### 3. THE CONCEPT OF INFLATION

3.1 The Basic Mechanism

Some of these difficulties of the standard FRW models may be approached by modifying the dynamics of the very early universe. The trick lies in introducing a temporary phase during which the universe expanded exponentially as in the classical de Sitter (35) model. Such an exponentially expanding phase is called inflation. The de Sitter model described an empty universe with expansion caused by negative stresses due to the -term. The inflationary universe also requires a -term; but here it arises and is supposed to last only during the transient stage in which the GUTs phase transition is taking place. We consider the actual mechanisms proposed for this purpose in Section 3.2. Here we outline the actual model that emerges and the way it can handle some of the awkward features of the standard model.

Consider a model for the universe in which the universe was radiation-dominated up to, say, t = ti, but expanded exponentially in the interval ti < t < tf:

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For t > tf, evolution is again radiation-dominated [a (t) t1/2] until t = teq 4.36 x 1010 ( h2)-2 s. Evolution becomes matter-dominated for teq < t < tnow = t0. Typical values for ti, tf, and H suggested in the literature are

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which give an overall inflation of about A exp N exp (70) 2.5 x 1030 to the scale factor in the period ti < t < tf. At t = ti, the temperature of the universe is about 1014 GeV. During this exponential inflation, the temperature drops drastically but the matter is expected to be reheated to the initial temperature ~ 1014 GeV at t tf. The reheating takes place when the phase transition is over, and the energy released in the process is passed on to the radiation content of the universe. The situation is analogous to the reheating that takes place when supercooled steam condenses and releases its latent heat. Thus, inflation effectively changes the value of S T (t) a (t) by a factor A = exp (70) 1030. Note that this quantity S is conserved during the noninflationary phases of the expansion.

Such an evolution, if it can be implemented dynamically, has several attractive features. Let us first explore how this evolution helps us to overcome some of the difficulties of the FRW models (48, 67, 107).

Consider first the ``flatness problem,'' which concerns the unusually small value of the curvature term (kc2 / a2) in the early epochs. Inflation of a (t) by a factor A decreases the value of this term by a factor A-2 10-60. Thus one can start with moderate values of (kc2 / a2) before inflation and bring it down to a very small value for t 10-33 s. This solves the flatness problem, interpreted as the smallness of (kc2 / a2). Notice that no classical process can change a k = ± 1 universe to a k = 0 universe, since doing so involves a change in topological properties. What inflation does is to decrease the importance of (kc2 / a2) so much that, for all practical purposes, we can ignore the dynamical effect of the curvature term thus having the same effect as setting k = 0.

Inflation also solves the horizon problem by bringing the entire observed region of the last scattering surface into a causally connected patch. The coordinate size of the region in the LSS from which we receive signals today is

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while the coordinate size of the horizon at t = tdec will be

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[We have used the relations t0 >> tdec, A >> 1, ti H-1 and adec = ai A (tdec / tf)1/2]. The ratio

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is far larger than unity for A 1030. Thus, all the signals we receive are from a causally connected domain in the last scattering surface (LSS). Note that, in the absence of inflation, l (tdec, 0) = (2ctdec / adec), so that R = 2/3 (tdec / t0)1/3 << 1. This value is amplified by the factor 2 Ati (tf tdec)-1/2 107 in the course of inflation.

Inflation can also reduce the density of any stable, relic particle, e.g. the magnetic monopoles, by a dilution factor A-3 10-90, provided these relics were produced before the onset of inflation. Likewise, any discontinuities, e.g. the domain walls etc., are expanded away from one another so that the chance of observing one by a typical observer is made negligibly small. Further, because of the reheating at the end of inflation, one arrives at the large entropy presently observed in the universe. By the same token, any relic of the early universe that survives to the present (like baryon number) must be generated after the end of inflation.

The most attractive feature of the inflationary model is, however, the possibility of generating the seed perturbations that can grow to form the large-scale structures (8, 49, 52, 111). This can be achieved in the following manner:

In the FRW models with a (t) tn (n < 1), the physical wavelengths (which grow as a tn) will be far larger than the Hubble radius [which grows as H (t)-1 t] in the early phases. This situation is drastically altered in an inflationary model. During inflation, physical wavelengths grow exponentially [ a exp Ht] while the Hubble radius remains constant. Therefore, a given length scale has the possibility of crossing the Hubble radius twice in the inflationary models. Consider, for example, a wavelength 0 ~ 2 Mpc today (which, according to (15) contains a mass of a typical galaxy 1.2 x 1012 ( h2) M]. This scale would have been

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at the end of inflation. (This is, of course, much larger than the typical Hubble radius at that epoch, cH-1 1.4 x 10-24 cms.) But at the beginning of inflation, its proper length would have been

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This is much smaller than the Hubble radius. This Hubble radius remains constant throughout the inflation while increases exponentially. In an interval of t = t - ti 18H-1, will grow as big as the Hubble radius.

 Figure 2. The figure illustrates the growths of the length scales of primordial fluctuations in relation to the Hubble radius during the inflationary and Freidman phases of expansion.

The situation is summarized in Figure 2. We see that the scales that are astrophysically relevant today were much smaller than the Hubble radius at the onset of inflation. (Therefore, causal processes could have operated at these scales.) During inflation, the proper wavelength grows, and becomes equal to the Hubble radius cH-1 at some time t = texit. For a mode labeled by a wave vector k, this happens at texit (k) where,

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That is, when (kc / aH) = 2. In the radiation-dominated era after the inflation, the proper length grows only as t1/2 whereas the Hubble radius grows as t; thus the Hubble radius ``catches up'' with the proper wavelength at some t = tenter (k). For t > tenter, this wavelength will be completely within the Hubble radius.

Inflationary models thus allow to be less than cH-1 at two different epochs: an early phase, t < texit (k) and a late phase, t > tenter (k). Any perturbations generated by physical processes at t < texit can be preserved intact during texit < t < tenter and can lead to formation of structures at t > tenter.

What can lead to perturbations at t < texit? Since the physical processes taking place in this epoch are quantum mechanical by nature, quantum fluctuations in matter fields are obvious candidates as seed perturbations. Therefore, in principle we can now generate (and compute) the density inhomogeneities in the universe - indeed a major achievement of inflation.

Notice that all the above conclusions only depend on the scale factor growing rapidly (by a factor 1030 or so) in a short time. For example, if the energy density (t) varies slowly during ti < t < tf, then one has near exponential expansion with

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where H2 (t) = [8 G (t) / 3c2]. This provides a general definition of N.