Rationality of Spectral Action for Robertson-Walker Metrics

TL;DR

This paper proves a conjecture about the rationality of spectral action coefficients for Robertson-Walker metrics using advanced calculus techniques and computes several terms in the expansion, confirming previous results.

Contribution

It introduces a new method to prove the rationality conjecture and computes higher-order terms in the spectral action expansion.

Findings

Confirmed the rationality of spectral action coefficients for general Robertson-Walker metrics.

Computed terms up to $a_{12}$ in the spectral action expansion.

Validated previous computations up to $a_{10}$ using a different method.

Abstract

We use pseudodifferential calculus and heat kernel techniques to prove a conjecture by Chamseddine and Connes on rationality of the coefficients of the polynomials in the cosmic scale factor $a(t)$ and its higher derivatives, which describe the general terms $a_{2n}$ in the expansion of the spectral action for general Robertson-Walker metrics. We also compute the terms up to $a_{12}$ in the expansion of the spectral action by our method. As a byproduct, we verify that our computations agree with the terms up to $a_{10}$ that were previously computed by Chamseddine and Connes by a different method.