Annu. Rev. Astron. Astrophys. 1993. 31:
689-716
Copyright © 1993 by . 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)*k*^{1} 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 ×
10^{4}*h*^{2}
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 *k*^{1} 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 = H*_{0} / 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 10^{9}
*M*_{} (cf
Figure 1*b*). 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).