### What does the density of vacua predict?

#### Posted by Robert H.

Imagine a room where the floor has the profile

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 $P(x)$, 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
## 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