On the Problem of Predicting Inflationary Perturbations

Abstract

We examine the theoretical foundations of standard methods for computing density perturbations in inflationary models. We find that: (1) the time-delay formalism (introduced by Guth and Pi, 1982) is only valid when inflation is well-described by the de Sitter solution and the equation-of-state is nearly unchanging; and, (2) the horizon-crossing/Bessel approximation extends to non-exponential inflation, but only if the equation-of-state is changing slowly. Integration of the gauge-invariant perturbation equations mode-by-mode is the only method reliable for general models. For models with rapidly varying equation-of-state, the correction leads to significantly different predictions for the microwave background anisotropy. An important corollary is that methods proposed for "reconstruction" of the inflaton potential from anisotropy data are unreliable for general models.

Figures (1)

• Figure 1: A comparison of the horizon-crossing and Bessel approximations to exact mode-by-mode integration for an inflaton potential in which the equation-of-state ($\epsilon$) is varying rapidly. The power spectrum has been computed and converted into a prediction of the CMB temperature anisotropy spectrum on large angular scales.