It is well known that the so called Bianchi IX spacetimes with -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 [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 -symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painlevé VI equations.
Il est bien connu que l’espace-temps de Bianchi IX avec symétrie du groupe montre, dans le voisinage du Big Bang, un comportement chaotique à trajectoires typiques dans le sens inverse du mouvement du temps. Ce comportement (modèle Mixmaster de l’univers) peut être codé par le décalage de fractions continues à deux côtés.
Exactement le même décalage code les suites d’intersections de géodésiques hyperboliques dont l’axe imaginaire pur se situe dans le demi-plan complexe supérieur, c’est-à-dire à flot géodésique dans une surface modulaire appropriée.
Une interprétation physique de cette coincidence a été suggérée dans [23] : en effet, le chaos Mixmaster est une description approchée du passage d’un univers quantique chaud au moment du Big Bang à l’univers classique refroidissant. Nous discutons et étayons cette suggestion ici, en regardant le modèle Mixmaster pour la deuxième classe d’espaces-temps de Bianchi IX : ceux avec une symétrie (métriques d’Einstein auto-duales). Nous l’étendons aussi au contexte plus général relié aux équations de Painlevé VI.
@article{AFST_2016_6_25_2-3_517_0,
author = {Yuri Manin and Matilde Marcolli},
title = {Symbolic {Dynamics,} {Modular} {Curves,} and {Bianchi} {IX} {Cosmologies}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {517--542},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {2-3},
year = {2016},
doi = {10.5802/afst.1503},
zbl = {1380.37071},
mrnumber = {3530167},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1503/}
}
TY - JOUR AU - Yuri Manin AU - Matilde Marcolli TI - Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 517 EP - 542 VL - 25 IS - 2-3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1503/ DO - 10.5802/afst.1503 LA - en ID - AFST_2016_6_25_2-3_517_0 ER -
%0 Journal Article %A Yuri Manin %A Matilde Marcolli %T Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 517-542 %V 25 %N 2-3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1503/ %R 10.5802/afst.1503 %G en %F AFST_2016_6_25_2-3_517_0
Yuri Manin; Matilde Marcolli. Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 517-542. doi : 10.5802/afst.1503. https://afst.centre-mersenne.org/articles/10.5802/afst.1503/
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