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Journal of High Energy Physics

, 2016:49 | Cite as

Scalar field as an intrinsic time measure in coupled dynamical matter-geometry systems. I. Neutral gravitational collapse

  • Anna NakoniecznaEmail author
  • Dong-han Yeom
Open Access
Regular Article - Theoretical Physics

Abstract

There does not exist a notion of time which could be transferred straightforwardly from classical to quantum gravity. For this reason, a method of time quantification which would be appropriate for gravity quantization is being sought. One of the existing proposals is using the evolving matter as an intrinsic ‘clock’ while investigating the dynamics of gravitational systems. The objective of our research was to check whether scalar fields can serve as time variables during a dynamical evolution of a coupled multicomponent matter-geometry system. We concentrated on a neutral case, which means that the elaborated system was not charged electrically nor magnetically. For this purpose, we investigated a gravitational collapse of a self-interacting complex and real scalar fields in the Brans-Dicke theory using the 2+2 spacetime foliation. We focused mainly on the region of high curvature appearing nearby the emerging singularity, which is essential from the perspective of quantum gravity. We investigated several formulations of the theory for various values of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke field and the matter sector of the theory. The obtained results indicated that the evolving scalar fields can be treated as time variables in close proximity of the singularity due to the following reasons. The constancy hypersurfaces of the Brans-Dicke field are spacelike in the vicinity of the singularity apart from the case, in which the equation of motion of the field reduces to the wave equation due to a specific choice of free evolution parameters. The hypersurfaces of constant complex and real scalar fields are spacelike in the regions nearby the singularities formed during the examined process. The values of the field functions change monotonically in the areas, in which the constancy hypersurfaces are spacelike.

