In 1997 Nakamura, Sasaki, Tanaka and Thorne estimated that if the Milky Way's MACHOs are roughly 0.5 solar-mass black holes formed in the early universe (black-hole MACHOs, now usually called primordial black holes, PBHs), then about 5×1085 \times 10^8 PBH binaries with coalescence times comparable to the age of the universe exist in the galactic halo, giving a rate of roughly 5×10−25 \times 10^{-2} events per year per galaxy that first-generation LIGO/Virgo could detect, so that gravitational waves could test the black-hole-MACHO hypothesis. They left open four linked questions: the binary distribution and coalescence rate were estimated only to order of magnitude, with N-body simulations flagged as the key follow-up; the required primordial density perturbations (variance δ(lBH)2∼4×10−3\delta(l_{\mathrm{BH}})^2 \sim 4 \times 10^{-3} on comoving scales ∼5×1017\sim 5 \times 10^{17} cm, implying a blue power-law tilt n≳1.6n \gtrsim 1.6 if extrapolated from CMB scales) had unexamined consistency with CMB spectral-distortion limits; roughly 1 solar-mass black-hole binaries whose chirp masses overlap those of neutron-star binaries lacked a demonstrated gravitational-wave discriminator (black-hole absorption entering at 2.5th post-Newtonian order was suggested); and the expected absence of gamma-ray bursts rested on the then-unconfirmed neutron-star-merger/short-GRB association.

Background

Massive compact halo objects (MACHOs) are dark, roughly star-sized bodies in the galactic halo. They came to attention in the 1990s when microlensing surveys toward the Large Magellanic Clouds (a microlensing event is the temporary, achromatic brightening of a background star when a compact body crosses its line of sight) suggested that a large fraction of the Milky Way halo could consist of objects of about 0.5 solar masses. Stellar-evolution arguments disfavored red dwarfs and white dwarfs as the full explanation, so Nakamura, Sasaki, Tanaka and Thorne (1997) analyzed the possibility that MACHOs are black holes of about 0.5 solar masses. Such black holes cannot be produced by stellar collapse and must instead be primordial black holes (PBHs): black holes formed in the early radiation-dominated universe when rare, exceptionally large primordial density perturbations re-enter the horizon and collapse, with the mass inside the horizon at formation fixing the black-hole mass (about 0.5 solar masses corresponds to formation at cosmic temperature near 1 GeV, around the QCD epoch, the time when the cooling universe passed the quark-hadron transition at which free quarks were first bound into protons and neutrons by the strong interaction). Nakamura et al. estimated the PBH-binary population: pairs of PBHs close enough to decouple from the cosmic expansion before matter-radiation equality (the moment when the matter and radiation energy densities were equal) become gravitationally bound, and the tidal torque from the nearest neighboring PBH prevents direct radial collapse, producing highly eccentric binaries whose distribution of semimajor axis a and eccentricity e can be computed statistically. They concluded that about 5×1085 \times 10^8 binaries with gravitational-wave-driven coalescence times of order the age of the universe exist in the halo, implying roughly 5×10−25 \times 10^{-2} coalescences per year per galaxy, detectable by the first generation of the LIGO/Virgo/TAMA/GEO interferometer network, and proposed that gravitational waves could thereby test the black-hole-MACHO hypothesis. A coalescing binary is characterized by its chirp mass, Mchirp=(M1M2)3/5/(M1+M2)1/5M_{\mathrm{chirp}} = (M_1 M_2)^{3/5}/(M_1 + M_2)^{1/5}, which is measured from the gravitational-wave phase to a fraction of a percent. Neutron-star binaries have components near 1.4 solar masses (chirp mass near 1.2), so PBH binaries with much smaller component masses are identifiable by chirp mass alone, but PBHs near 1 solar mass would overlap the neutron-star signal. The gravitational-wave inspiral phase is computed as a post-Newtonian (PN) expansion, an expansion in powers of orbital velocity; black holes absorb a small fraction of the outgoing wave energy through their horizons, a correction that enters at 2.5PN order for spinning (Kerr) black holes and at 4th order for non-spinning (Schwarzschild) ones, and has no direct analogue for material bodies. In modern work the corresponding discriminator is instead the tidal deformability (the dimensionless Love number measuring how easily a body is deformed by a companion's tidal field): a black hole has zero static tidal deformability, while a neutron star has a large one that grows steeply as its mass decreases. On the formation side, the PBH abundance is exponentially sensitive to the variance of density perturbations on the