We begin this chapter with a discussion of bulk flows in Section 7.1; this has been the major theme of much of the work in peculiar velocities in the last decade. Section 7.2 summarizes the history of peculiar velocity work with a discussion of the effort needed to put on a common basis the various samples available. The rest of the chapter discusses various statistics measurable from the observed flow field. The velocity correlation function and the Cosmic Mach Number are discussed in Section 7.3 and Section 7.4, respectively. Section 7.5 is devoted to derivations of the full three-dimensional velocity field from its radial component, with emphasis on the POTENT technique of Bertschinger & Dekel (1989) , and the results that have come from this work.

**7.1. A History of Observations of Large-Scale
Flow**

Since the mid-1970's coherent departures from uniform Hubble
flow ^{(31)}
have been observed on ever larger scales. A comprehensive
historical review of the subject through ~ 1989 is given by
Burstein (1990) .
Here we briefly outline this history from ~ 1976 to the
present, with an emphasis on recent years.

When one speaks of a coherent bulk flow, one requires a velocity frame
of reference. Prior to the mid-1970's, the most logical choice for
such a frame was that defined by the barycenter of the Local Group of
galaxies (LG), the bulk of whose mass is found in the Milky Way and
the Andromeda galaxy M31. Our own motion with respect to
such a frame
was thought to arise from LG dynamics and the rotation of the Milky
Way; the LG barycenter itself was considered at rest with respect to
the Hubble expansion. This reasoning changed following the discovery
of the CMB dipole anisotropy in 1976 (see
Section 5.7). The
CMB dipole was and is most naturally interpreted as due to the motion
of the LG with respect to the rest frame of the totality of mass
within our observable universe, in the direction
*l* = 276°,
*b* = 30°, and with amplitude 627
km s^{-1}. With this finding it
became clear that significant peculiar velocities exist in the
Universe: unless the LG is atypical, peculiar motions of hundreds of
km s^{-1} could no longer be considered unusual. The CMB dipole
also suggested a natural reference frame for the analysis of peculiar
motions.

The first claimed detection of large-scale streaming was that of
Rubin et al. (1976a,
b),
who assumed that giant Sc spiral galaxies are
standard candles. They studied a sample of 96 such galaxies in the
redshift range 3500-6500
km s^{-1}, and found a bulk flow relative to the
Local Group frame of ~ 600
km s^{-1} in the direction *l* = 160°,
*b* = - 10°. The Rubin et al. result provoked a great deal of
skepticism in the astronomical community, despite the (nearly
simultaneous) discovery of the CMB dipole. The direction of the Rubin
et al. bulk flow was nearly orthogonal to the LG velocity vector,
making the relationship of the two measurements difficult to
understand. Indeed, the measurement of Rubin et al. to this day has
neither been confirmed nor entirely refuted, but instead has simply
been supplanted by more modern data based on better distance
indicators.

In the late 1970s and early 1980s, peculiar velocity work focused on
the detection of infall to the Virgo cluster, which lies near the
North Galactic Pole at
270°,
*b* 75°, and is
the nearest large cluster to the Local Group. A number of workers
using TF, Faber-Jackson, and other techniques estimated the amplitude
of this motion at the Local Group to be in the range ~ 150-400
km s^{-1} (*e.g.*,
Schechter 1980
;
Aaronson et al. 1982a;
de Vaucouleurs &
Peters 1981
;
Tonry & Davis 1981a
,
b;
Hart & Davies 1982
;
Dressler 1984).
Even if the largest of these estimates were correct, the
misalignment between the Virgo direction and the CMB dipole meant that
Virgocentric infall could not be the sole cause of the LG motion
(Davis & Peebles
1983a
;
de Vaucouleurs &
Peters 1984
;
Sandage & Tammann
1984
;
Yahil 1985).
This led to suggestions that the remaining part of the
LG motion resulted from the pull of the relatively nearby
Hydra-Centaurus supercluster complex, which lies almost near the apex
of the microwave dipole at a redshift
*cz* 3000 *km*
*s*^{-1}
(Shaya 1984
;
Tammann & Sandage
1985
;
Davies &
Stavely-Smith 1985).
Lilje, Yahil, &
Jones (1986)
studied residuals from the Virgocentric flow solution of
Aaronson et al. (1982a)
and concluded they were best fit
by a quadrupolar term arising from the pull of Hydra-Centaurus. These
suggestions were consistent with the generally accepted view that
peculiar motions arose from relatively "local" (scales
5000 *km*
*s*^{-1}) mass density perturbations.

**7.1.2. 1986-1990: The "Great
Attractor"**

Confirmation of this view appeared to come from a long term study by
the Aaronson group. Using IRTF distances to ten clusters in the
4000-10,000 km s^{-1} redshift range,
Aaronson et al. (1986)
concluded
that these clusters exhibited no net motion relative to the
CMB ^{(32)} . They
argued, as a corollary, that the LG motion had to be entirely
generated by mass fluctuations within ~ 5000
km s^{-1}. The picture
that prevailed in the mid-1980's was thus one in which the Hubble flow
was unperturbed on scales
5000 *km*
*s*^{-1}. But within several
years the paradigm had changed radically, due mainly to the work of
the "7 Samurai" (7S). Applying the
*D*_{n}-
relation to an all-sky sample of over 400 bright (*m*_{B}
13 mag,
<*cz*> 3000
*km* *s*^{-1}) elliptical galaxies, the group reported
a bulk flow of amplitude
600 ± 100 *km* *s*^{-1} relative to the CMB
frame, in the direction *l* = 312 ± 11°,
*b* = 6 ± 10°
(Dressler et
al. 1987a).
They emphasized that galaxies in the Hydra-Centaurus
concentration participated in, and therefore could not be the source
of, the observed flow. They concluded that, contrary to the
conventional wisdom, the greater part of the LG motion relative to the
CMB had to be generated on scales
5000 *km*
*s*^{-1}.

