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A Future Percent-Level Measurement of the Hubble Expansion at Redshift 0.8 With Advanced LIGO

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Merging binary black holes can measure the expansion rate of the universe at redshift 0.8 to 2.9% after five years of Advanced LIGO/Virgo observations, using only the pair-instability supernova mass scale to break the mass-redshift…

desk verdict A clean, well-documented forecast of a new standard-siren route to H(z) at z~0.8 via the PISN mass cutoff, but the headline precision is hostage to an unquantified 1–2 solar mass redshift drift in that cutoff. read the letter →

arxiv 1908.09084 v2 pith:CGZHLFA3 submitted 2019-08-24 astro-ph.CO astro-ph.HE

Will M. Farr , Maya Fishbach , Jiani Ye , Daniel Holz This is my paper · ORCID
classification astro-ph.COastro-ph.HE
keywords gravitationalwavesbinaryblackholesstandardsirenspair-instabilitysupernovamass-redshiftdegeneracyHubbleexpansiondarkenergyequationofstatehierarchicalBayesianinference
topics Dark Energy
open problems Dark Energy
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

Gravitational waves from merging black holes carry the distance to the source, but not its redshift, because the same waveform is produced by a heavy source far away and a lighter source nearby. This paper identifies a way to break that degeneracy: the pair-instability supernova process is thought to cut off the black hole mass distribution near 45 solar masses, and that cutoff should be nearly the same at all redshifts. If so, the observed detector-frame cutoff, which shifts with redshift, locates each source along the distance-redshift relation with no distance ladder and no cosmological model. Simulating realistic Advanced LIGO/Virgo observations, the authors find the expansion rate $H(z)$ is measurable to 6.1% at $z\simeq0.8$ after one year of design-sensitivity operation and 2.9% after five years, with future third-generation detectors reaching sub-percent precision to high redshift. This would supply an independent, gravitational-wave-only cosmography at the redshifts where dark energy begins to dominate.

What carries the argument

The machinery is the pair-instability supernova (PISN) mass cutoff used as a redshift calibrator. In the detector frame a black hole's measured mass is $m_{\rm det} = (1+z) m_{\rm source}$, so a source-frame cutoff that is fixed across cosmic time appears as a sharp diagonal edge in the detector-frame mass-distance plane; matching that edge to a single source-frame mass converts each event's measured distance into a redshift. The quantitative engine is a censored Poisson-process hierarchical model whose population distribution tapers smoothly to zero above $m_h \simeq 45\,M_\odot$ (Equations A1-A2) and whose likelihood incorporates per-event measurement uncertainties and the $\rho>8$ detection threshold; the posterior over $H_0$, $\Omega_M$, and $w$ is obtained after marginalizing over event-level masses, distances, and orientations.

What would settle it

Take a future sample of black hole mergers with independently known redshifts, for example events with electromagnetic counterparts or host-galaxy identifications, and measure the source-frame upper edge of the primary mass distribution as a function of redshift; if that edge shifts by more than about 2 solar masses between $z=0$ and $z=1.5$, the PISN-inferred $H(z)$ measurement is biased by more than the quoted 2.9% uncertainty.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the population of merging binary black holes is a cosmological probe without a distance ladder: because general relativity is scale-free, the only thing linking measured gravitational-wave distances to redshifts is a mass scale, and the pair-instability supernova cutoff supplies exactly that scale. Treating the cutoff as a smooth taper around $m_h \simeq 45\,M_\odot$ in a hierarchical model that simultaneously fits the mass distribution, redshift evolution, selection effects, and a flat $w$CDM cosmology to synthetic catalogs, the paper finds the BBH population constrains $H(z)$ to 6.1% (68% credible interval) at the pivot redshift $z\simeq0.8$ after one year and 2.9% after five years at design sensitivity. The analysis also recovers the mass scale to $44.64^{+0.76}_{-0.81}\,M_\odot$ after five years, and interprets the measurement as an absolute distance scale at $z\simeq0.8$ that can calibrate Type Ia supernovae and the baryon acoustic oscillation sound horizon without external distance information. With informative priors on $H_0$ and matter density, the same population constrains the dark energy equation of state to 19% after one year and 12% after five years.

