On the Coherence of WMAP and Planck Temperature Maps
  The recent data release of ESA's Planck mission together with earlier WMAP
releases provide the first opportunity to compare high resolution full sky
Cosmic Microwave Background temperature anisotropy maps. To quantify the
coherence of these maps beyond the power spectrum we introduce Generalized
Phases, unit vectors in the (2l+1) dimensional representation spaces. For a
Gaussian distribution, Generalized Phases are random and if there is
non-Gaussianity, they represent most of the non-Gaussian information. The
alignment of these unit vectors from two maps can be characterized by their
angle, 0 deg expected for full coherence, and 90 deg for random vectors. We
analyze maps from both missions with the same mask and Nside=512 resolution,
and compare both power spectra and Generalized Phases. We find excellent
agreement of the Generalize Phases of Planck Smica map with that of the WMAP
Q,V,W maps, rejecting the null hypothesis of no correlations at 5 sigma for l's
l<700, l<900 and l<1100, respectively, except perhaps for l<10. Using
foreground reduced maps for WMAP increases the phase coherence. The observed
coherence angles can be explained with a simple assumption of Gaussianity and a
WMAP noise model neglecting Planck noise, except for low-intermediate l's there
is a slight, but significant off-set, depending on WMAP band. On the same
scales WMAP power spectrum is about 2.6% higher at a very high significance,
while at higher l's there appears to be no significant bias. Using our
theoretical tools, we predict the phase alignment of Planck with a hypothetical
perfect noiseless CMB experiment, finding decoherence at l > 2900; below this
value Planck can be used most efficiently to constrain non-Gaussianity.
