlinear regime for Gaussian initial conditions, the ratio
of its skewness to the square of its variance is sensitive to
(Bernardeau et al 1995,
1997);
and for a potential flow (as expected
on large scales for gravitationally induced motions), the velocity field
is fully described by
()
(Bertschinger & Dekel
1989).

The statistics of relative velocities of pairs of galaxies have been
extensively studied on small scales because of the possibility of
measuring using the
cosmic virial theorem
(Peebles 1980), which
relates these velocities to the two- and three-point correlation functions.
The relative velocity of a pair separated by
,
_{1} -
_{2} ,
is decomposed into a component along
and the
remainder perpendicular to it. The distribution of
*v*_{12}
^{.}
(_{1} -
_{2}) ,
in particular its mean <*v*_{12}> and standard deviation
_{12}, has been
studied extensively in numerical simulations,
initially by
Davis et al (1985)
and more recently by others, including
Zurek et al (1994),
Gelb & Bertschinger
(1994b),
Brainerd et al (1996),
Colin et al (1997).
The sensitivity of
_{12} to rich
clusters
makes it difficult to measure well but also provides discriminating power
among different structure formation models
(Somerville et al
1997).
Sheth (1996)
recently has provided analytical insight into the exponential form of
*f*(*v*_{12}) by using an extension of the
Press-Schechter theory
(Press & Schechter
1974; cf also
Mo et al 1996).