Complementary constraints on dark energy equation of state from strongly lensed gravitational wave

It has been shown that time delay of strong gravitational lensing is not only an effective cosmological probe to constrain the Hubble constant, the matter density, and the curvature of the universe but also useful for breaking the degeneracy of the dark energy equation of state and thus provide complementarity with other popular probes, such as Type Ia supernovae, baryon acoustic oscillations, and cosmic microwave background radiation. Interestingly, compared to traditional strong lensing systems where quasars act as sources, strongly lensed gravitational waves (GWs) from compact binary coalescence and their electromagnetic (EM) counterpart systems have been recently proposed to be more powerful for studying cosmology since GWs and their EM counterparts are transients. Short durations of GWs and their EM counterparts are very advantageous to time delay measurement and lens profile modelling. Here, in the framework of Chevalier-Polarski-Linder parametrization, we investigate improvement of constraining power on the dark energy equation of state due to including time delay measurements of strong lensed GW systems. It is suggested that, on the basis of the third generation ground-based detector, e.g. the Einstein Telescope, adding time delay of only 30 strong lensed GW systems to Type Ia supernovae and cosmic microwave background radiation can improve the dark energy figure of merit by a factor of 2. For the traditional standard siren method where the uncertainties of luminosity distances of GWs are similar to 10 per cent, a few x 10(4) events are expected to present similar constraints. In the precision cosmology era, this progress is of great significance for studying the nature of dark energy.