  Cosmic microwave background studies of non-Gaussianity involving higher-order
multispectra can distinguish between early universe theories that predict
nearly identical power spectra. However, the recovery of higher-order
multispectra is difficult from realistic data due to their complex response to
inhomogeneous noise and partial sky coverage, which are often difficult to
model analytically. A traditional alternative is to use one-point cumulants of
various orders, which collapse the information present in a multispectrum to
one number. The disadvantage of such a radical compression of the data is a
loss of information as to the source of the statistical behaviour. A recent
study by Munshi & Heavens (2009) has shown how to define the skew spectrum (the
power spectra of a certain cubic field, related to the bispectrum) in an
optimal way and how to estimate it from realistic data. The skew spectrum
retains some of the information from the full configuration-dependence of the
bispectrum, and can contain all the information on non-Gaussianity. In the
present study, we extend the results of the skew spectrum to the case of two
degenerate power-spectra related to the trispectrum. We also explore the
relationship of these power-spectra and cumulant correlators previously used to
study non-Gaussianity in projected galaxy surveys or weak lensing surveys. We
construct nearly optimal estimators for quick tests and generalise them to
estimators which can handle realistic data with all their complexity in a
completely optimal manner. We show how these higher-order statistics and the
related power spectra are related to the Taylor expansion coefficients of the
potential in inflation models, and demonstrate how the trispectrum can
constrain both the quadratic and cubic terms.
