logo AFST
Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 517-542.

Il est bien connu que l’espace-temps de Bianchi IX avec symétrie du groupe SO(3) 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 SU(2) (métriques d’Einstein auto-duales). Nous l’étendons aussi au contexte plus général relié aux équations de Painlevé VI.

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 [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). We also extend it to the more general context related to Painlevé VI equations.

@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},
     mrnumber = {3530167},
     zbl = {1380.37071},
     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
DA  - 2016///
SP  - 517
EP  - 542
VL  - Ser. 6, 25
IS  - 2-3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1503/
UR  - https://www.ams.org/mathscinet-getitem?mr=3530167
UR  - https://zbmath.org/?q=an%3A1380.37071
UR  - https://doi.org/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 Ser. 6, 25
%N 2-3
%I Université Paul Sabatier, Toulouse
%U https://doi.org/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, Série 6, Tome 25 (2016) no. 2-3, pp. 517-542. doi : 10.5802/afst.1503. https://afst.centre-mersenne.org/articles/10.5802/afst.1503/

[1] Artin (E.).— Ein mechanisches System mit quasiergodischen Bahnen. Abh. Math. Sem. Univ. Hamburg 3, no. 1, p. 170-175 (1924). | Article | MR 3069425 | Zbl 50.0677.11

[2] Babich (M. V.), Korotkin (D. A.).— Self-dual SU(2)-Invariant Einstein Metrics and Modular Dependence of Theta-Functions. Lett. Math. Phys. 46, p. 323-337 (1998). | Article | MR 1668577 | Zbl 0917.53016

[3] Boalch (Ph.).— Towards a nonlinear Schwarz’s list. arXiv:0707.3375, 27 pp. | Article

[4] Bogoyavlensky (O. I.).— Methods in the qualitative theory of dynamical systems in astrophysics and gas dynamics. Springer Series in Soviet Mathematics. Springer Verlag, Berlin, 1985. ix+301 pp. | Article | MR 810832

[5] Bogoyavlenskii (O. I.), Novikov (S. P.).— Singularities of the cosmological model of the Bianchi IX type according to the qualitative theory of differential equations. Zh. Eksp. Teor. Fiz. 64, p. 1475-1494 (1973).

[6] Chamseddine (A.H.), Connes (A.).— Spectral action for Robertson-Walker metrics, J. High Energy Phys. (2012) N.10, 101, 29 pp. | Article | MR 3033848 | Zbl 1397.81413

[7] Cirio (L.S.), Landi (G.), Szabo (R.J.).— Algebraic deformations of toric varieties. I. General constructions, Adv. Math. 246, p. 33-88 (2013). | Article | MR 3091799 | Zbl 1288.14003

[8] Connes (A.), Landi (G.).— Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221, p. 141-159 (2001). | Article | MR 1846904 | Zbl 0997.81045

[11] Eguchi (T.), Hanson (A.J.).— Self-dual solutions to Euclidean Gravity, Annals of Physics, 120, p. 82-106 (1979). | Article | MR 540896 | Zbl 0409.53020

[12] Eguchi (T.), Hanson (A.J.).— Gravitational Instantons, Gen. Relativity Gravitation 11, No 5, p. 315-320 (1979). | Article | MR 563971

[13] Estrada (C.), Marcolli (M.).— Noncommutative Mixmaster Cosmologies, International Journal of Geometric Methods in Modern Physics 10 (2013) 1250086, 28 pp. | Article | MR 2998331 | Zbl 1261.83044

[14] Fan (W.), Fathizadeh (F.), Marcolli (M.).— Spectral Action for Bianchi Type-IX Cosmological Models, J. High Energy Phys., no15, 085, 28pp. (2015). | Article | MR 3435554

[15] Furusawa (T.).— Quantum chaos of Mixmaster Universe. Progr. Theor. Phys., Vol. 75, No. 1, p. 59-67 (1986). | Article | MR 832832