comoving scale (the scale measured in coordinates that expand with the universe) that entered the horizon at PBH formation; Nakamura et al. required δ(lBH)2∼4×10−3\delta(l_{\mathrm{BH}})^2 \sim 4 \times 10^{-3} on a comoving scale lBH∼5×1017l_{\mathrm{BH}} \sim 5 \times 10^{17} cm, which a power-law spectrum P(k)∝knP(k) \propto k^n extrapolated from CMB scales would demand n≳1.6n \gtrsim 1.6. Damping of acoustic waves in the photon-baryon plasma (Silk damping) dissipates the energy of small-scale perturbations into the cosmic microwave background (CMB), distorting its blackbody spectrum into so-called mu-type distortions (chemical-potential-like deviations of the photon energy distribution from a blackbody); the FIRAS instrument (the Far-Infrared Absolute Spectrophotometer) aboard the COBE satellite (Cosmic Background Explorer) has measured the CMB spectrum precisely enough to bound these distortions, but the 1994 bound applied only to scales larger than the thermalization scale lD∼3×1020l_D \sim 3 \times 10^{20} cm, three orders of magnitude above the PBH-formation scale, so the consistency question was open. Finally, short gamma-ray bursts (GRBs) were hypothesized to arise from neutron-star mergers via relativistic fireball expansion, with a delay of order 1 second after the gravitational waves; PBH coalescences should produce no such counterpart. The modern literature reframes all of this under the PBH dark-matter program driven by LIGO/Virgo/KAGRA (LVK) observations: microlensing bounds (the EROS-2 survey, Tisserand et al. 2007) limit compact objects below 1 solar mass to at most about 10 percent of the halo; dedicated LVK searches for sub-solar-mass binaries constrain the PBH fraction; and PBH formation is expressed as a requirement on the curvature power spectrum (the variance of the primordial curvature fluctuations per logarithmic wavenumber interval, the seed of all later structure), PR∼10−2P_R \sim 10^{-2} at wavenumbers k∼106k \sim 10^6-10710^7 Mpc−1^{-1}, about seven orders of magnitude above its measured CMB-scale amplitude, subject to spectral-distortion limits and to limits from induced gravitational waves (a second-order gravitational-wave background generated by the same small-scale perturbations) on intermediate scales (Green and Kavanagh 2021).

Problem Statement

Under the hypothesis that a population of roughly 0.5 solar-mass primordial black holes (black-hole MACHOs) exists, forms binaries in the early universe, and coalesces at a measurable rate, determine whether this scenario can be predicted robustly and identified distinctly enough in gravitational-wave (with electromagnetic) observations to be tested. This single objective, stated by the source paper as 'the existence of black hole MACHOs can be tested... by gravitational waves', unifies four linked sub-questions posed there and kept here in their original generality. (1) Binary distribution and rate: compute the distribution of PBH-binary orbital parameters and the resulting coalescence rate to better than order-of-magnitude accuracy, originally posed as a challenge for N-body simulation of binary formation and now including whether early-formed binaries survive disruption by neighboring PBHs, compact PBH systems, or later-forming halos until merger. (2) Formation-spectrum consistency: determine whether the primordial density perturbations required for roughly 0.5 solar-mass PBH formation, namely δ(lBH)2∼4×10−3\delta(l_{\mathrm{BH}})^2 \sim 4 \times 10^{-3} on comoving scales ∼5×1017\sim 5 \times 10^{17} cm (a power-law tilt n≳1.6n \gtrsim 1.6 if extrapolated from CMB scales), are compatible with observational constraints, in particular CMB spectral distortions produced by Silk damping, noting that the distortion bound of the source era applied only down to the thermalization scale lD∼3×1020l_D \sim 3 \times 10^{20} cm, three orders of magnitude above the PBH-formation scale; in modern terms, whether any primordial spectrum achieving the required small-scale enhancement can evade the mu/y-distortion (and induced-gravitational-wave) limits on intermediate scales while matching the measured CMB-scale amplitude and tilt. (3) Gravitational-wave-only discrimination: establish whether coalescing roughly 1 solar-mass black-hole binaries can be distinguished from neutron-star binaries by gravitational waves alone in the regime where chirp masses overlap and quantify the signal-to-noise ratio or detector generation at which such discrimination becomes decisive. (4) Electromagnetic complement: establish whether the expected absence of electromagnetic emission (short gamma-ray bursts) from PBH coalescences, unlike neutron-star mergers, provides a usable discriminator, contingent on the neutron-star-merger/short-GRB association that later observation confirmed.

