22. About decoherence in quasi de Sitter space
DmitryMy discussion of disorder on the landscape with Lubos continues (L - Lubos, D - me).
L: So if I understand well, your degree of freedom is really the inflaton phi, and it is one degree of freedom per Hubble patch, with one expectation value and one variation. The value of the inflaton is assumed to directly control the Hubble constant and the size of the patch, according to the normal equations.
D: Yes, I am sure you understood me correctly. What I do is the following: taken a given Hubble patch, I focus only on the dynamics of IR degrees of freedom, that is, the ones with wavelength 
(so that I introduce a coarsegraining). Why can one do this? Physics at superhubble scales may be causally connected during inflation - both particle and event horizons in the de Sitter universe are exponentially larger than 
:
.
After the end of inflation one causally connected patch has the characteristic size , that’s where the notion of Hubble patches actually comes from.
L: So you neglect all higher-frequency modes of the inflaton and other fields which may or may not be a sane truncation.
D: There are two things here: a) I care only about light fields (more or less with 
) and do not care about the others, since they get quickly damped to zero during inflation and b) I do neglect UV modes of the inflaton, because they are in the vacuum state (Bunch-Davies or in-vacuum); particle creation takes place only after a given mode crosses the Hubble scale.
L: Moreover, when you create new causally disconnected bubbles, you probably suddenly increase the number of degrees of freedom of the system, with a new phi per every new patch, don’t you?
D:No, bubbles get causally disconnected only after the end of inflation, so I am happy with that, I don’t introduce anything artificially. It is not actually a new 
in every patch, it is more like IR fluctuation of the inflaton; expectation values of in each patch are averages over ensemble of patches (that is equivalent to spatial average, to average over randomly distributed quantum phase of ).
L: Such sudden jumps indicate that the original truncation was probably not 100% fair, was it?
D: It is unfare in the same sense as description in terms of kinetic equation (or Fokker-Planck) is unfare compared to full QFT description. So, pretty much fare as long as you understand what you are doing (say, you are not interested in exponentially small corrections to the distribution function ;-))
L: OK, let me accept it. It is some sort of minisuperspace approximation.
D: In fact, it is. You may not know it, but Hartle-Hawking wave function is the 
asymptotics of the probability distribution - solution of the Fokker-Planck equation I was talking about last time.
L: Then, it seems that your answer about the volume factors is a resounding No. If the Hubble radius (and patch) is large, it doesn’t make its value of the cosmological constant more likely. It is still some value of the inflaton, one observable which is the only degree of freedom, right? For the same reason, one doesn’t increase the probability of D3-branes deep inside throats. Isn’t it a prediction that inflation shouldn’t occur then?
D: No, as you can easily see by introducing a model with 
. In the limit 
there will be a distribution of effective cosmological constant values among the Hubble patches given by your favorite Hartle-Hawking wavefunction :-), but inflation will continue to run in each patch.
If one takes chaotic inflationary model, where inflation is actually allowed to end, then the probability for inflation to end is finite (say, one can calculate such things as the average number of efoldings which is not divergent).
L: The “funny thing” you are describing is the traditional freezing of the quantum fluctuations, or something else? Strictly, is it OK to assume that such a process is microscopically a result of decoherence?
D: Yes, it is. One neglects decaying superhubble adiabatic mode compared to the growing one. In practice, decaying part of the mode can be neglected after a couple of efoldings after the given mode leaves the Hubble scale.
L: I am confused about various additional kinds of tools, including superHubble fluctuations.
D: People usually say “superhorizon”, I don’t like it since horizon scale is exponentially larger than during inflation.
L: Those of us who tend to believe complementarity think that these modes don’t really exist because all the information behind the horizon is a scrambled gauge copy of the information inside the horizon (of course, it can be a scrambled copy of some degrees of freedom you omitted), much like in the black hole case. Is that OK to conclude that your picture assumes/implies that complementarity can’t hold?
