Gravitational lensing of gravitational waves (GWs) will become the next frontier in studying cosmology and gravity. While time-delay cosmography using quadruply lensed GW events associated with optical images of the lens systems can provide precise measurement of the Hubble constant ($H_0$), they are considered to be much rarer than doubly lensed events. In this work, we analyze time-delay cosmography with doubly lensed GW events for the first time. We generate mock doubly lensed GW events with the designed sensitivity of the LIGO-Virgo-KAGRA (LVK) O5 network, with the LIGO post-O5 upgrade, and with the Einstein Telescope $(\mathrm{ET})+\mathrm{Cosmic}\mathrm{Explorer}(\mathrm{CE})$ network, and we select the events that can be associated with future galaxy surveys. Over 1000 realizations, we find an average of $\sim 0.2(2.3)$ qualified events with the LVK O5 (post-O5) network, whereas with the $\mathrm{ET}+\mathrm{CE}$ network, we find an average of 80.9 qualified events over 100 realizations. Using the singular isothermal sphere (SIS) lens model, we forecast the constraints on cosmological parameters and modified GW propagation by combining time-delay cosmography and the standard siren approach. The average posterior gives a constraint on $H_0$ with relative uncertainties of 14%, 10%, and 0.42% in the $\mathrm{\Lambda }\mathrm{CDM}$ model for the LVK O5, LVK post-O5, and $\mathrm{ET}+\mathrm{CE}$ networks, respectively. While the LVK network gives uninformative constraints on the $(w_0,w_a)$ dynamical dark energy model, the $\mathrm{ET}+\mathrm{CE}$ network yields moderate constraints of $w_0=-1.02_{-0.25}^{+0.31}$ and $w_a=0.39_{-1.55}^{+1.01}$. In addition, our method can provide precise constraints on modified GW propagation effects jointly with $H_0$.

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• Alternative gravity theories
• Cosmological parameters
• Gravitational lenses
• Gravitational waves
• Astrophysical & cosmological simulations