Constraints on cosmic curvature with lensing time delays and gravitational waves

Assuming the $\mathrm{\Lambda }$CDM model, the CMB and BAO observations indicate a very flat Universe. Model-independent measurements are therefore worth studying. Time delays measured in lensed quasars provide the time delay distances. When compared with the luminosity distances from Supernova Ia observation, the measurements can provide the curvature information under the Distance Sum Rule of FLRW metric. This method is limited by the low redshifts of SNe Ia. In this work, we propose gravitational waves from the Einstein Telescope as standard sirens which reach higher redshifts covering the redshift range of lensed quasars from Large Synoptic Survey Telescope, could provide much more stringent constraints on the curvature. We first consider a conservative case where only 100 gravitational waves with electromagnetic counterparts are available, the $1\sigma$ uncertainty for the curvature parameter ${\mathrm{\Omega }}_k$ is 0.057. In an optimistic case with 1000 signals available, then ${\mathrm{\Omega }}_k$ uncertainty is 0.027. Combining with SNe Ia from Dark Energy Survey, ${\mathrm{\Omega }}_k$ can be further constrained to 0.027 and 0.018, respectively.