Quantum cosmological model of the inflationary universe

Ursula Carow; Satoshi Watamura

doi:10.1103/PhysRevD.32.1290

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Quantum cosmological model of the inflationary universe

Ursula Carow and Satoshi Watamura

Research Institute for Fundamental Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606, Japan

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Phys. Rev. D 32, 1290 – Published 15 September, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.1290

Abstract

A quantum cosmological model of the inflationary universe is investigated by solving the Wheeler-DeWitt equation. We consider a model with a minimally coupled scalar field, the potential of which is a simple double well. By applying the boundary condition of ‘‘no boundary,’’ we calculate the wave function of our model universe. We find that in a certain parameter range a big peak is formed near the maximum of the double-well potential of the scalar field, accompanied by a recession of the exponential behavior of the wave function. We show that this peak can be consistently interpreted as representing a high density of classical paths of generalized oscillating universes, and as a consequence of the constructive interference of quantum states corresponding to these classical paths by the WKB approximation. The cosmological scenario with nonvanishing, nearly critical ‘‘velocity’’ of the vacuum expectation value in the early universe, which is suggested by the behavior of the wave function, is discussed.

References (28)

A. H. Guth, Phys. Rev. D 23, 347 (1981); K. Sato, Mon. Not. R. Astron. Soc. 195, 467 (1981).
A. D. Linde, Rep. Prog. Phys. 47, 925 (1984), and references therein.
A. Vilenkin, Phys. Lett. 117 B, 25 (1982); Phys. Rev. D 27, 2848 (1983).
A. Vilenkin, Phys. Rev. D 30, 509 (1984).
A. D. Linde, Lett. Nuovo Cimento 39, 401 (1984).
Ideas of the quantum creation of the universe are discussed by E. P. Tryon [Nature 246, 396 (1973)]; R. Brout, F. Englert, and E. Gunzig [Ann. Phys. (NY) 115, 78 (1978)]; Atkatz and H. Pagels [Phys. Rev. D 25, 2065 (1982)]; and J. R. Gott [Nature 295, 304 (1982)].
J. A. Wheeler, in Battelle Rencontrés, edited by C. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).
B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
C. W. Misner, Phys. Rev. 186, 1319 (1969); in Magic Without Magic: John Archibald Wheeler, a Collection of Essays in Honor of his 60th Birthday, edited by J. R. Klauder (Freeman, San Francisco, 1972).
For a review see, e.g., M. Ryan, Hamiltonian Cosmology (Springer, New York, 1972) see , also M. A. H. MacCallum, in Quantum Gravity, An Oxford Symposium, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, New York 1975), and references therein.
D. J. Kaup and A. P. Vitello, Phys. Rev. D 9, 1648 (1974); W. F. Blyth and C. J. Isham, ibid. 11, 768 (1975); C. J. Isham and J. E. Nelson, ibid. 15, 3226 (1974).
S. W. Hawking, in Astrophysical Cosmology, proceedings of the Study Week on Cosmology and Fundamental Sciences, 1982, edited by H. A. Brück, G. V. Coyne, and M. S. Longair (Pontificia Academia Scientarium, Vatican City, 1982).
J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).
S. W. Hawking, in Relativity, Groups, and Topology II, Les Houches Lectures, 1983, edited by B. S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984), p. 333.
I. G. Moss and W. A. Wright, Phys. Rev. D 29, 1067 (1984).
S. W. Hawking, Nucl. Phys. B 239, 257 (1984).
S. W. Hawking and Z. C. Wu, Phys. Lett. 151 B, 15 (1985).
A. D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38, 149 (1983); Phys. Lett. 129 B, 177 (1983).
G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
See, e.g., R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation, edited by L. Witten (Wiley, New York, 1962), p. 227.
In the numerical calculation, we choose the lapse function as N=1.
S. Coleman, Phys. Rev. D 15, 2929 (1977); C. Callan and S. Coleman, ibid. 16, 1762 (1977); S. Coleman, in The Whys of Subnuclear Physics, proceedings of the International School of Subnuclear Physics, Erice, 1977, edited by A. Zichichi (Plenum, New York, 1979).
In the case of the de Sitter universe, the wave function shows rapid oscillations with a slowly varying amplitude. This wave function can be considered as a standing wave consisting of a superposition of quantum states describing expanding and contracting universes. It is shown that the envelope of the probability distribution $a^{p}$ | PSI (a) $|^{2}$ , i.e., the square of the slowly varying amplitude, is proportional to the distribution of the three-spheres in a in the de Sitter universe (Ref. 13). Thus in this context the envelope of the square of the wave function is to be understood as a square of the slowly varying amplitude. In other words, if we think of the semiclassical approximation, the values of this envelope are given by the square of the prefactor. The factor $a^{p}$ in the probability distribution is demanded by choosing the measure of the minisuperspace in such a way that the differential operator of the Wheeler-DeWitt equation becomes Hermitian. In our case the probability distribution is simply given by | PSI (x,y) $|^{2}$ in (x,y) variables.
S. W. Hawking and I. G. Moss, Phys. Lett. 110 B, 35 (1982).
The possibility of this type of solution is suggested by Hawking in a different context (Ref. 14) and discussed by Page (Ref. 26). However, in our case this type of solution is inevitable in order to understand the behavior of the wave function.
D. N. Page, Class. Quantum Grav. 1, 417 (1984).
At the bounce point of a classical path the corresponding classical universe reaches its minimal size and from there goes into a new expansion phase. From this point of view we can construct a cosmological model in which the bounce point represents the beginning of the evolution of a universe.
For a parameter value M = 0.32, all peaks within the whole size of the diagram (a <= 6) on the line x=0 are definitely higher than the neighboring wavelet structure. The phase for this parameter is comparable to that of the de Sitter universe.