Load-bearing premise

The measurement depends on the assumption that the pair-instability cutoff mass near 45 solar masses is essentially constant across cosmic time out to $z\sim1.5$, or can be calibrated to better than the 1-2 solar mass drift that stellar models allow; if the cutoff moves with redshift, inferred redshifts and hence $H(z)$ are biased.

Editorial extensions

If this is right

  • After one year of Advanced LIGO/Virgo at design sensitivity, the BBH population alone measures $H(z)$ to 6.1% at $z\simeq0.8$; after five years the constraint tightens to 2.9%.
  • The measurement is independent of the cosmic distance ladder and of any assumed cosmological model, relying only on general relativity and a mass scale that is fixed or calibrated across cosmic time.
  • Combining the absolute distance scale at $z\simeq0.8$ with Type Ia supernova or baryon acoustic oscillation data independently calibrates those standard candles and rulers, corresponding to an $H_0$ uncertainty of $\pm2.0\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ if mapped to $z=0$.
  • A sharper PISN cutoff than the smooth taper assumed here reduces the quoted uncertainties by roughly a factor of two.
  • Third-generation detectors, which see roughly 15,000 BBH mergers per month to $z\gtrsim10$, would yield sub-percent cosmography to $z\gtrsim4$ within one month of observation, provided the PISN mass scale is calibrated.

Reading between the lines

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

  • This reading suggests that any sharp, approximately redshift-invariant feature in the compact-object mass distribution, not only the PISN cutoff, could serve as a redshift calibrator; a future detection of a pile-up near the maximum mass, or a neutron-star maximum-mass feature, would provide additional independent scales.
  • Because the pivot redshift $z\simeq0.8$ sits near matter-dark-energy equality, the same population could be combined with a CMB-based high-redshift distance to constrain dark energy without relying on supernova standardization.
  • A testable extension is to split detected events into distance bins and verify that the inferred source-frame cutoff is constant; a drift of more than a few solar masses would indicate either metallicity-driven evolution or a breakdown of the assumption, and could itself be modeled and calibrated.
  • The method treats the PISN cutoff as a standardizable ruler, which suggests that redshift evolution of the mass scale could be measured jointly with cosmology rather than assumed, at the cost of some precision.
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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

2 major / 4 minor

Summary. The paper proposes a new method to measure the cosmic expansion history at redshift z≈0.8 using binary black hole (BBH) mergers as standard sirens without electromagnetic counterparts. The key idea is that the pair-instability supernova (PISN) process imprints a sharp upper mass scale m_h≈45 M⊙ on the source-frame black hole mass distribution; because the observed waveform depends on the detector-frame mass m_det=m(1+z), the measured detector-frame cutoff can serve as a redshift indicator. The authors simulate one and five years of Advanced LIGO/Virgo observations at design sensitivity using a population model with a smooth PISN taper (Eq. A1–A2), a simplified measurement and selection model (Appendix B), and a full hierarchical Bayesian analysis (Eq. C17) that jointly fits population and cosmological parameters (H0, ΩM, w). They find 6.1% and 2.9% uncertainty on H(z=0.8) after one and five years, respectively, and 19% and 12% on w when external H0 and ΩM priors are imposed. The paper explicitly acknowledges that the PISN mass scale may evolve by 1–2 M⊙ out to z≈1.5 and states that this 'must be calibrated,' but it does not propagate this systematic into the quoted precision.

