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A phase space description of the FLRW quantum cosmology in Hořava–Lifshitz type gravity

  • Rubén CorderoEmail author
  • Hugo García-Compeán
  • Francisco J. Turrubiates
Research Article
  • 42 Downloads

Abstract

Quantum cosmology of the Friedmann–Lemaître–Robertson–Walker model with cosmological constant in the Hořava–Lifshitz type gravity is studied in the phase space by means of the Wigner function. The modification of the usual general relativity description by the Hořava–Lifshitz type gravity induces a new scenario for the origin of the Universe with an embryonic era where the Universe can exist classically before the tunneling process takes place and which gives rise to the current evolution of the Universe. The Wigner functions corresponding to the Hartle–Hawking, Vilenkin and Linde boundary conditions are obtained by means of numerical calculations. In particular three cases were studied for the potential of the Wheeler–DeWitt equation: tunneling barrier with and without embryonic era and when the potential barrier is not present. The quantum behavior of these three cases are analyzed using the Wigner function for the three boundary conditions considered.

Keywords

Hořava–Lifshitz gravity Quantum cosmology Wigner functions 

Notes

Acknowledgements

We want to thank the referees for their comments and suggestions which allowed us to improve this work. The work of R. C., H. G.-C. and F. J. T. was partially supported by SNI-México, CONACyT research Grant: 128761. In addition R. C. and F. J. T. were partially supported by COFAA-IPN and by SIP-IPN Grants 20171168, 20171100, 20180735, 20180741, 20194924 and 20195330. We are indebted to Héctor Uriarte for all his help in the elaboration of the figures presented in the paper.

