Black Hole Binary Formation in the Expanding Universe --- Three Body Problem Approximation ---

Kunihito Ioka; Takahiro Tanaka; Takashi Nakamura; Takeshi Chiba

REVIEW

Black Hole Binary Formation in the Expanding Universe --- Three Body Problem Approximation ---

Kunihito Ioka, Takeshi Chiba, Takahiro Tanaka, Takashi Nakamura

Astrophysics · Submitted 2 Jul 1998 · 32 references

Pith's verdict: Real three-body numerics that confirm rather than overturn the 1997 estimate; the rate's error bar is a stack of worst cases, not a propagated uncertainty.

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CitationKunihito Ioka, Takahiro Tanaka, Takashi Nakamura, Takeshi Chiba. "Black Hole Binary Formation in the Expanding Universe --- Three Body Problem Approximation ---." arXiv:astro-ph/9807018 (1998). https://pith.science/paper/7HUH5PZP

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Bottom line: the integrations are real and honestly reported, but the headline rate is essentially the 1997 estimate with a numerical coefficient bolted on, and the quoted factor of two is a product of worst cases rather than an error analysis. That is fine for what the paper is trying to do, and the authors say as much themselves in Section 8 — the abstract's 'valid within ~50% error' is the actual claim, not a revision of the rate. Worth knowing before you cite the number as if the numerics moved it. What is new: Ioka et al. actually integrate the three-body problem in an expanding background, roughly 100 runs per third-body direction over 0.1 < x/xbar < 1 and 2 < y/x < 7, and check it by time reversal (coordinate error below 1e-7, velocity below 1e-5). They get a/xbar ~ 0.41 (x/xbar)^3.9 and b/a ~ 0.62-0.77 (x/y)^3.1, confirming the analytic powers of Nakamura et al. and fixing alpha ~ 0.4, beta ~ 0.8. The PDFs f(a,e), f_a(a), f_t(t) are a genuine addition. Note that f_t goes as beta^(-21/37) alpha^(-12/37), so the calibration barely moves the rate; the paper's own conclusion is that the old order-of-magnitude estimate survives. Modest, but honest. Soft spots, in order of weight. (1) The fits are extrapolated outside the checked box; Fig. 4 is an argument for coverage, not a verification of it. (2) gamma = 1 or 10 and delta ~ 0.57 are hand-set, and the 3-body-collision and mean-field corrections (0.60, 0.40) are order-unity judgments dressed as numbers. (3) The 'minimum' 1.2e-2 per year per galaxy in Section 8 multiplies worst cases; that is not an error bar. (4) 'Several events per year within 15 Mpc' needs a galaxy count in the Virgo volume that the paper never derives. On the stress-test note: clustering is genuinely unquantified, and the authors flag it in Section 8. But I would not lean on the one-in-eight binary MACHO fraction as evidence for an order-of-magnitude underestimate — that is a single event, and the paper itself notes the small-number caveat. Direction plausible, magnitude unsupported. Also, the record claims the authors state no limitations; that is wrong. Section 8 is explicit about N-body effects, clustering, and the delta-function mass function. Who should read it: anyone working on primordial black hole dark matter or gravitational-wave source populations. Yes to referees.

Referee report

Significance

The numerical calibration of α and β is a real advance over the order-of-magnitude arguments of Ref. [15], and the paper is unusually careful for its genre: the time-reversal check quoted in §4 (coordinate error below 10⁻⁷, velocity error below 10⁻⁵) is exactly the kind of internal validation that such scattering calculations usually lack, and the power-law indices are confirmed rather than assumed. The derivation is parameter-free in the sense that no observed event rate is used to fix α or β; the rate follows from the random-distribution PDF and the fitted coefficients. The prediction is falsifiable and stated in a form a gravitational-wave detector can test. The paper also does the reader the service of enumerating, one by one, the effects that could move the rate, and of stating in §8 and Appendix A what it has not done (N-body effects, clustering, unequal masses). These are genuine strengths and should be weighed alongside the concerns below.

