Symmetry behavior of the static Taub universe: Effect of curvature anisotropy

T. C. Shen; B. L. Hu; D. J. O’Connor

doi:10.1103/PhysRevD.31.2401

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Symmetry behavior of the static Taub universe: Effect of curvature anisotropy

T. C. Shen, B. L. Hu, and D. J. O’Connor

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

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Phys. Rev. D 31, 2401 – Published 15 May, 1985

DOI: https://doi.org/10.1103/PhysRevD.31.2401

Abstract

Using the static Taub universe as an example, we study the effect of curvature anisotropy on symmetry breaking of a self-interacting scalar field. The one-loop effective potential of a λ $φ^{4}$ field with arbitrary coupling (ξ) is computed by ζ-function regularization. It is expressed as a perturbative series in a small anisotropy parameter α measuring the deformation from the spherical Einstein universe with radius of curvature a. This result is used for analyzing the symmetry behavior of such a system as a function of the geometric (a,α) and field (ξ,λ) parameters. The result is also used to address the question of whether and how curvature anisotropy can affect the inflationary scenario, old or new. We find that for a massless scalar field conformally coupled to the background with a prolate configuration (negative scalar curvature) the phase transition is of second order, in which case inflation to the extent necessary for cosmological purposes becomes highly unlikely. For the massless minimally coupled scalar field, first-order phase transitions can occur for a certain range of the radius and deformation parameter. If the curvature radius in the axisymmetric direction is held fixed, increasing deformation can restore the symmetry, whereas if the shape is held constant but the size is allowed to vary, decreasing the radius of the universe can induce symmetry breaking. For the minimally coupled field in a closed universe with high curvature a term linear in the background field in the effective potential appears. The barrier thus generated in the effective potential replaces the broad plateau of the flat-space Coleman-Weinberg potential. The meaning and implication of these results are discussed.

References (33)

D. J. O'Connor, B. L. Hu and T. C. Shen, Phys. Lett. 130B, 31 (1983).
A. H. Taub, Ann. Math. 53, 472 (1951).
C. W. Misner, Phys. Rev. Lett. 22, 1071 (1969).
A. H. Guth, Phys. Rev. D 23, 347 (1981).
A. Linde, Phys. Lett. 108B, 382 (1982); A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
V. A. Belinsky, E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys. 19, 525 (1970); ibid. 31, 639 (1982).
R. M. Wald, Phys. Rev. 28, 2118 (1983); G. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977); W. Bocher and G. W. Gibbons, in Ref. 10.
J. D. Barrow and M. S. Turner, Nature 292, 35 (1981); G. Steigman and M. S. Turner, Phys. Lett. 128B, 295 (1983).
See, e.g., B. L. Hu, in Proceedings of the Tenth International Conference on General Relativity and Gravitation, edited by B. Bertotti, F. deFelice, and A. Pascolini (Consiglio Nazionale Delle Ricerche, Roma, 1983), in Proceedings of the Inner Space/Outer Space Conference, Fermilab, edited by E. Kolb, D. Schramm, and M. Turner (University of Chicago Press, Chicago, 1985).
For discussions on phase transition in the de Sitter universe see, for example, contributions in The Early Universe, edited by G. Gibbons, S. W. Hawking, and S. Siklos (Cambridge University Press, Cambridge, 1983).
See, e.g., V. Privman and M. E. Fisher, J. Stat. Phys. 33, 385 (1983).
B. L. Hu and D. J. O'Connor, Phys. Rev. D 30, 743 (1984).
B. L. Hu and D. J. O'Connor (unpublished).
B. L. Hu and T. C. Shen (unpublished).
S. W. Hawking, Commun. Math. Phys. 55, 133 (1977); J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224 (1976).
Note the difference between the two formulations in treating the divergent terms. See footnote 4 of Ref. 1.
B. L. Hu, Phys. Rev. D 8, 1048 (1973).
See, for example, J. D. Talman, Special Functions (Benjamin, New York, 1968).
R. Critchley and J. S. Dowker, J. Phys. A 14, 1943 (1981); ibid. 15, 157 (1982).
L. Ford and D. J. Toms, Phys. Rev. D 25, 1510 (1982).
K. Ishikawa, Phys. Rev. D 28, 2445 (1983); G. Denardo and E. Spallucci, Nuovo Cimento 83A, 35 (1984).
L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).
S. Coleman, Commun. Math. Phys. 31, 259 (1973).
S. Coleman, R. Jackiw and H. D. Politzer, Phys. Rev. D 10, 2491 (1974).
N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
The infrared problems can be seen by looking at the massless Green's functions in these dimensions: in one dimension the Green's function grows linearly with separation, in two dimensions it grows logarithmically, whereas for dimension d > 2 it falls off as | x - y $|^{2 - d}$ .
As pointed out by Coleman and Weinberg (Ref. 28), the $λ^{4}$ phi theory in flat spacetime has a fake minimum at λ ln ( φ / M ) = - case 16 over case 3 $π^{2}$ . This is so because it is too far from the origin, causing the ln ( φ / M ) term to be too large to be valid under one-loop perturbation theory. There is no phase transition associated with this minimum. In our case, the one-loop generalized CW potential does have a nonzero minimum at a nonzero value of phi which is within the validity range of perturbation theory. This is true for the case of minimum coupling, where classical curvature effect is absent. Phase transition induced by the quantum effects of curvature is thus possible.
S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).
G. Mazenko, W. G. Unruh and R. M. Wald, Phys. Rev. D 31, 273 (1984).
See, e.g., B. L. Hu, Phys. Lett. 103B, 331 (1982); ibid. 108B, 19 (1983).
T. Futamase, Phys. Rev. D 29, 2783 (1984).
D. J. Toms, Phys. Rev. D 21, 2805 (1980).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1980).

