Fast generation of weak lensing maps with analytical point transformation functions

Nonlinear cosmological fields like galaxy density and lensing convergence can be approximately related to Gaussian fields via analytic point transforms. The LogNormal (LN) transform has been widely used and is a simple example of a function that relates nonlinear fields to Gaussian fields. We consider more accurate general point-transformed Gaussian (GPTG) functions for such a mapping and apply them to convergence maps. We show that we can create maps that preserve the LN's ability to exactly match any desired power spectrum but go beyond LN by significantly improving the accuracy of the probability distribution function (PDF). With the aid of symbolic regression, we find a remarkably accurate GPTG function for convergence maps: its higher-order moments, scattering wavelet transform, Minkowski functionals, and peak counts match those of $N$-body simulations to the statistical uncertainty expected from tomographic lensing maps of the Rubin LSST 10 years survey. Our five-parameter function performs 2 to $5\ifmmode\times\else\texttimes\fi{}$ better than the LogNormal in terms of accuracy of tested non-Gaussian summary statistics. Since baryonic feedback has a strong influence on very small scales, we restrict our study to scales larger than 7 arcmin. We demonstrate that the GPTG can robustly emulate variations in cosmological parameters due to the simplicity of the analytic transform. This opens up several possible applications, such as field-level inference, rapid covariance estimation, and other uses based on the generation of arbitrarily many maps with laptop-level computation capability.