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TDCOSMO - XVI. Measurement of the Hubble constant from the lensed quasar WGD 2038–4008

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 obtain DΔ t = 1.68−0.38+0.40 Gpc 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 be H0 = 65−14+23 km s−1 Mpc−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 the Vera C. Rubin Observatory’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.Key words: gravitational lensing: strong / cosmological parameters / distance scale⋆ Corresponding author; kcwong19@gmail.com.⋆⋆ NHFP Einstein fellow.

79 ASTRONOMY AND ASTROPHYSICS

Cosmology from Planck CMB lensing and DESI DR1 quasar tomography

We present a measurement of the amplitude of matter fluctuations over the redshift range 0.8 ≤ z ≤ 3.5 from the cross correlation of over 1.2 million spectroscopic quasars selected by the Dark Energy Spectroscopic Instrument (DESI) across 7,200 deg 2 (∼ 170 quasars/deg 2 ) and Planck PR4 (NPIPE) cosmic microwave background (CMB) lensing maps. We perform a tomographic measurement in three bins centered at effective redshiftsz=1.44, 2.27 and 2.75, which have ample overlap with the CMB lensing kernel. From a joint fit using the angular clustering of all three redshift bins (auto and cross-spectra), and including an Q m prior from DESI DR1 baryon acoustic oscillations to break the $Ω_{m}-σ_{8}$ degeneracy, we constrain the amplitude of matter fluctuations in the matter-dominated regime to be $σ_{8}=0.929^{+0.059}_{-0.074}$ and $S_{8}≡σ_{8}(Ω_m/0.3)^{0.5} = 0.922^{+0.059}_{-0.073}$. We provide a growth of structure measurement with the largest spectroscopic quasar sample to date at high redshift, which is ∼ 1.5σ higher than predictions from ΛCDM fits to measurements of the primary CMB from Planck PR4. The cross-correlation between PR4 lensing maps and DESI DR1 quasars is detected with a signal-to-noise ratio of 21.7 and the quasar auto-correlation at 27.2 for the joint analysis of all redshift bins. We combine our measurement with the CMB lensing auto-spectrum from the ground-based Atacama Cosmology Telescope (ACT DR6) and Planck PR4 to perform a sound-horizon-free measurement of the Hubble constant, yielding $H_{0}=69.1^{+2.2}_{-2.6} {\rm \ km\ s}^{-1}{\rm Mpc}^{-1}$ through its sensitivity to the matter-radiation equality scale.

baryon acoustic oscillations

Constraining primordial non-Gaussianity from DESI DR1 quasars and Planck PR4 CMB lensing

We present the first measurement of local-type primordial non-Gaussianity from the cross-correlation between 1.2 million spectroscopically confirmed quasars from the first data release (DR1) of the Dark Energy Spectroscopic Instrument (DESI) and the Planck PR4 CMB lensing reconstructions. The analysis is performed in three tomographic redshift bins covering 0.8 < z < 3.5, covering a sky fraction of ∼20%. We adopt a catalog-based pseudo-C ℓ estimator and apply linear imaging weights validated on noiseless mocks. Compared to previous analyses using photometric quasar samples, our results benefit from the high purity of the DESI spectroscopic sample, the reduced noise of PR4 lensing, and the absence of excess large-scale power in the spectroscopic quasar auto-correlation. Fitting simultaneously for the non-Gaussianity parameter f NL and the linear bias amplitude in each redshift bin, we obtain f NL = 2 +28 -34 for a response parameter p = 1.6, and f NL = 6 +20 -24 for p = 1.0. These results improve the constraints on f NL by ∼35% compared to the previous analysis based on the Legacy Imaging Survey DR9. Additionally, we derive an optimal weighting scheme to maximize the constraining power. In this case, and assuming p = 1.6, we obtain f NL = 19 +25 -31 . Our results demonstrate the statistical power of DESI quasars for probing inflationary physics, and highlight the promise of future DESI data releases.

cosmological parameters from CMBR

Joint Modeling of Quasar Variability and Accretion Disk Reprocessing Using Latent Stochastic Differential Equations