Major objections (5)

[§5B and Abstract] The rate is quoted per galaxy, but the two numerical values given in §5B are computed from different halo masses: ~10¹² BHMACHOs out to the LMC gives ~10⁻¹ events/yr, while BHMACHOs extending 'half way to M31' gives ~6 × 10⁹ coalescing binaries and ~6 × 10⁻¹ events/yr. These differ by a factor of six because the assumed halo mass differs by a factor of six, not because the per-galaxy rate differs. The abstract's 5 × 10⁻² × 2^±1 events/yr/galaxy therefore has no unambiguous normalization. The authors should state the rate per unit BHMACHO halo mass (or per 10¹² M⊙) and then show explicitly how the per-galaxy value is obtained. Relatedly, the step from a per-galaxy rate to 'several events per year within 15 Mpc' is asserted rather than computed: no galaxy number density, volume integral, or statement about what fraction of galaxies host BHMACHO halos is given. A short explicit calculation (e.g., ~10⁻² Mpc⁻³ over the ~1.4 × 10⁴ Mpc³ within 15 Mpc) would make the headline claim checkable.
[§5C and Fig. 4] The numerical runs cover 0.1 < x/x̄ < 1 and 2 < y/x < 7, but the integrations in Eqs. (5.8), (6.3), (6.5), (6.16) and (6.29) run over 1 < y/x < 2 and over y/x > 7, using the fitted power laws outside the calibrated box. Fig. 4 is offered as evidence that the relevant region is covered, but the argument is made for a single parameter set (Ωh² = 0.1, M_BH = 0.5 M⊙, α = 0.4, β = 0.8) and for 0.1t₀ < t < 10t₀. The region y/x < 2 is precisely where the eccentricity approaches its maximum and the coalescence time is shortest, so it is not obviously negligible. The authors should either extend the runs toward y/x → 1 or quantify the fractional contribution of the uncovered region to f_t(t₀).
[§6 and §8] The 'minimum' event rate in §8 is constructed as a product of three separately-extreme reductions: 0.60 from the three-body collision cut (Eq. 6.6), 0.67 from the mean fluctuation field (footnote 2), and 0.30 from the maximum initial peculiar velocity (§6D), giving ~0.12 of the fiducial rate. Each factor is taken at its extreme value (γ = 1, v_f = 1), and the product is then presented in the abstract as a symmetric 2^±1 uncertainty band. This is a compounded worst case, not a plausible lower bound. The v_f = 1 case is in tension with §6D's own argument that peculiar velocities are damped to ~10⁻⁶(x/x̄)⁻¹ by the time R = 10⁻³(x/x̄), so the 30% factor may not be attainable at all. I recommend quoting a central value with one-sided sensitivities stated separately, rather than a product of independent extremes presented as an error bar.
[§5A and §8] The predicted fraction of BHMACHOs in ~10 AU binaries, ~0.8% for Ωh² = 0.1 with α = 0.4, is compared in §5A with the observed binary MACHO rate of roughly one event in eight [23]. The authors correctly note that small-number statistics prevent a definite statement, and §8 acknowledges that 'the high density region may have a strong correlation' and that clustering could increase the binary fraction. But the abstract's 2^±1 band does not include this effect, and the direction is not bounded: clustering could raise the formation rate by an order of magnitude if the §5A comparison is taken at face value, while enhanced nearby-body destruction (§6B, §6C) could lower it. Since the rate scales with the number of binaries formed, this is plausibly the dominant systematic. The authors should either bound it or state explicitly that the quoted uncertainty excludes clustering.
[§4, Eqs. (4.1)–(4.2)] The fits give exponents 3.9 and 3.1–3.2, while the analytic derivation of f_t(t) in §5B uses exactly 4 and 3. Because t ∝ x^{4n}(x/y)^{7m}, the fitted indices give exponents 15.6 and 21.7 in place of 16 and 21, which propagates into the slope 3/37 of f_t. The authors state they 'will not discuss the small deviation of n from 4' and similarly for m, but the sensitivity of the rate to this choice should be shown, since it is a systematic that is not obviously smaller than the factor-of-two band. In addition, no fit residuals or uncertainties on α and β are quoted anywhere; the abstract's '~50% error' is therefore not traceable to a numerical error budget. Quoting the scatter of the filled triangles in Figs. 2 and 3 would make the claim verifiable.

