The graviton one-loop effective action in cosmological space-times with constant deceleration

We consider the quantum Friedmann equations which include one-loop vacuum fluctuations due to gravitons and scalar field matter in a FLRW background with constant $\epsilon=-{\dot{H}}/{H^2}$. After several field redefinitions, to remove the mixing between the gravitational and matter degrees of freedom, we can construct the one loop correction to the Friedmann equations. Due to cosmological particle creation, the propagators needed in such a calculation are typically infrared divergent. In this paper we construct the graviton and matter propagators, making use of the recent construction of the infrared finite scalar propagators calculated on a compact spatial manifold in \cite{Janssen:2008px}. The resulting correction to the Friedman equations is suppressed with respect to the tree level contribution by a factor of $H^2/m_p^2$ and shows no secular growth.