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Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies / Farzad Fathi Zadeh, Yeorgia Kafkoulis, Matilde Marcolli

Annales Henri Poincaré, Volume: 21, Issue: 4, Pages: 1329 - 1382

Swansea University Author: Farzad Fathi Zadeh

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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...

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Published in: Annales Henri Poincaré
ISSN: 1424-0637 1424-0661
Published: Springer Science and Business Media LLC 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa55855
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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–Walkerspacetime, 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 fromthe poles (off the real line) of this zeta function.
Keywords: Robertson-Walker metrics, Dirac Laplacian, Heat kernel expansion, Feynman-Kac formula, Brownian bridge
College: College of Science
Issue: 4
Start Page: 1329
End Page: 1382