Minor objections (8)

[Abstract] The claim that the first LIGO/VIRGO interferometers 'will be able to verify whether the MACHOs are black holes or not' overstates what a detection can establish: a rate consistent with the prediction would support a BHMACHO population, but a null result at the predicted sensitivity would constrain rather than exclude it, and a detection at a different rate would not cleanly discriminate against other binary black hole channels. Softening this sentence would be appropriate.
[§2 and throughout] The paper assumes Ω = Ω_BHM throughout and states that the extension to a subdominant black hole fraction is 'not difficult' but does not give it. Since the rate scales with the number of BHMACHOs, a one-line statement of the scaling (linear in Ω_BHM/Ω) would let the reader rescale the result without redoing the calculation.
[§6A] The fit β ≃ 0.8 sin(2θ) is obtained with the exponent n fixed at 3 and averaged over the x and y/x range. The range over which it holds should be stated, and the behaviour as θ → 0 (where the tidal torque vanishes and the binary may merge promptly rather than form a long-lived system) should be discussed, since the θ integral in Eq. (6.3) extends to small θ.
[§6E] The dynamical friction estimate in Eq. (6.37) invokes a Coulomb logarithm but does not state b_max and b_min explicitly. Given that the conclusion is that the effect is negligible by many orders of magnitude, a sentence giving the adopted values would close the argument.
[§7] The conclusion that tidal perturbation from other bodies is negligible rests on sufficient conditions whose sign is uncertain, as the authors note at the end of the section. That caveat would be better placed at the start of §7, where the reader first meets the estimates in Eqs. (7.4) and (7.7).
[Fig. 6 caption] The caption reads 'We can see the dependence is week'; this should read 'weak'.
[Whole manuscript] The text contains numerous broken words from typesetting, including 'Jap an' and 'To yonaka' in the author list, 'extistence' in §2, 'hierarc hical' in §6B, and split tokens such as 'BHM ACHO' and 'pr obability' throughout. These should be repaired before publication.
[References] References [14] and [23] are listed as 'to be published' and 'unpublished' respectively, and [23] is the sole support for the observed binary MACHO fraction used in the §5A comparison. The status of these should be updated or the comparison qualified.

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The paper

The paper claims

Numerical three-body calculations confirm that black-hole MACHO binaries form with semimajor axes and eccentricities close to earlier order-of-magnitude estimates, giving a coalescence rate of about 0.05 events per year per galaxy.

The reading

If the dark matter in our galaxy's halo is made of black holes (the MACHO hypothesis), those black holes would have formed very early, about ten microseconds after the Big Bang. Most would fly apart with the cosmic expansion, but occasionally three would come close enough that two of them lose energy to the third and settle into a bound pair. The paper works out how tight and how eccentric those pairs would be. The authors solve the Newtonian three-body problem for black holes embedded in an expanding universe, running about a hundred numerical experiments for each direction of the third body. They find that the earlier back-of-envelope formulas are right in form: the pair's size scales as the fourth power of the initial separation, and its shape (how elongated the orbit is) scales as the cube of the ratio of the pair separation to the distance of the nearest neighbor. The numerical coefficients come out near one, so the earlier estimates were good to about 50 percent. Assuming the black holes are scattered at random, the authors then derive the full distribution of pair sizes, eccentricities, and merger times. They fold in corrections for the third body sometimes staying bound, for the gravitational pull of more distant black holes, and for the angle at which the third body arrives. The net estimate is about one merger per twenty galaxies per year, with a factor-of-two uncertainty either way. That is enough, they argue, for the first generation of gravitational-wave interferometers to see several events per year out to the Virgo cluster, which would settle whether the MACHOs are black holes.

Extended reading notes

Core claim

The coalescence rate of black hole MACHO binaries is about 5 x 10^-2 x 2^(+/-1) events per year per galaxy, implying at least several coalescences per year within 15 Mpc detectable by the first LIGO/VIRGO interferometers (abstract; Sections 5B and 8). If correct, the MACHO dark matter is made of black holes and gravitational-wave detectors will see a guaranteed source population.

