In ref.  a particular string theory vacuum with positive A was studied. One of the many interesting things that the authors found was that the vacuum is unstable with respect to tunnelling to other vacua. In particular, the vacuum can tunnel back to the supermoduli-space with vanishing cosmological constant. Using instanton methods, the authors calculated that the lifetime of the vacuum is less than the Poincare recurrence time. This is no accident. To see why it must always be so, let us consider the effective potential that the authors of ref.  derived. The only relevant modulus is the overall size of the compact manifold $. The potential is shown in Fig. 16.1. The de Sitter vacuum occurs at the point $ = $0. However, the absolute minimum of the potential occurs not at $0 but at $ = o. At this point the vacuum energy is exactly zero and the vacuum is one of the 10-dimensional vacua of the supermoduli-space. As was noted long ago by Dine and Seiberg , there are always runaway solutions like this in string theory. The potential on the supermoduli-space is zero and so it is always possible to lower the energy by tunnelling to a point on the supermoduli-space.
Suppose we are stuck in the potential well at $0. The vacuum of the causal patch has a finite entropy and fluctuates up and down the walls of the potential. One might think that fluctuations up the sides of the potential are
Boltzmann-suppressed. In a usual thermal system there are two things that suppress fluctuations. The first is the Boltzmann suppression by a factor exp(—f3E) and the second is entropy suppression by a factor exp(Sf — S), where S is the thermal entropy and Sf is the entropy characterizing the fluctuation, which is generally smaller than S. However, in a gravitational theory in which space is bounded (as in the static patch), the total energy is always zero, at least classically. Hence the only suppression is entropic. The phase point wanders around in phase space, spending a time in each region proportional to its phase space volume, i.e. exp(—Sf). Furthermore the typical time-scale for such a fluctuation to take place is of order
Now consider a fluctuation which brings the field 0 to the top of the local maximum at 0 = 01 in the entire causal patch. The entropy at the top of the potential is given in terms of the cosmological constant at the top. It is obviously positive and less than the entropy at 0O. Thus the time for the field to fluctuate to 01 (over the whole causal patch) is strictly less than the recurrence time Tr = exp S. But once the field gets to the top, there is no obstruction to its rolling down the other side to infinity. It follows that a de Sitter vacuum of string theory is never longer lived than Tr and furthermore we end up at a supersymmetric point of vanishing cosmological constant.
There are other possibilities. If the cosmological constant is not very small, it may tunnel over the nearest mountain pass to a neighbouring valley of smaller positive cosmological constant. This will also take place on a time-scale which is too short to allow recurrences. By the same argument, it will not stay in the new vacuum indefinitely. It may find a vacuum with yet smaller cosmological constant to tunnel to. Eventually it will have to make a transition out of the space of vacua with positive cosmological constant.5
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