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arxiv: 2604.25307 · v2 · submitted 2026-04-28 · 🌌 astro-ph.CO · astro-ph.IM· gr-qc

Recovering cosmological parameters from the mock gravitational wave data of the Einstein Telescope

Pinaki Roy , Tomasz Bulik This is my paper

Pith reviewed 2026-05-07 15:14 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.IMgr-qc
keywords Einstein Telescopegravitational wavesHubble constantspectral sirenscosmological parametersblack hole binariesmock catalog
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The pith

The Einstein Telescope can recover the Hubble constant to 1 percent precision using gravitational wave spectral sirens from black hole binaries after one year.

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

This paper generates a mock catalog of gravitational wave events expected from the Einstein Telescope to test whether cosmological parameters can be recovered. It introduces a technique that treats the known intrinsic chirp mass spectrum of stellar-mass black hole binaries as spectral sirens, allowing the observed redshifted masses to reveal the Hubble constant or matter density. A sympathetic reader would care because the method works with the telescope operating alone and without electromagnetic counterparts, offering an independent route to measure cosmic expansion. Under standard assumptions the approach reaches 1 percent uncertainty on the Hubble constant or 4 percent on the matter density parameter with one year of data.

Core claim

We generate a mock gravitational wave event catalog for the Einstein Telescope and show the recoverability of either the Hubble constant (H0) or the matter density parameter (Om). We present a simple, effective and fast technique for inferring H0 (or Om) using the intrinsic chirp mass spectrum of black hole binaries, and investigate the efficacy of the method assuming the standard model of cosmology. If only H0 has to be constrained, we find that at least one year of ET's observation will be required to achieve 1 percent uncertainty. With the same amount of observation, Om can be constrained to within 4 percent uncertainty. With ET operating as a standalone instrument, we show that the GW s

What carries the argument

The intrinsic chirp mass spectrum of black hole binaries, used as spectral sirens to convert the distribution of observed (redshifted) masses into constraints on cosmological parameters.

If this is right

  • ET can function as a standalone instrument to constrain the Hubble constant without electromagnetic counterparts or other facilities.
  • One year of ET observations yields 1 percent uncertainty on H0 when only that parameter is varied.
  • The same one-year dataset constrains the matter density parameter to 4 percent uncertainty.
  • The technique remains fast and effective even for large catalogs of stellar-mass black hole binaries.

Where Pith is reading between the lines

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

  • If the black hole mass spectrum proves stable across the ET catalog, the method could supply an independent cross-check on the current Hubble tension using purely gravitational-wave data.
  • The same spectral-siren approach could be extended to joint analyses with other third-generation detectors or with electromagnetic distance ladders.
  • Relaxing the fixed Lambda-CDM assumption in future mock studies would test how well the technique distinguishes between competing cosmological models.
  • Real ET data will allow iterative refinement of the assumed chirp-mass spectrum itself, turning the measurement into a self-calibrating cosmological probe.

Load-bearing premise

The intrinsic chirp mass spectrum of stellar-mass black hole binaries is known to sufficient accuracy and the standard Lambda-CDM cosmology holds for the mock universe.

What would settle it

If the actual distribution of intrinsic chirp masses inferred from real ET observations deviates from the assumed spectrum, the recovered values of H0 or Om will be systematically biased away from the true cosmological parameters.

Figures

Figures reproduced from arXiv: 2604.25307 by Pinaki Roy, Tomasz Bulik.

