Annu. Rev. Astron. Astrophys. 1991. 29:
325-362
Copyright © 1991 by Annual Reviews. All rights reserved |

**2.3 The Problem of Formation of Structures**

The above discussion is based on an idealized model of the universe,
populated by matter that is assumed to be homogeneous and isotropic;
the density and
pressure *p* are taken to be spatially uniform: (*t*,
**x**) = (*t*) and *p (t*, **x**) = *p (t)*. The
real universe, of course, is quite
different and has a very inhomogeneous distribution of matter. One
possible way of characterizing such inhomogeneities is to use the
``two-point-correlation function.'' Suppose is the constant mean
number density of galaxies in space. If the galaxies are distributed
at random, the chance of finding a galaxy in a given volume *V* is
simply *V*. If, however, the galaxies
are distributed inhomogeneously,
the chance of finding a galaxy in a volume element *V* at a distance *r*
from another galaxy, picked at random, may be expressed as

where (*r*) is
called the *covariance function or two-point correlation
function*. Detailed studies of the galaxy counts indicate that the
two-point correlation function for galaxies is (*r*) (*r* /
5*h*^{-1} Mpc)^{-1.8}
showing significant clustering of matter at scales *r* <
5*h*^{-1} Mpc
(28,
98).
On the other end, deep three-dimensional surveys indicate
evidence for *filaments* (100 *h*^{-1} Mpc long and
5*h*^{-1} Mpc across) and *voids*
(empty regions about 60 *h*^{-1} Mpc in diameter) at very
large scales
(6,
34,
47,
68,
70,
114).
The average density contrast in the universe
also shows significant variations depending on the coarse-graining
scale. It is about 10^{5} at galactic scales and about
10^{2} at the scale of clusters.

This raises the question: How do we reconcile the existence of small- scale structures - galaxies, groups, and clusters - with the overall homogeneity of the universe? In particular, how should the theoretical frame-work be modified to take these inhomogeneities into account?

Two approaches are possible regarding this issue. We may try to
obtain more realistic cosmological models in which the source energy
momentum tensor of matter *T _{ik} (t*,

The alternative point of view, which enjoys considerable popularity.
is the following: Let us assume that at some very early epoch *t* =
*t _{i}*
(say), the physical variables describing the universe had small
deviations around their mean values. The metric was

The above picture runs into several difficulties when the details are worked out. Of these, the following three appear to be the most serious ones:

1. To have any predictive power in this model. we need to know the
origin and magnitude of the initial density perturbations. In the
conventional big-bang model. there is no natural seed for these
perturbations. Truly primordial perturbations, e.g. those due to
quantum gravitational effects at *t* *t _{p}*, are likely to be of the order
of one [

2. The second difficulty is of technical nature. We define the
``density contrast''
(*t*, **x**) and its Fourier transform _{k} (*t*) by the
relations

where (*k / a*) is the physical wave number and | 1 | |*a (t)***x**| is the
proper distance. The Fourier transform separates a given inhomogeneity
into components of different characteristic sizes, which may grow at
different rates in the expanding universe. We can identify two effects
in the time evolution of
(*t*, **x**): (*a*) the growth of the amplitude
* _{k} (t)* due to
gravitational instability and (

To every wavelength _{0}, we can associate the mass scale:

which remains constant during expansion of the universe (since *a*^{-3}
and *a*). For example 1.88 Mpc corresponds to the galactic mass
of 10^{12} *M*_{}; of course, galaxies today are much smaller, because
overdense regions have collapsed gravitationally and ceased to expand
with the universe. Thus is
the hypothetical size containing galactic
mass, if it had continued with the cosmic expansion. This
parametrization turns out to be extremely useful. The wavelength of
this perturbation will be bigger than the Hubble radius at all
redshifts *z* > *z*_{enter} *(M)* where,

