We present a novel, purely geometric probe of cosmology based on measurements of differential time delays between images of strongly lensed quasars due to finite source effects. Our approach is solely dependent on cosmology via a ratio of angular diameter distances, the image separation, and the source size. It thereby entirely avoids the challenges of lens modelling that conventionally limit time delay cosmography, and instead entails the lensed reverberation mapping of the quasar Broad Line Region. We demonstrate that differential time delays are measurable with short cadence spectroscopic monitoring of lensed quasars, through the timing of kinematically identified features within the broad emission lines. This provides a geometric determination of an angular diameter distance ratio complementary to standard probes, and as a result is a potentially powerful new method of constraining cosmology.
The time delay between images of strongly gravitationally lensed quasars is an established cosmological probe. Its limitations, however, include uncertainties in the assumed mass distribution of the lens. We re-examine the methodology of a prior work presenting a geometric probe of cosmology independent of the lensing potential which considers differential time delays over images, originating from spatially-separated photometric signals within a strongly lensed quasar. We give an analytic description of the effect of the differential lensing on the emission line spectral flux for axisymmetric Broad Line Region geometries, with the inclined ring or disk, spherical shell, and double cone as examples. The proposed method is unable to recover cosmological information as the observed time delay and inferred line-of-sight velocity do not uniquely map to the three-dimensional position within the source.
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