Keywords

Classical Theories of Gravity Black Holes 

Notes

Open Access

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References

  1. [1]
    B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].
  2. [2]
    C. Rovelli, Time in quantum gravity: physics beyond the Schrödinger regime, Phys. Rev. D 43 (1991) 442 [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    C. Rovelli and L. Smolin, The physical Hamiltonian in nonperturbative quantum gravity, Phys. Rev. Lett. 72 (1994) 446 [gr-qc/9308002] [INSPIRE].
  4. [4]
    M. Domagala, K. Giesel, W. Kaminski and J. Lewandowski, Gravity quantized: loop quantum gravity with a scalar field, Phys. Rev. D 82 (2010) 104038 [arXiv:1009.2445] [INSPIRE].ADSGoogle Scholar
  5. [5]
    J. Lewandowski, M. Domagala and M. Dziendzikowski, The dynamics of the massless scalar field coupled to LQG in the polymer quantization, PoS(QGQGS 2011) 025 [INSPIRE].
  6. [6]
    S. Alexander, J. Malecki and L. Smolin, Quantum gravity and inflation, Phys. Rev. D 70 (2004) 044025 [hep-th/0309045] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    A. Ashtekar, T. Pawlowski and P. Singh, Quantum nature of the big bang, Phys. Rev. Lett. 96 (2006) 141301 [gr-qc/0602086] [INSPIRE].
  8. [8]
    A. Ashtekar, T. Pawlowski and P. Singh, Quantum nature of the big bang: improved dynamics, Phys. Rev. D 74 (2006) 084003 [gr-qc/0607039] [INSPIRE].
  9. [9]
    M.P. Dabrowski and A.L. Larsen, Quantum tunneling effect in oscillating Friedmann cosmology, Phys. Rev. D 52 (1995) 3424 [gr-qc/9504025] [INSPIRE].
  10. [10]
    P.W. Graham, B. Horn, S. Kachru, S. Rajendran and G. Torroba, A simple harmonic universe, JHEP 02 (2014) 029 [arXiv:1109.0282] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    A.T. Mithani and A. Vilenkin, Collapse of simple harmonic universe, JCAP 01 (2012) 028 [arXiv:1110.4096] [INSPIRE].CrossRefADSGoogle Scholar
  12. [12]
    A.T. Mithani and A. Vilenkin, Tunneling decay rate in quantum cosmology, Phys. Rev. D 91 (2015) 123511 [arXiv:1503.00400] [INSPIRE].ADSGoogle Scholar
  13. [13]
    L. Perlov, Wheeler-DeWitt equation for 4D supermetric and ADM with massless scalar field as internal time, Phys. Lett. B 743 (2015) 143 [arXiv:1412.4740] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    J.D. Brown and K.V. Kuchar, Dust as a standard of space and time in canonical quantum gravity, Phys. Rev. D 51 (1995) 5600 [gr-qc/9409001] [INSPIRE].
  15. [15]
    X. Zhang, Y. Ma and M. Artymowski, Loop quantum Brans-Dicke cosmology, Phys. Rev. D 87 (2013) 084024 [arXiv:1211.4183] [INSPIRE].ADSGoogle Scholar
  16. [16]
    C.R. Almeida, A.B. Batista, J.C. Fabris and P.R. L.V. Moniz, Quantum cosmology with scalar fields: self-adjointness and cosmological scenarios, Gravit. Cosmol. 21 (2015) 191 [arXiv:1501.04170] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    C. Brans and R.H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961) 925 [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  18. [18]
    V. Faraoni, Illusions of general relativity in Brans-Dicke gravity, Phys. Rev. D 59 (1999) 084021 [gr-qc/9902083] [INSPIRE].
  19. [19]
    C.M. Will, The confrontation between general relativity and experiment, Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377] [INSPIRE].Google Scholar
  20. [20]
    C. Romero and A. Barros, Does Brans-Dicke theory of gravity go over to the general relativity when ω → ∞?, Phys. Lett. A 173 (1993) 243 [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    B. Chauvineau, On the limit of Brans-Dicke theory when ω → ∞, Class. Quant. Grav. 20 (2003) 2617 [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    B. Chauvineau, A.D. A.M. Spallicci and J.-D. Fournier, Brans-Dicke gravity and the capture of stars by black holes: some asymptotic results, Class. Quant. Grav. 22 (2005) S457 [gr-qc/0412053] [INSPIRE].
  23. [23]
    B. Chauvineau, Stationarity and large ω Brans-Dicke solutions versus general relativity, Gen. Rel. Grav. 39 (2007) 297 [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  24. [24]
    B. Bertotti, L. Iess and P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature 425 (2003) 374 [INSPIRE].CrossRefADSGoogle Scholar
  25. [25]
    A. De Felice, G. Mangano, P.D. Serpico and M. Trodden, Relaxing nucleosynthesis constraints on Brans-Dicke theories, Phys. Rev. D 74 (2006) 103005 [astro-ph/0510359] [INSPIRE].
