On $i\epsilon$ Prescription in Cosmology

This is a technical note on the $i\epsilon$ prescription in cosmology where we consider a self-interacting scalar field in the Poincare patch of the de Sitter space whose Hamiltonian has explicit time dependence. We use both path integral and operator formalisms to work out the evolution of states from asymptotic past infinity with $i\epsilon$ prescription, which becomes nontrivial even in the free theory, and explicitly show how arbitrary states are projected onto the vacuum. We establish that in perturbation theory the $i\epsilon$ prescription can be implemented in Weinberg's commutator formula by just inserting $\epsilon$ dependent convergence factors that make the oscillating time integrals at infinity meaningful.