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A scalable hierarchical analysis of 800 simulated lensed AGN forecasts a dark-energy figure of merit of 6.7 in a w0waCDM cosmology.

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2026-08-03 21:43 UTC pith:OUHABJ7M

load-bearing objection The framework and honest caveats are welcome, but the headline DE FOM is a favorable seed selected for ground-truth alignment, not the expected constraining power. the 3 major comments →

arxiv 2511.13669 v3 pith:OUHABJ7M submitted 2025-11-17 astro-ph.CO

Investigating the Dark Energy Constraint from Strongly Lensed AGN at LSST-Scale

Sydney Erickson , Martin Millon , Padmavathi Venkatraman , Tian Li , Philip Holloway , Phil Marshall , Anowar Shajib , Simon Birrer
show 7 more authors
Xiang-Yu Huang Timo Anguita Steven Dillmann Narayan Khadka Kate Napier Aaron Roodman The LSST Dark Energy Science Collaboration
This is my paper
classification astro-ph.CO PACS 98.80.-k98.62.Sb
keywords time-delay cosmographydark energy figure of meritstrong gravitational lensinglensed AGNLSSThierarchical Bayesian inferencew0waCDMmass-sheet degeneracy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that the upcoming LSST sample of strongly lensed AGN—roughly 800 systems with measurable time delays—can be analyzed jointly with a new vectorized hierarchical inference scheme, and that this joint analysis simultaneously yields about a 2.5% constraint on the Hubble constant and a dark-energy figure of merit (inverse area of the w0–wa posterior) of 6.7. The decisive factor is sample size: adding 750 LSST-quality lenses with modest per-lens precision to a 50-lens high-fidelity sample roughly triples the dark-energy constraining power. The paper also isolates which follow-up investments matter: both IFU and single-aperture kinematics help, time-delay precision only pays off once it reaches about two days, and the lens redshift—not the source redshift—controls the dark-energy constraint. If the forecast holds, time-delay cosmography becomes an independent late-universe probe of dark energy using data LSST will collect anyway, complementing supernova and baryon-acoustic-oscillation measurements.

Core claim

Using a two-stage hierarchical Bayesian framework in which each lens is reduced to static data vectors—posterior samples over Fermat-potential differences, a Jeans-model quantity, orbital anisotropy, and external convergence, together with Gaussian summaries of time-delay and velocity-dispersion measurements—the authors evaluate the cosmological likelihood for hundreds of lenses simultaneously. With a baseline mix of 10 JWST-grade, 40 VLT-grade, and 750 LSST-grade lenses, fully marginalizing over mass-sheet and mass-anisotropy degeneracies, they forecast σ(H0)=1.7 km/s/Mpc and a dark-energy figure of merit of 6.7. Growing the sample from 50 to 800 lenses improves the DE FOM from 2.4 to 6.7.

What carries the argument

The central object is a vectorized hierarchical likelihood (the paper's fasttdc code) that evaluates the joint cosmological likelihood over hundreds of lenses at once. Each lens contributes importance samples from emulated posteriors, and the likelihood integral is evaluated by importance sampling over Fermat potential, Jeans model, anisotropy, mass-sheet, and external convergence; keeping Fermat potentials and kinematics correlated sample-by-sample is what lets the mass-sheet and mass-anisotropy degeneracies be constrained at the population level.

Load-bearing premise

The forecast depends on the assumption that the emulated mass-model posteriors, Gaussian time-delay errors (5-day baseline), and kinematic uncertainties accurately represent what LSST will actually measure, particularly that time-delay errors are independent Gaussians and that the neural mass-model posteriors reproduce image positions.

