Planck 2013 results. XXIII. Isotropy and Statistics of the CMB

The two fundamental assumptions of the standard cosmological model - that the initial fluctuations are statistically isotropic and Gaussian - are rigorously tested using maps of the CMB anisotropy from the \Planck\ satellite. The detailed results are based on studies of four independent estimates of the CMB that are compared to simulations using a fiducial $\Lambda$CDM model and incorporating essential aspects of the \Planck\ measurement process. Deviations from isotropy have been found and demonstrated to be robust against component separation algorithm, mask and frequency dependence. Many of these anomalies were previously observed in the \textit{WMAP} data, and are now confirmed at similar levels of significance (around $3\sigma$). However, we find little evidence for non-Gaussianity with the exception of a few statistical signatures that seem to be associated with specific anomalies. In particular, we find that the quadrupole-octopole alignment is also connected to a low observed variance of the CMB signal. The dipolar power asymmetry is now found to persist to much smaller angular scales, and can be described in the low-$\ell$ regime by a phenomenological dipole modulation model. Finally, it is plausible that some of these features may be reflected in the angular power spectrum of the data which shows a deficit of power on the same scales. Indeed, when the power spectra of two hemispheres defined by a preferred direction are considered separately, one shows evidence for a deficit in power, whilst its opposite contains oscillations between odd and even modes that may be related to the parity violation and phase correlations also detected in the data. Whilst these analyses represent a step forward in building an understanding of the anomalies, a satisfactory explanation based on physically motivated models is still lacking.