D: I think this is very much in the spirit of cristianity :-) Just add a God who sits on the horizon (or beyond it) and pulls ropes attached to us through dS/CFT (or AdS/CFT) :-) String theorists are Believers in their hearts :-)
Inflation was not described by de Sitter, it was quasi de Sitter, and the causality structure of quasi de Sitter is so much richer than that of de Sitter. My belief :-) is that it was the reason why dS/CFT was unsucessful.
But if by complimentarity you mean unitarity, even in quasi de Sitter, in the end, unitarity should hold; you just do not have an access to the information stored in deep IR modes. It is the same thing one has in kinetics - there, strictly speaking, thermal equilibrium is never reached, and thre are always exponentially small corrections to the Boltzmann law carrying information about all degrees of freedom. Should be the same in the case of black hole, isn’t it?
L: Sorry if I increased the confusion.
D: No, actually I did, I believe :-) Probably, your readers are bored to death at this point.
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Comments
Dmitry (Check me out!) Hi Lubos
Please don’t take my jokes on religion-physics interplay too seriously :-)
Thanks for the explanation of the meaning of complimentarity. Would you say that it holds only for spacetimes with event horizons? Does it hold in the presence of apparent horizons as well?
For example, suppose I am observer in Minkowski space living in the rocket. I than turn on the engine and start moving with acceleration. In my accelerated reference frame there is an apparent horizon and I feel thermal radiation from it. Is there doubling of information somewhere in this setup?
Cheers,
Dmitry.
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Dmitry (Check me out!) That’s the answer from Lubos:
Dear Dmitry, I try to take jokes both as jokes as well as potential inspirations for serious thoughts. Sometimes people tend to laugh at someone, pretending that it is obvious what they should think, but it is often far from obvious.
Various comparisons with religion may have some meaningful core but many other aspects of such analogies could be wrong. And vice versa. These analogies could be laughable in average but many aspects could still be legitimate.
Whether complementarity applies to the apparent horizons? Well, that’s a great question. It must apply to some different horizons than the canonical event horizons - because an evaporating black hole doesn’t really have the event horizon - but because I don’t know the exact algorithm saying how the different regional degrees of freedom depend on each other, I also can’t tell you which horizons had to exist for the overlapping effect to be nonzero.
If someone knows everything, she will surely also answer all of our questions.
I agree that there should be no doubling in your example constructed to have no doubling. But the absence of the phenomenon here might be related to your inability to sustain the acceleration for an arbitrarily long time. The fact that you can’t have enough fuel to sustain the acceleration for too long is a part of holography, I feel - too much fuel creates another black hole, too.
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Dmitry (Check me out!) That’s my thoughts :-)
In overall, I have an impression that in a general situation (apparent horizons, dynamical horizons) some information is doubled and some is not. Absolute complimentarity probably works in the static situation when incoming flow of particles is equal to outgoing flow (then, the doubling is trivial), and there is well defined static event horizon.
Somehow, it all reminds me Schwinger-Keldysh ;-) with its doubling degrees of freedom, but I am not going to make speculations about it right now.
Cheers,
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Dear Dmitry, it was very helpful, thanks! You taught me some things I should have organized in my skull before. Yes, I think you have justified and bounded the validity of your approximation etc.
In the future, I will try to avoid similar errors as misidentifying the horizon size during inflation etc.
Even though I obviously think the same thing as you do about the ultimate success or failure of dS/CFT, it’s the latter, your duality between Jesus Christ and complementarity is not quite clear to me. :-) In fact, I don’t even see any real link between complementarity and dS/CFT.
In my opinion, complementarity, at least in the black hole case, is a rather obvious qualitative feature of information in the black hole background. You can find slices through spacetime of an evaporating black hole at which you cross most of the Hawking radiation as well as most of the matter inside.
There seems to be a doubled information on the slice, and a natural answer apparently needed to avoid a quantum xerox is to say that the two copies of the information are not really independent. In fact, I don’t think that this argument has anything to do with string theory.
Because GR dictates that event horizons in any context should have similar properties, it is natural to extrapolate such a principle of complementarity, to dS, too.