Published in Astrophysical Journal, 486, 629, 1997.

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Maximum Likelihood Comparisons of Tully-Fisher and Redshift Data: Constraints on and Biasing

Jeffrey A. Willick ^a, Michael A. Strauss ^{b, e}, Avishai Dekel ^{c, f}, and Tsafir Kollat ^d

^a Department of Physics, Stanford University, Stanford, CA 94305-4060; jeffw@perseus.stanford.edu
^b Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544-1001; strauss@astro.princeton.edu
^c Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel; dekel@astro.huji.ac.il; and Center for Particle Astrophysics, 301 Le Conte Hall, University of California, Berkeley, CA 94720-3411.
^d Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; and University of California Observatories/Lick Observatory, Santa Cruz, CA 95064; tsafrir@ucolick.org.
^e Alfred P. Sloan Foundation Fellow.
^f Center for Particle Astrophysics, University of California, Berkeley, CA 94720

Abstract. We compare Tully-Fisher (TF) data for 838 galaxies within cz = 3000 km s^-1 from the Mark III catalog with the peculiar velocity and density fields predicted from the 1.2 Jy IRAS redshift survey. Our goal is to test the relation between the galaxy density and velocity fields predicted by gravitational instability theory and linear biasing, and thereby to estimate beta _I ident Omega ^0.6 / b_I, where b_I is the linear bias parameter for IRAS galaxies on a 300 km s^-1 scale. Adopting the IRAS velocity and density fields as a prior model, we maximize the likelihood of the raw TF observables, taking into account the full range of selection effects and properly treating triple-valued zones in the redshift-distance relation. This method is more general and correct than simply minimizing TF residuals with respect to the velocity field model. Extensive tests with realistic, simulated galaxy catalogs demonstrate that the method produces unbiased estimates of beta _I and its error. When we apply the method to the real data, we model the presence of a small but significant velocity quadrupole residual (~ 3.3% of Hubble flow), which we argue is due to density fluctuations incompletely sampled by IRAS. The method then yields a maximum likelihood estimate beta _I = 0.49 ± 0.07 (1 sigma error). We discuss the constraints on Omega and biasing that follow from this estimate of beta _I if we assume a COBE-normalized, cold dark matter power spectrum. Our model also yields the one-dimensional noise in the velocity field, including IRAS prediction errors, which we find to be 125 ± 20 km s^-1.

We define a chi ²-like statistic, chi ², that measures the coherence of residuals between the TF data and the IRAS model. In contrast to maximum likelihood, this statistic can identify poor fits but is relatively insensitive to the best beta _I. As measured by chi ², the IRAS model does not fit the data well without accounting for the residual quadrupole; when the quadrupole is added, the fit is acceptable for 0.3 leq beta _I leq 0.9. We discuss this in view of the Davis, Nusser, & Willick analysis that questions the consistency of the TF data and IRAS-predicted velocity field.