Strong lensing cosmography using binary-black-hole mergers: Prospects for the near future

TL;DR

This work evaluates the feasibility of using gravitational-wave strong lensing from binary-black-hole mergers to constrain cosmological parameters. It develops a Bayesian framework that jointly uses the lensing fraction and the distribution of lensing time delays, explicitly incorporating realistic detector-network selection effects across LVK upgrades and next-generation (XG) detectors. By modeling intrinsic BBH merger rates with two astrophysical scenarios and employing a singular-isothermal-sphere lens population, the authors forecast constraints on $H_0$, $\Omega_m$, and $\sigma_8$, finding that even modest numbers of lensed events can yield competitive precision, approaching current CMB and standard-siren capabilities in the XG era. The study highlights the practical roadmap for GW lensing cosmography, including the importance of combining lensed pairs observed in different runs and refining lens models and population assumptions as data accumulate.

Abstract

A small fraction of gravitational-wave (GW) signals from binary black holes (BBHs) will be gravitationally lensed by intervening galaxies and galaxy clusters. Strong lensing will produce multiple identical copies of the GW signal arriving at different times. Jana et al.~\cite{Jana_2023} recently proposed a method to constrain cosmological parameters using strongly lensed GW events detected by next-generation (XG) detectors. The idea is that the number of strongly lensed GW events and the distribution of their lensing time delays encode imprints of the cosmological parameters. From the observed number of lensed GW events (tens of thousands) and their time delay distribution, this method can provide a new probe of cosmology, obtaining information at intermediate redshifts. In this work, we explore the possibility of doing lensing cosmography using upcoming observations of the upgraded LIGO-Virgo-KAGRA (LVK) network. This requires incorporating the detector network selection effects in the analysis, which was neglected earlier. We expect dozens of lensed GW events to be detected by upgraded LVK detectors, potentially enabling modest constraints on cosmological parameters. Even with relatively modest numbers of lensed detections, we demonstrate the potential of lensing cosmography. For XG detectors, our revised forecasts are consistent with with the earlier forecasts that neglected the selection effects.

Figures (17)

• Figure 1: Left: Sensitivity curves creighton2012 for the Advanced LIGO Livingston detector for O3, O4a, and proposed upgrades (A+, A$^{\sharp}$, and Voyager). For the Voyager configuration, we adopt the projected sensitivity curve optimized for massive binary systems HLA-voyager. Right: Current and projected sensitivity curves for Virgo, along with projected sensitivity curves for KAGRA and XG detectors (CE and ET). Table \ref{['tab:networks']} describes how these detectors are combined into networks with varying sensitivities across different observing runs.
• Figure 2: The source-frame merger rate density per unit comoving volume as a function of redshift, as predicted by the Dominik model (left) and the MD without delay model (right). Both models are normalized such that the low-redshift merger rate is consistent with the constraints from GWTC-3 data ligo2023gwtc (black hatched regions). Each panel includes an inset comparing the model's distribution with GWTC-3 constraints. The dashed lines correspond to the extrapolation based on the median merger rate from GWTC-3, while the shaded regions represent the central $50\%$ credible bounds. The Dominik model exhibits relatively higher merger rates at higher redshifts, indicating a larger number of lensed GW events than the MD without delay model.
• Figure 3: Duty cycle of GW detector networks. The duty cycle equals $1$ when the detectors are operational and $0$ otherwise. Each observing segment is denoted by two time-stamps: for the $i^{\rm th}$ observing run, the run begins at $T_i^{s}$ and concludes at $T_i^{e}$.
• Figure 4: The selection function $S^{i}(z_s)$, defined as the fraction of BBH mergers (marginalized over all other source parameters) detectable at a given source redshift. We sample a large number ($10^{6}$) different BBH parameters at each constant redshift (or luminosity distances, equivalently), compute the cumulative probability distribution of their optimal network SNR at the threshold. At very small redshifts, almost all events can be detected, whereas at higher redshifts the detectable fraction smoothly decreases to zero, with the cutoff redshift depending on the sensitivity of the detector network. For O6, we additionally sample the luminosity distance (following MD without delay distribution), together with the remaining BBH parameters. We then compute the optimal network SNR for each event and compute the fraction of events exceeding the detection threshold. This explicit Monte Carlo sampling method (denoted by black scatters) agrees well with the averaged detection fraction.
• Figure 5: The absolute values of the magnification of the two images produced by an SIS lens when $y < 1$, as a function of the impact parameter. The green curve corresponds to the minima of the time-delay surface, which is always detected first, while the magenta curve corresponds to the saddle. Note that both images are always magnified ($|\mu_{\pm}| > 1$) when $y < 0.5$; otherwise, only the minima images are magnified.