Robust measurement of supernova $ u_e$ spectra with future neutrino detectors

Measuring precise all-flavor neutrino information from a supernova is crucial for understanding the core-collapse process as well as neutrino properties. We apply a chi-squared analysis for different detector setups to explore determination of $ u_{e}$ spectral parameters. Using a long-term two-dimensional core-collapse simulation with three time varying spectral parameters, we generate mock data to examine the capabilities of the current Super-Kamiokande detector and compare the relative improvements that gadolinium, Hyper-Kamiokande, and DUNE would have. We show that in a realistic three spectral parameter framework, the addition of gadolinium to Super-Kamiokande allows for a qualitative improvement in $ u_e$ determination. Significant improvements will be made by Hyper-Kamiokande and DUNE, allowing for much more precise determination of $ u_e$ spectral parameters.