A Robust Estimator for the Lyman-$α$ Forest 1D Power Spectrum at High Redshift

Abstract

The Lyman-$α$ (Ly$α$) forest flux power spectrum $P(k)$ is sensitive to a range of astrophysical and cosmological effects, including the thermal state of and large-scale fluctuations in the intergalactic medium and the nature of dark matter. Measuring $P(k)$ becomes increasingly challenging at high redshift, where fewer quasar spectra are available and higher Ly$α$ optical depths result in lower transmitted flux. In addition, observational effects and missing data complicate standard Fourier-transform estimators. Here, we validate a likelihood-based estimator for $P(k)$. The method constructs the pixel-level covariance matrix of the transmitted flux field using precomputed $k$-space integrals that account for instrumental resolution and pixelization, enabling direct maximum-likelihood inference of $P(k)$. We test the estimator using mock Ly$α$ forest spectra at $z \geq 5$, including analytic Gaussian realizations generated from known power spectra and forward-modeled simulation spectra incorporating observational effects relevant to high-redshift quasar surveys, such as finite spectral resolution, pixel sampling, noise, and missing data. We compare the recovered power spectra with both the input spectra and standard Fourier-transform estimates. We find that the likelihood-based estimator recovers the input power spectrum across the tested scales well within expected error and remains robust to the observational effects considered. This work establishes a framework for measuring Ly$α$ forest clustering with a robust, nearly optimal power spectrum estimator, with particular focus on quantifying large-scale fluctuations that may arise from ultraviolet background fluctuations or the late stages of reionization.