A General Analytic Formula for the Spectral Index of the Density Perturbations Produced during Inflation.

During inflation'> vacuum fluctuations on scales less than the Hubble radius in scalar fields with effective masses much less than the Hubble parameter**> are magnified into classical perturbations in the scalar fields on scales larger than the Hubble radius. These classical perturbations in the scalar fields can then change the number of e-folds of expansion and so lead to classical curvature/density perturba­ tions after inflation. These density perturbations are thought to be responsible for the formation of galaxies and the large scale structure of the observable Universe as well as, in combination with the gravitational waves produced during inflation, for the anisotropies in the cosmic microwave background. The standard calculation 7 H 1 > of the spectrum of density perturbations produced during inflation assumes that there is only one real dynamical degree of freedom during inflation. Although this is the case in most of the models of inflation con­ structed up to now, it is by assumption rather than prediction. When one tries to construct models of inflation 3 >,s>. 12 > that might arise naturally in realistic models of particle physics, such as the low energy effective supergravity theories derived from superstrings, one often gets more than one dynamical degree of freedom during inflation. The standard calculation is then generally not applicable. In this paper we derive general analytic formulae for the spectrum and spectral index of the density perturbations produced during inflation. This work is based on earlier work by Starobinsky. 13 > See also Ref. 14). While this work was in slow preparation three other related papers