if I understand your question correctly, the present stage of the evolution of our Universe is quasi-de Sitter (albeit the cosmological constant is extremely small). If so, there is en entropy associated with superhorizon degrees of freedom in this (quasi) de Sitter Universe.

Cheers,
Dmitry.

What if to assume that the entropy product is c.c.?

Dmitry.

I don’t know whether AdS/CFT really “proves” BH complementarity but in the pictures I can imagine, it directly incorporates it. It’s because a region on the boundary is only “dual” to some physics inside the spatially separated region in the bulk – from where it can receive returning signals, so to say.

And only the black hole exterior is connected with the boundary in this way. The boundary contains all the information about all the bulk, however, so the BH interior is encoded in the boundary observables, too. But we just said that the BH exterior bulk is equivalent to the boundary, which means that the BH interior data are encoded in the BH exterior data.

I am not aware of any more rigorous proof than the handwaving above but I surely do believe that their statement is true.

I have a question: DLS say that BH complementarity picture is supported by AdS/CFT. How exactly does it work in the AdS/CFT setup?

Cheers,
Dmitry.

de Sitter horizons are observed-dependent but that is a simple consequence of the absence of the asymptotic region in the dS space.

The “behind the cosmic horizon” volume is analogous to the black hole interior, while the observers outside the black hole become the observers inside the dS patch.

Two observers outside the BH agree about the BH horizon because they define it relatively to “scri+” where the radiation can eventually end. Can a point P send signals to “scri+”? If it cannot, it is inside the BH, below the horizon.

On the other hand, in dS, the horizon is defined by being able to send signals to a particular point at the top of the diagram which is horizontal, spacelike. There are many points like that, and the horizon only exists if you pick a specific one.

But it’s still true that things behind this horizon are causally disconnected and will never get connected. It seems to be the key thing that makes the two cases analogous.

So I feel that the dS space is bound to be “more fuzzy”, “more finite”, and “more undetermined” than the black hole horizon, but all the issues that existed in the latter also exist in the former.

These are speculations because we don’t have any full description of dS physics. Whether such a description has a manifestly finite number of degrees of freedom, or whether the number is infinite and only “effectively finite” remains to be seen. At any rate, certain unpredictivity of dS space is probably guaranteed to exist because you never know what is exactly radiated towards you from the cosmic horizon.

This problem is, of course, not a practical one because the dS horizon thermal radiation in the real cosmos is outrageously negligible.