Delta N formalism: non-linear evolution of scalar and tensor perturbations during inflation

Linear-perturbation theory has proven to be an extremely powerful tool to compare inflationary models with observational data. Recently, the newcoming high-precision observations call for predictions beyond linear perturbations. Such effects are known to be relevant for example in the production of primordial black holes or scalar-induced gravitational waves. The delta N formalism proposes to capture some of these non-linearities by describing the universe as a set of homogeneous patches evolving independently. After reviewing its formulation, I will apply it to the case of a single scalar field following a slow-roll evolution, briefly interrupted by a phase of ultra-slow roll where the potential is nearly flat. Taking care of the correlation of the patches, I will show how to obtain correlation functions of scalar perturbations. As a final note, I will briefly show how this procedure can also be applied to describe primordial gravitational waves.