Constraining cosmological parameters from strong lensing with DECIGO and B-DECIGO sources

Gravitational lensing has long been used to measure or constrain cosmology models. Although the lensing effect of gravitational waves has not been observed by LIGO/Virgo, it is expected that there can be a few to a few hundreds lensed events to be detected by the future Japanese space-borne interferometers DECIGO and B-DECIGO, if they are running for 4 years. Given the predicted lensed gravitational wave events, one can estimate the constraints on the cosmological parameters via the lensing statistics and the time delay methods. With the lensing statistics method, the knowledge of the lens redshifts, even with the moderate uncertainty, will set the tight bound on the energy density parameter $\Omega_M$ for matter, that is, $0.288\lesssim\Omega_M\lesssim0.314$ at best. The constraint on the Hubble constant $H_0$ can be determined using the time delay method. It is found out that at $5\sigma$, $|\delta H_0|/H_0$ ranges from $3\%$ to $11\%$ for DECIGO, and B-DECIGO will give less constrained results, $8\%-15\%$. In this work, the uncertainties on the luminosity distance and the time delay distance are set to be $10\%$ and $20\%$, respectively. The improvement on measuring these distances will tighten the bounds.