Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 073, 20 pages      arXiv:1402.2158      https://doi.org/10.3842/SIGMA.2014.073
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

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

Yuri I. Manin a and Matilde Marcolli b
a) Max-Planck-Institut für Mathematik, Bonn, Germany
b) California Institute of Technology, Pasadena, USA

Received March 01, 2014, in final form June 27, 2014; Published online July 09, 2014

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.

Key words: Big Bang cosmology; algebro-geometric blow-ups; cyclic cosmology; Mixmaster cosmologies; modular curves.

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References

  1. BICEP2 Collaboration, BICEP2 I: Detection of B-mode polarization at degree angular scales, Phys. Rev. Lett. 112 (2014), 241101, 25 pages, arXiv:1403.3985.
  2. Bognár J., Indefinite inner product spaces, Springer-Verlag, New York - Heidelberg, 1974.
  3. Bogoyavlenskiǐ O.I., Novikov S.P., Singularities of the cosmological model of the Bianchi $IX$ type according to the qualitative theory of differential equations, J. Exp. Theor. Phys. 64 (1973), 1475-1494.
  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.
  5. Bourguignon J.-P., Gauduchon P., Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581-599.
  6. Chang C.-H., Mayer D., Thermodynamic formalism and Selberg's zeta function for modular groups, Regul. Chaotic Dyn. 5 (2000), 281-312.
  7. Chen L., Gibney A., Krashen D., Pointed trees of projective spaces, J. Algebraic Geom. 18 (2009), 477-509, math.AG/0505296.
  8. Cirio L., Landi G., Szabo R.J., Algebraic deformations of toric varieties. II. Noncommutative instantons, Adv. Theor. Math. Phys. 15 (2011), 1817-1907, arXiv:1106.5708.
  9. Cirio L.S., Landi G., Szabo R.J., Algebraic deformations of toric varieties. I. General constructions, Adv. Math. 246 (2013), 33-88, arXiv:1001.1242.
  10. Cirio L., Landi G., Szabo R.J., Instantons and vortices on noncommutative toric varieties, arXiv:1212.3469.
  11. Connes A., Landi G., Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221 (2001), 141-159, math.QA/0011194.
  12. Connes A., Moscovici H., Type III and spectral triples, in Traces in number theory, geometry and quantum fields, Aspects Math., Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 57-71, math.OA/0609703.
  13. Donaldson S., Friedman R., Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity 2 (1989), 197-239.
  14. Estrada C., Marcolli M., Noncommutative Mixmaster cosmologies, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1250086, 28 pages, arXiv:1203.2679.
  15. Flores J.L., Herrera J., Sánchez M., On the final definition of the causal boundary and its relation with the conformal boundary, Adv. Theor. Math. Phys. 15 (2011), 991-1057, arXiv:1001.3270.
  16. Fukuma M., Kono Y., Miwa A., Effects of space-time noncommutativity on the angular power spectrum of the CMB, Nuclear Phys. B 682 (2004), 377-390, hep-th/0307029.
  17. Furusawa T., Quantum chaos of Mixmaster universe, Progr. Theoret. Phys. 75 (1986), 59-67.
  18. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  19. Geroch R., Kronheimer E.H., Penrose R., Ideal points in space-time, Proc. Roy. Soc. London Ser. A 327 (1972), 545-567.
  20. Greenfield M., Marcolli M., Teh K., Twisted spectral triples and quantum statistical mechanical systems, arXiv:1305.5492.
  21. Gurzadyan V.G., Penrose R., More on the low variance circles in CMB sky, arXiv:1012.1486.
  22. Gurzadyan V.G., Penrose R., On CCC-predicted concentric low-variance circles in the CMB sky, Eur. Phys. J. 128 (2013), 22, 17 pages, arXiv:1302.5162.
  23. Hajian A., Are there echoes from the Pre-Big Bang Universe? A search for low variance circles in the CMB sky, Astrophys. J. 740 (2011), no. 2, 52, 4 pages, arXiv:1012.1656.
  24. Harlow D., Shenker S.H., Stanford D., Susskind L., Tree-like structure of eternal inflation: a solvable model, Phys. Rev. D 85 (2012), 063516, 24 pages, arXiv:1110.0496.
  25. Khalatnikov I.M., Lifshitz E.M., Khanin K.M., Shchur L.N., Sinaǐ Y.G., On the stochasticity in relativistic cosmology, J. Statist. Phys. 38 (1985), 97-114.
