CMB lensing reconstruction biases in cross-correlation with large-scale structure probes

The cross-correlation between cosmic microwave background (CMB) gravitational lensing and large-scale structure tracers will be an important cosmological probe in the coming years. Quadratic estimators provide a simple and powerful (if suboptimal) way to reconstruct the CMB lensing potential and are widely used. For Gaussian fields, the cross-correlation of a quadratic-estimator CMB lensing reconstruction with a tracer is exactly unbiased if the power spectra are known and consistent analytic lensing mode response functions are used. However, the bispectrum induced by non-linear large-scale structure growth and post-Born lensing can introduce an additional bias term ($N_L^{(3/2)}$) in the cross-correlation spectrum, similar to the $N_L^{(3/2)}$ bias in the auto-spectrum demonstrated in recent works. We give analytic flat-sky results for the cross-correlation bias using approximate models for the post-Born and large-scale structure cross-bispectra, and compare with N-body simulation results using ray-tracing techniques. We show that the bias can be at the 5-15\% level in all large-scale structure cross-correlations using small-scale CMB temperature lensing reconstruction, but is substantially reduced using polarization-based lensing estimators or simple foreground-projected temperature estimators. The relative magnitude of these effects is almost three times higher than in the CMB lensing auto-correlation, but is small enough that it can be modelled to sufficient precision using simple analytic models. We show that $N_L^{(3/2)}$ effects in cross-correlation will be detected with high significance when using data of future surveys and could affect systematic effects marginalization in cosmic shear measurements mimicking galaxy intrinsic alignment.