145. Quantum and thermal decay in de Sitter space

Yesterday there was an interesting talk at Perimeter Institute – Adam Brown from Columbia University has discussed vacuum decay in de Sitter space. The title in the announcement immediately caught me, since I expected that Adam was actually going to discuss instability of de Sitter space, the subject I am currently working on. Then I have realized that Adam probably is to talk about his recent paper with Eric Weinberg, well, not decay of de Sitter space but still – very interesting.

So, what is this story about? The question Adam wants to ask is very simple. Let us consider a de Sitter space and a scalar field living in it.

The scalar field has a potential with two minima, 1 and 2, at and correspondingly separated by a potential barrier between them of the height H. Corresponding vacuum energies are different.

We prepare an initial condition such that the classical scalar field is situated near the minimum 1. What is the probability for a scalar field to tunnel from one minimum to the other?

We actually have two pictures giving two different answers. One is the picture of Coleman and de Luccia: the field tunnels quantum-mechanically (through the bubble nucleation). The rate of this tunneling is given by the Coleman-de Luccia instanton. Another picture is by Hawking and Moss (as well as Linde and Starobinsky). During one Hubble time fluctuations of the scalar field are generated with the characteristic amplitude of the order . These quantum fluctuations can pull the classical background value of the scalar field towards the top of the barrier, and the field will easily roll down to the minimum 2. The rate of corresponding tunneling is given by the Hawking-Moss instanton (different from the Coleman-de Luccia instanton).

Adam calls the second type of tunneling thermal and that is why. If instead of the planar patch you consider de Sitter in the static patch, the Hawking-Moss tunneling happens due to the thermal activation of the scalar field by its quanta (of Hawking radiation) coming from the de Sitter horizon. Both calculations – in static and planar patches – give the same expression for the Hawking-Moss instanton defiing the tunneling rate.

So, the question Adam wants to answer is the following: are both answers (Coleman-de Luccia and Hawking-Moss rates of tunneling) correct or only one of them is? If both of them are correct (of course, they are, but in different regimes – for example, if the barrier is very high, thermal activation is not sufficiently strong to provide fast tunneling, while the possibility of the CdL tunneling is always there), then how does one result  transform into the other?

The answer that Eric and Adam provide is interesting – there is an optimal point in the potential such that first is getting dragged by the thermal fluctuations up the barrier until it reaches the point . Then, the field tunnels according to the Coleman-de Luccia mechanism towards the minimum 2.  If the barrier is very high, then the optimal point lays really close to the position of the first minimum – and we just reproduce the standard CdL tunneling rate.