  This work uses Lorentz-signature in-in perturbation theory to analyze the
late-time behavior of correlators in time-dependent interacting massive scalar
field theory in de Sitter space. We study a scenario recently considered by
Krotov and Polyakov in which the coupling $g$ turns on smoothly at finite time,
starting from $g=0$ in the far past where the state is taken to be the (free)
Bunch-Davies vacuum. Our main result is that the resulting correlators (which
we compute at the one-loop level) approach those of the interacting
Hartle-Hawking state at late times. We argue that similar results should hold
for other physically-motivated choices of initial conditions. This behavior is
to be expected from recent quantum "no hair" theorems for interacting massive
scalar field theory in de Sitter space which established similar results to all
orders in perturbation theory for a dense set of states in the Hilbert space.
Our current work i) indicates that physically motivated initial conditions lie
in this dense set, ii) provides a Lorentz-signature counter-part to the
Euclidean techniques used to prove such theorems, and iii) provides an explicit
example of the relevant renormalization techniques.
