  We use a population synthesis approach to characterise, as a function of
cosmic time, the extragalactic close binary population descended from stars of
low to intermediate initial mass. The unresolved gravitational wave (GW)
background due to these systems is calculated for the 0.1-10 mHz frequency band
of the planned Laser Interferometer Space Antenna (LISA). This background is
found to be dominated by emission from close white dwarf-white dwarf pairs. The
spectral shape can be understood in terms of some simple analytic arguments. To
quantify the astrophysical uncertainties, we construct a range of evolutionary
models which produce populations consistent with Galactic observations of close
WD-WD binaries. The models differ in binary evolution prescriptions as well as
initial parameter distributions and cosmic star formation histories. We compare
the resulting background spectra, whose shapes are found to be insensitive to
the model chosen, and different to those found recently by Schneider et al.
(2001). From this set of models, we constrain the amplitude of the
extragalactic background to be 1E-12 < Omega(1 mHz) < 6E-12, in terms of
Omega(f), the fraction of closure density received in gravitational waves in
the logarithmic frequency interval around f.
