2. The Amount of Inflation

One can find the amount of inflation by considering the change of the scale factor. Considering the example of quasi-exponential expansion, meaning that the Hubble constant need not be constant. Then,

(76)

Using the slow-roll equations, the number of e-foldings can be expressed in terms of the inflaton potential. Dividing (73) by (72) yields,

(77)

Using this result, with the formula for N (76), one gets,

(78)

Here the fact that the SRA has been used is expressed using `' in (78).

For N > 60, which is needed to solve the initial value problems [22], we again find >> M_p. This can be seen from (78), where V' ~ V / using the SRA. This means that if one chooses a potential of 1/2 m^{2 2}, one must choose the coupling, m² to be small. Given a self interacting potential term, ⁴, the coupling must be extremely weak, << 1. This coupling agrees nicely with theories of supergravity and certain string theories, although other potentials are ruled out because of their couplings, such as the weak coupling. This leaves the question. Can inflation be considered without a theory of quantum gravity? As mentioned before, the inflationist is often not concerned with these initial stages of inflation. The common standpoint is that chaotic inflation can present an evolution and then one studies the predictions of this evolution. As aesthetically displeasing as this may be, it allows cosmology to progress further without a quantum theory of gravity. Ultimately this issue will be addressed to create a complete picture of the creation of the universe; however, Guth has recently shown this consideration may not need to be considered [44], [43].

Given the slow-roll conditions and the number of required e-foldings (N), one can test a model inflaton to see if it is compatible with an inflationary scenario. With this generic framework that has been set forth, one can construct particle theories and then test their validity within the context of inflation theory. However, the slow-roll approximation and initial value problems (N > 60) are not the only constraints on the inflaton and therefore particle theory. The inflaton is further restricted by the predicted large-scale structure of the universe, along with the mechanisms involved with reheating of the universe at the end of inflation. The large-scale structure is determined by density perturbations resulting from quantum fluctuations in the evolution of the inflaton field. This analysis can actually be done without the advent of quantum gravity; however, it is outside the scope of this paper. Instead, the author hopes to manifest the stringency of these parameters on the inflaton field by addressing the observational consequences and predictions that inflation offers. In the next section these observational tests will be explored.