Ever since Hubble, it is well known that the universe is expanding from a Big Bang. The size of the universe had gone up by many many orders of magnitude as space expanded. If the dimensionality of the quantum phase space is finite because of spatial cutoffs at the Planck scale, does it go up as space expanded? If yes, how can this be squared with unitarity? If no, would this lead to what Tegmark called the Big Snap where something has got to give sometime. What is that something which gives?
This is a deep mystery in quantum gravity, and it is discussed at length by Banks, and Susskind, among others, in connection with conjectural ds-CFT constructions. This is the central difficulty in describing expanding universes in string theory.
The quantum mechanical number of bits that a cosmological horizon can include seems heuristically to be limited by the area in Planck units. As the universe exapands, this area goes up, so the phase space itself seems to be getting bigger. This is very difficult to conceive, and it seems to be a true paradox, because quantum mechanical evolution can't make the state space bigger.
One way out was suggested by Susskind--- that deSitter space is unstable, and the correct degrees of freedom live on the space it eventually decays into. There is no agreed upon answer, and solving this problem likely requires a good new idea.
I have a feeling the dimensionality of the quantum state space is not conserved in an expanding universe. All our evidence for unitarity comes from experiments conducted at length and time scales far smaller than cosmological timescales. I really don't see any fundamental reason why we should assume unitarity applies to quantum cosmology. We have no right to extrapolate our experimental results to cosmological scales with no good reason. Even if unitarity is not a fundamental feature of our universe, at really short time scales, cosmologically speaking, unitarity might emerge as an approximation just as classical mechanics emerges when the action exceeds Planck's constant, or nonrelativistic mechanics emerges at speeds far less than the speed of light.
As an added bonus, the second law of thermodynamics has a trivial explanation and is no longer a mystery. The big bang was not a highly improbable configuration placed on a knife's edge as Penrose would have you believe. The second law is what comes naturally when the size of phase space goes up.
When the universe expands, the "new" modes that arise can be traced to transplanckian -- i.e. shorter than Planck length -- wavelength modes. What was once transplanckian now becomes redshifted adiabatically to subplanckian wavelengths. I stress adiabatically because this doesn't apply to the nonadiabatic Planck epoch. Of course, many physicists object to the existence of transplanckian modes. Supposedly, the Planck scale acts as an ultraviolet scale cut-off. Anyway, looking at the Wheeler-De Witt constraints, it can be shown the transplanckian modes aren't independent of the subplanckian modes. This ought to sidestep the problem.