Toward a Measurement of the Cosmological Geometry at z ∼ 2: Predicting Lyα Forest Correlation in Three Dimensions, and the Potential of Future Data Sets

The correlation between Lyα absorption in the spectra of quasar pairs can be used to measure the transverse distance scale at z ∼ 2, which is sensitive to the cosmological constant (ΩΛ) or other forms of vacuum energy. Using Hydro-PM simulations, I compute the three-dimensional power spectrum of the Lyα forest flux, PF (k), from which the redshift-space anisotropy of the correlation can be obtained. I find that box size ∼ 40h−1 Mpc and resolution ∼ 40h−1 kpc are necessary for convergence of the calculations to . 5% on all relevant scales, although somewhat poorer resolution can be used for large scales. I compute directly the linear theory bias parameters of the Lyα forest, potentially allowing simulation results to be extended to arbitrarily large scales. I investigate the dependence of PF (k) on the primordial power spectrum, the temperature-density relation of the gas, and the mean flux decrement, finding that the redshift-space anisotropy is relatively insensitive to these parameters. A table of results is provided for different parameter variations. I investigate the constraint that can be obtained on ΩΛ using quasars from a large survey. Assuming 13 (θ/1 ′) pairs at separation < θ, and including separations < 10′, a measurement to . 5% can be made if simulations can predict the redshift-space anisotropy with . 5% accuracy, or to . 10% if the anisotropy must be measured from the data. The Sloan Digital Sky Survey (SDSS) will obtain spectra for a factor ∼ 5 fewer pairs than this, so followup observations of fainter pair candidates will be necessary. I discuss the requirements on spectral resolution and signal-to-noise ratio (SDSS quality spectra are sufficient). Subject headings: cosmology: theory—intergalactic medium—large-scale structure of universe—methods: N-body simulations—quasars: absorption lines