Neil Turok has been among the men who have spent years by hyping theories about "cyclic universes" and related laymen's ideas that have never explained anything in physics and that are really not capable of explaining anything, for very good reasons. So this enterprise became a religious activity of a sort – he needs to publish new papers just to create the illusion that the previous papers weren't a stupid waste of time.
The latest, 5-pages-long paper by Boyle, Finn, and Turok is called
Concerning the anti-universe, I can never fail to quote a favorite joke of mine which I learned from my diploma adviser in Prague. Aside from the universe, there also exists the anti-universe where everything is anti-. For example, the hardest science over there is anti-physics and it's researched by anti-Semites. ;-)
OK, let us return to the Turok et al. paper.
First, if they want to really preserve the CPT, then the "anti-universe" must be a precise copy of the "universe". So it's just a redundancy, a mirror image added to the history of the "universe", and there's no extra information in it. So it should be removed. There is no analogy with particles and antiparticles because when a particle pair is created, the two members of the pair may undergo different fates.
Clearly, they must mean an "approximate anti-universe" which is just macroscopically similar but whose events aren't exact mirror images of the events in our "universe". So CPT is still violated, although in some macroscopic perspective, it's approximately preserved.
Second, there's a question whether the "anti-universe" should be drawn as some history for \(t\gt 0\), as the normal "universe", or as \(t\lt 0\). They explicitly propose the latter. But that can't be consistent with the claim that CPT is saved from spontaneous breaking. Why? Because the second law of thermodynamics (not to mention related consequences of the arrow of time) demands the entropy to increase with time.
So either they will have the entropy that is an even function of time\[
S(-t) = S(t)
\] in which case the entropy is decreasing at \(t\lt 0\) and they violate the second law of thermodynamics. Or\[
S(-t) \neq S(t)
\] in which case the evolution of the "anti-universe" violates the CPT and their claim about the preservation of CPT is wrong. You just can't obey the second law and "spontaneously unbroken CPT" simultaneously!
They included one short paragraph about the second law of thermodynamics:
Also note that density pertubations grow as we get further from the bang in either direction, and hence the thermodynamic arrow of time points away from the bang in both directions (to the future and past).This paragraph automatically includes a violation of the second law of thermodynamics. To allow the rules of the thermodynamics to be "reversed" in the "anti-universe" is really silly because the future must be defined as the side of the temporal axis where the entropy is higher than in the past. So if you have two parts of the universe where the entropy grows if you get further away from the Big Bang, then the correct way to draw these "two universe" on the \(t\)-axis is to draw both of them as branches of \(t\gt 0\).
That picture means that at \(t=0\), two universes are just created out of nothing and they may be considered anti-objects of each other. Well, it's different from how they want to spin it. On top of that, if some universal laws govern the pair creation, then the two universes are precise "anti-objects" to each other. They must be perfectly entangled and they're just two copies of the same Universe.
At any rate, the events in the "anti-universe" cannot possibly affect us because that "anti-universe" is just a parallel universe that is forever detached from ours. If you draw the "anti-universe" as a branch at \(t\gt 0\), it's clear that this branch in no meaningful way occurred "before our time", so the events in that branch can't be considered causes of anything we may observe.
To generalize the prefix "anti-" from antiparticles and anti-Semites to the universes could have been a good idea a priori. But if you look at the basic possibilities, you can see that it is actually not a good idea. While it's very useful to talk about antiparticles, antibranes, anti-instantons, and other things, it's not useful to talk about anti-universes.
A reason why "anti-" is silly for the whole universe is simply the fact that, as Richard Feynman and John Wheeler figured out, an antiparticle is a particle moving backwards in time (perhaps one with the negative energy, as it happens in the context of the Feynman diagrams). So to talk about "anti-", one needs a pre-existing notion of time and its arrow (which is needed either directly, for the directions of time, or indirectly, for the signs of energy, or both).
So only things that may be embedded within a spacetime with a well-defined arrow of time – e.g. particles and branes – may be associated with their anti-objects. The whole universe itself isn't one of these things because the universe doesn't exist as an object embedded within (another?) universe with a well-defined arrow of time.
But none of these things is really understood by most of the laymen. Turok et al. wrote a confusing package of some basic exciting notions – anti-stuff and Big Bang – and tons of stupid people will buy this piece of pop science regardless of the fact that it's complete junk.
Even if I return to the more acceptable notion of "history before the Big Bang", I think that the room for any meaningful physics of that kind is extremely limited. In other words, I am almost certain that all these ideas must be wrong. There's always a problem with the entropy before the Big Bang that should be even lower than the entropy at the moment of the Big Bang – and the latter is rather low if not zero. (Cyclic universes run into a conflict with the second law of thermodynamics, too.)
And then there's a problem with any predictive consequences of the pre-Big-Bang stage. One reason is that during the Big Bang, the spacetime curvature is huge, perhaps Planckian. That means that all gadgets – including all measurement apparatuses – break there. Because physical quantities are only meaningful to the extent to which they're measurable by apparatuses, I think it's right to say that even the continuity or predictability of the observables during the Big Bang era disappears. If all clocks break etc., the time itself becomes meaningless in the vicinity of the Big Bang, and that's why you shouldn't ask how the post-Big-Bang time is connected to another branch of time. It's a physically meaningless question.
A related fact is that the events are in the regime of "extreme quantum gravity" or "string theory" in the vicinity of the Big Bang. Quantum coherence and phases of the amplitudes matter a lot. It seems that all the people are imagining some simple union of classical geometries. That's almost certainly an inadequate description in that extreme epoch, in one way or another.
Stephen Hawking and Jim Hartle have proposed a much more promising point to address the initial conditions of the whole universe: their Hartle-Hawking state, the (especially initial) wave function of the universe. Whatever happens in the very early moments of the universe and/or "before that moment" should be incorporated in a state that may be defined a little bit after the Big Bang when the state of the world is no longer "insanely curved" and otherwise problematic.
I think that even if our Big Bang were just an event in some longer sequence of events within eternal inflation or something like that, there should exist a calculation of the probabilities that are relevant for our universe that simply "integrates out" all the conceivable pre-history. The Feynman path integral has the nice property that you may get the resulting probability amplitudes directly, through an explicit formula.
It only makes sense to talk about a "history in time" if the "time" is an uncontroversial continuation of the time that we actually observe in our world. If there is any other information or quantum information affecting us, that information and its impact on us should be converted to the variables that are compatible with the existence of time or spacetime of our type. In particular, all pre-Big-Bang influences should be convertible to a Hartle-Hawking state at the beginning of our time right after the Big Bang.
I think it's almost rigorously provable that no other treatment of the hypothetical pre-Big-Bang evolution of the universe may have defensible and calculable consequences for our post-Big-Bang evolution. But even if you doubt this ambitious statement of mine, it's a fact that no convincing explanation of any observable fact about the universe using a pre-Big-Bang (or cyclic) concepts has been proposed in the literature so far.