  This paper discusses the leading-order correction induced by cosmological
perturbations on the average expansion rate of an expanding spacetime,
containing one or many perfect fluids. The calculation is carried out up to the
second order in the perturbations, and is kept as general as possible. In
particular, no approximation such as a long-wavelength or a short-wavelength
limit is invoked, and all three types of perturbations (scalar, vector, and
tensor) are considered. First, the average value of the expansion rate is
computed over a three-dimensional space-like surface where the total density of
the fluids is constant. Then, a formula is derived relating that average value
to the one over any other surface, on which a different scalar property of the
fluids is constant. Moreover, the general formulas giving the correction to the
average expansion rate are applied, in particular, to the case of a spacetime
containing a single fluid with a constant equation of state. The sign and the
effective equation of state of the corresponding back-reaction effect in the
first Friedmann equation are examined.