Significance. If the forecast holds, the method would provide a genuinely new cosmological probe: an absolute distance-scale measurement at z≈0.8 that is independent of the cosmic distance ladder and of electromagnetic counterparts. The paper is significant because it identifies a concrete mechanism by which the PISN mass scale can break the mass–redshift degeneracy, and it backs this with an end-to-end simulation rather than a back-of-the-envelope estimate. Strengths include the full hierarchical analysis with selection effects, the forward-modeling anchored to GWTC-1 population constraints and to Vitale et al. measurement uncertainties, and the public availability of the code and data. The stress-test concern about the PISN mass-scale drift is valid and lands: the 6.1% and 2.9% numbers are purely statistical and are conditional on an uncalibrated 1–2 M⊙ astrophysical systematic that is of order the five-year statistical error. The paper should be revised to quantify this systematic before the headline precision can be accepted as stated.

major comments (2)
  1. [Main text, p. 6–7 ('Our simplistic analysis…'); Appendix A, Eq. (A1)–(A2); Eq. (C17)] The central result, 2.9% on H(z=0.8) after five years, is a purely statistical uncertainty computed from a simulated catalog generated with a constant m_h=45 M⊙. The paper explicitly acknowledges that the PISN mass scale may evolve by 1–2 M⊙ by z≈1.5 and that changes at that level 'are a systematic that must be calibrated,' but no term for this drift or its calibration uncertainty enters the model in Eq. (C17) or the quoted error budget. Because the redshift assignment is essentially m_h→m_det/(1+z), an unaccounted drift δm_h(z) maps to a fractional bias in (1+z) of order δm_h/m_h; for a linear drift reaching 1–2 M⊙ at z=1.5, the bias at the pivot z=0.8 is roughly 1–2.5%, i.e. of the same order as the 2.9% statistical error, and larger at higher redshift or for nonlinear drift. The five-year posterior on m_h is 44.64^{+0.76}_{-0.81} M⊙, so the admitted 1–2 M⊙ systematic is considerably larger than the internal statistical error on the mass scale. The authors should either add a redshift-dependent m_h(z) to the population model with a prior informed by stellar-evolution calculations and report how the H(z) uncertainty degrades, or specify the required calibration accuracy on m_h(z) and demonstrate that it can be met. Without this, the headline precision is conditional in a way that the abstract does not fully convey.
  2. [Appendix B and Fig. 1; §4 (precision claims)] The quoted 6.1% and 2.9% uncertainties are computed with a simplified measurement model in which the single-event likelihood is approximated by Gaussian uncertainties on chirp mass, symmetric mass ratio, and the angular amplitude factor, tuned to reproduce Vitale et al. (2017). The text states that this model reproduces the correlated mass measurements and typical distance uncertainties, but no direct comparison is shown. Because the statistical precision scales roughly as the inverse square root of the number of events that usefully constrain the mass cutoff, a mismatch between the approximate likelihood and full parameter estimation could change the forecast by a factor of order unity. Please provide a quantitative validation of the approximation (for example, a comparison of mass and distance uncertainties for a set of synthetic signals under this model versus a full parameter-estimation pipeline) and state how the headline numbers would change if the distance or mass uncertainties were, say, 20% larger or smaller.
minor comments (4)
  1. [Appendix B, Eqs. (B14)–(B16)] The notation says quantities are 'measured with uncertainty' followed by a Gaussian width, but it is not explicitly stated whether these widths are the standard deviations used directly in the likelihood; please state this explicitly.
  2. [Fig. 1 caption] The caption says 'Dots denote the mean and bars the 1σ width of the likelihood for each event,' but a likelihood has no mean without a prior; please clarify that the points are posterior means from a single-event analysis with a reference prior.
  3. [Main text, p. 6 (w constraint)] The sentence 'We do not obtain any meaningful constraint on the evolution of wDE with redshift when this parameter is allowed to vary' would be more informative if accompanied by the posterior width of the evolution parameter, so the reader can judge how much information is lost.
  4. [Appendix A, Eq. (A2) and main text, p. 2] The text says the taper acts over a characteristic scale of about 5 M⊙, while Eq. (A2) sets σ_h=0.1 in log mass; at m_h=45 M⊙, σ_h=0.1 in natural log corresponds to about 4.5 M⊙, so the '5 M⊙' is approximate; please align the wording and equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the forecast is a forward simulation against external population and measurement models, not a derivation that reduces to its own inputs.

