Why isn't Anti de Sitter space taken seriously as a model of reality? I asked a few questions here earlier regarding a physics model which could possibly point to Eternal Return, and was pointed towards Anti De Sitter space.
Arguments in theories for Eternal Return 
Could someone please explain to me why it is not taken seriously?
Are there serious flaws within the model?
Does experimental data simply not support it?
Is there a particular reason why it is not popular? If experimental data does in fact somehow support two models, what makes one model more viable than the other in Physics?
I apologize if this is not mainstream physics, but I did not know where else to ask such a question.
 A: AdS is taken seriously in the sense the AdS/CFT correspondence has been crucial to string theorists. Since the discovery that the universe's expansion is accelerating, de Sitter has looked more promising. You can in principle do string theory with the dS/CFT correspondence instead, but that's not as neat or well-understood.
A: agree with J.G. The reason AdS is an object of interest to theorists is not that it describes the universe we inhabit (for which there is no evidence at all) but that it can be linked to conformal field theory in a way that excites mathematicians and physicists who work on string theory. the AdS/CFT correspondence gives them new tools to apply to that task. 
A: Because of the accelerating expansion of the universe, it is now widely assumed that there is a positive cosmological constant, which implies that we live in de Sitter space. The mathematical alternatives are zero cosmological constant (flat space, Minkowski space) and negative cosmological constant (anti de Sitter space). 
There are still cosmologists studying alternative explanations for the accelerating expansion. So-called quintessence models come to mind, in which dark energy is due to a new field rather than to a cosmological constant. I think that particle physicists have perhaps embraced the idea of a nonzero cosmological constant because particle physics tends to predict a nonzero cosmological constant anyway (because of the vacuum energy of the fundamental fields), and it was always a struggle to explain why a c.c. wasn't seen. Now that the acceleration has been observed, they don't have to think of reasons why it is entirely absent, just why it is small, and they can adapt Weinberg's anthropic explanation and say that it's due to an almost-cancellation of negative and positive contributions to vacuum energy. 
However, if you could find a viable class of particle physics models in which the vacuum energy is zero after all, then maybe you could account for the cosmic acceleration with a quintessence field and space would be flat rather than de Sitter. Now, could you go further and suppose that spacetime is actually anti de Sitter, but a concentration of quintessence is producing a local expansion? 
Certainly I don't know. A literature search reveals a number of papers that have studied inflation in AdS space, in the context of the AdS/CFT duality (which says that physics in an AdS space, should be holographically equivalent to physics in a flat space of one less noncompact dimension). 
There is a claim that AdS inflation implies a "mixed" quantum state in the CFT dual. There has also been some interest in the idea that inflation could occur inside AdS black holes. Whether that would be compatible with standard black hole entropy calculations, or even AdS/CFT duality, I have no idea. It all touches on rather fundamental issues (like how the different vacua of string theory are connected); and I can't tell if these are all just wrong speculations which still have life only because less exciting and more sensible ideas aren't 100% proven yet, or if some of them may be windows on future knowledge. 
Since you express an interest in eternal return - something which most physicists won't be ideologically for or against, they'll just accept or reject it according to what theory as a whole says is possible or plausible - I will say that the creation of a region of expansion in an overall AdS space will be a problem for eternal return, if that expansion never stops; because never stopping means it doesn't go back to AdS, and eternal return would mean that it does. 
All of this says that the idea we actually live in anti de Sitter space is a very exotic one. For it to be true, either something is masking it (like a quintessence field), or we are in a part of it where the local geometry simply isn't AdS any more. 
However, there is one rather weird connection between the physics that we use to describe the world, and AdS, as discovered in 2007. Namely, that "the standard model plus gravity" - which is our empirically driven theory of the world (as opposed to unconfirmed speculative unified theories) - has a stable form in an anti de Sitter space, where one of the dimensions of space has a radius of 20 microns. From a macroscopic perspective, that's "flatland", there are only two large dimensions, and then there's a microscopic third dimension that's about 200,000 atoms thick. 
If, if, you took the idea of inflation in AdS space seriously, you might wonder e.g. if that AdS flatland is the ground state of the universe, and our expanding universe with its three large spatial dimensions represents some kind of fluctuation away from that ground state; maybe inside a black hole in that flatland, if the idea of "inflation inside a black hole" can be made to work. I suppose the idea would be that the black hole forms, an inflating dS-like universe takes shape inside, it expands until heat death, and then particles slipping across the dS cosmological horizon correspond back in the AdS3 flatland, to Hawking evaporation of the black hole. 
All that's total speculation. I only mention it because you asked, and because such ideas don't seem to be completely ruled out yet, so it's still a weird and interesting possibility. But I certainly wouldn't bet on it being true. 
