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TDCOSMO XXXI: New techniques in line-of-sight studies of time delay lenses

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Daniel Johnson, Devon Williams, Chris Fassnacht, Patrick Wells, Pierre Fleury, Allan Schweinfurth, Martin Millon, Aymeric Galan

Cosmology and Nongalactic Astrophysics · Submitted 10 Sep 2026 · 63 references

Pith's verdict: Solid incremental method paper with a genuine improvement (three-term κ_ext), but the central calibration assumption goes untested—the accuracy gain is plausible, not demonstrated.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2609.11652 (version 1)
Pith IDB4AT56BP, Pith's permanent record of this paper and its review
CitationAllan Schweinfurth, Aymeric Galan, Chris Fassnacht, Daniel Johnson, Devon Williams, Martin Millon, Patrick Wells, Pierre Fleury. "TDCOSMO XXXI: New techniques in line-of-sight studies of time delay lenses." arXiv:2609.11652 (2026). https://pith.science/paper/B4AT56BP
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The paper

The paper claims

Weighted number counts method for lens environment updates yields first full κ_ext estimates including foreground and background terms for time delay lenses

The reading

This paper updates the weighted number counts technique used to estimate the external convergence (κ_ext) affecting strong lensing time delays. The updates include using the Euclid Flagship Simulation instead of the Millennium Simulation, and computing κ_ext from three separate convergence contributions—observer-lens (κ_d), lens-source (κ_ds), and observer-source (κ_s)—rather than approximating with κ_s alone. The method is applied to 11 time delay lenses in the TDCOSMO 2026 sample, producing the first set of κ_ext estimates that account for all three line-of-sight contributions. Results are consistent with previous studies for most systems, with median κ_ext close to zero, suggesting no systematic over- or under-density bias across the sample. This work makes the correction for the external mass-sheet degeneracy more accurate, which is critical for unbiased Hubble constant inference from time delay cosmography.

What carries the argument

The central objects are the three convergence terms κ_d, κ_s, κ_ds (the line-of-sight projected mass overdensities between observer-lens, observer-source, and lens-source, respectively) and their combination into the external convergence κ_ext via the exact relation 1 - κ_ext = (1-κ_d)(1-κ_s)/(1-κ_ds). The method uses observed galaxy number counts (with distance and redshift weights) to reweight simulated lines of sight from the Euclid Flagship Simulation, matching the observed galaxy distribution to the known convergence fields. κ_ds is computed from the simulation's convergence maps using a discrete integral approximation to the second-derivative formula κ_ij = ∫ dχ (χW_ij) d²(χκ(χ))/dχ², avoiding the computational cost of full interpolation.

Extended reading notes

Core claim

The paper establishes that the weighted number counts method for estimating the external convergence κ_ext can be made more accurate by (a) replacing the Millennium Simulation with the larger-volume Euclid Flagship Simulation, which provides better resolution, more realistic galaxy modeling, and up-to-date cosmology; (b) computing the full κ_ext = 1 - (1-κ_d)(1-κ_s)/(1-κ_ds) rather than assuming κ_ext ≈ κ_s, thereby including the previously neglected observer-lens and lens-source contributions; and (c) excluding the modelled external shear as a weight, because it is often a poor proxy for the true line-of-sight shear. For the 11-lens TDCOSMO 2026 sample, the κ_ext posteriors are consistent with earlier κ_s-based results at 1σ for all but one system, indicating robustness, but the new method removes a potential bias in velocity-dispersion constraints and is essential for precision cosmology with double-plane lenses. This is the first application of a full three-term κ_ext estimate to a sample of time delay lenses.

Load-bearing premise

The method assumes that the correlation between galaxy number counts (weighted by distance and redshift) and the true external convergence in the Euclid Flagship Simulation is statistically well-calibrated, meaning that the simulation's galaxy formation model (halo occupation distribution and abundance matching) accurately represents the relationship between galaxies and dark matter at the relevant scales and redshifts.