References

  1. 1.
    Hořava, P.: JHEP 0903, 020 (2009). arXiv:0812.4287 [hep-th]ADSMathSciNetGoogle Scholar
  2. 2.
    Hořava, P.: Phys. Rev. D 79, 084008 (2009). arXiv:0901.3775 [hep-th]ADSMathSciNetGoogle Scholar
  3. 3.
    Mukohyama, S.: Class. Quantum Gravity 27, 223101 (2010). arXiv:1007.5199 [hep-th]ADSMathSciNetGoogle Scholar
  4. 4.
    Sotiriou, T.P.: J. Phys. Conf. Ser. 283, 012034 (2011). arXiv:1010.3218 [hep-th]Google Scholar
  5. 5.
    Gumrukcuoglu, A.E., Mukohyama, S.: Phys. Rev. D 83, 124033 (2011). arXiv:1104.2087 [hep-th]ADSGoogle Scholar
  6. 6.
    Lepe, S., Saavedra, J.: Astrophys. Sp. Sci. 350, 839 (2014)ADSGoogle Scholar
  7. 7.
    Vakili, B., Kord, V.: Gen. Relativ. Gravit. 45, 1313 (2013). arXiv:1301.0809 [gr-qc]ADSGoogle Scholar
  8. 8.
    Blas, D., Pujolas, O., Sibiryakov, S.: Phys. Rev. Lett. 104, 181302 (2010). arXiv:0909.3525 [hep-th]ADSMathSciNetGoogle Scholar
  9. 9.
    Blas, D., Pujolas, O., Sibiryakov, S.: JHEP 1104, 018 (2011). arXiv:1007.3503 [hep-th]ADSGoogle Scholar
  10. 10.
    Charmousis, C., Niz, G., Padilla, A., Saffin, P.M.: JHEP 0908, 070 (2009). arXiv:0905.2579 [hep-th]ADSGoogle Scholar
  11. 11.
    Blas, D., Pujolas, O., Sibiryakov, S.: JHEP 0910, 029 (2009). arXiv:0906.3046 [hep-th]ADSGoogle Scholar
  12. 12.
    Bogdanos, C., Saridakis, E.N.: Class. Quant. Gravity 27, 075005 (2010). arXiv:0907.1636 [hep-th]ADSGoogle Scholar
  13. 13.
    Koyama, K., Arroja, F.: JHEP 1003, 061 (2010). arXiv:0910.1998 [hep-th]ADSGoogle Scholar
  14. 14.
    Izumi, K., Mukohyama, S.: Phys. Rev. D 84, 064025 (2011).  https://doi.org/10.1103/PhysRevD.84.064025. arXiv:1105.0246 [hep-th]ADSCrossRefGoogle Scholar
  15. 15.
    Gumrukcuoglu, A.E., Mukohyama, S., Wang, A.: Phys. Rev. D 85, 064042 (2012).  https://doi.org/10.1103/PhysRevD.85.064042. arXiv:1109.2609 [hep-th]ADSCrossRefGoogle Scholar
  16. 16.
    Fukushima, M., Misonoh, Y., Miyashita, S., Sato, S.: Phys. Rev. D 99(6), 064004 (2019).  https://doi.org/10.1103/PhysRevD.99.064004. arXiv:1812.10295 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Sotiriou, T.P., Visser, M., Weinfurtner, S.: JHEP 0910, 033 (2009). arXiv:0905.2798 [hep-th]ADSGoogle Scholar
  18. 18.
    Papazoglou, A., Sotiriou, T.P.: Phys. Lett. B 685, 197 (2010). arXiv:0911.1299 [hep-th]ADSMathSciNetGoogle Scholar
  19. 19.
    Wang, A., Wu, Y.: JCAP 0907, 012 (2009). arXiv:0905.4117 [hep-th]ADSGoogle Scholar
  20. 20.
    Yamamoto, K., Kobayashi, T., Nakamura, G.: Phys. Rev. D 80, 063514 (2009). arXiv:0907.1549 [astro-ph.CO]ADSGoogle Scholar
  21. 21.
    Maeda, S., Mukohyama, S., Shiromizu, T.: Phys. Rev. D 80, 123538 (2009). arXiv:0909.2149 [astro-ph.CO]ADSGoogle Scholar
  22. 22.
    Carloni, S., Elizalde, E., Silva, P.J.: Class. Quantum Gravity 27, 045004 (2010). arXiv:0909.2219 [hep-th]ADSGoogle Scholar
  23. 23.
    Wang, A., Wands, D., Maartens, R.: JCAP 1003, 013 (2010). arXiv:0909.5167 [hep-th]ADSGoogle Scholar
  24. 24.
    Gao, X., Wang, Y., Xue, W., Brandenberger, R.: JCAP 1002, 020 (2010). arXiv:0911.3196 [hep-th]ADSGoogle Scholar
  25. 25.
    Dutta, S., Saridakis, E.N.: JCAP 1005, 013 (2010). arXiv:1002.3373 [hep-th]ADSGoogle Scholar
  26. 26.
    Saridakis, E.N.: Int. J. Mod. Phys. D 20, 1485 (2011).  https://doi.org/10.1142/S0218271811019670. arXiv:1101.0300 [astro-ph.CO]ADSCrossRefGoogle Scholar
  27. 27.
    DeWitt, B.S.: Phys. Rev. 160, 1113 (1967)ADSGoogle Scholar
