Measuring lensing ratios with future cosmological surveys

The ratio between the CMB lensing/galaxy counts and the galaxy shear/galaxy counts cross-correlations combines the information from different cosmological probes to infer cosmographic measurements that are less dependent on astrophysical uncertainties and can constrain the geometry of the Universe. We discuss the future perspectives for the measurement of this lensing ratio as previously introduced, i.e. with the use of the Limber and flat-sky approximations and neglecting all the effects on the galaxy survey from observing on the past lightcone. We then show how the cosmological information in this estimator is affected by the Limber approximation and by the inclusion of the redshift space distorsions (RSD) and magnification lensing contributions to the galaxy number counts. We find that the magnification lensing contribution induces a multipole dependence of the lensing ratio that we show to be detectable at a statistical significant level combining post-$Planck$ CMB surveys and a Euclid-like experiment. We propose an improved estimator which takes into account this angular scale dependence. Using this extended formalism, we present forecasts for upcoming and future cosmological surveys and we show at which extent the lensing ratio information can improve the CMB constraints on cosmological parameters. We get that for extended cosmological models where the neutrino mass, the spatial curvature and the dark energy equation of state are allowed to vary, the constraints from $Planck$ on these parameters and on $H_0$ can be reduced by $\sim 40\%$ with the inclusion of a single lensing ratio and by $\sim 60-70\%$ adding the joint measurement of 9 lensing ratios with a Euclid-like survey. We also find that neglegcting the contribution from magnification lensing can induce a bias on the derived cosmological parameters in a combined analysis.