Planck 2013 results. XXVI. Background geometry and topology of the Universe

Planck CMB temperature maps allow us to detect departures from homogeneity and isotropy on the largest scales. We search for topology with a fundamental domain (nearly) intersecting the last scattering surface (comoving distance X_r). For most topologies studied the likelihood maximized over the orientation relative to the observed map shows some preference for multi-connected models just larger than X_r. Since this effect is also present in simulated realizations of isotropic maps, we interpret it as the inevitable alignment of mild anisotropic correlations with chance features in a single sky realization; such a feature can also be present, in milder form, when the likelihood is marginalized over orientations. Thus marginalized, the limits on the radius R_i of the largest sphere inscribed in a topological domain (at log-likelihood-ratio -5) are: in a flat Universe, R_i>0.9X_r for the cubic torus (cf. R_i>0.9X_r at 99% CL for the matched-circles search); R_i>0.7X_r for the chimney; R_i>0.5X_r for the slab; in a positively curved Universe, R_i>1.0X_r for the dodecahedron; R_i>1.0X_r for the truncated cube; and R_i>0.9X_r for the octahedron. We perform a Bayesian search for an anisotropic Bianchi VII_h geometry. In a non-physical setting where the Bianchi parameters are decoupled from the standard cosmology, Planck data favour a Bianchi component with a Bayes factor of at least 1.5 units of log-evidence. Indeed, a Bianchi pattern is quite efficient at accounting for some large-scale anomalies seen in Planck data. However, the cosmological parameters are in strong disagreement with those found from CMB anisotropy data alone. In the physically motivated setting where the Bianchi parameters are fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VII_h cosmology and constrain the vorticity of such models at (w/H)_0