Binary neutron stars in the next-generation era: Multi-messenger detection prospects and constraints on the equation of state, mass distribution, and cosmology

Chris Van Den Broeck; Gilad Sadeh; Hauke Koehn; Mattia Bulla; Michael W. Coughlin; Peter T.H. Pang; Thibeau Wouters; Tim Dietrich

REVIEW 3 major objections 5 minor 300 references

Binary neutron stars in the next-generation era: Multi-messenger detection prospects and constraints on the equation of state, mass distribution, and cosmology

T0 review · 3 major / 5 minor · reviewed 2026-07-31 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Next-generation GW detectors plus kilonovae can pin the neutron-star radius to ~0.2 km and H0 to ~1 km/s/Mpc in an ideal year of multi-messenger BNS events.

desk verdict Solid end-to-end ET/CE multi-messenger forecast plus real hierarchical Bayesian joint inference; the ~0.2 km / ~1 km s−1 Mpc−1 numbers are explicitly ideal-scenario and should be read that way. read the letter →

arxiv 2607.28438 v1 pith:DQW74DN6 submitted 2026-07-30 astro-ph.HE astro-ph.COnucl-th

Hauke Koehn , Thibeau Wouters , Gilad Sadeh , Peter T.H. Pang , Mattia Bulla , Chris Van Den Broeck , Michael W. Coughlin , Tim Dietrich This is my paper · ORCID

classification astro-ph.HEastro-ph.COnucl-th

keywords binaryneutronstarsmulti-messengerastronomyEinsteinTelescopeCosmicExplorerequationofstatekilonovaHubbleconstanthierarchicalBayesianinference

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

This paper projects how many binary neutron star mergers next-generation gravitational-wave detectors (Einstein Telescope alone or with Cosmic Explorer) will catch together with electromagnetic counterparts, and how tightly those joint events can constrain neutron-star matter, the mass distribution, and cosmology. With a local merger rate of about 107 per cubic gigaparsec per year, ET alone yields roughly 40–100 identified optical/UV/IR counterparts per year; adding CE raises that to a few hundred. From the kilonova-bearing subset observed with ET, a fully Bayesian hierarchical analysis of gravitational-wave signals, light curves, and host redshifts recovers the essential shape of the mass distribution and, in an ideal setting with matched models, constrains the radius of a 1.4-solar-mass neutron star to roughly 0.2 km and the Hubble constant to roughly 1 km/s/Mpc. Light-curve data add little to the equation-of-state constraints once high-SNR gravitational-wave tides are in hand, but they help cosmological inference by tightening distance and inclination and mitigating selection bias toward face-on systems.

What carries the argument

Joint hierarchical Bayesian inference on equation-of-state, mass-distribution, and cosmological hyperparameters, with per-event posteriors represented by normalizing flows so the population likelihood can be evaluated efficiently and selection effects reweighted after sampling.

What would settle it

After one year of real ET (or ET+CE) multi-messenger binary neutron star detections, measure whether the hierarchical posterior on R1.4 is actually at the ~0.2 km level and H0 at the ~1 km s−1 Mpc−1 level when independent nuclear or X-ray radius priors and independent H0 anchors are compared, or whether systematics in waveforms and ejecta models broaden or bias those intervals beyond the ideal forecast.

Watch

Extended reading notes

Core claim

In an ideal injection-recovery campaign focused on ET, the multi-messenger sample of kilonova-associated binary neutron stars can constrain the canonical neutron-star radius R1.4 to within about 0.2 km and H0 to within about 1 km s−1 Mpc−1 while recovering the main features of the mass distribution; kilonova light-curve posteriors have negligible impact on the equation of state once gravitational-wave tidal information is included, but they improve cosmological parameter recovery.

Load-bearing premise

The quoted radius and Hubble precisions assume perfect knowledge of the gravitational-wave waveform and kilonova emission models, with injection and recovery done with identical models and fitting formulae, so real model mismatch is set to zero.

