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

Planck CMB temperature maps allow detection of large-scale departures from homogeneity and isotropy. We search for topology with a fundamental domain nearly intersecting the last scattering surface (comoving distance \$\backslash chi\_r\$). For most topologies studied the likelihood maximized over orientation shows some preference for multi-connected models just larger than \$\backslash chi\_r\$. This effect is also present in simulated realizations of isotropic maps and we interpret it as the alignment of mild anisotropic correlations with chance features in a single realization; such a feature can also exist, 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>0.9\backslash chi\_r\$ for the cubic torus (cf. \$R\_i>0.9\backslash chi\_r\$ at 99\% CL for a matched-circles search); \$R\_i>0.7\backslash chi\_r\$ for the chimney; \$R\_i>0.5\backslash chi\_r\$ for the slab; in a positively curved Universe, \$R\_i>1.0\backslash chi\_r\$ for the dodecahedron; \$R\_i>1.0\backslash chi\_r\$ for the truncated cube; \$R\_i>0.9\backslash chi\_r\$ for the octahedron. Similar limits apply to alternate topologies. We perform a Bayesian search for an anisotropic Bianchi VII\$\_h\$ geometry. In a non-physical setting where the Bianchi parameters are decoupled from cosmology, Planck data favour a Bianchi component with a Bayes factor of at least 1.5 units of log-evidence: a Bianchi pattern is efficient at accounting for some large-scale anomalies 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 standard cosmological parameters, we find no evidence for a Bianchi VII\$\_h\$ cosmology and constrain the vorticity of such models: \$(\backslash omega/H)\_0<8\backslash times10\^{}\{-10\}\$ (95\% CL).