- The probabilities in the landscape.
The probability distribution for different vacuum-depended quantities such as the cosmological constant is the product of
- P = Prior times SelectionFactor
Prof. Vilenkin explained that it is the SelectionFactor that drives many people up the wall. Well, it may be fair to say that I am driven up the wall by both of these factors. :-) The talk focused on the first one, the Prior.
The speaker showed beautiful colorful diagrams of the pocket universes. Each pocket universe starts as a bubble that expands, and inside this bubble, it looks like an FRW universe to the inhabitants but there can be many new bubbles inside each bubble - like M&M's inside bigger M&M's (or electrons inside electrons which are Universes, as Lenin believed).
In the causal diagram, you have the upper wedges attached to the far future (a horizontal line) representing the bubbles, and smaller bubbles inside them. The pictures are entirely hierarchical and there is no bubble collision in this model or any other causal relation between the different bubbles.
The task is to find the right measure that tells you which fraction of the volume of the branching bubbling multiverse contains one type of vacuum or another. The speaker (and Nima) explained that it is easy to create diffeomorphism invariant measures based on geodesics and proper distances, but there are infinitely many such definitions.
Their particular new breakthrough is based on the following construction. Take a minimal-surface 3D spatial slice across your universe at any moment. Fill it with many points. Draw geodesics from all these points directed into the future (that start as lines orthogonal to the slice). Each geodesic, as it continues to the future, will be visiting many types of universes and bubbles. You measure the relative abundance of a given type of a universe: Vilenkin argued that such a definition can be quantified with a finite length geodesic and the limit where the geodesic proper length goes to infinity exists.
Toby Wiseman did not like that the geodesics were treated as special trajectories. Well, I tend to think that geodesics are special and there are other laws of physics in which we have good reasons to think that the geodesics differ from other, more generic curves in spacetime. But I am not sure whether the anthropic probability distributions belong to the set of physical laws in which we know that geodesics play a special role. ;-)
Nima had some doubts about the existence of the limit. He was uncertain whether it is true that one half of integers are even (and thinking about other examples where the ratio could be ill-defined). Vilenkin argued that it is OK to say that one half of integers are even, and the limit in his construction is equally well-defined.
The final "prior" determining the weight of a given kind of vacuum in the probabilistic distribution (the prior) is obtained as the product of the relative abundance of a given vacuum along any chosen geodesic, explained above, and some reference scale volume "R^3" of the universe in a given bubble:
- Prior = RelativeAbundanceOnGeodesic times R^3.
The speaker has also mentioned other definitions of "Prior" that lead to different results.
Using the terminology from this blog's controversies, Cumrun Vafa protested and argued that one cannot operationally define these conjectured "probabilities" because they don't admit a frequentist definition. Since they are only based on untestable Bayesian reasoning, they should not be subject to scientific thinking.
One general "positivist" philosophical lesson of quantum mechanics, Cumrun argued, is that we should only talk about quantities that can be measured. As you can imagine, I knew very well where Cumrun was coming from, so that I can even add the right philosophical buzzwords to Cumrun's arguments :-), while Nima disagreed. Nima argued that we should replace the word "probability" by the word "confidence" and everything would be OK.
As far as the anthropic reasoning goes, I agree that such a confidence is calculable and the result that I obtained is zero. Don't worry: it is not a no-confidence vote; what we see is just a lack of confidence. ;-)
Vilenkin showed that their more refined analysis questions the uniform probability distribution of the cosmological constant in Weinberg's anthropically allowed anthropic interval "(-L,+500L)" where "L" is the observed value. The famous anthropic physicist from Tufts University argued that there could exist peaks and they could even exhibit a fractal structure.
Nima got a bit nervous because this was potentially bad news for the anthropic principle because this fact would naturally predict a higher cosmological constant than the observed one, and he immediately offered a one-sentence argument that everything is OK anyway :-), an argument that I cannot reproduce because I did not understand it.
Vilenkin said that by this single sentence, Nima has essentially scooped the speaker and presented the rest of the talk as Vilenkin planned it. ;-) On the other hand, Nima's argument was based on the hypothesis of a very large number of Universes, and Vilenkin said that this is not what the argument should be based upon.
Nima speculated that with a large number of universes the fractal structure can't arise because of the ergodic argument. In statistical mechanics, one must also assume the ergodicity as a first principle that can't be proved, so it is the same thing. Fortunately, when I mentioned that ergodicity can be proved for any particular ergodic statistical mechanical system - and when Vilenkin confirmed that point of mine - Nima agreed that the situation is not the same one after all. ;-)
Nima also added many interesting stories to this colorful seminar, including the observation that Nima was neither Chinese nor a bacterium, in a contradiction with the generic anthropic argument. For example, he also told us that there exists a program that can calculate whether the success of a physicist is a random fluctuation. The probability for Edward Witten is "10^{-192}" which is a rather small number. Given this non-generic fine-tuning, I asked Nima whether Witten should believe the anthropic principle even though the generic predictions fail so miserably for him. Nima essentially agreed that the answer is No. My next argument is - if Witten does not believe, why should the rest of us believe it? ;-)
Maybe the intelligent life should be defined to be just Edward Witten himself, which leads to predictions that differ by 192 orders of magnitudes from the anthropic predictions if some details are modified. There were other uncertainties in Vilenkin's calculation that could change the results by 240 orders of magnitude: for example, Vilenkin mentioned that some people do not include the volume into the probabilistic measure. A funny aspect of these discussions is that one can't quite distinguish which of the considerations are jokes and which of them are meant seriously. At least I can't distinguish them.
Earlier during the talk, Cumrun Vafa mentioned examples that lead to the same conclusion. He argued - and once again, I completely agree with him - that there are quantities whose values are non-generic because extreme values are needed for intelligent life. The amount of non-genericity of these quantities simply measures how lucky we are and good luck can't be used as a starting point for a quantitative calculation or justification of anything.
Vilenkin disagreed and responded with the example of the CMB quadrupole moment. It is just one number. Does it mean that it has no scientific value? Imagine that it is 20 sigma from the value predicted by your theory. Does it mean much to you? Needless to say, the answer of all colleagues who enjoy the Bayesian and anthropic reasoning is Yes. The answer of Cumrun and mine is essentially No. Science only becomes quantitative if its experiments are repeatable and its numerical predictions can be made increasingly more accurate. That's neither the case of the probability distribution in the landscape nor the case of a single multipole moment.
We just can't build the whole science on one number whose probability is "P", even if we would guess that a fundamental theory predicts a rather small value of "P". The reasoning based on one number is numerology. Science is about predicting a large number of quantities, preferrably an infinite number of quantities. If a theory only predicts one number, then it's not really a scientific one.
These debates have occured many times and most of us are becoming insensitive to the built-in controversies. The frequentist positivist people like Cumrun or me are convinced that this reasoning can't lead anywhere, and the anthropic people keep on trying. Good luck to them. ;-)