full rationale

The paper does not claim to derive cosmology from the PISN mass scale; it simulates a catalog from an assumed population (Eq. A1–A2) with external parameters (GWTC-1, Vitale et al. 2017, Planck) and then fits a hierarchical model to the simulated data. The target H(z) is not an input to the population model, and the mass-scale parameter m_h is fitted jointly with cosmology from the mock observations, with broad priors; recovering the injected cosmology is a self-consistency check rather than a circular reduction. Self-citations to Fishbach & Holz (2017) and Fishbach et al. (2018) supply empirical population inputs, not an unverified uniqueness theorem, and they do not by themselves force the result. The acknowledged 1–2 M⊙ redshift dependence of the PISN scale is a systematic uncertainty that must be calibrated externally; this limits the realism of the quoted precision but does not make the derivation circular.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The forecast rests on the assumed BBH population (rate, mass slopes, PISN cutoff location and width) and the simplified measurement and selection model. The PISN mass scale is an external prediction from stellar physics, not a free parameter invented for this paper. However, its assumed constancy with redshift is a critical input that would need calibration in practice.

free parameters (7)
  • BBH local merger rate R30 = 64.4 Gpc^-3 yr^-1 (assumed input)
    Sets the detection count, which directly drives the statistical precision of the forecast.
  • Primary mass power-law slope alpha = 0.75 (assumed input)
    Affects how many events fall near the PISN cutoff; taken from GWTC-1 population fits.
  • Mass ratio slope beta = 0.0 (assumed input)
    Assumed flat mass ratio distribution; affects fraction of events with informative mass measurements.
  • Redshift evolution exponent gamma = 3.0 (assumed input)
    Controls the redshift distribution of mergers; chosen to follow the star formation rate.
  • PISN cutoff mass m_h and taper width sigma_h = 45 M_sun, 0.1 (assumed input)
    The location and sharpness of the mass cutoff are the key astrophysical input; a sharper cutoff (smaller sigma_h) nearly halves the quoted uncertainty.
  • Lower mass cutoff m_l and sigma_l = 5 M_sun, 0.1 (assumed input)
    Lower edge of the mass range; less important for the central claim.
  • Cosmological parameters H0, Omega_M, w = Planck 2016 values (used to generate mock data)
    They set the true cosmology that the analysis recovers; the posterior widths on these parameters constitute the reported H(z) uncertainty.
assumptions (4)
  • domain assumption The mass distribution of merging BBHs has a cutoff around m_h = 45 M_sun due to the PISN process (Eq. A1-A2).
    Supported by stellar evolution models (Belczynski et al. 2016; Woosley 2017) and the observed rate drop in GWTC-1 (Fishbach and Holz 2017). The central claim requires this feature to be present in the source-frame mass distribution.
  • domain assumption The PISN mass scale is approximately constant with redshift (within 1 to 2 M_sun for z less than about 1.5).
    Stated in the text; if the scale evolves with redshift, the inferred redshifts are biased and the quoted H(z) precision is not achievable.
  • domain assumption The simplified measurement model in Appendix B (Gaussian uncertainties on log chirp mass, symmetric mass ratio, and amplitude, with SNR-dependent widths) reproduces the accuracy of full parameter estimation.
    The forecast precision depends on the assumed distance and mass measurement uncertainties; the model is tuned to reproduce Vitale et al. (2017) distance uncertainties.
  • domain assumption The detection selection function is approximated as a threshold on single-detector SNR rho greater than 8 (Eq. B11).
    Standard approximation for detectability in a three-detector network; the paper notes the analysis could use a full search selection function.

