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Dynamical Systematics for Time Delay Lenses and the Impact on the Hubble Constant

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper argues that the systematic uncertainties in measuring lens-galaxy velocity dispersions, not statistical noise, set the real floor on time-delay measurements of the Hubble constant, with biases in H0 typically exceeding the 2% pre

desk verdict A careful, transparent systematics study that makes a convincing broad case that kinematic modeling choices can break the 2% H0 budget; treat Table 2 as an illustrative warning, not exact corrections. read the letter →

arxiv 2602.03934 v2 pith:HUZVU663 submitted 2026-02-03 astro-ph.GA astro-ph.CO

R. Forés-Toribio , C. S. Kochanek , J. A. Muñoz This is my paper · ORCID
classification astro-ph.GAastro-ph.CO
keywords gravitationallensingtime-delaycosmographyHubbleconstantstellarvelocitydispersionorbitalanisotropypointspreadfunctionsystematicsearly-typegalaxiesJeansequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that stellar-dynamical systematics, not statistical noise, are the real limit on measuring the Hubble constant from time-delay lenses. Because H0 errors scale directly with errors in the squared velocity dispersion (ΔH/H0 ∝ Δσ²/σ²), biases of 2–40% in σ² translate into biases of several percent in H0, far above the 2% target. The paper identifies five sources: PSF modeling, the difference between the measured dispersion and the true mean-square velocity, choices of orbital-anisotropy models, mismatches between the true light profile and the assumed Hernquist profile, and population-level correlations. It also argues that early-type lens galaxies are dynamically homogeneous, so shared parameters like the anisotropy radius must be marginalized once for the whole sample; their uncertainties then do not shrink with more lenses. A sympathetic reader would take away that the current error budget for time-delay cosmography is dominated by systematics that averaging over lenses cannot remove.

What carries the argument

The driving mechanism is the sensitivity relation ΔH/H0 ∝ Δσ²/σ², which converts every fractional kinematic error into a proportional error in H0. The population-level machinery is the marginalization identity for a shared anisotropy radius: with N independent lenses the systematic variance shrinks as (σ_H² + α²σ_a²)/N, but with a homogeneous population it becomes σ_H²/N + α²σ_a², so the anisotropy contribution never averages away. The numerical work uses spherical Jeans models with a Hernquist stellar tracer inside a singular-isothermal-sphere mass distribution to compute how each systematic shifts the predicted velocity dispersion.

What would settle it

Measure the fourth Gauss–Hermite moment h4 of the eight lens galaxies; if early-type lenses have h4 near the typical 0.01–0.04 seen in local ellipticals, the claim that Osipkov-Merritt models underestimate anisotropy systematics is supported, and if h4 is near zero the up-to-18% anisotropy shifts are overestimated. Alternatively, compare measured dispersions of the same lenses fitted with Gaussian versus double-Moffat PSFs: the predicted 2–6% shifts, and the larger 5–40% shifts from light-profile choices, should appear in the data.

Watch

Extended reading notes

Core claim

The central claim is that most systematic errors in the kinematic measurements of the eight time-delay lenses can bias H0 by more than 2%, with some per-lens fractional changes in σ² reaching 40%. Specifically: a 10% error in PSF FWHM causes 0.2–2.6% changes; replacing a Gaussian PSF with Moffat wings causes up to 2–6%; the difference between the measured dispersion and the true mean-square velocity can be 2–8%; anisotropy model choice causes up to 5–18% shifts, with the commonly used Osipkov-Merritt model failing to produce h4 values typical of early-type galaxies; and changing the assumed stellar light profile from Hernquist to a Sérsic n=2 profile raises the predicted dispersion by 5–40%.

Load-bearing premise

The numerical amplitudes are computed with a spherical singular-isothermal-sphere mass distribution and a spherical Hernquist stellar tracer; if real lens mass distributions are power-law, composite, or triaxial, the percentages in Table 2 could change materially, even though the qualitative conclusion would likely survive — as the paper itself notes when discussing the nonphysical behavior of the SIS potential at the center.

