Solving the Hamilton-Jacobi equation for gravitationally interacting electromagnetic and scalar fields

TL;DR

This paper develops a method to solve the Hamilton-Jacobi equation for gravitational, electromagnetic, and scalar fields using spatial gradient expansion, providing explicit solutions up to fourth order in gradients for cosmological models.

Contribution

It extends the spatial gradient expansion technique to include electromagnetic fields and derives explicit solutions for cosmological perturbations with scalar fields.

Findings

Derived generating functional up to fourth order in spatial gradients.

Provided second-order perturbations for flat FRW cosmology with scalar fields.

Demonstrated application with exponential potential example.

Abstract

The spatial gradient expansion of the generating functional was recently developed by Parry, Salopek, and Stewart to solve the Hamiltonian constraint in Einstein-Hamilton-Jacobi theory for gravitationally interacting dust and scalar fields. This expansion is used here to derive an order-by-order solution of the Hamiltonian constraint for gravitationally interacting electromagnetic and scalar fields. A conformal transformation and functional integral are used to derive the generating functional up to the terms fourth order in spatial gradients. The perturbations of a flat Friedmann-Robertson-Walker cosmology with a scalar field, up to second order in spatial gradients, are given. The application of this formalism is demonstrated in the specific example of an exponential potential.