A Rigorous Proof of the Inflationary Spectrum

Fundamental formulae representing the density perturbation spectrums generated in inflationary scenarios are rigorously proved. Quantum fluctuations as initial conditions for structure generation in some inflationary era can be calculated in general scale. Thus, we could avoid the matching of the large and the small scale results at the horizon crossing epoch. This is possible because the perturbed scalar field equation in a special gauge has some remarkable nice properties. In the uniform-curvature gauge the contributions from the metric fluctuations effectively disappear in certain expansion stages, and also we can derive a simple general solution in the large scale limit. The equation reduces to the one neglecting the metric fluctuations in two main bases of inflation scenarios, namely, the exponential and power law type background expansion; this makes the quantum generation simple. The equation leads to a simple form of large scale solution valid for general scalar field potential; this makes the classical evolution simple. We also show that the same large scale integral form solution remains valid for a wide class of generalized gravity theories.