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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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first_indexed 2020-12-08T20:18:55Z
last_indexed 2021-01-26T04:19:57Z
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spelling 2021-01-25T12:31:07.2746187 v2 55855 2020-12-08 Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies c1859f5040a279bcd3fa991d8f6e7f21 Farzad Fathi Zadeh Farzad Fathi Zadeh true false 2020-12-08 SCS 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. Journal Article Annales Henri Poincaré 21 4 1329 1382 Springer Science and Business Media LLC 1424-0637 1424-0661 Robertson-Walker metrics, Dirac Laplacian, Heat kernel expansion, Feynman-Kac formula, Brownian bridge 1 4 2020 2020-04-01 10.1007/s00023-020-00894-5 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2021-01-25T12:31:07.2746187 2020-12-08T20:14:13.7644085 College of Science Mathematics Farzad Fathi Zadeh 1 Yeorgia Kafkoulis 2 Matilde Marcolli 3
title Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
spellingShingle Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
Farzad Fathi Zadeh
title_short Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
title_full Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
title_fullStr Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
title_full_unstemmed Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
title_sort Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
author_id_str_mv c1859f5040a279bcd3fa991d8f6e7f21
author_id_fullname_str_mv c1859f5040a279bcd3fa991d8f6e7f21_***_Farzad Fathi Zadeh
author Farzad Fathi Zadeh
author2 Farzad Fathi Zadeh
Yeorgia Kafkoulis
Matilde Marcolli
format Journal article
container_title Annales Henri Poincaré
container_volume 21
container_issue 4
container_start_page 1329
publishDate 2020
institution Swansea University
issn 1424-0637
1424-0661
doi_str_mv 10.1007/s00023-020-00894-5
publisher Springer Science and Business Media LLC
college_str College of Science
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hierarchy_top_title College of Science
hierarchy_parent_id collegeofscience
hierarchy_parent_title College of Science
department_str Mathematics{{{_:::_}}}College of Science{{{_:::_}}}Mathematics
document_store_str 0
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description 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.
published_date 2020-04-01T04:11:07Z
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