Skip to main content
SpringerLink
Log in

Resonant structure of the early-universe space-time

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

A new fully quantum method, describing the propagation of a packet from the internal well outside with its tunneling through a barrier of arbitrary shape used in problems of quantum cosmology, is presented. The method determines amplitudes of the wave function, penetrability T bar and reflection R bar (accuracy of the method is | T bar + R bar -1| < 10−15 , coefficient of penetration (i.e. probability of the packet propagating from the internal well outside), coefficient of oscillations (describing the oscillating behavior of the packet inside the internal well). Using this method, I study the evolution of the universe in the closed Friedmann-Robertson-Walker model with quantization in the presence of positive cosmological constant, radiation and the component of generalized Chaplygin gas. One established for the first time: 1) the oscillating dependence of the penetrability on the localization of the starting point of the packet; 2) the presence of resonant values of the radiation energy E bar , when the coefficient of penetration strongly increases. From the analysis of these results it follows that: 1) an initial condition should be introduced into the non-stationary and stationary quantum models; 2) the start of the expansion of the universe is most probable at some definite values for the scale factor; 3) the radius of the universe has changed non-continuously during its expansion, but it consequently passes through definite discrete values and tends to a continuous range of values at a later time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.B. Hartle, S.W. Hawking, Phys. Rev. D 28, 2960 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  2. Ya.B. Zeldovich, Sov. Astron. Lett. 7, 322 (1981).

    ADS  Google Scholar 

  3. A. Vilenkin, Phys. Lett. B 117, 25 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  4. A. Vilenkin, Phys. Rev. D 27, 2848 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  5. A. Vilenkin, Phys. Rev. D 30, 509 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  6. A. Vilenkin, Phys. Rev. D 33, 3560 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  7. A. Vilenkin, Phys. Rev. D 37, 888 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  8. A. Vilenkin, Phys. Rev. D 50, 2581 (1994) gr-qc/9403010

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Vilenkin, Predictions from quantum cosmology, in String Gravity and Physics at the Planck Energy Scale: Proceedings of the NATO Advanced Study Institute, Erice, Italy, 8--19 Sep 1995 (NATO Series C: Mathematical and Physical Sciences) edited by N. Sanchez, A. Zichichi (Kluwer Boston, MA, 1996) pp. 345--367

  10. J. Garriga, A. Vilenkin, Phys. Rev. D 56, 2464 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  11. J. Hong, A. Vilenkin, S. Winitzki, Phys. Rev. D 68, 023521 (2003).

    Article  MathSciNet  ADS  Google Scholar 

  12. A.D. Linde, JETP 60, 211 (1984).

    Google Scholar 

  13. A.D. Linde, Lett. Nuovo Cimento 39, 401 (1984).

    Article  ADS  Google Scholar 

  14. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. M. Bouhmadi-López, L.J. Garay, P.F. Gonzalez-Diaz, Phys. Rev. D 66, 083504 (2002).

    Article  MathSciNet  ADS  Google Scholar 

  16. M. Bouhmadi-López, P.V. Moniz, Phys. Rev. D 71, 063521 (2005).

    Article  ADS  Google Scholar 

  17. R. Casadio, F. Finelli, M. Luzzi, G. Venturi, Phys. Rev. D 71, 043517 (2005).

    Article  ADS  Google Scholar 

  18. R. Casadio, F. Finelli, M. Luzzi, G. Venturi, Phys. Rev. D 72, 103516 (2005).

    Article  ADS  Google Scholar 

  19. M. Luzzi, Semiclassical Approximations to Cosmological Perturbations, PhD thesis (Advisor: Prof. Giovanni Venturi, University of Bologna, 2007) 148 pages, arXiv:0705.3764

  20. S.P. Kim, J. Korean Phys. Soc. 45, S172 (2004).

    Google Scholar 

  21. E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, N.Y., 1990).

  22. E. Carugno, M. Litterio, F. Occhionero, G. Pollifrone, Phys. Rev. D 53, 6863 (1996).

    Article  ADS  Google Scholar 

  23. C. Bastos, O. Bertolami, N.C. Dias, J.N. Prata, Phys. Rev. D 78, 023516 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  24. R. Brustein, S.P. de Alwis, Phys. Rev. D 73, 046009 (2006).

