Loop coproduct in Morse and Floer homology

By a well-known theorem first proved by Viterbo, the Floer homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. We prove that, when restricted to positive Floer homology resp. loop space homology relative to the constant loops, this isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. The proof uses compactified moduli spaces of punctured annuli. We extend this result to reduced Floer resp. loop homology (essentially homology relative to a point), and we show that on reduced loop homology the loop product and coproduct satisfy Sullivan's relation. Along the way, we show that the Abbondandolo-Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism by using linear Hamiltonians and the square root of the energy functional.