Measuring the cosmological geometry from the Lyα forest along parallel lines of sight

The possibility of measuring the cosmological geometry using the redshift space correlation function of the Ly alpha forest in multiple lines of sight as a function of angular and velocity separation is discussed. The geometric parameter to be measured is f(z) = c(-1)H(z)D-A(z), where H(z) is the Hubble constant and D-A(z) the angular diameter distance at redshift z. The correlation function is computed in linear theory, assuming that the Ly alpha forest is a result of gravitational instability in a photoionized intergalactic medium. We describe a method to measure the correlation from observations with the Gaussianization procedure of Croft et al. to map the observed Ly alpha forest transmitted flux to an approximation of the linear density field. The effect of peculiar velocities on the shape of the recovered power spectrum is pointed out. We estimate the error in recovering the f(z) factor from observations due to the variance in the Ly alpha absorbers. We show that at least similar to 25 pairs of quasars (separations <3') are needed to distinguish a flat Omega(0) = 1 universe from a universe with Omega(0) = 0.2, Omega(Lambda) = 0.8. A second parameter that is obtained from the correlation function of the Ly alpha forest is beta similar or equal to Omega(z)(0.6)/b (affecting the magnitude of the peculiar velocities), where b is a linear theory bias of the Ly alpha forest. In the theory of the Ly alpha forest assumed here, the parameter beta can be predicted from numerical simulations; once beta is known, the number of quasar pairs needed to constrain f is reduced to about six. On small scales, where the correlation function is higher, f(z) should be measurable with fewer quasars, but nonlinear effects must then be taken into account. The anisotropy of the nonlinear redshift space correlation function as a function of scale should also provide a precise quantitative test of the gravitational instability theory of the Ly alpha forest.