  26. Lugo L.M., Chauvet P.A., BKL method in the Bianchi IX universe model revisited, Appl. Phys. Res. 5 (2013), no. 6, 107-117.
  27. Manin Yu.I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, Vol. 289, 2nd ed., Springer-Verlag, Berlin, 1997.
  28. Manin Yu.I., Sixth Painlevé equation, universal elliptic curve, and mirror of ${\mathbb P}^2$, in Geometry of Differential Equations, Amer. Math. Soc. Transl. Ser. 2, Vol. 186, Editors A. Khovanskii, A. Varchenko, V. Vassiliev, Amer. Math. Soc., Providence, RI, 1998, 131-151, alg-geom/9605010.
  29. Manin Yu.I., Marcolli M., Continued fractions, modular symbols, and noncommutative geometry, Selecta Math. (N.S.) 8 (2002), 475-521, math.NT/0102006.
  30. Manin Yu.I., Marcolli M., Modular shadows and the Lévy-Mellin $\infty$-adic transform, in Modular Forms on Schiermonnikoog, Editors B. Edixhoven, G. van der Geer, B. Moonen, Cambridge University Press, Cambridge, 2008, 189-238, math.NT/0703718.
  31. Manin Yu.I., Marcolli M., Moduli operad over ${\mathbb F}_1$, in Absolute Arithmetic and ${\mathbb F}_1$ Geometry, Editor K. Thas, Eur. Math. Soc., 2013, 331-364, arXiv:1302.6526.
  32. Marcolli M., Solvmanifolds and noncommutative tori with real multiplication, Commun. Number Theory Phys. 2 (2008), 421-476, arXiv:0711.2036.
  33. Marcolli M., Tedeschi N., Multifractals, Mumford curves and eternal inflation, p-Adic Numbers Ultrametric Anal. Appl. 6 (2014), 135-154, arXiv:1311.5458.
  34. Mayer D.H., Relaxation properties of the Mixmaster universe, Phys. Lett. A 121 (1987), 390-394.
  35. Moss A., Scott D., Zibin J.P., No evidence for anomalously low variance circles on the sky, J. Cosmol. Astropart. Phys. 2011 (2011), no. 4, 033, 7 pages, arXiv:1012.1305.
  36. Nautiyal A., Anisotropic non-gaussianity with noncommutative spacetime, Phys. Lett. B 728 (2014), 472-481, arXiv:1303.4159.
  37. Newman E.T., A fundamental solution to the CCC equations, Gen. Relativity Gravitation 46 (2014), no. 5, 1717, 13 pages, arXiv:1309.7271.
  38. Olive K.A., Peacock J.A., Big-Bang cosmology, available at http://pdg.lbl.gov/2005/reviews/bigbangrpp.pdf.
  39. Penrose R., Conformal treatment of infinity, in Relativité, Groupes et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys., Univ. Grenoble), Editors B. deWitt, C. deWitt, Gordon and Breach, New York, 1964, 565-584.
  40. Penrose R., Singularities and time-asymmetry, in General Relativity: an Einstein Centenary Survey, Editors S.W. Hawking, W. Israel, Cambridge University Press, Cambridge, 1979, 581-638.
  41. Penrose R., Twistor geometry of conformal infinity, in The Conformal Structure of Space-Time, Lecture Notes in Phys., Vol. 604, Editors J. Frauendiener, H. Friedrich, Springer, Berlin, 2002, 113-121.
  42. Penrose R., Cycles of time. An extraordinary new view of the universe, Alfred A. Knopf, Inc., New York, 2010.
  43. Series C., The modular surface and continued fractions, J. London Math. Soc. 31 (1985), 69-80.
  44. Shiraishi M., Mota D.F., Ricciardone A., Arroja F., CMB statistical anisotropy from noncommutative gravitational waves, arXiv:1401.7936.
  45. Takasaki K., Painlevé-Calogero correspondence revisited, J. Math. Phys. 42 (2001), 1443-1473, math.QA/0004118.
  46. Tod P., Penrose's circles in the CMB and a test of inflation, Gen. Relativity Gravitation 44 (2012), 2933-2938, arXiv:1107.1421.
  47. van Suijlekom W.D., The noncommutative Lorentzian cylinder as an isospectral deformation, J. Math. Phys. 45 (2004), 537-556, math-ph/0310009.
  48. Wehus I.K., Eriksen H.K., A search for concentric circles in the 7-year WMAP temperature sky maps, Astrophys. J. Lett. 733 (2011), L29, 6 pages, arXiv:1012.1268.
  49. Yamazaki H., Hara T., Dirac decomposition of Wheeler-DeWitt equation in the Bianchi Class A models, Progr. Theoret. Phys. 106 (2001), 323-337, gr-qc/0101066.
  50. Zaldarriaga M., BICEP2 results: a view from the outside, Lecture given at the ''Workshop on Primordial Gravitational Waves and Cosmology" (Burke Institute, Caltech, May 2014).

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