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Strong gravitational lenses with two background sources at widely separated redshifts are a promising independent probe of cosmological parameters. We can use these systems, known as double-source-plane lenses (DSPLs), to measure the ratio (β) of angular-diameter distances of the sources, which is sensitive to the matter density (Ω_m) and the equation-of-state parameter for dark-energy (w). However, DSPLs are rare and require high-resolution imaging and spectroscopy for detection, lens modeling, and measuringβ. Here, we report only the second DSPL ever used to measure cosmological parameters. We model the DSPLAGEL150745+052256 from the ASTRO 3D Galaxy Evolution with Lenses (AGEL) survey using Hubble Space Telescope/Wide-Field Camera 3 imaging and Keck Cosmic Web Imager spectroscopy. The spectroscopic redshifts for the deflector and two sources inAGEL1507 arez_defl= 0.594,z_S1 =  2.163, andz_S2= 2.591. We measure a stellar velocity dispersion ofσ_obs = 109 ± 27 km s⁻¹for the nearer source (S1). Usingσ_obsfor the main deflector (from literature) and S1, we test the robustness of our DSPL model. We measure $\beta =0.953_{-0.010}^{+0.008}$forAGEL1507 and infer Ω_m $=0.33_{-0.23}^{+0.38}$for ΛCDM cosmology. CombiningAGEL1507 with the published model of the Jackpot lens improves the precision on Ω_m(ΛCDM) andw(wCDM) by ∼10%. The inclusion of DSPLs significantly improves the constraints when combined with Planck’s cosmic microwave background observations, enhancing the precision onwby 30%. This paper demonstrates the potential constraining power of DSPLs and their complementarity to other standard cosmological probes. Tighter future constraints from larger DSPL samples discovered from ongoing and forthcoming large-area sky surveys would provide insights into the nature of dark energy. 

Double-source-plane strong gravitational lenses (DSPLs), with two sources at different redshifts, are independent cosmological probes of the dark energy equation of state parameterwand the matter density parameter Ω_m. We present the lens model for the DSPL AGEL035346−170639 and infer cosmological constraints from this system for flat Λ cold dark matter and flatwCDM cosmologies. From the joint posterior ofwand Ω_min the flatwCDM cosmology, we extract the following median values and 1σuncertainties: $w=-1.52_{-0.33}^{+0.49}$and ${\mathrm{\Omega }}_{\mathrm{m}}=0.192_{-0.131}^{+0.305}$from AGEL0353 alone. Combining our measurements with two previously analyzed DSPLs, we present the joint constraint on these parameters from a sample of three, the largest galaxy-scale DSPL sample used for cosmological measurement to date. The combined precision ofwfrom three DSPLs is higher by 15% over AGEL0353 alone. Combining DSPL and cosmic microwave background (CMB) measurements improves the precision ofwfrom CMB-only constraints by 39%, demonstrating the complementarity of DSPLs with the CMB. Despite their promising constraining power, DSPLs are limited by sample size, with only a handful discovered so far. Although ongoing and near-future wide-area sky surveys will increase the number of known DSPLs by up to two orders of magnitude, these systems will still require dedicated high-resolution imaging and spectroscopic follow-ups like those presented in this paper. Our ASTRO 3D Galaxy Evolution with Lenses collaboration is undertaking such follow-up campaigns for several newly discovered DSPLs and will provide cosmological measurements from larger samples of DSPLs in the future. 

Observed evolution of the total mass distribution with redshift is crucial to testing galaxy evolution theories. To measure the total mass distribution, strong gravitational lenses complement the resolved dynamical observations that are currently limited to z ≲ 0.5. Here we present the lens models for a pilot sample of seven galaxy-scale lenses from the ASTRO3D Galaxy Evolution with Lenses (AGEL) survey. The AGEL lenses, modeled using HST/WFC3-F140W images with Gravitational Lens Efficient Explorer (GLEE) software, have deflector redshifts in the range 0.3 < z defl < 0.9. Assuming a power-law density profile with slope γ, we measure the total density profile for the deflector galaxies via lens modeling. We also measure the stellar velocity dispersions (σ obs) for four lenses and obtain σ obs from SDSS-BOSS for the remaining lenses to test our lens models by comparing observed and model-predicted velocity dispersions. For the seven AGEL lenses, we measure an average density profile slope of ‑1.95 ± 0.09 and a γ–z relation that does not evolve with redshift at z < 1. Although our result is consistent with some observations and simulations, it differs from other studies at z < 1 that suggest the γ–z relation evolves with redshift. The apparent conflicts among observations and simulations may be due to a combination of (1) systematics in the lensing and dynamical modeling; (2) challenges in comparing observations with simulations; and (3) assuming a simple power law for the total mass distribution. By providing more lenses at z defl > 0.5, the AGEL survey will provide stronger constraints on whether the mass profiles evolve with redshift as predicted by current theoretical models. 