  28. 28.
    Tryon, E.P.: Nature (London) 246, 396 (1973)ADSGoogle Scholar
  29. 29.
    Fomin, P.I.: Dokl. Akad. Nauk Ukr. SSR 9A, 831 (1975)Google Scholar
  30. 30.
    Atkatz, D., Pagels, H.: Phys. Rev. D 25, 2065 (1982)ADSGoogle Scholar
  31. 31.
    Vilenkin, A.: Phys. Lett. B 117, 25 (1982)ADSGoogle Scholar
  32. 32.
    Grishchuk, L.P., Zel’dovich, Ya.B.: In: Duff, M., Isham, C. (eds.) Quantum Structure of Space and Time. Cambridge University Press, Cambridge (1982)Google Scholar
  33. 33.
    Hartle, J.B., Hawking, S.W.: Phys. Rev. D 28, 2960 (1983)ADSMathSciNetGoogle Scholar
  34. 34.
    Linde, A.D.: Lett. Nuovo Cimento 39, 401 (1984)ADSGoogle Scholar
  35. 35.
    Rubakov, V.A.: Phys. Lett. B 148, 280 (1984)ADSMathSciNetGoogle Scholar
  36. 36.
    Vilenkin, A.: Phys. Rev. D 30, 509 (1984)ADSMathSciNetGoogle Scholar
  37. 37.
    Vilenkin, A.: Phys. Rev. D 50, 2581 (1994)ADSMathSciNetGoogle Scholar
  38. 38.
    Bousso, R., Hawking, S.W.: Phys. Rev. D 54, 6312 (1996) ADSMathSciNetGoogle Scholar
  39. 39.
    Garriga, J., Vilenkin, A.: Phys. Rev. D 56, 2464 (1997)ADSMathSciNetGoogle Scholar
  40. 40.
    Linde, A.D.: Phys. Rev. D 58, 083514 (1998)ADSMathSciNetGoogle Scholar
  41. 41.
    Turok, N.G., Hawking, S.W.: Phys. Lett. 432, 271 (1998)MathSciNetGoogle Scholar
  42. 42.
    Vilenkin, A.: Phys. Rev. D 58, 067301 (1998)ADSMathSciNetGoogle Scholar
  43. 43.
    Brustein, R., de Alwis, S.P.: Phys. Rev. D 73, 046009 (2006). arXiv:hep-th/0511093 ADSMathSciNetGoogle Scholar
  44. 44.
    Bertolami, O., Zarro, C.A.D.: Phys. Rev. D 84, 044042 (2011). arXiv:1106.0126 [hep-th]ADSGoogle Scholar
  45. 45.
    Pitelli, J.P.M., Saa, A.: Phys. Rev. D 86, 063506 (2012). arXiv:1204.4924 [gr-qc]ADSGoogle Scholar
  46. 46.
    Christodoulakis, T., Dimakis, N.: J. Geom. Phys. 62, 2401 (2012). arXiv:1112.0903 [gr-qc]ADSMathSciNetGoogle Scholar
  47. 47.
    Obregón, O., Preciado, J.A.: Phys. Rev. D 86, 063502 (2012). arXiv:1305.6950 [gr-qc]ADSGoogle Scholar
  48. 48.
    Benedetti, D., Henson, J.: Class. Quantum Gravity 32, 215007 (2015). arXiv:1410.0845 [gr-qc]ADSGoogle Scholar
  49. 49.
    Kim, Y.S., Noz, M.E.: Phase Space Picture of Quantum Mechanics. Lecture Notes in Physics Series, vol. 40. World Scientific, Singapore (1991)Google Scholar
  50. 50.
    Zachos, C.K., Fairlie, D.B., Curtright, T.L.: Quantum Mechanics in Phase Space. An Overview with Selected Papers. World Scientific Series in 20th Century Physics, vol. 34. World Scientific, Singapore (2005)zbMATHGoogle Scholar
  51. 51.
    Weinbub, J., Ferry, D.K.: Appl. Phys. Rev. 5, 041104 (2018)ADSGoogle Scholar
  52. 52.
    Dragoman, D.: EURASIP J. Adv. Signal Process. 10, 1520 (2005)Google Scholar
  53. 53.
    Kurtsiefer, C., Pfau, T., Mlynek, J.: Nature 386, 150 (1997)ADSGoogle Scholar
  54. 54.
    Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., Grangier, P.: Nature 448, 784 (2007)ADSGoogle Scholar
  55. 55.
    Deleglise, S., et al.: Nature 455, 510 (2008)ADSGoogle Scholar
  56. 56.
    Cordero, R., García-Compeán, H., Turrubiates, F.J.: Phys. Rev. D 83, 125030 (2011). arXiv:1102.4379 [hep-th]ADSGoogle Scholar
  57. 57.
    Bernardini, A.E., Leal, P., Bertolami, O.: Quantum to classical transition in the Hořava–Lifshitz quantum cosmology. JCAP 1802(02), 025 (2018). arXiv:1711.02627 [gr-qc]ADSGoogle Scholar
  58. 58.
