179. Followup on ekpyrosis and phoenix universe

As follows from the title, this is a followup on the guest post “The return of the phoenix universe” by Jean-Luc Lehners. Jean-Luc kindly agreed to answer to a couple of questions related to physics behind ekpyrosis for NEQNET. In what follows D. – me, J. – Jean-Luc.

D. It is interesting that about a year ago I was talking to Andrei Linde (he was preparing his paper on ekpyrosis with Mukhanov at that time) and eventually asked him whether it is possible to construct an analogue of eternal inflation for ekpyrotic cosmology. His answer was negative at that time, but now it seems to be that your phoenix universe is exactly this analogue of eternal inflation – you are trying to find a balance between expanding (inflating in inflationary cosmology) and collapsing (AdS sinks in inflationary cosmology) patches. The only difference is that your phoenix universe seems to be dominated by large patches with small effective CC.

J. Yes, I guess you could say that in some sense what we did is the ekpyrotic equivalent to eternal inflation, however with a very different global structure as a result!

D. It looks like the actual distribution of patches will strongly depend on boundary conditions near the AdS crunch – in particular, it will determine the actual probability to live in a 60-efold-large patch at a given moment of time. Why do you choose them in such a way that the bounce is allowed at all ( at the point of bounce)? Can’t you choose them in some different way – for example, that evolution always ends up with an AdS crunch? The reason I ask is that once I have found a general solution of Einstein equations near the AdS crunch singularity. It turns out the asymptotic behavior of the metric components is quasi-isotropic – AdS crunch always happens in all points of 3d space, but in different moments of time.

J. Unfortunately, I don’t really understand what you are asking in your first question. We did not impose at the bounce as a boundary condition, although is much smaller for those trajectories corresponding to a long ekpyrotic phase than for general trajectories with a short and insufficient ekpyrotic phase.

D. Sorry for asking a confusing question and let me reformulate it better.

Let us foliate the ekpyrotic Universe with constant hypersurfaces (or constant , if you want). To quantitatively understand the causal structure of the Universe, I would like to learn how to calculate the probability that the given Hubble patch will have size (for given ) at a given moment of time t.

I understand that this probability depends on details of physics near the bounce. For example, if there is no in Eq. (6) of your paper, then there is simply no bounce – Universe will reach AdS Big Crunch in finite time (so after some time the probability will be equal to zero).

Now, what I know from my paper is that the solution with AdS crunch is a general solution of the Einstein equations in ekpyrotic models – that is, for any initial conditions and any ordinary matter present in the Universe apart of scalar fields driving ekpyrosis the AdS crunch should be achieved in finite time. This picture is different from yours, as you understand (you say instead that causal structure is dominated by large Hubble patches at late times).

So, my question is really is – what is the physics behind in your model? Do you have some not-so-ordinary matter (like, say, ghost condensate) that provides ?

J. arises due to ordinary matter on the negative-tension brane. Neil Turok and I have analyzed this in our paper. However, this reflection in moduli space (which is described in more detail in my papers with Turok and McFadden) occurs before the collision of the boundary orbifold branes, in other words, it does not correspond to the moment of the big bang. Probably I did not make this clear. In the 4d effective theory, the moment of the big bang is still singular, and from the higher-dimensional point of view it corresponds to the momentary shrinking to zero size of the orbifold direction. In the so-called “new” ekpyrotic models, this is different. There they put in a null energy-condition-violating type of matter so that the big bang is non-singular and corresponds to a bounce at a finite value of the scale factor.

D. Thanks! Did I understand correctly that you actually still need inflation (well, dark energy) in order for make patches large enough?

J. Yes, dark energy is absolutely essential in this minimal model in its role as a stabilizer. One could in fact imagine other stabilization mechanisms, such as for example a localizing potential for the s field at present times. In that case less efolds of dark energy would be needed. What we looked at here is the model without any additions, and for this model 60 efolds of dark energy domination are necessary to produce large habitable patches. However, I would not call this inflation, even though the mathematices is of course analogous: first of all, the relevant density perturbations are not produced by this dark energy phase. Secondly, the energy scale is 100 orders of magnitude lower than for inflation. This is crucial, as the large Hubble parameter during inflation is what causes all the problems of eternal inflation. Lastly, dark energy has been observed, and inflation not!

D. Thanks for the very interesting interview!