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Precision Joint Constraints on Cosmology and Gravity Using Strongly Lensed Gravitational Wave Populations

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Pith's one-line read A Bayesian forecast using only the number of strongly lensed gravitational-wave events and their time delays can jointly measure the Hubble constant to sub-percent precision and the post-Newtonian parameter $\gamma$ to a few percent.

desk verdict A clean, internally consistent forecast that adds PPN gamma to the lensed-GW population method, but its headline precision rests on optimistic assumptions about detection completeness and selection effects. read the letter →

arxiv 2505.09507 v2 pith:S5QYTX2X submitted 2025-05-14 gr-qc astro-ph.CO

Xinguang Ying , Tao Yang This is my paper · ORCID
classification gr-qcastro-ph.CO
keywords stronggravitationallensingwavesHubbleconstantpost-NewtonianparameterEinsteinTelescopeBayesianpopulationanalysistime-delaydistributioncosmology
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 the population statistics of strongly lensed binary black hole gravitational-wave events—specifically the total number detected and the distribution of time delays between lensed images—carry enough information to jointly measure the Hubble constant $H_0$ and the post-Newtonian parameter $\gamma$ far more precisely than existing jointly-constrained probes. Under a flat $\Lambda$CDM cosmology, a 10-year observation with the Einstein Telescope, and a Planck-informed prior on $\Omega_m$, the Bayesian forecast yields 68% credible intervals of roughly $0.5\%$ to $1\%$ on $H_0$ and $0.5\%$ to $3.3\%$ on $\gamma$. The method needs no electromagnetic counterpart, no waveform model, and no resolved stellar kinematics of the lens galaxy; it uses only event counts and time delays. If the forecast holds, lensed gravitational-wave statistics become a clean and independent probe of both cosmic expansion and gravity.

What carries the argument

The central object is the joint population likelihood $\mathcal{L}(N,\{\Delta t_i\}|\Omega,T_{\rm obs}) = \mathrm{Poisson}(N;\Lambda(\Omega,T_{\rm obs})) \times \prod_{i=1}^N p(\Delta t_i|\Omega,T_{\rm obs})$, where $\Omega = (h,\Omega_m,\gamma)$. The expected count $\Lambda$ is obtained from the PPN-corrected strong-lensing optical depth, and the time-delay distribution $p(\Delta t|\Omega)$ is generated by marginalizing over lens parameters. The PPN modification enters through the scaling $\psi_{\rm PPN} = (1+\gamma)\psi_{\rm GR}/2$, which rescales both image time delays and the lensing cross-section. This machinery works because the total event count and the shape of the time-delay distribution respond differently to $h$, $\Omega_m$, and $\gamma$, breaking the degeneracies that would limit a single statistic.

What would settle it

Run the same Bayesian analysis on simulated Einstein Telescope data that includes detector noise, waveform-based lensing identification, and realistic time-delay measurement errors, and compare the resulting 68% credible intervals on $h$ and $\gamma$ with the paper's quoted ranges. If the intervals widen beyond roughly $1\%$ on $h$ and approach the $8.7\%$ baseline on $\gamma$, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that a combined Poisson likelihood for the total number $N$ of lensed events and a product likelihood for the observed time delays $\{\Delta t_i\}$, built from a PPN-corrected singular-isothermal-sphere lens model, can jointly constrain the Hubble constant and the post-Newtonian $\gamma$ to sub-percent and few-percent precision respectively in the Einstein Telescope era. The quoted 68% credible intervals are roughly $0.4\%$--$1\%$ on $H_0$ (the abstract states $0.60\%$--$0.99\%$, while the body text varies between $0.42\%$ and $0.69\%$) and $0.53\%$--$3.3\%$ on $\gamma$, significantly better than previous joint constraints. The forecast assumes a flat $\Lambda$CDM cosmology, a known velocity dispersion function, and exact time-delay measurements; it requires no electromagnetic counterpart, no waveform modeling, and no kinematic modeling of the lens galaxy.

Load-bearing premise

The load-bearing premise is that the $N$ lensed events are identified with exact time delays $\{\Delta t_i\}$; if real time-delay measurement errors or lensed-event identification losses are comparable to the weeks-to-months delays, the quoted precision is optimistic.