What would settle it

Run the same hierarchical inference on a simulation where time-delay measurements are produced by realistic LSST light-curve modeling (with cadence, seeing, and correlated systematic errors) rather than assigned Gaussian errors; if the resulting w0–wa posterior widens by more than roughly 50% relative to the forecast, the DE FOM claim fails. Alternatively, ray-trace samples from the paper's neural mass-model posteriors and check whether the image positions are reproduced within astrometric tolerance; if not, the Fermat-potential posteriors are overconfident.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, time-delay cosmography from LSST-scale samples delivers a dark-energy figure of merit of about 6.7 in w0waCDM, an independent late-universe probe that does not rely on CMB or distance-ladder assumptions.
  • The same sample yields a simultaneous ~2.5% Hubble-constant constraint, so a single lens population can anchor both H0 and dark-energy parameters.
  • Follow-up strategy guidance: IFU kinematics on a small sample and aperture kinematics on many lenses are roughly equivalent for the DE FOM, while time-delay precision below ~2 days is the strongest lever identified.
  • Lens redshift is the dominant redshift variable; prioritizing low-redshift deflectors improves the DE and H0 constraints, whereas source redshift has a negligible effect.
  • Because the likelihood is built from static per-lens summaries, the framework can scale to thousands of lenses, and an analytic Gaussian limit (already derived) would make the evaluation even faster.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same vectorized, data-vector-based likelihood could be adapted to other cosmological probes with correlated per-object summaries (e.g., cluster strong lenses or supernova distances), potentially accelerating joint population inference across surveys.
  • The two-day time-delay precision threshold is a concrete target for multi-band light-curve modeling: effort spent improving LSST delay measurements from ~5 days to ~2 days is forecast to have more cosmological payoff than improving mass-model precision at the assumed kinematics quality.
  • The redshift-configuration result suggests a discovery and follow-up strategy that preferentially targets low-redshift deflectors, although a wide redshift range may still be needed to constrain wa or to break degeneracies when combining with other probes.
  • The framework's modularity—separating modeling posteriors from cosmology—makes it well suited for a 'round-trip' systematics test, where simulated images are processed through the full pipeline to validate calibration before real LSST data arrive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces fasttdc, a scalable hierarchical Bayesian likelihood code for time-delay cosmography, and uses it to forecast dark-energy constraints from a simulated LSST sample of 800 strongly lensed AGN. Individual lens observables (image-based mass models, time delays, stellar kinematics, external convergence) are emulated at various fidelities, and the joint inference marginalizes over mass-sheet and anisotropy population parameters. In a fiducial w0waCDM analysis, the authors report σ(H0)=1.7 km/s/Mpc (~2.5%) and a dark-energy figure of merit (DE FOM) of 6.7, roughly triple the DE FOM from a 50-lens IFU sample. Additional experiments compare follow-up strategies (IFU vs aperture kinematics, space-based imaging, forward modeling, improved time-delay precision) and redshift configurations.

Significance. If the quantitative forecasts are robust, this work would provide a valuable public tool (fasttdc is released) and a useful planning benchmark for LSST follow-up programs. The paper is careful to include several real physical degeneracies, uses a forward-modeling test from the OM10 catalog, and explicitly explores random-seed fluctuations in Appendix D. These are genuine strengths. However, the headline numbers are not robust because they are drawn from a single, selected random seed, and several key measurement errors are imposed rather than simulated. The qualitative conclusion that larger samples of lower-precision lenses add constraining power is plausible, but the specific DE FOM values and the 'factor of ~3' improvement should not be treated as established expectations.

