  We perform a definitive analysis of Bianchi VII_h cosmologies with WMAP
observations of the cosmic microwave background (CMB) temperature anisotropies.
Bayesian analysis techniques are developed to study anisotropic cosmologies
using full-sky and partial-sky, masked CMB temperature data. We apply these
techniques to analyse the full-sky internal linear combination (ILC) map and a
partial-sky, masked W-band map of WMAP 9-year observations. In addition to the
physically motivated Bianchi VII_h model, we examine phenomenological models
considered in previous studies, in which the Bianchi VII_h parameters are
decoupled from the standard cosmological parameters. In the two
phenomenological models considered, Bayes factors of 1.7 and 1.1 units of
log-evidence favouring a Bianchi component are found in full-sky ILC data. The
corresponding best-fit Bianchi maps recovered are similar for both
phenomenological models and are very close to those found in previous studies
using earlier WMAP data releases. However, no evidence for a phenomenological
Bianchi component is found in the partial-sky W-band data. In the physical
Bianchi VII_h model we find no evidence for a Bianchi component: WMAP data thus
do not favour Bianchi VII_h cosmologies over the standard Lambda Cold Dark
Matter (LCDM) cosmology. It is not possible to discount Bianchi VII_h
cosmologies in favour of LCDM completely, but we are able to constrain the
vorticity of physical Bianchi VII_h cosmologies at $(\omega/H)_0 < 8.6 \times
10^{-10}$ with 95% confidence.
