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

2.2. Linear Theory

As there are many excellent reviews available in the literature (e.g. Peebles 1980, Primack 1987, Bardeen et al 1986, Efstathiou 1991a), and, since there have been no dramatic recent developments, there is little need here for an extensive discussion.

Suffice it to say that current variants of the inflationary scenario are believed to provide support for the scale invariant Harrison-Zeldovich-Peebles spectrum of fluctuations, although there are plausible variants (cf Cen et al 1992 for a brief review) which adopt other spectra. Let us recall the original argument, which is elegant in its simplicity. What is the amplitude, ( / )_rms, of perturbations whose comoving wavelength at any given time t is equal to the horizon, = ct? First, assume that this amplitude can be expressed as a power in time t. Then, if < 0, the perturbations diverge in the past, and it is difficult to see why our piece of the universe did not collapse into a black hole at an early epoch; whereas, if > 0, this event occurs in the future and we are luckily living in that brief interval after perturbations were irrelevantly small and before they become catastrophically large. Thus, we must have 0 according to the HZP argument. Any non-power law dependence of amplitude on time selects out a preferred epoch. It is a very plausible argument, and, apart from small corrections like logarithmic terms, it is probably correct in a scale-free universe which does not in other respects have a preferred epoch. (However, in universes that are not flat, e.g. + 1, then there is, in any case, an epoch when one makes a transition from nearly flat to nearly empty, and one loses nothing, in terms of the "fine-tuning" issue, to have that as the epoch when perturbations become significant.) The statement of scale invariance can then be translated into a simple statement about the power spectrum (Fourier transform) of the amplitude of perturbations ( _(r) / ): P_k( / ) (const)k¹ at large redshift.

There is no accepted theory for determining the constant in this equation, which is thus set by matching with observations. The cleanest way of doing this is to detect the potential fluctuations directly in the linear regime at (or after) decoupling by measuring the Sachs-Wolfe fluctuations in the CBR at long wavelength. This has now been done by COBE with the normalization on the 10° scale being the most secure. Since the detected amplitude in (T / T) is of order 10^-5, the measurements are in the linear regime where the growth of perturbations can be calculated in a straight-forward way. Subsequent reionization and secondary perturbations (due to nonlinear effects), gravitational wave corrections and "cosmic variance," all have little effect for the standard CDM spectrum at this angular scale, so at present the amplitude of the CDM spectrum seems to be known to better than 25% (the formal one error is 17%) at long wavelengths.

The growth of perturbations at early epochs depends both on epoch (with a break at the era z_eq 2.5 × 10⁴h² after which matter dominates and growth accelerates) and scale, with kinematic growth greater for perturbations larger than the horizon in the radiation-dominated epochs. Convolving these effects, one derives the "transfer function" (cf e.g. Bardeen et al 1986) which, when multiplied by the primeval k¹ spectrum, determines the dark matter fluctuations after decoupling. Then, if matter is neutral (i.e. if radiation drag can be neglected), the baryonic component can respond to the potential fluctuations in the dark matter. The standard spectrum is shown in Figure 1 (for h = H₀ / 100 km s^-1 / Mpc = 0.5), and we see that, since perturbations are greatest on the smallest scales, small mass objects are expected to condense out first. But on very small scales the thermal energy (produced by even quite unimpressive heat sources) will produce Jeans' stability leading to a maximum in the growth rate at around 10⁹ M (cf Figure 1b). Since the amplitude (M / M)_rms is growing steadily [as (1 + z)^-1] with time, once perturbations at the peak become nonlinear (at a redshift of 6 for the COBE normalization), rapid growth of nonlinear structures occurs on ever larger (and smaller) scales.

Beyond this point, detailed numerical simulations are required to specify the predictions of the CDM scenario. For the dark matter, a collisionless code based on Newton's laws for noninteracting particles suffices, but for the gaseous component, which we ultimately observe as it is heated and emits X-rays or is transformed into stars and emits optical radiation, we require a true hydrodynamic simulation. Codes including both hydrodynamics and the detailed atomic processes needed to allow for both heating and cooling of gas are just now coming into use (cf Cen 1992 and references therein).