Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies

Manin, Yuri; Marcolli, Matilde

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 $SO\left(3\right)$ 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 $SU\left(2\right)$ (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 $SO\left(3\right)$-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 $SU\left(2\right)$-symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painlevé VI equations.

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

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

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

[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. | DOI | MR

[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. | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[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). | DOI | MR

[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] | DOI | MR | Zbl

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

[21] Manin (Y.).— Sixth Painlevé equation, universal elliptic curve, and mirror of ${\mathbf{P}}^2$. 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. | DOI | Zbl

[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 | DOI | MR

[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 | DOI

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

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

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

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

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

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

[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). | DOI | MR

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

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