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.
