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

Abstract

Abstract. 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.

Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies

“…A physical interpretation of this coincidence was suggested in [MaMar14]. Here we discuss and elaborate this suggestion, looking at the Mixmaster model from the perspective of Bianchi IX model with SU (2)-symmetry.…”

“…A physical interpretation of this coincidence was suggested in [MaMar14]: 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).…”

“…More precisely, according to [MaMar14], the Mixmaster "classical chaos" should be considered as an approximation to an unknown quantum description of the transition from the infinitely hot quantum Universe at the moment of Big Bang to the cooling Universe gradually fitting a classical Einsteinian model. Time axis at the moment of Big Bang is purely imaginary, and it becomes real during the observable history of Universe.…”

p-Adic mathematical physics: the first 30 years

“…Algebraic geometric models in cosmology related to the "boundaries" of spacetime: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons are introduced in [295]. It is suggested to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point.…”

Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

“…[5]. Out of all Bianchi type models of classes I -IX with vanishing cosmological constant, only Bianchi-IX has been found to exhibit a relationship with quasimodular forms [6,7]. Modular forms in physics are a consequence of duality properties, resulting either from an invariance or a relationship between two distinct theories.…”