Quantum phase space description of a cosmological minimal massive bigravity model


Bimetric gravity theories describes gravitational interactions in the presence of an extra spin-2 field. The Hassan–Rosen (HR) nonlinear massive minimal bigravity theory is a ghost-free bimetric theory formulated with respect a flat, dynamical reference metric. In this work the deformation quantization formalism is applied to a HR cosmological model in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in the minisuperspace. The Friedmann–Lemaître–Robertson–Walker model with flat metrics is worked out and the computation of the Wigner functions for the Hartle–Hawking, Vilenkin and Linde wavefunctions are done numerically and, in the Hartle–Hawking case, also analytically. From the stability analysis in the quantum minisuper phase space it is found an interpretation of the mass of graviton as an emergent cosmological constant and as a measure of the deviation of classical behavior of the Wigner functions. Also, from the Hartle–Hawking case, an interesting relation between the curvature and the mass of graviton in a cusp catastrophe surface is discussed.

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  1. 1.

    Einstein, A.: Die Grundlage der allgemeinen Relativitàtstheorie. Annalen der Physik 354(7), 769–822 (1916). https://doi.org/10.1002/andp.19163540702

  2. 2.

    Fierz, M., Pauli, W.: On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. A 173(953), 211–232 (1939). https://doi.org/10.1098/rspa.1939.0140

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Boulware, D.G., Desser, S.: Inconsistency of finite range gravitation. Phys. Lett. B 40(2), 227–229 (1972). https://doi.org/10.1016/0370-2693(72)90418-2

    ADS  Article  Google Scholar 

  4. 4.

    Boulware, D.G., Deser, S.: Can gravitation have a finite range? Phys. Rev. D 6(12), 3368–3382 (1972). https://doi.org/10.1103/PhysRevD.6.3368

    ADS  Article  Google Scholar 

  5. 5.

    de Rham, C., Gabadadze, G., Tolley, A.J.: Resummation of massive gravity. Phys. Rev. Lett. 106(23), 231101 (2011). https://doi.org/10.1103/PhysRevLett.106.231101

    ADS  Article  Google Scholar 

  6. 6.

    Hassan, S.F., Rosen, R.A.: Resolving the ghost problem in nonlinear massive gravity. Phys. Rev. Lett. 108(4), 041101 (2012). https://doi.org/10.1103/PhysRevLett.108.041101

    ADS  Article  Google Scholar 

  7. 7.

    de Rham, C., Fasiello, M., Tolley, A.J.: Stable FLRW solutions in generalized massive gravity. Int. J. Mod. Phys. D 23(13), 1443006 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Yamashita, Y., De Felice, A., Tanaka, T.: Appearance of Boulware–Deser ghost in bigravity with doubly coupled matter. Int. J. Mod. Phys. D 23(13), 1443003 (2014). https://doi.org/10.1142/S0218271814430032

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Hassan, S.F., Rosen, R.A., Schmidt-May, A.: Ghost-free massive gravity with a general reference metric. J. High Energy Phys. 2012(2), 26 (2012). https://doi.org/10.1007/JHEP02(2012)026

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Rosen, N.: Bimetric gravitation and cosmology. Astrophys. J. 211, 357–360 (1977). https://doi.org/10.1086/154941

    ADS  Article  Google Scholar 

  11. 11.

    Koennig, F., Akrami, Y., Amendola, L., Motta, M., Solomon, A.R.: Stable and unstable cosmological models in bimetric massive gravity. Phys. Rev. D 90, 124014 (2014). https://doi.org/10.1103/PhysRevD.90.124014

    ADS  Article  Google Scholar 

  12. 12.

