June 18th, 2012

## A Fourth Exception to the BVG Theorem

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.*

Time reversal at singularity

Example: Aguirre-Gratton

(Regarding BVG): The Intuitive reason why de Sitter inflation cannot be past eternal is that in the full de Sitter space, exponential expansion is preceded by exponential contraction.  Such a contracting phase is not part of standard inflationary models, and does not appear to be consistent with the physics of inflation.  If thermalized regions were able to form all the way to past infinity in the contracting spacetime, the whole universe would have been thermalized before inflationary expansion could begin.  In our analysis we will exclude the possibility of such a contracting phase by considering spacetimes for which the past region obeys an averaged expansion condition, by which we mean that the average expansion rate in the past is greater than zero: Havg > 0. (Borde, Guth, and Vilenkin 2003, p1)

June 15th, 2012

## A Third Exception to the BVG Theorem

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 third exception: Infinite Cyclicity (Havg=0)

Example: Baum-Frampton “phantom bounce”

These models suggest that the universe goes through a cycle in which it grows from zero (or non-zero) size to a maximum and then contracts back to its starting condition.  The average expansion rate would be a pure zero.

June 14th, 2012

## A Second Exception to the BVG Theorem

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.