Characterizing the Lyman-$\alpha$ flux probability distribution function using Legendre polynomials

TL;DR

This paper introduces a Legendre polynomial expansion method for analyzing the Lyman-alpha flux PDF, enabling more robust and efficient extraction of information from noisy data compared to traditional binning methods.

Contribution

It proposes a novel Legendre polynomial approach to measure PDF coefficients, improving noise resilience and data compression over standard binned analyses.

Findings

Finite Legendre coefficients are well measured with noise

Information recovery is highly non-linear with respect to spectral noise

Fewer high-quality spectra outperform many noisy measurements

Abstract

The Lyman-$\alpha$ forest is a highly non-linear field with a lot of information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polyonomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, $n$-th coefficient can be expressed as a linear combination of the first $n$ moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. We find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.