    Gümrükçüoğlu, A.E., Heisenberg, L., Mukohyama, S., Tanahashi, N.: Cosmology in bimetric theory with an effective composite coupling to matter. J. Cosmol. Astropart. Phys. 04, 008 (2015). https://doi.org/10.1088/1475-7516/2015/04/008

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Lüben, M., Mörtsell, E., Schmidt-May, A.: Bimetric cosmology is compatible with local tests of gravity. Class. Quantum Gravity 37(4), 047001 (2020). https://doi.org/10.1088/1361-6382/ab4f9b

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Vilenkin, A.: Creation of universes from nothing. Phys. Lett. B 117(1), 25–28 (1982). https://doi.org/10.1016/0370-2693(82)90866-8

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    DeWitt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160(5), 1113–1148 (1967). https://doi.org/10.1103/PhysRev.160.1113

    ADS  Article  MATH  Google Scholar 

  16. 16.

    Tryon, E.P.: Is the universe a vacuum fluctuation? Nature 246(5433), 396–397 (1973). https://doi.org/10.1038/246396a0

    ADS  Article  Google Scholar 

  17. 17.

    Rubakov, V.A.: Quantum mechanics in the tunneling universe. Phys. Lett. B 148(4), 280–286 (1984). https://doi.org/10.1016/0370-2693(84)90088-1

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Hartle, J.B., Hawking, S.W.: Wave function of the universe. Phys. Rev. D 28(12), 2960–2975 (1983). https://doi.org/10.1103/PhysRevD.28.2960

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Vilenkin, A.: Quantum creation of universes. Phys. Rev. D 30(2), 509–511 (1984). https://doi.org/10.1103/PhysRevD.30.509

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Vilenkin, A.: Approaches to quantum cosmology. Phys. Rev. D 50(4), 2581–2594 (1994). https://doi.org/10.1103/PhysRevD.50.2581

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Vilenkin, A., Shellard, E.P.S.: Cosmic Strings and Other Topological Defects. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  22. 22.

    Linde, A.D.: Quantum creation of the inflationary universe. Lett. Nuovo Cimento 39(17), 401–405 (1984). https://doi.org/10.1007/BF02790571

    ADS  Article  Google Scholar 

  23. 23.

    Pinto-Neto, N., Fabris, J.C.: Quantum cosmology from the de Broglie–Bohm perspective. Class. Quantum Gravity 30(14), 143001 (2013). https://doi.org/10.1088/0264-9381/30/14/143001

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kuchař, K.V., Ryan, M.P.: Is minisuperspace quantization valid?: Taub in mixmaster. Phys. Rev. D 40(12), 3982–3996 (1989). https://doi.org/10.1103/PhysRevD.40.3982

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Halliwell, J.J.: Introductory lectures on quantum cosmology. In: Coleman, S., Hartle, J.B., Piran, T., Weinberg, S. (eds.) Quantum Cosmology and BabyUniverses. Chap. 3, pp. 159–4243. World Scientific, Singapore (1991).https://doi.org/10.1142/9789814503501-0003

  26. 26.

    Halliwell, J.J.: How the quantum universe became classical. Contemp. Phys. 46(2), 93–104 (2005). https://doi.org/10.1080/0010751052000297588

    ADS  Article  Google Scholar 

  27. 27.

    Habib, S., Laflamme, R.: Wigner function and decoherence in quantum cosmology. Phys. Rev. D 42(12), 4056–4065 (1990). https://doi.org/10.1103/PhysRevD.42.4056

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40(2), 213–238 (1994). https://doi.org/10.4310/jdg/1214455536

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35–72 (1999). https://doi.org/10.1023/A:1007555725247

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Cordero, R., García-Compeán, H., Turrubiates, F.J.: Deformation quantization of cosmological models. Phys. Rev. D 83, 125030 (2011). https://doi.org/10.1103/PhysRevD.83.125030

    ADS  Article  Google Scholar 

  31. 31.

