The principles of Lorentz invariance, locality and unitarity highly constrain the physics at accessible high energy: the type of interactions allowed as well as most of the theorems known in particle physics are instances of these principles. This is neatly seen in the structure of scattering amplitudes in asymptotically flat space-times. However, cosmology suggests that such principles may be just approximate: Lorentz invariance is broken at cosmological scales and the accelerated expansion of the universe seems to prevent a full-fledge definition of quantum mechanical observables. If our fundamental ideas in particle physics become somehow approximate in cosmology, what are the fundamental rules governing cosmological processes? In this talk I will report on a recent program which aims to address this question, by bringing both philosophy and methods which have been successful for scattering amplitudes to the analysis of cosmological observables. In particular we investigate the analytic properties of the perturbative wavefunction of the universe, how fundamental physics is encoded into it, how the flat-space physics reflects into it, and how all these features are encoded into new mathematical structures, which can be used as a novel first principle definition of the perturbative wavefunction.

The analysis of the singularities in the S-matrix of particles has shown that perturbative structure field theories can be much simpler than what Feynman diagrams tell us. In particular, there exist classes of theories whose perturbative S-matrix is completely determined by a sub-set of its singularities. At tree level this reflect with the existence of recursion relations. At loop level, there are special singularities, the leading singularities, which are believed to determine a class of theories which N=4 SYM and N=8 supergravity belongs to. I will discuss both the tree and loop level, with a particular attention on the leading singularity technique and its application to the investigation of the perturbative N=8 supergravity.