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Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology?

Manin, Yuri I. and Marcolli, Matilde (2014) Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology? Symmetry, Integrability and Geometry, Methods and Applications (SIGMA), 10 . Art. No. 73. ISSN 1815-0659.

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We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from "the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon" is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.

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Additional Information:© 2014. The authors retain ownership of the copyright with respect to their papers published in SIGMA under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. Received March 01, 2014, in final form June 27, 2014; Published online July 09, 2014. This paper was conceived after the lecture in Bonn (November 2013), in which Sir Roger Penrose explained his fascinating ideas about cyclic cosmology. Ya. Sinai and O. Bogoyavlenskii made helpful remarks about BKLL treatment of the Bianchi IX model. We are grateful to them. Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel
Subject Keywords:Big Bang cosmology; algebro-geometric blow-ups; cyclic cosmology; Mixmaster cosmologies; modular curves
Classification Code:MSC 2010: 85A40; 14N05; 14G35
Record Number:CaltechAUTHORS:20140815-135716277
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Official Citation:Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology Yuri I. Manin and Matilde Marcolli SIGMA 10 (2014), 073, 20 pages
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:48624
Deposited By: Ruth Sustaita
Deposited On:19 Aug 2014 23:02
Last Modified:11 Sep 2015 19:32

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