Editorial extensions

If this is right

ET alone should deliver tens of multi-messenger BNS events per year at the assumed merger rate, enough for sub-kilometer radius and percent-level H0 constraints in the ideal case.
Adding Cosmic Explorer multiplies the multi-messenger yield by several times, mainly through tighter sky localizations that enable more successful follow-up.
Equation-of-state inference will be driven by the handful of highest-SNR events with clear tides, not by sheer event count.
Kilonova light curves are more valuable for cosmology (distance/inclination and bias control) than for further tightening the dense-matter equation of state once GW tides are precise.
Late-time radio surveys can still recover tens of afterglows even when early optical counterparts are missed.

Reading between the lines

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

If waveform or ejecta-model systematics remain at the levels the paper flags, the real R1.4 and H0 errors will be set by those systematics rather than by the statistical floor of ~0.2 km and ~1 km/s/Mpc.
The strong dependence of counterpart counts on the mass distribution (narrow versus wide, via prompt-collapse fraction) means an early multi-messenger sample will itself diagnose whether the Galactic narrow distribution or a broader one is closer to truth.
Because selection effects and inclination bias are hard to model exactly, cosmological analyses may need light-curve or jet information even when gravitational-wave SNRs are high.
Microphysical nuclear parameters (beyond bulk pressure near a few times saturation) will stay loosely constrained unless nuclear theory priors are folded in.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new piece here is not another rate table. It is the full pipeline: mock catalogues under two mass models and four ET/CE layouts, a staged follow-up algorithm, then joint hierarchical Bayesian recovery of microphysical EOS + mass hyperparameters + H0/Ω0 from the resulting KN events, with normalizing flows and exact TOV/Love solutions inside jester. That layer is genuinely useful and mostly missing from the Loffredo/Colombo/Branchesi-style forecasts.

What they do well is explicit. Rates (Table 4) and posteriors (Tables 6–7) are internally consistent with the stated setup. The GW-only vs GW+host-z vs full multi-messenger comparison is clean: light curves add almost nothing to R1.4 once high-SNR tides are in hand (differences ≲0.15 km, seed-scale), but they help break ι–dL bias for cosmology. Selection reweighting is attempted and its imperfections on the mass distribution are shown rather than hidden. Code path is public. Citations cover the right prior art.

Soft spots are real but already flagged by the authors. Absolute rates scale with the fixed 106.6 Gpc−3 yr−1, η0 is tuned to Fermi short-GRB fluences, ζ is ad-hoc, follow-up ignores impostors and uses Fisher sky areas, and—load-bearing for the headline precisions—inject and recover share the same waveform, possis geometry, and NR ejecta fits. Sec. 4.4 and the appendices say this out loud; the ~0.2 km / ~1 numbers are ideal-scenario forecasts, not calibrated real-data error bars. Mass-distribution hyperparameters remain sensitive to the coarse pdet model. None of that breaks the internal logic.

This is for people designing ET/CE science cases, Rubin ToO strategy, or hierarchical multi-messenger pipelines. It deserves a serious referee. I would cite the rate tables and the GW-vs-EM comparison, and I would bring the hierarchical section to reading group.

Referee Report

3 major / 5 minor

Summary. The paper forecasts multi-messenger BNS detection rates with ET and CE under two mass distributions and a fixed local merger rate of 106.6 Gpc^{-3} yr^{-1}, using a staged mock follow-up algorithm for prompt GRBs, UVOIR counterparts, and late afterglows. It then performs fully Bayesian hierarchical injection-recovery on the identified KN events (ET-only) to jointly constrain the EOS, mass distribution, and cosmology. In an ideal scenario with identical inject/recover models, the authors report R_{1.4} constrained to ~0.2 km and H_0 to ~1 km s^{-1} Mpc^{-1}, with KN light curves having negligible impact on the EOS but helping cosmological inference via inclination–distance constraints.

