Speaker
Description
The Lorentzian gravitational path integral over homogeneous and isotropic metrics in a closed universe with positive cosmological constant is dominated by complex saddle points, such as the Hartle–Hawking saddles. We ask how to compute cosmological correlators on such complex backgrounds, and how they change with respect to shifts of the kinematic variables — the kinematic flow — without tracking the bulk time evolution explicitly. We study a conformally coupled scalar test field with a cubic interaction around these saddles. We find that the wave function coefficients can be written as discrete spectral sums whose summand contains the corresponding coefficients in the Einstein static universe, which are obtainable in analogy with the cosmological polytope program for flat FLRW cosmologies: where the spatially flat case gives integrals over shifted kinematics, the discrete spectrum on the three-sphere gives sums. The coefficients depend on the final hypersurface through a phase fixed by its size, and we identify them at all orders with Srivastava–Daoust multi-variable hypergeometric functions. For the one- and two-site Feynman–Witten diagrams we derive difference equations in the kinematic variables which close on a finite set of sums, giving a discrete kinematic flow. We also discuss computation of three- and four-point correlators.