Editorial extensions

If this is right

  • A 1% bias in the squared velocity dispersion produces roughly a 1% bias in H0 for a typical lens, so kinematics must be known to better than 1% in σ (2% in σ²) to reach the 2% H0 target.
  • PSF FWHM must be measured to better than 10% accuracy; otherwise seeing errors alone can exceed the 2% budget.
  • If early-type galaxies are dynamically homogeneous, the uncertainty from the anisotropy radius does not decrease with sample size, so combining more lenses cannot cure this systematic.
  • Measured dispersions must be corrected to true mean-square velocities using h4 or full velocity distributions; for radially anisotropic galaxies this raises v_rms relative to σ*, typically shifting inferred H0 upward.
  • Using the real photometric profile (e.g., Sérsic n≈2) instead of the Hernquist profile changes predicted dispersions by up to 40% for some lenses, biasing H0 in either direction depending on the true profile.

Reading between the lines

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

  • A testable prediction of the homogeneity argument is that velocity-dispersion offsets between different template stars should be systematically correlated across early-type lenses of similar redshift; re-fitting archival spectra with matched templates and checking for lens-to-lens correlation would test this directly.
  • The numerical magnitudes in Table 2 rest on SIS plus spherical Hernquist models; applying the same systematics to the newer double-Sérsic, power-law, or composite mass models would likely change the numbers but not the qualitative conclusion that systematics exceed the 2% budget.
  • The population-level marginalization logic also applies to template-star choice and light-profile choice: posterior distributions should be computed for each global model assumption and then marginalized over assumptions, rather than averaging per-lens shifts.
  • The same homogeneous-population reasoning used here has been applied in cluster cosmology to model mass–richness relations; borrowing those correlation-marginalization tools could handle partially correlated lens populations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper assembles a systematic-error budget for the stellar dynamical constraints used in time-delay cosmography, applied to the eight lenses in the TDCOSMO/H0LiCOW joint analyses. Working mostly with spherical SIS total-mass models and Hernquist stellar tracers, the authors compute fractional biases in the squared velocity dispersion from PSF width and shape, aperture miscentering, the difference between the measured Gaussian dispersion and the Jeans rms velocity, the choice of orbital anisotropy, and the assumed stellar light profile. They conclude that most of these effects exceed the 2% σ² budget required for a 2% H0 measurement, and they argue that because early-type galaxies form a fairly homogeneous population, nuisance parameters such as the anisotropy radius must be marginalized globally rather than independently per lens. The paper also derives a population-weighted Jeans equation for mixed stellar populations and uses it to argue that the photometric profile entering dynamical models should be weighted by line equivalent width, not by broad-band flux.

Significance. If the quantitative conclusions hold, this is a useful and timely cautionary analysis for the TDCOSMO program. The paper is transparent about its approximations, provides analytic Jeans solutions and Monte Carlo line-of-sight velocity distributions, and makes a falsifiable point that Osipkov-Merritt models fail to reproduce the h4 values observed in early-type galaxies. The strongest contribution is the systematic side-by-side comparison in Tables 2 and 5, which gives the community a concrete checklist of where the 2% budget is violated. The population-level homogeneity argument in §5, though in need of empirical calibration, is conceptually important and could affect how future lens samples are averaged. The main weakness is that the quantitative amplitudes are computed for idealized spherical SIS/Hernquist models and are not propagated through an H0 posterior, so the mapping from Δσ²/σ² to ΔH0/H0 remains schematic.

