**6.4. Summary**

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 *cz*_{LG}
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} /
*b*_{I},
where *b*_{I} 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 *b*_{I} = 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
*H*_{0}
55 km
s^{-1} Mpc^{-1}. For scale-invariant, flat universes,
we find 0.16
0.34
for the same range of *H*_{0}. 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 (*H*_{0} 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 *b*_{I} 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 *D*_{n} -
(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.