Inflationary perturbation theory is geometrical optics in phase space

TL;DR

This paper reformulates inflationary perturbation theory using geometrical optics in phase space, enabling efficient computation of correlation functions and gauge transformations through raytracing techniques.

Contribution

It introduces a novel geometrical optics approach to inflationary perturbation theory, extending transport equations to all momentum space correlation functions.

Findings

Raytracing reproduces the delta N expansion.

Efficient differential equations for Taylor coefficients.

Compact expression for fNL in terms of principal curvatures.

Abstract

A pressing problem in comparing inflationary models with observation is the accurate calculation of correlation functions. One approach is to evolve them using ordinary differential equations ("transport equations"), analogous to the Schwinger-Dyson hierarchy of in-out quantum field theory. We extend this approach to the complete set of momentum space correlation functions. A formal solution can be obtained using raytracing techniques adapted from geometrical optics. We reformulate inflationary perturbation theory in this language, and show that raytracing reproduces the familiar "delta N" Taylor expansion. Our method produces ordinary differential equations which allow the Taylor coefficients to be computed efficiently. We use raytracing methods to express the gauge transformation between field fluctuations and the curvature perturbation, zeta, in geometrical terms. Using these results we give a compact expression for the nonlinear gauge-transform part of fNL in terms of the principal curvatures of uniform energy-density hypersurfaces in field space.