Back-reaction of long-wavelength cosmological fluctuations as measured by a clock field

We consider the back-reaction of cosmological fluctuations on the local expansion rate averaged over a space-like hypersurface of constant value of a clock field. We show that in the infrared limit, the fluctuations lead to a decrease in the average expansion rate, measured at a fixed value of the clock field, compared to what would be obtained in a homogeneous universe. We work in the context of Einstein gravity coupled to perfect fluid matter.