    Article  MathSciNet  ADS  Google Scholar 

  25. M. Bouhmadi-López, P.V. Moniz, J. Cosmol. Astropart. Phys. 5, 005 (2007).

    Article  ADS  Google Scholar 

  26. S. Lee, Class. Quantum. Grav. 24, 5247 (2007).

    Article  ADS  MATH  Google Scholar 

  27. S. Lee, Class. Quantum. Grav. 25, 055008 (2008).

    Article  ADS  Google Scholar 

  28. E.M. Lifshitz, Sov. J. Phys. 10, 116 (1945).

    MathSciNet  Google Scholar 

  29. E.M. Lifshitz, I.M. Khalatnikov, Adv. Phys. 12, 185 (1963).

    Article  MathSciNet  ADS  Google Scholar 

  30. P. Ginsparg, M.J. Perry, Nucl. Phys. B 222, 245 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  31. D. Levkov, C. Rebbi, V.A. Rubakov, Phys. Rev. D 66, 083516 (2002).

    Article  ADS  Google Scholar 

  32. B.S. DeWitt, Phys. Rev. 160, 1113 (1967).

    Article  ADS  MATH  Google Scholar 

  33. J.A. Wheeler Batelle Rencontres (Benjamin, N.Y., 1968).

    Article  Google Scholar 

  34. J. Acacio de Barros, E.V. Correa Silva, G.A. Monerat, G. Oliveira-Neto, L.G. Ferreira Filho, P. Romildo Jr., Phys. Rev. D 75, 104004 (2007).

    Article  ADS  Google Scholar 

  35. G.A. Monerat, G. Oliveira-Neto, E.V. Correa Silva, L.G. Ferreira Filho, P. Romildo Jr., J.C. Fabris, R. Fracalossi, S.V.B. Goncalves, F.G. Alvarenga, Phys. Rev. D 76, 024017 (2007).

    Article  ADS  Google Scholar 

  36. A.V. Yurov, S.D. Vereshchagin, Theor. Math. Phys. 139, 787 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Garcia, W. Guzman, M. Sabido, J. Socorro, Int. J. Theor. Phys. 45, 2529 (2006).

    Article  MathSciNet  Google Scholar 

  38. S.P. Maydanyuk, Eur. Phys. J. C 57, 769 (2008) arxiv.org:0707.0585

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. S.P. Maydanyuk, Int. J. Mod. Phys. D 19, 392 (2010) arXiv:0812.5081

    Article  MathSciNet  ADS  Google Scholar 

  40. A. Kobakhidze, L. Mersini-Houghton, Eur. Phys. J. C 49, 869 (2007).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. A.A. Starobinsky, JETP Lett. 30, 682 (1979).

    ADS  Google Scholar 

  42. A.A. Starobinsky, Phys. Lett. B 91, 99 (1980).

    Article  ADS  Google Scholar 

  43. R. Brout, F. Englert, E. Gunzig, Gen. Relativ. Gravit. 10, 1 (1979).

    Article  ADS  Google Scholar 

  44. A.D. Linde, Phys. Lett. B 129, 177 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  45. A.D. Linde, Lett. Nuovo Cimento 39, 401 (1984).

    Article  ADS  Google Scholar 

  46. A. Linde, Particle physics and inflationary cosmology (Harwood, Chur, Switzerland, 1990) 362 pp

  47. A. Linde, Lectures on inflationary cosmology, Lectures given at the School on Particle Physics and Cosmology (Lake Louise, Canada, Feb 1994), at the Marcel Grossmann Conference (Stanford, CA, Jul 24--27, 1994), at the Workshop on the Birth of the Universe (Rome, Italy, May, 18--25, 1994) and at the Symposyum on Elementary Particle Physics (Capri, Italy and at International School of Astrophysics).

  48. R. H. Brandenberger, Inflationary cosmology: progress and problems, Lectures at the International School on Cosmology (Kish Island, Iran, Jan. 22--Feb. 4 1999) Proceedings (Kluwer, Dordrecht, 2000) p. 48, hep-ph/9910410

  49. D.N. Page, Phys. Rev. D 56, 2065 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  50. P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. M. Bouhmadi-López, J.A.J. Madrid, J. Cosmol. Astropart. Phys. 5, 005 (2005).

    Article  ADS  Google Scholar 

  52. M. Bouhmadi-López, P.F. Gonzáles-Diaz, P. Martin-Moruno, Int. J. Mod. Phys. D 17, 2269 (2008).

    Article  ADS  MATH  Google Scholar 

  53. D. Bertacca, N. Bartolo, A. Diaferio, S. Matarrese, J. Cosmol. Astropart. Phys. 10, 023 (2008).

    Article  ADS  Google Scholar 

  54. I.G. Moss, C. Xiong, J. Cosmol. Astropart. Phys. 11, 023 (2008).

    Article  ADS  Google Scholar 

  55. J. Lu, Yu. Gui, L. xin Xu, Eur. Phys. J. C 63, 349 (2009).

    Article  ADS  Google Scholar 

  56. A.Y. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B 511, 265 (2001).

    Article  ADS  MATH  Google Scholar 

  57. S. Chaplygin, Sci. Mom. Moskow Univ. Math. Phys. 21, 1 (1904).

    Google Scholar 

  58. S.P. Maydanyuk, Time analysis of tunneling processes in nuclear collisions and decays, Ph.D. dissertation (Supervisor: Prof. V.S. Olkhovsky, Kiev, 2003) p. 147 (in Ukrainian).

