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We present a method to recover the shape and amplitude of the power spectrum of mass fluctuations, P(k), from observations of the high-redshift Lyα forest. The method is motivated by the physical picture that has emerged from hydrodynamic cosmological simulations and related semianalytic models, in which typical Lyα forest lines arise in a diffuse, continuous, fluctuating intergalactic medium. The thermal state of this low-density gas (δρ/ρ 10) is governed by simple physical processes, which lead to a tight correlation between the Lyα optical depth and the underlying matter density. To recover the mass power spectrum, we (1) apply a monotonic Gaussian mapping to convert the QSO spectrum to an approximate line-of-sight density field with arbitrary normalization, (2) measure the power spectrum of this continuous density field and convert it to the equivalent three-dimensional P(k), and (3) evolve cosmological simulations with this P(k) shape and a range of normalizations and choose the normalization for which the simulations reproduce the observed power spectrum of the transmitted QSO flux. Imposing the observed mean Lyα opacity as a constraint in step (3) makes the derived P(k) normalization insensitive to the choice of cosmological parameters, ionizing background spectrum, or reionization history. Thus, in contrast to estimates of P(k) from galaxy clustering, there are no uncertain "bias parameters" in the recovery of the mass power spectrum from the Lyα forest. We test the full recovery procedure on smoothed particle hydrodynamics (SPH) simulations of three different cosmological models and show that it recovers the true mass power spectrum of the models on comoving scales ~1-10 h-1 Mpc, the upper scale being set by the size of the simulation boxes. The procedure works well even when it is applied to noisy (S/N ~ 10), moderate-resolution (~40 km s-1 pixels) spectra. We present an illustrative application to Songaila & Cowie's Keck HIRES spectrum of Q1422+231; the recovered P(k) is consistent with that of an Ω = 1, h = 0.5, σ8(z = 0) 0.5 cold dark matter model. The statistical uncertainty in this result is large because it is based on a single QSO, but the method can be applied to large samples of existing QSO spectra and should thereby yield the power spectrum of mass fluctuations on small and intermediate scales at redshifts z ~ 2-4.


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