Inferring the Merger History of Primordial Black Holes from Gravitational-Wave data and the Stochastic Signatures

Aritra Basak; Maxim Khlopov; Maxim Krasnov; Sourav Roy Chowdhury

REVIEW 4 major objections 7 minor 1 cited by

Inferring the Merger History of Primordial Black Holes from Gravitational-Wave data and the Stochastic Signatures

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper argues that a log-normal mass spectrum of primordial black holes, inferred from the published binary black hole catalogs, gives a local merger rate consistent with observations and predicts a stochastic gravitational-wave…

desk verdict Standard PBH machinery, but the inference skips selection effects and event counts, the local-rate agreement is circular, and the SNR statement is inconsistent—so the quoted posteriors are not credible. read the letter →

arxiv 2507.21332 v1 pith:DKFUIKOO submitted 2025-07-28 gr-qc astro-ph.CO

Sourav Roy Chowdhury , Aritra Basak , Maxim Khlopov , Maxim Krasnov This is my paper · ORCID

classification gr-qcastro-ph.CO

keywords primordialblackholesgravitationalwavesstochasticgravitational-wavebackgroundbinaryholemergerslog-normalmassfunctionhierarchicalBayesianinferencemergerratedarkmatter

topics Dark Matter

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 sets out to show that the merger history of primordial black holes (PBHs) can be inferred from current gravitational-wave observations, and that the same population leaves an imprint in the stochastic gravitational-wave background. Assuming a log-normal mass function, a Bayesian likelihood built from the predicted merger rate returns a central mass of about $21.4$ solar masses, a width of $0.84$, a PBH abundance of $\log_{10} f_{\mathrm{PBH}} \approx -2.67$, and a redshift-evolution index of $2.19$. The resulting local merger rate, $23.5$–$30.3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$, sits inside the range estimated from the observed binary black holes. Integrating the same rate over cosmic history with a standard waveform model yields a stochastic background with mean energy density $\langle \Omega_{\mathrm{GW}}\rangle = 1.67\times10^{-6}$ peaking near $158$ Hz, within reach of a next-generation ground-based detector. If correct, the analysis connects a dark-matter candidate to both resolved merger statistics and an unresolved background that future instruments could measure.

What carries the argument

The central object is the differential merger rate density of PBH binaries, $R(t,m_i,m_j)$, built from the gravitational torque that a third PBH exerts on a newly forming pair. Together with the log-normal mass function $P(m)$ and a power-law redshift evolution $(1+z)^\alpha$, it fixes both the event likelihood (3.2) and the stochastic background integral (4.2). The load-bearing identity is the rate formula, which scales roughly as $f_{\mathrm{PBH}}^{53/37} m^{-21/37}$ and controls how the four population parameters map onto the observed events and the background spectrum.

What would settle it

Recompute the posterior including the published detection probability as a function of component masses and redshift. If the best-fit $\alpha$ or $\log_{10} f_{\mathrm{PBH}}$ moves by more than the quoted error bars, the reported population is not robust; alternatively, a next-generation detector that saw no background above the predicted level near $158$ Hz, after subtracting resolved events, would rule out the steep-redshift-evolution scenario.

Watch

Extended reading notes

Core claim

Under a log-normal PBH mass function, the paper's hierarchical Bayesian analysis of the published transient catalog yields $M_c = 21.44^{+0.79}_{-0.77}\,M_\odot$, $\sigma = 0.84^{+0.03}_{-0.03}$, $\log_{10} f_{\mathrm{PBH}} = -2.67^{+0.01}_{-0.01}$, and $\alpha = 2.19^{+0.16}_{-0.16}$. The model's local PBH binary merger rate, $23.5$–$30.3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$, matches the empirically estimated range. Propagating this population to the unresolved background gives a mean total energy density of $1.67\times10^{-6}$ with the spectrum peaking near $158$ Hz, and a predicted signal-to-noise ratio of about $3.24$ for a next-generation triangular detector with one year of observation, while current detectors fall well below threshold.

Load-bearing premise

The inference treats the observed merger events as an unbiased random draw from the model's predicted rate, ignoring the selection bias of the detector network toward heavy, nearby binaries; if that selection matters, the four inferred population parameters and their error bars would shift.

Editorial extensions

If this is right

PBHs with $\log_{10} f_{\mathrm{PBH}} \approx -2.67$ make up roughly $0.2\%$ of the dark matter, below the threshold where PBH clustering would matter.
The steep redshift index $\alpha \approx 2.2$ implies most PBH mergers happen at high redshift, before star formation begins.
The predicted stochastic background peaks near $158$ Hz and is accessible to a next-generation triangular detector, while current detectors are predicted to be about two orders of magnitude short.
The local PBH merger rate of $23.5$–$30.3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ is consistent with the observed binary black hole rate, so a substantial fraction of the detected events could be primordial in origin.
The spectral peak at $158$ Hz provides a distinguishing feature that could separate PBH mergers from the smoother astrophysical background.

