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Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies

Manin, Yuri I. and Marcolli, Matilde (2015) Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies. . (Submitted)

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It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical trajectories in the backward movement of time. This behaviour (Mixmaster Model of the Universe) can be encoded by the shift of two-sided continued fractions. Exactly the same shift encodes the sequences of intersections of hyperbolic geodesics with purely imaginary axis in the upper complex half-plane, that is geodesic flow on an appropriate modular surface. A physical interpretation of this coincidence was suggested in arXiv:1402.2158: namely, that Mixmaster chaos is an approximate description of the passage from a hot quantum Universe at the Big Bang moment to the cooling classical Universe. Here we discuss and elaborate this suggestion, looking at the Mixmaster Model from the perspective of the second class of Bianchi IX spacetimes: those with SU(2)-symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painlevé VI equations.

Item Type:Report or Paper (Discussion Paper)
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Record Number:CaltechAUTHORS:20170712-124502018
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79013
Deposited By: Tony Diaz
Deposited On:12 Jul 2017 21:28
Last Modified:03 Oct 2019 18:15

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