Metric Tests for Curvature from Weak Lensing and Baryon Acoustic Oscillations - Bernstein, G (The Astrophysical journal, 2006)
We describe a practical measurement of the curvature of the universe which, unlike current constraints, relies purely on the properties of the Robertson-Walker metric rather than any assumed model for the dynamics and content of the universe. The observable quantity is the cross-correlation between foreground mass and gravitational shear of background galaxies, which depends on the angular diameter distances d sub(A)(z sub(l)), d sub(A)(z sub(s)) and d sub(A)(z sub(s),z sub(l)) on the degenerate triangle formed by observer, source, and lens. In a flat universe, d sub(A)(z sub(l),z sub(s)) = d sub(A)(z sub(s))-d sub(A)(z sub(l)), but in curved universes an additional term 8 sub(k) appears and alters the lensing observables even if d sub(A)(z) is fixed. We describe a method whereby weak-lensing data can be used to solve simultaneously for d sub(A) and the curvature. This method is completely insensitive to the equation of state of the contents of the universe, or amendments to general relativity that alter the gravitational deflection of light or the growth of structure. The curvature estimate is also independent of biases in the photometric redshift scale. This measurement is shown to be subject to a degeneracy among d sub(A), sub(k), and the galaxy bias factors that may be broken by using the same imaging data to measure the angular scale of baryon acoustic oscillations. Simplified estimates of the accuracy attainable by this method indicate that ambitious weak-lensing + baryon-oscillation surveys would measure sub(k) to an accuracy -0.04f super(-) sub(s) super(1) sub(k) super(/) sub(y) super(2) (s sub(ln z)/0.04) super(1/2), where s sub(ln z) is the photometric redshift error. The Fisher-matrix formalism developed here is also useful for predicting bounds on curvature and other characteristics of parametric dark energy models. We forecast some representative error levels and compare ours to other analyses of the weak-lensing cross-correlation method. We find both curvature and parametric constraints to be surprisingly insensitive to the systematic shear calibration errors.