Reading between the lines

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

The likelihood omits detection probability; a selection-corrected re-analysis would likely widen the quoted uncertainties and could shift $\alpha$ and $f_{\mathrm{PBH}}$ toward different best-fit values, so the current error bars should be read as conditional on that omission.
The same rate machinery could be applied to subsolar-mass PBH mergers, where a stochastic background of overlapping events above a few hundred hertz might become a probe of lighter PBH populations.
If PBHs cluster in dark-matter halos even at $f_{\mathrm{PBH}} \sim 10^{-3}$, the background spectrum would develop a turnover at higher frequencies; the present paper's assumption of a Poisson spatial distribution is a testable simplification that future halo-occupation modelling could relax.
Because PBH mergers are rare and individually loud, the predicted background should be non-Gaussian and 'popcorn-like'; measuring the statistical distribution of the background could distinguish it from a smooth astrophysical component.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Dear [Name],

The paper is a PBH population fit to the GWTC black-hole merger catalog plus a stochastic-background forecast. The machinery is almost entirely assembled from existing work: merger rate with third-body torques from Liu et al. 2019/2023, redshift power-law from Mukherjee & Silk, SGWB integral from Christensen. That doesn't kill it, but it means the only new product is the specific posterior and the SNR numbers.

What it does well: the analysis is transparent, uses Bilby/dynesty, reports PPDs, and the SGWB part is a clean textbook implementation. The paper also flags the clustering caveat (low fPBH < 3e-3 justifies ignoring clustering). So it's a plausible template for this type of study.

The soft spots are severe, though. The likelihood in Eq. 3.2 treats the observed events as random draws from the model rate normalized only by the phase-space integral in Eq. 3.3. There is no detection probability, no Poisson term linking the total number of events to the model, and no measurement uncertainty on masses and redshift. The text says this is done 'without requiring ... detailed modelling of selection effects,' but that phrase does not make the likelihood correct. GWTC is heavily biased to high masses and low redshift, so all four posterior parameters are suspect. The reported uncertainties, e.g. log10 fPBH = -2.67 ± 0.01, reflect the prior and the normalization, not the data.

Related to that is the circularity. Section 3 restricts the prior to models with local merger rate in 17.9–44 Gpc^-3 yr^-1, and Section 5 calls the resulting 23.5–30.3 range 'excellent agreement.' That is agreement with the input, not with the data.

There are also smaller internal inconsistencies. The text says ET-B would reach SNR of O(10^1) but the mean SNR is 3.24, which is O(1); and it quotes LIGO/Virgo as O(10^-2) presumably. The mean total energy density 1.67e-6 should be compared carefully to the LVK upper limit they cite; if it's an integrated quantity, say so.

Who is this for? A reader who wants a worked exercise in PBH inference might look at it, but no one should cite the numbers. With a selection-corrected likelihood, inclusion of event counts, and corrected SNR wording, the pipeline could become a useful baseline. I would send it to a referee with a strong request to fix those issues, but as is, the central claims are not supported.

Referee Report

4 major / 7 minor

Summary. This paper proposes an inference of primordial black hole (PBH) population parameters from gravitational-wave transient catalog (GWTC) data, assuming a log-normal PBH mass function and a merger rate model that accounts for gravitational torques from surrounding PBHs. The authors perform a Bayesian analysis with a likelihood built from the model merger rate, impose a prior restriction that the local PBH merger rate lie within the LVK empirical range 17.9–44 Gpc^-3 yr^-1, and report posterior estimates M_c = 21.44 ± 0.79 M_sun, sigma = 0.84 ± 0.03, log10 fPBH = -2.67 ± 0.01, alpha = 2.19 ± 0.16, with a local rate of 23.5–30.3 Gpc^-3 yr^-1. They then compute the stochastic gravitational-wave background (SGWB) from this population and find a mean total energy density of 1.67e-6 peaking near 158 Hz, with an Einstein Telescope (ET-B) SNR of about 3.24, while current detectors would not see it.

Significance. The topic is timely and the paper addresses a question of current interest: whether PBH mergers can contribute to LVK observations and to the SGWB. The model is explicitly formulated and the SGWB computation follows standard methods. If the inference were sound, the result would provide an interesting constraint on the PBH abundance and merger history. However, as detailed below, the likelihood is not a correct model of the GWTC data, the fPBH constraint is effectively imposed by a prior box, and the consistency with LVK rates is circular. These problems are load-bearing, so the specific numerical claims are not supported.

major comments (4)

