  A key goal of many Cosmic Microwave Background experiments is the detection
of gravitational waves, through their B-mode polarization signal at large
scales. To extract such a signal requires modelling contamination from the
Galaxy. Using the Planck experiment as an example, we investigate the impact of
incorrectly modelling foregrounds on estimates of the polarized CMB, quantified
by the bias in tensor-to-scalar ratio r, and optical depth tau. We use a
Bayesian parameter estimation method to estimate the CMB, synchrotron, and
thermal dust components from simulated observations spanning 30-353 GHz,
starting from a model that fits the simulated data, returning r<0.03 at 95%
confidence for an r=0 model, and r=0.09+-0.03 for an r=0.1 model. We then
introduce a set of mismatches between the simulated data and assumed model.
Including a curvature of the synchrotron spectral index with frequency, but
assuming a power-law model, can bias r high by ~1-sigma (delta r ~ 0.03). A
similar bias is seen for thermal dust with a modified black-body frequency
dependence, incorrectly modelled as a power-law. If too much freedom is allowed
in the model, for example fitting for spectral indices in 3 degree pixels over
the sky with physically reasonable priors, we find r can be biased up to
~3-sigma high by effectively setting the indices to the wrong values.
Increasing the signal-to-noise ratio by reducing parameters, or adding
additional foreground data, reduces the bias. We also find that neglecting a 1%
polarized free-free or spinning dust component has a negligible effect on r.
These tests highlight the importance of modelling the foregrounds in a way that
allows for sufficient complexity, while minimizing the number of free
parameters.
