Markov chain Monte Carlo analysis of Bianchi VII_h models

We have extended the analysis of Jaffe et al. to a complete Markov chain Monte Carlo (MCMC) study of the Bianchi type ${\rm VII_h}$ models including a dark energy density, using 1-year and 3-year Wilkinson Microwave Anisotropy Probe (WMAP) cosmic microwave background (CMB) data. Since we perform the analysis in a Bayesian framework our entire inference is contained in the multidimensional posterior distribution from which we can extract marginalised parameter constraints and the comparative Bayesian evidence. Treating the left-handed Bianchi CMB anisotropy as a template centred upon the `cold-spot' in the southern hemisphere, the parameter estimates derived for the total energy density, `tightness' and vorticity from 3-year data are found to be: $\Omega_{tot} = 0.43\pm 0.04$, $h = 0.32^{+0.02}_{-0.13}$, $\omega = 9.7^{+1.6}_{-1.5}\times 10^{-10}$ with orientation $\gamma = {337^{\circ}}^{+17}_{-23}$). This template is preferred by a factor of roughly unity in log-evidence over a concordance cosmology alone. A Bianchi type template is supported by the data only if its position on the sky is heavily restricted. The low total energy density of the preferred template, implies a geometry that is incompatible with cosmologies inferred from recent CMB observations. Jaffe et al. found that extending the Bianchi model to include a term in $\Omega_{\Lambda}$ creates a degeneracy in the $\Omega_m - \Omega_{\Lambda}$ plane. We explore this region fully by MCMC and find that the degenerate likelihood contours do not intersect areas of parameter space that 1 or 3 year WMAP data would prefer at any significance above $2\sigma$. Thus we can confirm that a physical Bianchi ${\rm VII_h}$ model is not responsible for this signature.