Absorber Model: the Halo-like model for the Lyman-alpha forest

We present a semi-analytic model for the Lyman-$\alpha$ forest that is inspired by the Halo Model. This model is built on the absorption line decomposition of the forest. Flux correlations are decomposed into those within each absorption line (the 1-absorber term) and those between separate lines (the 2-absorber term), treating the lines as biased tracers of the underlying matter fluctuations. While the nonlinear exponential mapping between optical depth and flux requires an infinite series of moments to calculate any statistic, we show that this series can be resumed (truncating at the desired order in the linear matter overdensity). We focus on the $z=2-3$ line-of-sight power spectrum. Our model finds that 1-absorber term dominates the power on all scales, with most of its contribution coming from HI columns of $10^{14}-10^{15}\;\mathrm{cm^{-2}}$, while the smaller 2-absorber contribution comes from lower columns that trace overdensities of a few. The prominence of the 1-absorber correlations indicates that the line-of-sight power spectrum is shaped principally by the lines' number densities and their absorption profiles, with correlations between lines contributing to a lesser extent. We present intuitive formulae for the effective optical depth as well as the large-scale limits of 1-absorber and 2-absorber terms, which simplify to integrals over the HI column density distribution with different equivalent-width weightings. With minimalist models for the bias of absorption systems and their peculiar velocity broadening, our model predicts values for the density bias and velocity gradient bias that are consistent with those found in simulations.