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.Comment: 8 pages, 10 figures, significant update to include more accurate results and conclusions to match version accepted by MNRA

Fast and Exact Spin-s Spherical Harmonic Transforms

We demonstrate a fast spin-s spherical harmonic transform algorithm, which is flexible and exact for band-limited functions. In contrast to previous work, where spin transforms are computed independently, our algorithm permits the computation of several distinct spin transforms simultaneously. Specifically, only one set of special functions is computed for transforms of quantities with any spin, namely the Wigner d-matrices evaluated at {\pi}/2, which may be computed with efficient recursions. For any spin the computation scales as O(L^3) where L is the band-limit of the function. Our publicly available numerical implementation permits very high accuracy at modest computational cost. We discuss applications to the Cosmic Microwave Background (CMB) and gravitational lensing.Comment: 22 pages, preprint format, 5 figure