Load-bearing premise

The binary population is generated by Eq. 5.1, P(x,y) = 9 x^2 y^2 / xbar^6 exp(-y^3/xbar^3), which assumes black holes form at rest with respect to the Hubble flow, at random uncorrelated positions, with a delta-function mass function and Omega = Omega_BHM (Sections 2 and 5A). The entire event rate scales with the number of binaries this distribution produces; clustering or a broad mass function would change it by more than the quoted factor of two. The authors acknowledge this in Section 8 and sketch an unequal-mass extension in Appendix A but do not compute it.

Original abstract

We study black hole MACHO binary formation through three-body interactions in the early universe at $t\sim 10^{-5}$s. The probability distribution functions of the eccentricity and the semimajor axis of binaries as well as of the coalescence time are obtained assuming that the black holes are randomly formed in space. We confirm that the previous order-of-magnitude estimate for the binary parameters is valid within $\sim 50%$ error. We find that the coalescence rate of the black hole MACHO binaries is $\sim 5 \times 10^{-2} \times 2^{\pm 1}$ events/year/galaxy taking into consideration several possible factors which may affect this estimate. This suggests that the event rate of coalescing binary black holes will be at least several events per year within 15 Mpc. The first LIGO/VIRGO interferometers in 2001 will be able to verify whether the MACHOs are black holes or not.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.78 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.05 · states quantitative result 0.98 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study black hole MACHO binary formation through three-body interactions in the early universe at $t\sim 10^{-5}$s. The probability distribution functions of the eccentricity and the semimajor axis of binaries as well as of the coalescence time are obtained assuming that the black holes are randomly formed in space. We confirm that the previous order-of-magnitude estimate for the binary parameters is valid within $\sim 50%$ error. We find that the coalescence rate of the black hole MACHO binaries is $\sim 5 \times 10^{-2} \times 2^{\pm 1}$ events/year/galaxy taking into consideration several possible factors which may affect this estimate. This suggests that the event rate of coalescing binary black holes will be at least several events per year within 15 Mpc. The first LIGO/VIRGO interferometers in 2001 will be able to verify whether the MACHOs are black holes or not.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study black hole MACHO binary formation through three-body interactions in the early universe at $t\sim 10^{-5}$s.	contribution 1.00	0.89	0.05	asserted 0.87
s1	The probability distribution functions of the eccentricity and the semimajor axis of binaries as well as of the coalescence time are obtained assuming that the black holes are randomly formed in space.	method 0.42	0.91	0.06	asserted 0.98
s2	We confirm that the previous order-of-magnitude estimate for the binary parameters is valid within $\sim 50%$ error.	result 0.96	0.87	0.11	asserted 1.00
s3	We find that the coalescence rate of the black hole MACHO binaries is $\sim 5 \times 10^{-2} \times 2^{\pm 1}$ events/year/galaxy taking into consideration several possible factors which may affect this estimate.	result 1.00	0.94	0.45	asserted 1.00
s4	This suggests that the event rate of coalescing binary black holes will be at least several events per year within 15 Mpc.	interpretation 0.98	0.72	0.32	hedged possibility 1.00
s5	The first LIGO/VIRGO interferometers in 2001 will be able to verify whether the MACHOs are black holes or not.	future work 0.81	0.09	0.07	hedged possibility 0.67

What kind of paper the classifier reports this is: theoretical result 0.61 · evidence from simulation or computation 0.75.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Fig.1. The trajectories of the second body (thick curve) and

**Figure 5.** Fig.5. The angle dependence of

Showing 2 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9807018 by the authors.

References

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J. C. Niemeyer and K. Jedamzik, astro-ph/9709072 (unpu blished). 24 FIGURE CAPTION Fig.1. The trajectories of the second body (thick curve) and the third body (dotted curve) relative to the ﬁrst body for (a) x/¯x = 0 .3, y/x = 2 .0 and (b) x/¯x = 0 .3, y/x = 4 .0. θ = π/4. The coordinate is normalized by ¯ x. Fig.2. The semimajor axis a as a function of i...

