Enhanced Localization of Dark Lensed Gravitational Wave Events Enables Host Galaxy Identification and Precise Cosmological Inference

TL;DR

The paper demonstrates that third-generation gravitational-wave detectors will routinely observe strongly lensed GW events with three or more images, forming an effective detector network that shrinks localization to ~0.01 deg^2 and enables sub-arcsecond host localization via lens modelling. It introduces the dark lensed siren approach, which uses precise GW time delays, luminosity distances, and lens-model reconstructions of the lensed host to infer cosmological parameters without relying on electromagnetic counterparts. Through simulated mock populations and Fisher-Matrix/MCMC analyses, the authors forecast H0 constraints at the sub-percent level within ~2 years of 3G detector operation and show potential constraints on Ω_m, w, and Ω_k with larger samples, contingent on the rate of lensed events with identifiable hosts. They also assess rates, lens-model populations, and host-identification fractions for future surveys, highlighting the central image's role in achieving sub-arcsecond localization and robust cosmological inferences as a complementary approach to current probes.

Abstract

Lensed gravitational wave (GW) events are expected to be powerful new probes of cosmology, contingent on redshift measurement by electromagnetic observations. Host galaxy identification is thus crucial but challenging due to poor localization by GW signal alone. In this paper, we show that the third-generation ground-based GW detectors will detect a population of lensed events with three or more detectable images (including the central one), each arriving at distinct times and Earth locations in the space, forming an effective network that reduces the typical localization area to $\sim0.01$ deg$^2$. For at least $90\%$ (or $50\%$) of these events, the localization improves by more than a factor of $10$ (or $30$) comparing with unlensed cases. Such precise localization and multiple-image detections enable robust host-galaxy identification and, through lens modelling, further yield sub-arcsecond position. As ``dark lensed sirens", these events become powerful probes of cosmological parameters. Using simulated lensed compact-binary mergers, we show that two-year or longer observations with third-generation GW detectors can measure the Hubble constant to $\lesssim1$\% precision via ``dark lensed sirens" (even when relying solely on lensed stellar-mass binary black hole events), while simultaneously constraining other cosmological parameters. This approach will provide an independent, complementary avenue for measuring cosmological parameters.

Figures (8)

• Figure 1: A schematic diagram for the effective network of ground-based GW detectors for a lensed GW event with double images detected at different time. S$_1$ and S$_2$ are the double images of the event, L is the foreground lens galaxy, and $v_0$ denotes the velocity of the Sun in the Galaxy, $(\rm A_1,B_1,C_1$) and $(\rm A_2,B_2,C_2)$ represent the locations of three GW detectors in the space when they receive the signal from S$_1$ and S$_2$, respectively. Due to the rotation of the Earth, usually the baselines $\rm A_2B_2$, $\rm B_2C_2$, and $\rm C_2A_2$ are not parallel with $\rm A_1B_1$, $\rm B_1C_1$, and $\rm C_1A_1$. However, the baselines $\rm A_1A_2$, $\rm B_1B_2$, and $\rm C_1C_2$ are close to, though not exactly, parallel to each other due to the lengths of these baselines are much longer than those of $\rm A_1B_1$, $\rm A_1C_1$, and $\rm B_1C_1$.
• Figure 2: The expected localization uncertainties estimated for mock lensed sBBH GW events considering the effective network (EN) effects due to the motion of the Earth ($\Delta\Omega_{\rm EN}$) against those without considering it ($\Delta\Omega_{\rm nonEN}$). Left and right panels show the cases for those systems with triple and quadruple images, respectively. The color of each symbol represents the S/N of the faintest image with values indicated by the right colour bar. The blue dashed lines represent $\Delta\Omega_{\rm EN}=\Delta\Omega_{\rm nonEN}$. Our sample points are almost all located substantially below the blue dashed lines, which means considering the motion of the Earth in the space can improve the localization precision significantly. The black dotted lines represent the best fits to each sample, i.e., $\log(\Delta\Omega_{\rm s,EN}/\textrm{deg}^2)=0.80\log(\Delta\Omega_{\rm s,nonEN}/\textrm{deg}^2)-1.66$ (left panel) and $\log(\Delta \Omega_{\rm s,EN} /\textrm{deg}^2)=0.57\log(\Delta\Omega_{\rm s,nonEN}/\textrm{deg}^2)-3.07$ (right panel), which suggest the improvement by the effective network to the localization is roughly a factor of $10^{-1.66}$ and $10^{-3.07}$, as also indicated by the top and the right small panels for the one dimension distributions of $\Delta\Omega_{\rm s,EN}$ peaked at $\sim0.01$ deg$^2$ (or $\sim0.001$ deg$^2$) and $\Delta\Omega_{\rm s,nonEN}$ peaked at $\sim0.14$ deg$^2$ (or $\sim0.58$ deg$^2$).
• Figure 3: Flowchart of the "dark lensed siren" method for cosmological inference. In the left and right block, we show the basic recipes needed to obtain or infer from GW and EM (host galaxy) observation, respectively. We use a blue dotted pane to emphasize the key steps in such a method, i.e., (1) precisely localize the GW source via the effective network proposed and find the associated lensed host galaxy; (2) further sub-arcsecond localization of the GW source within the host galaxy via triple or more images (with central image). With the above two key steps, we may constrain the cosmological parameters via the time-delay and luminosity distance measurement from the lensed GW sources, together with the Fermat potential and magnification factor reconstructed by the EM image of the lensed host galaxies.
• Figure 4: Expected number of lensed GW events ($N_{\rm len}$; top panel), number of the lensed GW event with triple or more images detection with identifiable lensed host galaxies ($N_{\rm len}^{\rm host}$; middle panel), and the corresponding fractional error of the $H_0$ measurement obtained by using these lensed events together (bottom panel) as a function of time. Here we assume that the second/second and a half/third generation (2G/2.5G/3G) GW detectors, i.e., LIGO A+/Voyager/CE, start to work at the year of 2027/2030/2035. In the top panel, the dashed curves with diamond symbols show the results obtained for lensed merger events of sBBHs (red), BNSs (green), and NSBHs (blue) with double images (labelled as sBBH-2, BNS-2, and NSBH-2), respectively, while the solid curves with square symbols show those with triple or more images, labelled as sBBH-3+ (red), BNS-3+ (green), and NSBH-3+(blue), respectively. In the middle and bottom panels, the shaded regions around these curves indicate the uncertainties of the estimates introduced by the uncertainty in the constraints of the local merger rate densities. Note that the numbers for detectable sources shown in the top and middle panel are obtained by averaging those from a large number of realizations.
• Figure 5: Cumulative probability distribution functions of S/N ($\varrho_{{\rm GW},i_j}$) for all the images of the lensed GW sources ($\rm CDF(>\varrho_{\rm GW})$), normalized to $1$ at $\varrho_{\rm GW}=5$. The red, blue, green solid lines show the results of triple image cases of sBBH, BNS and NSBH, while the dashed lines correspondingly represent the cases of quadruple image.