BGV-theorem and Penrose's Conformal Cyclic Cosmology

[haushofer]

TL;DR

How does the BGV-theorem fit in with Penrose's Conformal Cyclic Cosmology?

Dear all,

Some time ago I stumbled upon the famous BGV-theorem,

- https://en.wikipedia.org/wiki/Borde–Guth–Vilenkin_theorem
- https://arxiv.org/abs/gr-qc/0110012

which states that on spacetimes which have, on average, a positive Hubble constant, one can find timelike geodesics which cannot be completed indefinitely in the past. This is often rephrased as "expanding universes must have had a beginning", although this statement is a bit subtle. My simple question is: How does the BGV-theorem fit in with Penrose's Conformal Cyclic Cosmology (CCC)? Does it imply that this CCC is also not past-complete? Can someone point to references in which this is explicitly treated? :) I can't find any references in Penroses' Cycles of Time.

 

[PeterDonis]

This is often rephrased as "expanding universes must have had a beginning", although this statement is a bit subtle.

Yes, it is. I would rephrase it as follows: An expanding region of spacetime (a region in which the averaged Hubble constant is positive) must either have an initial singularity, or must not be the entire spacetime.

How does the BGV-theorem fit in with Penrose's Conformal Cyclic Cosmology (CCC)?

I'm not sure, but I think the "conformal rescaling" that occurs at each boundary in CCC (where the future boundary of one expanding universe gets matched to the past "Big Bang" boundary of the next) breaks the "expanding" condition of the BGV theorem, mathematically speaking. Whether that "conformal rescaling" is actually a reasonable physical thing to happen is a different question.

 

[haushofer]

Hi Peter, thanks for your response! Yes, I guess the answer lies in how the gluing together of the endphase of one universe to the beginning of the next one is defined precisely. I don't see how expansion of space can be properly defined in a conformal theory anyway.

 

[PeterDonis]

I don't see how expansion of space can be properly defined in a conformal theory anyway.

Yes, that's a good point, since in a purely conformal theory there is no distance or time scale.

 

[haushofer]

Mmm, now I think of it: is the inflationary epoch also described by a conformally-invariant spacetime, as there are no massive particles yet? Then this issue of expansion should already hold in the usual inflationary scenarios.

 

[PeterDonis]

is the inflationary epoch also described by a conformally-invariant spacetime

I don't think so. I believe the only way to have a conformally invariant spacetime with nonzero stress-energy present (where I'm including a cosmological constant or the equivalent as nonzero stress-energy) is to have the stress-energy be pure null dust, i.e., equation of state $p = \rho / 3$, which is what I understand the stress-energy in the conformally invariant part of each cycle in CCC to be. The inflaton field, like a cosmological constant, has $p = - \rho$. But I haven't dug deeply into the math to confirm that my belief is correct.