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Constraints on Earth-mass primordial black holes from OGLE 5-year microlensing events
astro-ph.CO 2019-01 conditional novelty 6.0 of 10

Earth-mass primordial black holes at roughly one percent of the dark matter abundance can reproduce the six ultrashort OGLE microlensing events, and the same dataset sets the tightest upper bound on primordial black h...
Formation and Evolution of Primordial Black Hole Binaries in the Early Universe
astro-ph.CO 2018-12 conditional novelty 6.0 of 10

N-body simulations show that early-universe PBH binaries are largely disrupted when PBHs are all of dark matter, so analytical merger-rate estimates overestimate the LIGO signal unless the PBH fraction is below roughl...
Black hole formation from a general quadratic action for inflationary primordial fluctuations
astro-ph.CO 2018-11 conditional novelty 6.0 of 10

A time-varying sound speed or effective Planck mass can amplify inflationary perturbations into a black-hole-seeding peak, but the vanilla effective theory of inflation and solid inflation both become strongly coupled...
On the merger rate of primordial black holes: effects of nearest neighbours distribution and clustering
astro-ph.CO 2018-07 accept novelty 6.0 of 10

Including primordial-black-hole clustering and a corrected nearest-neighbour distribution leaves the standard LIGO/Virgo merger-rate bound on 10-solar-mass primordial black hole dark matter intact for narrow mass functions.
Black Holes' Dark Dress: On the merger rate of a subdominant population of primordial black holes
astro-ph.CO 2018-05 conditional novelty 6.0 of 10

Dark matter halos around primordial black holes shrink and circularise their binaries but change the merger rate by only about ten percent, tightening the LIGO limit on the primordial black hole dark matter fraction t...
Detecting the gravitational wave background from primordial black hole dark matter
astro-ph.CO 2016-10 conditional novelty 6.0 of 10

For primordial-black-hole dark matter models that reproduce the LIGO merger rate, the predicted stochastic gravitational-wave background is strong enough for LISA to detect, with a low-frequency turnover that distingu...
Implications for Pulsar Timing Arrays of Sub-solar Black Hole Detections: From LVK to Einstein Telescope and Cosmic Explorer
astro-ph.CO 2025-08 conditional novelty 5.0 of 10

A Bayesian analysis shows that a future sub-solar PBH detection would make the primordial SIGW interpretation of PTA data favored over the SMBH interpretation, but this preference is driven by the detection prior.
Scrutinizing the Primordial Black Holes Interpretation of PTA Gravitational Waves and JWST Early Galaxies
astro-ph.CO 2023-07 conditional novelty 5.0 of 10

Superheavy primordial black holes cannot simultaneously explain the pulsar-timing gravitational-wave background and JWST's massive early galaxies, because Hubble's UV galaxy luminosity function excludes the required a...
Gravitational waves induced from primordial black hole fluctuations: The effect of an extended mass function
astro-ph.CO 2022-07 conditional novelty 5.0 of 10

Extending the primordial-black-hole Poisson gravitational-wave calculation to an extended mass function yields a narrow allowed window for the running spectral index, alpha_s between about 3.316 and 3.355 times 10^-3,...
Hunt for Light Primordial Black Hole Dark Matter with Ultra-High-Frequency Gravitational Waves
astro-ph.CO 2022-05 conditional novelty 5.0 of 10

Resonant LC circuit detectors come within roughly one order of magnitude of the loudest single primordial black hole merger signal near 30 MHz, and a Gaussian beam detector could probe the stochastic background from a...
NANOGrav Signal from the End of Inflation and the LIGO Mass and Heavier Primordial Black Holes
astro-ph.CO 2022-02 conditional novelty 5.0 of 10

A low-scale double-field inflation model with a first-order phase transition can be tuned to match the NANOGrav gravitational-wave spectrum, and the same transition would produce LIGO-mass and heavier primordial black holes.
Scalar induced gravitational waves from primordial black hole Poisson fluctuations in $f(R)$ gravity
astro-ph.CO 2021-12 conditional novelty 5.0 of 10

