Annu. Rev. Astron. Astrophys. 1993. 31: 689-716 Copyright © 1993 by Annual Reviews. 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/H₀), 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 b₈ = (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 b₈ 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 b₈ = 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 b₈ 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 (b₈ ~ 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 b₈ 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 b₈ 0.9.