On in-in correlators for spinning theories and their shadow formulation

In-in correlators are the basic observables in cosmology and are traditionally computed using the Schwinger-Keldysh formalism. In this paper we revisit this formalism for photons, gluons, and gravitons coupled to scalars in four dimensional de Sitter space and provide a novel treatment of boundary gauge-fixing of the underlying path integral. We also derive effective actions in Euclidean Anti-de Sitter space whose Feynman rules compute the in-in correlators of these theories in de Sitter space, generalising the shadow formalism recently obtained for scalar theories. We illustrate this formalism in a number of examples at tree-level and 1-loop, demonstrating its relative simplicity compared to other approaches. Moreover, we initiate the study of color/kinematics duality and the double copy for in-in correlators using dressing rules which uplift flat space Feynman diagrams to de Sitter space.