In R² gravity, gravitational waves induced by primordial black hole Poisson fluctuations constrain the black hole abundance at formation to Ω_PBH,f < 5.5e-5 (1e9 g / m_PBH)^(1/4), which the authors claim is 45% tighte...
Search for subsolar-mass binaries in the first half of Advanced LIGO and Virgo's third observing run
astro-ph.CO 2021-09 accept novelty 5.0 of 10

No subsolar-mass binary mergers were detected in the first half of LIGO-Virgo's third observing run, yielding merger-rate upper limits that cap the primordial black hole dark matter fraction at about 6 percent.
Prospects for probing gravitational waves from primordial black hole binaries
astro-ph.CO 2021-07 conditional novelty 5.0 of 10

Forecasts show that ET, AEDGE, and LISA could probe primordial black hole dark matter down to 10^-5 solar masses via continuous gravitational waves from galactic binaries, beating current constraints across a wide mass range.
Two populations of LIGO-Virgo black holes
astro-ph.CO 2020-12 conditional novelty 5.0 of 10

Fitting LIGO-Virgo GWTC-2 mergers with a combined astrophysical plus primordial black hole model disfavours a primordial-only population and tightens the primordial black hole dark matter fraction to below 0.2 percent...
Bayesian analysis of LIGO-Virgo mergers: Primordial vs. astrophysical black hole populations
astro-ph.CO 2020-08 conditional novelty 5.0 of 10

Using ten LIGO-Virgo mergers, Bayesian evidence ratios disfavour a lognormal primordial black hole mass function relative to empirical astrophysical population models by roughly a factor of 1500, traced to an over-pre...
The Evolution of Primordial Black Holes and their Final Observable Spins
astro-ph.CO 2020-03 conditional novelty 5.0 of 10

Accretion drives primordial black holes heavier than about 30 solar masses to near-extremal spin while lighter ones stay non-spinning, producing a redshift-dependent mass-spin correlation that propagates into gravitat...
Initial clustering and the primordial black hole merger rate
astro-ph.CO 2019-10 conditional novelty 5.0 of 10

A small quadratic non-Gaussianity in the curvature perturbation makes primordial black holes cluster strongly, shifting their mass function to higher masses and raising the predicted binary merger rate enough that dar...
GUT-Scale Primordial Black Holes: Mergers and Gravitational Waves
astro-ph.CO 2019-03 accept novelty 5.0 of 10

Mergers between GUT-scale primordial black holes during a transient matter-dominated era produce a stochastic gravitational wave background peaking near 10^15 to 10^19 Hz with Omega_GW at most about 10^-12, well below...
The Spatial Clustering of Primordial Black Holes
astro-ph.CO 2018-06 conditional novelty 5.0 of 10

Short-range exclusion between primordial black holes leaves their large-scale power spectrum essentially Poissonian, and for a narrow spectral feature the initial clustering length stays below the mean PBH separation.
Gravitational Waves from Primordial Black Hole Mergers
astro-ph.CO 2017-07 conditional novelty 5.0 of 10

Extending the primordial black hole binary merger rate to a lognormal mass function and clustered initial conditions yields a gravitational-wave background bound on primordial black holes that the authors claim is the...
Primordial black hole constraints for extended mass functions
astro-ph.CO 2017-05 accept novelty 5.0 of 10

Widening the primordial black hole mass spectrum tightens rather than relaxes the observational limits on their share of dark matter, leaving at most about 10 percent once all bounds are counted.
New X-ray bound on density of primordial black holes
astro-ph.CO 2017-05 conditional novelty 5.0 of 10

Comparing predicted X-ray emission from gas-accreting primordial black holes with the observed luminosity function of X-ray binaries caps the primordial black hole fraction of dark matter at about four percent for 30 ...
Spin Distribution of Primordial Black Holes
gr-qc 2017-04 conditional novelty 5.0 of 10

Applying critical-collapse scaling laws for a rotating radiation fluid, the authors derive a spin distribution for primordial black holes and find that nearly all form with dimensionless spin below about 0.4.
Inflationary theory and pulsar timing investigations of primordial black holes and gravitational waves
astro-ph.CO 2016-12 conditional novelty 5.0 of 10

