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3 Cf. P. Anninos, R. A. Matzner, T. Rothman, M. P. Ryan Jr., Phys. Rev. D 43, 3821 (1991).

An alternative argument current in inflationary literature is that the patch of primordial matter that later became our visible universe (the 10-48 proton equivalent referred to earlier) had accidentally homogenized during the lull before inflation set in - homogenized, in fact, to a sufficient degree to justify Friedmanian evolution. The large horizons then encouraged even greater smoothing to occur. In this connection it may be of interest to note that in all FRW models the metric size of the particle horizon near the big bang is proportional to 1 / jp (cf. Exercise 18.6). This may be some indication of the relative ease of homogenization even out of chaos, if the density is sufficiently low. (But also recall Penrose's criticism on p. 380.)

A second problem often cited as having been successfully solved by inflation is the so-called 'flatness problem'. This refers to the perceived improbability, in standard cosmology, of finding our present universe so very nearly flat. Inflation, on the other hand, claims to predict flatness. According to inflation, the early universe expands exponentially, driven by something equivalent to a huge cosmological constant A. The pre-inflation energy density quickly dissipates and leaves essentially pure vacuum. So the possible Friedman solutions for this phase are the models (c), (d) and (e) of (18.31).

The corresponding [cf. (18.52)] are easily calculated; they are, respectively,

At the end of inflation, therefore, no matter which of the three models, ^a = 1, either exactly or to very high accuracy. All other density has dissipated by then, and so the flatness condition ^ + = 1 [cf. (18.55)] is satisfied to very high accuracy. Yet we cannot conclude k = 0. Eqn (18.54) still yields, not surprisingly, k = 0, 1, -1 in the respective three cases. It seems that the only inflational argument for flatness is not that k = 0 but that R is necessarily very big with inflation, so that the actual curvature k/R2 is is very small.

One crucial contribution that inflation has made, and which no other theory has so far matched, is to predict the fine structure of the temperature fluctuations in the cosmic microwave background across the surface of last scattering. Accordingly the results of the COBE satellite observations in 1992 were anxiously awaited. And, indeed, they seemed to confirm the predictions. Even stronger confirmation came from the later 'Boomerang' balloon experiment and from the WMAP satellite. These predictions, moreover, are of great importance in the problem of how inhomogeneities formed in the substratum that eventually led to the formation of structures like the galaxies.

We may also mention the solution of the 'monopole problem'. This has been called an internal problem, since to perceive it as a problem you must believe in grand unified theory to start with. But then indeed you have a problem explaining where all the magnetic monopoles went, that should have been created at ~ 1027 K. According to this theory, monopoles of enormous energy (~ 1016 GeV) should occur at a spacing of about one per particle-horizon volume at the end of inflation. If the particle horizons were as small as in standard cosmology (cf. Exercise 17.10), the mass density of monopoles would exceed that of baryons by a factor 1014! With inflation, on the other hand, there is at most one monopole in the universe we see.

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