    Cordero, R., Garcıa-Compeán, H., Turrubiates, F.J.: A phase space description of the FLRW quantum cosmology in Hořava–Lifshitz type gravity. Gen. Relativ. Gravit. 51(10), 138 (2019). https://doi.org/10.1007/s10714-019-2627-x

    ADS  Article  MATH  Google Scholar 

  32. 32.

    de Rham, C.: Massive gravity. Living Rev. Relativ. 17(1), 7 (2014). https://doi.org/10.12942/lrr-2014-7

    ADS  Article  MATH  Google Scholar 

  33. 33.

    Darabi, F., Mousavi, M.: Classical and quantum cosmology of minimal massive bigravity. Phys. Lett. B 761, 269–280 (2016). https://doi.org/10.1016/j.physletb.2016.08.031

    ADS  Article  MATH  Google Scholar 

  34. 34.

    Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys. 12(3), 498–501 (1971). https://doi.org/10.1063/1.1665613

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Cruz, M., Rojas, E.: Born–Infeld extension of Lovelock brane gravity. Class. Quantum Gravity 30(11), 115012 (2013). https://doi.org/10.1088/0264-9381/30/11/115012

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Hassan, S.F., Rosen, R.A.: On non-linear actions for massive gravity. J. High Energy Phys. 2011(7), 9 (2011). https://doi.org/10.1007/JHEP07(2011)009

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Hall, B.C.: Quantum Theory for Mathematicians. Number 257 in Graduate Texts in Mathematics. Springer, Berlin (2013)

    Book  Google Scholar 

  38. 38.

    Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367–387 (1948). https://doi.org/10.1103/RevModPhys.20.367

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978). https://doi.org/10.1016/0003-4916(78)90224-5

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Weyl, H.: Quantenmechanik und Gruppentheorie. Z. Phys. 46(1), 1–46 (1927). https://doi.org/10.1007/BF02055756

    ADS  Article  MATH  Google Scholar 

  41. 41.

    Groenewold, H.J.: On the principles of elementary quantum mechanics. Physica 12(7), 405–460 (1946). https://doi.org/10.1016/S0031-8914(46)80059-4

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Moyal, J.E.: Quantum mechanics as a statistical theory. Math. Proc. Camb. Philos. Soc. 45(1), 99–124 (1949). https://doi.org/10.1017/S0305004100000487

    ADS  MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Antonsen, F.: Deformation Quantisation of Gravity. arXiv:gr-qc/9712012 (1997)

  44. 44.

    Antonsen, F.: Deformation Quantisation of Constrained Systems. arXiv:gr-qc/9710021 (1997)

  45. 45.

    Hall, B.C.: Lie Groups, Lie Algebras, and Representations An Elementary Introduction. Number 222 in Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  46. 46.

    Schutz, B.F.: Perfect fluids in general relativity: velocity potentials and a variational principle. Phys. Rev. D 2(12), 2762–2773 (1970). https://doi.org/10.1103/PhysRevD.2.2762

    ADS  MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Vakili, B.: Quadratic quantum cosmology with Schutz’ perfect fluid. Class. Quantum Gravity 27, 025008 (2010). https://doi.org/10.1088/0264-9381/27/2/025008

