The observed distribution of galaxies from redshift surveys gives a prediction for the large-scale components of the bulk flow via the integral version of Eq. (6) . In particular, we observe from observations of the dipole anisotropy of the CMB (e.g., ) that the Local Group is moving with a velocity of 627 ± 22 km s-1 towards l = 276°, b = +30° (with 3° errors in each angular coordinate); this is indeed by far the most accurately measured peculiar velocity we know. One can predict this peculiar velocity from the observed galaxy distribution to be:
where (r) is a selection function, to correct for the decrease in density of galaxies as a function of distance in a flux-limited sample, and W(r) is a window function with cutoffs at large and small scales (cf., ). The cutoff is needed at large distance because any flux-limited sample has only finite depth, and therefore the dipole one calculates is missing contributions from large scales (; ; ). Indeed, one might think that the difference between the observed and predicted motion of the Local Group would be a direct measure of large-scale components of the velocity field. Fig. 2 shows the growth of the amplitude and direction of the predicted motion vLG(R) as a function of the redshift R out to which galaxies are included in the sum, for two redshift surveys: the IRAS 1.2 Jy redshift survey (; cf., ), and the Optical Redshift Survey (; ). Interestingly, the two curves have a very different amplitude, which has interesting things to tell us about the relative bias of IRAS and optically-selected galaxies, but discussing that would get us too far afield. For the moment, notice that both curves seem to converge to a constant value (both amplitude and direction) for cz > 4000 km s-1, implying that there is little contribution on larger scales. This in turn would imply that the sphere of radius 4000 km s-1 is at rest.
Figure 2. The amplitude (upper panel) and direction relative to the CMB dipole (lower panel) of the gravitational dipole of two surveys, the IRAS 1.2 Jy survey (solid lines) and the Optical Redshift Survey (dashed lines). Notice the apparent convergence of the dipole in both cases beyond roughly 4000 km s-1 (although the two differ quite a bit in amplitude). Notice also that the ORS is not quite as deep as the IRAS sample, and therefore the dipole calculation is cut off sooner.
Unfortunately, things are not so simple. First, as Juszkiewicz et al.  pointed out, the difference between the true peculiar velocity and vLG(R) depends on the position of the center of mass of the sample out to R:
where vbulk (R) is the quantity we're interested in in the current context, the mean bulk flow of the sphere out to radius R. One can calculate the rms position of the center of mass of a sample given a power spectrum from linear theory; one finds another integral over the power spectrum like Eq. (7), although with a different smoothing kernel. This term is quite small for small values of R, but becomes comparable to the expected rms bulk flow for values of R above 5000 km s-1 or so , and indeed, for the IRAS 1.2 Jy sample, rcenter of mass is of the order of 250 km s-1 for an outer radius of 10,000 km s-1 . More important than this, however, are all the additional effects which cause the quantity in Eq. (12) to differ from the theoretical ideal. Non-linear effects, shot noise, assuming the incorrect value of (which of course we don't know) and the smoothing on small scales all will contribute to the difference between the observed and predicted motion of the Local Group . The most pernicious effect, however, was pointed out by . With a redshift survey, one is measuring the density field in redshift space. However, as Eq. (2) makes clear, this differs from the bulk flow in real space by the effects of peculiar velocities, and to the extent that the peculiar velocity field shows coherence (which of course is what we're trying to get a handle on here), Eq. (13) is systematically biased. In particular, if one's estimate of the velocity of the Local Group itself is off (e.g., if one doesn't correct for the v(0) term in Eq. (2) at all), the positions of all galaxies in the sample are affected in a dipolar way, clearly affecting the predicted motion of the Local Group, and the apparent convergence, or lack thereof, of vLG(R). Strauss et al.  find that with their best correction of the density field for peculiar velocities, the IRAS dipole indeed seems to converge quite nicely, but even then, there is a very intriguing, large contribution to the dipole (albeit at the 2 level) between 17,000 and 20,000 km s-1. It will be very interesting to see whether this contribution remains with the just completed PSCZ survey of IRAS galaxies to 0.6 Jy (cf., Efstathiou, this volume).