Published in Astrophysical Journal, 486, 629, 1997.
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MAXIMUM LIKELIHOOD COMPARISONS OF
TULLY-FISHER AND REDSHIFT DATA: CONSTRAINTS ON
Ω 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
I
0.6 /
bI, where bI 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
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
I
= 0.49 ± 0.07 (1
error). We discuss the constraints on
and
biasing that follow from this estimate of
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
2-like
statistic, 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
I.
As measured
by 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
I
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.
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