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