Cosmological model-independent measurement on cosmic curvature using
  distance sum rule with the help of gravitational waves
  Although the cosmic spatial curvature has been precisely constrained in the
standard cosmological model using the observations of cosmic microwave
background anisotropies, it is still of great importance to independently
measure this key parameter using only the late-universe observations in a
cosmological model-independent way. The distance sum rule in strong
gravitational lensing (SGL) provides such a way, provided that the three
distances in the sum rule can be calibrated by other observations. Usually,
such a calibration can be performed by using the type Ia supernovae (SN Ia),
which, however, has some drawbacks, e.g., dependence on distance ladder, narrow
redshift range, etc. In this paper, we propose that gravitational waves (GWs)
can be used to provide the distance calibration in the SGL method, which can
avoid the dependence on distance ladder and cover a wider redshift range. Using
the simulated GW standard siren observation by the Einstein Telescope as an
example, we show that this scheme is feasible and advantageous. We find that
$\Delta\Omega_k\simeq 0.17$ with the current SGL data, which is slightly more
precise than the case of using SN Ia to calibrate. Furthermore, we consider the
forthcoming LSST survey that is expected to observe many SGL systems. We
simulate a realistic population of SGL systems that will be observed in the
near future, and we find that about $10^4$ SGL data could provide the precise
measurement of $\Delta\Omega_k\simeq 10^{-2}$ with the help of GWs. Our work
indicates that the observations of SGL and GWs by the next-generation
facilities would improve the late-universe measurement of cosmic curvature by
one order of magnitude.
