Optimal 1D Ly-$\alpha$ Forest Power Spectrum Estimation I: DESI-Lite Spectra

The 1D Ly-$\alpha$ forest flux power spectrum $P_{\mathrm{1D}}$ is sensitive to scales smaller than a typical galaxy survey, and hence ties to the intergalactic medium's thermal state, suppression from neutrino masses and new dark matter models. It has emerged as a competitive framework to study new physics, but also has come with various challenges and systematic errors in analysis. In this work, we revisit the optimal quadratic estimator for $P_{\mathrm{1D}}$, which is robust against the relevant problems such as pixel masking, time evolution within spectrum and quasar continuum errors. We further improve the estimator by introducing a fiducial power spectrum, which enables us to extract more information by alleviating the discreteness of band powers. We meticulously apply our method to synthetic DESI spectra and demonstrate how the estimator overcomes each challenge. We further apply an optimisation scheme that approximates the Fisher matrix to three elements per row and reduces computation time by 60%. We show that we can achieve percent precision in $P_{\mathrm{1D}}$ with 5-year DESI data in the absence of systematics and provide forecasts for different spectral qualities.