Quasars are bright active galactic nuclei powered by the accretion of matter around supermassive black holes at the center of galaxies. Their stochastic brightness variability depends on the physical properties of the accretion disk and black hole. The upcoming Rubin Observatory Legacy Survey of Space and Time (LSST) is expected to observe tens of millions of quasars, so there is a need for efficient techniques like machine learning that can handle the large volume of data. Quasar variability is believed to be driven by an X-ray corona, which is reprocessed by the accretion disk and emitted as UV/optical variability. We are the first to introduce an auto-differentiable simulation of the accretion disk and reprocessing. We use the simulation as a direct component of our neural network to jointly model the driving variability and reprocessing, trained with supervised learning on simulated LSST-like 10 yr quasar light curves. We encode the light curves using a transformer encoder, and the driving variability is reconstructed using latent stochastic differential equations, a physically motivated generative deep learning method that can model continuous-time stochastic dynamics. By embedding the physical processes of the driving signal and reprocessing into our network, we achieve a model that is more robust and interpretable. We demonstrate that our model outperforms a Gaussian process regression baseline and can infer accretion disk parameters and time delays between wave bands, even for out-of-distribution driving signals. Our approach provides a powerful framework that can be adapted to solve other inverse problems in multivariate time series.

Fagin, Joshua [City Univ. of New York (CUNY), NY (

TDCOSMO - XVII. New time delays in 22 lensed quasars from optical monitoring with the ESO-VST 2.6m and MPG 2.2m telescopes

We present new time delays, the main ingredient of time delay cosmography, for 22 lensed quasars resulting from high-cadence r-band monitoring on the 2.6 m ESO VLT Survey Telescope and Max-Planck-Gesellschaft 2.2 m telescope. Each lensed quasar was typically monitored for one to four seasons, often shared between the two telescopes to mitigate the interruptions forced by the COVID-19 pandemic. The sample of targets consists of 19 quadruply and 3 doubly imaged quasars, which received a total of 1918 hours of on-sky time split into 21 581 wide-field frames, each 320 seconds long. In a given field, the 5-σ depth of the combined exposures typically reaches the 27th magnitude, while that of single visits is 24.5 mag – similar to the expected depth of the upcoming Vera-Rubin LSST. The fluxes of the different lensed images of the targets were reliably de-blended, providing not only light curves with photometric precision down to the photon noise limit, but also high-resolution models of the targets whose features and astrometry were systematically confirmed in Hubble Space Telescope imaging. This was made possible thanks to a new photometric pipeline, lightcurver, and the forward modelling method STARRED. Finally, the time delays between pairs of curves and their uncertainties were estimated, taking into account the degeneracy due to microlensing, and for the first time the full covariance matrices of the delay pairs are provided. Of note, this survey, with 13 square degrees, has applications beyond that of time delays, such as the study of the structure function of the multiple high-redshift quasars present in the footprint at a new high in terms of both depth and frequency. The reduced images will be available through the European Southern Observatory Science Portal.Key words: methods: data analysis / surveys / distance scale

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND

Lens Modeling of STRIDES Strongly Lensed Quasars Using Neural Posterior Estimation

Strongly lensed quasars can be used to constrain cosmological parameters through time-delay cosmography. Models of the lens masses are a necessary component of this analysis. To enable time-delay cosmography from a sample of $\mathcal{O}(10^3)$ lenses, which will soon become available from surveys like the Rubin Observatory’s Legacy Survey of Space and Time and the Euclid Wide Survey, we require fast and standardizable modeling techniques. To address this need, we apply neural posterior estimation (NPE) for modeling galaxy-scale strongly lensed quasars from the Strong Lensing Insights into the Dark Energy Survey (STRIDES) sample. NPE brings two advantages: speed and the ability to implicitly marginalize over nuisance parameters. We extend this method by employing sequential NPE to increase precision of mass model posteriors. We then fold individual lens models into a hierarchical Bayesian inference to recover the population distribution of lens mass parameters, accounting for out-of-distribution shift. After verifying our method using simulated analogs of the STRIDES lens sample, we apply our method to 14 Hubble Space Telescope single-filter observations. We find the population mean of the power-law elliptical mass distribution slope, γ lens , to be $\mathcal{M}_γ$ lens = 2.13 ± 0.06. Our result represents the first population-level constraint for these systems. This population-level inference from fully automated modeling is an important stepping stone toward cosmological inference with large samples of strongly lensed quasars.

