Unified reconstruction of the Lyman-alpha power spectrum with Hamiltonian Monte Carlo

The complex geometry of the Ly$\alpha$ forest data has motivated the use of various two-point statistics as alternatives to the three-dimensional power spectrum ($P_{3\mathrm{D}}$), which carries cosmological information in Fourier space. On large scales, the three-dimensional correlation function (${\xi }_{3\mathrm{D}}$) has provided robust measurements of the baryon acoustic oscillation (BAO) scale at 150~Mpc. On smaller scales, the one-dimensional power spectrum, $P_{1\mathrm{D}}(k_{\mathrm{\parallel }})$, has been the primary tool for extracting information. At the same time, the cross-spectrum, $P_{\times }(\theta ,k_{\mathrm{\parallel }})$, has been introduced to incorporate angular information without the complications caused by survey window functions. We propose an analytical forward-modeling framework to reconstruct $P_{3\mathrm{D}}$ from all these observables, based on the mathematical relation between them and $P_{3\mathrm{D}}$. We demonstrate the performance of our method using a hypothetical mock data vector representative of future Dark Energy Spectroscopic Instrument (DESI) measurements. We show that the monopole of $P_{3\mathrm{D}}$ can be reconstructed in 25 $k$ bins between $0.07{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ and $1.8{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$, achieving an average precision of ${\sigma }_P\mathrm{/}P=13\mathrm{\%}$ across the bins. Our method can serve as an intermediary for consistency checks, though it is not intended to replace direct $P_{3\mathrm{D}}$ estimation.