QFT in nearly dS space-times and, more generally in FRW backgrounds, allows us to describe correlations at the end of inflation. However, how to extract fundamental physics out of them is still a challange: we do not even know how fundamental pillars such as causality and unitarity of time evolution constrain it. In this talk I will review recent progress in the understanding of the analytic structure of cosmological observables. Such progress is based on a novel geometrical approach which provides on one side novel computational tools and on the other side novel insights on the physics encoded into them and its ties to the flat-space physics. I'll finally comment on the issue of infrared divergence and a proof of concept on the possibility of defining infrared safe observables in perturbation theory.