Editorial extensions

If this is right

  • If the forecast is correct, lensed gravitational-wave population statistics will provide a joint $H_0$--$\gamma$ probe that beats current electromagnetic time-delay lensing constraints by a large margin, with no reliance on electromagnetic counterparts or waveform modeling.
  • The method is directly applicable to the binary-black-hole-dominated gravitational-wave catalog, which is exactly the source population third-generation detectors will record in large numbers.
  • The approach converts a previously nuisance-like feature of strongly lensed events—their time delays—into a precision cosmological and gravitational observable.
  • The framework is modular: it can be extended to more realistic lens models, other source redshift distributions, and alternative cosmologies such as $w$CDM, as the paper notes in its outlook.

Reading between the lines

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

  • The exact-time-delay assumption is likely optimistic; if realistic time-delay measurement errors (from detector noise and waveform-based identification) are comparable to the predicted delays of weeks to months, the quoted sub-percent precision on $H_0$ will degrade, so the next natural test is to rerun the forecast with injected measurement noise.
  • The separation between $h$ and $\gamma$ leans heavily on the strong Planck prior on $\Omega_m$; without it, the degeneracies visible in the paper's own Figures 1 and 2 would likely inflate the $\gamma$ error, meaning the method's power is coupled to external cosmological information.
  • In practice, the bottleneck may not be timing precision but identifying lensed events at all: when millions of unlensed binary-black-hole signals are present, waveform-based lensing identification becomes a necessary selection step, and the paper's clean statistical statement will need to be folded with that selection function.
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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

4 major / 4 minor

Summary. This paper presents a Bayesian framework for jointly constraining the Hubble constant (parameterized by h) and the post-Newtonian parameter γ using the population statistics of strongly lensed gravitational wave (GW) events from binary black hole mergers. The authors derive a PPN-modified time delay and optical depth for the singular isothermal sphere lens model, construct a likelihood from the total number of lensed events and the time-delay distribution, and simulate an Einstein Telescope observation to produce forecasts. Under a flat ΛCDM cosmology with various priors on the matter density, the paper reports 68% credible-level constraints of 0.60%–0.99% on H0 (abstract) or 0.42%–0.69% (main text) and 0.53%–3.3% on γ, claiming that these significantly outperform existing joint constraints from electro-magnetic lensing.

Significance. The method is conceptually novel: it avoids EM counterparts, waveform modeling, and resolved stellar kinematics, instead extracting cosmological and gravitational information from the population-level statistics of lensed GW events. The derivation of the PPN-modified time delay and optical depth is explicit, and the Bayesian likelihood is clearly laid out. If the idealized assumptions (exact time delays, perfect detection and identification, fixed merger rate and velocity dispersion function) hold, the forecast would establish lensed GW population statistics as a high-precision probe. The paper also benefits from a transparent discussion of limitations in the final section, listing several extensions for future work. However, because the headline precision depends on these idealizations, the significance claim must be qualified by a realistic treatment of detection efficiency and nuisance-parameter uncertainties.

