Annu. Rev. Astron. Astrophys. 1991. 29: 325-362
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3.2 The Epicycles of Inflation

Since the inflationary idea seemed to be quite attractive, several mechanisms were devised by which this idea can be implemented. Each of these models has some advantages and disadvantages and none of them is completely satisfactory. We briefly highlight three models.

For the universe to expand exponentially, the energy density should remain (at least approximately) constant. Various models of inflation differ in the process by which this is achieved. In most of them the quasi-constant energy density epsilon (t) is derived from phase transition at the GUTs epoch. Although in specific details one grand unified theory may differ from another, almost all of them involve gauge theories with a mediating role played by the Higgs scalar field phi. We need not go into the intricacies of how phi is related to the other matter fields. The feature of interest to us is that the potential energy density V of the scalar field phi depends on the ambient temperature T.

At any given temperature T that is higher than a critical temperature Tc, the minimum value of V is found to be at the expected zero of phi. We may term this minimum at phi = 0 as the vacuum state of phi. As the temperature is lowered, however, the minimum of V may no longer remain at phi = 0 but may shift to a finite value phi = sigma. This phase transition occurs at T = Tc and may be likened to the condensation of steam. Thus phi would tend to transit from phi = 0 to phi = sigma.

If phi were to condense immediately at Tc, all the excess energy could be released at once. In the more likely case of supercooling, however, phi may continue at phi = 0 and move to the true minimum phi = sigma later. During this transitional stage, the state phi = 0 is called the false vacuum state, since the true vacuum is now at phi = sigma. The original model for the inflation, due to Guth (48), invoked this temperature dependence of the potential energy of the Higgs field V (phi, T). [The potential energy has the form shown in Figure 3. Here Tc approx 3 x 1014 GeV.]

Figure 3

Figure 3. The potential energy of the Higgs field phi at various temperatures in the original model proposed by Guth.

At temperatures T >> Tc, the potential V has only one minimum (at phi = 0) with V(0) approx (1014 GeV)4. As the temperature is lowered to T ~ Tc, a second minimum appears at phi = sigma. For T << Tc, the phi = sigma minimum is the ``true'' minimum [i.e. V (sigma) approx 0 << V (0)]. Now consider what happens in the early universe as matter cools through T approx Tc. At T >> Tc, the minimum configuration corresponds to phi = 0 whereas for T ~ Tc it is phi = sigma. Matter in the universe does not instantaneously switch over from phi = 0 to phi = sigma, however. The universe can get ``stuck'' at phi = 0 (the false vacuum), with V = V (0), even at T < Tc, and will expand exponentially because the dominant energy density driving the expansion is the constant V (0) - V (sigma) approx V (0). Over the course of time, thermal fluctuations and quantum tunneling will induce a transition from the false vacuum phi = 0 to the true vacuum phi = sigma, thereby ending the inflation in localized regions (``bubbles''). The phase transition is expected to be completed by the expanding bubbles colliding, coalescing, and reheating the matter.

Detailed analysis, however, shows that this model does not work (50). In order to have sufficient amount of inflation, it is necessary to keep the ``false'' vacuum fairly stable. In such a case, the bubble nucleation rate is small and even the resulting bubbles do not coalesce together efficiently. The final configuration is very inhomogeneous and quite different from the universe we need.

The original model was soon replaced by a version based on a very special form for V (phi) called the Coleman-Weinberg potential (3, 74, 75). At zero temperature, this potential is given by (25)

Equation 26a
Equation 26b 26.

This potential is extremely flat for phi ltapprox sigma and drops rapidly near phi approx sigma. At finite temperatures, the potential picks up a small barrier near the origin [at phi appeq O (T)] with height O (T4), creating a local minimum at phi = 0 (see Figure 4). This false vacuum, however, is quite unstable when the temperature becomes O (109 GeV) (74). The scalar field rapidly tunnels to phi approx phi0 approx O (H), and starts ``rolling down'' the gently sloped potential toward phi = sigma. Since the potential is nearly flat in this region, the energy density driving the universe is approximately constant and about V (0) approx (3 x 1014 GeV)4. The evolution of the scalar field in this slow rollover phase can be approximated as

Figure 4

Figure 4. The Coleman-Weinberg potential that was used in the first major revision of the inflationary model.

