Predictions from Quantum Cosmology

  • Alexander Vilenkin
Part of the Mathematical and Physical Sciences book series (ASIC, volume 476)


If the cosmological evolution is followed back in time, we come to the initial singularity where the classical equations of general relativity break down. This led many people to believe that in order to understand what actually happened at the origin of the universe, we should treat the universe quantum-mechanically and describe it by a wave function rather than by a classical spacetime. This quantum approach to cosmology was initiated by DeWitt [1] and Misner [2], and after a somewhat slow start has become very popular in the last decade or so. The picture that has emerged from this line of development [3, 4, 6, 5, 7, 8, 9] is that a small closed universe can spontaneously nucleate out of nothing, where by ‘nothing’ I mean a state with no classical space and time. The cosmological wave function can be used to calculate the probability distribution for the initial configurations of the nucleating universes. Once the universe nucleated, it is expected to go through a period of inflation, which is a rapid (quasi-exponential) expansion driven by the energy of a false vacuum. The vacuum energy is eventually thermalized, inflation ends, and from then on the universe follows the standard hot cosmological scenario. Inflation is a necessary ingredient in this kind of scheme, since it gives the only way to get from the tiny nucleated universe to the large universe we live in today.


Wave Function Modulus Space Vacuum Energy Semiclassical Approximation Quantum Cosmology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    De Witt, B.S. (1967) Phys. Rev. 160, 1113.ADSCrossRefGoogle Scholar
  2. 2.
    Misner, C.W. (1972) in Magic Without Magic, Freeman, San Francisco.Google Scholar
  3. 3.
    Vilenkin, A. (1982) Phys. Lett. 117B, 25.MathSciNetGoogle Scholar
  4. 4.
    Hartle, J.B. and Hawking, S.W. (1983) Phys. Rev. D28, 2960.ADSMathSciNetGoogle Scholar
  5. 5.
    Linde, A.D. (1984) Lett. Nuovo Cim. 39, 401.ADSCrossRefGoogle Scholar
  6. 6.
    Zel’dovich, Y.B. and Starobinsky, A.A. (1984) Sov. Astron. Lett. 10, 135.ADSGoogle Scholar
  7. 7.
    Rubakov, V.A. (1984) Phys. Lett. 148B, 280.MathSciNetGoogle Scholar
  8. 8.
    Vilenkin, A. (1984) Phys. Rev. D30 509.ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    The idea that a closed universe could be a vacuum fluctuation was first suggested by E.P. Tryon (1973) Nature 246, 396, and independently by P.I. Fomin (1973) ITP Preprint, Kiev; (1975) Dokl Akad. Nauk Ukr. SSR 9A, 831. However, these authors offered no mathematical description for the nucleation of the universe. Quantum tunneling of the entire universe through a potential barrier was first discussed by D. Atkatz and H. Pagels (1982) Phys. Rev. D25, 2065.Google Scholar
  10. 10.
    Coleman, S. (1988) Nucl. Phys. B307, 867. Similar ideas were explored by Giddings, S.B. and Strominger, A. (1988) Nucl Phys. B307, 854 and by Banks, T. (1988) Nucl. Phys. B309, 493.ADSCrossRefGoogle Scholar
  11. 11.
    We note that calling something the greatest mistake of one’s life may be a mistake. For example, the introduction of the cosmological constant, which Einstein called the greatest mistake of his life, now appears to be not such a bad idea.Google Scholar
  12. 12.
    Halliwell, J.J. and Hawking, S.W. (1985) Phys. Rev. D31, 1777.ADSMathSciNetGoogle Scholar
  13. 13.
    Vachaspati, T. and Vilenkin, A. (1988) Phys. Rev. D37, 898.ADSMathSciNetGoogle Scholar
  14. 14.
    Vilenkin, A. (1989) Phys. Rev. D39, 1116.ADSGoogle Scholar
  15. 15.
    Lapchinsky, V. and Rubakov, V.A. (1979) Acta Phys. Polon. B10, 1041.MathSciNetGoogle Scholar
  16. 16.
    Banks, T. (1985) Nucl. Phys. B249, 332.ADSCrossRefGoogle Scholar
  17. 17.
