The Quest for a Realistic Cosmology

in the Landscape of String Theory

Andrew Chamblin, University of Louisville
Recent astronomical observations [1,2,3,4] would appear to indicate that the universe is accelerating. Assuming that these observations have been correctly interpreted, then it is clear that physicists today are faced with a number of mysteries which to date have defied any elegant and straightforward explanation. First of all, there is the obvious question: What is the nature of this mysterious `dark energy' which is driving the expansion? Evidently this vacuum energy is exactly isotropic and homogeneous at the present time - but what is it? In addition to this question, there is the legendary `cosmological constant problem': Whatever this dark energy is, why is it so incredibly small? Observationally, the dark energy density is 120 orders of magnitude smaller than the energy density associated with the Planck scale - the obvious cut off. Furthermore, the standard model of cosmology posits that very early on the universe experienced a period of inflation: A brief period of very rapid acceleration, during which the Hubble constant was about 52 orders of magnitude larger than the value observed today. How could the cosmological constant have been so large then, and so small now? Finally, there is the `coincidence problem': Why is the energy density of matter nearly equal to the dark energy density today? Considering all of these problems at once can be a humbling experience: It is clear that we are presently unable to explain several of the most basic experimental facts about this universe.

String theory is much vaunted as a fully consistent quantum theory of gravity. If this is the case, then we would expect string theory to tell us something about the acceleration of the universe. Ideally, string theory would provide clear mechanisms which resolve all of the above mentioned problems.

Remarkably, it is very difficult to get accelerating solutions directly through standard compactification techniques of the low-energy limit of string theory. More precisely, the different string theories are related to each other through dualities, special symmetries which ultimately involve the mysterious quantum M-theory in eleven dimensions. All of these theories have as a low- energy limit some supergravity theory: A classical theory consisting of gravity coupled to other fields. These theories should be thought of as special limits of some underlying, quantum M-theory. We do not know what the entire moduli space of this theory looks like, but we know what it looks like at these special limit points. A major triumph for string theory would be to show precisely `where' in the M-theory moduli space there exist solutions which actually look like our universe. For example, it would be nice if we could recover a realistic M-theory cosmology beginning with one of the supergravity theories. However, there is a `no-go theorem', due to Gibbons [5], Maldacena and Nuñez [6], and also de Wit, Dass and Smit [7], which basically asserts that if you compactify any string-derived supergravity on a smooth compact internal space, then you will never get de Sitter space. Since the universe is evidently both past and future de Sitter (albeit with vastly differing vacuum energies), this would seem to be a problem.

However, there are various ways around this particular no-go result. The theorem assumes time independence of the internal space, and so one may search for time-dependent solutions. Following this intuition, Townsend, Wohlfarth and others [8,9,10] have constructed a variety of time-dependent compactifications which describe a period of acceleration. The basic idea is simple: The internal space is described by certain scalar fields known as `moduli'. These moduli describe the size, shape and other basic properties of the internal space. These moduli typically have exponential potentials, with the property that as you flow to the minimum of the potential the universe decompactifies (i.e., for a given scalar field $\phi$, $V(\phi) ~{\sim}~ e^{-\phi}$). One can now imagine `bouncing' the universe off of this potential: The universe comes in from a period of being decompactified, and rolls up the exponential potential during the process of compactification. At some point there will have to be a `turnaround' point, where the universe stops compactifying and reenters a decompactification stage. At the turnaround, there is little kinetic energy for the moduli, and so all of the energy is dominated by the potential term, which can then act as a cosmological constant. Problems with this approach include the fact that it is difficult to get a very long period of inflation, unless one uses many moduli [11]. Furthermore, if the size of the extra dimensions vary, then there will be variations in Newton's constant and the fine structure constant. Strong experimental bounds on such variation place tight constraints on these models.

Another way to get around the no-go result involves beginning with rather exotic supergravities which may not necessarily have anything to do with string or M-theory. For example, Hull has championed the viewpoint that we may wish to consider supergravity theories with extra dimensions of time [12]. These supergravity actions come from perfectly well-defined superalgebras, and compactification of these theories can give de Sitter spacetime in any dimension. One drawback is that these theories typically contain ghosts: Gauge fields which have the wrong sign for the kinetic term. Furthermore there are the usual problems with causality: If you have two or more dimensions of time, then there exist closed timelike curves through every point. While the extra dimensions of time can be eliminated by applying certain duality transformations which yield another theory, one is often still left with ghosts.

One may also choose to `compactify' a string-derived supergravity on a non-compact space [13]. This gets around the no-go result because the internal space is non-compact. This may sound counterintuitive, but actually it is a well-defined procedure known as 'consistent truncation'. To perform a consistent truncation, one writes the full higher-dimensional space as a product (or in general warped product) of the non-compact directions and some space X. One then constructs a theory on X, with the property that any solution of that theory corresponds to a solution of the theory in the higher dimensions, and vice-versa. In this way one can obtain solutions where X is isometric to de Sitter. One obvious problem with this approach is that it is not clear how one should interpret the large extra dimension.