Significance. This is a substantial and timely contribution that unifies population synthesis, multi-messenger detection modeling, and hierarchical Bayesian inference in one framework. Strengths include the use of full Bayesian single-event posteriors (not FIM approximations) with normalizing flows, direct microphysical EOS sampling via jester/TOV solutions, publicly available analysis scripts, and an explicit comparison of GW-only versus multi-messenger hierarchical posteriors. The detection-rate tables and the ideal-scenario precision forecasts will be useful benchmarks for ET/CE science cases, provided the idealizations are kept clearly in view.

major comments (3)

[Abstract; Sec. 4.4; Apps. D–E] Sec. 4.4 and Apps. D–E state that GW waveforms and KN/ejecta models are identical at injection and recovery, and that the same NR-informed ejecta–mass fits used to build the catalogues are reused in the hierarchical likelihood (Eqs. 35–38). The headline R_{1.4} ≲ 0.2 km and H_0 ≲ 1 km s^{-1} Mpc^{-1} claims (abstract; Sec. 5) are therefore calibrated only under zero model mismatch. The paper already cites waveform systematics, ≳50% atomic/thermalization ejecta errors, and NR-fit differences as real-data failure modes. These caveats should be elevated into the abstract and conclusion so that the quoted intervals are not read as robust forecasts; a short quantitative stress test (e.g., one mismatched KN morphology or one alternate ejecta fit) would substantially strengthen the claim.
[Appendix G; Sec. 4.2.2; Tables 6–7] Appendix G approximates p_det(λ) with a neural-network SNR cut plus a single Rubin i-band magnitude threshold, then clips extreme weights. The text notes that this imperfect reweighting leaves residual bias in mass-distribution hyperparameters (e.g., α and m_u for the narrow model; m_max for the wide model; Tables 6–7). Because selection correction is load-bearing for population and cosmology inference (less so for R_{1.4}, which is dominated by a few high-SNR tides), the paper should either (i) demonstrate that EOS/H_0 intervals are stable under alternate p_det prescriptions, or (ii) more clearly separate which quoted constraints are robust to the App. G approximation.
[Sec. 2.3; Fig. 3; Table 4] Sec. 2.3 tunes the Blandford–Znajek fudge factor η_0 so that the synthetic Fermi/GBM fluence distribution matches observations (Fig. 3). GRB and afterglow detection counts in Table 4 and Fig. 5 therefore partly reflect this calibration rather than an ab initio jet model. The paper should state more explicitly which science conclusions (especially afterglow rates and the fraction of UVOIR counterparts that are pure GRB afterglows) inherit from this tuning, versus those driven by the KN channel alone.

minor comments (5)

[Table 4] Table 4 bracketed afterglow numbers (late-time surveys without prior UVOIR) are useful but easy to misread; a one-sentence clarification in the caption would help.
[Sec. 4.2.2; Fig. 6] The modest radius bias toward smaller R at fixed tight M–Λ (Sec. 4.2.2, Fig. 6) is attributed to CSE flexibility; a brief note on whether a different high-density extension would remove the bias would aid interpretation.
[Sec. 4.2.1; Eq. (39)] Eq. (39) replaces the Puecher & Dietrich classifier with a simple k_coll cut for jax compatibility. State the fraction of catalogue events for which the two criteria disagree, if available.
[Sec. 3.1; Sec. 4.3] Several typos and notation nits: “s (PSDs)” → “PSDs” (Sec. 3.1); “the proposed the proposed cosmology” (Sec. 4.3); inconsistent use of Ω_0 vs Ω_m.
[Fig. 5] Fig. 5 orange/green patches distinguishing KN vs GRB-afterglow contributions are valuable; ensure the 30% flux criterion is restated in the caption.

Circularity Check

1 steps flagged · score 2.0 of 10

Standard next-gen injection–recovery forecast; only mild circularity is η0 tuned to Fermi short-GRB fluences. Central R1.4/H0 widths are conditional ideal-scenario results, not forced by definition.

fitted input called prediction [Sec. 2.3, Eqs. (18)–(19), Fig. 3]
"Moreover, we have tuned the fudge factor in Eq. (19a) to η0 = 0.016, so that the resulting fluence distribution matches the Fermi/GBM short GRB samples (Colombo et al. 2022; Loffredo et al. 2025), more details are provided below. ... To ensure consistency between our ad-hoc prescriptions for the gamma-ray energy Eγ and the observed short GRB population, we tuned η0 in such a way that our results roughly agree with the fluence distribution of short Fermi/GBM GRBs"

