Spectral Action for Bianchi Type-IX Cosmological Models

TL;DR

This paper extends a rationality result for spectral actions from Robertson-Walker metrics to Bianchi type-IX cosmological models, using heat kernel methods and noncommutative residue techniques to express coefficients in terms of rational polynomials.

Contribution

It introduces a new efficient method for computing Seeley-de Witt coefficients using Wodzicki's residue, applicable to anisotropic cosmological models like Bianchi type-IX.

Findings

Seeley-de Witt coefficients are rational polynomials in cosmic evolution factors.

The new method confirms rationality of spectral action coefficients.

Efficient computation technique leverages symmetries of the Bianchi type-IX metric.

Abstract

A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors $w_1(t), w_2(t), w_3(t),$ and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients $a_0, a_2, a_4$ of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki's noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.