This result shook up the cosmology community considerably. Its
significance lay not so much in the amplitude as in the coherence
scale of the flow. Indeed, as initially reported by
Dressler et al. (1987a)
,
the true scale of the bulk flow was unconstrained,
potentially larger than the effective limit (~ 6000 km s^{-1})
of their sample. It was quickly realized (*e.g.*,
Vittorio, Juszkiewicz,
& Davis 1986)
that a bulk flow on this scale was inconsistent with essentially
all of the then-popular scenarios for large-scale structure formation,
including the standard cold and hot dark matter models. The power
spectrum *P*(*k*) for any of those models did not contain
sufficient power on large scales to generate flows of amplitude
~ 600 *km* *s*^{-1} on scales
5000 *km*
*s*^{-1} (Eq. 40).
Kaiser (1988)
showed that the situation for cosmological models was not as dire as
Vittorio et al. (1986)
implied, as the effective depth of the
7S sample was not as large as had been assumed. Indeed, the fitted
bulk flow was not inconsistent with the standard CDM model with high
enough normalization. However, in this case, the small-scale velocity
dispersion is larger than that observed;
Bertschinger &
Juszkiewicz (1988)
point out that there is no normalization of standard CDM which
can simultaneously match the 7S bulk flow and the observed small
velocity dispersion (cf., Section 7.4).

The controversial nature of the 7S result was at least partially
alleviated when the group undertook a reinterpretation of their data
in Lynden-Bell et al.
(1988a,
hereafter LB88). They showed that much
of the signal for the elliptical streaming motion was provided by
galaxies within ~ 2000
km s^{-1}; if they confined their analysis to
objects at distances
3000 *km*
*s*^{-1}, bulk flow was only
marginally detected. Moreover, in a conical region within
~ 60° of the apex of the apparent bulk flow, the derived amplitude
was nearly 1000
km s^{-1}. They interpreted this as due to infall into an
attracting point. They fit their data using a Method I approach
(Section 6.4) to a model in which the
flow is generated
primarily by a spherically symmetric density perturbation they called
the "Great Attractor" (GA) (cf., Section 4). In
addition to the GA distance and flow amplitude, which were treated as
free parameters, the model of LB88 included a Virgo infall motion with
a fixed amplitude of 250
km s^{-1} at the position of the LG. Their
maximum-likelihood fits placed the GA at a distance of
4350 ± 350 *km* *s*^{-1} in the direction
*l* = 307°, *b* = 9°, indicated that
it induced an infall velocity of
570 ± 60 *km* *s*^{-1} at the position of
the LG, and attributed to it a mass of ~ 5.4 × 10^{16}
*M*_{}.

A further elaboration of the GA model was provided by
Faber & Burstein
(1988,
FB88), who combined TF data for nearby spirals from
Aaronson et al. (1982b)
with the 7S elliptical data. FB88 fitted the
combined data set to a model in which the local velocity field was
fully described by infall motions into the GA and Virgo, but with the
parameters characterizing each motion allowed to vary independently
(see also
Han & Mould 1990).
They obtained consistent results using
Method I and Method II analyses. Two important conclusions resulted
from this work. First, the nearby TF spirals (*cz*
2500
*km* *s*^{-1})
mapped out a velocity field similar to that of the ellipticals.
Second, the effect of Virgo was found to be very small in comparison
with the tidal effect of the GA: the infall motion due to Virgo alone
at the position of the LG had an amplitude of ~ 100
km s^{-1}. This
contrasted strongly with earlier estimates (*e.g.*,
Aaronson et al. 1982a)
which had found a Virgocentric motion of ~ 300
km s^{-1} and
higher (Section 7.1,
Section 8.1.1). FB88 verified that the
discrepancy between their Virgo flow solution and that of Aaronson
et al. was due solely to the shear caused by the GA in the model
(cf.,
Lilje et al. 1986)
^{(33)} .

Apparent confirmation of the GA picture came from two
directions. First, three newly-completed redshift surveys
(da Costa et al. 1988
;
Dressler 1988
;
Strauss & Davis
1988)
showed that there
was in fact a very strong excess number of galaxies in the direction
of the proposed GA, peaking at
*cz* ~ 4000 *km* *s*^{-1} (cf.,
Fig. 6). The second source of
support for
the GA model came from further peculiar velocity studies of the
region.
Aaronson et al. (1989)
extended their infrared TF cluster
study to the Hydra, Centaurus, and other neighboring clusters and
confirmed large positive peculiar velocities for these clusters (see
also Mould et
al. 1991).
Dressler & Faber
(1990a,
b)
studied field
galaxies in the GA region using both the TF (using CCD photometry and
optical linewidths) and
*D*_{n}-
methods. Probing deeper than earlier
studies, they claimed to detect "back-side infall" into the GA,
*i.e.*, negative peculiar motions of galaxies at distances
5000 *km*
*s*^{-1}. Such motions would have to exist if the positive
velocities
detected in the foreground were indeed generated by the GA, and the GA
was an isolated structure. The detection of backside infall remains
controversial (cf.,
Burstein et al. 1990;
Roth 1993
;
Mathewson & Ford
1994),
given the large and uncertain correction for
inhomogeneous Malmquist bias. In addition, it is clear that the
velocity field within the Centaurus cluster is quite complex; it
consists of two pieces with substantially different distances and
peculiar velocities (e.g.,
Lucey, Currie, &
Dickens 1986;
Burstein et al. 1990).

**7.1.3. 1990-1994: Very Large-Scale
Streaming?**

Several independent peculiar velocity surveys came to fruition around
1990, with results that challenged the consensus which had emerged
around the GA picture.
Willick (1990,
1991;
hereafter W91) reported an
R band CCD TF study of 155 cluster spirals from the sample of
Aaronson et al. (1986)
,
and of 326 field spirals in the Perseus-Pisces
supercluster region (PP). One of the more prominent structures in the
nearby universe (Fig. 6), PP
consists of a dense filament of galaxies at redshift ~ 5000
km s^{-1}, centered on *l* ~ 110°,
*b* ~ - 30°. W91 obtained velocity widths for
PP galaxies from the published 21 cm data of
Giovanelli & Haynes
(1985,
1989) and
Giovanelli et
al. (1986).
The W91 PP sample was
concentrated in a relatively narrow (~ 15°) strip in
galactic latitude, stretching over ~ 90° in longitude. PP
lies nearly opposite on the sky from the GA at a similar distance, but
was not well sampled in the 7S study. The PP region thus represented
a significant uncharted volume in the velocity field as mapped by the
elliptical data.