  26. [26]
    J.C. Fabris, S.V.B. Goncalves and R. de Sa Ribeiro, Late time accelerated Brans-Dicke pressureless solutions and the supernovae type-IA data, Grav. Cosmol. 12 (2006) 49 [astro-ph/0510779] [INSPIRE].
  27. [27]
    L.-E. Qiang, Y. Gong, Y. Ma and X. Chen, Cosmological implications of 5-dimensional Brans-Dicke theory, Phys. Lett. B 681 (2009) 210 [arXiv:0910.1885] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    O. Hrycyna, M. Szydlowski and M. Kamionka, Dynamics and cosmological constraints on Brans-Dicke cosmology, Phys. Rev. D 90 (2014) 124040 [arXiv:1404.7112] [INSPIRE].ADSGoogle Scholar
  29. [29]
    A. Avilez and C. Skordis, Cosmological constraints on Brans-Dicke theory, Phys. Rev. Lett. 113 (2014) 011101 [arXiv:1303.4330] [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    Y.-C. Li, F.-Q. Wu and X. Chen, Constraints on the Brans-Dicke gravity theory with the Planck data, Phys. Rev. D 88 (2013) 084053 [arXiv:1305.0055] [INSPIRE].ADSGoogle Scholar
  31. [31]
    V. Acquaviva, C. Baccigalupi, S.M. Leach, A.R. Liddle and F. Perrotta, Structure formation constraints on the Jordan-Brans-Dicke theory, Phys. Rev. D 71 (2005) 104025 [astro-ph/0412052] [INSPIRE].
  32. [32]
    T. Damour and K. Nordtvedt, General relativity as a cosmological attractor of tensor scalar theories, Phys. Rev. Lett. 70 (1993) 2217 [INSPIRE].CrossRefADSGoogle Scholar
  33. [33]
    T. Damour and K. Nordtvedt, Tensor-scalar cosmological models and their relaxation toward general relativity, Phys. Rev. D 48 (1993) 3436 [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    T. Damour, F. Piazza and G. Veneziano, Violations of the equivalence principle in a dilaton runaway scenario, Phys. Rev. D 66 (2002) 046007 [hep-th/0205111] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    D. La and P.J. Steinhardt, Extended inflationary cosmology, Phys. Rev. Lett. 62 (1989) 376 [Erratum ibid. 62 (1989) 1066] [INSPIRE].
  36. [36]
    C. Mathiazhagan and V.B. Johri, An inflationary universe in Brans-Dicke theory: a hopeful sign of theoretical estimation of the gravitational constant, Class. Quant. Grav. 1 (1984) L29 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    M. Arik and M.C. Calik, Primordial and asymptotic inflation in Brans-Dicke cosmology, JCAP 01 (2005) 013 [gr-qc/0403108] [INSPIRE].
  38. [38]
    M. Arik, M.C. Calik and M.B. Sheftel, Friedmann equation for Brans-Dicke cosmology, Int. J. Mod. Phys. D 17 (2008) 225 [gr-qc/0604082] [INSPIRE].
  39. [39]
    S. Sen and A.A. Sen, Late time acceleration in Brans-Dicke cosmology, Phys. Rev. D 63 (2001) 124006 [gr-qc/0010092] [INSPIRE].
  40. [40]
    L.-E. Qiang, Y.-G. Ma, M.-X. Han and D. Yu, 5-dimensional Brans-Dicke theory and cosmic acceleration, Phys. Rev. D 71 (2005) 061501 [gr-qc/0411066] [INSPIRE].
  41. [41]
    J.P. de Leon, Late time cosmic acceleration from vacuum Brans-Dicke theory in 5D, Class. Quant. Grav. 27 (2010) 095002 [arXiv:0912.1026] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  42. [42]
    J. Cortez, G.A. Mena Marugan, J. Olmedo and J.M. Velhinho, A unique Fock quantization for fields in non-stationary spacetimes, JCAP 10 (2010) 030 [arXiv:1004.5320] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  43. [43]
    Y. Bisabr, Cosmic acceleration in Brans-Dicke cosmology, Gen. Rel. Grav. 44 (2012) 427 [arXiv:1110.3421] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    Y. Bisabr, On the chameleon Brans-Dicke cosmology, Phys. Rev. D 86 (2012) 127503 [arXiv:1212.2709] [INSPIRE].ADSGoogle Scholar
  45. [45]
    L.L. Samojeden, F.P. Devecchi and G.M. Kremer, Fermions in Brans-Dicke cosmology, Phys. Rev. D 81 (2010) 027301 [arXiv:1001.2285] [INSPIRE].ADSGoogle Scholar
  46. [46]
    D.-J. Liu, Dynamics of Brans-Dicke cosmology with varying mass fermions, Phys. Rev. D 82 (2010) 063523 [arXiv:1005.5508] [INSPIRE].ADSGoogle Scholar
  47. [47]
    S. Nojiri and S.D. Odintsov, Unifying phantom inflation with late-time acceleration: scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Rel. Grav. 38 (2006) 1285 [hep-th/0506212] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    S. Capozziello, S. Nojiri and S.D. Odintsov, Unified phantom cosmology: inflation, dark energy and dark matter under the same standard, Phys. Lett. B 632 (2006) 597 [hep-th/0507182] [INSPIRE].CrossRefADSGoogle Scholar
  49. [49]
