  We use large N-body simulations and empirical scaling relations between dark
matter halos, galaxies, and supermassive black holes to estimate the formation
rates of supermassive black hole binaries and the resulting low-frequency
stochastic gravitational wave background (GWB). We find this GWB to be
relatively insensitive ($\lesssim10\%$) to cosmological parameters, with only
slight variation between WMAP5 and Planck cosmologies. We find that uncertainty
in the astrophysical scaling relations changes the amplitude of the GWB by a
factor of $\sim 2$. Current observational limits are already constraining this
predicted range of models. We investigate the Poisson variance in the amplitude
of the GWB for randomly-generated populations of supermassive black holes,
finding a scatter of order unity per frequency bin below 10 nHz, and increasing
to a factor of $\sim 10$ near 100 nHz. This variance is a result of the rarity
of the most massive binaries, which dominate the signal, and acts as a
fundamental uncertainty on the amplitude of the underlying power law spectrum.
This Poisson uncertainty dominates at $\gtrsim 20$ nHz, while at lower
frequencies the dominant uncertainty is related to our poor understanding of
the astrophysical scaling relations, although very low frequencies may be
dominated by uncertainties related to the final parsec problem and the
processes which drive binaries to the gravitational wave dominated regime.
Cosmological effects are negligible at all frequencies.