Explicit inflationary models that produce LIGO-mass primordial black holes are tested against pulsar timing array limits on secondary gravitational waves, with broad-spectrum models already excluded and sharply peaked...
Gravitational waves from primordial black holes collisions in binary systems
astro-ph.CO 2016-04 conditional novelty 5.0 of 10

Including tidal forces from inflationary dark matter perturbations suppresses the predicted merger rate of primordial black hole binaries by a factor of roughly 1.5 to 2, and the rate matches LIGO if PBHs are about 5e...
The Minimum Testable Abundance of Primordial Black Holes at Future Gravitational-Wave Detectors
astro-ph.CO 2021-06 conditional novelty 4.0 of 10

Future gravitational-wave detectors could probe primordial black hole abundances as low as 10^-10 of the dark matter, but only if the black holes are strongly clustered at formation.
Prospective constraints on the primordial black hole abundance from the stochastic gravitational-wave backgrounds produced by coalescing events and curvature perturbations
astro-ph.CO 2019-03 conditional novelty 4.0 of 10

Future gravitational-wave detectors could constrain the primordial black hole fraction down to about 10^-13 in the subsolar mass range, using both merger and curvature-induced stochastic backgrounds.
Solving puzzles of GW150914 by primordial black holes
astro-ph.HE 2016-11 conditional novelty 4.0 of 10

A modified Affleck-Dine baryogenesis mechanism is argued to yield a log-normal primordial black hole mass distribution whose fitted parameters can accommodate GW150914, MACHO microlensing, and early supermassive black...
Primordial Black Holes
astro-ph.CO 2022-11 conditional novelty 3.0 of 10

This review consolidates the physics of primordial black hole formation, abundance statistics, spin, quantum corrections, observational constraints, and gravitational-wave signatures into a single reference document.
Black holes, gravitational waves and fundamental physics: a roadmap
gr-qc 2018-06 conditional novelty 1.0 of 10

A roughly 200-author roadmap argues that gravitational-wave observations of black holes will test General Relativity in the strong field, reveal how black holes form and grow, and may expose new fundamental fields.

How later papers used it

93 later papers cite it

Evidence: substantial. 90 of the citing papers share no author with it, and 21 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 166 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 1 used its method or tool and 2 used its data or values.

Uses its method or tool 1 of 8 papers with this label

Used this paper's expression for the probability of finding the relevant distances in given intervals, assuming a random distribution of PBHs.

Multi-messenger signal from QCD-Axion Ultracompact Minihalos (2026) · 0.84 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 4 papers with this label

Adopted c1=0.4 and c2=0.8 in alignment with this paper.

Gravitational Waves From More Attractive Dark Binaries (2023) · 0.95 · paraphrased: the licence does not allow quoting
Adopted c1 = 0.4 from this paper's numerical simulations of the gravity-only case.

Gravitational Waves From Dark Binaries With Finite-Range Dark Forces (2024) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

83 arXiv papers cite this in 143 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 96 sentences without a confident label

Builds on it 3 papers

= ∫t aμ[ 1/2𝐫̇^2 + ä/2a𝐫^2 + a m_12 /r - 1/2𝐫 ·𝐓 ·𝐫 ], can be derived along the lines of [ref] but by using the evolving mass [ref] instead of a constant one.

Cosmological black holes are not described by the Thakurta metric: LIGO-Virgo bounds on PBHs remain unchanged · Binary dynamics with growing mass · 0.91
Our analysis generalises the previous studies [ref] , as we consider extended PBH mass functions and estimate the effect of PBH clustering.

Gravitational Waves from Primordial Black Hole Mergers · Introduction · 0.98
To calculate the merger rate, we update the treatment in [ref] (although see further refinements in [ref] ) to account for radiation due to dQED, which dominates the gravitational radiation.

Primordial Extremal Black Holes as Dark Matter · Appendix I: binary merger rate · 0.94

Uses its method or tool 6 papers

The probability that a pair of black holes with unequal masses $M_1$ and $M_2=qM_1$, $q≤1$, coalesce within the time interval (t,t+dt) can be written as [ref] (see also [ref] ) [equation] In this model, the formation of a binary BH with masses $M_1, M_2$ is assisted by the collision with third BH $M_3$. $F(M)$ is the BH mass function which we assume to be of log-normal form ([ref]) that follows from the physical model suggested by [ref] , Eq. ([ref]).