[16] Hitchin (N.J.).— Harmonic spinors, Advances in Math. 14, p. 1-55 (1974). | Article | MR 358873 | Zbl 0284.58016

[17] Hitchin (N.J.).— Twistor spaces, Einstein metrics and isomonodromic deformations. J. Diff. Geo., Vol. 42, No. 1, p. 30-112 (1995). | Article | MR 1350695 | Zbl 0861.53049

[18] Khalatnikov (I. M.), Lifshitz (E. M.), Khanin (K. M.), Shchur (L. N.), and Sinai (Y. G.).— On the stochasticity in relativistic cosmology. Journ. Stat. Phys., Vol. 38, Nos. 1/2, p. 97-114 (1985). | Article | MR 784932

[19] Lecian (M. M.).— Reflections on the hyperbolic plane. International Journal of Modern Physics D Vol. 22, No. 14 (2013), 1350085 (53 pages) DOI: 10.1142/S0218271813500855. arXiv:1303.6343 [gr-qc] | Article | MR 3157534 | Zbl 1284.83186

[20] Lisovyy (O.), Tykhyy (Y.).— Algebraic solutions of the sixth Painlevé equation. arXiv:0809.4873 | Article | Zbl 1307.34135

[21] Manin (Y.).— Sixth Painlevé equation, universal elliptic curve, and mirror of P 2 . In: geometry of Differential Equations, ed. by A. Khovanskii, A. Varchenko, V. Vassiliev. Amer. Math. Soc. Transl. (2), vol. 186 (1998), p. 131-151. arXiv:alg-geom/9605010. | Article | Zbl 0948.14025

[22] Manin (Y.), Marcolli (M.).— Continued fractions, modular symbols, and non-commutative geometry. Selecta math., new ser. 8 (2002), 475-521. arXiv:math.NT/0102006 | Article | MR 1931172

[23] Manin (Y.), Marcolli (M.).— Big Bang, Blow Up, and Modular Curves: Algebraic Geometry in Cosmology. SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), Paper 073, 20 pp. Preprint arXiv:1402.2158 | Article

[24] Mayer (D.).— Relaxation properties of the Mixmaster Universe. Phys. Lett. A, Vol. 121, Nos. 8-9, p. 390-394 (1987). | Article | MR 894186

[25] Newman (E.), Tamburino (L.), Unti (T.).— Empty-space generalization of the Schwarzschild metric, Journ. Math. Phys. 4, p. 915-923 (1963). | Article | MR 152345 | Zbl 0115.43305

[26] Okumura (S.).— The self-dual Einstein-Weyl metric and classical solutions of Painlevé VI. Lett. in Math. Phys., 46, p. 219-232 (1998). | Article | MR 1661225 | Zbl 0917.53017

[27] Petropoulos (P.M.), Vanhove (P.).— Gravity, strings, modular and quasimodular forms, Ann. Math. Blaise Pascal 19, No. 2, p. 379-430 (2012). | Article | MR 3025139 | Zbl 1263.11117

[28] Series (C.).— The modular surface and continued fractions. J. London MS, Vol. 2, no. 31, 69-80 (1985). | Article | MR 810563 | Zbl 0545.30001

[29] Takasaki (K.).— Painlevé-Calogero correspondence revisited. Journ. Math. Phys., vol. 42, No 3, p. 1443-1473 (2001). | Article | Zbl 1016.34089

[30] Takhtajan (L.).— A simple example of modular forms as tau-functions for integrable equations, Teoret. Mat. Fiz. 93, no. 2, p. 330-341 (1992). | Article | MR 1233549

[31] Taub (A.H.).— Empty space-times admitting a three parameter group of motions, Annals of Mathematics 53, p. 472-490 (1951). | Article | MR 41565 | Zbl 0044.22804

[32] Tod (K. P.).— Self-dual Einstein metrics from the Painlevé VI equation. Phys. Lett. A 190, p. 221-224 (1994). | Article | Zbl 0960.83505

Cité par Sources :