A related but differing approach involves using the scale invariance of eleven-dimensional supergravity, which is the low-energy limit of M-theory. The equations of motion of eleven-dimensional supergravity admit a scale invariance, whereby rescaling the field content in a certain way simply rescales the action by an overall power of the scale parameter. Instead of compactifying the theory on a circle using `conventional' Kaluza-Klein boundary conditions (where the fields are periodic), one can use the scale symmetry to allow fields to be rescaled around the circle. Upon reduction to ten dimensions one obtains a new massive supergravity theory, which has the property that de Sitter space is the ground state. Intuitively, the apparent expansion of the universe is really an effect generated by the rescaling of the metric. This theory was first introduced by Howe, Lambert and West [14] and was obtained through consistent truncation by others [15]. It was further studied by this author and Lambert [16,17] where we dubbed the theory `MM-theory', for modified or massive M-theory. The main problem with MM-theory is that the scale invariance is an anomalous symmetry: Higher derivative corrections to the supergravity Lagrangian manifestly break the symmetry. Since the theory is anomalous in the ultraviolet, the only way it can make sense is if scale invariance is realized deep in the infrared. In this picture were correct, then the cosmological constant itself would be an infrared effect.

Ultimately, all of these classical approaches to cosmology seem a bit contrived: In order to get around the no-go theorem, one is forced to make rather unusual or unnatural assumptions. But of course, all of these expeditions are only probing the classical borders of the full landscape of string theory. The world is not classical: There is an underlying quantum reality, and we need to better understand the classical to quantum phase transition within the context of cosmology. Could it be that if we simply venture into the quantum wilderness of the string theory landscape, we will find a realistic cosmology? In fact, it is the case that quantum effects seem to lead in the right direction. In a recent paper, by Kachru, Kallosh, Linde and Trivedi (KKLT) [18] it was shown that if you carefully consider certain instanton corrections, you can construct solutions of string theory which exhibit a small, positive cosmological constant. Their example is an example of a `flux compactification' - crudely, a compactification in which certain fluxes are turned on. Typically, certain branes are the `sources' for a given flux. For example, just as the electron is the source for $F_{ab}$ (a two-form), so a membrane can act as the `electric' source for a four-form flux. In four dimensions, the equations of motion for a four-form will tell you that the form is locally just covariantly constant: The term $(F_{4})^2$ in the action will thus `look' like a cosmological constant. Membranes in such scenarios are thus surfaces across which the effective cosmological constant can jump. Neutralization of the cosmological constant through membrane nucleation was first studied by Brown and Teitelboim [19], and has been further explored in the context of string theory by others [20,21].

In the KKLT construction, the authors begin by compactifying six dimensions of space on a Calabi-Yau manifold - a complex manifold which has a special holonomy that leaves minimal supersymmetry ($N =
1$) in the effective four-dimensional theory obtained through the compactification. Certain background fluxes are turned on throughout the construction, and in the effective four-dimensional theory the spacetime is initially anti-de Sitter (adS). A Calabi-Yau manifold has certain moduli associated with the fact that it admits a complex structure, and these moduli need to be fixed. KKLT show that this is possible by arguing that quantum effects modify the superpotential [22] in such a way that they are able to explicitly demonstrate the existence of supersymmetric adS vacua with fixed complex and Kahler moduli.

Finally, and crucially, KKLT add in certain branes, known as (`anti') D3-branes. These branes have the effect of `lifting' the stable (and supersymmetric) adS vacuum to a de Sitter (dS) vacuum. By fine tuning various things, the authors are able to argue that the resulting dS vacuum can even have a very small cosmological constant 1. Furthermore, the inclusion of the three-branes breaks supersymmetry, and so it would seem that supersymmetry breaking and a positive cosmological constant always go `hand-in-hand' in these constructions. Finally, the de Sitter vacua are always metastable in these models, i.e., they are false vacua and therefore have some lifetime. In particular, KKLT argue that these vacua are resonances which can decay faster than the timescale for the Poincare recurrences which have bothered some people [23].

Now, the KKLT model is but a very special case of a huge class of more general compactifications. One can imagine solutions where only four dimensions are compactified, or indeed where none of the dimensions are compactified and the universe exhibits the full eleven dimensions of M-theory. One could imagine that other fluxes are turned on, or that no fluxes are turned on. If we think of the string theory landscape as a huge potential or functional which varies depending on all of the different possible moduli, then it is clear that the quantum wilderness of string theory is a vast, higher dimensional cornucopia of moutaintops, valleys and precipices. The moduli space of supersymmetric vacua is rather like a vast plain extending up to the mountains - one may move continuously between different vacua by varying certain moduli. Accelerating cosmologies correspond to isolated valleys which sit up between the mountain peaks and passes (i.e., one might imagine equating the magnitude of the dark energy with the altitude of the valley). For whatever reason, we live in a universe where four spacetime dimensions are compactified, and our `altitude' is just barely above sea level.

Of course, when one starts to think of the universe in these terms, it can have a profound impact on one's expectations and outlook. It starts to look like many things - the masses of the elementary particles, the values of the couplings, the value of the cosmological constant - are probably just accidents, random numbers that will never be calculated from first principles using string theory. But it is a short journey from this philosophy to that house of ill-repute known as the Anthropic Principle 2. For this reason, various people have begun to `count' [24] all of the different discrete valleys in the string theory landscape. After all, it may be that many of the vacua look somewhat like our universe - and even if only about 1% look like home, we will have still learned something about our world.

In summary, cosmology is now a science based on high precision measurements which are yielding very detailed information about the large-scale structure of the universe. For some time it was unclear that string theory could consistently explain the observed acceleration of the universe. This situation has now been rectified, and there is now a realization that there are likely many metastable, de Sitter like vacua in string theory. These are just the first tentative steps towards a fully realistic string cosmology, and the years ahead will no doubt bring even more twists, turns and surprises.

I thank N. Lambert and H. Reall for discussions, and the Kavli Institute for Theoretical Physics for hospitality while this work was completed.

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Jorge Pullin 2004-03-12