W91 used the cluster galaxies to calibrate the R band TF relation,
assuming (following the finding of the Aaronson group) that the
Northern Sky clusters were at rest in the CMB. When applied to the PP
galaxies, the TF data revealed a coherent motion directed toward the
LG, with a mean amplitude of ~ 350
km s^{-1}. Superposed on the coherent
streaming were compressional motions presumably due to the overdense
PP filament, but the bulk motion was the most prominent feature of the
data. Because of the location of PP, the streaming motion was
directed not only toward the LG, but also approximately toward the GA.
The amplitude of this streaming was much larger than what was
predicted at PP (~ 150
km s^{-1}) by the FB88 GA model. W91 suggested
that the residual motion might be due to very large-scale, low
amplitude density fluctuations, but noted that at the largest
distances in his sample (~ 10,000
km s^{-1}), objects appeared to be at rest in the CMB frame.

The small solid angle covered by the W91 PP sample raised the question
of whether the PP inflow extended over a larger region of the sky.
Confirmation that it did was provided by the R band CCD TF survey of
Courteau and Faber
(Courteau 1992
;
Courteau et al. 1993
;
CF), who
obtained data for 321 galaxies distributed over the Northern sky using
a dilute sampling approach. CF used optical
H rotation curves
to measure velocity widths. Because of their large sky coverage and
very uniform selection criteria, CF were able to compare different
regions of the sky reliably. Well away from PP their data suggested a
relatively quiescent velocity field, as indicated by the small scatter
of the TF relation when distances were modeled by pure Hubble
flow. However, within the PP region - now defined to cover an area
several times larger than the W91 sample - the scatter was larger and
the residuals systematically offset from Hubble flow. The implied
velocity was ~ 300-350
km s^{-1} toward the LG, in agreement with
W91. Further confirmation of a bulk motion came from
Han & Mould (1992)
,
who used the I band CCD TF relation to study five clusters
embedded in the PP supercluster. They found PP to be moving towards
the LG with amplitude ~ 400 *km* *s*^{-1}.

Han, Mould and collaborators also obtained I band TF data for 16 further clusters distributed over much of the sky (Han 1992 ; Mould et al. 1991). This sample of clusters largely overlapped those studied using the IRTF by Aaronson et al. (1986, 1989), but the I band relation was considered more accurate. Han & Mould (1992) fitted the combined data for 21 clusters to two models: a large-scale bulk streaming similar to the original model proposed by the 7S group (Dressler et al. 1987a); and a "bi-infall" model in which the local universe is dominated by competing gravitational pulls of the GA and Perseus-Pisces. They found that both models were consistent with the data.

Connolly (1993)
measured Tully-Fisher parameters for a sample of 218
Sc galaxies in both Galactic hemispheres, with mean redshift of 5500
km s^{-1}. He fits the data to a dipole and finds good
agreement with the analysis of a much larger sample by
Courteau et al. (1993)
.

The largest single contribution of Tully-Fisher data in recent years
has been that of Mathewson and collaborators
(Mathewson et
al. 1992a,
b,
hereafter MAT92;
Mathewson & Ford
1994).
Using I band CCD photometry and a combination of
H*I* and optical velocity widths from several
telescopes in Australia, MAT92 obtained TF data for 1355 Southern sky
galaxies. Roughly half of these were in the steradian around the GA
region as defined by the 7S group, and another half, thought of as a
control group, were outside. The sample was constructed to be roughly
complete to an ESO diameter limit of 1.7'. Most objects in the
sample have redshifts
4000 *km*
*s*^{-1}, but a substantial number have
5000 *cz*
7000 *km*
*s*^{-1}. MAT92 calibrated their TF
relation relative to ~ 40 galaxies with TF data from
Aaronson et al. (1989)
,
Willick (1991)
, and
Han & Mould (1992)
.

The redshift vs. TF distance diagram for galaxies in the "control"
region exhibited pure Hubble flow in the mean. Within the GA region,
MAT92 confirmed the large (~ 1000
km s^{-1}) outflowing motions in the GA
foreground that had been detected in the 7S elliptical data. However,
the spirals did not appear to trace a velocity field which fell to
zero at the distance (~ 4500 km s^{-1}) of the nominal Great
Attractor. Instead, the data indicated that the GA itself (*i.e.*,
galaxies in the redshift range ~ 4000-5000 km s^{-1}) possessed
a mean positive peculiar velocity of
~ 600 *km* *s*^{-1}. Furthermore, the
MAT92 data did not exhibit the "backside infall" which
Dressler & Faber (1990ab)
had claimed to detect: even at distances
000 *km*
*s*^{-1} a return to the Hubble line was not
apparent. MAT92 argued that their data, in combination with those of
Willick (1990)
,
suggested a vast bulk flow in the Supergalactic plane on scales well
in excess of 10,000 km s^{-1}.

However, a recent reanalysis of the MAT92 data, as well as further
data from the Mathewson group itself, have not lent support to this
hypothesis. MAT92 did not attempt a careful correction for selection
or Malmquist biases; in addition, their zeropoint calibration used
only a small number of independent comparisons. Using a careful
zeropointing technique which ties together the distance scale of
several TF samples,
Willick et al. (1995a,
b,
c)
have recalibrated the TF
relations used by MAT92 as well as other TF samples (see
Section 7.2). Preliminary analyses of the recalibrated
data
(Courteau et al. 1993
;
Faber et al. 1993
,
1994)
have derived modest (~ 300-350
km s^{-1}) bulk flows in the GA direction, as well as
evidence of a return to the Hubble line at the distance of the GA.
The backside infall, however, remains undetected in these recent
studies. Very recently,
Mathewson & Ford
(1994)
have enlarged their
Southern sky sample by ~ 1000 galaxies. The new sample is selected
to an ESO diameter of 1' and thus probes to greater distances
(*cz* 10, 000
*km* *s*^{-1}). Mathewson & Ford measure a bulk
flow in the GA direction of ~ 370
km s^{-1} for the combined sample, in good agreement with
Courteau et al. (1993)
,
and a return to the Hubble line at
*cz* 7000 *km*
*s*^{-1}. The new data have not been reanalyzed using
the techniques of Willick et al.; this will be carried out in the near
future.

