Spectral Action for Robertson-Walker metrics

TL;DR

This paper introduces a direct computational method for the spectral action of Robertson-Walker metrics, extending previous techniques and verifying high-order terms against known universal formulas.

Contribution

It develops a new direct approach to compute the spectral action for Robertson-Walker metrics, surpassing previous methods based on Poisson summation.

Findings

Validated expansion terms up to a_6 against Gilkey's formulas

Computed expansion up to a_{10} using the new method

Demonstrated the effectiveness of the direct computation approach

Abstract

We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson-Walker metrics. We check the terms of the expansion up to a_6 against the known universal formulas of Gilkey and compute the expansion up to a_{10} using our direct method.