Classical string backgrounds with cosmological constant

Since almost a century, we know that our universe is expanding, meaning growing. At the end of the 20th century, a major discovery was made and later celebrated with a Nobel Prize: the expansion of the universe is actually accelerating, i.e. it grows faster and faster! While this is what we observe today, some cosmological models, named inflation models, suggest that our universe has also been through such a phase of accelerated expansion in its early times. This acceleration would then have stopped at some point, and restarted more recently. To explain this acceleration, theoretical models propose the existence of a mysterious dark energy, whose nature is unknown. The dark energy in our current universe seems very compatible with a (positive) cosmological constant, namely a vacuum energy in the universe. The same could be true to some extent for inflation in the early universe. The reason and nature of this cosmological constant is nevertheless not at all understood. In this project, we want to use string theory to understand the nature of dark energy. String theory is a serious candidate to be a fundamental theory describing our world. In particular, it allows to unify all known forces (electromagnetism, nuclear interactions, gravity) in one formalism. It is natural to expect that a fundamental theory would explain the nature of dark energy. With that aim in mind, part of the project consists in looking for solutions of string theory that describe a universe in accelerated expansion as ours. More precisely, we look for a universe with a positive cosmological constant, called a de Sitter spacetime. If we find some, we could then question the string theory origin of this cosmological constant. However, it turns out to be extremely difficult to find solutions with such a universe in string theory! Rather, one obtains universes with a negative cosmological constant, corresponding to a contraction phase instead of an expansion: those are called anti-de Sitter spacetimes. No cosmological constant at all is also possible, but a positive one, as for our universe, is difficult to find! Recently, conjectures have even been put forward (in the so-called swampland program) to claim that a de Sitter spacetime cannot be obtained in string theory, at least under some conditions! But difficult does not necessarily mean impossible. Part of this project consists in looking for solutions of string theory with a de Sitter spacetime, using advanced numerical tools. If we find some, we further want to use them in cosmological models to describe our universe today, or in an early inflation phase. We also want to understand mathematically why finding such solutions is so difficult, thus possibly proving under some conditions the swampland conjectures. Another part of the project consists in studying solutions with an anti-de Sitter spacetime: their similarities with de Sitter ones still make them interesting to understand properties of all these solutions.