major comments (3)
  1. [§2.1–§2.3, Table 2, Eq. (1)] The quantitative core of the paper, Table 2, is computed for a spherical SIS total mass distribution and a Hernquist tracer with s=0.55 R_e q^{1/2}. The authors state in §2.1 that this is for simplicity and due to unavailability of the TDCOSMO double-Sérsic parameters, and §2.2 notes that the SIS potential produces 'nonphysical behavior at the center' that affects the NIRSpec comparisons. Since the actual TDCOSMO mass models are power-law or composite and often triaxial, the amplitudes in Table 2, especially the Sérsic n=2 rows (up to +40%) and Cuddeford rows (up to +18%), are model-dependent. The mapping to H0 is made only through the SIS-based sensitivity ψ of Eq. (1), not through an H0 posterior. The broad claim that systematics exceed 2% is likely robust because several independent effects already exceed 2% under the stated assumptions, but the quantitative amplitudes are not yet ful
  2. [§2.2, Fig. 4, Table 2] The double-Moffat entry for RX J1131 observed with NIRSpec has a positive sign only because, as the authors write, the changes in the numerator and denominator are affected by 'the nonphysical behavior at the center from using an SIS potential.' This is an artifact of the model, not a physical PSF effect. Reporting it in Table 2 without a caveat overstates the certainty of that particular systematic. The same inverted trends for other NIRSpec systems are flagged in the text, so the table should either recompute these entries with a cored or power-law potential or mark them as model-dependent.
  3. [§5, Eqs. (15)–(19)] The formal argument that a dynamically homogeneous population prevents the anisotropy radius uncertainty from averaging down with N is mathematically correct for a single shared r_a/s with a fixed Gaussian prior. However, the paper does not provide an empirical estimate of the correlation scale for r_a/s, stellar-template offsets, or light-profile choice across the lens sample. Early-type galaxies have mass and redshift trends, which the paper acknowledges, so the shared-value case is an upper limit. A partially correlated model, e.g. a correlation coefficient between lens-to-lens values, would interpolate between Eq. (17) and Eq. (19). The phrase 'must be marginalized over the lens sample as a whole' overstates the case as written; the homogeneous limit should be framed as one end of a correlation model, ideally quantified with simple test cases.
minor comments (5)
  1. [§5, text near Eq. (15)] The Gaussian prior is written as P(r_a) ∝ exp(−r_a/2σ_a²), which is an exponential, not a Gaussian. It should read exp(−r_a²/(2σ_a²)). The analytic marginalizations in Eqs. (17) and (19) rely on a true Gaussian prior, so this typo should be corrected.
  2. [Table 1] The RX J1131−1231 row is ambiguous: the seeing and aperture entries '0.15/0.96π×0.955 2' need a clearer layout specifying which values apply to the NIRSpec and KCWI observations. Also state explicitly that all FWHM and aperture values are in arcseconds.
  3. [§2.3, Figs. 5–6] The Monte Carlo distributions use 10^9 particles and show residual shot noise. Since some reported differences are only 0.7–1.6%, a quantitative estimate of the Monte Carlo noise floor would help the reader judge which entries in the velocity-DF rows of Table 2 are significant.
  4. [§2.2] The paper says LSF-related systematic errors were considered but are negligible compared with other sources, but no calculation is shown. Given that template/LSF issues are known to be non-negligible in some dispersion measurements, one sentence summarizing the magnitude or a pointer to the relevant check would be useful.
  5. [Fig. 9 caption] The caption says the weighted mean temperature is 'B-band luminosity weighted,' but the text says this is roughly the rest wavelength range usually modeled. Clarify whether the weight is a B-band filter or the actual spectral window used in the dispersion fits, since the two need not be identical.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: systematic shifts are forward-modeled; self-citations are contextual, not load-bearing.

full rationale

The paper's central quantitative output (Table 2) lists fractional changes in squared velocity dispersion computed by forward modeling: Jeans-equation solutions for specified mass/light/anisotropy models, Gaussian/Moffat PSF convolution, aperture averaging (Eq. 6), Monte-Carlo line-of-sight velocity distributions, and Sersic/Hernquist/Jaffe tracer comparisons. No H0 value is fit or predicted from these shifts; the mapping to H0 uses a toy-model sensitivity defined in Eq. 1, whose values are derived from the Appendix A Hernquist/SIS model, not from the target cosmological parameter. Section 5's population-homogeneity argument is an analytic marginalization calculation (Eqs. 15-19): if a single r_a/s is shared, its uncertainty cannot average down; this is a statistical consequence, not a fitted prediction. Self-citations (Kochanek 2002, 2006, 2020, 2021) establish the mass-sheet/xi-kappa_E degeneracy and prior accuracy estimates, but the paper's own amplitudes do not reduce to those citations. Assumptions of spherical SIS/Hernquist models are explicit and are flagged as producing 'nonphysical behavior at the center'; this is a modeling limitation, not circularity. No load-bearing step reduces by construction to its inputs. Score 2 reflects minor, non-load-bearing self-citations rather than circular derivation.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

