Geometrical constraints on curvature from galaxy-lensing cross-correlations

Accurate constraints on curvature provide a powerful probe of inflation. However, curvature constraints based on specific assumptions of dark energy may lead to unreliable conclusions when used to test inflation models. To avoid this, it is important to obtain constraints that are independent on assumptions for dark energy. In this paper, we investigate such constraints on curvature from the geometrical probe constructed from galaxy-lensing cross-correlations. We study comprehensively the cross-correlations of galaxy with magnification, measured from type Ia supernovae's brightnesses ("$g\kappa^{\rm SN}$"), with shear ("$g\kappa^{\rm g}$"), and with CMB lensing ("$g\kappa^{\rm CMB}$"). We find for the LSST and Stage IV CMB surveys, "$g\kappa^{\rm SN}$" , "$g\kappa^{\rm g}$" and "$g\kappa^{\rm CMB}$" can be detected with signal-to-noise ratio $S/N=104,\ 2291,\ 1842$ respectively. When combined with supernovae Hubble diagram ("SN") to constrain curvature, we find galaxy-lensing cross-correlation becomes increasingly important with more degrees of freedom allowed in dark energy. Without any priors, we obtain error on $\Omega_K$ of $0.723$ from "SN + $g\kappa^{\rm SN}$", $0.0417$ from "SN + $g\kappa^{\rm g}$", and $0.04$ from "SN + $g\kappa^{\rm g}$ + $g\kappa^{\rm CMB}$" for the LSST and Stage IV CMB surveys. The last one is more competitive than a Stage IV BAO survey ("BAO"). When galaxy-lensing cross-correlations are added to the combined probe of "SN + BAO + CMB", where "CMB" stands for Planck measurement for the CMB acoustic scale, we obtain constraint on $\Omega_K$ of $0.0013$, which is a factor of 7 improvement from "SN + BAO + CMB". We study improvements in these results from increasing the high redshift extension of supernovae.