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

Next Contents Previous


2.1 The Friedman-Robertson-Walker Model with a Hot Big Bang

The Friedman models (44, 45) were proposed as the simplest solutions of Einstein's equations without the Lambda-term. Robertson (104) and Walker (121) showed that global symmetry arguments of homogeneity and isotropy lead to a spacetime geometry described by the line element

Equation 1
Equation 1 1.

For k = 0, the expression in parentheses describes the Euclidean metric for three dimensional space in spherical polar coordinates. For k = + 1, the three-space has the closed topology of the surface of a hypersphere in four dimensions, whereas for k = - 1, the three-space is open. The (r, theta, phi) coordinates are the ``comoving coordinates'' of a typical ``fundamental observer'' who ``sees'' the universe as homogeneous and isotropic. The proper time of such an observer is measured by t, called the cosmic time, and the corresponding frame of reference is called the cosmic rest frame. The function a (t) denotes the typical length scale of the universe and, for an expanding model, it is called the expansion factor.

This function a (t) is determined by Einstein's equations if the energy momentum tensor of the physical contents of the universe are known. If epsilon is the energy density (ident rho c2) and p the pressure of these contents, these equations can be reduced to

Equation 2 2.

Equation 3 3.

The second equation is none other than that of energy conservation under adiabatic expansion.

The present state of the universe is matter-dominated in the sense that bulk of the contribution to rho comes from matter that is either at rest in the cosmic rest frame or is moving slowly (compared to c) relative to it. For such matter pm approx 0, and from Eq. 3 we get rho propto a-3. The simplest model of this kind is the flat model, for which rhom propto (adot / a)2. This leads therefore to a critical density rhoc appeq 2 x 10-29 h2 gcm-3 of the matter. For k = + 1 models, rho > rhoc, whereas for k = - 1 models, rho < rhoc, In general we write the density rho as Omega rhoc with Omega > 1 for k = 1, Omega < 1 for k = -1.

The small component of radiation present today was, however, more dominant in the past. This is because for radiation with the equation of state epsilonr = 3 pr, Eq. 3 gives epsilonr propto a-4. We denote by teq and aeq ident a (teq), the epoch and the expansion factor when epsilonm = epsilonr. Clearly,

Equation 4 4.

The microwave background radiation (MBR) energy density may be taken as a close approximation to (epsilonr)0. A simple calculation gives

Equation 5 5.

We may specify the redshift z of an epoch by the relation

Equation 6 6.

Clearly 1 + zeq approx 2.3 x 104 (Omega h2) theta-4; if we take rhomo approx 3 x 10-31 gcm-3, then zeq approx 103. This means that for z > zeq the universe is radiation-dominated, whereas for z < zeq it is matter-dominated.

At times t << teq (i.e. z >> zeq), the radiation dominated over matter, and we find that the function a (t) was approximately given by

Equation 7 7.

(This presupposes the neglect of the curvature term kc2 / a2 in Eq. 2 in comparison with adot / a. As discussed below, this assumption is nontrivial.) Since epsilon propto T4 for radiation, it follows that T propto t-1/2. This time-temperature relationship can be written, more quantitatively, as

Equation 8 8.

where g denotes the effective degrees of freedom of relativistic particles present in thermodynamic equilibrium at that temperature. The number varies between a value of about 102 (at 1020 MeV) and 3 or so (at present).

Working backwards chronologically, there are three significant epochs in the early universe. In the first epoch, electrons combined with ions to form neutral atoms. The characteristic energy was the binding energy (~ 13.6 eV) of the H-atom, and the temperature was about 3000-4000 K. We denote this epoch by t = tdec to indicate that the radiation decoupled from matter in the absence of free electrons as scatterers. During the second epoch, free neutrons and protons combined to form light nuclei at temperatures between 108-109 K. In the third epoch, that of the grand unified theories (GUTs), the breakdown of grand-unification symmetry at energies of about 1014 GeV led to the bifurcation of the electroweak interaction from the strong interaction. For this t ~ 10-35 s. (There is also a fourth epoch preceding the GUTs epoch, prior to which the universe was governed by the laws of quantum gravity. Known as the Planck epoch, it was at tp ~ 10-43 s. Classical general relativity could not be valid up to this epoch.)

The third epoch, at temperatures around 1014 GeV, is of interest to particle physicists. Some of the basic features of our universe - e.g. the photon-to-baryon-number ratio, which is presently observed to be (Ngamma / NB) = 3.52 x 107 (Omega h2)-1 theta3 - may have become frozen in at this epoch. Discrete structures (galaxies, clusters, superclusters, etc.) could conceivably grow from primordial seeds going back to this epoch. (For reviews and discussions of some of these problems, see 81, 84, 125).

The confidence with which physicists extrapolate their discussions to epochs as early as 10-35 s rests on the successes of the standard hot big-bang model at the first two stages, namely the prediction of relic abundances of light nuclei and the interpretation of MBR as the relic radiation. The standard theory encounters many problems of a fundamental nature, however, which require a radical rethinking of the very early scenario. This was the motivation for introducing the concept of inflation. Before considering the proposed remedy, it is appropriate to take a look at the problems.

Next Contents Previous