Singularities in inflationary cosmology: A review

International Journal of Modern Physics D

Volumn 5, Issue 6, 1996, Pages 813-824

  Borde, Arvind ^a,b   Vilenkin, Alexander ^a  

^a TUFTS UNIVERSITY   (United States)

^b LONG ISLAND UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030327428     PISSN: 02182718     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218271896000497     Document Type: Article

References (24)

1. The inflationary expansion is driven by the potential energy of a scalar field φ, while the field slowly "rolls down" its potential V(φ). When φ reaches the minimum of the potential this vacuum energy thermalizes, and inflation is followed by the usual radiation-dominated expansion. The evolution of the field φ is influenced by quantum fluctuations, and as a result thermalization does not occur simultaneously in different parts of the Universe.
2. A. Vilenkin, Phys. Rev. D, 21, 2848 (1983); A.D. Linde, Phys. Lett. B 175, 395 (1986).
3. A. Vilenkin, Phys. Rev. D, 21, 2848 (1983); A.D. Linde, Phys. Lett. B 175, 395 (1986).
4. M. Aryal and A. Vilenkin, Phys. Lett. B 199, 351 (1987); A.S. Goncharov, A.D. Linde and V.F. Mukhanov, Int. J. Mod. Phys. A 2, 561 (1987); K. Nakao, Y. Nambu and M. Sasaki, Prog. Theor. Phys. 80, 1041 (1988).
5. M. Aryal and A. Vilenkin, Phys. Lett. B 199, 351 (1987); A.S. Goncharov, A.D. Linde and V.F. Mukhanov, Int. J. Mod. Phys. A 2, 561 (1987); K. Nakao, Y. Nambu and M. Sasaki, Prog. Theor. Phys. 80, 1041 (1988).
6. M. Aryal and A. Vilenkin, Phys. Lett. B 199, 351 (1987); A.S. Goncharov, A.D. Linde and V.F. Mukhanov, Int. J. Mod. Phys. A 2, 561 (1987); K. Nakao, Y. Nambu and M. Sasaki, Prog. Theor. Phys. 80, 1041 (1988).
7. A. Linde, D. Linde and A. Mezhlumian, Phys. Rev. D, 49, 1783 (1994).
8. A. Borde and A. Vilenkin, Phys. Rev. Lett., 72, 3305 (1994).
9. A. Borde, Phys. Rev. D., 50, 3392 (1994).
10. A. Borde and A. Vilenkin, in Relativistic Astrophysics: The Proceedings of the Eighth Yukawa Symposium, edited by M. Sasaki, Universal Academy Press, Japan (1995).
11. A. Borde, Tufts Institute of Cosmology preprint (1995).
12. A. Vilenkin, Phys. Rev. D, 46, 2355 (1992).
13. A. Borde, Cl. and Quant. Gravity 4, 343 (1987).
14. S.W. Hawking and G.F.R. Ellis, The large scale structure of spacetime, Cambridge University Press, Cambridge, England (1973).
15. This means that the notions of "past" and "future" are globally well-defined.
16. A congruence is a set of curves in an open region of spacetime, one through each point of the region.
17. A weakening of the conditions under which θ diverges, was discussed by F.J. Tipler, J. Diff. Eq., 30, 165 (1978); Phys. Rev. D, 17, 2521 (1978); these results were extended in [10].
18. A weakening of the conditions under which θ diverges, was discussed by F.J. Tipler, J. Diff. Eq., 30, 165 (1978); Phys. Rev. D, 17, 2521 (1978); these results were extended in [10].
19. Similar behavior occurs, for instance, in the Einstein Universe, but it does not in the de Sitter Universe, nor in some closed Robertson-Walker Universes [11, 6].
20. We actually need to impose a further causality requirement, called the stable causality condition, in order for this definition be meaningful; see ref. [6] for the details.
21. Our results are also stronger than many standard singularity theorems - such as the Hawking-Penrose theorem [11] - because we do not assume the strong energy condition. This is crucially important when discussing the structure of inflationary spacetimes, because the condition is explicitly violated there [7].
22. The existence, or not, of such a curve is related to issues raised in A.D. Linde, D. Linde and A. Mezhlumian, Phys. Rev. D, 49, 1783 (1994).
23. It is possible to contrive examples in which this is not true - where, for instance, a timelike curve avoids singularities in the past, but no null ones in the boundary of its future do. In a physically reasonable spacetime, however, one would expect singularities to be visible to timelike and null curves alike.
24. 2, however, the t = 0 surface appears further and further in the past the further one moves away from x = 0. There is no upper bound here on the lengths of past-directed timelike curves from S.