1. Introduction

We used to think that roughly 13.7 billion years ago the Big Bang marked the dramatic creation of the universe from the supreme emptiness of that which did not exist, into the all encompassing fullness of that within which everything exists. From a virulent exploding fireball bursting with elementary particles, the universe evolved into the pitch-black expanse we observe today, majestically sprinkled with star-studded galaxies. Since the early ‘80’s, however, eternal inflation has revealed that this is not the full story…

Our entire observable universe is large… very large... “40 billion light years in diameter” large. We don’t know what Starbucks would call this size, but cosmologists sometimes call this our Hubble or horizon volume. Yet, according to eternal inflation, this is just a small fraction of an infinitely large universe which itself is only one out of an infinite number of other universes! Each of these “other” universes is the product of their own “local” big bang. Instead of being the ultimate creation event, our Big Bang merely marks the emergence of our local universe into the far grander “multiverse” much like a lone bubble nucleating in an infinite flute of champagne which is already brimming with other bubbles. Each bubble represents an infinite universe, and our cosmic champagne is home to an infinite number of bubbles!

So where do all the bubbles come from? Why couldn’t we just have one universe? After all, the inflationary cosmological paradigm in which the universe undergoes a period of inflation, which then gives way to the standard big bang evolution is spectacularly successful! Well, it turns out that if the universe ever underwent an inflationary stage (and all observational and theoretical signs indicate that it did), then, even if inflation ends somewhere in a big bang “thermalization” event, elsewhere inflation will continue. The mindboggling proliferation of universes comes from the remarkable fact that the theory of inflation is generically eternal - once it starts it never ends! To illustrate this, one can consider a simple 4 dimensional model (3 spatial dimensions and one time dimension) with a scalar field potential that has two metastable de Sitter minima separated by a barrier (see Fig. 1). (A de Sitter vacuum is one of the simplest solutions to Einstein’s equations. de Sitter space describes a universe which contracts from the eternal past until at some time (usually chosen to be t=0) it reaches it’s minimum radius (the de Sitter horizon radius), and then begins to expand exponentially into the eternal future. In the simplest inflationary scenarios, the universe starts at the minimum de Sitter horizon radius and then expands into the future.)

Figure 1. Double well potential energy density diagram for a scalar field φ. When the field tunnels from vacuum A to vacuum B, a bubble of vacuum B (red) nucleates within vacuum A (yellow). A bubbles can in turn nucleate in B vacua, so we get a multiverse with two types of vacua, as illustrated below the potential.

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 a self-sustaining everlasting cosmic production line (see Fig. 2)!

Figure 2. A “multiverse” with many different types of bubbles.

The “multiversal” panorama that has emerged from the inflationary paradigm, has found further support from string theory, which also suggests the existence of a multitude of vacua characterized by different values of the low-energy constants of Nature, and even different numbers of spacetime dimensions. String theory insists that we consider 10 or 11 dimensional spacetimes instead of the mundane 4 that we're used to! That seems like a stretch, but since string theory is our best candidate for a quantum theory of gravity, we need to take the idea of extra dimensions seriously. In fact, 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 5 dimensional (5d) gravity theory which precipitates Electromagnetism in a reduced 4d 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 (see Fig. 3). 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 4d reality).

Figure 3. A few different ways in which 2 dimensions can be compactified (figure reproduced with permission from Vilenkin, 2006). The sphere has the topology S₂ and the torus has topology S₁ x S₁.

It turns out that there are many ways to compactify extra dimensions (see Fig. 3). 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. These ingredients combine combinatorically to produce a huge number of different vacua – the so called "string landscape" of possible vacuum solutions (see Fig. 4).This is somewhat akin to creating a huge variety of different colors from only the three primary colors.

Figure 4. The string theory landscape: the altitude represents potential energy density and each valley is a possible vacuum solution with either positive, zero or negative cosmological constant. In this multidimensional potential energy density surface, only two out of hundreds of possible directions in parameter space are shown (figure reproduced with permission from Vilenkin, 2006).

In this article we will mostly discuss the results from Blanco-Pillado, Schwartz- Perlov and Vilenkin (2009a,b; Please see the references in these papers for citations to the original literature). In these papers we studied a 6-dimensional theory of gravity and electromagnetism, called Einstein-Maxwell theory (see also a closely related interesting paper by Sean Carroll, Matthew Johnson and Lisa Randall, 2009). This model serves as a toy string theory model: it includes some of the essential features of the string theory landscape and eternal inflation, while excluding some of the unnecessary (for our purposes) complexities of more realistic models.

