Secondary non-Gaussianity and Cross-Correlation Analysis

We develop optimised estimators of two sorts of power spectra for fields defined on the sky, in the presence of partial sky coverage. The first is the cross-power spectrum of two fields on the sky; the second is the skew spectrum of three fields. The cross-power spectrum of the Cosmic Microwave Background (CMB) sky with tracers of large-scale-structure is useful as it provides valuable information on cosmological parameters. Numerous recent studies have proved the usefulness of cross-correlating CMB sky with external data sets, which probes the Integrated Sachs Wolfe Effect (ISW) at large angular scales and the Sunyaev Zéldovich (SZ) effect from hot gas in clusters at small angular scales. The skew spectrum, recently introduced by Munshi & Heavens (2009), is an optimised statistic which can be tuned to study a particular form of non-gaussianity, such as may arise in the early Universe, but which retains information on the nature of non-gaussianity. In this paper we develop the mathematical formalism for the skew spectrum of 3 different fields. When applied to the CMB, this allows us to explore the contamination of the skew spectrum by secondary sources of CMB fluctuations. Considering the threepoint function, the study of the bispectrum provides valuable information regarding cross-correlation of secondaries with lensing of CMB with much higher significance compared to just the study involving CMB sky alone. After developing the analytical model we use them to study specific cases of cosmological interest which include cross-correlating CMB with various large scale tracers to probe ISW and SZ effects for cross spectral analysis. Next we use the formalism to study the signal-to-noise ratio for detection of the weak lensing of the CMB by cross-correlating it with different tracers as well as point sources for CMB experiments such as Planck.