Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble parameter are both constant with respect to time, so there must be new pocket universes created as time spans on (such that the number of universes is proportional to the volume). So lets look at a geodesic (a particle with mass) travelling in one direction. From the metric of the FRW universe it's acceleration equation should be

This is differentiated with respect to "proper time". We integrate acceleration to find velocity

Thus when velocity is zero, the time we use is equal to the time as measured on that geodesic, which can extend back indefinitely. These particles are geodesically complete but consider when velocity is non-zero then

The future is geodesically complete when time tends toward positive infinity but when time tends toward negative infinity the proper time interval on the right hand side of the equation is finite. Otherwise some observer will see the geodesic receding at a speed greater than the velocity of light. This is a variation of the argument used to show that inflation cannot be eternal into the past. BGV integrate over the Hubble parameter and it's below some finite bound. This is how I interpret Eq. (11) in their paper.

The argument applies to "almost all geodesics" the original geodesic with respect to which others have their relative velocity defined is allowed to extend back indefinitely into the past. The authors prove this only for spacetimes with a classical metric similar to the one I used above. In Eq. (2). this spacetime is not just expanding but inflating, so they state their assumption as "the average expansion rate in the past is greater than zero" in other words, the universe is on average accelerating, which is satisfied in eternally inflating models but we can evade the conclusion of the theorem for models which are contracting prior to expanding, so long as it does not expand for longer than it contracts.

Some models which expand constantly may evade the condition if the scale factor

These are all ways of avoiding or everting the condition of the theorem, but there are also models which tell us how to deal with the boundary if the condition is met. As BGV state the argument, they do not consider spacetimes with a "quantum metric" that are not described by the metric I gave, in principle it may be possible to carry over the argument into these class of models too, but the authors certainly do not consider it. Several proposals do have a "break down" in classical spacetime both eternal cosmologies like the Hartle-Hawking model or cosmologies with a beginning like Linde's or Vilenkin's own model or Gott & Li's proposal.

I say it is "uninteresting" because no serious position advocated by any mainstream cosmologist has really been undermined by the theorem with the possible exception of the Ekpyrotic Cyclic Universe. This is a really bizarre alternative to inflation that assumes two infinite homogenous sheet-like universes that continually fall, attract each other and collide but these models expand more than they contract meeting the condition of the theorem. The model was just too fantastic to be believed anyway. But I don't doubt that even this model can avoid serious repercussions of the BGV theorem if one measures time using the number of bounces rather than the proper time of a geodesic. Then the model is self contained.

No class of models is ruled out, although eternally inflating steady state models might loose much of their appeal. Normally they don't set any boundary or initial conditions, given that their model is in a "steady state" it only aims to show that the model satisfies the same state throughout it's entire evolution without any deviation. Thus there is no need to explain "why" inflation started, it's sufficient to simply assume this fact in order to get the model to work but now these models need to offer some kind of "boundary conditions" unless you reverse the arrow of time like Aguirre-Gratton, which of course you can do but really, that's about it.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble parameter are both constant with respect to time, so there must be new pocket universes created as time spans on (such that the number of universes is proportional to the volume). So lets look at a geodesic (a particle with mass) travelling in one direction. From the metric of the FRW universe it's acceleration equation should be

This is differentiated with respect to "proper time". We integrate acceleration to find velocity

The future is geodesically complete when time tends toward positive infinity but when time tends toward negative infinity the proper time interval on the right hand side of the equation is finite. Otherwise some observer will see the geodesic receding at a speed greater than the velocity of light. This is a variation of the argument used to show that inflation cannot be eternal into the past. BGV integrate over the Hubble parameter and it's below some finite bound. This is how I interpret Eq. (11) in their paper.

The argument applies to "almost all geodesics" the original geodesic with respect to which others have their relative velocity defined is allowed to extend back indefinitely into the past. The authors prove this only for spacetimes with a classical metric similar to the one I used above. In Eq. (2). this spacetime is not just expanding but inflating, so they state their assumption as "the average expansion rate in the past is greater than zero" in other words, the universe is on average accelerating, which is satisfied in eternally inflating models but we can evade the conclusion of the theorem for models which are contracting prior to expanding, so long as it does not expand for longer than it contracts.

Some models which expand constantly may evade the condition if the scale factor

**a(t)**starts out a certain way but slowly approaches a constant value. Other expanding models could reverse the arrow of time to avoid the conclusion of the theorem, Aguirre-Gratton consider an eternally inflating model (with the same metric equation as above) but map a second identical universe which grows toward minus infinity onto the "boundary" of our spacetime. These two universes are completely causally separated and are not "time orientated" in the sense that one universe is the past of the other. Both universes have been inflating for an infinite amount of time and are geodesically complete.These are all ways of avoiding or everting the condition of the theorem, but there are also models which tell us how to deal with the boundary if the condition is met. As BGV state the argument, they do not consider spacetimes with a "quantum metric" that are not described by the metric I gave, in principle it may be possible to carry over the argument into these class of models too, but the authors certainly do not consider it. Several proposals do have a "break down" in classical spacetime both eternal cosmologies like the Hartle-Hawking model or cosmologies with a beginning like Linde's or Vilenkin's own model or Gott & Li's proposal.

I say it is "uninteresting" because no serious position advocated by any mainstream cosmologist has really been undermined by the theorem with the possible exception of the Ekpyrotic Cyclic Universe. This is a really bizarre alternative to inflation that assumes two infinite homogenous sheet-like universes that continually fall, attract each other and collide but these models expand more than they contract meeting the condition of the theorem. The model was just too fantastic to be believed anyway. But I don't doubt that even this model can avoid serious repercussions of the BGV theorem if one measures time using the number of bounces rather than the proper time of a geodesic. Then the model is self contained.

No class of models is ruled out, although eternally inflating steady state models might loose much of their appeal. Normally they don't set any boundary or initial conditions, given that their model is in a "steady state" it only aims to show that the model satisfies the same state throughout it's entire evolution without any deviation. Thus there is no need to explain "why" inflation started, it's sufficient to simply assume this fact in order to get the model to work but now these models need to offer some kind of "boundary conditions" unless you reverse the arrow of time like Aguirre-Gratton, which of course you can do but really, that's about it.

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