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

Understand

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.