[c1] A. H. Guth, Phys. Rev. D 23, 347 (1981); K. Sato, Mon. Not. R. Astron. Soc. 195, 467 (1981).

[c2] A. D. Linde, Rep. Prog. Phys. 47, 925 (1984), and references therein.

[c3] A. Vilenkin, Phys. Lett. 117 B, 25 (1982); Phys. Rev. D 27, 2848 (1983).

[c4] A. Vilenkin, Phys. Rev. D 30, 509 (1984).

[c5] A. D. Linde, Lett. Nuovo Cimento 39, 401 (1984).

[c6] Ideas of the quantum creation of the universe are discussed by E. P. Tryon [Nature 246, 396 (1973)]; R. Brout, F. Englert, and E. Gunzig [Ann. Phys. (NY) 115, 78 (1978)]; Atkatz and H. Pagels [Phys. Rev. D 25, 2065 (1982)]; and J. R. Gott [Nature 295, 304 (1982)].

[c7] J. A. Wheeler, in Battelle Rencontrés, edited by C. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).

[c8] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).

[c9] C. W. Misner, Phys. Rev. 186, 1319 (1969); in Magic Without Magic: John Archibald Wheeler, a Collection of Essays in Honor of his 60th Birthday, edited by J. R. Klauder (Freeman, San Francisco, 1972).

[c10] For a review see, e.g., M. Ryan, Hamiltonian Cosmology (Springer, New York, 1972) see , also M. A. H. MacCallum, in Quantum Gravity, An Oxford Symposium, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, New York 1975), and references therein.

[c11] D. J. Kaup and A. P. Vitello, Phys. Rev. D 9, 1648 (1974); W. F. Blyth and C. J. Isham, ibid. 11, 768 (1975); C. J. Isham and J. E. Nelson, ibid. 15, 3226 (1974).

[c12] S. W. Hawking, in Astrophysical Cosmology, proceedings of the Study Week on Cosmology and Fundamental Sciences, 1982, edited by H. A. Brück, G. V. Coyne, and M. S. Longair (Pontificia Academia Scientarium, Vatican City, 1982).

[c13] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).

[c14] S. W. Hawking, in Relativity, Groups, and Topology II, Les Houches Lectures, 1983, edited by B. S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984), p. 333.

[c15] I. G. Moss and W. A. Wright, Phys. Rev. D 29, 1067 (1984).

[c16] S. W. Hawking, Nucl. Phys. B 239, 257 (1984).

[c17] S. W. Hawking and Z. C. Wu, Phys. Lett. 151 B, 15 (1985).

[c18] A. D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 38, 149 (1983); Phys. Lett. 129 B, 177 (1983).

[c19] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).

[c20] See, e.g., R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation, edited by L. Witten (Wiley, New York, 1962), p. 227.

[c21] In the numerical calculation, we choose the lapse function as N=1.

[c22] S. Coleman, Phys. Rev. D 15, 2929 (1977); C. Callan and S. Coleman, ibid. 16, 1762 (1977); S. Coleman, in The Whys of Subnuclear Physics, proceedings of the International School of Subnuclear Physics, Erice, 1977, edited by A. Zichichi (Plenum, New York, 1979).

[c23] In the case of the de Sitter universe, the wave function shows rapid oscillations with a slowly varying amplitude. This wave function can be considered as a standing wave consisting of a superposition of quantum states describing expanding and contracting universes. It is shown that the envelope of the probability distribution $a^{p}$ | PSI (a) $|^{2}$ , i.e., the square of the slowly varying amplitude, is proportional to the distribution of the three-spheres in a in the de Sitter universe (Ref. 13). Thus in this context the envelope of the square of the wave function is to be understood as a square of the slowly varying amplitude. In other words, if we think of the semiclassical approximation, the values of this envelope are given by the square of the prefactor. The factor $a^{p}$ in the probability distribution is demanded by choosing the measure of the minisuperspace in such a way that the differential operator of the Wheeler-DeWitt equation becomes Hermitian. In our case the probability distribution is simply given by | PSI (x,y) $|^{2}$ in (x,y) variables.

[c24] S. W. Hawking and I. G. Moss, Phys. Lett. 110 B, 35 (1982).

[c25] The possibility of this type of solution is suggested by Hawking in a different context (Ref. 14) and discussed by Page (Ref. 26). However, in our case this type of solution is inevitable in order to understand the behavior of the wave function.

[c26] D. N. Page, Class. Quantum Grav. 1, 417 (1984).

[c27] At the bounce point of a classical path the corresponding classical universe reaches its minimal size and from there goes into a new expansion phase. From this point of view we can construct a cosmological model in which the bounce point represents the beginning of the evolution of a universe.

[c28] For a parameter value M = 0.32, all peaks within the whole size of the diagram (a <= 6) on the line x=0 are definitely higher than the neighboring wavelet structure. The phase for this parameter is comparable to that of the de Sitter universe.

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Quantum cosmological model of the inflationary universe

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