η0 is adjusted until the synthetic bolometric fluence histogram reproduces the observed Fermi/GBM short-GRB sample (Fig. 3, including the 0.6 duty-cycle rescaling). Projected prompt-GRB and GRB-afterglow detection counts therefore partly inherit that calibration by construction rather than arising as independent first-principles forecasts. This does not force the paper’s headline R1.4 or H0 intervals, which come from KN-selected GW hierarchical runs, but it does make the GRB multi-messenger rate panels non-predictive for the tuned observable.

full rationale

The paper is a forward population-synthesis and hierarchical injection–recovery study. Detection counts and hierarchical posteriors are generated from explicit mock catalogues (fixed local rate 106.6 Gpc−3 yr−1, two mass models, QMC-RMF3 EOS) and a mock EM follow-up algorithm; the abstract and Sec. 4 explicitly label the R1.4 ≲ 0.2 km and H0 ≲ 1 km s−1 Mpc−1 figures as an “ideal scenario.” Recovered precision is not algebraically equal to any fitted input: it is set by network SNRs, number of KN-selected events, and tidal information content under matched inject/recover models (disclosed in Sec. 4.4 and Apps. D–E). Self-citations (possis/fiesta surrogates, jester, prior multi-messenger pipelines) supply tools and priors, not load-bearing uniqueness theorems. The sole clear fitted-input step is the Blandford–Znajek fudge factor η0, tuned so the synthetic short-GRB fluence histogram matches Fermi/GBM; that forces intermediate GRB rate/fluence plots (Fig. 3) but does not determine the EOS or H0 hierarchical widths, which are driven by GW tides plus host redshifts (and secondarily KN light curves). Selection-effect reweighting (App. G) is approximate and visibly biases some mass-distribution hyperparameters, but that is a modelling limitation, not a circular reduction of the central claims. Overall circularity is minor and proportionate to score 2.

Assumptions & free parameters 7 free parameters · 8 assumptions · 2 invented entities

Projections rest on a fixed local rate and delay-time model, one injected EOS, two parametric mass/spin models, NR-informed ejecta fits plus ad-hoc wind fraction and jet efficiency knobs, simplified telescope decision rules, and perfect inject=recover model identity. Hierarchical claims further assume 1% host-redshift errors, a coarse detection-probability reweighting, and meta-model+CSE EOS priors. These are mostly standard domain tools plus several paper-specific fudge factors.

free parameters (7)

local merger rate density Rn / R(0) = R(0)=106.6 Gpc−3 yr−1
Rn=50 Gpc−3 yr−1 and delay τ=3 Gyr set R(0)=106.6 Gpc−3 yr−1; all absolute detection counts scale linearly with this choice.
Blandford–Znajek fudge factor η0 = η0=0.016
Tuned in Eq. (19a) so synthetic Fermi/GBM fluences match the observed short-GRB sample (Sec. 2.3).
wind-to-disk mass fraction ζ distributions = prompt: TN(0.2,0.1); remnant: TN(0.5,0.2)
Ad-hoc truncated normals (Eqs. 15–16) control bright KN rates; paper notes most detectable KNe need large wind ejecta.
gamma-ray efficiency ηγ = U(0.01,0.15)
Drawn uniformly 0.01–0.15 (Sec. 2.3) with limited short-GRB empirical anchor.
ISM density prior for afterglows = U(−5,0) in log10 cm−3
log10(nism) ~ U(−5,0) (Eq. 26) directly sets afterglow peak flux/time and late-time detection counts.
instrument ΔΩ and dL follow-up thresholds = network-dependent cuts in App. B
Telescope- and network-specific cuts in App. B/C (e.g. Rubin ΔΩ≤10 deg2 with CE) are chosen by hand to keep telescope time plausible and dominate counterpart counts.
EM systematic magnitude floor σsys = U(0.3,1) mag
Uniform 0.3–1 mag broadening in light-curve PE (App. E) to mimic model error; affects how much EM informs hierarchy.

