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

Abstract

We obtain an explicit formula for the full expansion of the spectral action on Robertson-Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman-Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson-Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson-Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function. Contents

Gluing Non-commutative Twistor Spaces

“…In order to view the θdeformation S θ as a spectral triple, one can take as Hilbert space and Dirac operator the direct sum of the respective ones on the two copies of S 3 . This is analogous to the spectral triple construction used in the gluing of copies of smooth manifolds into fractal configurations, see [11], [23].…”

Fractality in Cosmic Topology Models with Spectral Action Gravity

“…We consider here a more realistic Robertson-Walker model of multifractal cosmology, where the individual manifolds in the fractal configuration are copies of 4-dimensional Robertson-Walker spacetimes. This is the same type of model considered, in the case of Apollonian packings of spheres, in [18]. A (Euclidean) Robertson-Walker metric on a spacetime X = R × S 3 is of the form ds 2 RW = dt 2 + a(t) 2 dσ 2 with dσ 2 is the metric on the unit sphere S 3 .…”

“…In this section we consider different candidate cosmic topology models, first in the spherical and then in the flat case, with two models of spacetime: a simplified static model based on a product Y × S 1 with circle compactification, and then a more realistic Robertson-Walker model on a cylinder Y × R with an expansion/contraction factor a(t). We compute the asymptotic expansion of the spectral action in all of these cases, along the lines of [2] for the static model and of [18] for the Robertson-Walker case. For every cosmic topology candidate we identify a corresponding possible fractal structure, based on a Sierpinski construction associated to the fundamental domain in the 3-sphere (or the 3-torus, respectively).…”

“…Cosmological and gravitational models based on the spectral action functional were considered, for instance, in [2], [4], [5], [10], [16], [17], [18], [19], [37], [38], [39], [40], [41], [42], [45], [46], [59], though this is certainly not an exhaustive list of references on the subject. The main goal of the present paper is to continue this investigation by combining the analysis of cosmic topology effects considered in [5], [41], [42], with the analysis of the effect of multifractal structures considered in [2] and [18]. 1.2.…”

“…In [2] and [18] the Packed Swiss Cheese Cosmology models are analyzed from the point of view of spectral action gravity and it is shown that the presence of fractality is detectable in the form of additional terms in the spectral action, coming from poles of the zeta function off the real line and from the non-integer Hausdorff dimension, as well as in a modification to the effective gravitational and cosmological constants of the terms of the action coming from the underlying Robertson-Walker cosmology. This analysis is done in [2] in the case of a static model, and in [18] in the more general case of a Robertson-Walker cosmology with a nontrivial expansion/contraction factor.…”