Cross-correlation between cosmic microwave background lensing and galaxy intrinsic alignment as a contaminant to gravitational lensing cross-correlated probes of the universe

We introduce here a cross-correlation term between CMB lensing and galaxy intrinsic alignment, noted hereby as $\phi I$. This effect acts as a contaminant to the cross-correlation between CMB lensing and galaxy lensing. The latter cross-correlation has recently been detected for the first time, and measurements will greatly improve as the area of overlap between galaxy and CMB surveys increases and measurements of the CMB polarization become more significant. This will constitute a powerful probe for studying the structure and evolution of the universe. The magnitude of the $\phi I$ term is found to be about 15% of the pure CMB lensing-galaxy lensing component and acts to reduce the magnitude of its measured spectrum. This offset in the spectrum will strongly impact its use for precision cosmological study if left unmitigated. We also propose here a method to calibrate this $\phi I$ contamination through use of a scaling relation that allows one to reduce the impact of $\phi I$ by a factor of 20 or more in all redshift bins, which would reduce its magnitude down to detection limits in almost all cases. This will allow the full use of this probe for precision cosmology.