Abstract Multiply imaged time-variable sources can be used to measure absolute distances as a function of redshifts and thus determine cosmological parameters, chiefly the Hubble Constant H$$_0$$ ${}_0$. In the two decades up to 2020, through a number of observational and conceptual breakthroughs, this so-called time-delay cosmography has reached a precision sufficient to be an important independent voice in the current “Hubble tension” debate between early- and late-universe determinations of H$$_0$$ ${}_0$. The 2020s promise to deliver major advances in time-delay cosmography, owing to the large number of lenses to be discovered by new and upcoming surveys and the vastly improved capabilities for follow-up and analysis. In this review, after a brief summary of the foundations of the method and recent advances, we outline the opportunities for the decade and the challenges that will need to be overcome in order to meet the goal of the determination of H$$_0$$ ${}_0$from time-delay cosmography with 1% precision and accuracy. 

Time-delay cosmography is a powerful technique to constrain cosmological parameters, particularly the Hubble constant (H₀). The TDCOSMO Collaboration is performing an ongoing analysis of lensed quasars to constrain cosmology using this method. In this work, we obtain constraints from the lensed quasar WGD 2038−4008 using new time-delay measurements and previous mass models by TDCOSMO. This is the first TDCOSMO lens to incorporate multiple lens modeling codes and the full time-delay covariance matrix into the cosmological inference. The models are fixed before the time delay is measured, and the analysis is performed blinded with respect to the cosmological parameters to prevent unconscious experimenter bias. We obtainD_Δt = 1.68_−0.38^+0.40Gpc using two families of mass models, a power-law describing the total mass distribution, and a composite model of baryons and dark matter, although the composite model is disfavored due to kinematics constraints. In a flat ΛCDM cosmology, we constrain the Hubble constant to beH₀ = 65₋₁₄⁺²³km s⁻¹Mpc⁻¹. The dominant source of uncertainty comes from the time delays, due to the low variability of the quasar. Future long-term monitoring, especially in the era of theVera C. RubinObservatory’s Legacy Survey of Space and Time, could catch stronger quasar variability and further reduce the uncertainties. This system will be incorporated into an upcoming hierarchical analysis of the entire TDCOSMO sample, and improved time delays and spatially-resolved stellar kinematics could strengthen the constraints from this system in the future. 

Strong-lensing time delays enable the measurement of the Hubble constant ( H 0 ) independently of other traditional methods. The main limitation to the precision of time-delay cosmography is mass-sheet degeneracy (MSD). Some of the previous TDCOSMO analyses broke the MSD by making standard assumptions about the mass density profile of the lens galaxy, reaching 2% precision from seven lenses. However, this approach could potentially bias the H 0 measurement or underestimate the errors. For this work, we broke the MSD for the first time using spatially resolved kinematics of the lens galaxy in RXJ1131−1231 obtained from the Keck Cosmic Web Imager spectroscopy, in combination with previously published time delay and lens models derived from Hubble Space Telescope imaging. This approach allowed us to robustly estimate H 0 , effectively implementing a maximally flexible mass model. Following a blind analysis, we estimated the angular diameter distance to the lens galaxy D d  = 865 −81 +85 Mpc and the time-delay distance D Δt  = 2180 −271 +472 Mpc, giving H 0  = 77.1 −7.1 +7.3 km s −1 Mpc −1 – for a flat Λ cold dark matter cosmology. The error budget accounts for all uncertainties, including the MSD inherent to the lens mass profile and line-of-sight effects, and those related to the mass–anisotropy degeneracy and projection effects. Our new measurement is in excellent agreement with those obtained in the past using standard simply parametrized mass profiles for this single system ( H 0  = 78.3 −3.3 +3.4 km s −1 Mpc −1 ) and for seven lenses ( H 0  = 74.2 −1.6 +1.6 km s −1 Mpc −1 ), or for seven lenses using single-aperture kinematics and the same maximally flexible models used by us ( H 0  = 73.3 −5.8 +5.8 km s −1 Mpc −1 ). This agreement corroborates the methodology of time-delay cosmography. 

