Cosmological Polytopes and the Wavefuncton of the Universe for Light States

We extend the investigation of the structure of the late-time wavefunction of the universe to a class of toy models of scalars with time-dependent masses and polynomial couplings, which contains general massive scalars in FRW cosmologies. We associate a universal integrand to each Feynman diagram contributing to the wavefunction of the universe. For certain (light) masses, such an integrand satisfies recursion relations involving differential operators, connecting states with different masses and having, as a seed, the massless scalar (which describes a conformally coupled scalar as a special case). We show that it is a degenerate limit of the canonical form of a generalisation of the cosmological polytopes describing the wavefunction for massless scalars. Intriguingly, the flat-space scattering amplitude appears as a higher codimension face: it is encoding the leading term in the Laurent expansion as the total energy is taken to zero, with the codimension of the face providing the order of the total energy pole. The same connection between the other faces and the Laurent expansion coefficients holds for the other singularities of the wavefunction of the universe, all of them connectable to flat-space processes. As the degenerate limit is taken, some of the singularities of the canonical form of the polytope collapse onto each other generating higher order poles. Finally, we consider the mass as a perturbative coupling, showing that the contribution to the wavefunction coming from graphs with mass two-point couplings can be identified with a degenerate limit of the canonical form of the cosmological polytope, if the perturbative expansion is done around the conformally coupled state; or as double degenerate limit of the canonical form of the extension of the cosmological polytopes introduced in the present paper, if the perturbative expansion is done around minimally coupled states.