assumptions (8)

domain assumption Source-frame merger rate follows Madau–Dickinson SFR convolved with exponential delay-time distribution independent of mass/spin.
Sec. 2.1 Eqs. (3)–(4); standard but not unique delay model.
domain assumption Single EOS (QMC-RMF3) and perfect spin–orbit alignment for all catalogue events.
Sec. 2.1; radii/tides and ejecta inherit this fixed nuclear model.
domain assumption Prompt collapse and ejecta masses given by specified NR fits / ML classifier (or k_coll MTOV proxy in inference).
Secs. 2.2, 4.2.1; different literature fits can change ejecta substantially (acknowledged).
domain assumption GW detection if network SNR>12; 90% sky area from Fisher matrix (gwfish), 100% duty cycle.
Sec. 3.1; FIM and full duty cycle known to bias localization/rates.
ad hoc to paper Counterpart 'detected' after ≥2 filter/epoch/instrument flags with no impostor contamination and random tiling order.
Sec. 3.2 and App. B; authors note this overcounts relative to real classification.
domain assumption Host cosmological redshift known to 1% Gaussian error; flat ΛCDM with only H0, Ω0 free in cosmology runs.
Secs. 4.2.1, 4.3; peculiar-velocity floor and two-parameter cosmology.
ad hoc to paper Hierarchical likelihood uses NF density estimates of single-event posteriors and approximate p_det reweighting with clipping.
Sec. 4.1, App. G; selection treatment is coarse and affects mass-distribution recovery.
ad hoc to paper Injected and recovered GW/KN models are identical (no waveform or radiative-transfer mismatch).
Sec. 4.4; explicitly idealizes away the dominant real-world systematic for the precision claims.

invented entities (2)

Mock multi-messenger follow-up algorithm (staged UVOIR + late afterglow surveys)
purpose: Map GW localizations and light curves to yearly counterpart and afterglow counts per instrument/network.
Not a physical entity but a paper-specific operational model (Fig. 4, Apps. B–C) that defines the detection catalogues feeding hierarchical inference.
Extended possis ML surrogate geometry for wide ejecta parameter space
purpose: Generate KN light curves across the catalogue ejecta range (App. A).
Engineering extension of prior possis/Kawaguchi geometry; validated only within the paper's simulation set.

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Cite this review

Pith. "Pith review of Binary neutron stars in the next-generation era: Multi-messenger detection prospects and constraints on the equation of state, mass distribution, and cosmology." pith.science (2026). https://pith.science/paper/DQW74DN6

@misc{pith2026260728438,
  author       = {Pith},
  title        = {Pith review of: Binary neutron stars in the next-generation era: Multi-messenger detection prospects and constraints on the equation of state, mass distribution, and cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQW74DN6}},
  note         = {Machine review of arXiv:2607.28438}
}

abstract

Next-generation gravitational-wave (GW) observatories will provide crucial insights into the nature of neutron star (NS) matter and the cosmological expansion history. We estimate the number of multi-messenger detections from binary neutron stars (BNS) with the Einstein Telescope (ET) and Cosmic Explorer (CE), and project the resulting constraints on the equation of state (EOS), BNS mass distribution, and cosmology via joint hierarchical Bayesian inference. Assuming a local merger rate of 106.6 Gpc$^{-3}$ yr$^{-1}$ and considering two different mass functions, a narrow one centred around 1.4 $M_\odot$ and a wide one ranging between 1.1--2 $M_\odot$, we find that for ET, our mock follow-up algorithm results in at least $\sim40$ and up to $\sim100$ successfully identified electromagnetic counterparts per year, depending on the detector layout and mass distribution. In a joint network with CE, the number of multi-messenger detections can range from $\sim 200$ to $\sim500$. Additionally, several more afterglows from gamma-ray bursts or KNe could be found with dedicated late-time observations. Based on the identified multi-messenger events, we perform an injection campaign to hierarchically constrain the EOS, mass distribution, and cosmology in a fully Bayesian framework. Focussing on ET alone, we show how in an ideal scenario, GW signals, KNe, and host galaxy redshifts can constrain the canonical NS radius $R_{1.4}$ within $\sim 0.2$ km and the Hubble constant $H_0$ within $\sim 1$ km s$^{-1}$ Mpc$^{-1}$, while recovering the essential features of the mass distribution. By comparing inference results that rely solely on GW data and those that incorporate light curve information, we find that while KN light-curve posteriors have a negligible impact on the EOS constraints, they can benefit the inference of cosmological parameters.