[Sec. 3, Eqs. (3.2)-(3.3)] The likelihood in Eq. (3.2) is not a correct model of the GWTC data. It evaluates the model rate at point estimates of masses and redshift, omits the detection probability P_det(m1,m2,z), and omits the Poisson event-count term exp(-N_expected) that links the total number of observed events to the model rate. Because the GWTC catalog is strongly selection-biased toward high masses and low redshifts, these omissions bias all inferred parameters. The statement at the end of Section 3 that the analysis proceeds 'without requiring ... detailed modelling of selection effects' is precisely the problem, not a simplification. Additionally, the normalization integral in Eq. (3.3) integrates R_PBH over z without the proper comoving-volume and (1+z) weighting that converts a comoving merger rate density into a detector-frame event count. The quoted posteriors in Table 1 and all downstream results rest on this incorrect likelihood.
[Sec. 3 prior restriction; Sec. 5 Fig. 6] The use of the empirical local merger rate is circular. In Section 3, the parameter space is explicitly restricted to models whose local merger rate lies in the range 17.9–44 Gpc^-3 yr^-1 (the LVK empirical range), and in Section 5 the posterior local rate of 23.5–30.3 Gpc^-3 yr^-1 is presented as 'excellent agreement' with LVK. This is not an independent validation. Moreover, because the model rate in Eq. (2.6) depends on fPBH only through an overall multiplicative factor (fPBH^{53/37}), and the normalized likelihood in Eq. (3.2) divides by N(Λ), fPBH cancels exactly from the likelihood. The reported posterior log10 fPBH = -2.67 ± 0.01 is therefore entirely a consequence of the imposed local-rate prior box, not a constraint from the GWTC events.
[Fig. 1 caption] The caption of Figure 1 states that 'Orange vertical and horizontal lines indicate the injected parameter values.' In a Bayesian analysis of real observational data, there are no injected parameter values; such wording is only appropriate for a mock-injection study. This raises a serious question about whether the reported posteriors come from real GWTC data or from a simulation. The manuscript must clarify this point, because the central claim of the paper is the inference of PBH parameters from actual GWTC events. If the results are from an injection study, they do not constitute empirical constraints.
[Secs. 4-5] The predicted SGWB is never compared with the LVK O3 upper limit Omega_GW(25 Hz) <= 1.04e-9 quoted in the Introduction. Since the inferred local PBH merger rate (23.5–30.3 Gpc^-3 yr^-1) is comparable to the full observed BBH rate, the predicted background, with mean total energy density 1.67e-6, should be checked against this existing limit. As written, the spectrum appears to overshoot the limit unless the spectral shape is extremely unusual, and the absence of any such comparison leaves the consistency claim in Section 5 incomplete.

minor comments (7)

[Sec. 2, Eq. (2.2)] The factor 0.85 in the abundance formula is not derived or referenced; please clarify its origin (it appears to be Omega_cdm/(Omega_cdm+Omega_b)) or provide a citation.
[Sec. 3, Table 1] The hard restriction that the local merger rate lie in 17.9–44 Gpc^-3 yr^-1 is a prior constraint and should be listed explicitly as part of the prior, since it is the dominant constraint on fPBH.
[Sec. 4, Eq. (4.3)] The redshift-evolution power law (1+z)^alpha is introduced in Eq. (4.3), but Section 3's likelihood in Eq. (3.2) does not state how alpha enters the model rate used for inference; please specify the full rate model used in the Bayesian analysis.
[Sec. 4, Eq. (4.6)] The text sets the overlap reduction function to 1 for 'co-located detector pairs (such as Advanced LIGO and Virgo)', but Advanced LIGO and Virgo are not co-located; this choice is inappropriate and affects the reported LIGO/Virgo SNR values.
[Sec. 5] The text reports an ET-B 'SNR of O(10^1)' while giving a mean SNR of about 3.24; 3.24 is O(1), not O(10), so please correct the order-of-magnitude statement.
[Sec. 3, Fig. 2] The description of the posterior predictive distribution as 'the probability of parameter values theta given the observed data d' is incorrect; a posterior predictive distribution describes predicted future observations, not parameter values.
[Reproducibility] The manuscript does not specify which GWTC events were used, how many, how the point estimates of masses and redshift were obtained, or the values of mmin, mmax, and zmax in Eq. (3.3); this information is needed to reproduce the analysis.

Circularity Check

1 steps flagged · score 6.0 of 10

The local-rate 'agreement with LVK' is enforced by a prior in Sec. 3 and then reported as an independent prediction in Sec. 5; the SGWB computation is not circular.

fitted input called prediction [Section 3 (prior restriction, p. 5) and Section 5 (results, p. 10)]
"To maintain consistency with observational constraints, we restrict the parameter space to combinations that produce a local merger rate within the empirically established range of 17.9–44 Gpc−3 yr−1 at redshift z = 0.2 [10], thereby ensuring the physical viability of the model."

