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

TL;DR

This work investigates cosmological and gravity constraints from strongly lensed gravitational waves that can be associated with host galaxies seen in upcoming galaxy surveys. By adopting a singular isothermal sphere lens model and jointly analyzing time-delay cosmography with standard sirens, the authors forecast constraints across LVK O5, post-O5, and ET+CE networks using mock realizations. They find that next-generation detectors (ET+CE) can achieve sub-percent precision on $H_0$ in $\Lambda$CDM and provide moderate constraints on dynamical dark energy and on GW propagation parameters, including the $Ξ_0,n$ and $α$-basis parameterizations, while LVK networks yield only modest gains due to fewer events. The approach demonstrates the potential of lensed-GW cosmography as an independent probe of the Hubble tension and modified gravity, particularly when augmented by large galaxy surveys like LSST and Euclid, though careful treatment of selection effects and lens modeling remains important.

Abstract

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 $Λ$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 $H_0$.

Figures (9)

• Figure 1: Schematic figure for strong-lensing with the thin-lens approximation.
• Figure 2: Posterior samples from joint re-analysis of one of the mock doubly lensed GW events detected by LVK O5 network using nessai nested sampling in Bilby. The estimated parameters include chirp mass ${\cal M}_c[M_\odot]$, mass ratio $q$, GW luminosity distance $d_L^{\rm GW}$[Gpc], inclination angle $\iota$, and impact parameter $y$. The orange lines show the injected values for parameters.
• Figure 3: The figure shows the forecasted posterior on cosmological parameters in the $\Lambda$CDM model by jointly analyzing all lensed GW events in each realization, and averaging over all realizations, with different detector networks. The grey dashed lines indicate the parameter values in the fiducial $\Lambda$CDM model.
• Figure 4: The figure shows the forecasted posterior on cosmological parameters in the $w_0w_a$CDM model by jointly analyzing all lensed GW events in each realization, and averaging over all realizations, with different detector networks. The grey dashed lines indicate the parameter values in the fiducial $\Lambda$CDM model.
• Figure 5: Fractional 1$\sigma$ uncertainty of $H_0$ in the $\Lambda$CDM model (blue) and $w_0w_a$CDM model (orange) with different detector networks. The upper and lower bars show the upper and lower $1\sigma$ uncertainty bounds respectively, and the circles represent their means.