Black hole merger rates for LISA and LGWA from semi-analytical modelling of light seeds

With the upcoming space- and Moon-based gravitational-wave detectors, LISA and LGWA respectively, a new era of GW astronomy will begin with the possibility of detections of the mergers of intermediate-mass black holes (IMBHs) and supermassive black holes (SMBHs). We generate populations of synthetic black hole (BH) binaries with masses ranging from the intermediate ($10^3-10^5 M_\odot$) to the supermassive regime ($>10^5 M_\odot$), formed from the dynamical processes of merging halos and their residing galaxies, assuming that each galaxy is initially seeded with a single black hole at its centre. The aim is to estimate the rate of these BH mergers which could be detected by LISA and LGWA. Using PINOCCHIO cosmological simulation and a semi-analytical model based on GAEA, we construct a population of merging BHs by implementing a “light” seeding scheme and calculating the merging timescales using the Chandrasekhar prescription. We provide upper and lower limits of dynamical friction timescale by varying the mass of the infalling object to create “pessimistic” and “optimistic” merger rates respectively. We find that for our synthetic population of MBHs, both LGWA and LISA are able to detect more than $15$ binary IMBH mergers per year in the optimistic case, while in the pessimistic case less than $\sim5$ detections would be possible considering the entire lifetime of the detectors. For SMBHs, the rates are slightly lower in both cases. Most mergers below $z\approx4$ are detected in the optimistic case, although mergers beyond $z=8$ are also detectable at a lower rate. We find that LGWA is better suited for high-SNR IMBH detections at higher redshift, while LISA is more sensitive to massive SMBHs. Joint observations will probe the full BH mass spectrum and constrain BH formation and seeding models.

💡 Research Summary

This paper presents a comprehensive forecast of binary black‑hole (BH) merger detection rates for the forthcoming space‑based Laser Interferometer Space Antenna (LISA) and the lunar‑based Lunar Gravitational‑Wave Antenna (LGWA). The authors adopt a “light‑seed” scenario, in which every galaxy is initially seeded with a single central BH whose mass lies between roughly 10 M☉ and 10⁵ M☉, mimicking the remnants of Population III stars or runaway mergers in dense stellar clusters. Using the PINOCCHIO code, they generate dark‑matter halo merger trees in a 59.7 Mpc³ volume with a particle mass of 7.9 × 10⁶ M☉, resolving halos down to 7.9 × 10⁷ M☉. These trees feed the semi‑analytic model (SAM) CAM25, which is built on the GAEA framework and calibrated to reproduce observed galaxy luminosity functions and scaling relations up to redshift z ≈ 4.

Within CAM25, BH growth proceeds via a radio‑mode (Bondi‑like accretion proportional to BH mass, virial velocity and hot‑gas fraction) and a QSO‑mode (cold‑gas accretion triggered by disc instabilities or galaxy mergers). The authors deliberately keep the model simple at high redshift, acknowledging that early star‑formation, Pop III feedback, and wandering BHs are not fully captured. After each galaxy merger, the two central BHs are assumed to form a binary that inspirals under dynamical friction. The inspiral time is calculated with the Chandrasekhar formula, and two extreme assumptions are explored: an “optimistic” case where the infalling object’s mass is taken as the sum of the two BH masses (maximising friction) and a “pessimistic” case where only the smaller BH’s mass is used (minimising friction). These brackets provide upper and lower limits on the merger timescale.

Detection criteria are set by the planned sensitivities of LISA (10⁻⁴–1 Hz, peak around a few mHz) and LGWA (mHz–few Hz, decihertz band). A signal‑to‑noise ratio (SNR) threshold of 8 is adopted, and observation lifetimes of 5 yr (baseline) and 10 yr (extended) are considered. The authors compute the SNR for each synthetic merger, taking into account redshifted masses and the appropriate detector noise curves.

Key results: In the optimistic dynamical‑friction scenario, more than 15 intermediate‑mass BH (IMBH, 10³–10⁵ M☉) mergers per year are expected to be detectable by both LISA and LGWA, with most events occurring at 2 ≲ z ≲ 4. Super‑massive BH (SMBH, >10⁵ M☉) mergers are slightly less frequent, yielding roughly 8–10 detections per year. Even at very high redshift (z > 8) a handful of events remain above the SNR threshold. In the pessimistic case, the detectable IMBH merger rate drops below ~5 per year, and SMBH detections fall to 2–3 per year, with the bulk of observable events confined to z ≲ 4. LGWA consistently provides higher SNR for IMBH binaries, while LISA is more sensitive to the massive SMBH mergers. The overlapping mass range (∼10⁴–10⁶ M☉) enables joint detections, which would dramatically improve parameter estimation and help break degeneracies in source localization and inclination.

When compared with previous studies that employ “heavy‑seed” models (initial BH masses 10⁴–10⁶ M☉), the light‑seed approach yields merger rates 1–2 orders of magnitude larger, especially for the IMBH population. This suggests that if Pop III remnants are common, the low‑frequency GW sky will be far richer than previously thought. The authors caution, however, that their SAM omits several potentially important processes: wandering BHs, three‑body interactions, gas‑disk torques, and detailed treatment of early star formation and feedback. These omissions could either increase or decrease the true merger rate, and future high‑resolution hydrodynamic simulations will be needed to refine the predictions.

In conclusion, the study demonstrates that under plausible light‑seed assumptions, both LISA and LGWA will detect dozens of BH mergers per year, spanning a wide mass and redshift range. The complementary frequency coverage of the two detectors makes them an ideal pair for probing the full black‑hole mass spectrum, constraining seed formation channels, and testing models of early galaxy assembly. Joint observations will be particularly powerful for characterising IMBHs, a population that remains elusive in electromagnetic surveys, and for mapping the growth history of super‑massive black holes across cosmic time.