[c1] D. J. O'Connor, B. L. Hu and T. C. Shen, Phys. Lett. 130B, 31 (1983).

[c2] A. H. Taub, Ann. Math. 53, 472 (1951).

[c3] C. W. Misner, Phys. Rev. Lett. 22, 1071 (1969).

[c4] A. H. Guth, Phys. Rev. D 23, 347 (1981).

[c5] A. Linde, Phys. Lett. 108B, 382 (1982); A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).

[c6] V. A. Belinsky, E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys. 19, 525 (1970); ibid. 31, 639 (1982).

[c7] R. M. Wald, Phys. Rev. 28, 2118 (1983); G. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977); W. Bocher and G. W. Gibbons, in Ref. 10.

[c8] J. D. Barrow and M. S. Turner, Nature 292, 35 (1981); G. Steigman and M. S. Turner, Phys. Lett. 128B, 295 (1983).

[c9] See, e.g., B. L. Hu, in Proceedings of the Tenth International Conference on General Relativity and Gravitation, edited by B. Bertotti, F. deFelice, and A. Pascolini (Consiglio Nazionale Delle Ricerche, Roma, 1983), in Proceedings of the Inner Space/Outer Space Conference, Fermilab, edited by E. Kolb, D. Schramm, and M. Turner (University of Chicago Press, Chicago, 1985).

[c10] For discussions on phase transition in the de Sitter universe see, for example, contributions in The Early Universe, edited by G. Gibbons, S. W. Hawking, and S. Siklos (Cambridge University Press, Cambridge, 1983).

[c11] See, e.g., V. Privman and M. E. Fisher, J. Stat. Phys. 33, 385 (1983).

[c12] B. L. Hu and D. J. O'Connor, Phys. Rev. D 30, 743 (1984).

[c13] B. L. Hu and D. J. O'Connor (unpublished).

[c14] B. L. Hu and T. C. Shen (unpublished).

[c15] S. W. Hawking, Commun. Math. Phys. 55, 133 (1977); J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224 (1976).

[c16] Note the difference between the two formulations in treating the divergent terms. See footnote 4 of Ref. 1.

[c17] B. L. Hu, Phys. Rev. D 8, 1048 (1973).

[c18] See, for example, J. D. Talman, Special Functions (Benjamin, New York, 1968).

[c19] R. Critchley and J. S. Dowker, J. Phys. A 14, 1943 (1981); ibid. 15, 157 (1982).

[c20] L. Ford and D. J. Toms, Phys. Rev. D 25, 1510 (1982).

[c21] K. Ishikawa, Phys. Rev. D 28, 2445 (1983); G. Denardo and E. Spallucci, Nuovo Cimento 83A, 35 (1984).

[c22] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).

[c23] S. Coleman, Commun. Math. Phys. 31, 259 (1973).

[c24] S. Coleman, R. Jackiw and H. D. Politzer, Phys. Rev. D 10, 2491 (1974).

[c25] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).

[c26] The infrared problems can be seen by looking at the massless Green's functions in these dimensions: in one dimension the Green's function grows linearly with separation, in two dimensions it grows logarithmically, whereas for dimension d > 2 it falls off as | x - y $|^{2 - d}$ .

[c27] As pointed out by Coleman and Weinberg (Ref. 28), the $λ^{4}$ phi theory in flat spacetime has a fake minimum at λ ln ( φ / M ) = - case 16 over case 3 $π^{2}$ . This is so because it is too far from the origin, causing the ln ( φ / M ) term to be too large to be valid under one-loop perturbation theory. There is no phase transition associated with this minimum. In our case, the one-loop generalized CW potential does have a nonzero minimum at a nonzero value of phi which is within the validity range of perturbation theory. This is true for the case of minimum coupling, where classical curvature effect is absent. Phase transition induced by the quantum effects of curvature is thus possible.

[c28] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).

[c29] G. Mazenko, W. G. Unruh and R. M. Wald, Phys. Rev. D 31, 273 (1984).

[c30] See, e.g., B. L. Hu, Phys. Lett. 103B, 331 (1982); ibid. 108B, 19 (1983).

[c31] T. Futamase, Phys. Rev. D 29, 2783 (1984).

[c32] D. J. Toms, Phys. Rev. D 21, 2805 (1980).

[c33] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1980).

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Symmetry behavior of the static Taub universe: Effect of curvature anisotropy

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