Secondary non-Gaussianity and cross-correlation analysis

We develop optimized 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 the CMB sky with external data sets, which probes the integrated Sachs–Wolfe (ISW) effect at large angular scales and the Sunyaev–Zel’dovich (SZ) effect from hot gas in clusters at small angular scales. The skew spectrum, recently introduced by Munshi & Heavens, is a powerful statistic, as it is optimized to study particular forms of non-Gaussianity, such as may arise in the early Universe, but in addition, it retains information on the nature of non-Gaussianity. As such, it allows a robust statistical analysis, where contributions from primordial and contaminating non-Gaussianity can be estimated. In this paper we develop the mathematical formalism for the skew spectrum of three different fields. When applied to the CMB, this allows us to explore the contamination of the skew spectrum by secondary sources of CMB fluctuations, in the case where the foreground contamination and the primary signal are not independent. After developing the analytical model we use them to study specific cases of cosmological interest which include cross-correlating the CMB with various large-scale tracers to probe the ISW and SZ effects for cross-spectral analysis. Next we use the formalism to study the signal-to-noise ratio for the 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.