79 ASTRONOMY AND ASTROPHYSICS

The First Identification of Lyα Changing-look Quasars at High Redshift in DESI

We present two cases of Lyα changing-look (CL) quasars (J1306 and J1512) along with two additional candidates (J1511 and J1602), all discovered serendipitously at z > 2 through the Dark Energy Spectroscopic Instrument and the Sloan Digital Sky Survey. It is the first time to capture CL events in Lyα at high redshift, which is crucial for understanding the underlying mechanisms driving the CL phenomenon and the evolution of high-redshift quasars and galaxies. We find that the accretion rate in the dim state for these CL objects corresponds to a relatively low value ($\mathop{\,}\limits^{{\mathscr{M}}}\,\approx 2\times 1{0}^{-3}$), which suggests that the inner region of the accretion disk might be in transition between the advection dominated accretion flow ($\dot{\,{\mathscr{M}}}\,\lt 1{0}^{-3}\sim 1{0}^{-2}$) and the canonical accretion disk (optically thick, geometrically thin). However, unlike in C IV CL quasars in which broad Lyα remained, the broad C IV may still persist after a CL event occurs in Lyα, making the physical origin of the CL and ionization mechanism event more puzzling and interesting.

accretion

DESI 2024 III: baryon acoustic oscillations from galaxies and quasars

We present the DESI 2024 galaxy and quasar baryon acoustic oscillations (BAO) measurements using over 5.7 million unique galaxy and quasar redshifts in the range 0.1 < z < 2.1. Divided by tracer type, we utilize 300,017 galaxies from the magnitude-limited Bright Galaxy Survey with 0.1 < z < 0.4, 2,138,600 Luminous Red Galaxies with 0.4 < z < 1.1, 2,432,022 Emission Line Galaxies with 0.8 < z < 1.6, and 856,652 quasars with 0.8 < z < 2.1, over a ∼ 7,500 square degree footprint. The analysis was blinded at the catalog-level to avoid confirmation bias. All fiducial choices of the BAO fitting and reconstruction methodology, as well as the size of the systematic errors, were determined on the basis of the tests with mock catalogs and the blinded data catalogs. We present several improvements to the BAO analysis pipeline, including enhancing the BAO fitting and reconstruction methods in a more physically-motivated direction, and also present results using combinations of tracers. We employ a unified BAO analysis method across all tracers. We present a re-analysis of SDSS BOSS and eBOSS results applying the improved DESI methodology and find scatter consistent with the level of the quoted SDSS theoretical systematic uncertainties. With the total effective survey volume of ∼ 18 Gpc3, the combined precision of the BAO measurements across the six different redshift bins is ∼0.52%, marking a 1.2-fold improvement over the previous state-of-the-art results using only first-year data. We detect the BAO in all of these six redshift bins. The highest significance of BAO detection is 9.1σ at the effective redshift of 0.93, with a constraint of 0.86% placed on the BAO scale. We find that our observed BAO scales are systematically larger than the prediction of the Planck 2018-ΛCDM at z < 0.8. We translate the results into transverse comoving distance and radial Hubble distance measurements, which are used to constrain cosmological models in our companion paper.

79 ASTRONOMY AND ASTROPHYSICS

The variability structure function of the highest luminosity quasars on short time-scales

ABSTRACT The stochastic photometric variability of quasars is known to follow a random-walk phenomenology on emission time-scales of months to years. Some high-cadence rest-frame optical monitoring in the past has hinted at a suppression of variability amplitudes on shorter time-scales of a few days or weeks, opening the question of what drives the suppression and how it might scale with quasar properties. Here, we study a few thousand of the highest luminosity quasars in the sky, mostly in the luminosity range of $L_{\rm bol}$$=[46.4, 47.3]$ and redshift range of $z=[0.7, 2.4]$. We use a data set from the NASA/Asteroid Terrestrial-impact Last Alert System facility with nightly cadence, weather permitting, which has been used before to quantify strong regularity in longer term rest-frame-UV variability. As we focus on a careful treatment of short time-scales across the sample, we find that a linear function is sufficient to describe the UV variability structure function. Although the result can not rule out the existence of breaks in some groups completely, a simpler model is usually favoured under this circumstance. In conclusion, the data are consistent with a single-slope random walk across rest-frame time-scales of $\Delta t=[10, 250]$ d.

