Strongly Lensed Gravitational Waves and Electromagnetic Signals as Powerful Cosmic Rulers

In this paper, we discuss the possibility of using strongly lensed gravitational waves (GWs) and their electromagnetic (EM) counterparts as powerful cosmic rulers. In the EM domain, it has been suggested that joint observations of the time delay ($\Delta\tau$) between lensed quasar images and the velocity dispersion ($\sigma$) of the lensing galaxy (i.e., the combination $\Delta\tau/\sigma^{2}$) are able to constrain the cosmological parameters more strongly than $\Delta\tau$ or $\sigma^{2}$ separately. Here, for the first time, we propose that this $\Delta\tau/\sigma^{2}$ method can be applied to the strongly lensed systems observed in both GW and EM windows. Combining the redshifts, images and $\sigma$ observed in the EM domain with the very precise $\Delta\tau$ derived from lensed GW signals, we expect that accurate multimessenger cosmology can be achieved in the era of third-generation GW detectors. Comparing with the constraints from the $\Delta\tau$ method, we prove that using $\Delta\tau/\sigma^{2}$ can improve the discrimination between cosmological models. Furthermore, we demonstrate that with $\sim50$ strongly lensed GW-EM systems, we can reach a constraint on the dark energy equation of state $w$ comparable to the 580 Union2.1 Type Ia supernovae data. Much more stringent constraints on $w$ can be obtained when combining the $\Delta\tau$ and $\Delta\tau/\sigma^{2}$ methods.