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Published October 2015 | Published + Submitted
Journal Article Open

Spectral Action for Bianchi Type-IX Cosmological Models


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.

Additional Information

© 2015 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: June 30, 2015; Revised: August 21, 2015; Accepted: September 20, 2015; Published: October 13, 2015. The first author is supported by a Summer Undergraduate Research Fellowship at Caltech. The third author is partially supported by NSF grants DMS-1201512 and PHY-1205440 and by the Perimeter Institute for Theoretical Physics.

Attached Files

Published - art_10.1007_JHEP10_2015_085.pdf

Submitted - 1506.06779v1.pdf


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