**6.3. What is the Value of
?**

Measuring
= ^{0.6} /
*b*
is an important objective of velocity analysis. Of course, the more
important objective is the determination of
itself. There are, broadly speaking, two ways to proceed.

One may attempt to break the degeneracy between
and
biasing by extending the gravitational instability theory to the nonlinear
dynamical regime. In an earlier phase of the VELMOD project, we attempted
to do this; very preliminary results of this effort were described in
SW,
Section 8.1.2. In brief, the *IRAS*
reconstruction was done as described in
Appendix A, but
a nonlinear generalization of equation (1),

(27) |

was used to derive peculiar velocities from the redshift survey density
field _{g}.
In equation (27), *a* = 0.28 and
<^{2}>
is the mean square value of
(G. Ganon et al. 1995, private communication; cf.
Nusser et al. 1991).
Note that the mass fluctuation
is
written as a generic function of
_{g}
and *b*, rather than simply as
_{g} /
*b*. This
is because once we generalize to nonlinear dynamics, we must allow for the
possibility of *nonlinear* *biasing* as well. There are many ways
one might imagine doing this
(SW, Section 2.5;
Fry & Gaztañaga
1993).
Generically, however, all these complications can be expanded to second
order to yield a correction to equation (3):

(28) |

where parameterizes the combined effects of nonlinear dynamics and nonlinear biasing.

We carried out a suite of VELMOD runs
using predicted peculiar velocities based on equation (28) for a range
of values of
,
both positive and negative. Our hope was that the VELMOD likelihood
statistic would be significantly lower for some value of
than
for the pure linear case. However, to our surprise, we found that the
linear dynamics/linear biasing reconstruction
( =
0) gives the best likelihood of all. We are not certain why this is the
case. Nonlinear dynamics must enter to some degree, because we know for a
fact that _{g}
is not everywhere << 1 and
indeed can be quite large with our small smoothing. (Of course, we do not
know whether or not nonlinear biasing is important.) Nevertheless, the
small scatter between the true and *IRAS*-predicted peculiar
velocity fields for the mock catalogs
(Section 3.2) confirms that the linear
*IRAS* velocity field, smoothed on
a 300 km s^{-1} scale, is a good match to actual peculiar
velocities that arise from gravitational instability, at least in an
*N*-body simulation.

A possible explanation for this seeming
contradiction is as follows. Our method for predicting peculiar velocities
(Appendix A) entails assigning a smooth,
continuous density field from discrete redshift survey data - a
procedure that takes into account the probability distribution of distance
given redshift (eq. [A2]),
smoothes the data with a 300 km s^{-1} Gaussian,
and applies a Wiener filter - and
thus this reduces small-scale density enhancements. In doing so, this
procedure mimics qualitatively the effects of nonlinear corrections on
the velocity-density relation. The good match between the *IRAS*
predictions and the actual peculiar velocities suggests that this mimicry
is in fact fortuitously good, to the degree that formal nonlinear
corrections are unnecessary. Unfortunately, if this is true, it is unlikely
that nonlinear analyses will enable us to
determine
and *b* separately.

**6.3.2. Constraining
from Independent
Estimates of b _{I}**

The second way to estimate
,
given our measurement
of _{I},
is to constrain *b*_{I} using independent information.
If equation (2) with constant *b* holds exactly, then
*b*_{I} is the ratio of the rms fluctuations of
*IRAS* galaxies on an 8 *h*^{-1} Mpc scale,
_{8}(*IRAS*),
to the corresponding mass density fluctuations,
_{8}.
Fisher et al. (1994a)
found that
_{8}(*IRAS*)
= 0.69 ± 0.04 in real space. It follows
that _{I}
can be viewed as a prediction
of _{8}
for a given value of :

(29) |

An entirely independent (although highly model-*dependent*) way to
predict _{8}
as a function of
is
to use *COBE*-normalized power spectra for a range of cosmological
parameters. Liddle et al.
(1996a,
1996b)
have presented fitting functions
that provide the normalization of CDM power spectra, in open and flat
cosmologies, as a function of
,
_{},
the Hubble parameter *h*
*H*_{0} / (100
km s^{-1} Mpc^{-1}), and the primordial power spectrum
index *n*, based on the four-year *COBE* observations
(Bennett et al. 1996;
Górski et
al. 1996).
Eke, Cole, & Frenk
(1996) used these fitting functions to obtain
_{8}
by direct integration of the Liddle et al. power spectra and have kindly
provided us with their code for doing this calculation. Thus, we may
constrain _{8}
by comparing the VELMOD and *COBE*/CDM predictions of its value
and requiring that they agree to within the errors. This will be the case
only for a limited range of
(the
"concordance range").
We emphasize, however, that the discussion to follow depends on two
uncertain assumptions: first, that the CMB fluctuations measured by
*COBE* can be reliably extrapolated down to 8
*h*^{-1} Mpc scales, and second, that the bias parameter is
scale independent from 3 to 8 *h*^{-1} Mpc.