The same LVK empirical local rate range is first imposed as a hard truncation on the allowed R_PBH(0) values (17.9–44 Gpc^-3 yr^-1). The paper later reports the posterior R_PBH(0) = 23.5–30.3 Gpc^-3 yr^-1 as 'demonstrating excellent agreement with empirical measurements from the LVK collaborations.' Because the prior already guarantees R_PBH(0) lies inside the LVK interval, the posterior is forced to be compatible at the prior level. The agreement is therefore not an independent confirmation of the model; it is a re-statement of the input constraint. The parameter posteriors and SGWB computation retain independent content, so the circularity is partial.

full rationale

The inference chain is: (i) adopt a log-normal PBH mass function and the merger-rate model from Refs. [51,76]; (ii) evaluate the likelihood in Eq. (3.2) on GWTC point estimates under a prior that truncates parameter space to models whose local rate lies in the LVK empirical range 17.9–44 Gpc^-3 yr^-1; (iii) report the resulting posterior local rate 23.5–30.3 Gpc^-3 yr^-1 as 'excellent agreement' with LVK. Step (iii) is circular for the agreement claim because step (ii) imposed the very range being 'confirmed'. The SGWB spectrum of Sec. 4 is computed by integrating the inferred R_PBH(z) with the Ajith waveform and detector sensitivities; that calculation is a forward prediction from the fitted parameters and is not circular. The likelihood's omission of detection probabilities, the event-count Poisson term, and measurement uncertainties (explicitly done 'without requiring ... detailed modelling of selection effects') is a serious correctness risk but not a circularity. No load-bearing self-citation chain is present: self-references [22,24,46] are background or unrelated SGWB calculations. Overall, the central 'agreement' prediction reduces by construction, so a partial circularity score of 6 is appropriate.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on four fitted parameters, a log-normal mass function, a Poisson spatial distribution, the Liu et al. merger rate formula, a single power-law redshift evolution, and the assumption that GWTC events are unbiased by detectability. The empirical local rate is imposed as a prior rather than predicted. No new entities such as particles or forces are introduced.

free parameters (4)

Mc = 21.44 +/- 0.79 Msun
Central mass of the log-normal mass function, fitted to GWTC data with prior U(10,50).
sigma = 0.84 +/- 0.03
Width of the log-normal mass function, fitted with prior U(0,1).
log10 fPBH = -2.67 +/- 0.01
Logarithm of the PBH abundance relative to dark matter, fitted with prior log10 U(-4,-2).
alpha = 2.19 +/- 0.16
Redshift evolution index in R_PBH(z) proportional to (1+z)^alpha, fitted with prior U(1,3).

assumptions (6)

domain assumption PBH spatial distribution is Poisson after matter-radiation equality
Sec 2 assumes a random Poisson distribution to derive the binary formation rate, following Refs [35,75]; clustering is neglected.
domain assumption Merger rate formula Eq 2.6 from Liu et al. is correct
The rate density including third-body torques is taken from Refs [51,76] and is the basis of the likelihood.
domain assumption PBH mass function is log-normal
Eq 3.1 assumes a log-normal form, motivated by inflationary power spectrum models.
domain assumption Redshift evolution is a single power law (1+z)^alpha
Eq 4.3 adopts this phenomenological form following Ref [69], with no turnover or break at high redshift.
ad hoc to paper GWTC events are an unbiased sample of the merger population
Eq 3.2 builds the likelihood without a detection probability or selection function; the paper acknowledges this simplification in Sec 3.
domain assumption Local merger rate must lie in 17.9 to 44 Gpc^-3 yr^-1
Sec 3 restricts the parameter space to match the empirical range from Ref [10], used to enforce physical viability.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Inferring the Merger History of Primordial Black Holes from Gravitational-Wave data and the Stochastic Signatures." pith.science (2026). https://pith.science/paper/DKFUIKOO

@misc{pith2026250721332,
  author       = {Pith},
  title        = {Pith review of: Inferring the Merger History of Primordial Black Holes from Gravitational-Wave data and the Stochastic Signatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKFUIKOO}},
  note         = {Machine review of arXiv:2507.21332}
}

abstract

Primordial black holes (PBHs) are well-motivated candidates for cold dark matter and may also account for a fraction of the binary black hole mergers observed by the LIGO-Virgo-KAGRA Collaboration. In this study, we investigate the gravitational-wave signatures of PBHs, with a particular focus on evaluating their integrated contribution to the stochastic gravitational-wave background arising from binary mergers over a broad range of redshifts. We perform a Bayesian analysis of gravitational-wave events following all Gravitational-Wave Transient Catalog data, assuming a log-normal PBH mass function. We compute the merger rate distribution of PBH binaries by accounting for gravitational torques from the surrounding PBH. To constrain this rate, we employ the latest limits from the third observing run of LIGO/Virgo. Owing to their primordial origin, PBHs exhibit enhanced merger activity at high redshifts, prior to the onset of stellar formation. Our analysis yields a relatively weak inference on the redshift evolution index of the PBH merger rate, with $\alpha = 2.19^{+0.16}_{-0.16}$ at 68\% confidence level. The local merger rate of PBH binaries is found with posterior estimates lying in the range $23.5-30.3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$, reflecting a high degree of statistical precision in the inferred distribution. Additionally, we emphasize the potential of stochastic gravitational-wave background observations to probe the cumulative history of PBH mergers across cosmic time.

Discussion (0). Continue with ORCID to comment.

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

A population study on the effect of metallicity on ZAMS to the merger
astro-ph.HE 2024-11 conditional novelty 4.0 of 10

Assigning individual metallicities to ZAMS stars via a mass-redshift-metallicity relation in COSMIC produces first-generation BBH total masses of 8-86 solar masses and BH-NS masses of 6-31 solar masses at z~4, with su...

