Cross bispectra and trispectra of the non-linear integrated Sachs-Wolfe effect and the tracer galaxy density field

In order to investigate possibilities to measure non-Gaussian signatures of the non-linear iSW effect, we study in this work the family of mixed bispectra <tau^q gamma^(3-q)> and trispectra <tau^q gamma^(4-q)> between the integrated Sachs-Wolfe (iSW) temperature perturbation tau and the galaxy over-density gamma. We use standard Eulerian perturbation theory restricted to tree level expansion for predicting the cosmic matter field. As expected, the spectra are found to decrease in amplitude with increasing q. The transition scale between linear domination and the scales, on which non-linearities take over, moves to larger scales with increasing number of included iSW source fields q. We derive the cumulative signal-to-noise ratios for a combination of Planck CMB data and the galaxy sample of a Euclid-like survey. Including scales down to l_max = 1000 we find sobering values of sigma = 0.83 for the mixed bispectrum and sigma = 0.19 in case of the trispectrum for q=1. For higher values of q the polyspectra <tau^2 gamma> and <tau^3 gamma> are found to be far below the detection limit.