In the last two sections of this book we report on two rather speculative topics, which the reader (according to taste) can either accept or reject. But since they form part of the present-day cosmological debate, it is well to be informed about them. The first of these topics is inflation. It affects only the first 10-35 s (!!) of the universe. Thereafter the inflationary model agrees kinematically with the standard FRW model, except for one important detail: the size of the particle horizons (creation light fronts) around each fundamental particle, both now and at all earlier times, are many, many orders of magnitude larger with inflation than without it. And this is one of its often cited attractions: inflation, it is claimed, solves the smoothness problem with its large horizons. But it also has important implications for cosmogony.

Though the basic idea, like most novel ideas, had precursors (Zeldovich, Starobin-sky, Sato, etc.), inflation in its modern form was first put forward by Guth in 1980. Since then many cosmologists (especially those coming to cosmology from particle physics) have strongly rallied to this theory, even though it is still very largely based on pure hypothesis. Others, perhaps remembering the equally strong attachment that many people (possibly they themselves) felt for the equally hypothetical steady state theory during the fifties of the last century, have remained more skeptical.1

Inflationary cosmology is based on not yet fully established 'grand unified theories' (or GUTs). According to these theories, the quantum vacuum is stable at the highest temperatures, but then, some 10-37 s after the big bang, when the temperature had dropped to a critical value of ~ 1027 K, the vacuum became unstable but still highly energetic. This 'false' vacuum is described by an energy tensor having precisely the form of the cosmological term in Einstein's field equations but with a huge A, some 100 orders of magnitude greater than today's 'cosmological' A. (Cf. Section 18.2E.) At this stage, or soon thereafter, the vacuum energy begins to dominate that of any other matter or radiation present, and causes a rapid expansion of the universe, by a factor of ~ 1043, in a mere 10-35 s. The expansion is exponential, just like that of the de Sitter model [cf. (18.31)(c)], and for the same reason, namely the (here temporary) presence of a A term in the field equations, albeit with a different interpretation. The positive vacuum density corresponding to the A term remains constant throughout this expansion [cf. (18.22)]. As a consequence, the matter-energy content of each comoving volume increases in proportion to that volume. In Guth's phrase, this was 'the ultimate free lunch'. In inflationary cosmology the big bang itself was a mere 'big whimper', most of the matter-energy of the universe being created during the inflationary phase. When this phase comes to an end, the false vacuum becomes a real vacuum and its energy is converted into radiation. From here on the universe is kinematically and dynamically indistinguishable from one that could have been created by a standard big bang of the right strength.

1 For two recent and sympathetic accounts of inflation, see, for example, A.H.Guth, Physics Reports 333-334,555 (2000) and A. Linde, ibid., p. 575. For a strong critique, see R. Penrose, loc. cit. (Footnote 1 on p. 380.)



Fig. 18.6, which is obviously not drawn to scale, schematically illustrates the relation between standard Friedman and inflationary cosmology2; BB stands for big bang, BW for big whimper, and BI and El for beginning and end of inflation. We know that the present conditions of our universe (age, density, expansion, acceleration) in principle determine a unique Friedman model—and inflationists have no quarrel with that. We can follow this model back in time. Somewhere around decoupling time, for increased accuracy, we can replace the dust model by a radiation model, and still, inflationists have no quarrel with that. Only after we come to a radius of about

10 25 R0 do the backward continuations diverge: the inflationary R drops somewhat more gently. But since the divergence of the two models in that miniscule earliest stretch of time has essentially no effect on the calculated age of the universe, inflation is irrelevant in the empirical determination of the correct Friedman model along the lines of the preceding section.

It is perhaps worth taking a closer look at the join between the two models at iEI. For dynamical reasons R must be continuous at the end of inflation, as can be seen from eqn (18.9): During inflation A is constant and p essentially zero: whatever non-vacuum p was present before inflation is quickly dissipated by the expansion. At the end of inflation the term -A/3 [or -8nGpVac/3c2, cf. (18.22)] shifts to the RHS,

2 I am grateful to Professor John Peacock for drawing my attention to a significant error in my previous version of this Figure.

with pvac -—> p: R and k do not change. (The 'cosmological' A-term is still totally negligible then.) So at R = REI the accelerating branch of inflation smoothly joins the decelerating branch of standard cosmology, and so must lie to the left of it, and be slower.

There is actually a conceptual problem with the sudden appearance, at the end of inflation, of a material (mainly radiative) substratum performing Hubble expansion on what was essentially, up to that time, a featureless Lorentz-invariant vacuum. How does each newly created particle know its required momentum at birth, or, in other words, the Hubble flow at its location? During inflation, which enlarged the universe by a linear factor of ~1043, the density remained constant, so that only one part in 103x43 of the final matter-energy was present initially. Our observable universe of some 1011 galaxies thus grew out of a patch of pre-inflationary matter-energy equivalent to only 10-118 of a galaxy or 10-48 of a proton. The rest came into being at tEI. It is hard to see how that miniscule remnant of primordial matter-energy could have provided at tEI a matrix for the new substratum; and if not that, then what?3

As we have already mentioned, there is one main difference between the universe that emerges from inflation at tEI and the corresponding FRW big-bang universe at that juncture. The former has vastly more inclusive past light cones at each point. To see why, consider the inflationary epoch from tBI to tEI and during that time let R = A exp(Ht), with A, H constant, assuming for simplicity that k = 0 [cf.(18.31)(c)]. Then for the 'conformal' time TBI reckoned backward from tEI [cf.(17.7)(ii)] we have

And this can be made arbitrarily large by simply choosing RBI small enough.

Qualitatively the reason for the enlarged particle horizon is to be found in the relatively shallow part that all exponential curves have, namely the initial slow expansion phase. In terms of the balloon picture, the beetles (photons) can cover large comoving angular distances f at small radius before that radius swells in earnest and frustrates much further f -progress. It is sometimes said that inflationary horizons are big merely because the 'huge expansion' stretches them; but, of course, the comparative Friedman models must expand every bit as hugely.

So by the time of recombination our visible universe could well have had time to thermalize. But how this solves the smoothness problem is unclear on two counts. First, there was so little real matter in the universe to thermalize during inflation. And secondly, the above analysis of the inflationary period, including the horizon stretching, was based on FRW formalism, and thus on the assumption of already achieved homogeneity-isotropy!

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