Figures

Figures reproduced from arXiv: 2607.28438 by the authors.

**Figure 1.** Narrow (green) and wide (blue) mass distributions used to generate our BNSs. The solid lines show the marginalized mass distribution of the heavier NS and the dashed lines of the lighter one. stars formed at redshift 𝑧 ′ will result in merging BNSs at redshift 𝑧 R (𝑧) = Rn × ∫ ∞ 𝑧 𝜓(𝑧 ′ )D (𝑧 | 𝑧 ′ ) 𝑑𝑧′ . (3) We adopt a simple analytical expression (Du et al. 2025) D (𝑧 | 𝑧 ′ ) = 1 𝜏 exp − 𝑡(𝑧) − 𝑡(𝑧 ′ ) 𝜏 𝑑𝑡 𝑑… view at source ↗

**Figure 2.** Ejecta properties for the BNS catalogues with the narrow (green) and wide (blue) mass distributions. The histograms with no filling and lighter colors represent BNSs that undergo prompt collapse. The distributions of 300 apparent 𝑖 band light curves for the narrow (penultimate panel) and wide (last panel) distributions are also shown. There, black (orange) solid lines are the median magnitude for KNe within 𝑧 ≤ 1.5 … view at source ↗

**Figure 3.** Top: Distribution of the observable bolometric fluence between 0.1–1 MeV for the narrow (green) and wide (blue) BNS mass distribution. For comparison, we show the fluence distribution of the short GRBs observed by FERMI-GBM as grey dashed line. The rates from our catalogues are scaled to a duty cycle factor of 0.6. Bottom: Isotropic kinetic energy-equivalent. The grey band indicates the 95% credibility interval from… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Schematic overview of the mock detection algorithm that we use to determine for which BNS GW signals an EM counterpart will be detected. Starting from the GW detection, we check whether a KN or UVOIR afterglow can be found in the first days, and, depending on the resul…

**Figure 5.** Figure 5: Redshift distribution of our projected BNS detections in one year. The rows correspond to a specific next-generation GW detector network. The left and right columns are for the narrow or wide mass distribution respectively. Different types of detections (GW signal, GW …

**Figure 6.** Figure 6: Joint hierarchical inference of the EOS and mass distribution from the KN events in the narrow+ETL catalogue. The posteriors are obtained using either only the GW data (purple) or the full multi-messenger data (orange). The top panels show the EOS posterior equivalentl…

**Figure 7.** Figure 7: EOS posteriors from the KN events in narrow+ETL when the full multi-messenger information is used (orange) and when only based on the GW signals and host galaxy redshifts (blue). The top panel shows the 95% credibility intervals on the relative error for the 𝑀-Λ relati…

**Figure 8.** Figure 8: EOS and mass distribution inference from the KN events in the wide+ETL catalogue. The layout is equivalent to [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: EOS and mass distribution inference from the KN events in the narrow+ETΔ catalogue. The layout is equivalent to [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 11.** Figure 11: Posterior on the nuclear empirical parameters from the multimessenger inferences. We show the 95% credibility contours in different colours according to the legend. The values from our injected QMC-RMF3 EOS are marked in red. events, has only a negligible impact on t…

**Figure 12.** Figure 12: Cosmological constraints from the KN events in all our catalogues, compared in different colours according to the legend. We show the 95% credibility region for 𝐻0 and Ω0, with the injected Planck18 parameters shown marked by red lines. as hyperparameters in our hiera…

**Figure 13.** Figure 13: Joint hierarchical inference of the EOS, mass distribution, 𝐻0, and Ω0 from the KN events in the narrow+ETL catalogue. We show both the multi-messenger posterior (orange) and the posterior using the GW data and host galaxy redshift, but no light curve posteriors (blue…

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