Cosmologists Vilenkin and Mithani tackle the question of whether the universe had a beginning in their short paper, ** Did the universe have a beginning? **Vilenkin and Mithani simply argue that there is no cosmological model currently that is able to avoid an absolute beginning to the universe. They state in their abstract:

“We discuss three candidate scenarios which seem to allow the possibility that the universe could have existed forever with no initial singularity: eternal inflation, cyclic evolution, and the emergent universe. The first two of these scenarios are geodesically incomplete to the past, and thus cannot describe a universe without a beginning. The third, although it is stable with respect to classical perturbations, can collapse quantum mechanically, and therefore cannot have an eternal past.”

Vilenkin, Borde, Guth had proved in another paper, ** Inflationary spacetimes are not past-complete** that any universe that is on average expanding will be finite in the past. Or rather as they put it in technical terms:

“Our argument shows that null and timelike geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion conditionHav>0 holds along these past-directed geodesics.”

What is amazing about this theorem is how general it is. It does not assume that space-time is described by Einstein’s Theory of General Relativity. This means that even if space-time is described by some yet unknown theory of quantum gravity which is the physics that will explain the behaviour of the universe as its volume falls below sub-atomic levels (planck length). The theorem makes no assumptions about the physics governing space-time but only that on average it is expanding. Vilenkin gives an excellent account of this theorem in an online video. In another excellent essay, The beginning of the universe, Vilenkin explains the concept of geodesics:

In general relativity, the trajectories of matter are represented by straight lines in space-time, or geodesics. If space-time is free of singularities, all geodesics should have infinite extent. A geodesic encountering a singularity cannot be further extended.

They begin this paper by looking at inflationary models of the universe and whether they avert the absolute beginning of the universe, and conclude inflationary models cannot be eternal in the past.

“A more general incompleteness theorem was proved recently that does not rely on energy conditions or Einstein’s equations. Instead, it states simply that past geodesics are incomplete provided that the expansion rate averaged along the geodesic is positive:Hav>0. This is a much weaker condition, and should certainly apply to the past of any inflating region of spacetime. Therefore, although inflation may be eternal in the future, it cannot be extended indefinitely to the past.”

They briefly turn their attention to a cycling model and summarise why it to fails to avert an absolute beginning:

“Another possibility could be a universe which cycles through an infinite series of big bang followed by expansion, contraction into a crunch that transitions into the next big bang. A potential problem with such a cyclic universe is that the entropy must continue to increase through each cycle, leading to a “thermal death” of the universe. This can be avoided if the volume of theuniverse increases through each cycle as well, allowing the ratioS=Vto remain finite. But if the volume continues to increase over each cycle,Hav>0, meaning that the universe is past-incomplete.”

They then turn to the main focus of their paper, an emergent universe scenario.

“In the emergent universe model, the universe is closed and static in the asymptotic past…ThenHav= 0 and the incompleteness theorem does not apply. This universe can be thought of as a “cosmic egg” that exists forever until it breaks open to produce an expanding universe. In order for the model to be successful, two key features are necessary. First, the universe should be stable, so that quantum fluctuations will not push it to expansion or contraction. In addition, it should contain some mechanism to exit the stationary regime and begin inflation.”

Vilenkin and Mithani describe the emergent universe as a quantum wave and show that it cannot remain in its state for eternity. In other words if the universe could remain in its stable pre-inflation for eternity past it would remain in that state for eternity into the future as well. But seeing how it has and in fact it cannot, it cannot be past eternal as well.

Since the tunneling probability is nonzero, the simple harmonic universe cannot last forever.

* *To the question, did the universe have a beginning? Vilenkin and Mithani conclude:

“At this point, it seems that the answer to this question is probably yes.Here we have addressed three scenarios which seemed to offer a way to avoid a beginning, and have found that none of them can actually be eternal in the past. Both eternal inflation and cyclic universe scenarios haveHav>0, which means that they must be past-geodesically incomplete. We have also examined a simple emergent universe model, and concluded that it cannot escape quantum collapse. Even considering more general emergent universe models, there do not seem to be any matter sources that admit solutions that are immune to collapse.”

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In previous posts I did a survey of different cosmological models.

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**References**

- Vilenkin, A. Mithani. 2012.
*Did the universe have a beginning?*Tufts University [arXiv:1204.4658v1] - Borde, A.H. Guth, A. Vilenkin. 2003.
*Inflationary spacetimes are not past-complete.*Tufts University [arXiv:gr-qc/0110012v2] - Inference-review. (2015) Alexander Vilenkin. [Online] Available from http://inference-review.com/article/the-beginning-of-the-universe [Accessed 20 December 2015]