Building Cosmological Models via Noncommutative Geometry

Marcolli

2015

Self Cite

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.

Section: Jhep10(2015)085mentioning

confidence: 99%

Spectral action for Bianchi type-IX cosmological models

Fan

Marcolli

2015

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“…We start by an explanation of the spectral action functional [5,9], which is based on the Dirac operator and provides a modified Euclidean gravity model. Implications of this action as a source of new early universe models and inflationary mechanisms in cosmology has been studied in recent years [3,31,38,39,40,41,42,43,44,18]. We recall briefly some basic facts about the Dirac operator, the heat kernel method and pseudo-differential calculus, and how they can be employed to compute the terms in the asymptotic expansion of the spectral action in the energy scale.…”

Section: Introductionmentioning

confidence: 99%

Modular forms in the spectral action of Bianchi IX gravitational instantons

Fan

Marcolli

2019

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Following our result on rationality of the spectral action for Bianchi type-IX cosmological models, which suggests the existence of a rich arithmetic structure in the action, we study the arithmetic and modular properties of the action for an especially interesting family of metrics, namely SU (2)-invariant Bianchi IX gravitational instantons. A variant of the previous rationality result is first presented for a time-dependent conformal perturbation of the triaxial Bianchi IX metric which serves as a general form of Bianchi IX gravitational instantons. By exploiting a parametrization of the gravitational instantons in terms of theta functions with characteristics, we then show that each Seeleyde Witt coefficientã 2n appearing in the expansion of their spectral action gives rise to vector-valued and ordinary modular functions of weight 2. We investigate the modular properties of the first three termsã 0 ,ã 2 andã 4 by preforming direct calculations. This guides us to provide spectral arguments for general termsã 2n , which are based on isospectrality of the involved Dirac operators. Special attention is paid to the relation of the new modular functions with well-known modular forms by studying explicitly two different cases of metrics with pairs of rational parameters that, in a particular sense, belong to two different general families. In the first case, it is shown that by running a summation over a finite orbit of the rational parameters, up to multiplication by the cusp form ∆ of weight 12, each termã 2n lands in the 1-dimensional linear space of modular forms of weight 14. In the second case, it is proved that, after a summation over a finite orbit of the parameters and multiplication by G 4 4 , where G 4 is the Eisenstein series of weight 4, eachã 2n lands in the 1-dimensional linear space of cusp forms of weight 18.Mathematics Subject Classification (2010). 58B34, 11F37, 11M36.

“…Thus various methods have been developed for computing the terms in the expansion in the energy scale Λ of the spectral action [3,6,8,9,20,21]. Potential applications of noncommutative geometry in cosmology have recently been carried out in [22,25,26,27,28,29,30,31,16].…”

mentioning

confidence: 99%

Rationality of spectral action for Robertson-Walker metrics

Ghorbanpour

Khalkhali

2014

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.