One of the most important tasks facing
observational cosmology is determination of the density parameter
.
Along with the Hubble constant *H*_{0} and the cosmological
constant ,
the density parameter fixes the global structure of spacetime. One approach
to the problem uses the classical cosmological tests of the geometry of the
universe, such as the apparent magnitudes as a function of the redshift of
standard candles (e.g., Type Ia supernovae;
Perlmutter et al. 1997).
While promising, this approach is sensitive to the possible evolution of the
standard candles with redshift. Moreover, it is difficult to disentangle
the effects of
and
in such tests
(Dekel, Burstein, &
White 1997a).
Alternatively, one may carry out *dynamical* measurements
of
in the local
(*z*
0.05) universe, in which both evolution and the geometrical effects of the
cosmological constant may be safely neglected.

Low-redshift tests of
are
based on dynamical measurements of the mass of gravitating matter on some
characteristic size scale. For example, measurements of rotation curves
(Rubin 1983)
or the motions of satellite galaxies
(Zaritsky et al. 1993)
yield the masses of ordinary spirals within
~ 10-200 kpc of their centers. The velocity dispersions
(Carlberg et al. 1996),
X-ray temperatures
(White, Efstathiou, &
Frenk 1993), and gravitational lensing effects
(Tyson & Fischer
1995;
Squires et al. 1996)
of rich clusters of galaxies provide mass estimates on ~ 1 Mpc
scales. In general, these and other dynamical analyses of matter in the
highly clustered regime have pointed to a mass density corresponding to
0.2 ± 0.1 (e.g.,
Bahcall, Lubin, &
Dorman 1995).
This value exceeds that implied by known sources of luminosity
(_{lum}
0.01; Peebles 1993)
or inferred from primordial nucleosynthesis
(_{baryon}
0.05; Turner et
al. 1996),
and thus points to the existence of nonbaryonic dark matter. However, it
is well below the Einstein-de Sitter value of
= 1
that is favored by simplicity and coincidence arguments (e.g.,
Dicke 1970).
The natural expectation from
the inflation scenario is that the universe is flat,
+
_{} = 1,
where _{}
/
3*H*_{0}^{2} is
the effective energy density contributed by a cosmological constant
(Guth 1981;
Linde 1982;
Albrecht & Steinhardt
1982).
However, if
0.2,
this inflationary prediction requires
_{}
0.8, which conflicts with upper limits obtained from studies
of gravitational lensing
(Carroll, Press, &
Turner 1992;
Maoz & Rix 1993;
Kochanek 1996).

It is possible, however, that could be close to or exactly equal to unity, despite evidence to the contrary from dynamical tests on ~ 1 Mpc scales. This could occur if the dark matter is poorly traced by dense concentrations of luminous matter such as galaxies and galaxy clusters. If so, dynamical tests on scales 10 Mpc are necessary to obtain an unbiased estimate of . Such tests involve measurements of the coherent, large-scale peculiar velocities of galaxies. According to gravitational instability theory (cf. eq. [1]), these motions are related in an -dependent way to the large-scale distribution of mass. If the latter, in turn, can be inferred from the observed distribution of galaxies on large scales, one might hope to derive an estimate of that is free from the pitfalls of small-scale dynamical analyses.

This program requires a comparative
analysis of two types of data sets. The first consists of radial velocities
and redshift-independent distance estimates for large samples of galaxies.
The largest such compilation to date is the Mark III catalog
(Willick et al. 1997),
which contains distance estimates for ~ 3000
spiral galaxies from the Tully-Fisher (TF;
Tully & Fisher 1977)
relation and 544 elliptical galaxies from
the *D*_{n} -
relation
(Djorgovski & Davis
1987;
Dressler et al. 1987).
The second type of data set is a full-sky redshift survey with
well-understood selection criteria. Several large redshift surveys exist
(cf. Strauss & Willick
1995,
hereafter SW, and
Strauss 1996
for reviews); the one that
most nearly meets the requirements of full-sky coverage and
well-understood selection is the *IRAS* 1.2 Jy
survey (Fisher et
al. 1995).
The basic idea behind the comparison is as follows. In the linear regime
(mass density fluctuations
/
_{0}
<< 1), the global relationship between the peculiar velocity
field **v(r)**
and the mass density fluctuation
field (**r**)
is given by gravitational instability theory:

(1) |

where *f*()
^{0.6}
(Peebles 1980).
^{(1)}
If mass density fluctuations are equal to galaxy number density
fluctuations, at least on the scales
( few
megaparsecs) over which it is possible to define continuous density fields,
then the redshift survey data yield a map of
(**r**)
(after correction for peculiar velocities;
Appendix A).
By equation (1), one then derives a predicted peculiar velocity
field **v(r)** as a function of
. The
TF or *D*_{n} -
data provide the observed peculiar velocities. The best estimate of
is
the one for which the predicted and observed peculiar velocities best agree.

Two obstacles make this comparison a
difficult one. The first, already alluded to, is fundamental: one observes
galaxy number density
(_{g})
rather than mass density
() fluctuations.
A model is required for relating the first to the second. The simplest
approximation is *linear* biasing,

(2) |

in which the bias parameter *b* is assumed to be spatially constant,
and no scatter around the relation in equation (2) is
assumed. Substituting equation (2) in equation (1) yields

(3) |

where
*f*() / *b*.
Thus, under the dual assumptions of linear dynamics and linear biasing,
comparisons of peculiar velocity and redshift survey data, by themselves,
can yield the
parameter
but not . One
might hope to break
the -*b*
degeneracy by generalizing equation (1) to the nonlinear dynamical
regime (cf.
Dekel 1994,
Section 2,
or Sahni & Coles 1996
for a review).
However, such generalizations are difficult to implement in practice;
furthermore, nonlinear extensions to equation (2) will enter in the same
order as nonlinear dynamics
(we discuss this issue further in
Section 6.3.1). Thus, without a more
realistic a priori model of the relative
distribution of galaxies versus mass, it is prudent to limit the goals of
the peculiar velocity-redshift survey comparison to testing
gravitational instability theory and determining
. One
may then adduce external information on the value of *b* to place
constraints on itself.

