Auto from cross: CMB lensing power spectrum without noise bias

Upcoming surveys will measure the cosmic microwave background (CMB) weak lensing power spectrum in exquisite detail, allowing for strong constraints on the sum of neutrino masses among other cosmological parameters. Standard CMB lensing power spectrum estimators aim to extract the connected non-Gaussian trispectrum of CMB temperature maps. However, they are generically dominated by a large Gaussian noise bias which thus needs to be subtracted at high accuracy. This is currently done with realistic map simulations of the CMB and noise, whose finite accuracy currently limits our ability to recover the CMB lensing on small-scale. In this paper, we propose a novel estimator which instead avoids this large Gaussian bias. This estimator relies only on the data and avoids the need for bias subtraction with simulations. Thus our bias avoidance method is (1) insensitive to misestimates in simulated CMB and noise models and (2) avoids the large computational cost of standard simulation-based methods like "realization-dependent N^(0)" (RDN^(0)). We show that our estimator is as robust as standard methods in the presence realistic inhomogeneous noise (e.g. from scan strategy) and masking. Moreover, our method can be combined with split-based methods, making it completely insensitive to mode coupling from inhomogeneous atmospheric and detector noise. We derive the corresponding expressions for our estimator when estimating lensing from CMB temperature and polarization. Although in this paper we specifically consider CMB weak lensing power spectrum estimation, we illuminate the relation between our new estimator, RDN^(0) subtraction, and general optimal trispectrum estimation. Through this discussion we conclude that our estimator is applicable to analogous problems in other fields which rely on estimating connected trispectra/four-point functions like large-scale structure.