## September 21, 2005

### What does the density of vacua predict?

#### Posted by Robert H.

Imagine a room where the floor has the profile

(1)$z\left(x,y\right)=\mathrm{sin}\left(\pi /x\right).$

Now, you throw a ball into that room. Where will it end up?

Obviously, the ball will settle into one of the minima of the floor profile, so $1/x$ will be an odd integer. We can consider the density of these minima (a.k.a. vacua) and this diverges for $x\to 0$. So it is most likely that the ball will end up at small $x$.

You will have noticed that this reasoning is flawed: At the same rate as the vacua get denser, the basin of attraction of each vacuum gets smaller, it is bounded by $1/x$ being an even integer. So if we do not have any other a priori knowledge of the distribution $P\left(x\right)$, we should conclude that on the average, small $x$ is as likely as large $x$.

I would expect a similar reasoning also to apply to the vacua of string theory.

My understanding of the philosophy of all this anthropic reasoning is that either by quantum probablity (given in terms of some Hartle-Hawking type wave function) the universe pops into being or via some eternal inflation type process there happens to be a region of the universe that expands exponetially. In both cases, the moduli are randomly distributed and from that point on you make statistical predictions.

(Alternatively, we could live in a world described by a Theory of Really Everything that only has one single state. And that state describes all phenomena including the layout of the blog. As explained here I do consider this unlikely.)

So, our fundamental theory is going to have many states. A small fraction of them will be ground states (still, this might be a tricky concept in a theory containing gravity) where the potential energy has a local minimum. So why assume that the random process sends us to one of these? If it doesn’t excess energy will be radiated off and the expansion of the universe also acts like a dissipating system with the Hubble constant giving the amount of friction. So eventually, the universe will settle into (or close to) one of the ground states.

But as the example of the ball above shows, they are not equally likely. In general, a greater density will be counter balanced by smaller regions in parameter space that will settle to that ground state. As long as one does not have a precise understanding of the detailed dynamics of the settling down process, I would assume that by far the most important factor in the determination of the probability distribution of the physical parameters (moduli) is the a priori distribution of states (which is of course unknown) and not the distribution of vacua.

Posted at September 21, 2005 2:03 PM UTC

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### Re: What does the desity of vacua predict?

“My understanding of the philosophy of all this anthropic reasoning is that either by quantum probablity (given in terms of some Hartle-Hawking type wave function) the universe pops into being”

Umm…I don’t think that Prof Tye will be too pleased about this. His objective is precisely to counter the anthropic principle using his version of the HH wavefunction…see
http://arxiv.org/abs/hep-th/0406107

Posted by: Jack on September 22, 2005 2:14 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

I haven’t read that paper. But I have very briefly looked at it. Is it correct that the main point is that the authors take into account cosmological decoherence induced by fluctuating modes and use that to argue that the tunneling probabilities between minima of the landscape are less than might be naively expected?

Posted by: Urs on September 22, 2005 3:48 PM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

Something even stronger than that I think: Tye wants to compute which spot in the landscape you are most likely to find yourself. If the probability is very sharply peaked, and if you don’t find yourself in the place predicted, then your theory is falsified. See Tye’s talk at Strings05. The decoherence stuff is mainly to get the HH wavefunction to say something sensible.

Posted by: jack on September 23, 2005 3:43 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

Imagine a room where the floor has the profile $z\left(x,y\right)=\mathrm{sin}\left(\pi /x\right)$. Now, you throw a ball into that room. Where will it end up?

Looking more closely at that ball, I see that it vibrates. Hence I wait a while after throwing it. After that I expect it to end up in a region of size

(1)$\Delta =\sqrt{\frac{{\int }_{0}^{1}\phantom{\rule{thinmathspace}{0ex}}\left(x-⟨x⟩{\right)}^{2}\mathrm{exp}\left(-\mathrm{sin}\left(\pi /x\right)\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}}{{\int }_{0}^{1}\mathrm{exp}\left(-\mathrm{sin}\left(\pi /x\right)\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}}}$

around

(2)$⟨x⟩=\frac{{\int }_{0}^{1}\phantom{\rule{thinmathspace}{0ex}}x\mathrm{exp}\left(-\mathrm{sin}\left(\pi /x\right)\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}}{{\int }_{0}^{1}\mathrm{exp}\left(-\mathrm{sin}\left(\pi /x\right)\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}}\phantom{\rule{thinmathspace}{0ex}}.$
Posted by: Urs on September 22, 2005 11:14 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

Why do you integrate from 0 to 1? I had in mind the room to have $x$ from 0 to [size of the room]$\gg 1$.