Equation 27 27.

where we have ignored the phiddot term and H = (4 pi BG sigma / 3c2) approx 2 x 1010 GeV (in energy units). If the slow roll over lasts when phi varies from phistart appeq O (H) to some phiend ltapprox O (sigma), then

Equation 28 28.

For the typical values of the Co1eman-Weinberg potential, this number can easily be about 102, thus ensuring sufficient inflation.

As phi0 approaches sigma, the field ``falls down'' the potential and oscillates around the minimum at phi = sigma with the frequency omega2 = V"(sigma) approx (2 x 1014 GeV)2 >> H2. These oscillations are damped by the decay of phi into other particles (with some decay time Gamma-1, say); and by the expansion of the universe. If Gamma-1 << H-1, the coherent-field energy (1/2 phidot2 + V) will be converted into relativistic particles in a timescale Deltatreheat approx Gamma-1 << H-1. This will allow the universe to be reheated to a temperature of about Treheat approx omega 2 x 1014 GeV approx Tinitial. The decay width of several Coleman-Weinberg models can be about Gamma-1 approx 1013 GeV >> H. This ensures good ``reheating'' of the universe (2, 4, 36). Because the field has already tunneled out of the false vacuum before the onset of inflation, we do not face the problems that plagued the original inflation. Instead of several bubbles having to collide, coelesce, and make up the whole observable universe of today, we have one huge bubble encompassing everything observable now.

Though it is an improvement on the original version, this model, too, is not free from problems. The field should start its slow roll over from a value phis approx H to ensure sufficient inflation. The quantum fluctuations in the scalar field are about Deltaphi appeq (H / 2pi) (75, 120). Since phis ~ Deltaphi, the entire analysis based on semiclassical V (phi) is of doubtful validity. The second and more serious difficulty stems from the calculation of density perturbations in this model: They are too large by a factor of about 106, unless the parameter B is artificially reduced by a factor 10-12 or so! (See Section 4.)

The original model for inflation used a strongly first order phase transition whereas the second model may be considered to be using a weakly first order (or even second order) phase transition. It is possible to construct inflationary scenarios in which no phase transition is involved. The idea of chaotic inflation, suggested by Linde, falls in this class (76). In this model, the potential has a very simple form: V (phi) = lambda phi4. inflation results because of the rather slow motion of phi from some initial value phi0 toward the minimum. (The initial nonzero value of the phi0 is supposed to be due to ``chaotic'' initial conditions.) This model can also lead to sufficient inflation but suffers from two other difficulties: (a) To obtain the correct value for the density perturbation, it is necessary to fine-tune lambda to very small values: lambda propto 4 x 10-14. (b) In order for the inflation to take place, the kinetic energy of the scalar field has to be small compared to its potential energy. Detailed calculation shows that this requires the field to be uniform over sizes bigger than the Hubble radius, a requirement completely against the original spirit of inflation!

A further epicycle in the saga of inflation envisages a universe whose origin was without a big bang (77). In this version, the de Sitter type inflationary phase is self-reproducing in a chaotic set up with the help of large scale quantum fluctuations of a scalar field phi. The bubbles of FRW models are nucleated in it at random points of space and time through quantum phase transitions.

An attractive feature of the de Sitter expansion is that because of its rapidity, the universe loses all information on initial conditions. This is a conjecture known as the cosmic baldness hypothesis (10). It has not been rigorously proved but looks plausible. On the basis of this hypothesis, one can assert that whatever the initial conditions, the universe will eventually reach the de Sitter state.

A solution to the bubble nucleation and coelescence problem of the original Guth model (sometimes referred to as the graceful exit problem) was proposed in yet another way by La & Steinhardt (71). In their extended inflationary cosmology, these authors used the Brans-Dicke theory of gravity (see 20) instead of general relativity as the background theory for the early universe. The inflationary phase in this model has a power law type of expansion factor instead of the exponential one, thus allowing the inflationary phase to end gracefully through bubble nucleation.

Nevertheless, this idea also ran into trouble with distortions of the MBR and was changed to hyper-extended inflation. The background theory of gravity for this model differs from the Brans-Dicke theory through the inclusion of higher order couplings of the scalar field with gravity (113). In a rapidly changing subject in which the half-life of a theory is one year, passing judgment on the merits of this scenario is difficult.

The schemes and shortcomings discussed above are typical of several other models suggested in the literature. The most serious constraint on inflationary scenarios arises from the study of density perturbations (discussed in detail in Section 4). No single model for inflation suggested so far can be considered completely satisfactory.

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