Annu. Rev. Astron. Astrophys. 1993. 31:
689-716
Copyright © 1993 by . All rights reserved |

**2.4. Input Parameters: By Hypothesis
_{tot} = 1
and = 0 in
the Standard Model**

In this scenario, the age of the universe is *t*_{H} =
(2/3)(1/*H*_{0}), so that a small
value of the Hubble constant is required for consistency with Galactic
stellar ages. By convention, a value of *h* = 0.5 is usually
adopted to give
*t*_{H} = 13 billion years - which is barely larger than
the estimated ages of
the globular clusters. The amplitude of the spectrum is conventionally
specified by the value of the mass fluctuations in a top-hat window of
8 *h*^{-1} Mpc scale:

(3) |

Using this definition, we note from Equation 2 that *b*_{8}
= (*N /
N*)_{8} /
_{8}, and
since (*N /
N*)_{8} is estimated observationally to be close to unity (cf
Loveday et al 1992,
and discussion in
Cen & Ostriker 1992b),
we have *b*_{8} approx 1 /
_{8};
there is an inverse relationship between the required bias and the amplitude
of the input spectrum. At first, before bias was introduced, the CDM
modelers adopted *b*_{8} = 1. Then to satisfy the dynamical
constraints and
the _{tot} =
1 requirement, a value of 1 /
_{8} = 2-2.5 was
typically utilized
in the work of DEFW with a correspondingly low amplitude
_{8} < 0.5
implied for the input spectrum. The large value of *b*_{8}
also helped to produce
"contrasty" pictures of large-scale structure with rather empty voids. But
this amplitude was somewhat too low to account for the large-scale flows
which required higher amplitude and lower bias (*b*_{8} ~
1.5). Now, a still higher value of
_{8} is required
to fit the *COBE* measurements:
_{8} = 1.1
± 0.2
(Efstathiou et al 1992,
with *b*_{8}
0.9 implied).

Nonstandard variants on CDM, which we will discuss subsequently,
have somewhat more flexibility. In general, they typically have a higher
ratio of large-scale power to small-scale power than the standard model
[either by lowering
*h* or *n*
or by adding an admixture of Hot Dark Matter
(HDM)]. Thus, they can keep the large-scale normalization to *COBE* but
allow a lower amplitude at the 8 *h*^{-1} Mpc scale so as
to have _{8}
0.5.

Before turning to the observational tests of the CDM scenario, it may
be appropriate to add a further word on nonstandard models. In addition
to open variants
(Blumenthal et al 1988,
Efstathiou 1992),
"tilted" models with *n* < 1
(Cen et al 1992,
Vittorio, et al 1988,
Adams et al 1993,
Salopek 1992),
and mixed dark matter models
(Davis et al 1992),
there are also
CDM variants which keep all the standard assumptions except that of
Gaussian perturbations. Thus, texture, black hole, or string-seeded models
have been computed.

The consequences of non-Gaussian perturbations are manifold. Since
the time-dependent potentials associated with seed formation will affect
the CBR, there will be a smaller value of
_{8} in these
models for given CBR
fluctuation measurements. Second, there will at all times be small volumes
of space in which fluctuations are in the nonlinear region in these
scenarios.
Thus, there is an earlier formation of structure for the same value of
(*M /
M*)_{rms} and physical bias will be larger. There is also the
possibility
that shocks, from nonlinear structures forming before decoupling, could
produce small-scale entropy perturbations in a natural way in these
models. We will return to a discussion of the non-Gaussian variants of
CDM in the concluding section
(Section 4.1), but to summarize this
section
and look ahead to the next, the *standard* CDM model has amplitude
_{8}
1.0-1.1 and
*b*_{8}
0.9.