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The argument for eternal inflation is we have some patch of metastable vacuum with positive cosmological constant, and so it expands exponentially a la de Sitter. Most of the patch decays to something which expands more slowly, but only a tiny fraction continues to expand at that fast exponential rate. The magic of exponentiality is despite only being a tiny fraction, it still dominates, and this process continues forever.

OK, but add in causal patch complementarity, and well, only the causal patch counts, right? The fraction counting is no longer admissible because we can only measure the fraction within the causal patch, and any observer will always exit the inflationary phase with probability 1. What gives?

The spatial multiverse presupposes there is such a thing as outside the patch. Is that a no no? The multiverse of different string compactifications, well, how many of them are there inside a causal patch simultaneously? As for many worlds multiverses, is many worlds even true?

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No they are not compatible. Eternal inflation is dead because of this, but this is not acknowledged by eternal inflation advocates. This is a good thing, because eternal inflation has impossible measure paradoxes, noted by Guth, which are the cosmological analogs of the black hole information paradox. If you have an infinite amount of universe outside the patch, shouldn't we be in a generic point, and so maximize the time of inflation? Blah blah blah. Causal patch complementarity fixes these issues. There's nothing outside the horizon.

This point of view is sort of 90s folklore, at least, it was clear to a lot of people back then, just from the holographic principle, but it was never popular with cosmologists. Susskind says it now (quietly). Polchinski even made a replacement for the eternal inflation measure which is holographically ok. He measure the probability of us being vacuum by the probability of a given worldline ending on it, not by the measure of the volume it occupies.

EDIT: I found this informative article by Tom Banks (Lubos Motl's advisor, and they are very similar in spirit, although Lubos seems to disagree with him on eternal inflation): http://blogs.discovermagazine.com/cosmicvariance/2011/10/24/guest-post-tom-banks-contra-eternal-inflation-2/

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The whole point of "complementarity" is that there are two (or more) equally valid ways of looking at something. Take a black hole, for instance. There is an outside perspective which describes in-falling matter as getting squished on the surface and then there is the equally valid perspective of the in-falling matter. Thus, we have two different descriptions of the same Hilbert space. In the case of inflation, "we" as observers can only measure things in a given universe, but "we" with the godlike perspective afforded by mathematics may describe the multiverse in other, more democratic ways. Although it has been suggested that the holographic principle implies that the hilbert space of the universe inside a de Sitter horizon should be equal to that outside, I don't think that it can be said that this is a necessary conclusion, and, in any case, this wouldn't invalidate the complementary perspective of viewing universes from the "outside". Indeed, inflation, which is well established by CMB spectra, essentially predicts the existence of regions of the universe outside our causal patch. It just so happens that for the sake of developing a probability measure some people have found it more consistent to work entirely within a single causal patch. This doesn't necessarily make the fraction counting idea inadmissible; indeed, any measure is simply a more formal kind of 'fraction counting'. I don't think one can say, however, that this issue is closed since there is still a debate about which measure is the most appropriate.

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Inflation predicts no such thing--- it just predicts that the universe had a large CC in the past of our causal patch, making that patch a small deSitter thing way back when. The statement that deSitter horizons need to be treated like black hole horizons follows from the Nariai limit, where the two switch roles. It is also strongly suggested by the thermalizing and bounded character of dS, which does not allow you to do an unbounded computation inside. It's really right, and the current point of view you hear from most cosmologists is really bullshit. –  Ron Maimon Oct 26 '12 at 20:58
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