Forecast Cosmological Constraints with the 1D Wavelet Scattering Transform and the Lyman-$\alpha$ forest

We make forecasts for the constraining power of the 1D Wavelet Scattering Transform (WST) in the context of Lyman-$\alpha$ forest cosmology. Using mock simulations and a Fisher matrix, we show that there is considerable cosmological information in the scattering transform coefficients. We estimate mock covariance matrices assuming uncorrelated Gaussian pixel noise for each quasar, at a level drawn from a simple lognormal model. The extra information comes from a smaller estimated covariance in the first-order wavelet power, and from second-order wavelet coefficients which probe non-Gaussian information in the forest. Forecast constraints on cosmological parameters from the WST are as much as an order of magnitude tighter than for the power spectrum. Should these constraints be confirmed on real data, it would substantially improve cosmological constraints on, for example, neutrino mass.