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 which 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 $\sim 50\%$ 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.
