We have described a new maximum likelihood method, VELMOD, for comparing Tully-Fisher data with predicted peculiar velocity fields from redshift surveys. We implemented the method for a czLG 3000 km s-1 TF subsample from the Mark III catalog (Willick et al. 1997) and velocity fields predicted from the 1.2 Jy IRAS redshift survey (Fisher et al. 1995). The velocity field prediction is dependent on the value of I 0.6 / bI, where bI is the bias parameter for IRAS galaxies at 300 km s-1 Gaussian smoothing. We maximized likelihood with respect to I, the parameters of the TF relation, and several other velocity parameters.
We applied our method to 20 mock Mark III and IRAS catalogs constructed to mimic the properties of the real data. The mock catalogs were drawn from an = 1 N-body simulation and were constructed to ensure bI = 1. Thus, the mock catalogs satisfy I = 1. Our VELMOD runs with the 20 mock catalogs returned a mean value of I = 0.984 ± 0.018, consistent with the statement that VELMOD yields an unbiased value of I. In addition, our mock catalog tests enabled us to assign reliable 1 errors to our estimates of I and showed that our other derived parameters, including those of the TF relation and the small-scale velocity noise, are also unbiased. Because the mock catalogs came from an = 1 universe, triple-valued zones in the mock Virgo region were strong but were handled properly by the VELMOD analysis.
When VELMOD was applied to the real Mark III data, a considerably smaller value of I was derived. If we assume that the IRAS-predicted velocity field fully describes the actual one, we obtain I = 0.563 ± 0.074. However, the residuals from this fit were large and coherent; fitting them by a quadrupolar flow gave a maximum likelihood value of I = 0.492 ± 0.068. The quadrupole points toward the Ursa Major cluster and has an rms amplitude of 3.3% of the Hubble flow. In Appendix B, we show analytically that a quadrupole of this amplitude is expected, given the way that we smooth the density field; its presence is not a sign that the IRAS galaxies do not trace the mass responsible for the local flow field. An analysis of the fit residuals demonstrated that the IRAS-predicted peculiar velocity field, with the external quadrupole, provides a statistically acceptable fit to the TF data within 3000 km s-1. The data are thus consistent with the hypothesis that the peculiar velocities are due to the gravitational effects of a mass density field that is proportional to the IRAS galaxy distribution when smoothed on a 300 km s-1 scale, although our analysis does not rule out alternative biasing relations. We also find that the data are consistent with a very quiet flow field; the one-dimensional rms noise in the velocity field relative to the IRAS model is 125 ± 20 km s-1.
The value of I obtained here also may be thought of as a measurement of the rms mass density fluctuations 8 as a function of . Similarly, COBE-normalized CDM power spectra predict a value of 8 as a function of and other cosmological parameters. If we require that the VELMOD and COBE-normalized calculations agree, we can constrain the value of . For scale-invariant, = 0 universes, we derive the constraints 0.28 0.46 for 85 H0 55 km s-1 Mpc-1. For scale-invariant, flat universes, we find 0.16 0.34 for the same range of H0. The constraints on shift to higher values (Section 6.3.2) if the primordial power spectra are "tilted," n < 1, and if tensor fluctuations are present. However, both extreme tilt (n 0.7) and a Hubble constant at the lowest end of the observationally allowed range (H0 50 km s-1 Mpc-1) would be required to reconcile these results with an Einstein-de Sitter universe.
The conclusions of the previous paragraph all rest, of course, on the validity of our measurement of I. Tests with mock catalogs show that, subject to our basic assumptions, this measurement is reliable to within the quoted errors. We have identified two ways these assumptions can break down. First, the effective bias factor bI could depend on scale. In that case, our measurement of I, which reflects a 300 km s-1 Gaussian smoothing scale, might not be the same as a measurement obtained at larger smoothing; it then would not be valid to equate the estimate of 8 obtained from equation (29) with the COBE/CDM prediction. Second, although we have found agreement between the predicted and observed peculiar velocities within 3000 km s-1, DNW found disagreement on larger scales. If the DNW result is validated by future observations (Strauss 1997) aimed at improved TF calibration across the sky, our present claim of TF-IRAS agreement will be undermined.
There are several areas for further work. One, alluded to in several places in this paper, is to extend our analysis to larger redshift, using both the forward and inverse forms of the TF relation. This can be done with both the Mark III data and the extensive new TF (Mathewson & Ford 1994; Giovanelli et al. 1997a, 1997b) and Dn - (Saglia et al. 1997) samples that are being compiled. We also should consider extending this work to other distance indicators; surface brightness fluctuation galaxies (Tonry et al. 1997), with their accurate sampling of the nearby velocity field, are natural candidates for the VELMOD analysis. On the modeling side, this work has left us with several conundrums, the most puzzling of which is why the linear IRAS model does so well with a smoothing scale of 300 km s-1. More work is needed with N-body simulations to understand this. Finally, we will not have a coherent picture of the relationship between the velocity and density fields until we can understand the different values of I obtained by VELMOD and POTIRAS.
We thank Marc Davis, Carlos Frenk, and Amos Yahil for extensive discussions of various aspects of this project, as well as the support of the entire Mark III team: David Burstein, Stéphane Courteau, and Sandra Faber. We also thank the referee, Alan Dressler, for an insightful report that improved the quality of the paper. J. A. W. and M. A. S. are grateful for the hospitality of the Hebrew University in Jerusalem, Lick Observatory at the University of California, Santa Cruz, and the Astronomy Department of the University of Tokyo for visits while we worked on this paper. M. A. S. gratefully acknowledges the support of an Alfred P. Sloan Foundation Fellowship. This work was supported in part by the US National Science Foundation grant PHY-91-06678, the US-Israel Binational Science Foundation grants 92-00355 and 95-00330, and the Israel Science Foundation grant 950/95.