6.2. Do IRAS and TF Velocity Fields Agree?
An important conclusion of this paper is that the agreement between the predicted and observed peculiar velocity fields is satisfactory (Section 5), as it must be if the resulting estimate of I is to be believed. This agreement is consistent with the hypothesis that the gravitational instability theory describes correctly the relationship between the peculiar velocity and mass density fields. It also suggests that the linear biasing model, equation (2), is a reasonable description of the relative distribution of IRAS galaxies and all gravitating matter Gaussian-smoothed at 300 km s-1.
6.2.1. Comparison with Davis, Nusser, & Willick
DNW reached a different conclusion. Comparing the IRAS and TF velocity fields with a Method II approach (ITF; cf. Section 2.1), DNW found that the fields do not agree at a statistically acceptable level. In particular, a 2 statistic resulting from a mode-by-mode comparison of the IRAS and ITF velocity fields was found to be 100 for 55 dof. (15) DNW argued that the excessive value of their 2 statistic resulted primarily from a dipole in the TF velocity field that grows with scale, a feature not seen in the IRAS predictions. They cautioned that, as a result, their maximum likelihood value of I ~ 0.5 was not necessarily meaningful.
Why do we find agreement between the TF and the IRAS data, while the ITF analysis of DNW did not? We cannot answer this question with assurance, but we can suggest two likely causes of the discrepancy. First, the ITF analysis requires that the raw magnitude and velocity width data of the different samples be placed on a single, uniform system. This was achieved by applying linear transformations to the magnitudes and widths of each sample (Willick et al. 1997). Such a procedure in effect links together the TF zero points of samples that probe different volumes. Any systematic error in matching the data sets will manifest itself in spurious large-scale motions; in particular, the scale-dependent, dipolar flow found by DNW (see, for example, their Figs. 12 and 13) is fully degenerate, with a zero-point error in the relative TF calibrations of southern and northern sky samples. Second, DNW extended their ITF analysis to 6000 km s-1, whereas we have restricted our analysis to czLG 3000 km s-1. In so doing, they (like POTIRAS) incorporated several Mark III TF samples (W91CL, HMCL, W91PP, and CF) not included in the VELMOD analysis. The possible zero-point errors mentioned above could affect mainly those Mark III samples used by DNW but not included here, given the agreement we found between the MAT and A82 distances with the VELMOD calibrations (Section 4.7). Some support for this conjecture comes from the fact that, when limited to the A82 and MAT samples within 3000 km s-1, the ITF velocity field does not exhibit the scale-dependent dipole found by DNW (M. Davis 1996, private communication).
Since we believe that the DNW discrepancy between the IRAS and TF velocity fields may well be a result of systematic errors incurred in matching data sets, an effect to which VELMOD is insensitive, we are inclined to give more weight to our present conclusion that the IRAS-TF agreement is satisfactory. However, if in fact the matching of data sets by DNW is validated by ongoing observations aimed at providing reliable north-south homogenization (cf. Strauss 1997), it will be difficult to escape their conclusion that the predicted and observed velocity fields do not agree on large scales. In that case, it will be necessary to reexamine the conclusions of this paper with regard to the value of I.
6.2.2. The Role of Quadrupole
Our conclusion that the predicted and observed velocity fields agree also depends on the validity of our adopted external quadrupole. Figure 19 shows that only with the quadrupole does our goodness of fit statistic 2 take on acceptable values. We argue in Appendix B that the 3.3% residual quadrupole we see is mostly due to the systematic difference between the true and the Wiener-filtered IRAS density field on large scales. The residual quadrupole in the mock catalogs is appreciably smaller, less than 1%, but this can be understood in terms of the different amount of power on intermediate scales (2 / k 100 h-1 Mpc) in the mock catalog and the real universe. Thus, the presence of the quadrupole residual is not evidence for a breakdown of our assumptions of the gravitational instability theory and linear biasing.
15 Note that unlike this paper, DNW assumed a TF scatter a priori, which allows them to define a goodness of fit directly from their 2 (cf. the discussion in Section 2.2.2). Back.