Hubble's Law, now spectacularly confirmed by the work of [27], [35], and [39], tells us that the distances of galaxies are proportional to their observed recession velocities, at least at low redshifts:

(1) |

However, this is not exactly correct. Galaxies have *peculiar
velocities *above and beyond the Hubble flow
indicated by Eq. (1). We denote the peculiar velocity **v**(**r**)
at every point in space; the observed redshift in
the rest frame of the Local Group is then:

(2) |

where the peculiar velocity of the Local Group itself is
**v**(**0**), and
is the unit vector to the galaxy in
question. In practice, we will measure distances in units of km
s^{-1}, which means that *H*_{0}
1, and the
uncertainties in the value of *H*_{0} discussed by Freedman
and Tammann in this volume are not an issue.
Thus measurements of redshifts *cz*, and of redshift-independent
distances via standard candles, yield
estimates of the radial component of the velocity field.
What does the resulting velocity field tell us? On scales large enough
that the rms density
fluctuations are small, the equations of gravitational instability can
be linearized, yielding a direct
proportionality between the divergence of the velocity field and the
density field at late times
[33],
[34]:

(3) |

This equation is easily translated to Fourier space:

(4) |

which means that if we define a *velocity* power spectrum
*P*_{v}(*k*) ~
<^{2}(**k**)> in
analogy with the usual density power spectrum *P*(*k*), we
find that

(5) |

There are several immediate conclusions that we can draw from
this. Peculiar velocities are tightly coupled to the *matter*
density field
(**r**). Therefore, peculiar velocities are a probe of
the *matter* power
spectrum; any bias of the distribution of galaxies relative to that of
matter is not an issue. Moreover, Eq. (5) shows that it is in principle
easier to probe large spatial scales with peculiar velocities than
with the density field, because of the two extra powers of *k*
weighting for the velocity power spectrum.

Eq. (3) shows that a comparison of the velocity field with the
*galaxy* density field
_{gal} allows a
test of gravitational instability theory. However, in order to do this,
one must assume a relation between the
galaxy density field (which is observed via redshift surveys) and the
mass density field (which does the
gravitating). The simplest and most common assumption (other than simply
assuming the two are
identical) is that they are proportional (*linear biasing*),
i.e., _{gal} =
*b*. If this is the
case, then we can rewrite Eq. (3) to give:

(6) |

Thus if the observed density and velocity field are consistent with one another and Eq. (6), we can hope to measure . This, and other approaches to via peculiar velocities are reviewed in Dekel’s contribution to this volume; cf., the reviews by [11] and [45].