15. Anderson localization. Just crossed my mind…

I personally consider Anderson localization as one of the most prominent dynamic vacuum selection principles on the string theory landscape. Let me remind you the main idea behind it.

In condensed matter theory, Anderson localization was first discovered as a phenomenon of electron wave function localization inside a semiconductor when disorder (for example, impurities or defects) of the effective potential electrons are moving in is sufficiently strong. Physics of Anderson localization in a semiconductor is related to existence of impurities with potential so strong that they bind the Bloch wave function of the electron propagating in a disordered medium. Mechanism responsible for the localization of the single electron wave function near such impurities is interference between Bloch waves scattered by impurities (defects).

In a few words, in a semiconductor the probability to find an electron near the localization center (impurity with strong attracting potential) is much higher than to find it somewhere in the bulk far away from impurities.

Likewise, the wave function of the Universe on the string theory landscape can be localized near some vacua if disorder on the landscape is sufficiently strong. In particular, there is a chance that it will be localized in vacua with very small positive cosmological constant , so that smallness of the value of CC is dynamically preferable, and one does not need to follow the logic of anthropic principle.

Now, this idea was discussed in details by Henry Tye and Laura Mersini in the context of WdW equation with random potential (the latter defines behavior of the wave function of the Universe); also, I have discussed it with Kari Enqvist in the context of eternal inflation.

Henry’s idea is the following. Suppose we have an inflaton in some very complicated potential with several minima (and stochastic force is weak, so that inflation is not in the eternal selfreproducing regime). If barriers are high between the minima are high, inflaton can quantum-mechanically tunnel between the minima (the rates of tunneling are given by Coleman-de Luccia instantons). If the potential is significantly disordered, there could be effect analogous to Anderson localization, and the inflaton will be localized near the deepest minimum of its potential (localization center).

Now, in Henry’s picture everything happens in the slow roll deterministic regime, since Henry does not like eternal inflation. However, the question is whether it is necessary to have it to reproduce our reality with small cosmological constant and flat power spectrum of inflationary perturbations.

I have just realized it is actually necessary to have eternal inflation. In slow roll deterministic regime inflaton of course can tunnel between the minima of its potential, but if a tunneling happens within last 60 efoldings of inflaton, it gives rise to features in the power spectrum. Indeed,

and suddenly changes during the tunneling. As a result, one gets a jump in power spectrum at , where is the value of scale factor coresponding to the moment of tunneling. Of course, we don’t see such features in the power spectrum.

The bottom line is that evolution of the inflaton should be smooth during last 60 efoldings, i.e., no tunnelings are allowed.