ABSTRACT Astrometric precision and knowledge of the point spread function are key ingredients for a wide range of astrophysical studies including time-delay cosmography in which strongly lensed quasar systems are used to determine the Hubble constant and other cosmological parameters. Astrometric uncertainty on the positions of the multiply-imaged point sources contributes to the overall uncertainty in inferred distances and therefore the Hubble constant. Similarly, knowledge of the wings of the point spread function is necessary to disentangle light from the background sources and the foreground deflector. We analyse adaptive optics (AO) images of the strong lens system J 0659+1629 obtained with the W. M. Keck Observatory using the laser guide star AO system. We show that by using a reconstructed point spread function we can (i) obtain astrometric precision of <1 mas, which is more than sufficient for time-delay cosmography; and (ii) subtract all point-like images resulting in residuals consistent with the noise level. The method we have developed is not limited to strong lensing, and is generally applicable to a wide range of scientific cases that have multiple point sources nearby. 

ABSTRACT Strongly lensed quasars can provide measurements of the Hubble constant (H0) independent of any other methods. One of the key ingredients is exquisite high-resolution imaging data, such as Hubble Space Telescope (HST) imaging and adaptive-optics (AO) imaging from ground-based telescopes, which provide strong constraints on the mass distribution of the lensing galaxy. In this work, we expand on the previous analysis of three time-delay lenses with AO imaging (RX J1131−1231, HE 0435−1223, and PG 1115+080), and perform a joint analysis of J0924+0219 by using AO imaging from the Keck telescope, obtained as part of the Strong lensing at High Angular Resolution Program (SHARP) AO effort, with HST imaging to constrain the mass distribution of the lensing galaxy. Under the assumption of a flat Λ cold dark matter (ΛCDM) model with fixed Ωm = 0.3, we show that by marginalizing over two different kinds of mass models (power-law and composite models) and their transformed mass profiles via a mass-sheet transformation, we obtain $$\Delta t_{\rm BA}=6.89\substack{+0.8\\-0.7}\, h^{-1}\hat{\sigma }_{v}^{2}$$ d, $$\Delta t_{\rm CA}=10.7\substack{+1.6\\-1.2}\, h^{-1}\hat{\sigma }_{v}^{2}$$ d, and $$\Delta t_{\rm DA}=7.70\substack{+1.0\\-0.9}\, h^{-1}\hat{\sigma }_{v}^{2}$$ d, where $$h=H_{0}/100\,\rm km\, s^{-1}\, Mpc^{-1}$$ is the dimensionless Hubble constant and $$\hat{\sigma }_{v}=\sigma ^{\rm ob}_{v}/(280\,\rm km\, s^{-1})$$ is the scaled dimensionless velocity dispersion. Future measurements of time delays with 10 per cent uncertainty and velocity dispersion with 5 per cent uncertainty would yield a H0 constraint of ∼15 per cent precision. 

Time-delay cosmography with gravitationally lensed quasars plays an important role in anchoring the absolute distance scale and hence measuring the Hubble constant, H 0 , independent of traditional distance ladder methodology. A current potential limitation of time-delay distance measurements is the mass-sheet transformation (MST), which leaves the lensed imaging unchanged but changes the distance measurements and the derived value of H 0 . In this work we show that the standard method of addressing the MST in time-delay cosmography, through a combination of high-resolution imaging and the measurement of the stellar velocity dispersion of the lensing galaxy, depends on the assumption that the ratio, D s / D ds , of angular diameter distances to the background quasar and between the lensing galaxy and the quasar can be constrained. This is typically achieved through the assumption of a particular cosmological model. Previous work (TDCOSMO IV) addressed the mass-sheet degeneracy and derived H 0 under the assumption of the ΛCDM model. In this paper we show that the mass-sheet degeneracy can be broken without relying on a specific cosmological model by combining lensing with relative distance indicators such as supernovae Type Ia and baryon acoustic oscillations, which constrain the shape of the expansion history and hence D s / D ds . With this approach, we demonstrate that the mass-sheet degeneracy can be constrained in a cosmological model-independent way. Hence model-independent distance measurements in time-delay cosmography under MSTs can be obtained. 

The local expansion rate of the Universe is parametrized by the Hubble constant, H 0 , the ratio between recession velocity and distance. Different techniques lead to inconsistent estimates of H 0 . Observations of Type Ia supernovae (SNe) can be used to measure H 0 , but this requires an external calibrator to convert relative distances to absolute ones. We use the angular diameter distance to strong gravitational lenses as a suitable calibrator, which is only weakly sensitive to cosmological assumptions. We determine the angular diameter distances to two gravitational lenses, 810 − 130 + 160 and 1230 − 150 + 180 megaparsec, at redshifts z = 0.295 and 0.6304. Using these absolute distances to calibrate 740 previously measured relative distances to SNe, we measure the Hubble constant to be H 0 = 82.4 − 8.3 + 8.4 kilometers per second per megaparsec.