[Paper Review] Merger rates of intermediate-mass black hole binaries in nuclear star clusters

Giacomo Fragione, Abraham Loeb+2 more|arXiv (Cornell University)|Apr 7, 2022

Pulsars and Gravitational Waves Research92 references33 citations

TL;DR

This paper models the formation and merger rates of intermediate-mass black hole (IMBH) binaries in nuclear star clusters (NSCs) via repeated stellar-mass black hole mergers, using semi-analytical simulations that account for varying metallicities, initial seed masses, and black hole spins. It predicts IMBH merger rates of 0.01–10 Gpc⁻³ yr⁻¹, with several detectable mergers per year for LISA, DECIGO, ET, and LIGO for IMBHs with masses ≤1000 M⊙, and tens per year for DECIGO, ET, and LIGO alone.

ABSTRACT

Repeated mergers of stellar-mass black holes (BHs) in dense star clusters can produce intermediate-mass black holes (IMBHs). In particular, nuclear star clusters at the centers of galaxies have deep enough potential wells to retain most of the BH merger products, in spite of the significant recoil kicks due to anisotropic emission of gravitational radiation. These events can be detected in gravitational waves (GWs), which represent an unprecedented opportunity to reveal IMBHs. In this paper, we analyze the statistical results of a wide range of numerical simulations, which encompass different cluster metallicities, initial BH seed masses, and initial BH spins, and we compute the merger rate of IMBH binaries. We find that merger rates are in the range $0.01$-$10$\,Gpc$^{-3}$\,yr$^{-1}$ depending on IMBH masses. We also compute the number of multi-band detections in ground-based and space-based observatories. Our model predicts that a few merger events per year should be detectable with LISA, DECIGO, ET, and LIGO for IMBHs with masses $\lesssim 1000\msun$, and a few tens of merger events per year with DECIGO, ET, and LIGO only.

Motivation & Objective

• To determine the cosmic merger rate of intermediate-mass black hole (IMBH) binaries formed via repeated mergers in nuclear star clusters (NSCs).
• To assess the detectability of these mergers across multi-band gravitational wave (GW) observatories, including LISA, DECIGO, ET, and LIGO.
• To evaluate how initial seed mass, black hole spin, and cluster metallicity influence IMBH formation and merger rates.
• To provide quantitative predictions for future GW detections to guide observational campaigns and constrain NSC and IMBH formation models.

Proposed method

• Uses a semi-analytical framework based on the code from Fragione et al. (2022) to simulate black hole growth and merger dynamics in NSCs.
• Samples galaxy masses from a Schechter function and derives NSC properties (mass, half-mass radius, escape velocity) using empirical scaling relations.
• Models black hole seed formation via stellar evolution (Kroupa IMF) and accounts for natal kicks, retaining seeds only if vnatal < vesc.
• Simulates repeated mergers of stellar-mass black holes with growing IMBH seeds, tracking mass evolution and recoil kicks from anisotropic gravitational wave emission.
• Computes merger rates across a range of initial seed masses (50–100 M⊙), initial spins (χ = 0.0 to 0.8), and metallicities.
• Estimates signal-to-noise ratios (SNR) and detectability of IMBH binaries across LISA, DECIGO, ET, and LIGO, using detector sensitivity curves and redshift-dependent signal evolution.

Experimental results

Research questions

• RQ1 What is the expected merger rate of intermediate-mass black hole binaries formed via repeated mergers in nuclear star clusters?
• RQ2 How do initial seed mass and black hole spin influence the formation and merger rate of IMBHs in NSCs?
• RQ3 Which gravitational wave detectors—LISA, DECIGO, ET, or LIGO—can detect IMBH binaries, and how many events per year are expected for different IMBH masses?
• RQ4 How does the redshift distribution of detectable IMBH mergers vary with detector sensitivity and source mass?
• RQ5 To what extent do NSC properties (e.g., escape velocity, metallicity) affect the retention of merger products and the resulting merger rate?

Key findings

• The merger rate of intermediate-mass black hole binaries in nuclear star clusters ranges from 0.01 to 10 Gpc⁻³ yr⁻¹, depending on the final IMBH mass.
• For IMBHs with masses ≤1000 M⊙, the model predicts a few detectable merger events per year with LISA, DECIGO, ET, and LIGO.
• For the same mass range, DECIGO, ET, and LIGO alone are expected to detect a few tens of merger events per year.
• LISA can detect IMBH binaries with masses ≥800 M⊙ and secondary masses ≥30 M⊙ at redshifts z ≤ 0.5 with SNR ≳10.
• DECIGO is expected to detect binaries with SNR ≳10³ even at redshift z = 2, particularly for more massive systems.
• The detectability of IMBH mergers is highest for lower IMBH masses and higher secondary masses, with ET outperforming LIGO at high redshifts due to lower noise.