2. The 6d Einstein‐Maxwell Theory

As detailed in Blanco-Pillado et al., (2009a,b), we analyzed a 6d Einstein-Maxwell theory with a positive 6d cosmological constant term. We assumed that either 4 or 2 spacetime dimensions are large, and that the remaining 2 or 4 spatial dimensions are compactified on either a 2- sphere (denoted S₂ ) or a 4-sphere (denoted S₄), respectively. The model includes a magnetic flux which permeates the extra dimensions and is described by a Maxwell field strength tensor of rank 2 or 4.

The landscape of this theory includes one 6 dimensional de Sitter vacuum (denoted dS₆ ), a variety of metastable 4 dimensional de Sitter and anti-de Sitter vacua (dS₄ and AdS₄ ) with two extra dimensions compactified on S₂, and a number of AdS₂ vacua with four extra dimensions compactified on S₄. (Note: Anti‐de sitter space is like de Sitter space only the energy density of the vacuum is negative instead of positive. Sooner or later anti‐de Sitter vacua undergo a catastrophic collapse into a big crunch singularity). The model also includes some perturbatively unstable dS₄ x S₂ and dS₂ x S₄ vacua. (The notation dS_n x S_m indicates that we have an n‐dimensional de Sitter space with m spherically compactified dimensions at each spacetime point. Vacua can be divided into those which have a positive effective cosmological constant, the so-called recyclable vacua, and into those which have a zero or negative effective cosmological constant, the so-called terminal vacua. Transitions can take place to and from recyclable states. Transitions can also take place from recyclable to terminal vacua, but terminal vacua do not produce other vacua. In our landscape, the de Sitter vacua are recyclable and the anti-de Sitter vacua are terminal. Thus tunneling can take place, for example, from dS₆ to dS₄ x S₂ , from dS₆ to AdS₄ x S₂ , from dS₄ x S₂ to dS₆, but not from AdS₄ x S₂ to dS₆. We studied quantum tunneling transitions between vacua with the same and with different effective dimensionality. Let’s first discuss the topology-preserving transitions, where the parent and daughter vacua have the same number of effective dimensions.

2.1. Topology‐Preserving Flux Tunneling

We assumed that a 2-form magnetic flux permeates an extra dimensional 2- sphere. For our metric ansatz we assumed a maximally symmetric 4d Riemannian manifold, and compactified the extra dimensions on a 2-sphere. While the model can be studied directly in 6d it is beneficial to dimensionally reduce the model to an effective 4d potential as shown in Fig. 5. Each minimum in the figure corresponds to a metastable vacuum with a given quanta of the Maxwell field flux, n. For each n there is also a set of maxima which correspond to unstable dS₄ x S₂ vacua. The set of minima with different values of n constitute a "small" landscape. The three minima shown in this potential, from the bottom to the top, have topology AdS₄ x S₂ , Minkowski space x S₂ and dS₄ x S₂.

Figure 5. Plot of the 4d effective potential, in Planckian energy density units (Mp⁴ ), as a function of the modulus field ψ. The potential is shown for 3 different values of the flux quantum: n = 180, 200, 220 (figure reproduced with permission from Blanco-Pillado, Schwartz-Perlov and Vilenkin, 2009a).

We can describe "flux tunneling" from a (A)dS₄ x S₂ configuration with a given value of n to a neighboring (A)dS₄ x S₂ minimum with flux quantum n – 1 (upward jumps may also be allowed if the initial vacuum is dS₄ x S₂ - that is, it must have positive effective vacuum energy). We showed that this process of vacuum decay occurs through the nucleation of magnetically charged 2 dimensional membranes (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. This flux tunneling nucleation event is analogous to the Schwinger process of particle-anti-particle production in a background electric field. We calculated the corresponding transition rates and compared them to another tunneling channel – that of decompactification to which we now turn.

2.2. Topology‐Changing Decompactification Transitions

For any given value of the flux n, the effective 4d potential has stable vacua under small perturbations in the compactification radius, also known as the modulus field ψ (see Fig. 5). However, we see that the potential tends to zero for large values of the radius/modulus field ψ. This in turn means that positive-energy vacua can decay by tunneling through the barrier, resulting in a decompactification of space. This seems to be a generic situation for 4 dimensional effective potentials for moduli fields that represent the size of internal manifolds and that are stabilized at non-negative values of the 4d cosmological constant.

When a decompactification transition (dS₄ x S₂ -> dS₆) takes place, the effective dimension of the daughter vacuum is higher than that of the parent vacuum (see Fig. 6). Observers in the parent vacuum see the nucleation and subsequent expansion of spherical bubbles, as in ordinary topology non-changing Coleman de Luccia vacuum decay. The role of the bubble wall is played by a spherical magnetically charged black brane. The bubble interior is initially anisotropic, but as the compact dimensions expand, it approaches local isotropy, with the metric approaching that of 6d de Sitter space.