On mass distribution of coalescing black holes · The model of binary PBH coalescences · 0.84
We followed the picture detailed in [ref] .

Detecting the gravitational wave background from primordial black hole dark matter · Synthetizing a population of PBH binaries · 0.99
We compute it following the derivation of [ref] , which builds on several earlier works [ref] .

Dancing in the dark: detecting a population of distant primordial black holes · Primordial black holes · 0.96

Uses its data or values 4 papers

Numerical 3-body calculations (for a static mass) indicate that $α≈0.4$ and $β≈0.8$ [ref] are suitable corrections for the semi-analytic approximation here.

Eliminating the LIGO bounds on primordial black hole dark matter · Introduction · 0.96
We adopt $c_1=0.4$ and $c_2=0.8$, in alignment with [ref] .

Gravitational Waves From More Attractive Dark Binaries · Merger Rate and Stochastic GW Spectrum · 0.95
A detailed investigation of the dynamics of the binary formation suggests $A=0.4$ and $B=0.8$ [ref] .

Effects of the merger history on the merger rate density of primordial black hole binaries · Single-merger events · 0.99

Compares with it 6 papers

The decoupling condition, Eq.([ref]), is very similar to the one used previously [ref] , but differs in one important aspect: it involves the approximate time-dependent Misner-Sharp mass, $ma=m/(1+z)$, instead of the static ADM mass $m$.

Eliminating the LIGO bounds on primordial black hole dark matter · Introduction · 0.84
Solving the equation ([ref]) and ([ref]) for any $x$ (different values of $y$ are taken into account in ([ref]) by the scaling of $ξ$) and comparing with ([ref]), we obtain the following correction factors in ([ref]) [equation] In our calculations these factors are independent of $x$ with high accuracy, however [ref] obtained $α≃0.4$, $β≃0.8$ and weak dependence on the fraction $x/y$.

Gravitational waves from primordial black holes collisions in binary systems · The evolution of binaries at the stage of radiation domination · 0.98
This implies an angular momentum j ≈1.5 x^3/y^3 (x̂ ·ŷ) (x̂ ×ŷ), with magnitude $j ≈0.8 (x/y)^3 sin(2 θ)$, where $θ$ is the angle between $x̂$ and $ŷ$, consistent with the results of Ref. [ref] .

The merger rate of primordial-black-hole binaries · Torques by other PBHs · 0.95

Cites it as support 2 papers

The coalescence of PBH binaries can successfully explain this merger rate if PBHs constitute a small fraction of dark matter [ref] .

Implications for Dark Matter Direct Detection in the Presence of LIGO-Motivated Primordial Black Holes · Introduction · 0.98
Indeed, it was shown that this process typically takes place before the matter-radiation equality and it is due to gravitational attraction leading to PBH pairs decoupling from the Hubble flow [ref] .

Constraining high-redshift stellar-mass primordial black holes with next-generation ground-based gravitational-wave detectors · Primordial Black Hole mergers · 0.96

Cites it as background 20 papers

Binary PBHs can be formed in the early Universe (early formation channel) or at the matter-dominated stage due to dynamical interactions (late formation stage) [ref] .

Effect of post-recombination accretion on primordial binary black hole mergers within virialized dark matter halos · Introduction · 0.98
Merger rate of primordial black hole binaries.— We give a brief overview of the formation mechanism of the binary PBH mergers proposed in Ref. [ref] and revisited by Refs. [ref] to study the merger rate of PBH binaries and the SGWB from PBH binary merger.

Constraints on the Primordial Black Hole Abundance from the First Advanced LIGO Observation Run Using the Stochastic Gravitational-Wave Background · Introduction · 0.91
In this scenario, PBHs are not clustered at formation [ref] and primordial BBHs are assembled via gravitational decoupling from the Hubble flow before the matter-radiation equality [ref] (see [ref] for reviews).

Searching for a subpopulation of primordial black holes in LIGO/Virgo gravitational-wave data · Introduction · 0.87

Discussion

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