References

  1. Takashi Nakamura, Misao Sasaki, Takahiro Tanaka, Kip S. Thorne, 'Gravitational Waves from Coalescing Black Hole MACHO Binaries', The Astrophysical Journal 487, L139-L142 (1997), DOI 10.1086/310886, https://doi.org/10.1086/310886 (arXiv:astro-ph/9708060)
  2. Kunihito Ioka, Takahiro Tanaka, Takashi Nakamura, 'Low frequency gravitational waves from black hole MACHO binaries', Physical Review D 60, 083512 (1999), DOI 10.1103/PhysRevD.60.083512, https://doi.org/10.1103/PhysRevD.60.083512
  3. P. Tisserand et al. (EROS-2 collaboration), 'Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds', Astronomy & Astrophysics 469, 387-404 (2007), DOI 10.1051/0004-6361:20066017, https://doi.org/10.1051/0004-6361:20066017 (arXiv:astro-ph/0607207)
  4. Martti Raidal, Ville Vaskonen, Hardi Veermäe, 'Gravitational Waves from Primordial Black Hole Mergers', Journal of Cosmology and Astroparticle Physics 1709, 037 (2017), DOI 10.1088/1475-7516/2017/09/037, https://doi.org/10.1088/1475-7516/2017/09/037 (arXiv:1707.01480)
  5. Yacine Ali-Haïmoud, Ely D. Kovetz, Marc Kamionkowski, 'Merger rate of primordial black-hole binaries', Physical Review D 96, 123523 (2017), DOI 10.1103/PhysRevD.96.123523, https://doi.org/10.1103/PhysRevD.96.123523 (arXiv:1709.06576)
  6. Martti Raidal, Christian Spethmann, Ville Vaskonen, Hardi Veermäe, 'Formation and Evolution of Primordial Black Hole Binaries in the Early Universe', Journal of Cosmology and Astroparticle Physics 1902, 018 (2019), DOI 10.1088/1475-7516/2019/02/018, https://doi.org/10.1088/1475-7516/2019/02/018 (arXiv:1812.01930)
  7. Zu-Cheng Chen, Qing-Guo Huang, 'Merger Rate Distribution of Primordial Black Hole Binaries', The Astrophysical Journal 864, 61 (2018), DOI 10.3847/1538-4357/aad6e2, https://doi.org/10.3847/1538-4357/aad6e2
  8. Lang Liu, Zong-Kuan Guo, Rong-Gen Cai, 'Effects of the surrounding primordial black holes on the merger rate of primordial black hole binaries', Physical Review D 99, 063523 (2019), DOI 10.1103/PhysRevD.99.063523, https://doi.org/10.1103/PhysRevD.99.063523 (arXiv:1812.05376)
  9. Misao Sasaki, Teruaki Suyama, Takahiro Tanaka, Shuichiro Yokoyama, 'Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -', Classical and Quantum Gravity 35, 063001 (2018), DOI 10.1088/1361-6382/aaa7b4, https://doi.org/10.1088/1361-6382/aaa7b4 (arXiv:1801.05235)
  10. Anne M. Green, Bradley J. Kavanagh, 'Primordial Black Holes as a dark matter candidate', Journal of Physics G: Nuclear and Particle Physics 48, 043001 (2021), DOI 10.1088/1361-6471/abc534, https://doi.org/10.1088/1361-6471/abc534 (arXiv:2007.10722)
  11. Jens Chluba, Adrienne L. Erickcek, Ido Ben-Dayan, 'Probing the inflaton: Small-scale power spectrum constraints from measurements of the cosmic microwave background energy spectrum', The Astrophysical Journal 758, 76 (2012), DOI 10.1088/0004-637X/758/2/76, https://doi.org/10.1088/0004-637X/758/2/76 (arXiv:1203.2681)
  12. David Zegeye, Keisuke Inomata, Wayne Hu, 'Spectral Distortion Anisotropy from Inflation for Primordial Black Holes', Physical Review D 105, 103535 (2021), DOI 10.1103/PhysRevD.105.103535, https://doi.org/10.1103/PhysRevD.105.103535 (arXiv:2112.05190)
  13. B. P. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, Fermi Gamma-Ray Burst Monitor, and INTEGRAL), 'Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A', The Astrophysical Journal Letters 848, L13 (2017), DOI 10.3847/2041-8213/aa920c, https://doi.org/10.3847/2041-8213/aa920c (arXiv:1710.05834)