**7.1.4. The Lauer-Postman Result**

The most dramatic apparent detection of very large-scale streaming to
date comes from the work of Lauer & Postman
(1994,
hereafter
LP). Using the BCG *L*-
relation (Section 6.3.2), LP
obtained redshift-independent distances to a volume-limited sample of
119 Abell clusters in the redshift range
*cz* < 15, 000 *km* *s*^{-1}. The
typical distance error from the method was estimated by LP as
~ 16%, so that the peculiar velocity error for
a typical sample cluster is ~ 2000
km s^{-1}, much too large to estimate
individual cluster peculiar velocities. However, the data do allow the
determination of a mean bulk velocity vector for the sample.

LP defined an "Abell Cluster Inertial Frame," or ACIF, and measured
the velocity of the LG **L** relative to it. They solved for
**L** by minimizing the
^{2}-like
quantity ^{(34)}

(200) |

where:

_{i}is the -value for galaxy*i*;*z*_{i}its redshift in the LG frame;*M*(_{i}) is its residual (in magnitudes) from the*L*- relation;-
_{i}is a unit vector toward the galaxy; and **L**is the motion of the LG with respect to the ACIF.

The minimization is carried out with respect to the parameters
specifying the *L*-
relation (which determine the quantity
*M*()), and the three components of
**L**. The quantity
(2 - ) which appears both in
the weighting term and in
the exponent arises from the fact that both *L* and
must be
measured within *metric* apertures; as a result the effective
scatter of the relation is actually given by
2_{M} /
(2 - ) where
_{M} = 0.24 mag is
the scatter about the relation. The
*z*^{-2} term gives greatest weight to the nearby clusters,
although LP show the solution is insensitive to the redshift weighting.

LP found that the LG motion relative to the ACIF was in the direction
*l* = 220°, *b* = - 28° (total angular uncertainty
± 27°). That the LG would have a significant motion relative to
an inertial rest frame is of course no surprise; we already know that
the LG moves at ~ 600
km s^{-1} relative to the CMB. However, the latter
motion is in the direction *l* = 276°,
*b* = 30° (with
negligible angular uncertainty). Consequently, according to LP the
ACIF differs significantly from the frame defined by the CMB
dipole. Stated another way, if the CMB defines the true cosmic rest
frame, the LP data indicate that the Abell clusters out to 15,000
km s^{-1} possess a net peculiar motion of
689 ± 178 *km* *s*^{-1} in the direction
*l* = 343°, *b* = + 52 (total angular uncertainty
± 23°). Aware of the controversial nature of their result, LP
performed a suite of tests aimed at identifying systematic effects in
their analysis. Among other things, they considered the effects of
cutting their sample at low and high redshifts, and at low and high
values of , of removing
prominent superclusters (such as the
Hydra-Centaurus complex) from their sample, of using a linear rather
than a quadratic *L*-
relation, and of removing galaxies with
very large residuals. They found that none of these tests produced
significant changes in their solution. They did identify one
non-negligible systematic effect called "geometry bias," which
arises due to coupling between the dipole moment of the sky
distribution of the sample and the distance indicator relation. LP
calibrated and corrected for this effect using Monte Carlo
experiments; in practice, it was rather small (< 100 *km*
*s*^{-1}) since their sample has excellent sky coverage.

One could take the view that a ~ 700
km s^{-1} bulk motion of a volume ~ 30,000
km s^{-1} in diameter is highly implausible, and attribute it
instead to an incorrect choice of cosmic rest frame. One would then be
obliged to jettison the conventional view that the CMB dipole is
kinematic in origin; possible explanations of a *cosmological*
origin for the CMB dipole have been suggested by
Gunn (1988)
,
Paczynski & Piran
(1990)
,
and Turner (1991)
.
While the
possibility that the CMB does not define a cosmic rest frame cannot be
excluded, such a view has become increasingly difficult to uphold in
light of the discovery of small but measurable CMB anisotropies by the
COBE satellite (Smoot
et al. 1992). These COBE measurements have
corroborated most theoretical ideas about the nature of the CMB.

Alternatively, if one accepts the kinematic origin of the CMB dipole
and the resultant CMB-defined rest frame, the LP data indicate a ~
5% perturbation to the Hubble flow on a 15,000
km s^{-1} scale. This is
difficult to reconcile with bulk flow observations from the TF and
*D*_{n}-
relations.
Courteau et al. (1993)
used the Mark III dataset
(Section 7.2) to estimate the bulk flow within 6000
km s^{-1} to have an amplitude of
360 ± 40 *km* *s*^{-1} in a nearly orthogonal
direction, toward
300°,
*b* + 10°. It is
difficult to understand how the amplitude of the bulk flow could grow
with scale in a universe which approaches uniformity in the galaxy
distribution in the large, especially given the apparent convergence
of the dipole in the galaxy distribution between 6000 and 15,000
km s^{-1}
(Section 5.7).

The LP result also poses severe problems for a number of currently popular theories for the growth of large-scale structure. Strauss et al. (1995) have generated mock LP-like BCG samples in N-body simulations based on six different cosmogonic scenarios, including CDM, HDM, and PIB models. These mock samples were constructed to mimic all the observational errors in the actual LP sample. Strauss et al. found that no more than 2-5% of the realizations of any given model exhibited an apparent flow as large as that measured by LP. Similar results were found by Feldman & Watkins (1994) using an analytic approach. Jaffe & Kaiser (1995) suggest comparing not just the bulk flow, but the full sample of peculiar velocities, to the predictions of various models. They find better agreement between the LP data and models than do Strauss et al. and Feldman & Watkins. However, as Jaffe & Kaiser ignore the correlations between the measured peculiar velocities of different clusters, which arise from the self-calibrating nature of the LP sample, the validity of their analysis is uncertain.

The very careful treatment by LP makes it unlikely that the observed flow will "go away" with a future reanalysis of the data. Rather, it seems likely that future measurements on comparably large scales (Section 9.7) will determine the ultimate evaluation of LP, as was the case with the Rubin et al. (1976a, b) result. If LP is confirmed, it will need to be reconciled with the discrepant flows seen on smaller scales, and theoretical models are likely to require significant revision.