Reference graph

Works this paper leans on

82 extracted references · 79 canonical work pages · cited by 1 Pith paper

[1]

LIGO Scientific, Virgo collaboration, ASTROPHYSICAL IMPLICATIONS OF THE BINARY BLACK HOLE MERGER GW150914, The Astrophysical Journal818 (2016) L22

2016
[2]

LIGO Scientific, Virgo collaboration, Observation of gravitational waves from two neutron star–black hole coalescences, The Astrophysical Journal Letters915 (2021) L5

2021
[3]

LIGO Scientific, Virgo collaboration, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Physical Review Letters119 (2017) 161101

work page 2017
[4]

LIGO Scientific, Virgo collaboration, Multi-messenger observations of a binary neutron star merger*, The Astrophysical Journal Letters848 (2017) L12

work page 2017
[5]

LIGO Scientific, Virgo collaboration, GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Physical Review X9 (2019) 031040

work page 2019
[6]

LIGO Scientific, Virgo collaboration, GW190412: Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses, Physical Review D102 (2020) 043015

work page 2020
[7]

LIGO Scientific, Virgo collaboration, Population Properties of Compact Objects from the Second LIGO–Virgo Gravitational-Wave Transient Catalog, The Astrophysical Journal Letters 913 (2021) L7

work page 2021
[8]

LIGO Scientific, Virgo collaboration, GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo during the First Half of the Third Observing Run, Physical Review X11 (2021) 021053. – 12 –

work page 2021

Show all 82 references

[9]

LIGO Scientific, Virgo collaboration, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run, Physical Review X13 (2023) 041039

2023
[10]

LIGO Scientific, Virgo collaboration, Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3, Physical Review X13 (2023) 011048

2023
[11]

Marchant, K.M.W

P. Marchant, K.M.W. Pappas, M. Gallegos-Garcia, C.P.L. Berry, R.E. Taam, V. Kalogera et al.,The role of mass transfer and common envelope evolution in the formation of merging binary black holes, Astronomy & Astrophysics650 (2021) A107

2021
[12]

Zevin and S.S

M. Zevin and S.S. Bavera,Suspicious siblings: The distribution of mass and spin across component black holes in isolated binary evolution, The Astrophysical Journal933 (2022) 86

2022
[13]

D. Park, C. Kim, H.M. Lee, Y.-B. Bae and K. Belczynski,Black hole binaries dynamically formed in globular clusters, Monthly Notices of the Royal Astronomical Society469 (2017) 4665

2017
[14]

Chatterjee, C.L

S. Chatterjee, C.L. Rodriguez, V. Kalogera and F.A. Rasio,Dynamical Formation of Low-mass Merging Black Hole Binaries like GW151226, The Astrophysical Journal Letter836 (2017) L26

2017
[15]

de Mink, M

S.E. de Mink, M. Cantiello, N. Langer, O.R. Pols, I. Brott and S. Yoon,Rotational mixing in massive binaries - detached short-period systems, Astronomy & Astrophysics497 (2009) 243

2009
[16]

Mandel and S.E

I. Mandel and S.E. de Mink,Merging binary black holes formed through chemically homogeneous evolution in short-period stellar binaries, Monthly Notices of the Royal Astronomical Society 458 (2016) 2634

2016
[17]

Ali-Haïmoud, E.D

Y. Ali-Haïmoud, E.D. Kovetz and M. Kamionkowski,Merger rate of primordial black-hole binaries, Physical Review D96 (2017) 123523

2017
[18]

Raidal, C

M. Raidal, C. Spethmann, V. Vaskonen and H. Veermäe,Formation and evolution of primordial black hole binaries in the early universe, Journal of Cosmology and Astroparticle Physics 02 (2019) 018

2019
[19]

Christensen,Measuring the stochastic gravitational-radiation background with laser-interferometric antennas, Physical Review D46 (1992) 5250

N. Christensen,Measuring the stochastic gravitational-radiation background with laser-interferometric antennas, Physical Review D46 (1992) 5250

1992
[20]

Christensen,Stochastic gravitational wave backgrounds, Reports on Progress in Physics82 (2018) 016903

N. Christensen,Stochastic gravitational wave backgrounds, Reports on Progress in Physics82 (2018) 016903

2018
[21]

Buonanno, G

A. Buonanno, G. Sigl, G.G. Raffelt, H.-T. Janka and E. Müller,Stochastic gravitational-wave background from cosmological supernovae, Physical Review D72 (2005) 084001

2005
[22]

Chowdhury and M

S.R. Chowdhury and M. Khlopov,Stochastic gravitational wave background due to core collapse resulting in neutron stars, Physical Review D110 (2024) 063037

2024
[23]

Ciolfi and L

R. Ciolfi and L. Rezzolla,Twisted-torus configurations with large toroidal magnetic fields in relativistic stars, Monthly Notices of the Royal Astronomical Society: Letters435 (2013) L43