All central quantitative claims rest on spherical, SIS, and homogeneous-population approximations plus a simplified spectral-weighting model; these are the upstream costs. The paper introduces no new physical entities and fits no constants to the Hubble constant itself.

free parameters (6)
  • Hernquist scale radius s = 0.55 Re q^{1/2} = per lens in Table 1 (0.147-1.127 arcsec)
    Sets the stellar tracer for all dynamical model calculations; chosen because double-Sersic parameters are unavailable, not fitted to H0.
  • constant anisotropy beta = scanned 0 to 1
    PSF, sigma*/vrms and photometric-model systematics are reported as maxima over this scan.
  • Osipkov-Merritt anisotropy radius ra/s = 1, 3, 10
    Spans the range marginalised over in previous time-delay analyses; determines h4 and sigma*/vrms offsets in Section 2.3.
  • Cuddeford central anisotropy beta0 = 0.1, 0.3, 0.5 at ra/s=1
    Expands the allowed sigma-squared range and drives the largest anisotropy-model entries in Table 2.
  • Moffat PSF parameters = single eta=4; double 80% eta=7 + 20% eta=2
    Realistic non-Gaussian PSF models; produce up to 5.7% sigma-squared bias in Table 2.
  • Sersic index n for photometric comparison = n=2,3,4 and Jaffe
    Used to quantify light-profile model systematics relative to Hernquist; n=2 yields up to 40% differences.
assumptions (6)
  • domain assumption Spherical, non-rotating lens galaxy for dynamical modeling.
    Stated at the start of Section 2 and deferred in Section 6; ellipticity and rotation introduce sigma-squared changes on the scale of the potential ellipticity.
  • domain assumption SIS total mass profile for the systematic calculations.
    Used throughout Sections 2-4; Section 2.2 notes the SIS potential creates nonphysical central behavior affecting NIRSpec results.
  • domain assumption Early-type lens galaxies form a homogeneous population in dynamics and stellar populations.
    Basis of Section 5; if false, the no-1/sqrt(N) averaging conclusion weakens to a partial-correlation statement.
  • standard math Measured Gaussian width sigma* differs from vrms, with the offset set by Gauss-Hermite h4.
    Standard van der Marel and Franx 1993 expansion; used throughout Section 2.3 and Section 3.
  • ad hoc to paper Mixed stellar populations each satisfy the Jeans equation, and a single Gaussian absorption line can model the weighting.
    Simple model in Section 4 used to conclude the correct light profile is equivalent-width weighted; not validated against full spectral template fits.
  • standard math H0 scales as H0 proportional to 1 - kappa_E and kappa_SIS = 1/2.
    Standard lensing mass-sheet degeneracy from Kochanek 2002; used for the psi sensitivity estimate and Figure 1.

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Pith. "Pith review of Dynamical Systematics for Time Delay Lenses and the Impact on the Hubble Constant." pith.science (2026). https://pith.science/paper/HUZVU663

@misc{pith2026260203934,
  author       = {Pith},
  title        = {Pith review of: Dynamical Systematics for Time Delay Lenses and the Impact on the Hubble Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUZVU663}},
  note         = {Machine review of arXiv:2602.03934}
}
abstract

While time-delay lenses can independently probe $H_0$, the estimates are degenerate with the convergence of the lens near the Einstein radius. Velocity dispersions, $\sigma$, can be used to break the degeneracy, with uncertainties $\Delta H/H_0 \propto \Delta\sigma^2/\sigma^2$ ultimately limited by systematic uncertainties in the kinematic measurements - measuring $H_0$ to 2\% requires $\Delta\sigma^2/\sigma^2 < 2\%$. Here we explore a broad range of potential systematic uncertainties affecting eight time-delay lenses used in cosmological analyses. We find that: (1) The characterization of the PSF in both absolute scale and shape is important, with biases in $\Delta\sigma^2/\sigma^2$ up to $1$-$5\%$ for ground-based observations. Small miscenterings of the lens are less important. (2) The difference between the measured velocity dispersion and the mean square velocity needed for the Jeans equations is important, with up to $\Delta\sigma^2/\sigma^2 \sim 2$-$6\%$. (3) The choice of anisotropy models is important with maximum changes of $\Delta\sigma^2/\sigma^2 \sim2$-$18\%$. Biases may be minimized by using models that reproduce the $h_4$ velocity moments typical of early-type galaxies. (4) Small differences between the true stellar mass distribution and the model light profile matter ($\Delta\sigma^2/\sigma^2 \sim 1$-$10\%$), with radial color gradients further complicating the problem. The Jeans equations for mixed stellar populations imply that the correct profile is a population line equivalent width weighting corresponding to no broad band filter profile. Finally, the homogeneity of the early-type galaxy population means that many dynamically related parameters must be marginalized over the lens sample as a whole and not over individual lenses.