Figure 1
Figure 1. Figure 1: Intrinsic chirp mass distribution of the merging NS-BH and BH￾BH systems. The chirp mass range of the injected sources is 1.6–40 M⊙. on dL and Mz which may yield Mi jk outside the intrinsic chirp mass range. We use the chirp mass range [0.1, 150] M⊙. If P is the true probability distribution and Q is an approxi￾mating probability distribution, the KL divergence (Kullback & Leibler 1951), also known as rela… view at source ↗
Figure 2
Figure 2. Figure 2: Scatter plot of KL divergences for a certain iteration for Case I (left), and for a certain iteration for Case II (right) with the dashed line showing the minimum. The H0 value corresponding to this is taken to be the best H0 for these iterations view at source ↗
Figure 3
Figure 3. Figure 3: Localization for (ϕ, θ∗ ) and (ψ, ι∗ ) recovered for a certain event with ρeff = 82.66. The coordinates, θ ∗ = 90° − θ and ι ∗ = 90° − ι are used instead of θ and ι, respectively. The stars denote the injected coordinates of the source view at source ↗
Figure 4
Figure 4. Figure 4: Plot of relative error dL versus effective SNR (left), and relative error dL versus SNR asymmetry (right), for dataset 1. Both effective SNR and SNR asymmetry decide the accuracy of the recovered dL. Article number, page 5 of 10 view at source ↗
Figure 5
Figure 5. Figure 5: Plot of recovered dL versus injected dL for dataset 1 to check the sanity of mock data. The white dashed line corresponds to 45° slope representing ideal recovery. No evident bias is seen view at source ↗
Figure 6
Figure 6. Figure 6: Histograms of the H0 values obtained from 10000 iterations for case I (left) and for case II ((right). The binsize is 0.5 km s−1 Mpc−1 . The dashed line shows the injected value. The middle dotted line shows the mode of the distribution. The left and the right dotted lines show the lower and upper bounds, respectively. The smooth curve is the kernel density estimate view at source ↗
Figure 7
Figure 7. Figure 7: Distribution and cumulative distribution of recovered luminosity distance of all the NS-BH and BH-BH sources in dataset 1. 3. Results Once an event in the set of injected GW events is classified as de￾tected based on the SNR threshold, it is taken up for localization and subsequently, distance estimation. An example of source lo￾calization is shown in view at source ↗
Figure 8
Figure 8. Figure 8: Histograms of the Ωm values obtained from 10000 iterations for case III (left) and for case IV ((right). The binsize is 0.005. The dashed line shows the injected value. The middle dotted line shows the mode of the distribution. The left and the right dotted lines show the lower and upper bounds, respectively. The smooth curve is the kernel density estimate view at source ↗
Figure 9
Figure 9. Figure 9: Histograms of the H0 values for case IX (left) and of the Ωm values (right) for case X (right) obtained from 10000 iterations using dataset 5+6. The binsize is 0.005. The dashed line shows the injected value. The middle dotted line shows the mode of the distribution. The left and the right dotted lines show the lower and upper bounds, respectively. The smooth curve is the kernel density estimate view at source ↗
Figure 10
Figure 10. Figure 10: Plot of q ς 2 net − ς 2 stat versus the systematic uncertainty, ςsys, for H0 (left) and for Ωm (right). The values for N ≈ N0 are the averages of the values for datasets 5 and 6. The values for N ≈ 2N0 are the values for dataset 5+6. Article number, page 7 of 10 view at source ↗
read the original abstract

Einstein Telescope (ET) is a third-generation gravitational wave (GW) detector with tenfold better sensitivity compared to the advanced LIGO detectors. It will be capable of observing copious stellar mass binary black hole mergers up to a redshift of 10 which will make it especially useful for cosmography. We generate a mock gravitational wave event catalog for the Einstein Telescope and show the recoverability of either the Hubble constant ($H_0$) or the matter density parameter ($\Omega_{\rm m}$). We present a simple, effective and fast technique for inferring $H_0$ (or $\Omega_{\rm m}$) using the intrinsic chirp mass spectrum of black hole binaries, and investigate the efficacy of the method assuming the standard model of cosmology. If only $H_0$ has to be constrained, we find that at least one year of ET's observation will be required to achieve 1% uncertainty. With the same amount of observation, $\Omega_{\rm m}$ can be constrained to within 4% uncertainty. With ET operating as a standalone instrument, we show that the GW spectral sirens detected by it can constrain the Hubble constant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generates mock gravitational-wave catalogs for stellar-mass binary black hole mergers detectable by the Einstein Telescope and presents a spectral-siren technique to recover either H0 or Ωm from the observed (redshifted) chirp-mass distribution, assuming a fixed, known intrinsic chirp-mass spectrum and standard ΛCDM cosmology. With one year of ET data the authors report 1% precision on H0 and 4% on Ωm when the other parameter is fixed.

