Coupling of gravity to matter, spectral action and cosmic topology

Abstract

Abstract. We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat-kernel computation. Contents

Noncommutative Mixmaster Cosmologies

“…This action functional has an asymptotic expansion at high energies (see (5.10) below). Thus, one can approach the computation of this action functional either by explicit information on the spectrum and a computation of the series defined by Tr(f (D/Λ)) (as in [10], [11], [9], [31], [32]), or else through its asymptotic expansion and the computation via heat kernel methods and local expressions in curvature tensors of the various terms in the expansion, as, for instance, in [9], [12], [22], [24].…”

Asymptotic Safety, Hypergeometric Functions, and the Higgs Mass in Spectral Action Models

“…These potentials have been studied as possible models of inflationary cosmology. A derivation of the non-perturbative spectral action and the slow-roll potential in terms of heat-kernel techniques is also given in [7]. For more general geometries, in adopting the form (4.2), as opposed to the more general (2.3), we are neglecting the additional gravitational term given by the Weyl curvature.…”

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

“…It was shown in [2], [30], [31] that one can incorporate in the spectral action functional a scalar field, seen as a perturbation of the Dirac operator. The action functional then determines a potential for this scalar field that has the shape of a slowroll potential, suitable for a cosmological inflation scenario.…”

“…The action functional then determines a potential for this scalar field that has the shape of a slowroll potential, suitable for a cosmological inflation scenario. This slow-roll potential was used in [2], [30], [31] to study how an action functional model of gravity can address the cosmic topology question. For an overview of spectral action models in cosmology see [29].…”