Quantum tunneling in flux compactifications

I am very happy to find myself writing a blog about a recent paper written by Jose Juan Blanco-Pillado, Alex Vilenkin and myself, and titled “Quantum tunneling in flux compactifications“. In this paper we studied bubble nucleation rates in a 6-dimensional Einstein-Maxwell theory. The two extra dimensions are compactified into a 2-sphere, and their radius is stabilized by a magnetic flux through that sphere. We picked this toy model because it is simple enough to allow a quantitative analysis, yet it also includes some of the essential features of string theory compactifications (a related paper by Sean Carroll, Matthew Johnson and Lisa Randall was posted on the same day as ours!).

But why do we care about bubble nucleations, extra dimensional theories and compactification? To help set the mood, let’s come down to earth and think about a vanilla galaxy. Mmm, nice. How about the Hubble deep field? Mmmm – very nice! We can see thousands of galaxies effortlessly suspended in space billions of light years away. The universe we see is undeniably beautiful and incomprehensibly vast. But still, we can’t help but notice that our laws of physics are telling us that there may be more to the story – infinitely more!

For over two decades eternal inflation has been telling us that our entire observable universe is just a small part of an infinitely large universe (that was spawned some 13.7 billion years ago in the Big Bang), which itself is only one out of an infinite number of other universes (each the product of their own “local” big bang). Here’s the thing: once inflation starts it never ends! To illustrate this, let’s think about a simple model with a scalar field potential that has two metastable de Sitter minima separated by a barrier. Let vacuum A have a larger cosmological constant than vacuum B. If the universe starts in vacuum A, bubbles of vacuum B can nucleate and begin to expand at a speed approaching that of light within A. However, vacuum A is itself expanding, always leaving room for new bubbles to form. Furthermore, since B also has a positive vacuum energy, it can itself become a parent vacuum to type A bubbles. This simple recycling universe is an example of a “multiverse” which gets populated by the two possible vacua in the theory. This idea generalizes to theories with many different vacua – all possible types of bubbles are nucleated one within the other in an everlasting effervescent cosmic extravaganza!

More recently String theory has echoed the same sentiment, suggesting the existence of a multitude of vacua characterized by different values of the low-energy constants of Nature. String theory (currently our best candidate for a quantum theory of gravity) demands that we consider  or  dimensional spacetimes instead of the mundane  that we’re used to! The idea of extra dimensions predates string theory, going back roughly a century to Kaluza-Klein theory, which attempted to unify gravity and electromagnetism by considering a  gravity theory which precipitates electromagnetism in a reduced  perspective. At any rate, it does seem as though we live in a  4d world, so where do all the extra dimensions go? This is where the idea of compactification comes in. Physicists have been able to show that if we start with a higher dimensional world, some of the extra dimensions can be “compactified” so that we don’t “experience” them directly (although what’s going on in those compactified dimensions does influence our effective  reality).

It turns out that there are many ways to compactify extra dimensions. In string theory, the role of scalar fields is played by the moduli that characterize the sizes and other geometric aspects of these extra dimensions. String theory vacua also involve additional objects, such as fluxes and branes. There are so many ways to combine these ingredients (we’re talking numbers in the googols here!) to produce different vacua, that we land up with a “string landscape” of possible vacuum solutions. When the string landscape is combined with inflationary cosmology, the picture of an eternally inflating “multiverse”, populated by all possible types of vacua comes into sharper focus. The calculation of bubble nucleation rates is an essential part of the irresistible task of quantitatively understanding the multiverse and it’s evolution.

Plot of the effective potential, in units, as a function of the modulus field . We show the potential for 3 different values of the flux quantum .

So now that we have motivated why we study bubble nucleation rates, let’s get back to the  Einstein-Maxwell model we investigated in the paper. Our action included a  cosmological constant term and we assumed that a 2-form magnetic flux permeates the extra dimensional  sphere. For our metric ansatz we assumed a maximally symmetric  [/tex] Riemannian manifold, and compactified the extra dimensions on a 2-sphere. While the model can be studied directly in  it is easier to see what’s going on when we dimensionally reduce our model so that we have an effective  potential as shown in the figure. Each minimum in the figure corresponds to a metastable vacuum with a given quanta of the Maxwell field flux  . The set of minima with different values of  , constitute a “small” landscape.