The myriad of bulk flow analyses we have discussed here have used a
wide variety of samples, methodologies and techniques, and have been
carried out during a time when our knowledge of the properties of
distance indicators, and the various biases which plague them, was
evolving rapidly (as indeed it still is). The bulk flows measured by
different samples necessarily probe different volumes of space; a
direct comparison between them is problematic, and we have not
attempted one in this review. Nevertheless, there is a rough consensus
between the different workers in the field as to the nature of the
bulk flow within 6000 km s^{-1} (cf., the review by
Postman 1995),
both in its amplitude and in its direction. Even the Lauer-Postman bulk
flow, which is on appreciably larger scales than the others, agrees very
roughly in direction. The surprising thing about this result is the
fact that it also agrees roughly in amplitude, on such a large scale.

One of our aims in measuring bulk flows is to compare with theoretical
predictions, as given for a given model by Eq. (40).
This equation refers to the *volume-weighted* average of the
three-dimensional velocity of particles within a spherical volume,
which can differ substantially from the *galaxy- and
error-weighted* average of radial velocities that is usually
measured. Indeed, as
Juszkiewicz et al. (1990)
and Strauss et
al. (1995)
point out, one can have a bulk flow within a volume even in the
absence of density fluctuations on larger scales, if the center of
mass of the volume does not coincide with its geometric center. There
are a number of such subtleties which are only now being appreciated
in this game; further theoretical analyses along the lines of
Kaiser (1988;
1991) and
Feldman & Watkins
(1994)
are needed.

Of the samples discussed here, only that of
Lauer & Postman
(1994)
approached being volume-limited (cf.,
Roth 1993
,
1994).
However, one
can measure an effective volume-limited bulk flow if one weights the
galaxies appropriately. The POTENT technique (to be discussed in
detail in Section 7.5) does exactly that, and
Bertschinger et
al. (1990)
quote volume-limited bulk flows measured within a series
of spheres centered on the Local Group. More recent results using
POTENT applied to a more complete data set by
Dekel et al. (1995)
show a bulk flow of 400
km s^{-1} within a sphere of radius 6000 km s^{-1},
which is in fact consistent with standard CDM normalized to COBE.
Again, the Lauer-Postman result, which probes appreciably larger
scales, is in disagreement with this and all other viable models;
we suspect that measurements of bulk flows on large scales will
continue to be a very active area of research for at least the next
five years.

**7.2. Homogeneous Peculiar Velocity Catalogs**

Tully-Fisher and
*D*_{n}- data
have been collected, at an increasing
rate, over the past fifteen years. There now exists a large number of
independent data sets potentially useful in peculiar velocity surveys.
This trend will accelerate in the coming years, as most ordinary
spiral and elliptical galaxies out to ~ 10,000
km s^{-1} acquire quality
photometric and spectroscopic measurements. There is and will be a
need for uniform catalogs containing the TF and
*D*_{n}- observable
data, and the corresponding estimated distances and peculiar
velocities. Distances obtained using the newer DIs such as the PNLF,
SBF, BCG, and SNe methods will eventually be cataloged as well.

The greatest problem in constructing a peculiar velocity catalog is
ensuring the *homogeneity* of the separate samples. The originally
published inferred distances to galaxies found in two or more
independent data sets are almost sure to differ systematically, for
two basic reasons. First, observers measure the relevant
quantities - apparent magnitudes, diameters, and internal velocity
widths - differently; they not only use different observational
methods, but apply different corrections (*e.g.* for internal
extinction) to the raw observables. As a result, there are as many TF
relations (say) as there are data acquisition and reduction
techniques. Nonetheless, a TF relation derived for one data set is
sometimes heedlessly applied to another. Second, observers determine
the appropriate TF relation (say) from their data in different ways.
Some fit an inverse relation, some a forward relation; in each case,
the treatment (if any) of selection bias can differ. A naïvely
constructed peculiar velocity catalog, in which inferred distances are
simply appropriated from the published literature, will thus be
plagued with systematic errors within and between data sets.

Until recently, attempts to produce uniform peculiar velocity catalogs
from the literature have been largely the work of D. Burstein, a
member of the 7S collaboration. Two early versions known as the Mark I
and Mark II, were distributed privately to the community by electronic
mail in 1987 and 1989 respectively. Burstein tabulated the
*D*_{n}-
elliptical galaxy data of the 7S group,
Lucey & Carter
(1988)
and
Dressler & Faber
(1990a)
,
as well as the infrared TF spiral data of
the Aaronson group. Burstein was able to make direct comparisons of the
*D*_{n}- data
from the different groups. However, it was
difficult to guarantee that the spiral and elliptical data were on the
same system; comparison of TF and
*D*_{n}-
*group* distances was
used to make small adjustments, but these comparisons suffered from
the inevitable uncertainty associated with the spatial segregation of
spiral and elliptical galaxies. Although not optimal, the Mark II
catalog played an important role in some of the major statistical
studies of peculiar velocity in recent years (e.g.,
Bertschinger et al. 1990
;
Dekel et al. 1993).

With the advent of new large samples of peculiar velocity data, there
has been a strong need to extend Burstein's effort, particularly in
view of the more complete understanding of bias effects acquired in
recent years (Section 6.5). A group led
by one of the present
authors (J.W.) has recently completed this task. The basic
methodology is described in
Willick et al. (1995a,
b);
the *Mark III
Catalog of Galaxy Peculiar Velocities* is presented in
Willick et al. 1995c,
and is available for electronic distribution as
well. The Mark III consists of the CCD TF samples of Han, Mould, and
collaborators
(Han 1991
,
1992;
Han & Mould 1992
;
Mould et al. 1991,
1993;
collectively, HM);
Willick (1991,
W91);
Courteau and Faber
(Courteau 1992
;
CF); and
Mathewson et al. 1992b;
and of the infrared TF sample of
Aaronson et al. (1982b)
,
as recently reanalyzed by Tormen & Burstein
(1994,
1995),
who corrected previously unrecognized
systematic errors in the diameter system used by the Aaronson group
(see Section 6.1.1). The Mark III
catalog will also include expanded versions of the elliptical
*D*_{n}-
samples found in the Mark II.

