Measuring Distance Ratios with CMB-Galaxy Lensing Cross-correlations

We propose a method for cosmographic measurements by combining gravitational lensing of the cosmic microwave background (CMB) with cosmic shear surveys. We cross-correlate the galaxy counts in the lens plane with two different source planes: the CMB at $z \sim 1100$ and galaxies at an intermediate redshift. The ratio of the galaxy count/CMB lensing cross-correlation to the galaxy count/galaxy lensing cross correlation is shown to be a purely geometric quantity, depending only on the distribution function of the source galaxies. By combining Planck, ADEPT and LSST the ratio can be measured to $\sim 4%$ accuracy, whereas a future polarization based experiment like CMBPOL can make a more precise ($\sim 1%$) measurement. For cosmological models where the curvature and the equation of state parameter are allowed to vary, the direction of degeneracy defined by the measurement of this ratio is different from that traced out by Baryon Acoustic Oscillation (BAO) measurements. Combining this method with the stacked cluster mass reconstruction cosmography technique as proposed by Hu, Holz and Vale (2007), the uncertainty in the ratio can be further reduced, improving the constraints on cosmological parameters. We also study the implications of the lensing-ratio measurement for early dark energy models, in context of the parameterization proposed by Doran and Robbers (2006). For models which are degenerate with respect to the CMB, we find both BAO and lensing-ratio measurements to be insensitive to the early component of the dark energy density.