Patches of normal space randomly condense out of the false vacuum of inflating space. What is the average distance between these patches? Another words, what is the average number of bubble patches per unit volume of inflating space?

I know we can't see other bubbles. I'm asking what Eternal Inflation Theory predicts.

I know false vacuum decays at random, but can you give me an average distance between bubbles, or an upper and lower bound on the distance?


1 Answer 1


The short answer is we don't know.

Firstly saying anything about the landscape is difficult because it's so complicated. So everything in this answer is based on the random Gaussian field approximation. If you've not encountered this before, it takes the landscape and approximates it as a random Gaussian field. This random Gaussian field is in turn the N-dimensional generalization of Gaussian distributions. If we have a hundred random, independent fields and add them up, by the central limit theorem, they tend towards a random Gaussian field. Note there is no need for the landscape to resemble a random Gaussian field, it just provides a baseline.

Under this approximation, there hasn't yet been a study (to my knowledge) of the distance between minima, although a few studies in particular have come close:

Charting an Inflationary Landscape with Random Matrix Theory establishes that inflation does not last long. Of course, inflation happens when one stationary point is a saddle, not a minimum, so it's not exactly an indication of the distance between two minima.

Vacuum statistics and stability in axionic landscapes establishes that minima with high energy density are inherently unstable. This result is intuitive, because the higher the energy density the fewer minima there are, and the minima that do exist would be shallow (and therefore unstable). But it still doesn't say what the distance between two minima is.

Hessian eigenvalue distribution in a random Gaussian landscape and The Distribution of Vacua in Random Landscape Potentials both touch on the expected eigenvalues (i.e. slopes) at minima. Presumably the distance to the next minimum is directly related to the slope out of the current one, although the exact relation remains to be worked out.

Finally, Counting Vacua in Random Landscapes gives some details on the number density of minima and saddles. The greater the density, the less the distance to the next minimum. However, it does not reach 100 dimensions.


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