Summary
The flatness problem posed by Dicke is studied within the framework of the pulsating model. In contrast with the simple solution proposed by the inflationary models of Guth and others,i.e. Ω = 1 always, it is found that Ω is a U-shaped function of the epoch: Ω = ∞ at the bounce as well as at the recollapse point, and has a minimum, which may be close to but exceeding unity, that occursafter the expansion rate a has reached a maximum. The values of Ω, the deceleration parameter, the Hubble constant, and the look-back time for several key values of the expansion epoch are given in closed form. Importantly, in contrast with the standard model, forq = 1/2, one has Ω > 1. It is proposed that the reason why the Universe is apparently near the Ω-minimum at an epoch at which terrestrial science has developed to a point it can make such a determination is because there are relations between the parameters governing the cosmological-expansion and terrestrial-evolution rates, which generate this synchronism. There are two appendices: the first shows in detail that there is no particle horizon problem for the pulsating Universe, and the second presents a space-time imbedding argument which clarifies how the expansion rate a can exceed the numerical speed of light because it is analogous to a Minkowski proper velocity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. H. Dicke:Gravitation and the Universe: The Jayne Lectures for 1969 (American Philosophical Society, Philadelphia, Penn., 1970), p. 62.
R. H. Dicke andP. J. E. Peebles: inGeneral Relativity: An Einstein Centenary Survey edited byS. W. Hawking andW. Israel (Cambridge University Press, Cambridge, 1979); for a review see:A. Lightman andO. Gingrich:Science,255, 690 (1991).
A. Guth:Phys. Rev. D,23, 347 (1981). For an update and review see:S. K. Blau andA. H. Guth: inThree Hundred Years of Gravitation, edited byS. Hawking andW. Israel (Cambridge University Press, Cambridge, 1987), Chapt. 12.
A. Albrecht andP. J. Steinhardt:Phys. Rev. Lett,48, 1220 (1982).
A. D. Linde:Phys. Lett. B,108, 389 (1982).
A. D. Linde:JETP Lett,38, 176 (1983);Phys. Lett. B,129, 177 (1983).
D. La andP. J. Steinhardt:Phys. Rev. Lett,62, 376 (1989);T. Damour, G. W. Gibbons andC. Grundlach:Phys. Rev. Lett,64, 123 (1990);R. Holman, E. W. Kolb andY. Wang:Phys. Rev. Lett,64, 123 (1990);K. Freese, J. A. Frieman andA. V. Olinto:Phys. Rev. Lett.,65, 3233 (1990);M. Yoshimura:Phys. Rev. Lett,66, 1559 (1991);H. Murayama, H. Suzuki, T. Yanagida andJ. Yokoyama:Phys. Rev. Lett,70, 1912 (1993).
F. R. Tangherlini:Ann. N. Y. Acad. Sci,422, 385 (1984);Lett. Nuovo Cimento,38, 573 (1983).
F. R. Tangherlini:Nuovo Cimento B,108, 911 (1993).
See,e.g,R. C. Tolman:Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford, 1934), pp. 270–71.
F. R. Tangherlini:Nuovo Cimento B,106, 123 (1991);103, 311 (1989).
F. R. Tangherlini:Nuovo Cimento B,91, 209 (1986).
S. J. Lilly:Astrophys. J.,333, 161 (1988).
P. R. McCarthy, M. Dickinson, A. V. Filippenko, H. Spinrad andW. J. M. van Breugel:Astrophys. J.,328, L29 (1988);D. P. Schneider, M. Schmidt andJ. E. Gunn:Astron. J.,102, 837 (1991).
W. M. Smart:Celestial Mechanics (Longmans Green, London, 1953), p. 17.
G. de Vaucouleurs:Nature,299, 303 (1982);G. De Vaucouleurs andG. Bollingen:Astrophys. J.,233, 433 (1979).
B. F. Madore andW. L. Freedman:Publ. Astron. Soc. Pacific,103, 933 (1991).
J. L. Tonry:Astrophys. J.,373, L1 (1991);G. H. Jacoby, R. Ciardullo andH. C. Ford:Astrophys. J.,356, 332 (1990).
A. R. Sandage andG. Tammann:Astrophys. J.,210, 7 (1976).
A. R. Sandage, A. Saha, G. A. Tammann, N. Panagia andD. Macchetto:Astrophys. J.,401, L7 (1992). See also,M. J. Pierce, M. E. Ressler andM. S. Shure:Astrophys. J.,390, L45 (1992).
G. Rhee:Nature,350, 211 (1991);D. H. Roberts, J. Lehar, J. N. Hewitt andB. F. Burke:Nature,352, 43 (1991).
A. Renzini: inGalaxy Distances and Deviations from Universal Expansion, edited byB. F. Madore andR. B. Tully (Reidel, Dordrecht, 1986), p. 177.
A. S. Eddington:The Philosophy of Physical Science (Cambridge University Press, Cambridge, 1939), Chapt. XI.
J. D. Barrow andF. J. Tipler:The Anthropic Cosmological Principle (Clarendon Press, Oxford, 1986).
P. Ehrenfest:Proc. Amsterdam Acad.,20, 200 (1917);Ann. Phys. (Leipzig),61, 440 (1920).
G. J. Whitrow:The Structure and Evolution of the Universe (Harper & Brothers, New York, N.Y., 1959);Brit. J. Philos. Sci,6, 13 (1955).
F. R. Tangherlini:Nuovo Cimento,27, 636 (1963).
L. Gurevich andV. Mostepanenko:Phys. Lett. A,35, 201 (1971).
F. Crick:Life Itself (Simon and Schuster, New York, N.Y., 1981), pp. 13–16; 155–160.
S. L. Jaki:Science and Creation (Scottish Academic Press, Edinburgh, 1986).
D. T. Wilkinson:Science,232, 1517 (1986).
G. F. Smoot et al.:Astrophys. J.,396, L1 (1992).
S. Weinberg:Gravitation and Cosmology (John Wiley, New York, N.Y., 1972).
W. Rindler:Mon. Not. R. Astron. Soc,116, 662 (1956).
C. Misner:Phys. Rev. Lett,22, 1071 (1969).
E. W. Kolb andM. S. Turner:The Early Universe (Addison-Wesley, Reading, Mass., 1990).
F. R. Tangherlini:Bull. Amer. Phys. Soc,35, 1547 (1990).
R. M. Wald:General Relativity (University of Chicago Press, Chicago, Ill., 1984), pp. 116–117.
A. Einstein:Sitzungsber. Preuss. Akad. Wiss., 142 (1917).
T. Kaluza:Sitzungsber. Preuss. Akad. Wiss., 966 (1921).
O. Klein:Z. Phys.,37, 895 (1926). See also,H. Mandel:Z. Phys.,39, 136 (1926);V. Fock:Z. Phys.,39, 226 (1926).
A. Friedmann:Z. Phys.,10, 377 (1922).
Author information
Authors and Affiliations
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/BF02849849.
Rights and permissions
About this article
Cite this article
Tangherlini, F.R. The flatness problem and the pulsating universe. Nuov Cim B 108, 1253–1273 (1993). https://doi.org/10.1007/BF02741278
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02741278