REVIEW
Gravitational Waves from Coalescing Black Hole MACHO Binaries
Astrophysics · Submitted 7 Aug 1997 · 40 references
Pith's verdict: The paper that made 'PBH dark matter is testable with LIGO' a real claim — the binary distribution is postulated rather than derived, but the authors know that and the prediction is falsifiable.
T0 · AI review
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Referee report
The paper
The paper claims
If the halo's dark matter is half-solar-mass black holes, their binaries should be audible in gravitational waves within a few years.
The reading
What carries the argument
Two pieces of machinery carry the argument. The first is the birth model for binaries: two black holes separated by comoving distance $x$ decouple from the cosmic expansion when their own mean density exceeds the radiation density, giving a semimajor axis $a = x^4/ē^3$, and the tidal pull of the nearest neighbour at distance $y$ supplies enough angular momentum to set the eccentricity through $e = √(1-(x/y)^6)$. With $x$ and $y$ taken as uniform, this yields the distribution $f(a,e)$ and, after integrating over eccentricity, the semimajor-axis distribution $f_a(a)$. The second is the gravitational-wave clock: the Peters-Mathews formula converts each orbit into a coalescence time, so the same distribution predicts both the population of wide binaries seen as microlensing binaries today and the short-period binaries merging now. That single curve is what links an observed microlensing binary fraction of order ten per cent to a merger rate of order $10^{-2}$ per galaxy per year.
Extended reading notes
Core claim
The paper's central claim is that a halo made of $0.5 M_☉$ primordial black holes is a loud gravitational-wave source. Assuming the holes are born with comoving separations spread uniformly out to the mean inter-hole spacing, the authors derive a joint distribution of binary semimajor axis and eccentricity, $f(a,e) = (3/2) a^{1/2} ē^{-3/2} e (1-e^2)^{-3/2}$, and then convert it, via the Peters-Mathews inspiral time $t = t_0 (a/a_0)^4 (1-e^2)^{7/2}$, into a distribution of coalescence times. Integrating that distribution gives about $5$ times $10^{8}$ binaries with coalescence times near a Hubble time in the halo out to $50$ kpc for $Ω h^2 = 0.1$, hence a merger rate near $5$ times $10^{-2}$ per year per galaxy, rising to $0.3$ per year if the black holes extend halfway to M31. For equal $0.5 M_☉$ components the chirp mass is about $0.44 M_☉$, and the characteristic strain at $100$ Hz puts such mergers within reach of the first LIGO/VIRGO interferometers out to roughly $15$ Mpc, where several galaxies like ours sit. The authors present this as a way to settle the MACHO identity question within about five years.
Load-bearing premise
The estimate rests on assuming that the black holes are born with separations spread evenly from zero out to the mean spacing between holes, and that the nearest neighbour's distance is likewise evenly spread; nothing in the paper derives this from a formation mechanism, and a clumpier birth distribution would move the merger rate by orders of magnitude.
Editorial extensions
If this is right
- First-generation LIGO/VIRGO should detect black-hole-MACHO mergers out to about $15$ Mpc, giving a few events per year if the halo is made of such holes.
- A planned factor-of-ten sensitivity improvement would extend reach to about $150$ Mpc and would still see a few events per year even if the true rate is a thousand times smaller than estimated.
- Because the chirp mass of a $0.5 + 0.5 M_☉$ binary is about $0.44 M_☉$, well below the $1.2 M_☉$ expected for double neutron stars, a measured chirp mass can separate the two populations.
- Unlike coalescing neutron stars, black hole mergers are not expected to produce gamma-ray bursts, so the presence or absence of a coincident burst is a second discriminator.
- The same birth model predicts that roughly one per cent to ten per cent of MACHOs sit in wide binaries of order $2$ times $10^{14}$ cm, consistent with the one binary microlensing event seen among eight events.
Reading between the lines
- Because the merger rate scales steeply with the assumed initial separation distribution, a null result from a detector that reaches $15$ Mpc would not by itself kill the black-hole-MACHO idea; it would constrain the birth model rather than the dark-matter identity.
- The same counting logic applies to any primordial object that clusters gravitationally, so the rate estimate is a template for constraining primordial black holes across a range of masses, not only at $0.5 M_☉$.
- If the rate estimate holds, the stochastic background from unresolved distant mergers would be a second, independent signature, potentially detectable by cross-correlating two detectors even before individual events are resolved.
- The paper's internal consistency check against the CMB spectral distortion limit pushes the required primordial density spectrum to $n ≳ 1.6$, which makes the scenario a joint constraint on inflation as well as on dark matter.
Original abstract
If MACHOs are black holes of mass about 0.5 solar mass, they must have been formed in the early universe when the temperature was about 1 GeV. We estimate that in this case in our galaxy's halo out to about 50kpc there exist about half billion black hole binaries whose coalescence times are comparable to the age of the universe, so that the coalescence rate will be about five hundredth events/year/galaxy. This suggests that we can expect a few events/year within 15Mpc. The gravitational waves from such coalescing black hole MACHOs can be detected by the first generation of interferometers in the LIGO/VIRGO/TAMA/GEO network. Therefore, the existence of black hole MACHOs can be tested within the next five years by gravitational waves.