major comments (3)
  1. [Appendix D / Fig. 6b / Table 6 / Abstract] The headline DE FOM=6.7 and the factor-of-~3 improvement over the 50-lens sample are reported from a single random seed chosen, as Appendix D states, 'whose central values align with the ground truth ΛCDM values usually assumed when using the DE FOM metric.' Figure 6b shows that DE FOM across ten seeds spans roughly 2.5 to 20 and is correlated with the median w0 of each realization. Because the DE FOM is a nonlinear functional of the posterior and the seed is selected on posterior medians, 6.7 is a favorable draw, not an expectation or median. This affects every experiment comparison in Table 6 and the conclusions drawn from them. I request that all headline numbers and Table 6 comparisons be recomputed over a fixed set of seeds and reported as medians with scatter, or that an alternative seed-independent metric be used.
  2. [Section 4.2 / Section 6.5 / Table 4] The mass-model emulation relies on NPE posteriors whose covariance is rescaled to emulate HST-FM and JWST-FM precision. Section 6.5 concedes that samples from these posteriors 'are not guaranteed to produce lensing configurations where the lens model and source position reproduce the image positions exactly.' The conclusions from Experiments 2.1-2.3, including the claim in Section 5.3 that 'the mass model precision is not a limiting factor,' depend on this unvalidated emulation. If the rescaling factors or the NPE covariance shapes are not representative of true forward modeling, the error budget changes. Please either validate the emulation against a small set of full forward models or explicitly frame the quantitative results as conditional on this approximation, including in the abstract.
  3. [Section 4.3 / Section 5.4 / Section 6.2] Time-delay measurement precision is assigned directly as a Gaussian error (5-day baseline, 3% for monitored lenses, and 2-day upgrades) without simulating light curves or accounting for correlated errors, seasonal gaps, or the fraction of lenses that yield usable delays. The paper's strongest follow-up recommendation — that a 2-day LSST time-delay precision produces significant gains (Experiment 3.4, DE FOM=10.1) while 3- and 4-day precisions do not — is therefore driven by the assumed error model. Correlated or non-Gaussian time-delay errors, or a reduced yield of measurable delays, could materially weaken this threshold. Please test sensitivity to correlated errors and detection yield, or temper the recommendation accordingly.
minor comments (5)
  1. [Section 5.4] Typo: 'the the importance' should read 'the importance' in the sentence 'This again demonstrates the the importance of assessing constraining power across all parameters simultaneously.'
  2. [Sections 5.2, 5.4, 6.2] The phrase 'improving off of the baseline' should be 'improving on the baseline' (appears in Experiment 1.1 discussion and elsewhere).
  3. [Appendix A.1] Typo: 'futher work' should be 'further work' in the sentence 'Exploration of photometric redshifts is left for futher work.'
  4. [Table 6] The reported DE FOM values are approximate and fluctuate with seed, as the authors note in Appendix D; this caveat should appear directly in the Table 6 caption to avoid the appearance of fixed, reproducible numbers.
  5. [Abstract / Section 7] Given the seed-dependence documented in Appendix D, the abstract and conclusion should state that the quoted DE FOM values are from one illustrative realization, not a guaranteed expectation, until the multi-seed analysis is incorporated.

Circularity Check

1 steps flagged

Headline DE FOM=6.7 is post-selected from a favorable noise seed; otherwise the forward-model derivation is not circular.

specific steps
  1. fitted input called prediction [Appendix D, Figure 6; used for baseline result in Table 6]
    "When assessing the DE FOM across these 10 seeds, we noticed a large range of values. The DE FOM is correlated with the median values of w0 and wa in each posterior... To account for this effect, we choose a baseline seed for our experiments (brown) whose central values align with the ground truth ΛCDM values usually assumed when using the DE FOM metric."

    The paper presents DE FOM = 6.7 as the baseline forecast and as the basis for the 'factor of ~3' improvement from 2.4 to 6.7. But the measurement-noise seed was not drawn at random for that headline: it was explicitly selected so that the posterior median (w0, wa) matches the ΛCDM ground truth that the DE FOM metric assumes. The paper's own Figure 6b shows that the FOM fluctuates widely with seed, roughly 2.5 to 20, and that these fluctuations are correlated with the posterior central values. Thus the headline FOM is partly an input/selection choice rather than an expected value of the stochastic simulation. This is postselection bias rather than full construction-level circularity, because the same-seed comparisons between experiments and the ground-truth recovery tests remain informative

full rationale

The paper is a forward-modeling forecast: it generates a simulated LSST lens population, emulates measurements under a stated ΛCDM ground truth, runs the hierarchical likelihood of Equations 13-14, and checks whether the inference recovers the input cosmology. The likelihood derivation is self-contained and follows from Bayes' theorem and standard time-delay cosmography equations; it is not defined in terms of the cosmological answer. The claimed 'predictions' are precision forecasts, so they naturally depend on assumed measurement precisions, but that is the normal content of a forecast rather than circular reduction. The main legitimate concern is the baseline seed choice in Appendix D: the DE FOM=6.7 headline is from a seed selected because its posterior central values align with the ground truth, and the FOM is strongly correlated with those central values, so the reported number is a favorable realization, not the expected value. There are also self-citations (TDCOSMO25 for the λint/βani population scatter and time-delay precision; Erickson et al. 2025 for NPE mass models; Venkatraman et al. 2025 for catalog preparation), and Section 6.5 concedes that the NPE mass-model posteriors are not guaranteed to reproduce image positions. These are real limitations and correctness risks, but they are input assumptions with stated motivations rather than load-bearing arguments that make the output equal to the input by construction. The central derivation and the qualitative conclusions about sample-growth and follow-up strategies survive even if the exact FOM is seed-dependent. Hence no significant construction-level circularity; the score reflects the post-selection of the headline seed.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 0 invented entities

