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

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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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Acknowledgements

The first author acknowledges the support from the Marie Curie/SER Cymru II Cofund Research Fellowship 663830-SU-008 and thanks the Perimeter Institute for Theoretical Physics for their hospitality in July 2018 and their excellent environment where this work was partially carried out. The second author was partially supported by a Gantvoort Scholarship, a Mr. and Mrs. Robert C. Loschke Summer Undergraduate Research Fellowship, and a Taussky-Todd Prize. The third author was partially supported by NSF Grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement Grant RGPAS-2018-522593, and by the Perimeter Institute for Theoretical Physics.

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Authors and Affiliations

  1. Department of Mathematics, Computational Foundry, Swansea University, Bay Campus, Swansea, SA1 8EN, UK

    Farzad Fathizadeh

  2. Max Planck Institute for Biological Cybernetics, 72076, Tübingen, Germany

    Farzad Fathizadeh

  3. Mathematics Department, Stony Brook University, Stony Brook, NY , 11794-3651, USA

    Yeorgia Kafkoulis

  4. Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA , 91125, USA

    Yeorgia Kafkoulis & Matilde Marcolli

  5. Department of Mathematics, University of Toronto, Toronto, ON , M5S 2E4, Canada

    Matilde Marcolli

  6. Perimeter Institute for Theoretical Physics, Waterloo, ON , N2L 2Y5, Canada

    Matilde Marcolli

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  1. Farzad Fathizadeh
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  2. Yeorgia Kafkoulis
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  3. Matilde Marcolli
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Correspondence to Farzad Fathizadeh.

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Communicated by Carlo Rovelli.

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Fathizadeh, F., Kafkoulis, Y. & Marcolli, M. Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies. Ann. Henri Poincaré 21, 1329–1382 (2020). https://doi.org/10.1007/s00023-020-00894-5

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