Efficient estimation of rotation-induced bias to reconstructed CMB lensing power spectrum

The cosmic microwave background (CMB) lensing power spectrum is a powerful probe of the late-time universe, encoding valuable information about cosmological parameters such as the sum of neutrino masses and dark energy equation of state. However, the presence of anisotropic cosmic birefringence can bias the reconstructed CMB lensing power spectrum using CMB polarization maps, particularly at small scales, and affect the constraints on these parameters. Upcoming experiments, which will be dominated by the polarization lensing signal, are especially susceptible to this bias. We identify the dominant contribution to this bias as an $N_L^{(1)}$-like noise, caused by anisotropic rotation instead of lensing. We show that, for an CMB-S4-like experiment, a scale-invariant anisotropic rotation field with a standard deviation of 0.05 degrees can suppress the small-scale lensing power spectrum ($L\gtrsim 2000$) at a comparable level to the effect of massive neutrino with $\sum_i m_{\nu_{i}}=50~\rm{meV}$, making rotation field an important source of degeneracy in neutrino mass measurement for future CMB experiments. We provide an analytic expression and a simulation-based estimator for this $N_L^{(1)}$-like noise, which allows for efficient forecasting and mitigation of the bias in future experiments. Furthermore, we investigate the impact of a non-scale-invariant rotation power spectrum on the reconstructed lensing power spectrum and find that an excess of power in the small-scale rotation power spectrum leads to a larger bias. Our work provides an effective numeric framework to accurately model and account for the bias caused by anisotropic rotation in future CMB lensing measurements.