Tang, Ji-Jia (ORCID:0000000218600886)

Modelling the impact of quasar redshift errors on the full-shape analysis of correlations in the Lyman-α forest.

In preparation for the first cosmological measurements from the full shape of the Lyman-α (Lyα) forest from DESI, we must carefully model all relevant systematics that might bias our analysis. It was shown in Youles et al. (2022) that random quasar redshift errors produce a smoothing effect on the mean quasar continuum in the Lyα forest region. This, in turn, gives rise to spurious features in the Lyα autocorrelation and its cross-correlation with quasars. Using synthetic data sets based on the DESI survey, we confirm that the impact on BAO measurements is small, but that a bias is introduced to parameters which depend on the full shape of our correlations. We combine a model of this contamination in the cross-correlation (Youles et al. 2022) with a new model we introduce here for the auto-correlation. These are parametrised by 3 parameters, which, when included in a joint fit to both correlation functions, successfully eliminate any impact of redshift errors on our full-shape constraints. We also present a strategy for removing this contamination from real data, by removing ∼0.3% of correlating pairs.

cosmology

The Cosmic Evolution of C IV Absorbers at 1.4 < z < 4.5: Insights from 100,000 Systems in DESI Quasars

We present the largest catalog to date of triply ionized carbon (C IV ) absorbers detected in quasar spectra from the Dark Energy Spectroscopic Instrument. Using an automated matched-kernel convolution method with adaptive signal-to-noise thresholds, we identify 101,487 C IV systems in the redshift range 1.4 < z < 4.5 from 300,637 quasar spectra. Completeness is estimated via Monte Carlo simulations, and the catalog is 50% complete at EW C IV ≥ 0.4 Å. The differential equivalent width frequency distribution declines exponentially and shows weak redshift evolution. The absorber incidence per unit comoving path increases by a factor of 2–5 from z ≈ 4.5 to z ≈ 1.4, with stronger redshift evolution for strong systems. Using column densities derived from the apparent optical depth method, we constrain the cosmic mass density of C IV , Ω C IV , which increases by a factor of ∼3.8 from (0.82 ± 0.05) × 10 −8 at z ≈ 4.5 to (3.16 ± 0.2) × 10 −8 at z ≈ 1.4. From Ω C IV , we estimate a lower limit on intergalactic medium metallicity ${\mathrm{log}}({Z}_{{\rm{IGM}}}/{Z}_{\odot })\gtrsim -3.25$ at z ∼ 2.3, with a smooth decline at higher redshifts. These trends trace the cosmic star formation history and He II photoheating rate, suggesting a link between C IV enrichment, star formation, and UV background over ∼3 Gyr. The catalog also provides a critical resource for future studies connecting circumgalactic metals to galaxy evolution, especially near cosmic noon.

79 ASTRONOMY AND ASTROPHYSICS

Structured ionized winds shooting out from a quasar at relativistic speeds

Evidence indicates that supermassive black holes (SMBHs) exist at the centres of most galaxies. Their mass correlates with the galactic bulge mass, suggesting a coevolution with their host galaxies, most likely through powerful winds. X-ray observations have detected highly ionized winds outflowing at sub-relativistic speeds from the accretion disks around SMBHs. However, the limited spectral resolution of present X-ray instruments has left the physical structure and location of the winds poorly understood, hindering accurate estimates of their kinetic power. Here, in this study, the first X-Ray Imaging and Spectroscopy Mission (XRISM) observation of the luminous quasar PDS 456 is reported. The high-resolution spectrometer Resolve aboard XRISM enabled the discovery of five discrete velocity components outflowing at 20–30% of the speed of light. This demonstrates that the wind structure is highly inhomogeneous, which probably consists of up to a million clumps. The mass outflow rate is estimated to be 60–300 solar masses per year, with the wind kinetic power exceeding the Eddington luminosity limit. Compared with the galaxy-scale outflows, the kinetic power is more than three orders of magnitude larger, whereas the momentum flux is ten times larger. These estimates disfavour both energy-driven and momentum-driven outflow models. This suggests that such wind activity occurs in less than 10% of the quasar phase and/or that its energy/momentum is not efficiently transferred to the galaxy-scale outflows owing to the clumpiness of the wind and the interstellar medium.