No new particles, forces, or phenomenological entities are introduced; the ledger is entirely populated by assumptions about survey precision, population scatter, Gaussianity, and the realism of the emulated mass models. The most consequential free parameters are the per-tier error budgets (time-delay, kinematics, Fermat potential, κ_ext) and the population scatters, all of which are taken from prior TDCOSMO/forecast literature or hand-assigned.

free parameters (7)
  • Population scatter σ(λ_int)=0.1 = 0.1
    Assigned from TDCOSMO Collaboration et al. (2025) to give realistic scatter in the mass-sheet parameter; directly controls how much the population hyperparameters absorb cosmology.
  • Population scatter σ(β_ani)=0.1 = 0.1
    Assigned anisotropy scatter from TDCOSMO25; wider scatter makes the DE forecast more conservative.
  • External convergence uncertainty σ(κ_ext) = 0.05
    Assumed Gaussian LOS posterior width for every lens, stated in Section 4.5 as 'typical of the current precision'; not a measured value for the simulated sample.
  • LSST time-delay precision = 5 days
    Baseline Gaussian time-delay error for LSST-only lenses; authors label it 'conservative' and test 4/3/2-day variants; no light-curve simulation.
  • Monitored time-delay precision = 3%
    Assigned precision for long-term monitored lenses based on TDCOSMO25; directly sets the D_dt precision anchor.
  • Kinematic precision / bin structure = 5% per bin; 10 bins JWST, 3 bins MUSE
    Assumed based on Birrer & Treu (2021) and Knabel et al. (2025); the resolved-kinematics power for λ_int/β_ani is calibrated by these assumptions.
  • Fermat potential precision per imaging tier = 2% (JWST-FM), 4% (HST-FM), 11% (HST-NPE), 18% (LSST-NPE)
    Median Fermat-potential precisions produced by NPE and covariance re-scaling; re-scaling is tuned so HST-FM matches TDCOSMO25's 0.04 power-law slope precision.
axioms (7)
  • domain assumption Gaussian likelihoods for time delays, kinematics, and lens-model posteriors (normal distributions in Eqs. A22–A23)
    Section A.3 assumes measurement uncertainties are summarized by Gaussian mean and covariance; real light-curve delays and Jeans modeling can be non-Gaussian.
  • domain assumption Flat w0waCDM with Chevallier-Polarski-Linder parameterization, flat spatial geometry
    Equation 3 and Section 2 assume a specific dark-energy parameterization; forecasts are conditional on that model.
  • domain assumption OM10 catalog redshifts/velocities are representative of LSST's lensed AGN population, with random selection across the catalog except for stated selection cuts
    Section 4.1 starts from OM10 and Venkatraman et al. (2025) modifications; any catalog incompleteness (e.g., lensed AGN with faint hosts) propagates into the forecast.
  • domain assumption Gaussian population distributions for λ_int and β_ani (normal hyperparameters in Eq. 8)
    Section 2.4 assumes a Gaussian population model; Section 6.5 notes λ_int radial dependence is neglected.
  • ad hoc to paper The NPE mass-model posteriors are unbiased and their sample spread is a valid uncertainty estimate, with covariance re-scaling faithfully emulating dedicated forward modeling
    Section 4.2 re-assigns posterior means to ground truth; Section 6.5 concedes samples are 'not guaranteed to produce lensing configurations' that reproduce image positions exactly, so the emulated posterior width may not equal true modeling uncertainty.
  • domain assumption Perfect spectroscopic redshifts for all 800 lenses
    Section 2.4 states redshifts are not marginalized over, citing ChANGES; catastrophic photo-z failures or incomplete spectroscopy would bias D_dt.
  • standard math Independent lenses with independent measurement errors (Eq. A5)
    Standard hierarchical assumption; breaks if shared calibration errors or correlated LOS structures are present.

pith-pipeline@v1.3.0-alltime-deepseek · 28115 in / 9310 out tokens · 74093 ms · 2026-08-03T21:43:02.337910+00:00 · methodology