    M.R. Setare, The holographic dark energy in non-flat Brans-Dicke cosmology, Phys. Lett. B 644 (2007) 99 [hep-th/0610190] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  50. [50]
    M.R. Setare and M. Jamil, Holographic dark energy in Brans-Dicke cosmology with chameleon scalar field, Phys. Lett. B 690 (2010) 1 [arXiv:1006.0658] [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    H. Farajollahi, J. Sadeghi, M. Pourali and A. Salehi, Stability analysis of agegraphic dark energy in Brans-Dicke cosmology, Astrophys. Space Sci. 339 (2012) 79 [arXiv:1201.0007] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  52. [52]
    S. Chattopadhyay, A. Pasqua and M. Khurshudyan, New holographic reconstruction of scalar field dark energy models in the framework of chameleon Brans-Dicke cosmology, Eur. Phys. J. C 74 (2014) 3080 [arXiv:1401.8208] [INSPIRE].CrossRefADSGoogle Scholar
  53. [53]
    O. Hrycyna and M. Szydlowski, Dynamical complexity of the Brans-Dicke cosmology, JCAP 12 (2013) 016 [arXiv:1310.1961] [INSPIRE].CrossRefADSGoogle Scholar
  54. [54]
    A. Paliathanasis, M. Tsamparlis, S. Basilakos and J.D. Barrow, Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid, arXiv:1511.00439 [INSPIRE].
  55. [55]
    A. Einstein and E.G. Straus, The influence of the expansion of space on the gravitation fields surrounding the individual stars, Rev. Mod. Phys. 17 (1945) 120 [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  56. [56]
    N. Sakai and J.D. Barrow, Cosmological evolution of black holes in Brans-Dicke gravity, Class. Quant. Grav. 18 (2001) 4717 [gr-qc/0102024] [INSPIRE].
  57. [57]
    M. Novello and S.E.P. Bergliaffa, Bouncing cosmologies, Phys. Rept. 463 (2008) 127 [arXiv:0802.1634] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  58. [58]
    J.C. Fabris, R.G. Furtado, N. Pinto-Neto and P. Peter, Regular cosmological solutions in low-energy effective action from string theories, Phys. Rev. D 67 (2003) 124003 [hep-th/0212312] [INSPIRE].ADSMathSciNetGoogle Scholar
  59. [59]
    D.A. Tretyakova, A.A. Shatskiy, I.D. Novikov and S. Alexeyev, Non-singular Brans-Dicke cosmology with cosmological constant, Phys. Rev. D 85 (2012) 124059 [arXiv:1112.3770] [INSPIRE].ADSGoogle Scholar
  60. [60]
    D.A. Tretyakova, B.N. Latosh and S.O. Alexeyev, Wormholes and naked singularities in Brans-Dicke cosmology, Class. Quant. Grav. 32 (2015) 185002 [arXiv:1504.06723] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  61. [61]
    M. Artymowski, Y. Ma and X. Zhang, Comparison between Jordan and Einstein frames of Brans-Dicke gravity a la loop quantum cosmology, Phys. Rev. D 88 (2013) 104010 [arXiv:1309.3045] [INSPIRE].ADSGoogle Scholar
  62. [62]
    A. Nakonieczna and J. Lewandowski, Scalar field as a time variable during gravitational evolution, Phys. Rev. D 92 (2015) 064031 [arXiv:1508.05578] [INSPIRE].ADSGoogle Scholar
  63. [63]
    R. Torres and F. Fayos, Singularity free gravitational collapse in an effective dynamical quantum spacetime, Phys. Lett. B 733 (2014) 169 [arXiv:1405.7922] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  64. [64]
    R. Torres and F. Fayos, On the quantum corrected gravitational collapse, Phys. Lett. B 747 (2015) 245 [arXiv:1503.07407] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  65. [65]
    C. Vaz, Quantum gravitational dust collapse does not result in a black hole, Nucl. Phys. B 891 (2015) 558 [arXiv:1407.3823] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  66. [66]
    R. Gambini and J. Pullin, An introduction to spherically symmetric loop quantum gravity black holes, AIP Conf. Proc. 1647 (2015) 19 [arXiv:1312.5512] [INSPIRE].CrossRefADSGoogle Scholar
  67. [67]
    M.A. Scheel, S.L. Shapiro and S.A. Teukolsky, Collapse to black holes in Brans-Dicke theory. 1. Horizon boundary conditions for dynamical space-times, Phys. Rev. D 51 (1995) 4208 [gr-qc/9411025] [INSPIRE].
  68. [68]
    M.A. Scheel, S.L. Shapiro and S.A. Teukolsky, Collapse to black holes in Brans-Dicke theory. 2. Comparison with general relativity, Phys. Rev. D 51 (1995) 4236 [gr-qc/9411026] [INSPIRE].
  69. [69]
    D.-I. Hwang and D.-H. Yeom, Responses of the Brans-Dicke field due to gravitational collapses, Class. Quant. Grav. 27 (2010) 205002 [arXiv:1002.4246] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  70. [70]
