Annu. Rev. Astron. Astrophys. 1991. 29: 325-362
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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 rho and pressure p are taken to be spatially uniform: rho(t, x) = rho(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 nbar 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 deltaV is simply nbardeltaV. If, however, the galaxies are distributed inhomogeneously, the chance of finding a galaxy in a volume element deltaV at a distance r from another galaxy, picked at random, may be expressed as

Equation 13 13.

where xi(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 xi2(r) approx (r / 5h-1 Mpc)-1.8 showing significant clustering of matter at scales r < 5h-1 Mpc (28, 98). On the other end, deep three-dimensional surveys indicate evidence for filaments (100 h-1 Mpc long and 5h-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 105 at galactic scales and about 102 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 Tik (t, x) is inhomogeneous and anisotropic with an average value equal to that in a typical Friedman-Robertson-Walker (FRW) universe. The exact spacetime metric gik (t, x) obtained as a solution to this problem will provide a more realistic description of our universe. Because Einstein's equations are nonlinear, no simple relation exists between the (large-scale) average value of the metric <gik (t, x)> and the average value of Tik. Thus, this model for the universe may show significant departures from the FRW universe, even at large scales (39). This idea is extremely difficult to put into practice, however, because of our inability to find inhomogeneous solutions to Einstein's equations.

The alternative point of view, which enjoys considerable popularity. is the following: Let us assume that at some very early epoch t = ti (say), the physical variables describing the universe had small deviations around their mean values. The metric was gik (t, x) = gbarik (t) + hik (t, x) and the source was Tik (t, x) = Tbarik (t) + tauik (t, x), where the quantities with an overhead bar denote the FRW values and the terms hik and tauik are small corrections to these mean values. Since these terms are small, one can linearize Einstein's equations in these variables and study the linear growth of metric and matter perturbations. These studies show that matter perturbations do grow under favorable circumstances, and will reach values comparable to their mean value in finite time (73, 98, 122). When this happens, the over-dense region (effectively) decouples from the expansion of the universe and collapses further to form a condensation. It is generally believed that the structures in the universe arose by this process, that is, the gravitational growth of small, primordial seed perturbations.

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 approx tp, are likely to be of the order of one [O(1)] and will be difficult to estimate. Thus, the theory lacks predictive power.

2. The second difficulty is of technical nature. We define the ``density contrast'' delta (t, x) and its Fourier transform deltak (t) by the relations

Equation 14 14.

where (k / a) is the physical wave number and | 1 | ident |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 delta (t, x): (a) the growth of the amplitude deltak (t) due to gravitational instability and (b) the kinematic ``stretching'' of the proper wave lengths (2 pi / k) a (t) due to the overall expansion of the universe. Thus, a perturbation at a characteristic physical scale lambda0 today would correspond to a proper length of lambda0 [a (t) / a0] propto tn in the past, if we take a (t) propto tn. The characteristic expansion scale of the universe, on the other hand, is given by the Hubble radius c H-1(t) = c(adot / a)-1 = cn-1t. In realistic cosmological models, n < 1 and hence the ratio [lambda (t) / cH-1(t)] increases as we go to the earlier epochs. In other words, lambda (t) would have been larger than the Hubble radius at sufficiently high redshifts.

To every wavelength lambda0, we can associate the mass scale:

Equation 15 15.

which remains constant during expansion of the universe (since rhobar propto a-3 and lambda propto a). For example lambda approx 1.88 Mpc corresponds to the galactic mass of 1012 Msun; of course, galaxies today are much smaller, because overdense regions have collapsed gravitationally and ceased to expand with the universe. Thus lambda 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 > zenter (M) where,

Equation 16 16.

It is usual to say that the perturbation carrying mass M enters the Hubble radius at z = zenter or that the perturbation with mass M was outside the Hubble radius at z > zenter. 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 teq = 4.36 x 1010 (Omega h2)-2 theta 6 s with zeq = 2.32 x 104 (Omega h2)theta -4. A scale of lambdaeq appeq 13 Mpc (Omega h2)-1 theta 2 enters the Hubble radius at teq, carrying the mass Meq appeq 3.19 x 1014 Msun. Scales with lambda < lambdaeq (M < Meq) enter the Hubble radius before (z > zeq), when a (t) propto t1/2, whereas the scales lambda > lambdaeq enter later (z < zeq), when a (t) propto t2/3.

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 106. 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 = ti could have only operated at scales smaller than cH-1 (ti), but most of the relevant astrophysical scales (corresponding to clusters, groups, galaxies, etc.) were much bigger than cH-1 (t) for reasonably early epochs! Therefore, if we want the seed perturbations to have originated in the very early universe, it is difficult to understand how any physical process could have contributed to it. (1)

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 lambda (t) propto a (t) in a matter-dominated universe from z = zdec up to a redshift of about (Omega-1 - 1) (see e.g. 98). Since zdec appeq 1.1 x 103 theta -1, these perturbations could have grown only by a factor of about Omega zdec since then. Today delta = (delta rho / rho) is of order unity at scales larger than lambda approx 8h-1 Mpc. Therefore, in a baryonic universe, these perturbations must have been at least delta approx 0.8 x 10-3 OmegaB-1 gtapprox 5 x 10-3, at decoupling if we take into account the constraint OmegaB < 0.16 coming from the primordial nucleosynthesis calculations (130). In the simplest models, this will lead to a temperature anisotropy of MBR of (DeltaT / T) approx 0.3delta approx 1.7 x 10-3 at angular scales of about 4.4'. (The angular scale of anisotropy theta and the linear scale lambda of the perturbation producing the anisotropy are related by theta approx 0.55' Omega h(lambda / 1 Mpc).] No such anisotropy has been discovered. The bounds on (DeltaT / T) (less than 3 x 10-5 at 4.5' (119)] suggest that delta 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 deltaphik appeq (lambdaphy c-1 H) deltak]. It is possible to think of another kind of perturbation (isothermal) in which delta rho = 0 but the p = p (rho) 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. Back.

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