[Paper Review] Signatures of hierarchical mergers in black hole spin and mass distribution

Hiromichi Tagawa, Zoltán Haiman+3 more|arXiv (Cornell University)|Jan 1, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy142 references64 citations

TL;DR

This paper proposes that hierarchical black hole mergers—where black holes form from prior mergers—leave distinct signatures in the distributions of effective spin (χeff), precession spin (χp), and chirp mass (mchirp). Using a toy model and Bayesian analysis of LIGO/Virgo O1–O3a data, it shows that hierarchical mergers produce a monotonic rise in the typical spin magnitude ¯χtyp ≡ (χ²eff + χ²p)^1/2 with mchirp up to a maximum mass, followed by a plateau at ~0.6. The observed data favor this trend at ~2σ confidence, suggesting that future detections of >100 events will robustly confirm the plateau and constrain the maximum mass and spin of first-generation black holes.

ABSTRACT

Recent gravitational wave (GW) observations by LIGO/Virgo show evidence for hierarchical mergers, where the merging BHs are the remnants of previous BH merger events. These events may carry important clues about the astrophysical host environments of the GW sources. In this paper, we present the distributions of the effective spin parameter (χeff), the precession spin parameter (χp), and the chirp mass (mchirp) expected in hierarchical mergers. Under a wide range of assumptions, hierarchical mergers produce (i) a monotonic increase of the average of the typical total spin for merging binaries, which we characterize with χ¯typ≡(χ2eff+χ2p)1/2⁠, up to roughly the maximum mchirp among first-generation (1g) BHs, and (ii) a plateau at χ¯typ∼0.6 at higher mchirp. We suggest that the maximum mass and typical spin magnitudes for 1g BHs can be estimated from χ¯typ as a function of mchirp. The GW data observed in LIGO/Virgo O1–O3a prefers an increase in χ¯typ at low mchirp, which is consistent with the growth of the BH spin magnitude by hierarchical mergers at ∼2σ confidence. A Bayesian analysis using the χeff, χp, and mchirp distributions suggests that 1g BHs have the maximum mass of ∼15–30M⊙ if the majority of mergers are of high-generation BHs (not among 1g–1g BHs), which is consistent with mergers in active galactic nucleus discs and/or nuclear star clusters, while if mergers mainly originate from globular clusters, 1g BHs are favoured to have non-zero spin magnitudes of ∼0.3. We also forecast that signatures for hierarchical mergers in the χ¯typ distribution can be confidently recovered once the number of GW events increases to ≳ O(100).

Motivation & Objective

• To identify distinctive signatures of hierarchical black hole mergers in gravitational wave (GW) data, particularly in spin and mass distributions.
• To determine how the effective spin (χeff), precession spin (χp), and chirp mass (mchirp) evolve in hierarchical merger scenarios.
• To constrain the maximum mass and typical spin of first-generation (1g) black holes using observed GW distributions.
• To assess the detectability of hierarchical merger signatures with increasing GW event statistics.

Proposed method

• Modeling hierarchical mergers using a toy population synthesis framework with varying assumptions on 1g black hole spin and mass distributions.
• Deriving the expected distribution of ¯χtyp = (χ²eff + χ²p)^1/2 as a function of mchirp under different merger scenarios.
• Performing Bayesian model comparison using χeff, χp, and mchirp distributions from LIGO/Virgo O1–O3a data to assess support for hierarchical merger models.
• Using mock GW data with N = 44, 100, and 1000 events to test the robustness of the ¯χtyp profile reconstruction and detectability of the plateau.
• Applying the Akaike information criterion and Bayes factors to evaluate the statistical significance of the plateau and bending point in the ¯χtyp vs. mchirp profile.
• Estimating the critical chirp mass (mcrit) at which the transition from rising to plateaued ¯χtyp occurs, and relating it to the maximum 1g black hole mass (mmax).

Experimental results

Research questions

• RQ1 Does the observed distribution of χeff, χp, and mchirp in LIGO/Virgo O1–O3a data show a monotonic increase in ¯χtyp with mchirp, consistent with hierarchical mergers?
• RQ2 Can the existence of a plateau in ¯χtyp at ~0.6 be statistically confirmed with current and future GW event samples?
• RQ3 What constraints can be placed on the maximum mass and typical spin of first-generation black holes using the observed ¯χtyp profile?
• RQ4 How do different astrophysical environments—such as active galactic nucleus disks, nuclear star clusters, or globular clusters—affect the predicted spin and mass distributions of hierarchical mergers?
• RQ5 At what event count (N) does the statistical significance of the plateau in the ¯χtyp profile become robustly detectable?

Key findings

• The observed LIGO/Virgo O1–O3a data show a ~2σ preference for an increase in ¯χtyp at low mchirp (≲15–50 M⊙), consistent with spin growth via hierarchical mergers.
• A plateau in ¯χtyp at ~0.6 is predicted for high-mass hierarchical mergers, and this feature can be confidently recovered with ≳100 GW events.
• The maximum mass of first-generation black holes is constrained to mmax ≈ 15–30 M⊙ if hierarchical mergers are frequent, consistent with formation in active galactic nucleus disks or nuclear star clusters.
• If mergers originate primarily from globular clusters (with mmax ≈ 45 M⊙), first-generation black holes are favored to have non-zero spin magnitudes of ~0.3.
• The critical chirp mass mcrit marking the transition from rising to plateaued ¯χtyp is estimated at 25+6−3 M⊙ for N = 100, and 24.9+0.9−0.8 M⊙ for N = 1000, implying mmax ≈ 1.2 × mcrit.
• Bayes factor analysis shows that the broken-line model (with plateau) is strongly preferred over a single-line model, with log Bayes factor increasing from 1.5 (N=44) to 24 (N=1000), indicating high statistical confidence in the plateau at large N.