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arXiv · 1811.02972

Bell polynomials and Brownian bridge in Spectral Gravity models on multifractal Robertson-Walker cosmologies

Farzad Fathizadeh·Yeorgia Kafkoulis·Matilde Marcolli
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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.

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Farzad Fathizadeh, Yeorgia Kafkoulis, Matilde Marcolli. 2018-11-07. Bell polynomials and Brownian bridge in Spectral Gravity models on multifractal Robertson-Walker cosmologies. https://arxiv.org/abs/1811.02972

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