Tensor Minkowski Functionals: first application to the CMB
  Tensor Minkowski Functionals (TMFs) are tensorial generalizations of the
usual Minkowski Functionals which are scalar quantities. We introduce them here
for use in cosmological analysis, in particular to analyze CMB maps. They
encapsulate information about the shapes and the orientation of structures. We
focus on one of the TMFs, namely $W_2^{1,1}$, which is the generalization of
the genus. The ratio of the eigenvalues of the average of $W_2^{1,1}$ over all
structures, $\alpha$, encodes the net orientation; and the average of the
ratios of the eigenvalues of $W_2^{1,1}$ for each structure, $\beta$, encodes
the net anisotropy. We have developed a code that computes $W_2^{1,1}$, and
from it $\alpha$ and $\beta$, for a set of structures on the plane. We compute
$\alpha$ and $\beta$ as functions of threshold levels for simulated Gaussian
and isotropic CMB fields. We obtain $\alpha$ to be one for both temperature and
$E$ mode, which means that we recover the input statistical isotropy of density
fluctuations in the simulations. The level of net anisotropy of hotspots and
coldspots in the CMB fields is quantified by $\beta\sim 0.62$. Then we compute
$\alpha$ and $\beta$ for data from PLANCK. We find that the temperature field
agrees with the standard LCDM prediction of no net orientation within
$3-\sigma$. However, we find that $E$ mode data shows a net orientation that
deviates from the theoretical expectation at $14-\sigma$. The possible origin
of this deviation may be instrumental effects or other sources and needs to be
investigated further. For the net anisotropy we obtain values of $\beta$ for
both temperature and $E$ mode that are consistent with the expectations from
the standard LCDM simulations. Accurate measurements of $\alpha$ and $\beta$
can be used to test the standard model of cosmology and to search for
deviations from it.
