12. SOME COSMOLOGICAL IMPLICATIONS

We shall not discuss all of the cosmological implications here. The ones that are perhaps the most important are a detailed picture of the spontaneous birth of the Universe and the perturbations that led to the present-day structure (galaxies, clusters, and other density fluctuations). The spontaneous birth of the Universe, the inflationary expansion, and the subsequent evolution of the Universe remain unclear and warrant another review (or rather several reviews). So what follows is two short remarks having a direct bearing on the scalar field theory. The first concerns the hypothesis of an eternally bouncing Universe. Consider a closed Universe whose expansion is limited by some a_max greater than 10²⁸ cm (the region that is now observable). In the simplest case, collapse will follow the expansion. At some instant of time (long after 2 x 10¹⁰ years, the present age of the Universe), the collapse becomes catastrophic and gives rise to infinite density. Can this collapse be reversed and give birth to a new cycle, similar to that which we now observe? There are several arguments (beginning with Tolman's remark concerning the irreversible growth of entropy) against this picture of the Universe. We shall not repeat these arguments here. But new hope for this arose in connection with the existence of formal symmetric bouncing solutions in scalar field theory. In particular, the Higgs field with

(12.1)

yields a solution

A reasonable V_max (corresponding to very heavy X mesons), on the order of (10¹⁴ GeV)⁴ in = c = 1 units, would lead to an H₀ = 10⁹.GeV = 10³⁵ s^-1 and an a_min = c/H₀ = 10^-25 cm in the usual units.

But an analysis of the behavior of the scalar field during compression and expansion shows that this picture is very improbable.

= 0, V = V_max is the only solution in the compression stage. The general solution is ² >> V(), p = +, which leads to irreversible collapse (a(t) = (t₀ - t)^1/3), at least in the classical, homogeneous case. ⁽¹²⁾

The argument against periodic repeating cycles of expansion and collapse is an argument in favor of the spontaneous birth of the Universe. For a detailed discussion of the spontaneous birth of the Universe, we refer the reader to the proceedings of the October 1984 Moscow symposium (Quantum Gravity 1984); for an earlier attempt, see Grishchuk and Zel'dovich (1982). Here, we shall only stress that inflationary expansion is necessary in the overall picture. This is a third argument, closely related to the two arguments of Guth (1981) (horizon and flatness), but not identical.

Assume that the Universe was born with a radius equal to the Planck radius (10^-33 cm) and a density equal to the Planck density (10⁹³ g cm^-3). Then assume that the expansion proceeds without inflation, like a plasma (_r a^-4) or simple dust (_d a^-3).

What is the density we should have at present, when a 10²⁸ cm? Simple arithmetic yields

These numbers are completely unacceptable - the present density of the Universe is on the order of 10^-30 g cm^-3.

This is a crude but weighty argument that shows that an inflationary phase with constant (or approximately constant) density during very strong expansion (a factor of ~ 10³⁰) is necessary. This can be provided by the scalar field, with its peculiar - equation of state. ⁽¹³⁾

This is why an astronomer should read this oversimplified story of the scalar field. The responsibility for the English remains my own.

Thanks to Professors L. P. Grishchuk and L. B. Okun' for discussions and to L. V. Rozhanskii for assistance.

Let us now return to the astronomer described in the introduction who works with telescopes but wishes to have a broader understanding. I do not pretend to give you the ultimate answer - it is not yet known. But perhaps you now know something about the language highbrow theoreticians use to talk about the early Universe. But even this modest goal has only been partially achieved. Alas, the ultimate topic - the very birth of the Universe - remains out of reach of even the highest brow.

¹² High temperature helps maintain = 0, V = V_max during expansion. But it does not help prevent collapse during compression. Back.

¹³ Another variant exists, where the negative pressure is due to the polarization of the vacuum (see (Starobinskii 1980)). This subject is too complicated to treat here, but we hope to return to it in another review paper. Back.