Figure 6. A lower-dimensional snapshot of the decompactification spacetime. The compact dimension blows up to become large. The junctions marked as “2- branes” should be 2-spheres in the full 5 spatial dimensions (figure reproduced with permission from Blanco-Pillado, Schwartz-Perlov and Vilenkin, 2009b).

In Blanco-Pillado, Vilenkin,Schwartz-Perlov (2009a) we estimated the decay rate of such vacua towards decompactification and compared it with the flux tunneling decay rates. We found that for light and extremal branes7 flux tunneling proceeds far more rapidly than decompactification tunneling, while for superheavy branes the two tunneling rates are comparable. The tension and charge of extremal branes are simply related.

2.3. Topology‐Changing Compactification Transitions

Let us now consider compactification transitions in which the parent vacuum has higher dimensionality than the daughter vacuum. These transitions are the reverse of the decompactification transitions.

2.3.1. Compactifying 2 dimensions

Transitions from dS₆ to dS₄ x S₂ and AdS₄ x S₂ proceed through the nucleation of spherical, magnetically charged black 2-branes. The process is analogous to the nucleation of spherical domain walls during 3+1 dimensional inflation. The nucleating brane starts out with a radius equal to the 6d de Sitter horizon. This radius is then stretched by the exponential expansion of the universe, while the transverse dimension of the brane (which can be identified with its horizon radius) remains fixed. Behind the horizon, in the black brane interior, the spacetime is effectively 4-dimensional, with the topology (A)dS₄ x S₂.

2.3.2. Compactifying 4 Dimensions

There is another flux compactification sector of our 6d theory where we have a four form field strength permeating a compactified 4-sphere. One can find solutions with two large spacetime dimensions having de Sitter, Minkowski, or antide Sitter geometry, and with the remaining 4 dimensions compactified on a S4. We can study tunneling processes between different values of the flux number on the 4-sphere or we can switch to the Maxwell description where the 4-form flux along the internal dimensions gets dualized to an electric field along the large spatial dimension. Doing so makes it transparent that the tunneling between vacua in this sector proceeds via the Schwinger decay of the electric field. We found that compactification transitions from dS₆ to AdS₂ x S₄ occur through the nucleation of pairs of electrically charged black holes. The charge of these black holes is quantized in units of the elementary charge of the theory, and their mass is determined by the condition that the temperature at the black hole horizon is the same as that at the cosmological horizon in the corresponding Reissner- Nordstrom-de Sitter solution. These black holes can be thought of as 0-branes of the theory.

3. Conclusions

String theory offers a huge menu of different types of possible universes and eternal inflation sees to it that the multiverse actually gets populated by each option on the menu via an ongoing process of nested bubble nucleations. In this paper we have discussed some recent research on bubble nucleation rates in a toy string theory landscape - the 6d Einstein-Maxwell model. This model is rich enough to include the possibility of both topology-preserving and topologychanging (“transdimensional”) tunneling processes, resulting in a multiverse populated by vacua of all possible dimensionalities.

For topology-preserving flux tunneling events, parent and daughter vacua have the same number of large dimensions. This type of vacuum decay can occur via the nucleation of magnetically charged black 2-branes. From the 4d viewpoint, these branes resemble expanding bubbles which have their magnetic flux on the inside reduced by one unit compared to that on the outside. In Blanco-Pillado, Schwart- Perlov and Vilenkin (2009 a) it was shown that the nucleation rates for flux tunneling are generally greater than those in the decompactification decay channel.

Topology-changing transdimensional transitions were also discussed. When 2 dimensions are compactified on a 2-sphere, decompactification (dS4 x S2 to dS6) and compactification (dS₆ to dS₄ x S₂ and dS₆ to AdS₄ x S₂ ) tunnelings also proceed via the nucleation of magnetically charged black 2-branes. When 4 dimensions are compactified on a 4-sphere, transdimensional tunneling from dS₆ to AdS₂ x S₄ proceeds via the nucleation of electrically charged black holes. Transitions from dS6 to the unstable maxima dS₄ x S₂ (or dS₂ x S₄ ) are also possible. However small perturbations cause these vacua to decay into regions of AdS₄ x S₂ ( or AdS₂ x S₄ ) and back to dS₆.

The simple toy model we have discussed here, gives rise to a structurally rich landscape teeming with vacua where some constants of nature are variable, and something as basic as the number of spacetime dimensions, turns out to be variable too! With so many diverse environments to choose from, we can ask, for example, what are the chances that we happen to find ourselves in a universe like ours? These types of questions are not easy to answer in the multiverse because of the infinite number of universes. The challenge of extracting predictions in the multiverse, is known as the “measure problem”, and is an area of ongoing research. Cosmologists are actively trying to explore the measure problem in the context of both regular 3+1 d eternal inflation, and transdimensional eternal inflation. We should be busy for a while…