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 $\chi_r$).
For most topologies studied the likelihood maximized over orientation shows
some preference for multi-connected models just larger than $\chi_r$. 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, $R_i>0.9\chi_r$
for the cubic torus (cf. $R_i>0.9\chi_r$ at 99% CL for a matched-circles
search); $R_i>0.7\chi_r$ for the chimney; $R_i>0.5\chi_r$ for the slab; in a
positively curved Universe, $R_i>1.0\chi_r$ for the dodecahedron;
$R_i>1.0\chi_r$ for the truncated cube; $R_i>0.9\chi_r$ 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: $(\omega/H)_0<8\times10^{-10}$ (95%
CL). [Abridged]
