Squared temperature-temperature power spectrum as a probe of the CMB bispectrum

It is now well known that mode-coupling effects associated with certain secondary effects generate higher order correlations in cosmic microwave background (CMB) temperature anisotropies, beyond the two-point function. In order to extract the non-Gaussian signal at the three-point level, we suggest a two-point statistic in the form of an angular power spectrum involving correlations between squared temperature and temperature anisotropies. This power spectrum contains compressed information from the bispectrum and can be easily measured in data with the same techniques that have been considered for the measurement of the usual temperature-temperature anisotropy power spectrum. We study the proposed power spectrum resulting from the non-Gaussian signal generated by correlations involved with gravitational lensing angular deflections in CMB and the Sunyave-Zel'dovich (SZ) effect due to large scale pressure fluctuations. Using the Planck frequency cleaned CMB and SZ maps, the CMB^2-SZ power spectrum provides a direct estimate of the cross-power between lensing angular deflections and the SZ effect. Through an optimal filter applied to the squared CMB map, the proposed statistic allows one to obtain all information from the lensing-SZ bispectrum. The observational measurement of the lensing-SZ cross-correlation is useful to understand the relation between large scale structure pressure and dark matter fluctuations.