    Vilenkin, A. (1985) Nucl. Phys. B252, 141.ADSCrossRefGoogle Scholar
  18. 18.
    Teitelboim, C. (1982) Phys. Rev. D25, 3159.ADSMathSciNetGoogle Scholar
  19. 19.
    Halliwell, J.J. and Hartle, J.B. (1990) Phys. Rev. D41, 1815.ADSMathSciNetGoogle Scholar
  20. 20.
    Vilenkin, A. (1986) Phys. Rev. D33, 3560.ADSMathSciNetGoogle Scholar
  21. 21.
    Vilenkin, A. (1988) Phys. Rev. D37, 888.ADSMathSciNetGoogle Scholar
  22. 22.
    Vilenkin, A. (1994) Phys. Rev. D50, 2581.ADSMathSciNetGoogle Scholar
  23. 23.
    Hawking, S.W. (1984) Nucl. Phys. B239, 257.ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    Linde, A.D. (1990) Particle Physics and Inflationary Cosmology, Harwood Academic, Chur.Google Scholar
  25. 25.
    I am grateful to Slava Mukhanov for pointing out to me that Linde’s contour rotation and the tunneling boundary condition give different wave functions and to Andrei Linde for a discussion of this point.Google Scholar
  26. 26.
    Hawking, S.W. and Page, D.N. (1986) Nucl. Phys. B264, 185.ADSCrossRefMathSciNetGoogle Scholar
  27. 27.
    Grishchuk, L.P. and Rozhansky, L.V. (1988) Phys. Lett. B208, 369.ADSMathSciNetGoogle Scholar
  28. 28.
    Barvinsky, A.O. and Kamenshchik, A.Y. (1994) Phys. Lett. B332, 270.ADSGoogle Scholar
  29. 29.
    Binnetruy, P. and Gaillard, M.K. (1986) Phys. Rev. D34, 3069.ADSGoogle Scholar
  30. 30.
    Banks, T. et. al. (1994) Modular Cosmology, Rutgers Preprint RU-94–93.Google Scholar
  31. 31.
    Thomas, S. (1995) Moduli Inflation from Dynamical Supersymmetry Breaking, SLAC Preprint SLAC-PUB-95–6762.Google Scholar
  32. 32.
    The most general factor ordering consistent with reparametrization in variance allows an extra term ξa-2ℜ(ϕ) in Eq.(49). Here, ℜ is the scalar curvature of the moduli space and ξ is a numerical coefficient. Addition of such a term modifies the potential U(α, ϕ) but does not change the conclusions of Sec. 4.2.Google Scholar
  33. 33.
    Horne, J. and Moore, G. (1994) Nucl. Phys. B432, 109.ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Gell-Mann, M. (1994) The Quark and the Jaguar, Freeman, New York.zbMATHGoogle Scholar
  35. 35.
    Strominger, A. (1995) Massless Black Holes and Conifolds in String Theory, hepth/9504047.Google Scholar
  36. 36.
    The probability distribution for a Brans-Dicke field (which is similar to the dilaton of superstring theories) was discussed, using a different approach, by Garcia-Bellido and Linde [37, 38].Google Scholar
  37. 37.
    Garcia-Bellido, J. and Linde, A.D. (1995) Phys. Rev. D51, 429.ADSMathSciNetGoogle Scholar
  38. 38.
    Garcia-Bellido, J., Linde, A.D. and Linde, D.A. (1994) Phys. Rev. D50, 730ADSGoogle Scholar
  39. 39.
    Vilenkin, A. (1995) Phys. Rev. Lett. 74, 846.ADSCrossRefGoogle Scholar
  40. 40.
    This and the following sections are partly based on my papers [39, 55]. Related ideas were discussed by Albrecht [41] and by Garcia-Bellido and Linde [37]Google Scholar
  41. 41.
    Albrecht, A. (1995) in The Birth of the Universe and Fundamental Forces, ed. by F. Occhionero, Springer-Verlag.Google Scholar
  42. 42.