Posted by: Robert on September 22, 2005 11:20 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

I had in mind the room to have $x$ from $0$ to [size of the room] $\gg 1$.

Whatever you like. (As long as these integrals exist, at least.) More generally one should also allow for an ‘inverse temperature’ coefficient $\beta$ in the exponent.

I guess my point was that mechanisms like ‘eternal inflation’ are somehow expected to lead to some kind of thermalization, which makes our expectation of which vacuum is chosen independent of the precise initial state.

So when you write

I would assume that by far the most important factor in the determination of the probability distribution of the physical parameters (moduli) is the a priori distribution of states (which is of course unknown) and not the distribution of vacua.

what you are addressing seems to be the question if or if not a ‘thermalization’-like process plays a role in vacuum selection or not.

Last time we talked about this, I think Aaron was emphasizing that we can expect all we want, if the ball ends up in any one vacuum that’s how it is, no matter which probability we assign to that outcome.

That’s of course true, as for every statistical theory.

Posted by: Urs on September 22, 2005 11:30 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

In the example given you generically lose global information (eg what other vacua exist) when you settle for some local minimum and you most certainly have no idea what P(x) is.

One would presumably invoke the weak anthropic principle to justify the fact that the situation has been settled, and thats more or less reasonable (although it sucks for predictivity).

What is less reasonable is to then try to work your way backwards (the strong anthropic principle) and try to reconstruct bits and pieces of P(x) or global characteristics of f(x) given your knowledge that you ended up somewhere special (the place that gives life). Certainly a clash with the copernican principle, but indeed this is what people have been doing in order to make *anthropic* predictions.

Of course, its more elegant if you guess P(x) in a way that is mechanistically driven or that satisfies some additional desirable attributes (this would be a loose analogy to the HH process). Then of course we will feel good as theorists if it generically picks out, or gives a high probability that we are in the correct vacua. But I mean no such extra *features* need exist a priori.

Posted by: Haelfix on September 22, 2005 3:57 PM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

One would presumably invoke the weak anthropic principle to justify the fact that the situation has been settled, and thats more or less reasonable (although it sucks for predictivity).

So to stay in the example, let us assume some experimenter has thrown that ball into that room. Now he makes measurements to determine its position.

But his measurement apparatus is imperfect. For one, he knows that it will be able to detect the ball at all only if that comes to sit at some $x\in {I}_{1}\subset ℝ$ for some subset ${I}_{1}$ of the positive reals.

Hence he states his version of the anthropic principle as

The mere fact that I have detected a ball in that room at all means it must be in a minimum such that it can be detected by my measurement apparatus.

Next, he wants to gain more information on the position of that ball. While his current measurement results are not precise enough to identify exactly the minimum that the ball is sitting in, they are consistent with the ball being, say, in the subset ${I}_{2}\subset ℝ$.

But a better measurement device will be delivered soon, so he wants to make an educated guess, or even a prediction, where in ${I}_{1}\cap {I}_{2}$ the ball is most likely to be found.

Now what?

Posted by: Urs on September 22, 2005 4:34 PM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

My point was that some physicist in the ball in room situation could come up with a fundamental theory that predicts the profile of the floor (and as we know gets it right).

Then he might conclude that in his theory the smaller $x$ part in the observable interval is prefered as there the desity of vacua is larger. He would call the conclusion “we will observe small $x$” a prediction of his fundamental theory.

As I argued in the original post, this conclusion is wrong and the profile does not prefer small or large $x$ as long as $1/x$ is an odd integer.

Posted by: Robert on September 23, 2005 10:00 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

So you would say, given the above scenario, that the ball is equally likely to be found in any minimum inside ${I}_{1}\cap {I}_{2}$.

I was trying to understand what Haelfix’s comments would mean in this toy example.

Posted by: Urs on September 23, 2005 10:09 AM | Permalink | Reply to this

### Re: What does the desity of vacua predict?

I would greatly appreciate your response.