A Second Exception to the BVG Theorem

by Max Andrews

The Borde-Vilenkin-Guth Theorem states that any universe, which has, on average, a rate of expansion greater 1 that system had to have a finite beginning. This would apply in any multiverse scenario as well.  There are four exceptions to the theorem.*

For a greater context please see the first exception to the BVG theorem, which is Initial Contraction (Havg<0).

The second exception: Asymptotically static (Havg=O)

Example: asymptotically static universe is an emergent model class.

An asymptotically static space is one in which the average expansion rate of the universe over its history is equal to zero, since the expansion rate of the universe “at” infinity is zero.  The problem is that we observe expansion today and if at any moment there is expansion then the Havg must be greater than 0.

A possibility is that the initial Einstein static universe is created from “nothing” by some quantum tunneling process.  Indeed, finiteness of the tunneling action requires that the universe created through instantonic tunneling be closed. It is not implasusible, then, that through spontaneous quantum fluctuations, a closed universe could be created in a long lived but transient Einstein static state, which then makes a transition to a finite lifetime de Sitter and subsequent marginally closed FRW phase along the lines as mentioned above.

There is a paper written by George Ellis, J. Murugan, and C.G. Tsagas, “The Emergent Universe: An Explicit Construction,” which deals with the asymptotic universe.

Of course, an immediate question arises as to a viable mechanism that realizes the initial Einstein static universe that our model emerges from. At this point several avenues present themselves. Among the more promising, we note two; the first of these is the observation of [29] that the Einstein static universe is one of only two asymptotic solutions of the Ramond-Ramond sector superstring cosmology field equations [41]. The size of the positively curved Einstein universe in this picture is controlled by the level number of the Kac-Moody algebra of the conformal fields living in the compactified internal space. Indeed, this observation leads quite naturally to the tantalizing possibility of realizing an Emergent-like universe within a string cosmology context [30]. A second, equally intriguing, possibility is that the initial Einstein static universe is created from “nothing” by some quantum tunneling process [31, 32]. Indeed, finiteness of the tunneling action requires that the universe created through instanonic tunneling be closed [33].

[29] I. Antoniadis, C. Bachas, J. Ellis and D.V. Nanopolous, Phys. Lett. B 211, 4 (1988)
[30] J. Murugan, G.F.R. Ellis and A. Weltman, In progress
[31] E.P. Tryon, Nature(London) 246, 396 (1973)
[32] A. Vilenkin, Phys. Rev. D 32, 10 (1985)
[33] S. Gratton, A. Lewis and N. Turok, Closed Universes from Cosmological Instantons,Phys. Rev. D 65, 043513 (2002), astro-ph/0111012
[41] The other being a linearly expanding Milne universe

The main problem with this exception is that it does not allow for an expanding or evolutionary universe.  This model cannot be true.  The best evidence and empirical observations indicate that the universe is not static; rather, it is expanding and evolving.  This might have been a great model under Newton but not since Einstein’s field equation concerning the energy-momentum of the universe.


*This information is primarily from and available in William Lane Craig and James Sinclair’s “The Kalam Cosmological Argument,”  inThe Blackwell Companion to Natural Theology Eds. William Lane Craig and J.P. Moreland (Oxford, UK: Blackwell, 2009), 143-147. Diagram on 146.

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