    Carter, B. (1974) in I.A.U. Symposium, Vol 63, ed. by M.S. Longair, Reidel, Dordrecht; (1983) Philos. Trans. R. Soc. London A310, 347; Carr, B.J. and Rees, M.J. (1979) Nature (London) 278, 605; Barrow, J.D. and Tipler, F.J. (1986) The Anthropic Cosmological Principle, Clarendon, Oxford.Google Scholar
  43. 43.
    Vilenkin, A. and Shellard, E.P.S. (1994) Cosmic Strings and Other Topological Defects, Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  44. 44.
    Lazarides, G., Panagiotakopoulos, C. and Shafi, Q.(1986) Phys. Rev. Lett. 56, 432; (1987) Phys. Lett. 183B, 289.ADSCrossRefGoogle Scholar
  45. 45.
    Linde, A.D. (1994) Phys. Rev. D49, 748; Copeland, E.J. et. al. (1994) Phys. Rev. D49, 6410.ADSGoogle Scholar
  46. 46.
    I am grateful to Andrei Linde for pointing out to me that hybrid inflation can give sufficiently large density fluctuations, even with flat potentials.Google Scholar
  47. 47.
    Rees, M.J. (1983) Philos. Trans. R. Soc. London A310, 311.ADSCrossRefGoogle Scholar
  48. 48.
    Weinberg, S. (1987) Phys. Rev. Lett. 59, 2607; (1989) Rev. Mod. Phys. 61, 1.ADSCrossRefGoogle Scholar
  49. 49.
    see, e.g., Carroll, S.M., Press, W.H. and Turner, E.L. (1992) Ann. Rev. Astron. Astrophys. 30, 499.ADSCrossRefGoogle Scholar
  50. 50.
    This argument assumes that the probability distribution for ρv in the range of interest is nearly flat. It is possible, however, that the ‘fundamental’ variable that has a flat distribution at sub-Planckian scales is the characteristic energy scale η = tex. Then the discrepancy between the anthropic and observational bounds on η is only by a factor ~ 2.Google Scholar
  51. 51.
    More exactly, we look for the values of ρv that achieve a balance between fine-tuning and maximizing the amount of matter in galaxies. To make this quantitative, let w(ρv)dpv be the probability distribution for ρv for the nucleating universes, and let f(pv) be the fraction of baryonic matter that ends up in galaxies at a given value of ρv. (Here I assume that ρv has a continuous spectrum). Then the most probable values of pv are found by maximizing the product fv)w(ρvv.Google Scholar
  52. 52.
    Vilenkin, A. (1983) Phys. Rev. D27, 2848.ADSMathSciNetGoogle Scholar
  53. 53.
    Linde, A.D. (1986) Phys. Lett. B175, 395.ADSGoogle Scholar
  54. 54.
    Linde, A.D., Linde, D.A. and Mezhlumian, A. (1994) Phys. Rev. D49, 1783.ADSGoogle Scholar
  55. 55.
    Vilenkin, A. (1995) Making Predictions in Eternally Inflating Universe, gr-qc/9505031.Google Scholar
  56. 56.
    Starobinsky, A.A. (1986) in Current Topics in Field Theory, Quantum Gravity and Strings, ed. by H.J. de Vega and N. Sanchez, Springer, Heidelberg.Google Scholar
  57. 57.
    Aryal, M. and Vilenkin, A. (1987) Phys. Lett. B199, 351.ADSMathSciNetGoogle Scholar
  58. 58.
    Borde, A. and Vilenkin, A. (1994) Phys. Rev. Lett. 72, 3305; Borde, A. (1994) Phys. Rev. D50, 3392.ADSCrossRefGoogle Scholar
  59. 59.
    A combined approach using both quantum cosmology and eternal inflation would be necessary only if {αj} split into groups, such that transitions between different groups are disallowed, and the absolute minimum of γ(α) is attained in more than one group.Google Scholar
  60. 60.
    Linde, A.D. (1994) Phys. Lett. B327, 208; Vilenkin, A. (1994) Phys. Rev. Lett. 72, 3137.ADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Alexander Vilenkin
    • 1
  1. 1.Institute of Cosmology, Department of Physics and AstronomyTufts UniversityMedfordUSA

Personalised recommendations