**6.1. What is the Value of**
_{I}?

VELMOD recovers the correct answer,
_{I}
= 1, to less than 10% accuracy when applied to the mock catalogs.
At _{I}
= 1, the velocity field in the mock Virgo region is significantly
triple-valued. Thus, VELMOD, despite being close in spirit to Method II,
properly treats triple-valuedness. If the strong triple-valuedness one sees
at _{I}
= 1 were present in the real universe, VELMOD would not assign it an unduly
small likelihood. Nonetheless, when VELMOD is applied to the real universe,
it returns a value
of _{I}
= 0.492 ± 0.068 (quadrupole modeled). This value is quite insensitive
to two other quantities treated as free parameters in the velocity field
model, the Local Group random velocity
**w**_{LG} and the small-scale velocity dispersion
_{v}
(Section 4.5). Tests with the mock catalogs
demonstrated that we obtain an unbiased
_{I}
using a 300 km s^{-1}-smoothed
*IRAS* reconstruction (Section 3.1).
However, we found that changing to a 500 km s^{-1}-smoothed
reconstruction makes relatively little difference in
_{I}
(Section 4.6). Finally, neglecting the
quadrupole causes
_{I}
to change by only ~ 1
. Our
conclusion that
_{I}
0.5 ± 0.07 is thus robust against systematic effects internal to our
method.

The VELMOD result is consistent with the
relatively low estimates of
_{I}
obtained from the Method II analyses of
Hudson (1994),
Roth (1994),
Shaya et al. (1995),
^{(14)}
DNW,
and Schlegel (1995),
as well as those derived from
comparisons of the *IRAS* density field with the motion of the Local
Group (Strauss et
al. 1992b) and from some analyses of
the redshift-space anisotropy of the *IRAS* density field (e.g.,
Hamilton
1993,
1995;
Fisher et al. 1994a,
1994b;
Cole, Fisher, &
Weinberg 1995;
Fisher & Nusser 1996).
However, it is apparently inconsistent with estimates of
_{I} near
unity, as have been found by the POTIRAS analysis
(Sigad et al. 1997),
measurements of the POTENT fluctuation amplitude
(Kolatt & Dekel 1997;
Zaroubi et al. 1997),
and redshift-space
distortions of spherical harmonic expansions of the density field
(Fisher, Scharf, &
Lahav 1994c;
Fisher 1994;
Heavens & Taylor
1995).

**6.1.1. Why Do VELMOD and POTIRAS Yield Different
Values of**
_{I}?

We do not yet have a satisfactory
explanation of why VELMOD and standard Method II
analyses characteristically yield smaller values
of _{I}
than the Method I POTIRAS approach. One possibility is that the differences
stem from the Method I/Method II distinction. However, VELMOD corrects the
principal drawback of Method II, the inability to deal with multivalued or
flat zones in the redshift-distance relation. Thus, if the Method I/Method
II distinction is at the root of the discrepancy, the reason must be
more subtle than the drawbacks of standard Method II.
Sigad et al. (1997)
test for biases in POTIRAS using
the *same* mock catalogs as this paper; they also find
their determination of
_{I}
to be essentially unbiased. The problem could lie with the Malmquist bias
corrections that are so crucial to Method I (cf. the discussion
in Willick et al. 1997).
If these corrections are underestimated for any reason - e.g.,
the TF scatter is larger than estimated, or the density fluctuations are
larger than modeled - a Method
I approach will produce too strong velocity gradients and thus overestimate
_{I}.
However, the TF scatters used by
Sigad et al. (1997)
are consistent with those obtained in this paper, and the large
POTENT smoothing limits the effect of Malmquist bias in any case. Thus, it
is unlikely that improper Malmquist bias corrections strongly affect the
value of
_{I}
obtained from POTIRAS.

An important difference between VELMOD and
POTENT is the Gaussian smoothing scales employed, 300 and 1200 km
s^{-1}, respectively. These very different smoothings could result
in different values of
_{I}
if the effective bias parameters on these scales are different. In order to
reconcile VELMOD and POTIRAS, we would need the effective bias parameter to
decrease by a factor of 1.7 between scales of 300 and 1200 km
s^{-1}. Such a scale-dependent biasing has been suggested by the
galaxy formation models of
Kauffman, Nusser, &
Steinmetz (1996),
but Weinberg (1995)
and Jenkins et al. (1996)
do not find these trends. A recent analysis by
Nusser & Dekel (1997)
that fits the
*IRAS* and peculiar velocity data simultaneously finds
_{I}
= 1.0 for 1200 km s^{-1} smoothing, but only 0.6 for 600
km s^{-1} smoothing, approaching the value we have found in this
paper. Such a change of
_{I}
with smoothing scale could signal scale-dependent biasing.

Still another difference is the volume
considered. We have restricted this analysis to *cz*
3000 km
s^{-1} (Section 4), whereas the analysis
of Sigad et al. (1997)
extends to 6000 km s^{-1}; only ~ 1/3 of
the points used fall within 3000 km s^{-1}. If, for whatever
reason, *b*_{I} differed locally from its global value,
the VELMOD result could be biased low. In a future paper, we will extend
the VELMOD analysis to larger distances; however, our preliminary results
do not show an increase
in _{I}
when we do so. In addition to probing a larger volume, the Sigad et al.
analysis uses the full Mark III sample, ellipticals included; the
possibility of systematic differences between the TF subset we have used in
this paper and the full sample is difficult to rule out. Finally, it is
conceivable that the requirement of precalibrating TF relations (POTENT),
as opposed to calibrating them simultaneously with fitting the velocity
field (VELMOD and Method II generally), accounts for part of the
discrepancy. However, fixing the VELMOD TF parameters at their Mark III
values has essentially no effect on the derived value
of _{I}
(Section 4.7). This argues strongly
against the notion
that a major difference between VELMOD and POTIRAS is the TF
relations themselves.

**6.1.2. Effect of Cosmic Scatter**

The sphere out to 3000 km s^{-1}
is small; the rms value of density fluctuations within spheres of this
radius is 20% for *COBE*-normalized cold dark matter (CDM). However,
this does not propagate to a cosmic scatter error on our derived
_{I}
for two reasons. First, the *IRAS* velocity field is determined within
a sphere of radius 12,800 km s^{-1}, within which the rms
fluctuations are only a few percent. Thus, the predicted peculiar velocity
field is subject to very little cosmic scatter. Second, this scatter
manifests itself primarily as a monopole term (cf. the discussion
in Section 4.4) and therefore is fully
absorbed into the zero points of the TF relations
(Section 3.3),
having *no* effect on the derived value
of _{I}.

We have assumed that the bias relation,
equation (2), is deterministic, without any scatter or variation in
the effective value of *b*_{I} with position. Such a
model must be unrealistic at some level, and we look forward to
realistic galaxy formation models that can quantify how large this form of
cosmic scatter might be, and how they might affect statistics such as
(cf. the discussion in
Dekel 1997).

^{14} The
Hudson and Shaya et al. papers actually derive
_{opt},
which must be multiplied by ~ 1.3 to obtain an equivalent
_{I}
(cf. footnote 2). Back.