Several basic principles guide the construction of the Mark III
catalog. The first is the adoption of a uniform set of rules for
correcting the raw observable quantities, in particular apparent
magnitude and velocity width, for the effects of extinction,
inclination, and redshift. Internal extinction corrections have been a
source of particular controversy in the past (see
Burstein et al. 1995
for a review); the Mark III compilation adopts bandpass-specific
internal extinctions which minimize TF scatter. The second principle
is to re-derive a Tully-Fisher relation for each sample separately,
taking full account of and correcting for selection biases in the
manner suggested by
Willick (1994;
cf. Section 6.5). An
important result of this re-derivation is that the TF relations in the
Mark III compilation can differ markedly from those of the original
authors. Third, the zeropoints (the constant *A* in
Eq. 152) of the individual TF relations are mutually
adjusted to yield the maximum agreement in the inferred distances for
several hundred galaxies which are common to two or more individual
samples. Finally, the global catalog zeropoint is determined by
assuming that in the sample as a whole, the volume-weighted radial
peculiar velocity component vanishes in the mean.

A representation of the Mark III velocity field is shown in
Fig. 16, which shows the measured radial peculiar
velocities of all galaxies within 22.5° of the Supergalactic
plane. The point is drawn at the measured distance of the galaxy,
while the line is drawn to its redshift, in the CMB rest frame.
Positive peculiar velocities are drawn with solid points and solid
lines, while negative peculiar velocities use open points and dashed
lines. When possible, galaxies are grouped in order to decrease the
errors and reduce the Malmquist bias; points representing groups of
more than three galaxies are drawn somewhat larger. The zone of
avoidance is apparent as the missing wedge out of the middle of the
figure. Compare this figure with the *IRAS* density field of
Fig. 6; we have labeled the major
structures as we did in that figure. The bulk flow into the GA is
apparent, as is the coherence of the flow even back to the
Pisces-Perseus supercluster. It is difficult, however, to draw
quantitative conclusions from this figure alone about bulk flows or
infalls into specific structures. In the remainder of this chapter, we
discuss various statistical analyses of peculiar velocity data,
culminating in reconstruction methods of the full three-dimensional
velocity field in Section 7.5.

**7.3. Velocity Correlation Function**

One of the most striking features of the observed large-scale velocity
field is its coherence. One way to quantify this is by fitting bulk
flows to the data (Section 7.1). Another approach was
suggested by
Górski (1988)
:
the correlation function of the velocity
field. Górski defines the correlation function between the *i*th
and *j*th Cartesian component of the velocity field as:

(201) |

where
_{} and
_{||} are the transverse
and radial
correlation functions. The second equality holds if the velocity field
is homogeneous and isotropic. If the velocity field is derivable from
a potential, then these two are not independent:

(202) |

In linear perturbation theory, one can derive simple expressions for these quantities:

(203) |

and

(204) |

compare with Eq. (40). Of course, we observe only the
radial component of the velocity field.
Górski et
al. (1989)
and
Groth, Juszkiewicz,
& Ostriker (1989)
have suggested methods to
determine the velocity correlation function from observational data. The
first of these papers defines a quantity
from a sample with
radial peculiar velocities *u*_{i}:

(205) |

where the sum is over all pairs of galaxies with separations between
*r* and *r* + *r*, and
_{12} is the angle
between galaxies 1
and 2 on the sky. A uniformly selected full-sky sample exhibiting a
bulk flow of amplitude *v* will have a velocity correlation function
_{1} =
*v*^{2} / 3. The second equality of Eq. (201)
implies that (*r*) is a
linear combination of
_{||}(*r*) and
_{}(*r*), with coefficients
depending on the spatial distribution of galaxies in the sample.

The velocity correlation function at zero lag is a measure of the
root-mean-square velocity dispersion of galaxies, while the scale on
which it drops to zero is a measure of the coherence length of the
velocity field.
Górski et
al. (1989)
looked at two datasets: the spiral galaxies of
Aaronson et al. (1982)
and the elliptical galaxies of the 7 Samurai. The correlation function
of both drop to zero
at separations of 2000 km s^{-1}. The amplitude of the elliptical
galaxy dataset has a much larger amplitude than that of the spirals,
although this amplitude was not robust: deleting a small number of
galaxies in the Great Attractor region caused the amplitude to drop by
more than a factor of two. These results were
compared to *N*-body simulations of CDM and PBI models. The CDM models
fit the observed correlation length well, and required a normalization
_{8} > 0.5 to
match the amplitude. The PBI models tended to show
*more* coherence than is seen in the real data.

Groth et al. (1989)
used the linear relation between
and
_{},
_{||} to solve for the
latter. They emphasized the
fact that in the frame comoving with the bulk flow of the
Aaronson et al. (1982)
data, the velocity correlations were essentially zero;
the flow was very cold. This is consistent with the small value of the
pairwise velocity dispersion deduced from redshift surveys
(Section 5.2.1). Seen in this light,
the correlation length of 2000 km s^{-1} reported by
Górski et
al. (1989)
is probably at least
partly an edge effect due to a finite sample. It is time to revisit
the velocity correlation function now that the data samples have
improved and expanded. More work is needed to characterize the effects
of survey geometry, and in particular, peculiar velocity errors, on
this statistic. It has the potential to place strong
constraints on cosmological models, once these various effects are
understood better.

Juszkiewicz & Yahil (1989) point out that the comparison of the velocity and the spatial correlation function yields an estimate of ; in particular, in linear theory:

(206) |

Note that the velocity correlations in this form on scale *r* depend
only on the correlation functions on scales smaller than *r*. Thus
accurate measurements of the velocity correlation function and the
spatial correlation function have the potential to yield a measurement
of . Moreover, the extent to
which the two sides of
Eq. (206) agree with one another as a function of *r*
on linear scales is a test of gravitational instability
theory. Unfortunately, existing data are not yet at the stage to allow
this test to be done.

