Dark energy, which makes up seventy-two percent of the energy in our universe today, is a simple parameter in the classical equations of general relativity. The parameter is called the cosmological constant, and it is positive. However, theories in which we unify quantum mechanics with general relativity tend to be much easier to formulate on space-times which have a zero or negative cosmological constant. There have been many attempts to formulate theories of quantum gravity on space-times with positive cosmological constant but there still isn't an elegant and attractive framework for such theories. We have made our own attempt, which also has serious drawbacks.

Costas Kounnas (Ecole Normale Superieure), Nikolaos Toumbas (Cyprus University) and Jan Troost (Ecole Normale Superieure) have recently described how to realize a space-time with the characteristics of a positive cosmological constant space-time as a background in string theory. The serious drawbacks of the model are on the one hand that it only has a positive cosmological constant in two space-time directions (instead of four), and that the coupling constant of the theory changes in time.

Despite the contradiction with phenomenology, the model gives a rare instance of a solution to string theory that is under calculational control (even at high curvature) and as such can serve as a laboratory for quantum theories of gravity on cosmological space-times.

We have made use of the laboratory to embed the concept of the wave-function of the universe as developped by Hartle and Hawking within the framework of general relativity into string theory. In particular our cosmology has a time-reversal symmetric spatial slice, and it moreover allows for an analytic continuation to a compact euclidean solution. These two conditions allow for the definition of a euclidean path-integral with boundary conditions at the spatial slice (indicated by the horizontal line in the lower picture), which provides for a definition of the wave-function of the universe at some initial point in time. The wave-function can then be evolved to later points in time through a unitary Hamiltonian evolution (from the horizontal line upwards into the future). The authors have been able to construct models in which they can prove that the wave-function of this particular universe is normalizable.

NASA/WMAP

To reach these results we needed to apply recent insights into a new sort of spaces that generalize manifolds. These spaces allow for gluings over common patches of an atlas with a group which includes not only diffeomorphisms but also symmetries of string theory. The generalization of the notion of classical geometry to stringy geometry was crucial to the existence of the model, as well as to the sensible definition of the wave-function of the universe.