It is usual to say that the perturbation carrying mass *M enters* the
Hubble radius at *z* = *z*_{enter} or that the
perturbation with mass *M* was
*outside* the Hubble radius at *z* >
*z*_{enter}. The discontinuity in the two
forms for Eq. 16 arises because the universe changes from the
radiation-dominated phase to the matter-dominated phase at some
*t*_{eq} =
4.36 x 10^{10} (
*h*^{2})^{-2} ^{6} s with *z*_{eq} = 2.32 x
10^{4} (
*h*^{2}) ^{-4}. A scale of _{eq}
13 Mpc (
*h*^{2})^{-1} ^{2} enters the Hubble radius at
*t*_{eq}, carrying the mass *M*_{eq}
3.19 x 10^{14}
*M*_{}. Scales with
< _{eq} (*M* <
*M*_{eq}) enter the Hubble
radius before (*z* > *z*_{eq}), when *a (t)* *t*^{1/2}, whereas
the scales > _{eq}
enter later (*z* < *z*_{eq}), when *a (t)* *t*^{2/3}.

Notice that according to the above calculation, a galactic mass
perturbation was bigger than the Hubble radius for redshifts larger
than a moderate value of about 10^{6}. This result leads to a major
difficulty in conventional cosmology. It is usually assumed that
physical processes can act coherently only over sizes smaller than-the
Hubble radius (see e.g.
8,
17).
Thus any physical processes leading to
small density perturbations at some early epoch *t* =
*t _{i}* could have only
operated at scales smaller than

3. A further major difficulty in the study of structure formation is
the following: It can be shown that the linear baryonic density
perturbations (at scales smaller than the Hubble radius) grew as (*t*)
*a (t)* in a
matter-dominated universe from *z* = *z*_{dec} up to a
redshift of about (^{-1} - 1) (see e.g.
98).
Since *z*_{dec}
1.1 x 10^{3}
^{-1}, these
perturbations could have grown only by a factor of about *z*_{dec} since
then. Today = ( / ) is of order unity at scales larger than
8*h*^{-1} Mpc. Therefore, in a baryonic universe, these
perturbations must
have been at least 0.8 x 10^{-3} _{B}^{-1} 5 x 10^{-3}, at
decoupling if we
take into account the constraint _{B} < 0.16 coming from the primordial
nucleosynthesis calculations
(130).
In the simplest models, this will
lead to a temperature anisotropy of MBR of (*T / T*)
0.3 1.7 x 10^{-3}
at angular scales of about 4.4'. (The angular scale of anisotropy
and the linear scale of the
perturbation producing the anisotropy
are related by 0.55' *h*( / 1 Mpc).] No such anisotropy has been
discovered. The bounds on (*T
/ T*) (less than 3 x 10^{-5} at 4.5'
(119)]
suggest that is less than at
least 10^{-4} at the time of decoupling.
This density contrast is insufficient to produce the observed
structures in a purely baryonic universe.

The perturbations described here, and elsewhere in this review, are
called *adiabatic perturbations or curvature perturbations*. They
actually change the curvature of space-time (locally, the equivalent
Newtonian potential changes by _{k}
(_{phy} *c*^{-1} *H*) _{k}]. It is possible to
think of another kind of perturbation (*isothermal*) in which = 0 but
the *p* = *p* () equation changes from place to place. We do not discuss
them here because it seems unlikely that they are able to account for
structures in the universe. (For a more detailed discussion of this
and related issues, see
(7,
101,
118).]

^{1} The conclusion above based on
*cH*^{-1} *(t)*as a distance scale limiting
causal interaction is accepted by most physicists, though it is very
difficult to prove in general terms and cannot be proved by causality
arguments because *cH*^{-1} *(t)* and the horizon size
can be quite different
in a general FRW model (see
40,
93).
Normally, the particle horizon at
an epoch limits the range of causal communication from the past, while
the event horizon limits the range of such communication to the
future. The radiation-dominated era has a particle horizon of the
order of Hubble radius, whereas the de Sitter universe has an event
horizon of the same order. In the above argument both the horizons are
used to limit causal communication in the same sense. Moreover, for
its limited duration, the inflationary phase cannot claim even to have
an event horizon! Thus one must consider the Hubble radius in the
above argument only as a hand-waving device.
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