Metric Tests for Curvature from Weak Lensing and Baryon Acoustic Oscillations

We describe a practical measurement of the curvature of the Universe whichrelies purely on the properties of the Robertson-Walker metric rather than anymodel for the dynamics or content of the Universe. The cross-correlationbetween foreground mass and gravitational shear of background galaxies dependsupon the angular diameter distances d_A(z_l), d_A(z_s), and d_A(z_s,z_l) on thedegenerate 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. Weaklensing data may be used to solve simultaneously for d_A and the curvature.This method is completely insensitive to: the equation of state; amendments tothe General Relativity formulae for the deflection of light or the growth ofstructure; or biases in the photometric redshift scale. There is however adegeneracy among d_A, $\Omega_k$ and the galaxy bias factors, that can bebroken by using measurements of baryon acoustic oscillations with the sameimaging data. Ambitious weak-lensing + baryon-oscillation surveys would measure$\Omega_k$ to an accuracy $\approx0.04 f_{\rm sky}^{-1/2} (\sigma_{\lnz}/0.04)^{1/2}$, where $\sigma_{\ln z}$ is the photometric redshift error. Wealso predict bounds on curvature and other parameters in the context ofspecific dark-energy models, and compare to other analyses of the weak lensingcross-correlation method. We find both curvature and parametric constraints tobe surprisingly insensitive to systematic shear calibration errors.Comment: 26 pages, accepted to ApJ. New notation and minor change