Ostriker & Suto (1990),
struck by the coldness of the velocity field observed in the
Aaronson et al. (1982)
data, suggested a new statistic
to quantify this coldness: the Cosmic Mach Number. The Mach number in
standard usage is the ratio of the flow velocity in some medium to the
sound velocity in that medium. In the cosmological context, the
equivalent of the sound velocity is the small-scale velocity
dispersion of galaxies. Following Eq. (40), the
characteristic bulk velocities in a given cosmological model measure
the large-scale component of the power spectrum, while the small-scale
velocity dispersion depends on the power on small scales. Their ratio
is thus independent of the *amplitude* of the power spectrum (at
least in linear theory), and is a diagnostic of its *shape*.
Strauss, Cen, & Ostriker
(1993;
cf.
Suto, Cen, &
Ostriker 1992)
fit bulk flows to observed datasets; all components of the
velocity field on smaller scales were attributed to incoherent
small-scale peculiar velocities. Subtracting off the estimated errors
in quadrature from the rms of the residuals allowed them to define a
small-scale velocity dispersion, and thus a Mach number.
Strauss et al. (1993)
compared the Mach number results from three different
datasets to the distribution of Mach numbers observed in Monte-Carlo
simulations of the observational data. The
Aaronson et al. (1982)
sample, with its relatively small errors, gave the strongest
constraints on models; standard CDM was ruled out at the 95%
confidence level by this statistic. This is largely a consequence of
the fact that it greatly over-predicts the velocity dispersion on small
scales, as was discussed in
Section 5.2.1. Other models
with less power on small scales relative to large, including tilted
CDM and HDM, fared much better by this statistic.

**7.5. Reconstructing the Three-Dimensional Velocity
Field**

The bulk flows discussed in Section 7.1 represent one
quantitative statistic that can be extracted from observations of
peculiar velocities. One would like a method to characterize all the
information available in the velocity field. In particular, given
gravitational instability theory (Eq. 30 or quasi-linear
extensions thereof), the observed velocity field gives a measure of
the gravitating density field, independent of any assumptions about
the relative distribution of galaxies and dark matter.
Bertschinger & Dekel
(1989)
have developed a technique they call *POTENT* (cf.
Dekel 1994
for a review) which starts from the basic
assumption that the observed velocity field is derivable from a
potential (**r**) such that

(207) |

This is valid to the extent that the velocity field is curl-free;
Kelvin's circulation theorem implies that vorticity is generated only
in regions of shell-crossing. Moreover, initial vorticity decays in an
expanding universe just as do initial peculiar velocities. Thus we
expect that at the present time, if we
smooth on large enough scales, the vorticity is likely to
be negligible. In this case, the radial component of the velocity field
(which is all that is observable) determines the full
three-dimensional velocity field. If
*u*(*r*, ,
) is the
observed radial velocity field, then the potential (normalized to zero
at the origin) is given by:

(208) |

differentiation via Eq. (207) then yields the full three-dimensional velocity field. Indeed, there is no reason to restrict the integration in Eq. (208) to radial rays; Simmons et al. (1994) discuss optimal integration paths for recovering the potential.

Given the three-dimensional velocity field, linear theory gives a simple relation to the density field (Eq. 30). The POTENT method as currently implemented uses a non-linear generalization of this following Nusser et al. (1991) :

(209) |

where ** I** is the unit matrix; Eq. (209) reduces to
^{.} **v** in the
linear limit.

Of course, we do not observe a radial velocity field *u*(**r**);
we have noisy data for a non-uniform and sparsely sampled set
of galaxies. Thus another crucial part of the POTENT technique
involves turning the data available into a continuous radial velocity
field. This smoothing has several features:

- It allows the radial velocity field to be defined at every point in space;
- It smoothes over small-scale vorticity in the velocity field due to shell-crossing;
- It smoothes over very non-linear effects that are not modeled by Eq. (209);
- It averages together the very noisy individual peculiar velocity measurements.

Dekel, Bertschinger, & Faber (1990) use a tensor window function smoothing that takes into account the radial nature of the observed velocity field. There are three particularly pernicious sources of error in the resulting smoothed velocity field:

- Statistical noise in the velocity field due to the errors in the individual peculiar velocities. One minimizes this noise by weighting each galaxy by the inverse square errors;
- Malmquist bias, both homogeneous and inhomogeneous, resulting from the peculiar velocity errors (Section 6.5.2); POTENT uses peculiar velocities derived via Method I applied to forward DIs;
- Sampling gradient bias, due to the inhomogeneous sampling of the velocity field within a smoothing window. This is minimized using equal volume weighting.

Note that the effects of items (i) and (iii) are minimized with
different weightings; one can never minimize both simultaneously.
In practice, the effects of these sources of noise and biases for any
given dataset and smoothing scheme are calculated using extensive
Monte-Carlo simulations. First results of the POTENT technique are
presented in
Bertschinger et
al. (1990)
,
using the Mark II data. The resulting
density field, smoothed with a 1200
km s^{-1} Gaussian window, clearly
shows the Great Attractor, and the void in the foreground of the
Pisces-Perseus supercluster. The smoothed velocity field and resulting
density field as gained from the Mark III data
()
are shown in Fig. 17.

The POTENT method and results have been used for a number of other studies, now using the Mark III data described in Section 7.2. We have already referred to its use to trace the density field at low Galactic latitudes (Section 3.8), and to compare the velocity fields as traced by elliptical and spiral galaxies separately (Section 6.2). Seljak & Bertschinger (1994) have used a maximum-likelihood method to fit the amplitude of fluctuations in the Mark II POTENT density field, assuming a given power spectrum. The analysis is complicated by the fact that data points are coupled not just by real correlations, but also by noise, necessitating a full Monte-Carlo approach to the covariance matrix. A summary of their results for a range of plausible CDM-like models is

(210) |

where
_{8, v} is the rms
value of ^{.} **v** in
spheres of radius 8 *h*^{-1} Mpc. This result is in good
agreement with standard CDM, normalized to the COBE quadrupole.

**7.5.1. The Initial Density Distribution
Function**

In Section 5.2.2, we described a
method developed by
Nusser & Dekel
(1993)
which uses the Zel'dovich approximation to reconstruct
the initial density distribution function from the eigenvalues of the
spatial derivatives of the velocity field. It can be shown that this
reconstruction is independent of
_{0} when
reconstructing from
the velocity field *predicted* from a redshift survey, while the
reconstruction from the *observed* velocity field is
_{0}-dependent; in
particular, the shape of the derived initial
density distribution function will depend on the value of
_{0} used.
Nusser & Dekel
(1993)
carried out a preliminary analysis of the
Mark II POTENT velocity field using this technique, and found that
they needed values of
_{0} close to unity
in order to match the
Gaussian distribution function seen in the reconstruction from the
density field. That is, for small values of
_{0}, the
distribution function reconstructed from the observed density and
velocity fields did not agree at all.
Nusser & Dekel
(1993)
used this approach to rule out
_{0} < 0.3 models
at the 4-6
confidence level; the data are consistent with
_{0} = 1. Further
tests are needed, however, to test whether very low
_{0} models,
in which the derived errors become large, can be ruled out as
well. The Mark III data should give much superior results.

