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Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton-scalaron duality and its applications

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Abstract

We briefly describe the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections and their reductions to two-dimensional dilaton-vecton gravity field theories. The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with an essentially nonlinear coupling to the dilaton gravity. We emphasize that the vecton field in dilaton-vecton gravity can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to the dilaton gravity. With this vecton-scalaron duality, we can use the methods and results of the standard dilaton gravity coupled to usual scalars in more complex dilaton-scalaron gravity theories equivalent to dilaton-vecton gravity. We present the dilaton-vecton gravity models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories and also in general one-dimensional dilaton-scalaron gravity models. We focus on general and global properties of the models, seeking integrals and analyzing the structure of the solution space. In integrable cases, it can be usefully visualized by drawing a “topological portrait” resembling the phase portraits of dynamical systems and simply exposing the global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations, we also propose an integral equation well suited for iterations.

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References

  1. V. Sahni and A. Starobinsky, Internat. J. Mod. Phys. D, 15, 2105–2132 (2006); arXiv:astro-ph/0610026v3 (2006).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. E. J. Copeland, M. Sami, and S. Tsujikawa, Internat. J. Mod. Phys. D, 15, 1753–1935 (2006); arXiv:hep-th/0603057v3 (2006).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. A. D. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic, Chur, Switzerland (1990); arXiv:hep-th/0503203v1 (2005).

    Google Scholar 

  4. V. Mukhanov, Physical Foundations of Cosmology, Cambridge Univ. Press, New York (2005).

    Book  MATH  Google Scholar 

  5. S. Weinberg, Cosmology, Oxford Univ. Press, Oxford (2008).

    MATH  Google Scholar 

  6. V. A. Rubakov and D. S. Gorbunov, Introduction to the Theory of the Early Universe: Hot Big Bang Theory [in Russian], URSS, Moscow (2008); English transl., World Scientific, Hackensack, N. J. (2011).

    Google Scholar 

  7. D. Langlois, “Lectures on inflation and cosmological perturbations,” arXiv:1001.5259v1 [astro-ph.CO] (2010).

    Google Scholar 

  8. A. H. Guth and Y. Nomura, Phys. Rev. D, 86, 023534 (2012); arXiv:1203.6876v2 [hep-th] (2012).

    Article  ADS  Google Scholar 

  9. D. Mulryne and J. Ward, Class. Q. Grav., 28, 204010 (2011); arXiv:1105.5421v3 [astro-ph.CO] (2011).

    Article  MathSciNet  ADS  Google Scholar 

  10. S. R. Green, E. J. Martinec, C. Quigley, and S. Sethi, Class. Q. Grav., 29, 075006 (2012); arXiv:1110.0545v4 [hep-th] (2011).

    Article  MathSciNet  ADS  Google Scholar 

  11. N. Mavromatos, “The issue of dark energy in string theory,” in: The Invisible Universe: Dark Matter and Dark Energy (Lect. Notes Phys., Vol. 720, L. Papantonopoulos, ed.), Springer, Berlin (2007), pp. 333–374; arXiv:hep-th/0507006v1 (2005).

    Chapter  Google Scholar 

  12. R. Bousso, “The cosmological constant problem, dark energy, and the landscape of string theory,” arXiv:1203.0307v2 [astro-ph.CO] (2012).

    Google Scholar 

  13. S. Nojiri and S. D. Odintsov, Phys. Rep., 505, 59–144 (2011); arXiv:1011.0544v4 [gr-qc] (2010).

    Article  MathSciNet  ADS  Google Scholar 

  14. T. Clifton, P. Ferreira, A. Padilla, and C. Skordis, Phys. Rep., 513, 1–189 (2012); arXiv:1106.2476v3 [astro-ph. CO] (2011).

    Article  MathSciNet  ADS  Google Scholar 

  15. D. Rodrigues, F. de Salles, I. Shapiro, and A. Starobinsky, Phys. Rev. D, 83, 084028 (2011); arXiv:1101.5028v2 [gr-qc] (2011).

    Article  ADS  Google Scholar 

  16. Ph. Brax, Acta Phys. Polon. B, 43, 2307–2329 (2012); arXiv:1211.5237v1 [hep-th] (2012).

    Article  MathSciNet  Google Scholar 

  17. J. E. Lidsey, D. Wands, and E. J. Copeland, Phys. Rep., 337, 343–492 (2000); arXiv:hep-th/9909061v2 (1999).

    Article  MathSciNet  ADS  Google Scholar 

  18. A. T. Filippov, “On Einstein-Weyl unified model of dark energy and dark matter,” arXiv:0812.2616v2 [gr-qc] (2008).

    Google Scholar 

  19. A. T. Filippov, Theor. Math. Phys., 163, 753–767 (2010); arXiv:1003.0782v2 [hep-th] (2010).

    Article  MATH  Google Scholar 

  20. A. T. Filippov, Proc. Steklov Inst. Math., 272, 107–118 (2011); arXiv:1008.2333v3 [hep-th] (2010).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. T. Filippov, “An old Einstein-Eddington generalized gravity and modern ideas on branes and cosmology,” in: Gribov-80 Memorial Volume (Yu. L. Dokshitzer, P. Levai, and J. Nyiri, eds.), World Scientific, Singapore (2011), pp. 479–495; arXiv:1011.2445v1 [gr-qc] (2010).