2013
[24]

Chowdhury and M

S.R. Chowdhury and M. Khlopov,The Stochastic Gravitational Wave Background from Magnetars, Universe 7 (2021) 381

2021
[25]

Siemens, V

X. Siemens, V. Mandic and J. Creighton,Gravitational-Wave Stochastic Background from Cosmic Strings, Physical Review Letters98 (2007) 111101

2007
[26]

Camargo Neves da Cunha, C

D. Camargo Neves da Cunha, C. Ringeval and F.R. Bouchet,Stochastic gravitational waves from long cosmic strings, Journal of Cosmology and Astroparticle Physics2022 (2022) 078

2022
[27]

Y. Cui, M. Lewicki and D.E. Morrissey,Gravitational Wave Bursts as Harbingers of Cosmic Strings Diluted by Inflation, Physical Review Letters125 (2020) 211302

2020
[28]

Caldwell, Y

R. Caldwell, Y. Cui and H.K. Guo et al.,Detection of early-universe gravitational-wave signatures and fundamental physics., Gen Relativ Gravit54 (2022) 156. – 13 –

2022
[29]

van Remortel, K

N. van Remortel, K. Janssens and K. Turbang,Stochastic gravitational wave background: Methods and implications, Progress in Particle and Nuclear Physics128 (2023) 104003

2023
[30]

Reardon, A

D.J. Reardon, A. Zic, R.M. Shannon, G.B. Hobbs, M. Bailes, V. Di Marco et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, The Astrophysical Journal Letters951 (2023) L6

2023
[31]

Agazie, A

G. Agazie, A. Anumarlapudi, A.M. Archibald, Z. Arzoumanian, P.T. Baker, B. Bécsy et al., The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, The Astrophysical Journal Letters951 (2023) L8

2023
[32]

Search for gravitational wave signals, Astronomy & Astrophysics 678 (2023) A50

EPTA Collaboration and InPTA Collaboration collaboration, The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals, Astronomy & Astrophysics 678 (2023) A50

2023
[33]

H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Research in Astronomy & Astrophysics23 (2023) 075024

2023
[34]

S. Bird, I. Cholis, J.B. Muñoz, Y. Ali-Haïmoud, M. Kamionkowski, E.D. Kovetz et al.,Did LIGO Detect Dark Matter?, Physical Review Letters116 (2016) 201301

2016
[35]

Sasaki, T

M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914, Physical Review Letters117 (2016) 061101

2016
[36]

Chen and Q.-G

Z.-C. Chen and Q.-G. Huang,Merger Rate Distribution of Primordial Black Hole Binaries, The Astrophysical Journal864 (2018) 61

2018
[37]

Liu, Z.-K

L. Liu, Z.-K. Guo and R.-G. Cai,Effects of the surrounding primordial black holes on the merger rate of primordial black hole binaries, Physical Review D99 (2019) 063523

2019
[38]

Z.-C. Chen, C. Yuan and Q.-G. Huang,Confronting the primordial black hole scenario with the gravitational-wave events detected by LIGO-Virgo, Physics Letters B829 (2022) 137040

2022
[39]

Zel’dovich and I.D

Y.B. Zel’dovich and I.D. Novikov,The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model, Sov. Astron. 10 (1967) 602

1967
[40]

Hawking,Gravitationally Collapsed Objects of Very Low Mass, Monthly Notices of the Royal Astronomical Society 152 (1971) 75

S. Hawking,Gravitationally Collapsed Objects of Very Low Mass, Monthly Notices of the Royal Astronomical Society 152 (1971) 75

1971
[41]

Carr and S.W

B.J. Carr and S.W. Hawking,Black Holes in the Early Universe, Monthly Notices of the Royal Astronomical Society 168 (1974) 399

1974
[42]

Nikulin, M.A

V.V. Nikulin, M.A. Krasnov and S.G. Rubin,Compact extra dimensions as the source of primordial black holes, Frontiers in Astronomy and Space Sciences9 (2022)

2022
[43]

Krasnov and V.V

M.A. Krasnov and V.V. Nikulin,Mechanisms of Producing Primordial Black Holes and Their Evolution, Particles 6 (2023) 580

2023
[44]

Sasaki, T

M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,Primordial black holes—perspectives in gravitational wave astronomy, Classical and Quantum Gravity35 (2018) 063001

2018
[45]

Green and B.J

A.M. Green and B.J. Kavanagh,Primordial black holes as a dark matter candidate, Journal of Physics G: Nuclear and Particle Physics48 (2021) 043001

2021
[46]

Khlopov,Primordial black holes, Research in Astronomy & Astrophysics10 (2010) 495

M.Y. Khlopov,Primordial black holes, Research in Astronomy & Astrophysics10 (2010) 495

2010
[47]

García-Bellido and E

J. García-Bellido and E. Ruiz Morales,Primordial black holes from single field models of inflation, Physics of the Dark Universe18 (2017) 47

2017
[48]

Belotsky, V.I

K.M. Belotsky, V.I. Dokuchaev, Y.N. Eroshenko, E.A. Esipova, M.Y. Khlopov, L.A. Khromykh et al.,Clusters of Primordial Black Holes, European Physical Journal C79 (2019) 246

2019
[49]

Franciolini, I

G. Franciolini, I. Musco, P. Pani and A. Urbano,From inflation to black hole mergers and back again: Gravitational-wave data-driven constraints on inflationary scenarios with a – 14 – first-principle model of primordial black holes across the QCD epoch, Physical Review D106 (2...

