Markov chain Monte Carlo analysis of Bianchi VII_h models
Abstract
We have extended the analysis of Jaffe et al. to a complete Markov chain Monte Carlo (MCMC) study of the Bianchi type
VI
I
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:
Ω
tot
=0.43±0.04
,
h=
0.32
+0.02
−0.13
,
ω=
9.7
+1.6
−1.5
×
10
−10
with orientation
γ=
337
∘
+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
Ω
Λ
creates a degeneracy in the
Ω
m
−
Ω
Λ
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
2σ
. Thus we can confirm that a physical Bianchi
VI
I
h
model is not responsible for this signature.