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Gravitational lensing can provide pure geometric tests of the structure ofspace-time, for instance by determining empirically the angular diameterdistance-redshift relation. This geometric test has been demonstrated severaltimes using massive clusters which produce a large lensing signal. In thiscase, matter at a single redshift dominates the lensing signal, so the analysisis straightforward. It is less clear how weaker signals from multiple sourcesat different redshifts can be stacked to demonstrate the geometric dependence.We introduce a simple measure of relative shear which for flat cosmologiesseparates the effect of lens and source positions into multiplicative terms,allowing signals from many different source-lens pairs to be combined. Applyingthis technique to a sample of groups and low-mass clusters in the COSMOSsurvey, we detect a clear variation of shear with distance behind the lens.This represents the first detection of the geometric effect using weak lensingby multiple, low-mass systems. The variation of distance with redshift ismeasured with sufficient precision to constrain the equation of state of theuniverse under the assumption of flatness, equivalent to a detection of a darkenergy component Omega_X at greater than 99% confidence for anequation-of-state parameter -2.5 < w < -0.1. For the case w = -1, we find avalue for the cosmological constant density parameter Omega_Lambda =0.85+0.044-0.19 (68% C.L.), and detect cosmic acceleration (q_0 < 0) at the 98%C.L.. We consider the systematic uncertainties associated with this techniqueand discuss the prospects for applying it in forthcoming weak-lensing surveys.

Measuring the Geometry of the Universe from Weak Gravitational Lensing behind Galaxy Groups in the HST COSMOS survey