Annu. Rev. Astron. Astrophys. 1991. 29: 325-362
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4.3 A Critique of Inflation: Expectations and Performance

The concept of inflation enjoyed considerable popularity in the first half of the last decade, though this enthusiasm seems to have died down somewhat in recent years. Taking stock of the expectations raised by inflationary scenarios in the light of its actual performance is therefore worthwhile.

Such a discussion falls conveniently into two types of questions:

1. How serious are the original problems that the inflation was invoked to solve? What alternative explanations are possible for these difficulties and how well does inflation perform vis a vis the other solutions?

2. What are new features, good and bad, that inflationary scenarios have introduced into cosmology?

Let us start with the first set of questions. In the original version, inflation was suggested as a possible solution to the horizon, flatness, and monopole problems. Of these, the monopole problem has received considerably less attention in recent literature. The currently favored unified field theories do not lead to a monopole problem. Thus, we concentrate on the flatness and horizon problems.

It is indeed true that inflation does solve these problems if these problems were stated in a particular form: inflation can suppress the value of (kc2 / a2) term; it can also make the observed region of the last scattering surface (LSS) a causally connected domain. One should not ignore, however, certain disturbing features in the inflationary solution to these problems:

Both these problems deal with the initial conditions for our universe. Since Einstein's equations permit k = - 1, 0 or + 1 in a FRW solution, the value of k needs to be supplied as an extra input to classical theory. But the creation of the universe (i.e. physics at t < tp) needs to be understood quantum mechanically rather than classically! [It is quite possible that quantum gravitational effects make a k = 0 model highly probable (see e.g. 87]. Inflation thus tries to provide a classical answer to an inherently quantum gravitational question.

This anomaly is quite striking when we consider the horizon problem. The horizon problem exists because the integral

Equation 48 48.

is finite. But do we know that it is finite? To make such a claim we have to assume that there was a singularity at t = 0 and that we know the behavior of a (t) arbitrarily close to t = 0. For t < tp, quantum gravitational effects will modify the behavior of a (t) and will probably eliminate the singularity problem. Then, for almost all a (t) (except for a class of functions of measure zero), the above integral will diverge, automatically solving the horizon problem. In other words, flatness and horizon problems owe their existence to our using classical physics beyond its domain of validity. There are several quantum gravitational models in which these problems are solved as an offshoot of elimination of the singularity problem (see e.g. 82, 83, 88). This is a possibility that did not exist in classical cosmology.

Even within the context of the inflationary models, the solutions to these problems work only for a limited time (40, 41, 93). For example, Ellis has shown how the flatness problem will resurface in the late-time behavior of a k neq 0 universe. [This should be clear from the fact that at sufficiently late times, both the k = + 1 and k = - 1 models will behave very differently irrespective of the present value of the (kc2 / a2) term.] In the case of the horizon problem, it can be shown that LSS will appear homogeneous only for times t < tcrit when

Equation 49 49.

[For t approx 10-34 s, tf approx 100ti, H approx (109 GeV), tcrit approx 3 x 1023 s, which is far larger than t0 approx 3 x 1017 s; so, right now, t0 < tcrit.] This quantity tcrit is determined once and for all by microscopic physics at t approx 10-34 s! If we wait long enough, t will be larger than tcrit, and the horizon problem will resurface. Thus inflation offers only a temporary, though long, relief from these problems, and one has to invoke the anthropic principle (11) to justify its success. In contrast, solutions based on quantum gravitational models solve these problems permanently.

Let us now consider the second question, that is, the new features that inflation has brought into cosmology. The major success of inflation in our opinion lies in these factors.

First, inflation has provided a mechanism that allows one, in principle, to compute the spectrum of density inhomogeneities from fundamental physics. This must be considered a success because this is the first time we have a computable mechanism for producing density inhomogeneities.

Also, a definite prediction emerges from inflationary models. The same mechanism that produces the density inhomogeneities also produces gravitational wave perturbations. These perturbations also have a scale invariant power spectrum and an r.m.s. amplitude of about (H / 1019 GeV). The energy density of the gravitational waves contributes a fraction Omegagrav approx 10-5 (H / mp)2 h-2 to the critical density, where mp is the Planck mass (1, 5, 43, 105, 129). Such perturbations can induce a quadrupole anisotropy in the MBR. The present bounds on this anisotropy (ltapprox 10-4) suggest that H < 1015 GeV. The value of Omegagrav can also be restricted by the timing measurements of the millisecond pulsar; the present bound is Omegagrav (lambda ~ 1pc) leq 3 x 10-7. A positive detection of quadrupole anisotropy in MBR or a direct detection of relic gravitational radiation will certainly go a long way toward boosting confidence in inflation. (The Laser Interferometer Gravity Wave Observatory (LIGO) and similar projects can, in principle, reach a sensitivity of Omegagrav ~ 10-11.]

Second, several investigations have suggested that inflation could be a generic feature of cosmology; in other words, for almost any kind of V (phi) and for a large class of initial conditions, the universe will undergo a phase of exponential expansion (63, 64, 112, 116). The amount of inflation, time of occurrence, etc. can all be quite varied, but the physical phenomenon is probably here to stay (see, however, the discussion in Section 6).

The above two features also have a negative side, discussed in Section 4.2: Inflation produces too much density inhomogeneity. Every unworkable model for inflation rules out a parameter range for the potential V (phi) and can, in principle, constrain particle physics models. This is probably the most important unsolved problem in the physics of the early universe.

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