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 (Ω_Λ) or other forms of vacuum energy. Using hydro-PM simulations, I compute the three-dimensional power spectrum of the Lyα forest flux, P_F(), from which the redshift-space anisotropy of the correlation can be obtained. 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 P_F() 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 Ω_Λ using quasars from a large survey. Assuming 13 ² pairs at a separation of less than θ and including separations less than 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 of ~5 fewer pairs than this, so follow-up observations of fainter pair candidates will be necessary. I discuss the requirements on the spectral resolution and signal-to-noise ratio (SDSS quality spectra are sufficient). I find that a box size of ~40 h^-1 Mpc and a resolution of ~40 h^-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.