D-term Inflation: the Good, the Bad and the Ugly

Prologo. It is by now commonly accepted that inflation looks more natural in supersymmetric theories rather in non-supersymmetric ones. This is because the necessity of introducing very small parameters to ensure the extreme flatness of the inflaton potential seems very unnatural and fine-tuned in most non-supersymmetric theories, while this naturalness is achieved in supersymmetric models. The nonrenormalization theorems in exact global supersymmetry guarantee that we can fine-tune any parameter at the tree-level and this fine-tuning will not be destabilized by radiative corrections at any order in perturbation theory. This is the advantage of invoking supersymmetry. There is, however, a severe problem one has to face when dealing with inflation model building in the context of supersymmetric theories. The generalization of supersymmetry from a global to a local symmetry automatically incorporates gravity and, therefore, inflation model building must be considered in the framework of supergravity theories. The supergravity potential is rather involved, but it can still be written as a D-term plus an F -term, and it is usually supposed that the D-term vanishes during inflation. Now, for models where the D-term vanishes, the slow-roll parameter η = M PlV /V generically receives various contributions of order ±1. This is the so-called η-problem of supergravity theories. In supergravity theories, supersymmetry breaking is transmitted by gravity interactions and the squared mass of the inflaton becomes naturally of order of V/M Pl ∼ H. The perturbative renormalization of the Kähler potential is therefore crucial for the inflationary dynamics due to a non-zero energy density which breaks supersymmetry spontaneously during inflation. How severe the problem is depends on the magnitude of η and