We set out to describe “flux tunneling” from a configuration with a given value of  to a neighboring minimum with flux quantum  (upward jumps may also be allowed if the initial vacuum has positive vacuum energy ). We showed that this process of vacuum decay occurs through the nucleation of magnetically charged 2-branes, which look like expanding spherical bubbles in the large 3 spatial dimensions and are localized in the extra 2 dimensions. The vacuum inside the bubble has its extra-dimensional magnetic flux reduced by one unit compared to that of the vacuum outside. We estimated the instanton action corresponding to this flux tunneling nucleation event, and used it to calculate transition rates.

While the effective  potential has stable vacua under small perturbations in the compactification radius (modulus field  ) for any given value of the flux  , we see that it tends to zero for large values of the radius/modulus field  . This in turn means that positive-energy vacua should be able to decay by tunneling through the barrier, leading effectively to decompactification of space. This seems to be a generic situation for four dimensional effective potentials for moduli fields that represent the size of internal manifolds and that are stabilized at non-negative values of the  cosmological constant. We estimated the decay rate of the above vacua towards decompactification and compared it with the flux tunneling decay rates.

We found that for light and extremal branes (extremal branes have a simple relation between their tension and charge), flux tunneling proceeds far more rapidly than decompactification tunneling, while for superheavy branes the two tunneling rates are comparable.

There is another flux compactification sector of our  theory. The existence of this branch of the landscape is more easily understood in the dual picture, where we have a four-form field flux that one could turn on in the four sphere. One can then find solutions of this model with two large spacetime dimensions (having de Sitter, Minkowski, or anti-deSitter geometry) and with the remaining 4 dimensions compactified on a  . We can study tunneling processes between different values of the flux number on the 4-sphere or go to the Maxwell description where the 4-form flux along the internal dimensions gets dualized to an electric field along the large spatial dimension. It is easy to see then that one can understand the tunneling between vacua in this sector as the Schwinger decay of this electric field.

We are currently investigating whether or not there is an instanton that interpolates between the two sectors in this model. This would probably be a more complicated instanton than the ones we have already studied, as it should involve a topology change to be able to interpolate between the different compactification schemes. This is an important point, since the existence of this type of instanton is necessary in order for the multiverse to explore all the sectors of the landscape.

Another interesting area for ongoing research involves bubble collisions. It is usually assumed that when two bubbles of the same vacuum collide, their domain walls annihilate in the vicinity of the collision point, with great energy release, and the two bubbles merge. At late times after the collision, the resulting configuration has the form of two expanding spheres which are joined along a circle of ever expanding radius. In the case of bubbles with different vacua, a similar configuration is formed, but now the colliding walls merge to produce a new wall that separates the two vacua inside the bubbles.

In contrast, branes separating flux vacua in different bubbles are generally localized at different points in the internal manifold and will therefore miss one another in the colliding bubbles. So the branes will not merge or annihilate, and the bubbles will simply propagate into one another, forming a new vacuum in the overlap region. This new type of behavior could have important phenomenological consequences for the observable signatures of bubble collisions.

To summarize: we set out to study bubble nucleation rates in a toy string theory landscape – the  Einstein-Maxwell model. We showed that vacuum decay can occur via the nucleation of magnetically charged 2-branes. From the  viewpoint, these branes look like expanding bubbles which have their magnetic flux on the inside reduced by one unit compared to that on the outside. We calculated the instanton action for this flux tunneling and compared it to the decompactification decay channel.

We also emphasized that the expanding bubbles resulting from flux tunneling are bounded by co-dimension  branes, which are generally localized at different points in the internal dimensions. We expect, therefore, that in bubble collisions, the branes will generally miss one another and the bubbles will continue expanding into each other’s interior, forming a new vacuum in the overlap region. This may have interesting observational implications, which we hope to explore in the future.

References to the literature can be found in our paper: Jose Juan Blanco-Pillado, Delia Schwartz-Perlov and Alex Vilenkin, “Quantum tunneling in flux compactifications” arXiv:0904.3106v1 [hep-th].

Some suggested further reading includes:

(1) the popular book by Alex Vilenkin, Many Worlds in One – the search for other universes.
(2) a Scientific American magazine article by Raphael Bousso and Joseph Polchinski, The string theory landscape, September 2004.