Metric Tests for Curvature from Weak Lensing and Baryon Acoustic Oscillations

We describe a practical measurement of the curvature of the Universe which relies purely on the properties of the Robertson-Walker metric rather than any model for the dynamics or content of the Universe. The cross-correlation between foreground mass and gravitational shear of background galaxies depends upon the angular diameter distances d_A(z_l), d_A(z_s), and d_A(z_s,z_l) on the degenerate triangle formed by observer, source, and lens. In a flat Universe, d_A(z_l,z_s) = d_A(z_s)-d_A(z_l), but in curved Universes an additional term $\propto\Omega_k$ alters the lensing observables even if d_A(z) is fixed. Weak lensing data may be used to solve simultaneously for d_A and the curvature. This method is completely insensitive to: the equation of state; amendments to the General Relativity formulae for the deflection of light or the growth of structure; or biases in the photometric redshift scale. There is however a degeneracy among d_A, $\Omega_k$ and the galaxy bias factors, that can be broken by using measurements of baryon acoustic oscillations with the same imaging data. Ambitious weak-lensing + baryon-oscillation surveys would measure $\Omega_k$ to an accuracy $\approx0.04 f_{\rm sky}^{-1/2} (\sigma_{\ln z}/0.04)^{1/2}$, where $\sigma_{\ln z}$ is the photometric redshift error. We also predict bounds on curvature and other parameters in the context of specific dark-energy models, and compare to other analyses of the weak lensing cross-correlation method. We find both curvature and parametric constraints to be surprisingly insensitive to systematic shear calibration errors.