The second obstacle is the sheer technical
difficulty of the problem. The random errors in the redshift-independent
distances obtained from methods such as TF are large (~ 20%;
Willick et al. 1996)
and are subject to potential systematic errors due to statistical bias
effects
(Dekel 1994;
SW,
Section 6).
Furthermore, we measure the galaxy density field
_{g}
in *redshift* space, whereas it is the real-space density that yields
peculiar velocities via equation (3). The relationship between the two
depends on the peculiar velocity field itself. Self-consistent methods, in
which **v**_{p}
is both the desired end product and a necessary intermediate ingredient in
the calculation, must therefore be developed for predicting peculiar
velocities from redshift surveys (Appendix A).
For these
reasons, reliable comparisons of peculiar velocity and redshift survey data
require extremely careful statistical analyses.

This problem has inspired a number of
independent approaches in recent years. The POTENT method
(Dekel, Bertschinger,
& Faber 1990;
Dekel 1994;
Dekel et al. 1997b)
was the first effort at a rigorous treatment of peculiar velocity data.
Dekel et al. (1993)
compared the POTENT reconstruction of the Mark II peculiar velocity data
(D. Burstein 1989, private communication [privately circulated computer
files]) to the *IRAS* 1.936 Jy redshift survey
(Strauss et al. 1992b),
finding
_{i}
= 1.28_{-0.59}^{+0.75} at
95% confidence. ^{(2)} An improved
treatment using the Mark III peculiar velocities
(Willick et al. 1997)
and the *IRAS* 1.2 Jy survey
(Fisher et al. 1995)
yields
_{I}
= 0.86 ± 0.15
(Sigad et al. 1997,
hereafter POTIRAS).
Hudson et al. (1995)
compared the optical redshift survey data of
Hudson (1993)
with the POTENT reconstruction based on a preliminary version of the
Mark III catalog, finding
_{opt}
= 0.74 ± 0.13 (1
errors). These results from POTENT were obtained using 1200
km s^{-1} Gaussian smoothing. A distinct approach, which differs
from POTIRAS in the statistical biases to which it is
vulnerable (SW)
and which typically uses much
smaller smoothing, is to predict galaxy peculiar velocities and thus
distances from the density field, and then to use these predictions to
minimize the scatter in the TF or *D*_{n} -
relations (Strauss 1989;
Hudson 1994;
Roth 1994;
Schlegel 1995;
Shaya, Peebles, &
Tully 1995;
Davis, Nusser, &
Willick 1996,
hereafter DNW).
This second kind of analysis has produced estimates
of _{I}
in the range ~ 0.4-0.7,
lower than the values obtained from POTIRAS. We further clarify the
distinction between the two methods in
Section 2.1,
and discuss possible reasons for the discrepancies
in Section 6.1.

In this paper, we present a new maximum
likelihood method for comparing TF data with the predicted peculiar
velocity and density fields in order to estimate
. Its
chief strength is an improved treatment of nearby galaxies (*cz*
3000 km
s^{-1}), and we limit the analysis to this range. The TF data that
we use comprise a subset of the Mark III catalog of
Willick et al. (1997).
The predicted peculiar
velocities are obtained using new reconstruction methods
(Appendix A) from the *IRAS* 1.2 Jy
redshift survey. ^{(3)} The
outline of this
paper is as follows. In Section 2, we first
review the
strengths and weaknesses of existing approaches, and then describe our new
method in detail. In Section 3, we present
tests of the
method using mock catalogs. In Section 4, we
apply the method to the Mark III catalog and obtain an estimate
of _{I}.
In Section 5, we analyze residuals from our
maximum likelihood solution in order to assess whether *IRAS*
predictions give a statistically acceptable fit to the Mark III data.
In Section 6, we further discuss and summarize
our principal results. This paper is the product of nearly 3 years work
and contains considerable detail. We recommend that readers
interested primarily in results and interpretation skim
Section 2, and then read
Sections 3.1, 4.4,
4.5, 5.1,
5.2, and 6.4.

^{1}
We measure distances **r** in
velocity units (km s^{-1}). In such a system of units, the Hubble
constant is equal to unity by definition and does not affect the amplitude
of predicted peculiar velocities. Back.

^{2} Because
the bias parameter can differ for different galaxy samples, the value of
can
differ as well. We will
use _{I}
for the *IRAS* redshift survey and
_{opt}
for an optical survey. Because optical galaxies are about 30% more
clustered than *IRAS* galaxies
(SW), the conversion
is _{I}
1.3_{opt}.
When speaking generically about the velocity-density relation, we
will place no subscript on
.
Back.

^{3} The
original *IRAS* 1.936 Jy survey was presented in a series of six
papers
(Strauss et al. 1990;
Yahil et al. 1991;
Davis, Strauss, &
Yahil 1991;
Strauss et al. 1992a,
1992b,
1992c),
numbered I, II, III, IV, V, and
VII, respectively. The missing Paper VI was to be the comparison of the
observed and predicted velocities, to be based on
Strauss (1989, chap. 3).
However, it has taken us until now to come up with
statistically rigorous ways of doing this comparison. Thus, the long-lost
*IRAS* Paper VI has been incorporated into
Dekel et al. (1993),
DNW,
Sigad et al. (1997),
and especially this paper. Back.