2022
[50]

Braglia, J

M. Braglia, J. García-Bellido and S. Kuroyanagi,Tracking the origin of black holes with the stochastic gravitational wave background popcorn signal, Monthly Notices of the Royal Astronomical Society 519 (2023) 6008

2023
[51]

Liu, Z.-K

L. Liu, Z.-K. Guo and R.-G. Cai,Effects of the merger history on the merger rate density of primordial black hole binaries, European Physical Journal C79 (2019) 717

2019
[52]

R.-G. Cai, S. Pi, S.-J. Wang and X.-Y. Yang,Pulsar timing array constraints on the induced gravitational waves, Journal of Cosmology and Astroparticle Physics2019 (2019) 059

2019
[53]

Cai, Z.-K

R.-G. Cai, Z.-K. Guo, J. Liu, L. Liu and X.-Y. Yang,Primordial black holes and gravitational waves from parametric amplification of curvature perturbations, Journal of Cosmology and Astroparticle Physics 2020 (2020) 013

2020
[54]

Jedamzik,Primordial black hole dark matter and the LIGO/Virgo observations, Journal of Cosmology and Astroparticle Physics2020 (2020) 022

K. Jedamzik,Primordial black hole dark matter and the LIGO/Virgo observations, Journal of Cosmology and Astroparticle Physics2020 (2020) 022

2020
[55]

B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,Constraints on primordial black holes, Reports on Progress in Physics84 (2021) 116902

2021
[56]

Carr and F

B. Carr and F. Kühnel,Primordial Black Holes as Dark Matter: Recent Developments, Annual Review of Nuclear and Particle Science70 (2020) 355

2020
[57]

Liu, X.-Y

L. Liu, X.-Y. Yang, Z.-K. Guo and R.-G. Cai,Testing primordial black hole and measuring the Hubble constant with multiband gravitational-wave observations, Journal of Cosmology and Astroparticle Physics 01 (2023) 006

2023
[58]

Wang, Y.-l

X. Wang, Y.-l. Zhang, R. Kimura and M. Yamaguchi,Reconstruction of power spectrum of primordial curvature perturbations on small scales from primordial black hole binaries scenario of LIGO/VIRGO detection, Sci. China Phys. Mech. Astron.66 (2023) 260462

2023
[59]

De Luca, V

V. De Luca, V. Desjacques, G. Franciolini, P. Pani and A. Riotto,GW190521 Mass Gap Event and the Primordial Black Hole Scenario, Physical Review Letters126 (2021) 051101

2021
[60]

Vaskonen and H

V. Vaskonen and H. Veermäe,Did NANOGrav See a Signal from Primordial Black Hole Formation?, Physical Review Letters126 (2021) 051303

2021
[61]

De Luca, G

V. De Luca, G. Franciolini and A. Riotto,NANOGrav Data Hints at Primordial Black Holes as Dark Matter, Physical Review Letters126 (2021) 041303

2021
[62]

Hütsi, M

G. Hütsi, M. Raidal, V. Vaskonen and H. Veermäe,Two populations of LIGO-Virgo black holes, Journal of Cosmology and Astroparticle Physics03 (2021) 068

2021
[63]

Marcoccia, G

P. Marcoccia, G. Nardini and M. Pieroni,Probing primordial black holes at high redshift with future gravitational wave detectors, Monthly Notices of the Royal Astronomical Society531 (2024) 4444

2024
[64]

K.K.Y. Ng, G. Franciolini, E. Berti, P. Pani, A. Riotto and S. Vitale,Constraining High-redshift Stellar-mass Primordial Black Holes with Next-generation Ground-based Gravitational-wave Detectors, The Astrophysical Journal Letters933 (2022) L41

2022
[65]

Sah and S

M.R. Sah and S. Mukherjee,Non-stationary astrophysical stochastic gravitational-wave background: a new probe to the high-redshift population of binary black holes, Monthly Notices of the Royal Astronomical Society527 (2023) 4100

2023
[66]

Escrivà, E

A. Escrivà, E. Bagui and S. Clesse,Simulations of PBH formation at the QCD epoch and comparison with the GWTC-3 catalog, Journal of Cosmology and Astroparticle Physics2023 (2023) 004

2023
[67]

Jangra, B.J

P. Jangra, B.J. Kavanagh and J. Diego,Impact of dark matter spikes on the merger rates of Primordial Black Holes, Journal of Cosmology and Astroparticle Physics2023 (2023) 069. – 15 –

2023
[68]

