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 (“$g{\kappa }^{\mathrm{SN}}$”), with shear (“$g{\kappa }^{\mathrm{g}}$”), and with CMB lensing (“$g{\kappa }^{\mathrm{CMB}}$”). We find for the LSST and Stage IV CMB surveys, “$g{\kappa }^{\mathrm{SN}}$”, “$g{\kappa }^{\mathrm{g}}$” and “$g{\kappa }^{\mathrm{CMB}}$” 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 ${\mathrm{\Omega }}_K$ of 0.723 from “$\mathrm{SN}+g{\kappa }^{\mathrm{SN}}$”, 0.0417 from “$\mathrm{SN}+g{\kappa }^{\mathrm{g}}$”, and 0.04 from “$\mathrm{SN}+g{\kappa }^{\mathrm{g}}+g{\kappa }^{\mathrm{CMB}}$” 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 “$\mathrm{SN}+\mathrm{BAO}+\mathrm{CMB}$”, where “CMB” stands for Planck measurement for the CMB acoustic scale, we obtain constraint on ${\mathrm{\Omega }}_K$ of 0.0013, which is a factor of 7 improvement from “$\mathrm{SN}+\mathrm{BAO}+\mathrm{CMB}$”. We study improvements in these results from increasing the high redshift extension of supernovae.

Figure 1

The comoving angular diameter distances to the lens $r_L$, to the source $r_S$, and from the lens to the source $r_{LS}$ provide a probe of curvature altogether. Shown in this plot is the dependence on ${\mathrm{\Omega }}_K$ for the ratio of $(r_L+r_{LS})$ to $r_S$, while $r_L$, $r_S$ fixed at 1500 and $3000h^{-1}\mathrm{Mpc}$ respectively. Here, we have used the approximation for $r_{LS}$ when $|{\mathrm{\Omega }}_K|\ll 1$ [34].

Figure 2

Normalized redshift distributions of type Ia supernovae (histogram) and galaxies (solid line) for the LSST.

Figure 3

The power spectra for the galaxy-supernovae cross-correlation (“$g{\kappa }^{\mathrm{SN}}$”, black), galaxy-galaxy lensing (“$g{\kappa }^{\mathrm{g}}$”, red), and galaxy-CMB lensing cross-correlation (“$g{\kappa }^{\mathrm{CMB}}$”, green). Error bands are forecasted for the LSST and Stage IV CMB experiment. We have divided the galaxy-supernovae cross power and its errors by a factor of $5/\ln (10)$. The lens galaxies are in the redshift bin of $\mathrm{z}:[0.4,0.5]$, while the supernovae and source galaxies are in $\mathrm{z}:[0.5,0.6]$. The error bands for the galaxy-CMB lensing cross power are plotted up to $\ell =3000$.

Figure 4

Forecast $1\sigma$ constraints on curvature as a function of the maximal redshift of supernovae for “SN”(black), “$\mathrm{SN}+g{\kappa }^{\mathrm{SN}}$”(red), “$\mathrm{SN}+g{\kappa }^{\mathrm{g}}+g{\kappa }^{\mathrm{CMB}}$”(green) and “$\mathrm{SN}+g{\kappa }^{\mathrm{g}}+g{\kappa }^{\mathrm{CMB}}$+BAO+CMB”(blue, denoted as “All”), respectively. The total number of supernovae is kept fixed at $4\times {10}^5$. The dark energy equation of state parameter $w$ is assumed to be a binned function parametrized by its values in 50 bins from $a=0$ to $a=1$.