Editorial extensions

If this is right

  • Time delay cosmography can now incorporate full κ_ext estimates that include foreground and background terms, reducing a potential systematic bias in H0 inference from the mass-sheet degeneracy.
  • The new method enables more accurate line-of-sight corrections for double-plane lens systems, where κ_ext differs more significantly from κ_s.
  • The use of the Euclid Flagship Simulation provides a larger statistical sample, better angular resolution, and an updated cosmological model, improving the reliability of κ_ext posteriors.
  • Excluding the modelled external shear as a weight eliminates a source of systematic error linked to lens-model degeneracies, leading to more homogeneous analyses across different lens systems.
  • The sample's median κ_ext is close to zero, supporting the assumption that the TDCOSMO lens sample is not systematically biased by line-of-sight over- or under-densities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If full κ_ext differs significantly from κ_s for some systems, previous H0 estimates relying on the κ_s approximation may require revision, particularly as statistical precision improves.
  • The method's accuracy depends on the galaxy formation model in the Euclid Flagship Simulation; residual biases in halo occupation distribution or abundance matching could propagate into κ_ext estimates, warranting systematic tests with alternative simulations.
  • The technique can be extended to larger lens samples from stage-IV surveys such as Euclid, where homogeneous photometry will reduce observational uncertainties and allow population-level studies of lens environments and selection effects.
Original abstract

The distribution of matter along the same line of sight but external to the main lens, quantified by $\kappa_\mathrm{ext}$, is a key source of uncertainty in time delay cosmography and other applications of strong lensing, and must be independently estimated to avoid biasing the inferred value of the Hubble constant ($H_0$). We present advancements and standardisations in the weighted number counts techniques used to constrain $\kappa_\mathrm{ext}$, in particular the use of the Euclid Flagship Simulation and a breakdown of the line-of-sight contributions between the observer, lens and source. As part of the TDCOSMO 2026 milestone analysis, we apply this updated method to the sample of 11 time delay lenses used in that study. Our estimates for certain systems are sensitive to these methodological changes, but are nonetheless consistent within $1\sigma$ for all but one of the systems which had been studied previously, with the median $\kappa_\mathrm{ext}$ across those systems changing from $-0.002$ to $-0.006$ following this new analysis. This work represents the first estimate of $\kappa_\mathrm{ext}$ values which includes the contribution of the observer-lens and lens-source terms, and largest standardised analysis of time delay lens environments to date.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,639 input tokens.

What the abstract itself states, as probabilities: claims first 0.92 · states limitation 0.81 · human participants 0.02 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.96 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

