New Approaches to Probing Minkowski Functionals
  We generalize the concept of the ordinary skew-spectrum to probe the effect
of non-Gaussianity on the morphology of Cosmic Microwave Background (CMB) maps
in several domains: in real-space (where they are commonly known as
cumulant-correlators), and in harmonic and needlet bases. The essential aim is
to retain more information than normally contained in these statistics, in
order to assist in determining the source of any measured non-Gaussianity, in
the same spirit as Munshi & Heavens (2010) skew-spectra were used to identify
foreground contaminants to the CMB bispectrum in Planck data. Using a
perturbative series to construct the Minkowski Functionals (MFs), we provide a
pseudo-Cl based approach in both harmonic and needlet representations to
estimate these spectra in the presence of a mask and inhomogeneous noise.
Assuming homogeneous noise we present approx- imate expressions for error
covariance for the purpose of joint estimation of these spectra. We present
specific results for four different models of primordial non-Gaussianity local,
equilateral, orthogonal and enfolded models, as well as non-Gaussianity caused
by unsubtracted point sources. Closed form results of next-order corrections to
MFs too are obtained in terms of a quadruplet of kurt-spectra. We also use the
method of modal decomposition of the bispectrum and trispectrum to reconstruct
the MFs as an alternative method of reconstruction of morphological properties
of CMB maps. Finally, we introduce the odd-parity skew-spectra to probe the
odd-parity bispectrum and its impact on the morphology of the CMB sky. Although
developed for the CMB, the generic results obtained here can be useful in other
areas of cosmology.