Audard, Marc [Univ. of Geneva, Versoix (Switzerlan

Using active learning to improve quasar identification for the DESI spectra processing pipeline

The Dark Energy Spectroscopic Instrument (DESI) survey uses an automatic spectral classification pipeline to classify spectra. QuasarNET is a convolutional neural network used as part of this pipeline originally trained using data from the Baryon Oscillation Spectroscopic Survey (BOSS). In this paper we implement an active learning algorithm to optimally select spectra to use for training a new version of the QuasarNET weights file using only DESI data, with the goal of improving classification accuracy. This active learning algorithm includes a novel outlier rejection step using a Self-Organizing Map to ensure we label spectra representative of the larger quasar sample observed in DESI. We perform two iterations of the active learning pipeline, assembling a final dataset of 5600 labeled spectra, a small subset of the approximately 1.3 million quasar targets in DESI's Data Release 1. When splitting the spectra into training and validation subsets we achieve similar performance to the previously trained weights file in completeness and purity calculated on the validation dataset but do so with less than one tenth of the amount of training data. The new weights also more consistently classify objects in the same way when used on unlabeled data compared to the old weights file. In the process of improving QuasarNET's classification accuracy we discovered a systemic error in QuasarNET's redshift estimation and used our findings to improve our understanding of QuasarNET's redshifts.

Machine learning

Assembly bias and local Primordial non-Gaussianity from DESI DR1 quasars

The analysis of the large-scale clustering of quasars (QSO) observed by the Dark Energy Spectroscopic Instrument (DESI) represents a promising avenue for constraining local Primordial non-Gaussianity (PNG), parameterized by f NL . The signal to be constrained is the scale-dependent bias induced in the 2-point clustering of the considered tracer sample. The resulting constraints on f NL , however, are fully degenerate with the local PNG bias parameter b ϕ , dependent on the assembly bias parameter p. Using IllustrisTNG hydrodynamical simulations, we select a QSO sample reflecting the selection criteria and properties of DESI QSOs, and provide a robust prior for p, and thus for b ϕ , building on the findings of Fondi et al. 2024. We find a distribution with mean p̅ ≃ 1.4 with weak redshift dependence, stable to selection noise and consistent with the expected recent merger history typical of quasar-hosting halos. By comparing with the CAMELS simulations we demonstrate that this prior is robust to astrophysical assumptions and cosmic variance. Finally, applying this prior to the DESI DR1 dataset, we derive updated constraints on local PNG, obtaining f NL = -3.3±9.2.

cosmological parameters from LSS

Validation of the DESI DR2 measurements of baryon acoustic oscillations from galaxies and quasars

The Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) galaxy and quasar clustering data represents a significant expansion of data from Data Release 1 (DR1), providing improved statistical precision in baryon acoustic oscillation (BAO) constraints across multiple tracers, including bright galaxies, luminous red galaxies, emission line galaxies, and quasars. In this paper, we validate the BAO analysis of DR2. We present the results of robustness tests on the blinded DR2 data and, after unblinding, consistency checks on the unblinded DR2 data. All results are compared with those obtained from a suite of mock catalogs that replicate the selection and clustering properties of the DR2 sample. We confirm the consistency of DR2 BAO measurements with DR1 while achieving a reduction in statistical uncertainties due to the increased survey volume and completeness. The combined BAO precision, including both statistical and systematic errors, improves from ∼0.52% in DR1 to 0.30% in DR2—a factor of 1.7 gain. We assess the impact of analysis choices, including different data vectors (correlation function vs power spectrum), modeling approaches and systematics treatments, and an assumption of the Gaussian likelihood, finding that our BAO constraints are stable across these variations and assumptions with a few minor refinements to the baseline setup of the DR1 BAO analysis. We summarize a series of pre-unblinding tests that confirmed the readiness of our analysis pipeline, the final systematic errors, and the DR2 BAO analysis baseline. The successful completion of these tests led to the unblinding of the DR2 BAO measurements, ultimately leading to the DESI DR2 cosmological analysis, with their implications for the expansion history of the Universe and the nature of dark energy presented in the DESI key paper (companion paper).

