Extragalactic foreground contamination in temperature-based CMB lens reconstruction

We discuss the effect of unresolved point source contamination on estimates of the CMB lensing potential, from components such as the thermal Sunyaev-Zel'dovich effect, radio point sources, and the Cosmic Infrared Background. We classify the possible trispectra associated with such source populations, and construct estimators for the amplitude and scale-dependence of several of the major trispectra. We show how to propagate analytical models for these source trispectra to biases for lensing. We also construct a "source-hardened" lensing estimator which experiences significantly smaller biases when exposed to unresolved point sources than the standard quadratic lensing estimator. We demonstrate these ideas in practice using the sky simulations of Sehgal et al., for cosmic-variance limited experiments designed to mimic ACT, SPT, and Planck. We find that for radio sources and SZ the bias is significantly reduced, but for CIB it is essentially unchanged. However, by using the high-frequency, all-sky CIB measurements from Planck and Herschel it may be possible to suppress this contribution