**7.5.2. Higher-Order Moments of the Velocity
Field**

In linear perturbation theory with Gaussian initial conditions, we saw
that the density field showed a Gaussian distribution. Given the
direct proportionality between
and
^{.} **v** (Eq. 30),
will also have a Gaussian
distribution. However, just as the distribution of
develops
skewness in second-order perturbation theory, so does
.
Bernardeau (1994a,
b;
cf. Bouchet et al. 1993)
calculates the
higher-order moments of in
perturbation theory for top-hat smoothing (cf.
Bernardeau et al. 1994
and Lokas et al. 1994
for Gaussian smoothing), and shows that the ratio of the skewness to the
variance squared is given by:

(211) |

(compare with Eq. 124; like that result, this is
valid only in the range
-3
_{1} <
1). Unlike the case of the
density field, the skewness of the velocity field is strongly
dependent on _{0}.
Unlike the density-velocity comparisons
discussed below, galaxy biasing does not affect the results (at least
for equal volume weighting).
Bernardeau et al. (1994)
have made a
tentative measurement of the skewness of the POTENT density field from
the Mark III data, and conclude that the data are consistent with
_{0} = 1, with
_{0} = 0.3 ruled out
at the 2
level. Because the skewness is so heavily weighted by the tails of the
distribution, and the substantial sources of errors and biases in
POTENT will strongly affect these tails, this result remains
tentative, and will require extensive Monte-Carlo simulations to test
it thoroughly. Higher-order moments remain unmeasurable from current data.

**7.5.3. Voids in the Reconstructed Density
Field**

The dimensionless density field
has a firm lower limit, -1,
corresponding to the absence of matter. In linear theory,
is
proportional to the divergence of the velocity field
, which
says that also has a lower
limit, depending only on the proportionality constant
*f* (_{0}).
Comparison with lowest observed
point in the POTENT maps thus puts a lower limit on
*f* (_{0}), again
independent of galaxy biasing.
Dekel & Rees (1994)
have applied this
idea to the Sculptor Void seen in the Mark III POTENT data (cf.,
Fig. 6), and find that
_{0} > 0.3 at the
2.4 level. The systematics
of this method have not yet been
properly treated, however. In particular, as there is a strong
correlation between
_{galaxies} and
, the
voids that one wants to use for this test are in those regions where
there are fewest galaxies, and thus the noise in the POTENT maps are
highest. In addition, these are regions which potentially suffer from
strong inhomogeneous Malmquist bias, precisely because the galaxy
density field shows strong gradients in voids.

**7.5.4. Other Approaches to Reconstructing the
Velocity Field**

The POTENT approach can be thought of as a parameterized fit to the velocity field, in which the velocity field in each smoothing volume is fit to a bulk flow (plus shear terms; cf. the discussion in Dekel et al. 1994). An alternative approach is to expand the velocity field in Fourier modes. Kaiser & Stebbins (1991) and Stebbins (1994) use a Bayesian approach, regularizing their solution for the Fourier coefficients by assuming that the velocities are drawn from a Gaussian distribution with a given power spectrum. Indeed, this regularization is equivalent to applying a Wiener filter to the data, and thus has the same feature as we saw above in Section 3.7: the derived density field goes to zero in regions of poor data. The results they get are a strong function of the power spectrum assumed; we await a detailed exposition of their technique in refereed journals.

As a Method I technique (Table 3)
the POTENT method has the serious
drawback that it assumes an *a priori* distance indicator
relation, calibrated independently of the data in question. A small
error in the slope of the Tully-Fisher relation, for example, will
cause systematic errors in the derived velocity field. An alternative
approach is to solve simultaneously for the parameters of the distance
indicator relation and the velocity field, in a Method II approach.
In the first paper that takes into account all the selection effects
in such a problem,
Han & Mould (1990)
fit the
Aaronson et al. (1982)
peculiar velocity data to a model involving infall into
the Virgo cluster and the Great Attractor (cf.,
Faber & Burstein
1988).
We describe a generalization of their technique in
Section 8.1.3 using the *IRAS* predicted
velocity field.

Nusser & Davis
(1994b)
suggest an expansion of the radial velocity
field in spherical harmonics and radial spherical Bessel
functions. They find linear combinations of the basis functions that
are orthonormal at the positions of the galaxies for which data exist,
allowing them to find an analytic solution for the Tully-Fisher
parameters and the coefficients of these orthonormalized functions
*as measured in redshift space* by minimizing the scatter in the
inverse Tully-Fisher relation (thereby eliminating selection bias).
Small-scale noise and triple-valued zones eliminate the one-to-one
mapping between real space and redshift space, causing a bias in the
derived velocity field, although this seems to be a small effect with
real data. This offers an alternative method to smooth peculiar
velocity data, and may be the ideal way to compare with the predicted
velocity field from redshift surveys using the
Nusser & Davis
(1994a)
approach (cf.,
Davis & Nusser
1995).
The applicability of the derived
velocity field to points other than those where data exist remains
unclear, and the method is limited to data characterized by a single
distance indicator relation (or at least is analytic only in this
limit). These problems are not fatal by any means, and this method
holds great promise.

^{31} On large scales, these flows are
referred to as as
"large-scale streaming motions," "large-scale flow," "bulk
flow," among other terms. We use all these terms interchangeably
here.
Back.

^{32} Willick (unpublished) has shown,
however, that the
limited sky coverage of the Aaronson et al. sample means that its
ruling out of large-scale bulk flows is not definitive.
Back.

^{33} The very small Virgo infall amplitude
found by FB88 remains controversial.
Tonry et al. (1992)
have applied
the SBF technique to local ellipticals and found an infall velocity of
340 ± 80 *km* *s*^{-1} at the LG. Work on the
subject will undoubtedly
continue in the coming years.
Back.

^{34}
Colless (1995)
derives a more accurate version of Eq. (200); his reanalysis
of their data gives results differing only slightly from those of
Lauer & Postman
(1994) .
Back.