The theory of inflation, which postulates an epoch of super-accelerated expansion at some time in the past, purports to explain a wide range of cosmological observations. We know that our universe today is very flat, isotropic, and homogenous. We know that the CMB (Cosmic Microwave Background) is the same temperature within one part in 10^5 throughout the sky, and we have not observed any of the magnetic monopoles predicted by Grand Unified Theories. Inflation produces these features naturally, because a generic initial state will be diluted by the cosmic expansion. However, the theory is almost too good, because it leaves essentially no information about the state of the universe before the inflationary epoch.
So, what happened before inflation?
Before we tackle this, one question that must be addressed is: how did the universe get from a state of super-accelerated expansion to the slower expansion we observe today? Another question that must be answered is: what kind of stuff can produce the accelerated expansion? The original model of inflation, put forward by Guth, provided a possible answer to these two questions by postulating a scalar field theory with two phases. The scalar field in this model is the "stuff" of inflation. It has the peculiar feature of producing negative pressure, and so if it permeates the universe (i.e. has a vacuum expectation value), it will cause accelerated expansion (ie inflation, and/or the slower accelerated expansion we observe today). The model has two phases, one where the vacuum energy of the scalar field is high (called the false vacuum) and one where it is low (called the true vacuum). A phase transition can occur (originally postulated by Coleman), which proceeds by the nucleation of bubbles of true vacuum in a (inflating) universe of false vacuum. The original model of Guth was proven to be phenomenologically unsatisfactory, but this sort of phase transition offers a nice answer to the two questions we posed above.
This simple model may also provide some insight into our original question, which can be rephrased as: how did the universe end up in the false vacuum? When the true vacuum has a positive, non-zero vacuum energy (cosmological constant) there is a clear-cut answer to this question: create bubbles of false vacuum in a universe of true vacuum. So, in this simple model there is a mechanism which allows for the transition between universes with different cosmological constants (expansion rates) by the nucleation of vacuum bubbles. Taking this model seriously suggests that the very-ancient history of the universe may have had many inflationary epochs. Indeed, the universe we observe may just be one part of a multiverse, containing many spatiotemporal regions where parameters, including the cosmological constant, take on different values.
There have been three seemingly different mechanisms discussed in the literature for the fluctuation of true and false vacuum bubbles: the Farhi-Guth-Guven mechanism (FGG), the nucleation of Coleman-De Luccia (CDL) bubbles, and thermal activation (Garriga and Megevand). All of these processes assume that the bubble wall is very thin compared to its radius.
The FGG mechanism consists of a few steps. First, a small vacuum bubble (which would classically reach a turning point and then collapse into a black hole) is formed in its expanding phase. At the turning point, this bubble quantum tunnels into a larger bubble which exists on the other side of a worm hole and houses a universe of true or false vacuum. An external observer in the old phase will see a black hole after this process has been completed. This mechanism is rather puzzling, and has some interesting connections with quantum cosmology and the wave function of the universe.
Coleman-De Luccia vacuum bubbles are solutions which have zero mass, with the bubble wall energy canceling the volume energy exactly. These bubbles are formed in the background phase by a quantum tunneling event and expand, eating up the old phase as they do so.
Thermally activated bubbles are massive bubbles formed by fluctuations in a spacetime with a positive cosmological constant. These solutions are static, nucleating in unstable equilibrium between expansion and collapse.
A. Aguirre and I have recently submitted a paper introducing a new mechanism, whose zero-mass limit corresponds to CDL, and whose high-mass limit corresponds to thermal activation. This new mechanism avoids many of the problems with FGG, and so may be an alternative to this process.
Another project I am working on is studying thick-wall instantons with T. Banks and A. Aguirre. The nucleation of zero-mass (CDL) bubbles can be described by the evolution of a field in an inverted (Euclidean) potential as shown in the figure above. Solutions where the field interpolates between the neighborhood of the true vacuum well and the neighborhood of the false vacuum well (and satisfy a number of other boundary conditions) are called instantons.
T. Banks and I have been exploring instantons with a false vacuum having positive vacuum energy (described by de Sitter space) and a true vacuum having negative vacuum energy. We originally claimed (hep-th/0512141) that as the false vacuum energy density is taken to zero, there is no instanton which approaches the flat space solution. We claimed further that there was an instanton which interpolates between the potential wells, but the probability to make this transition goes to zero as the false vacuum energy density goes to zero. In a subsequent publication (hep-th/0603107), we showed (and Bousso et. al. showed simultaneously) that the above claims are false, but in an interesting way. For each potential, we have found that there is a critical value of the parameter controlling the importance of gravity above which the tunneling rate goes to zero in the flat space limit and below which the tunneling rate smoothly approaches that of the flat space instanton. Exactly at the critical parameter, we find that there is a static domain wall solution. Since we have tuned only one paramter, there is a "Great Divide" in the space of potentials of codimension one. We are currently exploring the implications of this scenario.
I have performed a numerical study of these instantons, the details of which can be found in the pdf below (Note: These results are only for potentials above the Great Divide). Also linked below is a Mathematica notebook, which contains a few sample calculations.
A. Aguirre, S. Gratton, and M. C. Johnson, Hurdles for Recent Measures in Eternal Inflation, hep-th/0611221 (2006).
T. Banks and M. C. Johnson, Regulating Eternal Inflation, hep-th/0512141 (2005).
T. Bogdan et al., Waves in the magnetized solar atmosphere II: Waves from localized sources in magnetic flux concentrations, ApJ, 599, 626 (2003).
T. Bogdan et al., Waves in magnetic flux concentrations, Astronomische Nachrichten, 323, 196 (2002).
I will be the Head TA for the physics department beginning Winter quarter, 2006.
The handouts distributed during this year's TA training can be found below:
Some helpful links:
A link to the physics department's narrative eval site.
I will not be teaching Summer/Fall 2006.