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.