79 ASTRONOMY AND ASTROPHYSICS

Probing the Coherence of Metal Absorbers toward Background Quasar Pairs

Abstract We investigate the spatial coherence of metal absorption lines in the circumgalactic medium using 4115 quasar pairs from Sloan Digital Sky Survey DR16. We identify 184 Mg ii (15 with dual detections) and 50 C iv (11 with dual detections) absorber pairs. The fraction of pairs with fractional equivalent width difference δW r ≤ 0.5 declines with projected separation: for Mg ii , it drops from ∼52% at 0–25 kpc to ∼7% at 100–200 kpc, while for C iv , it decreases from ∼100% to ∼21% over 0–200 kpc. In every separation bin, C iv absorbers with δW r ≤ 0.5 maintain a higher fraction than Mg ii absorbers. Similar results are obtained when the nondetections are treated as 2 σ W r upper limits in a survival analysis. The transverse autocorrelation function shows strong Mg ii clustering within 0–25 kpc, whereas the C iv excess remains relatively flat out to ∼100 kpc. These findings suggest that high-ionization C iv gas retains coherence over larger transverse scales than low-ionization Mg ii gas, in agreement with previous lensed quasar and high-resolution studies.

Li, Jing [Guangxi University] (ORCID:0009000148751

Measuring the Mean Free Path of H i Ionizing Photons at 3.2 ≤ z ≤ 4.6 with DESI Y1 Quasars

The mean free path of ionizing photons in the intergalactic medium (IGM; λ $^{912}_{mpf}$ ) is a crucial quantity in modeling the ionization state of the IGM and the extragalactic ultraviolet background, and it is widely used in hydrodynamical simulations of galaxies and reionization. We construct the largest quasar spectrum data set to date—12,595 signal-to-noise ratio > 3 spectra—using the Y1 observations of the Dark Energy Spectroscopic Instrument, to make the most precise model-independent measurement of the mean free path at 3.2 ≤ z ≤ 4.6. By stacking the spectra in 17 redshift bins and modeling the Lyman-continuum profile, we get a redshift evolution λ $^{912}_{mpf}$ ∝ ( 1 + z ) − 4.27 at 2 ≤ z ≤ 5, which is much shallower than the previous estimate, λ $^{912}_{mpf}$ ∝ ( 1 + z ) − 5.4 . We then explore the sources of systematic bias, including the choice of the intrinsic quasar continuum, the consideration of Lyman-series opacity and Lyman-limit opacity evolution, and the definition of λ $^{912}_{mpf}$ . Combining our results with estimates of λ $^{912}_{mpf}$ at higher redshifts, we conclude at high confidence that the evolution in λ $^{912}_{mpf}$ steepens at z ≈ 5. We interpret this inflection as the transition from the end of H i reionization to a fully ionized plasma that characterizes the IGM of the past ∼10 billion yr.

79 ASTRONOMY AND ASTROPHYSICS

JWST lensed quasar dark matter survey – II. Strongest gravitational lensing limit on the dark matter free streaming length to date

This is the second in a series of papers in which we use JWST Mid Infrared Instrument multiband imaging to measure the warm dust emission in a sample of 31 multiply imaged quasars, to be used as a probe of the particle nature of dark matter. We present measurements of the relative magnifications of the strongly lensed warm dust emission in a sample of nine systems. The warm dust region is compact and sensitive to perturbations by populations of haloes down to masses $\sim 10^6$ M$_{\odot }$. Using these warm dust flux-ratio measurements in combination with five previous narrow-line flux-ratio measurements, we constrain the halo mass function. In our model, we allow for complex deflector macromodels with flexible third- and fourth-order multipole deviations from ellipticity, and we introduce an improved model of the tidal evolution of subhaloes. We constrain a WDM model and find an upper limit on the half-mode mass of $10^{7.6}\, {\rm M}_\odot$ at posterior odds of 10:1. This corresponds to a lower limit on a thermally produced dark matter particle mass of 6.1 keV. This is the strongest gravitational lensing constraint to date, and comparable to those from independent probes such as the Ly $\alpha$ forest and Milky Way satellite galaxies.

dark matter