The distribution of matter along the same line of sight but external to the main lens, quantified by $\kappa_\mathrm{ext}$, is a key source of uncertainty in time delay cosmography and other applications of strong lensing, and must be independently estimated to avoid biasing the inferred value of the Hubble constant ($H_0$). We present advancements and standardisations in the weighted number counts techniques used to constrain $\kappa_\mathrm{ext}$, in particular the use of the Euclid Flagship Simulation and a breakdown of the line-of-sight contributions between the observer, lens and source. As part of the TDCOSMO 2026 milestone analysis, we apply this updated method to the sample of 11 time delay lenses used in that study. Our estimates for certain systems are sensitive to these methodological changes, but are nonetheless consistent within $1\sigma$ for all but one of the systems which had been studied previously, with the median $\kappa_\mathrm{ext}$ across those systems changing from $-0.002$ to $-0.006$ following this new analysis. This work represents the first estimate of $\kappa_\mathrm{ext}$ values which includes the contribution of the observer-lens and lens-source terms, and largest standardised analysis of time delay lens environments to date.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The distribution of matter along the same line of sight but external to the main lens, quantified by $\kappa_\mathrm{ext}$, is a key source of uncertainty in time delay cosmography and other applications of strong lensing, and must be independently estimated to avoid biasing the inferred value of the Hubble constant ($H_0$). background 1.00 0.05 0.07 asserted 1.00
s1 We present advancements and standardisations in the weighted number counts techniques used to constrain $\kappa_\mathrm{ext}$, in particular the use of the Euclid Flagship Simulation and a breakdown of the line-of-sight contributions between the observer, lens and source. contribution 0.77 0.88 0.11 asserted 0.99
s2 As part of the TDCOSMO 2026 milestone analysis, we apply this updated method to the sample of 11 time delay lenses used in that study. method 0.54 0.91 0.04 asserted 1.00
s3 Our estimates for certain systems are sensitive to these methodological changes, but are nonetheless consistent within $1\sigma$ for all but one of the systems which had been studied previously, with the median $\kappa_\mathrm{ext}$ across those systems changing from $-0.002$ to $-0.006$ following this new analysis. result 1.00 0.95 0.34 asserted 0.95
s4 This work represents the first estimate of $\kappa_\mathrm{ext}$ values which includes the contribution of the observer-lens and lens-source terms, and largest standardised analysis of time delay lens environments to date. contribution 0.96 0.92 0.27 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from analysis of existing data 0.73.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The observer-to-source weak lensing kernel 𝐾os and the effective kernel 𝐾ext of the external convergence 𝜅ext, a combination of contribu￾tions from 𝜅s , 𝜅d and 𝜅ds, for a lens located at 𝜒 = 0.5𝜒s . include 𝜅s for a range of source redshifts, while 𝜅ds is not readily available. This choice, however, tends to under (over) estimate the contribution of foreground (background) objects, as shown in
Figure 2. The Subaru/SuprimeCam 𝑖-band data of the RXJ1131−1231 field. The crosshairs indicate the location of the lens system. The galaxies that contribute to the number count calculations are located in the annulus between the inner (𝑟in = 5 ′′) and outer (𝑟out = 120′′) circles
Figure 4. The mean 𝜅ext and 𝜅s values calculated with the method described in Section 3 for our sample of time delay lenses, with the 16-84th percentile interval plotted as error bars and the source redshift shown as the point colours. Points corresponding to lenses whose environments were constrained without photometric redshifts are shown as open circles. The histograms of sampled 𝜅ext values for our sample of lenses are shown in
Figure 5. Convergence measurements for our sample of time delay lenses. In red we show the 𝜅ext values obtained using the number counts method described in Section 3, our main results. In green, we show the 𝜅s values obtained using the same method. The medians and 1-sigma confidence intervals of these distributions are shown in the legend. In blue, we plot the 𝜅ext distribution obtained for the same lens and source redshift (given in the subplot titles), but sampled randomly from FS2, without using weighted number count information. sampled for our lenses from random lines of sight. We see…
Figure 6. The foreground (left) and background (right) convergence terms, 𝜅d and 𝜅ds, plotted as a function of lens redshift, with source redshift indicated by colour. The points indicate the mean of the distributions resulting from the weighted number counts method described in Section 3, with error bars corresponding to the 16-84th percentile interval. Points plotted as an open circle as those for which we do not use photometric redshifts in our weighted number count contraints. lens redshift. 𝜅ds has more scatter, with a standard deviation of mean 𝜅ds values of 0.011 versus just 0.003 for…
Figure 7. The difference between the 𝜅s values estimated in this work, and selected values reported in the literature. Lenses are ordered by the publication date of the comparison work. Triangular markers correspond to literature values which include the modelled external shear as a constraint, circle markers for those which do not. Filled markers correspond to literature values which include partial or complete spectroscopic redshift information for the environment. The error bars are the quadratic sum of the standard deviations of our distributions, and the uncertainties reported in the…
Figure 8. A comparison between our 𝜅s estimates (on the x-axis) and selected values found in other works (on the y-axis) and their reported uncertainties. The line 𝜅s = 𝜅s (literature) is shown as a dotted black line. The markers and linestyles are set as

Showing 7 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.11652 by the authors.

References

Discussion

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