Figures

Figures reproduced from arXiv: 2602.03934 by the authors.

Figure 1
Figure 1. Observed central velocity dispersion σ 2 /σ2 SIS (top) and surface density at the Einstein radius κE/κSIS normalized by the values for an SIS lens model. The heav￾ier lines indicate the changes that will increase H0 by 8% from the SIS model. The curves are for the eight time-delay lenses with the parameter values from [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Absolute values of the fractional changes, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Fractional changes, ∆σ 2 d/σ2 d, for apertures mis￾centered by half a pixel from the center of the galaxy as a function of anisotropy, β. Each system is plotted in a differ￾ent color and line style, and we assumed a Gaussian PSF for all the cases. ity dispersion decreases with radius, miscentering the aperture leads to a lower measured value than if it is placed at the center. Hence, the fraction ∆σ 2 d /σ2 d ≡ (σ 2… view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: The differences [P(v)−G(σ∗)]/P(v = 0) between the los velocity distribution (P(v)) and the best fit Gaussian model (G(σ∗)) normalized by the peak of the los velocity dis￾tribution at zero velocity, P(v = 0), for SDSS J1206+4332. The anisotropy radii considered are ra/s…
Figure 6
Figure 6. Figure 6: The same as in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Mean squared los velocity dispersion inside ra￾dius R, σ 2 los(< R), for the Hernquist profile embedded in an SIS mass model normalized by the squared velocity disper￾sion of the SIS model, σ 2 SIS. The solid lines are for the con￾stant anisotropy model with β = 0 (bot…
Figure 8
Figure 8. Figure 8: Mean squared enclosed los velocity dispersion, σ 2 los(< R)/σ2 SIS, for various stellar density distributions normalized to the same effective radius Re embedded in an SIS mass distribution and using a constant anisotropy model. The solid lines and the shaded region be…
Figure 9
Figure 9. Figure 9: B-band luminosity weighted mean temperatures ⟨log T⟩ as a function of age and metallicity for the standard PARSEC isochrones. The metallicities are roughly twice so￾lar (lowest), solar, half solar and quarter solar (top). Vertical lines indicate the redshift correspond…
Figure 10
Figure 10. Figure 10: The line-of-sight velocity dispersion profiles σ 2 los of isotropic Hernquist models and a Hernquist model in the potential of an SIS (solid). The Hernquist models are normalized to have the same projected mass as the SIS model inside 3s (top red dashdotted line), 2s …
Figure 11
Figure 11. Figure 11: Fractional changes in σ 2 , as a function of anisotropy, β, due to changes in the seeing FWHM (left), miscentering the aperture by 0. ′′1 (center), and changes in the PSF profile (right). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 v/ SIS 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.…
Figure 12
Figure 12. Figure 12: The differences [P(v) − G(σ∗)]/P(v = 0) between the los velocity distribution (P(v)) and the best fit Gaussian model (G(σ∗)) normalized by the peak of the los velocity distribution at zero velocity, P(v = 0), for SDSS J1206+4332 (left) and WGD 2038−4008 (right). The a…
Figure 13
Figure 13. Figure 13: Left: Mean squared los velocity dispersion inside radius R, σ 2 los(< R), for the Hernquist profile embedded in an SIS mass model normalized by the squared velocity dispersion σ 2 SIS. The solid lines are for the O-M model with ra/s = 10 (bottom) and ra/s = 1 (top) an…

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