Reconstructing spatially-varying multiplicative bias for Stage IV weak lensing galaxy surveys with a quadratic estimator

We present a quadratic estimator that detects and reconstructs spatially-varying multiplicative ($m-$) bias in weak lensing shear measurements, by exploiting the $EB$ mode coupling that it generates. The method combines $E$ and $B$ modes with inverse-variance weights, to yield an unbiased reconstruction of $m(\boldsymbol{\theta})$ to first order. We study the ability of future Stage IV surveys to obtain an unbiased reconstruction of the $m$-bias in differing scenarios, considering differing bias morphologies, and characteristic scales, as well as differing metrics to quantify the signal-to-noise ratio of the reconstructed map. Considering an $m$ pattern repeating on $\sim 1^\circ\times1^\circ$ sky patches, as might be the case for an $m$ field caused by focal-plane systematics. With a Euclid-like redshift distribution, we find that $\sim5\%$ rms variations in $m$-bias may be detected at the 20$\sigma$ level, after stacking between $\sim400$ and $\sim1000$ patches (rising to between $\sim2800$ and $\sim7600$ for $1\%$ rms variations, data volumes that are becoming available with upcoming surveys), depending on the morphology of the $m$ pattern. We show that these results are robust against the cosmological model assumed in the reconstruction, as well as the presence of intrinsic alignments or baryonic effects, and that the method shows no spurious response to additive ($c-$) bias. These results demonstrate that percent-level, spatially-varying $m-$bias can be detected at high significance, enabling diagnosis and mitigation in the Stage IV weak lensing era.