21. About probabilities in cosmology (and general) and in stochastic inflation in particular
We have a great discussion on the Lubos Motl’s blog about probabilities on the landscape and our recent paper. Since Lubos’ questions require a long answer, I decided to give it here instead of his blog (I hope Lubos will forgive me for exploiting his traffic a bit :-))
Discussion was initiated by the question “what is the physical meaning of time 
in vacuum dynamics equations” that was smoothly transformed into the question “what is the meaning of probability 
in the vacuum dynamics equations”. My answer was that is the world time for the 4-dim observer and is the probability for him to measure a given value of the cosmological constant. Lubos replied:
“But it is still difficult for me to follow the sentence “If P is the probability to measure a given value of the cosmological constant in a given 4d Hubble patch”. Why? Because if one considers a “given” 4D Hubble patch, it should already have a well-defined i.e. “given” curvature, shouldn’t it? What does it mean to have a “given” patch whose curvature (and volume and radius) are uncertain i.e. “non-given”? It sounds like a letter from a car shop saying “once we *give* you a car for your money, there is some probability that you will actually receive the car and some probability that it will be a Trabant.” That’s not how I imagine giving a car, especially after they were given the money. This comment of mine may sound as a silly game with words. 
”
The example with car does not quite work because dynamics of the inflaton field in the regime we discuss is stochastic. Imagine we are in QFT framework (do not know anything about string theory). There is a single inflaton field 
with some potential 
satisfying slow roll conditions. During one Hubble time 
the IR (i.e., superHubble) fluctuation of the inflaton value is generated, it has characteristic value 
. On the other hand, inflaton is affected by the classical force (gradient of its potential), and its value is decreased during one Hubble time because of the action of this classical force by 
.
Now, the question is: what happens in the regime when 
? Quantum fluctuation can kick the inflaton value both towards higher and lower values of . (This regime is called eternal inflation, as you know.)
Moreover, this IR contribution 
into the propagator of is nearly classical (you can calculate commutator with 
for the IR mode and see that it rapidly approaches zero while 
). So, what is the actual value of 
since it is classical?
It turns out that funny thing happens: is classical but stochastically distributed with the distribution width 
(the latter of course changes with time because 
is the function of ). To say more accurately, the expectation value of the inflaton 
is governed by the Langevin equation where stochastic force is fluctuations which go outside Hubble volume and become classical. The moment for a given mode of to cross the Hubble scale is random (since the given mode is quantum, and its phase is rapid variable). After crossing the Hubble scale quantum phase of the mode freezes, and this leads to stochastic description above.
What happens physically? Suppose inflation starts from some hypersurface of constant . Then, it enters eternal regime; in different Hubble volumes (the ones that get causally disconnected after the end of inflation) behavior of is different, so is behavior of which is the function of . That is the ensemble we average over.
Now, since behavior of is determined by the Langevin equation, you can immediately write down associated Fokker-Planck equation for the probability to measure a given value of for a given Hubble patch. If you take a look on this Fokker-Planck equation, you will easily see that it is nothing but a continuous version of the vacuum dynamics equations we discussed.
I hope I did not introduce too many technicalities 
“Except that I think that the uncertainty hiding in the definition of “given” and “probability” includes an uncertain factor of the Hubble volume and similar factors that lead to ramification of various anthropic papers. Are you claiming that you can settle these questions and decide whether your P should be proportional to the Hubble volume in Planck units or not? What are the rules that decide about the presence of the volume factor here”
I think the rules are not actually quite established. In the Fokker-Planck I discussed above there is no volume factor, the theory itself gives you FP for non-volume weighted measure (note that in our paper we don’t use volume weighting). Volume-weihting was introduced later by Sasaki et al. pretty much by hands. The logic was the following: we should take somehow the volume of exponentially expanding patches into account, since the largest patches clearly cover the most volume of the Universe.
There is a serious associated danger here. It is due to the fact that patches with Planckian energy density of the inflaton field expand most rapidly, and according to volume-weighted logic should cover the most of the Universe. Since description of inflation in terms of QFT breaks down when 
, you, yourself, by your own hands put your description outside the region where it is valid by introducing the volume-weighted measure.
As I understand, there are other sins associated with volume weighted measure. For example, is generally speaking not normalizable (the main contribution to normalization integral comes from large and large energies where you have no idea what happens), probability is not conserved (there is a flow towards high energies) etc.
“Concerning the last point, sure, I understand that you get inflation but my real question is what happens with the probability distribution if the inflationary era begins. This question is related to the previous one because inflation will lead to an exponential increase of various volumes so I wonder whether this will make the states with D3 branes deeply immersed in the throats much (exponentially?) more likely or not. It is still about the way how the Hubble volume and related quantities influence the “probability” of having various values of Lambda. It seems that there is no unity among the researchers on similar issues and so far I tend to think that no justifiable method to settle this huge uncertainty is known.”
I would expect that states deeply immersed outside the throats are much more likely. Eternal inflation in KKLMMT happens at higher , i.e., closer to the base of the throat. So, that is where the most volume (for 4d observer) is gained. Also, stochastic force I was talking above wants to kick D3 out of the throat.
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