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

The correlation between Lyman-alpha 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 (Omega_Lambda) or other forms of vacuum energy. Using Hydro-PM simulations, I compute the three-dimensional power spectrum of the Lyman-alpha forest flux, P_F(k,mu), from which the redshift-space anisotropy of the correlation can be obtained. I find that box size ~40 Mpc/h and resolution ~40 Kpc/h 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 Lyman-alpha forest, potentially allowing simulation results to be extended to arbitrarily large scales. I investigate the dependence of P_F(k,mu) 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 Omega_Lambda using quasars from a large survey. Assuming 13 (theta/1')^2 pairs at separation <theta, 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).

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