    J. Hansen and D.-H. Yeom, Charged black holes in string-inspired gravity: I. Causal structures and responses of the Brans-Dicke field, JHEP 10 (2014) 040 [arXiv:1406.0976] [INSPIRE].
  71. [71]
    J. Hansen and D.-H. Yeom, Charged black holes in string-inspired gravity: II. Mass inflation and dependence on parameters and potentials, JCAP 09 (2015) 019 [arXiv:1506.05689] [INSPIRE].
  72. [72]
    T. Koivisto and D.F. Mota, Vector field models of inflation and dark energy, JCAP 08 (2008) 021 [arXiv:0805.4229] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  73. [73]
    K. Becker, M. Becker and J.H. Schwarz, String theory and M-theory. A modern introduction, Cambridge University Press, Cambridge U.K. (2007) [INSPIRE].
  74. [74]
    T.P. Sotiriou and V. Faraoni, f (R) theories of gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].
  75. [75]
    M. Gasperini, Elements of string cosmology, Cambridge University Press, Cambridge U.K. (2007).Google Scholar
  76. [76]
    L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].
  77. [77]
    J. Garriga and T. Tanaka, Gravity in the brane world, Phys. Rev. Lett. 84 (2000) 2778 [hep-th/9911055] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  78. [78]
    H. Kim, B.-H. Lee, W. Lee, Y.J. Lee and D.-H. Yeom, Nucleation of vacuum bubbles in Brans-Dicke type theory, Phys. Rev. D 84 (2011) 023519 [arXiv:1011.5981] [INSPIRE].ADSGoogle Scholar
  79. [79]
    R.S. Hamadé and J.M. Stewart, The spherically symmetric collapse of a massless scalar field, Class. Quant. Grav. 13 (1996) 497 [gr-qc/9506044] [INSPIRE].
  80. [80]
    A. Borkowska, M. Rogatko and R. Moderski, Collapse of charged scalar field in dilaton gravity, Phys. Rev. D 83 (2011) 084007 [arXiv:1103.4808] [INSPIRE].ADSGoogle Scholar
  81. [81]
    A. Nakonieczna and M. Rogatko, Dilatons and the dynamical collapse of charged scalar field, Gen. Rel. Grav. 44 (2012) 3175 [arXiv:1209.3614] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  82. [82]
    A. Nakonieczna, M. Rogatko and R. Moderski, Dynamical collapse of charged scalar field in phantom gravity, Phys. Rev. D 86 (2012) 044043 [arXiv:1209.1203] [INSPIRE].ADSGoogle Scholar
  83. [83]
    A. Nakonieczna, M. Rogatko and L. Nakonieczny, Dark sector impact on gravitational collapse of an electrically charged scalar field, JHEP 11 (2015) 012 [arXiv:1508.02657] [INSPIRE].CrossRefADSGoogle Scholar
  84. [84]
    S. Hod and T. Piran, Mass inflation in dynamical gravitational collapse of a charged scalar field, Phys. Rev. Lett. 81 (1998) 1554 [gr-qc/9803004] [INSPIRE].
  85. [85]
    S. Hod and T. Piran, The inner structure of black holes, Gen. Rel. Grav. 30 (1998) 1555 [gr-qc/9902008] [INSPIRE].
  86. [86]
    E. Sorkin and T. Piran, The effects of pair creation on charged gravitational collapse, Phys. Rev. D 63 (2001) 084006 [gr-qc/0009095] [INSPIRE].
  87. [87]
    E. Sorkin and T. Piran, Formation and evaporation of charged black holes, Phys. Rev. D 63 (2001) 124024 [gr-qc/0103090] [INSPIRE].
  88. [88]
    Y. Oren and T. Piran, On the collapse of charged scalar fields, Phys. Rev. D 68 (2003) 044013 [gr-qc/0306078] [INSPIRE].
  89. [89]
    J. Hansen, A. Khokhlov and I. Novikov, Physics of the interior of a spherical, charged black hole with a scalar field, Phys. Rev. D 71 (2005) 064013 [gr-qc/0501015] [INSPIRE].
  90. [90]
    A. Doroshkevich, J. Hansen, D. Novikov, I. Novikov and A. Shatskiy, Physics of the interior of a black hole with an exotic scalar matter, Phys. Rev. D 81 (2010) 124011 [arXiv:0908.1300] [INSPIRE].ADSGoogle Scholar
  91. [91]
    S.E. Hong, D.-I. Hwang, E.D. Stewart and D.-H. Yeom, The causal structure of dynamical charged black holes, Class. Quant. Grav. 27 (2010) 045014 [arXiv:0808.1709] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  92. [92]
    D.-I. Hwang and D.-H. Yeom, Internal structure of charged black holes, Phys. Rev. D 84 (2011) 064020 [arXiv:1010.2585] [INSPIRE].ADSGoogle Scholar
  93. [93]
    J. Hansen, B.-H. Lee, C. Park and D.-H. Yeom, Inside and outside stories of black-branes in anti de Sitter space, Class. Quant. Grav. 30 (2013) 235022 [arXiv:1307.0266] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Institute of PhysicsMaria Curie-Sklodowska UniversityLublinPoland
  2. 2.Institute of AgrophysicsPolish Academy of SciencesLublinPoland
  3. 3.Leung Center for Cosmology and Particle AstrophysicsNational Taiwan UniversityTaipeiTaiwan

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