Significance. If the assumptions hold, the work demonstrates that ET operating alone can deliver competitive H0 constraints from GW spectral sirens without electromagnetic counterparts, which would be valuable for independent cosmography. The manuscript uses mock catalogs and a straightforward inference method, both of which are strengths.

major comments (2)
  1. [Abstract and results section] The recovery exercise (abstract and results) is performed exclusively on mocks generated under the exact same fixed intrinsic chirp-mass spectrum and redshift-independent ΛCDM assumptions used in the inference; successful recovery is therefore expected once the likelihood is correctly coded, but no robustness tests against even modest misspecification or mild redshift evolution of the mass spectrum are reported, which directly affects the claimed 1% H0 precision.
  2. [Methods and discussion] The central claim that ET standalone spectral sirens suffice for H0 at the 1% level (abstract) rests on perfect knowledge of the source-frame chirp-mass distribution; the manuscript provides no propagation of uncertainty from plausible variations in that distribution (e.g., metallicity-driven evolution) into the posterior on H0.
minor comments (2)
  1. [Abstract] The abstract states that either H0 or Ωm can be recovered but does not specify whether the quoted precisions assume the other parameter is fixed or jointly fitted.
  2. [Throughout] Notation for the observed versus intrinsic chirp mass should be introduced once and used consistently throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential value of our spectral-siren approach with the Einstein Telescope. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Abstract and results section] The recovery exercise (abstract and results) is performed exclusively on mocks generated under the exact same fixed intrinsic chirp-mass spectrum and redshift-independent ΛCDM assumptions used in the inference; successful recovery is therefore expected once the likelihood is correctly coded, but no robustness tests against even modest misspecification or mild redshift evolution of the mass spectrum are reported, which directly affects the claimed 1% H0 precision.

    Authors: We agree that the absence of robustness tests limits the strength of the claimed precisions. In the revised manuscript we will add a dedicated subsection presenting tests with mild redshift evolution in the chirp-mass spectrum (motivated by metallicity effects) and with modest misspecifications of the distribution. These tests will quantify the resulting biases and degradation in the recovered H0 and Ωm, thereby providing a more realistic assessment of the method's performance under the stated assumptions. revision: yes

  2. Referee: [Methods and discussion] The central claim that ET standalone spectral sirens suffice for H0 at the 1% level (abstract) rests on perfect knowledge of the source-frame chirp-mass distribution; the manuscript provides no propagation of uncertainty from plausible variations in that distribution (e.g., metallicity-driven evolution) into the posterior on H0.

    Authors: We acknowledge that the current analysis treats the intrinsic chirp-mass distribution as perfectly known. In the revision we will extend the Bayesian framework to marginalize over hyperparameters of the mass spectrum and will include a sensitivity study that varies the distribution within ranges allowed by current population-synthesis models. The resulting broadened posteriors on H0 (and Ωm) will be reported, giving a more complete error budget that incorporates uncertainty in the source-frame mass distribution. revision: yes

Circularity Check

0 steps flagged

No circularity: standard mock recovery validation under explicit assumptions

full rationale

The paper generates mock catalogs under a fixed source-frame chirp-mass spectrum plus standard LCDM, then recovers the input H0 (or Om) from the redshifted observed distribution. This is a conventional forward-simulation test of an inference pipeline; the recovery is expected when the likelihood matches the generative model, but the paper does not claim a derivation that reduces to its inputs by construction, nor does it smuggle an ansatz or rely on self-citation for uniqueness. The central claim is explicitly conditioned on the mass spectrum being known, which is stated as an assumption rather than derived. No load-bearing step equates a fitted quantity to a prediction or renames a known result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of the assumed black-hole mass distribution and the validity of standard cosmology for the simulated universe; no new entities are introduced.

free parameters (1)
  • parameters of the intrinsic chirp mass spectrum
    The shape and normalization of the black-hole mass distribution must be specified or fitted; these enter the statistical model used to recover cosmology.
axioms (2)
  • domain assumption Standard Lambda-CDM cosmology governs the mock universe
    The recovery exercise assumes the input cosmology is Lambda-CDM when generating and fitting the mock data.
  • domain assumption The intrinsic chirp mass distribution of stellar-mass black hole binaries is known to sufficient precision
    The method uses this distribution to break the mass-redshift degeneracy.

pith-pipeline@v0.9.0 · 5504 in / 1470 out tokens · 94751 ms · 2026-05-07T15:14:54.327804+00:00 · methodology

discussion (0)

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Reference graph

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