Reconstructing small-scale lenses from the cosmic microwave background temperature fluctuations

Cosmic microwave background (CMB) lensing is a powerful probe of the matter distribution in the Universe. The standard quadratic estimator, which is typically used to measure the lensing signal, is known to be suboptimal for low-noise polarization data from next-generation experiments. In this paper, we explain why the quadratic estimator will also be suboptimal for measuring lensing on very small scales, even for measurements in temperature where this estimator typically performs well. Though maximum likelihood methods could be implemented to improve performance, we explore a much simpler solution, revisiting a previously proposed method to measure lensing that involves a direct inversion of the background gradient. An important application of this simple formalism is the measurement of cluster masses with CMB lensing. We find that directly applying a gradient inversion matched filter to simulated lensed images of the CMB can tighten constraints on cluster masses compared to the quadratic estimator. While the difference is not relevant for existing surveys, for future surveys it can translate to significant improvements in mass calibration for distant clusters, where galaxy lensing calibration is ineffective due to the lack of enough resolved background galaxies. Improvements can be as large as ${\sim } 50{{\ \rm per\ cent}}$ for a cluster at z = 2 and a next-generation CMB experiment with 1 $\mu$K arcmin noise, and over an order of magnitude for lower noise levels. For future surveys, this simple matched filter or gradient inversion method approaches the performance of maximum likelihood methods, at a fraction of the computational cost.