Challenges in Supersymmetric Cosmology

We discuss the possibility that inflation is driven by supersymmetry breaking with the scalar component of the goldstino superfield (sgoldstino) playing the role of the inflaton and charged under a gauged U(1) R-symmetry. Imposing a linear superpotential allows us to satisfy easily the slow-roll conditions, avoiding the so-called η-problem, and leads to an interesting class of small field inflation models, characterised by an inflationary plateau around the maximum of the scalar potential near the origin, where R-symmetry is restored with the inflaton rolling down to a minimum describing the present phase of the Universe. Inflation can be driven by either an F- or a D-term, while the minimum has a positive tuneable vacuum energy. The models agree with cosmological observations and in the simplest case predict a rather small tensor-to-scalar ratio of primordial perturbations. We propose a generalisation of Fayet-Iliopoulos model as a microscopic model leading to this class of inflation models at low energy.