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Cite this review

Pith. "Pith review of A Future Percent-Level Measurement of the Hubble Expansion at Redshift 0.8 With Advanced LIGO." pith.science (2026). https://pith.science/paper/CGZHLFA3

@misc{pith2026190809084,
  author       = {Pith},
  title        = {Pith review of: A Future Percent-Level Measurement of the Hubble Expansion at Redshift 0.8 With Advanced LIGO},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGZHLFA3}},
  note         = {Machine review of arXiv:1908.09084}
}
abstract

Simultaneous measurements of distance and redshift can be used to constrain the expansion history of the universe and associated cosmological parameters. Merging binary black hole (BBH) systems are standard sirens---their gravitational waveform provides direct information about the luminosity distance to the source. Because gravity is scale-free, there is a perfect degeneracy between the source masses and redshift; some non-gravitational information is necessary to break the degeneracy and determine the redshift of the source. Here we suggest that the pair instability supernova (PISN) process, thought to be the source of the observed upper-limit on the black hole (BH) mass in merging BBH systems at $\sim 45 \, M_\odot$, imprints a mass scale in the population of BBH mergers and permits a measurement of the redshift-luminosity-distance relation with these sources. We simulate five years of BBH detections in the Advanced LIGO and Virgo detectors with realistic assumptions about the BBH merger rate, a mass distribution incorporating a smooth PISN cutoff, and measurement uncertainty. We show that after one year of operation at design sensitivity (circa 2021) the BBH population can constrain $H(z)$ to $6.1\%$ at a pivot redshift $z \simeq 0.8$. After five years (circa 2025) the constraint improves to $2.9\%$. This measurement relies only on general relativity and the presence of a cutoff mass scale that is approximately fixed or calibrated across cosmic time; it is independent of any distance ladder or cosmological model. Observations by future ``third-generation'' gravitational wave detectors, which can see BBH mergers throughout the universe, would permit sub-percent cosmographical measurements to $z \gtrsim 4$ within one month of observation.

Figures

Figures reproduced from arXiv: 1908.09084 by the authors.

Figure 1
Figure 1. Masses and luminosity distances for a simulated population of BBH mergers detected by an Advanced LIGO/Virgo network. Blue circles denote one year of observations, orange circles five years of observations. The solid black line shows the redshifting of the PISN detector-frame BH mass scale corresponding to the cosmology used to generate the events (Planck Collaboration et al. 2016, TT, TE, EE + lowP + lensing + ext)… view at source ↗
Figure 2
Figure 2. Inferred cosmological expansion history and distance scale. (Left) The local expansion rate, H(z), inferred from an analysis of the one year (blue) and five year (orange) simulated populations using a mass distribution model with a parameterized cutoff mass (see text). The black line gives the cosmology used to generate the simulated population (Planck Collaboration et al. 2016, TT, TE, EE + lowP + lensing + ext). T… view at source ↗
Figure 3
Figure 3. Inferred masses and redshifts, and maximum BH mass, for one year of observa￾tion. The points show the posterior mean and 1σ (68%) credible ranges for the source-frame primary BH masses and redshifts after one year of BBH merger observations. The horizontal line is the posterior median of the maximum black hole mass set by the PISN process; the dark and light bands correspond to the 1σ and 2σ (68% and 95%) credible i… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Posterior on the dark energy equation of state parameter after imposing additional cosmological constraints. If we impose a 1% measurement of H0 (Chen et al. 2017; Mortlock et al. 2018; Di Valentino et al. 2018) and the constraints on ΩMh 2 from existing observations o…
Figure 5
Figure 5. Figure 5: Mass distributions. The joint and marginal mass distributions for the masses in merging BBH systems implied by the merger rate density in Eq. (A1) and the parameter choices in Eq. (A3). The turnover at m ' 45 M due to the PISN mass scale is apparent in the primary mass…

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Forward citations

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.