Stasenko,Merger History of Clustered Primordial Black Holes, 2506.11290

V. Stasenko,Merger History of Clustered Primordial Black Holes, 2506.11290
[69]

Mukherjee and J

S. Mukherjee and J. Silk,Can we distinguish astrophysical from primordial black holes via the stochastic gravitational wave background?, Monthly Notices of the Royal Astronomical Society 506 (2021) 3977

2021
[70]

Mukherjee, M.S.P

S. Mukherjee, M.S.P. Meinema and J. Silk,Prospects of discovering subsolar primordial black holes using the stochastic gravitational wave background from third-generation detectors, Monthly Notices of the Royal Astronomical Society510 (2021) 6218

2021
[71]

Atal, J.J

V. Atal, J.J. Blanco-Pillado, A. Sanglas and N. Triantafyllou,Constraining changes in the merger history of BH and PBH binaries with the stochastic gravitational wave background, Physical Review D105 (2022) 123522

2022
[72]

Bagui, S

E. Bagui, S. Clesse, V. De Luca, J.M. Ezquiaga, G. Franciolini, J. García-Bellido et al., Primordial black holes and their gravitational-wave signatures, Living Reviews in Relativity28 (2025) 1

2025
[73]

Cholis, E.D

I. Cholis, E.D. Kovetz, Y. Ali-Haïmoud, S. Bird, M. Kamionkowski, J.B. Muñoz et al.,Orbital eccentricities in primordial black hole binaries, Physical Review D94 (2016) 084013

2016
[74]

Z.-C. Chen, F. Huang and Q.-G. Huang,Stochastic Gravitational-wave Background from Binary Black Holes and Binary Neutron Stars and Implications for LISA, The Astrophysical Journal 871 (2019) 97

2019
[75]

Nakamura, M

T. Nakamura, M. Sasaki, T. Tanaka and K.S. Thorne,Gravitational Waves from Coalescing Black Hole MACHO Binaries, The Astrophysical Journal487 (1997) L139

1997
[76]

Liu, Z.-Q

L. Liu, Z.-Q. You, Y. Wu and Z.-C. Chen,Constraining the merger history of primordial-black-hole binaries from GWTC-3, Physical Review D107 (2023) 063035

2023
[77]

Ashton, M

G. Ashton, M. Hübner, P.D. Lasky, C. Talbot, K. Ackley, S. Biscoveanu et al.,Bilby: A User-friendly Bayesian Inference Library for Gravitational-wave Astronomy, The Astrophysical Journal Supplement Series 241 (2019) 27

2019
[78]

Allen and J.D

B. Allen and J.D. Romano,Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities, Physical Review D59 (1999) 102001

1999
[79]

et al.,Planck 2018 results - VI

Planck Collaboration, Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C. et al.,Planck 2018 results - VI. Cosmological parameters, Astronomy & Astrophysics641 (2020) A6

2020
[80]

Ajith, S

P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A.M. Sintes et al.,Template bank for gravitational waveforms from coalescing binary black holes: Nonspinning binaries, Physical Review D 77 (2008) 104017

2008
[81]

Howell, T

E. Howell, T. Regimbau, A. Corsi, D. Coward and R. Burman,Gravitational wave background from sub-luminous GRBs: prospects for second- and third-generation detectors, Monthly Notices of the Royal Astronomical Society410 (2011) 2123

2011
[82]

X.-J. Zhu, E. Howell, T. Regimbau, D. Blair and Z.-H. Zhu,Stochastic Gravitational Wave Background from Coalescing Binary Black Holes, The Astrophysical Journal739 (2011) 86. – 16 –

2011

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

LIGO Scientific, Virgo collaboration, ASTROPHYSICAL IMPLICATIONS OF THE BINARY BLACK HOLE MERGER GW150914, The Astrophysical Journal818 (2016) L22

2016

[2] [2]

LIGO Scientific, Virgo collaboration, Observation of gravitational waves from two neutron star–black hole coalescences, The Astrophysical Journal Letters915 (2021) L5

2021

[3] [3]

LIGO Scientific, Virgo collaboration, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Physical Review Letters119 (2017) 161101

work page 2017

[4] [4]

LIGO Scientific, Virgo collaboration, Multi-messenger observations of a binary neutron star merger*, The Astrophysical Journal Letters848 (2017) L12

work page 2017

[5] [5]

LIGO Scientific, Virgo collaboration, GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Physical Review X9 (2019) 031040

work page 2019

[6] [6]

LIGO Scientific, Virgo collaboration, GW190412: Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses, Physical Review D102 (2020) 043015

work page 2020

[7] [7]

LIGO Scientific, Virgo collaboration, Population Properties of Compact Objects from the Second LIGO–Virgo Gravitational-Wave Transient Catalog, The Astrophysical Journal Letters 913 (2021) L7

work page 2021

[8] [8]

LIGO Scientific, Virgo collaboration, GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo during the First Half of the Third Observing Run, Physical Review X11 (2021) 021053. – 12 –

work page 2021