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Aghanim, N., Akrami, Y., Arroja, F., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A.J., Barreiro, R.B., Bartolo, N., Basak, S., Battye, R., Benabed, K., Bernard, J.-P., Bersanelli, M., Bielewicz, P., Bock, J.J., Bond, J.R., Borrill, J., Bouchet, F.R., Boulanger, F., Bucher, M., Burigana, C., Butler, R.C., Calabrese, E., Cardoso, J.-F., Carron, J., Casaponsa, B., Challinor, A., Chiang, H.C., Colombo, L.P.L., Combet, C., Contreras, D., Crill, B.P., Cuttaia, F., de Bernardis, P., de Zotti, G., Delabrouille, J., Delouis, J.-M., Désert, F.-X., Di Valentino, E., Dickinson, C., Diego, J.M., Donzelli, S., Doré, O., Douspis, M., Ducout, A., Dupac, X., Efstathiou, G., Elsner, F., Enßlin, T.A., Eriksen, H.K., Falgarone, E., Fantaye, Y., Fergusson, J., Fernandez-Cobos, R., Finelli, F., Forastieri, F., Frailis, M., Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K., Génova-Santos, R.T., Gerbino, M., Ghosh, T., González-Nuevo, J., Górski, K.M., Gratton, S., Gruppuso, A., Gudmundsson, J.E., Hamann, J., Handley, W., Hansen, F.K., Helou, G., Herranz, D., Hildebrandt, S.R., Hivon, E., Huang, Z., Jaffe, A.H., Jones, W.C., Karakci, A., Keihänen, E., Keskitalo, R., Kiiveri, K., Kim, J., Kisner, T.S., Knox, L., Krachmalnicoff, N., Kunz, M., Kurki-Suonio, H., Lagache, G., Lamarre, J.-M., Langer, M., Lasenby, A., Lattanzi, M., Lawrence, C.R., Le Jeune, M., Leahy, J.P., Lesgourgues, J., Levrier, F., Lewis, A., Liguori, M., Lilje, P.B., Lilley, M., Lindholm, V., López-Caniego, M., Lubin, P.M., Ma, Y.-Z., Macías-Pérez, J.F., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Marcos-Caballero, A., Maris, M., Martin, P.G., Martinelli, M., Martínez-González, E., Matarrese, S., Mauri, N., McEwen, J.D., Meerburg, P.D., Meinhold, P.R., Melchiorri, A., Mennella, A., Migliaccio, M., Millea, M., Mitra, S., Miville-Deschânes, M.-A., Molinari, D., Moneti, A., Montier, L., Morgante, G., Moss, A., Mottet, S., Münchmeyer, M., Natoli, P., Nørgaard-Nielsen, H.U., Oxborrow, C.A., Pagano, L., Paoletti, D., Partridge, B., Patanchon, G., Pearson, T.J., Peel, M., Peiris, H.V., Perrotta, F., Pettorino, V., Piacentini, F., Polastri, L., Polenta, G., Puget, J.-L., Rachen, J.P., Reinecke, M., Remazeilles, M., Renault, C., Renzi, A., Rocha, G., Rosset, C., Roudier, G., Rubiño-Martín, J.A., Ruiz-Granados, B., Salvati, L., Sandri, M., Savelainen, M., Scott, D., Shellard, E.P.S., Shiraishi, M., Sirignano, C., Sirri, G., Spencer, L.D., Sunyaev, R., Suur-Uski, A.-S., Tauber, J.A., Tavagnacco, D., Tenti, M., Terenzi, L., Toffolatti, L., Tomasi, M., Trombetti, T., Valiviita, J., Van Tent, B., Vibert, L., Vielva, P., Villa, F., Vittorio, N., Wandelt, B.D., Wehus, I.K., White, M., White, S.D.M., Zacchei, A., Zonca, A.: Planck 2018 results—I. Overview and the cosmological legacy of Planck. A&A 641, A1 (2020)

  49. 49.

    Cordero, R., Turrubiates, F.J., Vera, J.C.: On a phase space quantum description of the spherical \$2\$-brane. Phys. Scr. 89(7), 075001 (2014). https://doi.org/10.1088/0031-8949/89/7/075001

    ADS  Article  Google Scholar 

  50. 50.

    Zwillinger, D., Moll, V., Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 8th edn. Academic Press, Boston (2015)

    MATH  Google Scholar 

  51. 51.

    Zachos, C., Fairlie, D., Curtright, T.: Quantum Mechanics in Phase Space: An Overview with Selected Papers. World Scientific, Singapore (2005)

    Book  Google Scholar 

  52. 52.

