How to estimate the 3D power spectrum of the Lyman-$\alpha$ forest

TL;DR

This paper presents a new algorithm for efficiently estimating the 3D power spectrum of the Lyman-alpha forest flux fluctuations, leveraging the data's geometry and approximations to enable scalable analysis of large datasets.

Contribution

The authors develop a novel, computationally efficient algorithm that exploits the geometry of Ly-alpha forest data to estimate the 3D power spectrum, suitable for large upcoming surveys.

Findings

Algorithm accurately estimates the 3D power spectrum.

Method is scalable and suitable for large datasets.

Provides a likelihood approximation for the power spectrum.

Abstract

We derive and numerically implement an algorithm for estimating the 3D power spectrum of the Lyman-$\alpha$ (Ly-$\alpha$) forest flux fluctuations. The algorithm exploits the unique geometry of Ly-$\alpha$ forest data to efficiently measure the cross-spectrum between lines of sight as a function of parallel wavenumber, transverse separation and redshift. The key to fast evaluation is to approximate the global covariance matrix as block-diagonal, where only pixels from the same spectrum are correlated. We then compute the eigenvectors of the derivative of the signal covariance with respect to cross-spectrum parameters, and project the inverse-covariance-weighted spectra onto them. This acts much like a radial Fourier transform over redshift windows. The resulting cross-spectrum inference is then converted into our final product, an approximation of the likelihood for the 3D power spectrum expressed as second order Taylor expansion around a fiducial model. We demonstrate the accuracy and scalability of the algorithm and comment on possible extensions. Our algorithm will allow efficient analysis of the upcoming Dark Energy Spectroscopic Instrument dataset.