Probing dark energy with the shear-ratio geometric test

We adapt the Jain-Taylor (2003) shear-ratio geometric lensing method to measure the dark energy equation of state, w = pv/ρv and its time derivative from dark matter haloes in cosmologies with arbitrary spatial curvature. The full shear-ratio covariance matrix is calculated for lensed sources, including the intervening large-scale structure and photometric redshift errors as additional sources of noise, and a maximum likelihood method for applying the test is presented. Decomposing the lensing matter distribution into dark matter haloes we calculate the parameter covariance matrix for an arbitrary experiment. Combining with the expected results from the cosmic microwave background (CMB) we design an optimal survey for probing dark energy. This shows that a targeted survey imaging 60 of the largest clusters in a hemisphere with five-band optical photometric redshifts to a median galaxy depth of zm = 0.9 could measure w0 ≡ w(z = 0) to a marginal 1σ error of Δw0 = 0.5. We marginalize over all other parameters including wa, where the equation of state is parametrized in terms of scalefactor a as w(a) = w0 + wa(1 - a). For higher accuracy a large-scale photometric redshift survey is required, where the largest gain in signal arises from the numerous ~1014Msolar haloes corresponding to medium-sized galaxy clusters. Combined with the expected Planck Surveyor results, such a near-future five-band survey covering 10000 deg2 to zm = 0.7 could measure w0 to Δw0 = 0.075 and Δwa = 0.33. A stronger combined constraint is put on w measured at the pivot redshift zp = 0.27 of Δw(zp) = 0.0298. We compare and combine the geometric test with the cosmological and dark energy parameters measured from planned baryon acoustic oscillation (BAO) and supernova Type Ia experiments, and find that the geometric test results combine with a significant reduction in errors due to different degeneracies. A combination of geometric lensing, CMB and BAO experiments could achieve Δw0 = 0.047 and Δwa = 0.111 with a pivot redshift constraint of Δw(zp) = 0.020 at zp = 0.62. Simple relations are presented that show how our lensing results can be scaled to other telescope classes and survey parameters.