of structure formation. Schlegel *et al*. (6) measure deviations from Hubble flow of only 60 km/s for galaxies within 5 Mpc of the Local Group. Govertano *et al.* (7) show that candidate Local Groups within high-resolution *N*-body models never exhibit such cold local flows, either in O=1 or O=0.3 models. Typical local group candidates in their simulations have much higher local velocity dispersions, including objects with blue-shifts, which are not observed for the nearby galaxies outside our own Local Group.

The small-scale “coldness” of the galaxy distribution is well known by alternative expressions: the high “mach number” of the cosmic velocity field (8) or the thinness of the observed sheets of galaxies in redshift space (9). We want to emphasize that the problems presented by such low amplitude peculiar velocities are real and that they indicate a serious gap in our understanding of structure formation.

On scales greater than 5*h*^{-1} Mpc the deviations from Hubble flow are expected to be smaller than the Hubble flow itself, so that in most regions galaxies are physically expanding from each other. In such a situation it is reasonable to expect the velocity *field* to be largely irrotational. On slightly larger scales, one can expect linear perturbation theory to be a reasonably accurate guide to the expected peculiar velocity field. This is a fortunate circumstance because it allows comparison of the observed deviations from Hubble flow with the flow predicted on the basis of full-sky redshift catalogs of galaxies. The most recently compiled large datasets used in these analyses are the peculiar velocity data of the Mark III consortium (10) and the sample of Tully-Fisher galaxies presented by Giovanelli and colleagues (11). For comparison to the gravity field predicted by the observed galaxy distribution, most recent work has used the full-sky 1.2-Jy IRAS flux limited redshift survey (12), but an optically selected sample of galaxies combined with the IRAS survey has also been used in recent work (J.Baker, M.D., M.Strauss, and O.Lahav, unpublished work).

Recent reviews and methodological details are given by Dekel (13, 14) and by Strauss and Willick (15, 16). Although there are many uses for the peculiar velocity data, the most powerful, model-independent, tests are those that keep phase information. There are two broad categories of tests of this sort, the first of which compares the observed galaxy density field with the mass density field inferred from the divergence of the observed velocity field [also known as POTENT (13, 14)], and the second of which compares the observed velocity field with the gravity field inferred from the galaxy distribution [e.g., the VELMOD analysis (15–17), the inverse Tully-Fisher (ITF) method (18), or the least-action method (19)]. The search for reflex dipole flows [e.g., the local motion relative to brightest cluster galaxies (20) or relative to nearby supernovae (21)] are similar in spirit to the second form of these tests.

The density-density comparison of POTENT is inherently a local comparison, which is a tremendous virtue, because neither the galaxy density field nor the peculiar velocity field are well known at large distance. The velocity-gravity comparison, on the other hand, suffers from coherent errors induced by nonlocality, since the estimated peculiar gravity field at a given point in space is computed by effectively summing over neighbors at all distances, weighted inversely by the square of the distance. Poisson shot noise of the galaxy distribution in one locale therefore generates a coherent error in the gravity field over all space. This nonlocality makes the statistics of the field comparison quite complicated.

It has been argued that this sensitivity to the uncertain far-field galaxy distribution precludes the second class of tests as a useful diagnostic of gravitational instability or as a measure of the cosmic density parameter. For example, comparison of the gravitational dipole for the cosmic microwave background radiation (CMBR) dipole is a difficult, dangerous game (22, 23). The amplitude and direction of the CMBR dipole are rather precisely known, but the coherence of the flow is completely uncertain. How large a region is flowing at 620 km/s along with the Local Group of galaxies? Should a full reflex of this dipole be detectable within 50, 100, or 3,000 Mpc? Without an answer to this question—i.e., an assumption of the nature of the large-scale power spectrum—it is not possible to infer the cosmological density parameter by comparing the CMBR dipole to the gravitational dipole inferred from galaxy catalogs such as the 1.2-Jy IRAS survey.

However, the nonlocality does not apply to all aspects of the velocity-gravity field comparison. Recall Einstein’s gedanken experiment of an observer within an elevator. He cannot distinguish whether he is in an accelerating frame or is stationary in an external gravitational field. If the elevator goes into freefall, he can detect the presence of an external gravitational field only by its tidal influence on the matter within the elevator. Exactly the same considerations apply for the gravitational field estimated from an imperfectly sampled galaxy distribution. A poorly sampled, distant cluster of galaxies will induce coherent errors in the nearby gravity field, but they will be tidal in nature. Working in the Local Group frame of reference rather than the cosmic microwave background frame is conceptually cleaner for this analysis. Recall Newton’s iron sphere theorem, which states that for spherical symmetry, the acceleration of a point at radius *r* is sensitive only to the mass interior to that point. In terms of a spherical harmonic expansion of the external gravity field, the *l*=0 component of the field at radius *r* is insensitive to the mass distribution at *R> r.* The general solution of the Poisson equation for an exterior mass at radius *R* leads to an acceleration proportional to *(r/R)*^{l-}^{1}*/R*^{2}*.* Tidal effects are described by *l*=2, and grow linearly with *r*, as we know well. But we tend to forget that for the dipole term, *l*=1, the external field produces an acceleration independent of *r*, which means that by moving to the freely falling Local Group frame, the *l*=1 gravity field at radius *r* is sensitive only to the mass distribution interior to that point (24). This is a critical point, demonstrating why a comparison of the peculiar velocity field with the gravity field, if limited to the monopole and dipole terms, is effectively a local test that is totally insensitive to the matter distribution at large distances.

The ITF method of Nusser and Davis (24) is ideal for this type of comparison, because it allows expansion of the gravity and velocity fields in terms of whatever functional expansion is most convenient. This method expands the radial component of the peculiar velocity field in terms of a set of orthogonal functions, characterized by coefficients that are determined by minimizing the scatter in the linewidth direction for a uniformly calibrated set of Tully-Fisher data. Details are given by Davis *et al.* (18). The method is merely a filtering tool that can smooth both the gravity and velocity fields to the same resolution, which can be dependent on position; it smoothes the data without binning it, and the derived coefficients of both the gravity and velocity fields, for which one can compute a full covariance matrix, are a complete description of the field to a given resolution. In a typical application, 50–100 coefficients are fit. Tests demonstrate that the method works and recovers the true velocity field with minimal bias. These tests with mock data show that the residual field, the difference of the inferred velocity and gravity fields, has negligible dipole errors, and is dominated by *l*=2, 3 components due to the dilutely sampled exterior mass field.

As previously discussed in detail (18), the comparison of the IRAS gravity field with the Mark III Tully-Fisher data is not nearly as successful. For no value of the density parameter is it possible to eliminate the dipole residual. The comparison of the modal coefficients demonstrates that for no value of the density parameter are the gravity and velocity fields statisti