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.
The abstract, sentence by sentence
If MACHOs are black holes of mass about 0.5 solar mass, they must have been formed in the early universe when the temperature was about 1 GeV. We estimate that in this case in our galaxy's halo out to about 50kpc there exist about half billion black hole binaries whose coalescence times are comparable to the age of the universe, so that the coalescence rate will be about five hundredth events/year/galaxy. This suggests that we can expect a few events/year within 15Mpc. The gravitational waves from such coalescing black hole MACHOs can be detected by the first generation of interferometers in the LIGO/VIRGO/TAMA/GEO network. Therefore, the existence of black hole MACHOs can be tested within the next five years by gravitational waves.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | If MACHOs are black holes of mass about 0.5 solar mass, they must have been formed in the early universe when the temperature was about 1 GeV. | background 0.81 | 0.15 | 0.05 | asserted 0.72 |
| s1 | We estimate that in this case in our galaxy's halo out to about 50kpc there exist about half billion black hole binaries whose coalescence times are comparable to the age of the universe, so that the coalescence rate will be about five hundredth events/year/galaxy. | result 0.97 | 0.89 | 0.10 | asserted 0.66 |
| s2 | This suggests that we can expect a few events/year within 15Mpc. | interpretation 0.79 | 0.72 | 0.16 | hedged possibility 0.99 |
| s3 | The gravitational waves from such coalescing black hole MACHOs can be detected by the first generation of interferometers in the LIGO/VIRGO/TAMA/GEO network. | interpretation 0.69 | 0.23 | 0.18 | hedged possibility 0.64 |
| s4 | Therefore, the existence of black hole MACHOs can be tested within the next five years by gravitational waves. | interpretation 0.92 | 0.41 | 0.76 | hedged possibility 0.51 |
What was examined, and what was not
- arXiv listing abstract v1
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- relations to other papers
- P abstract
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References
How later papers used it
185 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool.
Uses its method or tool 2 of 11 papers with this label
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Followed this paper's approach to estimate the merger rate in more detail.
GUT-Scale Primordial Black Holes: Mergers and Gravitational Waves (2019) · 1.00 · paraphrased: the licence does not allow quoting
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Used the dynamical mechanism proposed in this paper and further elaborated in another work.
On the primordial binary black hole mergings in LIGO-Virgo-Kagra data (2023) · 0.93 · paraphrased: the licence does not allow quoting
How the literature uses it
153 arXiv papers cite this in 218 sentences
Papers citing it · 154 sentences without a confident label
Contests it 2 papers
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Consequently, binary dynamics should be affected by the cosmological expansion, leading to a dramatically different merger rate than has been reported previously [ref] .
Eliminating the LIGO bounds on primordial black hole dark matter · Introduction · contest 0.79
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Fig. 3 illustrates that any conclusion of limits on PBH dark matter relying on the distributions computed in [ref] is highly suspect.
Primordial Black Hole Dark Matter and the LIGO/Virgo observations · Introduction · contest 0.79
Builds on it 2 papers
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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
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This derivation of the EMBH merger rate follows similar derivations for the merger rate of uncharged black hole binaries [ref] , though modified to account for effects from the extremal magnetic fields of the black holes.
Constraints on Relic Magnetic Black Holes · Derivation of the EMBH Binary Merger Rate · 0.91
Uses its method or tool 9 papers
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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
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To estimate the merger rate in more detail we follow the approach of Ref. [ref] .
GUT-Scale Primordial Black Holes: Mergers and Gravitational Waves · Mergers in the Early Universe · 1.00
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To compute the rate at which primordial black holes merge within the observable volume of $(0.1 pc)^3$, we follow references [ref] .
Were recently reported MHz events planet mass primordial black hole mergers? · Merger rate · 0.97
Uses its data or values 1 paper
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The initial binary properties were taken from the distribution calculated in [ref] .
Primordial Black Hole Dark Matter and the LIGO/Virgo observations · Introduction · 0.97
Compares with it 2 papers
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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
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As a comparison, the angular momentum only accounting for tidal torquing by the nearest PBH with mass of $m_l$ (the comoving distance is $x'$) is estimated as [ref] [equation] which is physically natural only for a (or nearly) monochromatic mass function.
Merger rate of supermassive primordial black hole binaries · Merger rate · 0.97
Cites it as support 2 papers
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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
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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 37 papers
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Two decades ago, Nakamura et al. [ref] (hereafter, NSTT) pointed out that PBHs would form binaries in the early Universe, which would then slowly shrink through gravitational wave radiation, and eventually coalesce.
The merger rate of primordial-black-hole binaries · Introduction · 0.81
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There are several models of double PBH formation [ref] .
On the primordial binary black hole mergings in LIGO-Virgo-Kagra data · Chirp mass distribution of LVK binary black holes · 0.99
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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
Discussion
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