major comments (4)
  1. [Section III, Eq. (11); Section IV, Fig. 1] The expected number of lensed events Λ in Eq. (11) contains no detection or identification probability: it is the total lensing rate integrated over the source and time-delay populations. The simulation then draws N from Poisson(Λ) with Λ on the order of 2×10^4–7×10^4 (Fig. 1), so the count term alone yields roughly 0.4% statistical errors. In a realistic Einstein Telescope catalog, only a fraction f of lensed pairs will have both images above the SNR threshold and be identified as lensed; for f=0.1 the uncertainties grow by roughly a factor of 3, pushing the h and γ constraints to a few percent and to order 10%, comparable to or worse than existing joint constraints. This is load-bearing for the claim of significantly outperforming existing probes. The forecast should either include a detection/identification efficiency in Eq. (11) or present the constraints as an explicit function of the completeness fraction f.
  2. [Section III, Eq. (9); Section IV, Eqs. (12)–(13)] The paper explicitly assumes that N lensed events have been detected 'with exact time delays' and treats the measured Δt_i as noiseless in the likelihood. Real time-delay measurements for GW images will carry uncertainties, and the identification process itself may require waveform-based selection that is not modeled. While GW time-delay metrology may be precise, the assumption of exactness is an idealization that should be stated as such and, ideally, relaxed with a realistic error model to demonstrate that the quoted precision is not inflated by this idealization.
  3. [Abstract and Section IV] There is a direct internal inconsistency in the headline numbers. The abstract reports H0 precision of 0.60%–0.99%, while the main text (Section IV, after Fig. 5) reports 0.42%–0.69%. From the stated 68% intervals (σ_h = 0.0042 and 0.0069 at h = 0.7), the correct relative uncertainties are 0.6% and 0.99%, so the main text's 0.42% appears to be a numerical error. The comparison baselines also differ: the abstract says existing joint constraints 'typically achieve 2% precision on H0 and 20% precision on γ', whereas Section IV quotes '1.5% and 8.7%' from references [25–27]. These discrepancies must be reconciled before the quantitative claims can be evaluated.
  4. [Section IV, Eq. (11) and VDF parameters] The forecast fixes the BBH merger rate R = 5×10^5 yr^-1 and uses velocity dispersion function parameters from Ref. [39] without marginalizing over their uncertainties. Because Λ in Eq. (11) is directly proportional to R and is sensitive to the VDF parameters, the sub-percent constraints on h and γ are conditional on these astrophysical inputs being known exactly. A robust forecast should either include priors on R and the VDF parameters or explicitly state that the quoted precision is statistical only, conditional on fixed values of these nuisance parameters.
minor comments (4)
  1. [Figure 2] The axis labels in Figure 2 appear transposed or mislabeled: the x-axis is labeled 'log10 Tobs' but the plotted quantity is the time-delay distribution, and the y-axis label 'log10 (Δt [hrs])' is also unclear. Please clarify which variable is on each axis.
  2. [Section II, Eq. (2)] The dimensionless impact parameter y is introduced without an explicit definition; please state its normalization (e.g., y = β/θ_E) and the range assumed in the simulations.
  3. [Section IV, zmax rescaling] The text states that zmax must be rescaled for different h and Ωm to preserve the same maximum detectable luminosity distance, but the implementation of this rescaling is not shown. A short description or formula would improve reproducibility.
  4. [Section V] The sentence 'no need the EM counterparts and GW waveform knowledge' is grammatically awkward; suggest rewording to 'does not require EM counterparts or GW waveform knowledge'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a transparent forecast whose simulated data are generated from the same likelihood used for inference, with no fitted parameter renamed as a prediction and no load-bearing self-citation.

full rationale

The derivation chain is self-contained. The likelihood (Eqs. 9-13) is taken from the literature [15] and extended with the PPN-gamma scaling (Eqs. 2, 5, 6); the mock catalog is generated from exactly these equations (Section IV: 'We use Eqs. (11) and (13) to simulate the observed event number N and time delays {Delta_t_i} respectively, assuming the true parameters (h, Omega_m, gamma) = (0.7, 0.3, 1)'), and the same expressions are then evaluated to compute posteriors. That is a standard injection-recovery forecast, not a hidden reduction: the quoted 68% intervals are the expected statistical precision of the likelihood under the assumed priors, not a measurement that is claimed to validate the model. No parameter is fitted to external data and then re-expressed as a prediction; the VDF parameters are taken from Ref. [39], the merger rate and observation time are stated assumptions, and the Omega_m prior is taken from Planck [12]. The PPN scaling is borrowed from Ref. [34], not from the present authors. Self-citations ([25], [32]) appear only as background and comparison context; the comparison baseline of '1.5% and 8.7%' is attributed to Refs. [25-27] as a group and does not enter the derivation. Forecast realism concerns (exact time delays in Eq. 9, neglected selection effects, VDF uncertainties) are acknowledged in Section V and affect robustness, not circularity. No equation reduces to its own input by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The forecast rests on externally fitted galaxy population parameters (VDF), hand-chosen survey assumptions (R, Tobs, zmax, ymax), and a prior on Omega_m. The PPN implementation is a simplified rescaling of the lensing potential. No new physical entities are introduced.

