Spectral action models of gravity on packed swiss cheese cosmology

Abstract

Abstract. We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the Packed Swiss Cheese Cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of 3-dimensional spheres inside a 4-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of a divergent sum which involves the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of the points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.

Spectral action for Bianchi type-IX cosmological models

“…It is defined in terms of the spectrum of the Dirac operator, and it recovers the usual Einstein-Hilbert action with cosmological term, along with modified gravity terms (Weyl curvature conformal gravity, Gauss-Bonnet gravity), in an asymptotic expansion in the energy scale. As a model of gravity, the spectral action was used in recent years as a possible source of new early universe models and inflationary mechanisms, [4,22,36,[42][43][44][45][46][47][48].…”

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

“…In this second part of the paper we turn to consider the multifractal cases of Robertson-Walker cosmologies, where the spatial sections are obtained as an arrangement of 3-spheres such as an Apollonian packing, generalizing the static cases considered in [1].…”

“…In particular, the shape of a slow-roll potential for inflation derived from the spectral action model was shown in [1] to be also affected by these terms, so that corrections due to the presence of fractality also appear in the slow-roll coefficients, which in principle are detectable via observational data. The regularity assumptions on the fractal geometry used in [1] were aimed at obtaining sufficiently good analytic properties of the corresponding zeta functions, in the sense of [26]. However, the model of spectral action on multifractal swiss-cheese type cosmologies considered in [1] is not entirely realistic, because the relevant spacetime is assumed to be a product of a fractal packing of spheres (or of spherical manifolds) times a compactified Euclidean time dimension S 1 , so that, in particular, the scaling factor of the spatial sections remains constant.…”

“…We obtain the full expansion of the spectral action on these multifractal Packed Swiss Cheese Cosmologies in terms of the expansion for a single Robertson-Walker metric obtained in the first part of the paper, using a Mellin transform argument. The resulting expansion has two series of term, one that corresponds to the terms in the expansion of a single Robertson-Walker metric, where the coefficients are modified by a zeta regularized sum of powers of the packing radii, so that the coefficients are no longer rational numbers but they contain the zeta values ζ L (4 − 2M), for M ∈ N. The second series of terms corresponds to the poles of the fractal string zeta function ζ L (z) and give rise to a series of log-periodic terms as already observed in the simpler model of [1]. In this case the coefficients are values of the zeta function of the Dirac operator of the underlying model Robertson-Walker metric.…”

“…We first illustrate a lower dimensional example based on a special class of Apollonian circle packings, the Ford circles, where we show explicitly the terms arising in the spectral action that detect the fractal structure, which are expressible in terms of zeros of the Riemann zeta function. This example also illustrates the fact that the very restrictive condition on the sphere packing used in the simplified model of [1], based on an approximation by self-similar fractal strings with lattice property, is too strong for the general setting we need to consider here. The method we use in this paper to obtain the full asymptotic expansion of the spectral action is independent of this approximation assumption and only requires a milder condition on the fractal string L = {a n,k } of the sphere packing, namely the property that the zeta function ζ L (z) admits analytic continuation to a meromorphic function on C with simple poles located away from the set of integers less than or equal to 4.…”