    Google Scholar 

  22. A. T. Filippov, “General properties and some solutions of generalized Einstein-Eddington affine gravity I,” arXiv:1112.3023v1 [math-ph] (2011).

    Google Scholar 

  23. S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge Univ. Press, New York (1998).

    MATH  Google Scholar 

  24. E. Witten, “Three-dimensional gravity revisited,” arXiv:0706.3359v1 [hep-th] (2007).

    Google Scholar 

  25. A. T. Filippov, “Some unusual dimensional reductions of gravity: Geometric potentials, separation of variables, and static-cosmological duality,” arXiv:hep-th/0605276v2 (2006).

    Google Scholar 

  26. S. Chandrasekhar, The Mathematical Theory of Black Holes (Ser. Monogr. Phys., Vol. 69), Oxford Univ. Press, Oxford (1983).

    MATH  Google Scholar 

  27. P. Breitenlohner, G. Gibbons, and D. Maison, Commun. Math. Phys., 120, 295–333 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Univ. Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  29. A. T. Filippov, Modern Phys. Lett. A, 11, 1691–1704 (1996); Internat. J. Mod. Phys. A, 12, 13–22 (1997); arXiv:gr-qc/9612058v1 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. A. T. Filippov, “Integrable models of horizons and cosmologies,” arXiv:hep-th/0307266v1 (2003).

    Google Scholar 

  31. V. de Alfaro and A. T. Filippov, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 140, 139–145 (2007); arXiv:hep-th/0307269v1 (2003).

    Google Scholar 

  32. V. de Alfaro and A. T. Filippov, “Integrable low dimensional models for black holes and cosmologies from high dimensional theories,” arXiv:hep-th/0504101v1 (2005).

    Google Scholar 

  33. V. de Alfaro and A. T. Filippov, Theor. Math. Phys., 162, 34–56 (2010); arXiv:0902.4445v1 [hep-th] (2009).

    Article  MATH  Google Scholar 

  34. A. T. Filippov and D. Maison, Class. Q. Grav., 20, 1779–1786 (2003); arXiv:gr-qc/0210081v1 (2002).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. R. Arnowitt, S. Deser, and C.W. Misner, “The dynamics of general relativity,” in: Gravitation: An Introduction to Current Research (L. Witten, ed.), Wiley, New York (1962), pp. 227–264; arXiv:gr-qc/0405109v1 (2005).

    Google Scholar 

  36. A. T. Filippov, Theor. Math. Phys., 146, 95–107 (2006); arXiv:hep-th/0505060v2 (2005).

    Article  MATH  Google Scholar 

  37. V. De Alfaro and A. T. Filippov, Theor. Math. Phys., 153, 1709–1731 (2007); arXiv:hep-th/0612258v2 (2006).

    Article  MATH  Google Scholar 

  38. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton, N. J. (2008).

    Google Scholar 

  39. D. Grumiller, W. Kummer, and D. Vassilevich, Phys. Rep., 369, 327–430 (2002); arXiv:hep-th/0204253v9 (2002).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. R. Jackiw, Theor. Math. Phys., 92, 979–987 (1992).

    Article  MathSciNet  Google Scholar 

  41. C. Callan Jr., S. Giddings, J. Harvey, and A. Strominger, Phys. Rev. D, 45, R1005–R1009 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  42. V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic Concepts and New Developments (Fund. Theories Phys., Vol. 96), Springer, Berlin (1998).

    Book  MATH  Google Scholar 

  43. E. A. Davydov and A. T. Filippov, “Dilaton-scalar models in context of generalized affine gravity theories: Their properties and integrability,” arXiv:1302.6969v2 [hep-th] (2013).

    Google Scholar 

  44. L. H. Ford, Phys. Rev. D, 40, 967–972 (1989).

    Article  ADS  Google Scholar 

  45. M. Cavaglia, V. de Alfaro, and A. T. Filippov, Internat. J. Mod. Phys. D, 4, 661–672 (1995); arXiv:gr-qc/9411070v2 (1994); 5, 227–250 (1996); arXiv:gr-qc/9508062v1 (1995).

    Article  ADS  Google Scholar 

  46. M. Cavaglia, V. de Alfaro, and A. T. Filippov, Internat. J. Mod. Phys. A, 10, 611–633 (1995); arXiv:gr-qc/9402031v1 (1994).

    Article  ADS  MATH  Google Scholar 

  47. H. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer, Berlin (2009).

    Google Scholar 

  48. D. Oriti, ed., Approaches to Quantum Gravity: Toward a New Understanding of Space, Time, and Matter, Cambridge Univ. Press, Cambridge (2009).

    Book  Google Scholar 

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  1. Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia

    A. T. Filippov

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 2, pp. 323–352, November, 2013.

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Filippov, A.T. Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton-scalaron duality and its applications. Theor Math Phys 177, 1555–1577 (2013). https://doi.org/10.1007/s11232-013-0122-1

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