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The first conception of the inflationary stage of evolution Guth (1981) was based on a definite physical picture formulated by Kirzhnits and then rigorously developed by Kirzhnits and Linde (1972). They used the idea of a Higgs scalar field with a maximum V(0) = Vmax > 0. In this state, dV / dphi|phi=0 = 0, so that phi = 0 = const and phidot = phiddot = 0 is a solution. Obviously, p = -epsilon in this case. The corresponding solution to the cosmological equations in the preceding sections, (8.2) and (8.3), are

Equation 9.1 (9.1)


Equation 9.1

for the open, flat, and closed universe cases, respectively (Fig. 3). These three functions are identical for H0t >> 1 when we are dealing with the exponential "inflationary" stage in the expansion.

Figure 3
Figure 3.

The first idea of the authors was that one begins at the very high temperature Th at which the thermal oscillations in the scalar field go over the barrier (Fig. 4): phibar2 > phi2min. Obviously, due to symmetry, the average phibar = 0 at the high temperature Th. The low-temperature (Tl) equilibrium state, in which there are small oscillation about the minima, is show in Fig. 5. We have not included the excess energy due to the surfaces separating domains with phi = phimin and phi = -phimin in this figure. The existence of the domain energy is physically obvious: there is a contribution to the energy that is proportional to (grad phi)2, i.e., the square of the spatial derivatives.

Figure 4
Figure 4.

Figure 5
Figure 5.

A total breaking of symmetry at low temperature, with all space being one domain (with phi = phimin, for example) is thermodynamically preferred. However, this cannot be achieved in a system with finite cooling time: the sign of the average phi is not correlated in different regions separated by a large distance. However, supercooling can occur via the stabilizing effect of the energy connected with the gradient of phi. The energy in the fluctuations and the other fields excited at high temperature fall off rapidly (proportional to a-4, i.e., e-4H0t) during the expansion, so that they become completely negligible compared with V(0) very soon after the supercooling begins. The oscillations are damped out during the supercooling by the expansion, while phi fluctuates near the maximum (Fig. 4). This is the unstable situation that gives rise to the inflationary expansion.

Qualitatively, the instability is a blessing: the present-day situation requires that the inflationary stage end at some point! We need the decrease in V(phi) from V(0) to Vmin(phimin) in order to reheat the Universe, to give rise to the relic radiation, protons, neutrons, galaxies, stars, and life.

However, attempts at quantitative calculations of the perturbations created during the transition from phi = 0 to phi = phimin give bad results. For the mathematically oriented reader: the basic solution for a closed universe,

Equation 9.2 (9.2)


Equation 9.2

is called the "closed de Sitter universe." It has the peculiar property of total homogeneity: all world points, including points with different t, are equivalent in all respects. This is not easy to see from the metric written above: it would seem that the section t = 0 would have the smallest volume, V3 = 2pi2a3(0) = 2pi2c3 / H30. However, this is not true, since there is a coordinate transformation that makes the other world points lie on other minimal-volume hypersurface t' = 0.

The flat and hyperbolic de Sitter solutions are not complete. One can think of them as parts of the complete closed solution with another choice of t = const hypersurface. This also means that the transformation of the phi = 0 unstable state into hot plasma is purely probabilistic in this model. These important details can be found in the paper by Grishchuk and the author in the proceedings of the Moscow conference on quantum gravity (Grishchuk and Zel'dovich 1982).

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