  14. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), 'Search for Subsolar-Mass Binaries in the First Half of Advanced LIGO's and Advanced Virgo's Third Observing Run', Physical Review Letters 129, 061104 (2022), DOI 10.1103/PhysRevLett.129.061104, https://doi.org/10.1103/PhysRevLett.129.061104 (arXiv:2109.12197)
  15. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), 'Search for subsolar-mass black hole binaries in the second part of Advanced LIGO's and Advanced Virgo's third observing run', Monthly Notices of the Royal Astronomical Society 524, 5984-5992 (2023), DOI 10.1093/mnras/stad588, https://doi.org/10.1093/mnras/stad588 (arXiv:2212.01477)
  16. Marine Prunier, Gonzalo Morras, Jose Francisco Nuno Siles, Sebastien Clesse, Juan Garcia-Bellido, Ester Ruiz Morales, 'Analysis of the subsolar-mass black hole candidate SSM200308 from the second part of the third observing run of Advanced LIGO-Virgo', Physics of the Dark Universe 46, 101582 (2024), DOI 10.1016/j.dark.2024.101582, https://doi.org/10.1016/j.dark.2024.101582 (arXiv:2311.16085)
  17. F. Crescimbeni, G. Franciolini, P. Pani, A. Riotto, 'Can we identify primordial black holes? Tidal tests for subsolar-mass gravitational-wave observations', Physical Review D 109, 124063 (2024), DOI 10.1103/PhysRevD.109.124063, https://doi.org/10.1103/PhysRevD.109.124063 (arXiv:2402.18656)
  18. Francesco Crescimbeni, Gabriele Franciolini, Paolo Pani, Massimo Vaglio, 'Cosmology and nuclear-physics implications of a subsolar gravitational-wave event', Physical Review D 111, 083538 (2025), DOI 10.1103/PhysRevD.111.083538, https://doi.org/10.1103/PhysRevD.111.083538 (arXiv:2408.14287)
  19. Maxence Corman, William E. East, Jocelyn S. Read, 'Binary neutron star mergers with a subsolar mass star', arXiv:2603.25102 (2026), https://arxiv.org/abs/2603.25102
  20. Nicholas Vieira et al., 'Search For a Counterpart to the Subsolar Mass Gravitational Wave Candidate S251112cm', arXiv:2603.17009 (2026), https://arxiv.org/abs/2603.17009
  21. Alberto Magaraggia, Nico Cappelluti, 'Implications for Primordial Black Hole Dark Matter from a Single Subsolar Mass Gravitational-wave Detection in LVK O1-O4', The Astrophysical Journal 1000, 262 (2026), DOI 10.3847/1538-4357/ae48f9, https://doi.org/10.3847/1538-4357/ae48f9 (arXiv:2602.21295)
  22. Oriol Pujolas, Ville Vaskonen, Hardi Veermäe, 'Prospects for probing gravitational waves from primordial black hole binaries', Physical Review D 104, 083521 (2021), DOI 10.1103/PhysRevD.104.083521, https://doi.org/10.1103/PhysRevD.104.083521 (arXiv:2107.03379)
  23. Martti Raidal, Ville Vaskonen, Hardi Veermäe, 'Formation of Primordial Black Hole Binaries and Their Merger Rates', in 'Primordial Black Holes', Springer Series in Astrophysics and Cosmology, pp. 393-427 (2025), DOI 10.1007/978-981-97-8887-3_16, https://doi.org/10.1007/978-981-97-8887-3_16
  24. Xander J. Hall et al., 'Electromagnetic Follow-up of the Sub-Solar Mass Gravitational Wave Candidate S251112cm: Kilonova Constraints and a Coincident IIb Supernova', arXiv:2605.10940 (2026), https://arxiv.org/abs/2605.10940
  25. Gregory S. H. Paek, Felipe Olivares E., Willem B. Hoogendam, 'A Two-month, Galaxy-targeted NIR Follow-up of the Sub-solar-mass Gravitational Wave Candidate S251112cm: Probing Electromagnetic Counterpart Scenarios', arXiv:2608.22775 (2026), https://arxiv.org/abs/2608.22775