Forecasting Constraints on Cosmology and Modified Gravitational-wave Propagation by Combining Strongly Lensed Gravitational Waves and Galaxy Surveys

Gravitational lensing of gravitational wave (GW) 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 designed sensitivity of the LIGO-Virgo-KAGRA (LVK) O5 network, with LIGO post-O5 upgrade, and with Einstein Telescope (ET) + Cosmic Explorer (CE) respectively, and select the events that can be associated with future galaxy surveys. Over 1000 realizations, we find an average of 0.2(2.4) qualified events with the LVK O5(post-O5) network. Whereas with the ET+CE network, we find an average of 73.2 qualified events over 100 realizations. Using the Singular Isothermal Sphere (SIS) lens model, we jointly estimate waveform parameters and the impact parameter with doubly lensed GW signals, and then 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 a relative uncertainty of $14\%$, $10\%$ and $0.42\%$ in the $\Lambda$CDM model for the LVK O5, LVK post-O5, and ET+CE network, respectively. While the LVK network gives uninformative constraints on the $(w_0,w_a)$ dynamical dark energy model, the ET+CE network yields a moderate constraint of $w_0=-1.02^{+0.31}_{-0.22}$ and $w_a=0.48^{+0.99}_{-1.54}$. In addition, our method can provide precise constraints on modified GW propagation effects jointly with $H_0$.