free parameters (6)
  • Merger rate R = 5×10^5 yr^-1 (assumed)
    Assumed BBH merger rate for ET; directly sets the expected lensed event count Lambda in Eq. (11).
  • Observation duration Tobs = 10 years (assumed)
    Duration of the simulated survey; enters the Poisson mean and the (Tobs - Delta_t) selection factor.
  • Maximum source redshift zmax = 20 at true cosmology (h=0.7, Omega_m=0.3)
    Assumed ET detection horizon, rescaled for other cosmologies by matching luminosity distance.
  • VDF parameters (n*, alpha, beta, sigma*) = Values from Montero-Dorta et al. 2017 (Ref [39])
    Velocity dispersion function parameters entering the modified Schechter function; uncertainties are not propagated.
  • Maximum impact parameter ymax = 1 (y in [0,1])
    Assumed uniform source position within the Einstein radius in Eq. (15); sets the cross-section in Eq. (5).
  • Planck Omega_m prior width sigma = 0.0056
    Gaussian prior on Omega_m in Fig. 5, from Planck 2018; tight prior boosts the gamma constraint.
assumptions (5)
  • domain assumption The merger rate is uniform in comoving volume and the source redshift distribution follows pb(zs) proportional to Vc(zs)/(1+zs).
    Section III, Eq. (14)-(15); ignores evolution of the BBH merger rate with redshift.
  • domain assumption The strong lensing probability is well approximated by the optical depth: Pl(zs) approximate tau(zs).
    Section III, near Eq. (13); valid for small tau.
  • domain assumption The lens galaxies are singular isothermal spheres (SIS) with a modified Schechter velocity dispersion function.
    Section II, Eqs. (2)-(4); real galaxy density profiles are not modeled.
  • ad hoc to paper The PPN parameter gamma only rescales the lensing potential as psi_PPN=(1+gamma)/2 psi_GR and leaves the distance-redshift relation unchanged.
    Section II, Eq. (2); yields Delta_t_PPN=(1+gamma)/2 Delta_t_GR, but the geometric time delay does not generally scale with (1+gamma)/2.
  • domain assumption The background cosmology is flat LambdaCDM with parameters h and Omega_m.
    Throughout; extensions such as wCDM are deferred to future work.

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Cite this review

Pith. "Pith review of Precision Joint Constraints on Cosmology and Gravity Using Strongly Lensed Gravitational Wave Populations." pith.science (2026). https://pith.science/paper/S5QYTX2X

@misc{pith2026250509507,
  author       = {Pith},
  title        = {Pith review of: Precision Joint Constraints on Cosmology and Gravity Using Strongly Lensed Gravitational Wave Populations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S5QYTX2X}},
  note         = {Machine review of arXiv:2505.09507}
}
abstract

We present a Bayesian framework to jointly constrain the Hubble constant ($H_0$) and the post-Newtonian parameter ($\gamma$), a key indicator of deviations from general relativity, using the population characteristics of strongly lensed gravitational wave (GW) events from binary black hole mergers. Our method extracts cosmological and gravitational information directly from the statistical properties of lensed GW populations, without relying on electromagnetic counterparts of the GW events, waveform modeling, and resolved stellar kinematics of the lens galaxy. This establishes lensed GW statistics as a clean and independent probe of cosmic expansion and gravitational physics. Assuming a flat $\Lambda$CDM cosmology and simulating a GW population observed by the third-generation detector Einstein Telescope, we demonstrate that this method can achieve precision levels of $0.60\%\sim0.99\%$ for $H_0$ and $0.53\%\sim3.3\%$ for $\gamma$ with various priors of matter density, significantly outperforming existing joint constraints, which typically achieve $2\%$ precision on $H_0$ and $20\%$ precision on $\gamma$. These results highlight the potential of lensed GW population statistics as a robust and efficient tool for probing both the expansion history of the Universe and the nature of gravity.

Figures

Figures reproduced from arXiv: 2505.09507 by the authors.

Figure 1
Figure 1. FIG. 1. Distributions of the lensed event number [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Observed PDF of the time delays with different values [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Posterior distributions ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Forecasting Constraints on Cosmology and Modified Gravitational-wave Propagation by Combining Strongly Lensed Gravitational Waves and Galaxy Surveys

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.