0 comments
read the original abstract

Strongly lensed Active Galactic Nuclei (AGN) with an observable time delay can be used to constrain the expansion history of the Universe through time-delay cosmography (TDC). As the sample of time-delay lenses grows to statistical size, with $\mathcal{O}$(1000) lensed AGN forecast to be observed by the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), there is an emerging opportunity to use TDC as an independent probe of dark energy. To take advantage of this statistical sample, we implement a scalable hierarchical inference tool which computes the cosmological likelihood for hundreds of strong lenses simultaneously. With this new technique, we investigate the cosmological constraining power from a simulation of the full LSST sample. We start from individual lenses, and emulate the full joint hierarchical TDC analysis, including image-based modeling, time-delay measurement, velocity dispersion measurement, and external convergence prediction. We fully account for the mass-sheet and mass-anisotropy degeneracies. We assume a sample of 800 lenses, with varying levels of follow-up fidelity based on existing campaigns. With our baseline assumptions, within a flexible $w_0w_a$CDM cosmology, we simultaneously forecast a $\sim$2.5% constraint on H0 and a dark energy figure of merit (DE FOM) of 6.7. We show that by expanding the sample from 50 lenses with IFU kinematics to include 750 lenses with plausible LSST time-delay measurements, we improve the forecasted DE FOM by nearly a factor of 3, demonstrating the value of incorporating this portion of the sample. We also investigate different follow-up campaign strategies, and find significant improvements in the DE FOM with additional stellar kinematics measurements and higher-precision time-delay measurements. We also demonstrate how the redshift configuration of time-delay lenses impacts constraining power in $w_0w_a$CDM.

Figures

Figures reproduced from arXiv: 2511.13669 by Aaron Roodman, Anowar Shajib, Kate Napier, Martin Millon, Narayan Khadka, Padmavathi Venkatraman, Philip Holloway, Phil Marshall, Simon Birrer, Steven Dillmann, Sydney Erickson, The LSST Dark Energy Science Collaboration, Tian Li, Timo Anguita, Xiang-Yu Huang.

Figure 1
Figure 1. Figure 1: Diagram of the vectorized likelihood evaluation, isolated to the time-delay likelihood only (the kinematic likelihood is treated similarly). In step 1, samples of the modeling inputs (λint,κext,∆ϕ) are condensed into samples of the predicted time￾delay, with a value tracked for every importance sample, across every lens. In step 2, the likelihood of the predicted time-delay is evaluated against the observe… view at source ↗
Figure 2
Figure 2. Figure 2: We demonstrate how growing the sample size impacts the cosmological inference from the baseline experiment configuration ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cosmological posteriors from four experiments, testing additional follow-up compared to the baseline. The baseline posterior is shown in grey. On the top left, we test adding IFU observation on 72 lenses (dark green). On the top right, we test adding aperture kinematics on 300 lenses (light green). On the bottom left, we test a conservative long-term monitoring campaign, increasing the time-delay precision… view at source ↗
Figure 4
Figure 4. Figure 4: Cosmological constraint from a sample of 10 lenses, changing the redshift populations. For each plot, the source redshift population is constant, with µ(zsrc)=2.0, and the lens redshift changes from µ(zlens)=1.0 (red) to µ(zlens)=0.5 (purple) to µ(zlens)=0.2 (blue). The left plot has a uniform Ωm prior, the right plot has an informative Ωm prior. First, we make a more conservative assumption for lenses wit… view at source ↗
Figure 5
Figure 5. Figure 5: We compare cosmological contours from the forecast in Shajib et al. (2025b) (orange), to our baseline experiment (grey), and our experiment 3.1 with extra long-term time-delay monitoring (blue). As expected, results fluctuate about the ground truth values. When assessing the DE FOM across these 10 seeds, we noticed a large range of values. The DE FOM is correlated with the median values of w0 and wa in eac… view at source ↗
Figure 6
Figure 6. Figure 6: We assess the how stochasticity from measurement errors and lens selection impacts the inference by running the baseline experiment with ten random seeds. Each color corresponds to a run of the experiment with a different random seed. The baseline seed is shown in brown. On the left, we plot the difference between the inferred value and the ground truth, divided by the 1σ width of the posterior, for each p… view at source ↗

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