    Akhundova, E.A., Dodonov, V.V.: Wigner function for a particle in delta-potential and in a box. J. Russ. Laser Res. 13(4), 312–318 (1992). https://doi.org/10.1007/BF01371397

    Article  Google Scholar 

  53. 53.

    Bernardini, A.E., Leal, P., Bertolami, O.: Quantum to classical transition in the Hořava–Lifshitz quantum cosmology. J. Cosmol. Astropart. Phys. 2018(02), 025–025 (2018). https://doi.org/10.1088/1475-7516/2018/02/025

    Article  Google Scholar 

  54. 54.

    Rashki, M., Jalalzadeh, S.: The quantum state of the universe from deformation quantization and classical–quantum correlation. Gen. Relativ. Gravit. 49(2), 14 (2017). https://doi.org/10.1007/s10714-016-2178-3

    ADS  MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Calzetta, E., Hu, B.L.: Wigner distribution function and phase-space formulation of quantum cosmology. Phys. Rev. D 40(2), 380–389 (1989). https://doi.org/10.1103/PhysRevD.40.380.

    ADS  MathSciNet  Article  Google Scholar 

  56. 56.

    Berry, M.V., Ziman, J.M.: Semi-classical mechanics in phase space: a study of Wigner’s function. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 287(1343), 237–271 (1977). https://doi.org/10.1098/rsta.1977.0145

    ADS  MathSciNet  Article  MATH  Google Scholar 

  57. 57.

    Lötkenhaus, N., Barnett, S.M.: Nonclassical effects in phase space. Phys. Rev. A 51(4), 3340–3342 (1995). https://doi.org/10.1103/PhysRevA.51.3340

    ADS  Article  Google Scholar 

  58. 58.

    Kenfack, A., Źyczkowski, K.: Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B Quantum Semiclass. Opt. 6(10), 396–404 (2004). https://doi.org/10.1088/1464-4266/6/10/003

    ADS  MathSciNet  Article  Google Scholar 

  59. 59.

    Sasaki, M., Yeom, D., Zhang, Y.: Hartle–Hawking no-boundary proposal in dRGT massive gravity: making inflation exponentially more probable. Class. Quantum Gravity 30(23), 232001 (2013). https://doi.org/10.1088/0264-9381/30/23/232001

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. 60.

    Levin, J.: Topology and the cosmic microwave background. Phys. Rep. 365(4), 251–333 (2002). https://doi.org/10.1016/S0370-1573(02)00018-2

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    Luminet, J.P.: The status of cosmic topology after Planck data. Universe 2(1), 1 (2016). https://doi.org/10.3390/universe2010001

    ADS  Article  Google Scholar 

  62. 62.

    Zeldovich, Y.B., Starobinskii, A.A.: Quantum creation of a universe with nontrivial topology. Sov. Astron. Lett. 10, 135–137 (1984)

    ADS  Google Scholar 

  63. 63.

    de Lorenci, V.A., Martin, J., Pinto-Neto, N., Soares, I.D.: Topology change in canonical quantum cosmology. Phys. Rev. D 56(6), 3329–3340 (1997). https://doi.org/10.1103/PhysRevD.56.3329

    ADS  MathSciNet  Article  Google Scholar 

  64. 64.

    Steuernagel, O., Kakofengitis, D., Ritter, G.: Wigner flow reveals topological order in quantum phase space dynamics. Phys. Rev. Lett. 110(3), 030401 (2013). https://doi.org/10.1103/PhysRevLett.110.030401

    ADS  Article  Google Scholar 

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The author want to thank the referee for the comments and suggestions which allowed to improve this work. The author is indebted to Rubèn Cordero and Marta Costa for all his help and fruitful discussions.

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Vera-Hernández, J.C. Quantum phase space description of a cosmological minimal massive bigravity model. Gen Relativ Gravit 53, 55 (2021). https://doi.org/10.1007/s10714-021-02822-2

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  • Bigravity models
  • Quantum Cosmology
  • deformation quantization
  • Wigner functions