  59. V.S. Olkhovsky, S.P. Maydanyuk, Ukr. Phys. J. 45, 1262 (2000).

    Google Scholar 

  60. S.P. Maydanyuk, V.S. Olkhovsky, A.K. Zaichenko, J. Phys. Stud. 6, 1 (2002).

    Google Scholar 

  61. S.P. Maydanyuk, V.S. Olkhovsky, S.V. Belchikov The method of multiple internal reflections in description of nuclear decay, Probl. At. Sci. Tech. (Voprosi atomnoi nauki i tehniki, RFNC-VNIIEF, Sarov, Russia) 1, 16--19 (2002), nucl-th/0409037

    Google Scholar 

  62. F. Cardone, S.P. Maidanyuk, R. Mignani, V.S. Olkhovsky, Found. Phys. Lett. 19, 441 (2006).

    Article  MATH  Google Scholar 

  63. S.P. Maydanyuk, S.V. Belchikov, Method of determination of the most probable coordinate of formation of α-particle in α-decay, 25 p., arXiv: 0805.4165

  64. S.P. Maydanyuk, S.V. Belchikov, J. Mod. Phys. 2, 572 (2011).

    Article  Google Scholar 

  65. J.H. Fermor, Am. J. Phys. 34, 1168 (1966).

    Article  ADS  Google Scholar 

  66. K.W. McVoy, L. Heller, M. Bolsterli, Rev. Mod. Phys. 39, 245 (1967).

    Article  ADS  Google Scholar 

  67. A. Anderson, Am. J. Phys. 57, 230 (1989).

    Article  ADS  Google Scholar 

  68. S. Esposito, Phys. Rev. E 67, 016609 (2003).

    Article  ADS  Google Scholar 

  69. S.A. Gurvitz, G. Kälbermann, Phys. Rev. Lett. 59, 262 (1987).

    Article  ADS  Google Scholar 

  70. B. Buck, A.C. Merchant, S.M. Perez, At. Data. Nucl. Data. Tablles 54, 53 (1993).

    Article  ADS  Google Scholar 

  71. V.Yu. Denisov, H. Ikezoe, Phys. Rev. C 72, 064613 (2005).

    Article  ADS  Google Scholar 

  72. V.Yu. Denisov, A.A. Khudenko, At. Data. Nucl. Data. Tablles 95, 815 (2009).

    Article  ADS  Google Scholar 

  73. A. Sobiczewski, K. Pomorski, Prog. Part. Nucl. Phys. 58, 292 (2007).

    Article  ADS  Google Scholar 

  74. B. Buck, A.C. Merchant, S.M. Perez, Phys. Rev. C 45, 1688 (1988).

    Article  ADS  Google Scholar 

  75. S. Åberg, P.B. Semmes, W. Nazarewicz, Phys. Rev. C 56, 1762 (1997).

    Article  ADS  Google Scholar 

  76. S.A. Gurvitz, P.B. Semmes, W. Nazarewicz, T. Vertse, Phys. Rev. A 69, 042705 (2004).

    Article  ADS  Google Scholar 

  77. http://www-nds.iaea.org/RIPL-2/

  78. L.D. Landau, E.M. Lifshitz, Quantum mechanics, course of Theoretical Physics, Vol. 3 (Nauka, Mockva, 1989) p. 768 — [in Russian

  79. V.A. Rubakov, Klassicheskie kalibrovochnie poliya: bosonnie teorii (Moskva, KomKniga, 2005) p. 296 (in Russian).

Download references

Author information

Authors and Affiliations

  1. Institute for Nuclear Research, National Academy of Sciences of Ukraine, 03680, Kiev, Ukraine

    S. P. Maydanyuk

Authors
  1. S. P. Maydanyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. P. Maydanyuk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maydanyuk, S.P. Resonant structure of the early-universe space-time. Eur. Phys. J. Plus 126, 76 (2011). https://doi.org/10.1140/epjp/i2011-11076-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2011-11076-x

Keywords

Search

Navigation