    Davidson, A., Karasik, D., Lederer, Y.: Class. Quantum Gravity 16, 1349 (1999)ADSGoogle Scholar
  59. 59.
    Maeda, K.I., Misonoh, Y., Kobayashi, T.: Phys. Rev. D 82, 064024 (2010).  https://doi.org/10.1103/PhysRevD.82.064024. arXiv:1006.2739 [hep-th]ADSCrossRefGoogle Scholar
  60. 60.
    Barvinsky, A.O., Blas, D., Herrero-Valea, M., Sibiryakov, S.M., Steinwachs, C.F.: Phys. Rev. D 93(6), 064022 (2016).  https://doi.org/10.1103/PhysRevD.93.064022. arXiv:1512.02250 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    Sotiriou, T.P., Visser, M., Weinfurtner, S.: Phys. Rev. Lett. 102, 251601 (2009).  https://doi.org/10.1103/PhysRevLett.102.251601. arXiv:0904.4464 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Ann. Phys. NY 111, 61 (1978)ADSGoogle Scholar
  63. 63.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Ann. Phys. NY 111, 111 (1978)ADSGoogle Scholar
  64. 64.
    Fedosov, B.V.: J. Differ. Geom. 40, 213 (1994)Google Scholar
  65. 65.
    Kontsevich, M.: Lett. Math. Phys. 66, 157 (2003). arXiv:q-alg/9709040 [q-alg]ADSMathSciNetGoogle Scholar
  66. 66.
    Dito, G., Sternheimer, D.: Deformation quantization: genesis, developments and metamorphoses. Proc. Mtg Between Mathematicians and Theoretical Physicists, Strasbourg 2001. IRMA Lectures in Math. Theoret. Phys., vol. 1, pp. 9–54. de Gruyter, Berlin (2002)Google Scholar
  67. 67.
    Page, D.N.: In: Proceedings of the Eleventh Marcel Grossmann Meeting, Berlin, Germany, 23–29 July 2006, pp. 1928–1932 (2008). arXiv:hep-th/0612194
  68. 68.
    Calcagni, G., Kiefer, C., Steinwachs, C.F.: J. Phys. Conf. Ser. 626(1), 012003 (2015).  https://doi.org/10.1088/1742-6596/626/1/012003. arXiv:1503.08770 [gr-qc]CrossRefGoogle Scholar
  69. 69.
  70. 70.
    Feldbrugge, J., Lehners, J.L., Turok, N.: Universe 4(10), 100 (2018).  https://doi.org/10.3390/universe4100100. arXiv:1805.01609 [hep-th]ADSCrossRefGoogle Scholar
  71. 71.
    Feldbrugge, J., Lehners, J.L., Turok, N.: Phys. Rev. Lett. 119(17), 171301 (2017).  https://doi.org/10.1103/PhysRevLett.119.171301. arXiv:1705.00192 [hep-th]ADSCrossRefGoogle Scholar
  72. 72.
    Vilenkin, A., Yamada, M.: Phys. Rev. D 98(6), 066003 (2018).  https://doi.org/10.1103/PhysRevD.98.066003. arXiv:1808.02032 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    Vilenkin, A., Yamada, M.: Phys. Rev. D 99(6), 066010 (2019).  https://doi.org/10.1103/PhysRevD.99.066010. arXiv:1812.08084 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    Di Tucci, A., Feldbrugge, J., Lehners, J.L., Turok, N.: Phys. Rev. D 100(6), 063517 (2019).  https://doi.org/10.1103/PhysRevD.100.063517. arXiv:1906.09007 [hep-th]ADSCrossRefGoogle Scholar
  75. 75.
    Magueijo, J., Smolin, L.: Class. Quantum Gravity 21, 1725 (2004).  https://doi.org/10.1088/0264-9381/21/7/001. arXiv:gr-qc/0305055 ADSCrossRefGoogle Scholar
  76. 76.
    Garattini, R., Saridakis, E.N.: Eur. Phys. J. C 75(7), 343 (2015).  https://doi.org/10.1140/epjc/s10052-015-3562-y. arXiv:1411.7257 [gr-qc]ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de FísicaEscuela Superior de Física y Matemáticas del Instituto Politécnico Nacional, Unidad Adolfo López MateosMexico CityMexico
  2. 2.Departamento de FísicaCentro de Investigación y de Estudios Avanzados del IPNMexico CityMexico
  3. 3.División de Ciencias e Ingenierías, Departamento de FísicaUniversidad de Guanajuato, Campus LeónLeónMexico

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