In Figure 20, we compare the two constraints on
_{8}
for a scale-invariant (*n* = 1) power spectrum.
Figure 20*a* shows results for an open (i.e.,
=
0) universe, and Figure 20*b* for a
spatially flat (
+ _{}
= 1) universe. The *COBE*/CDM predictions (*solid lines
labeled with the values of the
Hubble constant*) and the constraint from equation (29) (*shaded
region*) scale very differently
with ,
so that the two together give strong constraints
on _{8}
and thus . The
shaded region represents the combined VELMOD error on
_{I}
and the error
in _{8}(*IRAS*)
from Fisher et
al. (1994a).
We do not show
corresponding error regions for the *COBE*/CDM predictions that result
from uncertainty in the *COBE* normalization, because the error in the
predicted _{8}
is in fact dominated by the allowed range of *H*_{0}, which we
take to be 55
*H*_{0}
85 km
s^{-1} Mpc^{-1} based on a number of recent measurements
(Sandage et al. 1996;
Freedman 1997;
Riess, Press, &
Kirshner 1996;
Mould et al. 1996;
Tonry et al. 1997;
Kundic et al. 1997).

Figure 20 gives the
following constraints for *n* = 1. For an open model, the concordance
range is =
0.28-0.46
with the low (high) value corresponding to the highest (lowest) value
of *H*_{0} considered. For the flat model, it
is = 0.16-0.34.
Expressed in terms of the *IRAS* bias parameter, these ranges
correspond to *b*_{I} = 0.92-1.38
(open) and *b*_{I} = 0.68-1.11
(flat). We also considered *n*
1 flat
models. For example, with *n* = 0.9, the concordance ranges are
= 0.19-0.40 and
0.21-0.45,
depending, respectively, on whether tensor fluctuations are not, or are,
included in the *COBE* normalization
(Liddle et al. 1996b).
The corresponding bias parameters are *b*_{I} =
0.74-1.21 and *b*_{I} = 0.80-1.29.

Two salient points follow from this
comparison. First, if *H*_{0}
60 km
s^{-1} Mpc^{-1}, the concordance range for the
flat, *n* = 1 models requires
0.30,
implying _{}
0.70.
However, studies of gravitational lensing have placed an upper limit
of _{}
0.65 at 95% confidence
(Maoz & Rix 1993;
Kochanek 1996),
while a recent analysis of intermediate-redshift Type Ia supernovae
(Perlmutter et al. 1997)
indicates _{}
0.50 at
95% confidence (both of these constraints apply when a flat universe is
assumed). This contradiction constitutes evidence against a flat universe
with a scale-invariant primordial power spectrum index
and *H*_{0}
60 km
s^{-1} Mpc^{-1}. If *n* < 1.0, one can more
easily accommodate flat universes with
_{}
< 0.65, provided the Hubble constant is
70 km
s^{-1} Mpc^{-1}. The second point is that the combined
VELMOD and *COBE*/CDM predictions of
_{8}
are extremely difficult to reconcile with an Einstein-de
Sitter universe for most reasonable values of the remaining cosmological
parameters. If one assumes *n*
0.9, a
Hubble constant 30
km s^{-1} Mpc^{-1}, far below current observational limits,
would be required for the concordance range to include
= 1.
Alternatively, if *H*_{0} = 50 km s^{-1}
Mpc^{-1}, one would require a primordial power spectrum index
*n* = 0.7 and tensor fluctuation contributions to the CMB
anisotropies. Such a power spectrum index is at the lowest end of the range
currently considered plausible in inflationary universe scenarios (e.g.,
Steinhardt 1996).

Another independent constraint on the bias
parameter comes from models of structure formation that predict
_{8} as
a function of
from
the number density of locally observed rich clusters (e.g.,
White et al. 1993;
Eke et al. 1996).
The _{8}()
relation obtained by these authors is quite similar in form to equation
(29), so we can merely test for qualitative agreement with
our VELMOD results, not use it to place strong constraints on
. For example, if
= 0.3, equation (6.1) of
Eke et al. (1996) gives
_{8}
= 0.85 ± 0.07, while the VELMOD result,
_{I}
= 0.492, yields _{8}
= 0.70 ± 0.11. Thus, the two estimates agree to within
~ 1.2
for
= 0.3; the agreement is improved for smaller
. For
= 1,
the Eke et al. formula gives
_{8}
= 0.50 ± 0.04, whereas the VELMOD result yields
_{8}
= 0.34 ± 0.05; they disagree at about the 2.5
level. (In each case, the error bar for the comparison is obtained as the
quadrature sum of the individual error bars.) A recent analysis of the
cluster gas mass function
(Shimasaku 1997)
yields a
value _{8}
= 0.80 ± 0.15 that is *independent* of
. The
VELMOD result for
_{8}
is in good agreement with this provided 0.15
0.4.