The quantity we hoped to get a handle on from the convergence of the density dipole, Eq. (12), is the average peculiar velocity of galaxies within a sphere of radius R centered on the Local Group. One approach is to measure it directly from a full-sky peculiar velocity survey. It is one of the lowest-order statistics one might imagine measuring from a peculiar velocity sample, but it is maximally sensitive to systematic errors in observations between different areas of the sky.
In particular, most peculiar velocity surveys carried out to date have been done over a relatively limited area of the sky. If they are calibrated externally (as they usually are), zero-point differences between the calibrators and the sample will give rise to false bulk flow measurements. Moreover, Malmquist bias can give an artificial signature of outflow .
To avoid these problems, we would like to have measurements of peculiar velocities over the full sky. We have already made reference above to the Mark III dataset of , , , which combines one Dn - , and six Tully-Fisher ; ; ,  (cf., ); ;  peculiar velocity samples. A great deal of work has been done to make these datasets consistent by matching them where they overlap. The bulk flow of the resulting full-sky sample has been calculated by , and more recently by Dekel et al. (in preparation). I describe the latter calculation here.
The peculiar velocity data are noisy and sample the field sparsely and inhomogeneously. The data can be smoothed if one assumes that the velocity field is derivable from a potential; this allows the calculation of a unique three-dimensional velocity field from observations of radial peculiar velocities (the POTENT method; ; ; ; Dekel, this volume). Calculating the bulk flow is then straightforward, and the results are shown in Fig. 3.
Figure 3. Determinations of the bulk flow of galaxies on various scales from the literature. The three panels give the components of the quoted bulk flows along the Galactic X, Y , and Z directions in km s-1 , as a function of the depth of the various surveys. Error bars are as quoted by each paper, and do not take into account the covariance between the different directions (i.e., due to error ellipsoids whose principal axes are not aligned with the Galactic Cartesian directions). Adapted from Postman 1995.
This approach has the advantage that the bulk flow that is calculated is close to the theorist's ideal, the volume-weighted bulk flow. Indeed, the straight fit of individual peculiar velocities in a sample to a bulk flow will not be equivalent to the volume-weighted bulk flow, both because of clustering within the sample (cf., the discussion in ) and because of the increasing peculiar velocity errors with distance (cf., ).
However, measuring a bulk flow on large scales requires tremendous control over systematic photometric errors. Indeed, a 0.10 mag difference in the photometric zero-points of the Mark III sample from one end of the sky to another would translate into an artificial 300 km s-1 bulk flow at 6000 km s-1 from the Local Group. Davis et al.  have carried out a multipole comparison of the Mark III peculiar velocity field with that predicted from the IRAS 1.2 Jy redshift survey, and found that there are indeed discrepancies between the two fields beyond 4000 km s-1 of roughly 300 km s-1 amplitude. It remains unclear whether this is the signature of the gravitational influence of dark matter whose distribution has nothing to do with that of galaxies, a sign that peculiar velocities are not wholly due to the process of gravitational instability, or more prosaically, that there are systematic errors in the Mark III data which are unaccounted for.
In the latter regard, Fig. 3 compares various determinations of the bulk flows of galaxies within spheres centered on us that have been published in the literature. This figure is an updated version of one shown by . Error bars are as given by each author, and do not take into account any misalignment between the error ellipsoids and the the Cartesian axes chosen. The current confused situation is reflected in the large number of non-overlapping error bars in this figure. However, note that the bulk flow within 6000 km s-1 of Dekel et al. (from the POTENT analysis of the Mark III data) and of  (from the POTENT-like analysis of the Giovanelli et al. data; cf., Giovanelli in this volume) are in excellent agreement, despite almost completely independent data (they do share the Mathewson et al.  data in common). It will be very interesting to see if they agree this well shell-by-shell.