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Elusive Plunges and Heavy Intermediate-mass-ratio Inspirals from Single and Binary Supermassive Black Holes

Lazaros Souvaitzis, Antti Rantala, Thorsten Naab

TL;DR

The paper addresses how a central $M_\bullet=10^9\,M_\odot$ SMBH surrounded by a cluster of ten $10^5\,M_\odot$ IMBHs evolves, and how a second SMBH from a major merger alters merger channels. It uses 300 direct $N$-body simulations with relativistic corrections up to 3.5PN to classify mergers into direct plunges and GW-driven heavy IMRIs, and to evaluate GW observability with LISA and PTAs. The main findings are that a companion SMBH increases the total merger rate by a factor of about 2–5, with plunges dominating in binary-SMBH runs and inspirals more prevalent in single-SMBH runs; most IMRIs occur at relatively high eccentricities, while DPs cluster at $e\gtrsim0.9$. For GW detectability, heavy IMRIs with $M_\bullet\sim10^9\,M_\odot$ are largely undetectable by LISA/PTAs, whereas IMRIs with $M_\bullet\sim10^7$–$10^8\,M_\odot$ could be observable by LISA at modest redshifts, highlighting a pathway to constrain IMBH populations in galactic nuclei.

Abstract

The most massive galaxies in the Universe also host the largest supermassive black holes (SMBHs), with masses of $10^9 \: \mathrm{M_{\odot}}$ and above. During their hierarchical assembly, these galaxies have experienced only a few major mergers at low redshift, but have accreted many low-mass galaxies across cosmic time, possibly hosting intermediate mass black holes (IMBHs). If some of these IMBHs migrate to the galactic center, they may form compact subsystems around the central SMBH. We investigate the evolution of such subsystems, consisting of ten $10^5 \: \mathrm{M_{\odot}}$ IMBHs at three different concentrations around a $10^9 \: \mathrm{M_{\odot}}$ SMBH. We evolve these systems both in isolation and in the presence of a companion SMBH, using \texttt{MSTAR}, a regularized integration method including relativistic effects up to post-Newtonian order 3.5PN. Our analysis focuses on gravitational--wave--driven intermediate--mass--ratio inspirals (heavy IMRIs) and direct plunges. We show that perturbations from a secondary SMBH enhance the number of IMBH direct plunges by more than a factor of two, making them the dominant merger channel. These plunges and IMRIs with a central $10^9 \: \mathrm{M_{\odot}}$ SMBH will contribute to SMBH growth but will likely evade detection with future gravitational-wave interferometers and pulsar timing arrays (PTAs). However, for galaxies with lower--mass SMBHs ($M_\bullet \lesssim 10^8 \:\mathrm{M_{\odot}}$), heavy IMRIs will be detectable with the Laser Interferometer Space Antenna (LISA) and can provide direct observational constraints on the existence of IMBHs, while the more numerous plunges will still remain hidden.

Elusive Plunges and Heavy Intermediate-mass-ratio Inspirals from Single and Binary Supermassive Black Holes

TL;DR

The paper addresses how a central SMBH surrounded by a cluster of ten IMBHs evolves, and how a second SMBH from a major merger alters merger channels. It uses 300 direct -body simulations with relativistic corrections up to 3.5PN to classify mergers into direct plunges and GW-driven heavy IMRIs, and to evaluate GW observability with LISA and PTAs. The main findings are that a companion SMBH increases the total merger rate by a factor of about 2–5, with plunges dominating in binary-SMBH runs and inspirals more prevalent in single-SMBH runs; most IMRIs occur at relatively high eccentricities, while DPs cluster at . For GW detectability, heavy IMRIs with are largely undetectable by LISA/PTAs, whereas IMRIs with – could be observable by LISA at modest redshifts, highlighting a pathway to constrain IMBH populations in galactic nuclei.

Abstract

The most massive galaxies in the Universe also host the largest supermassive black holes (SMBHs), with masses of and above. During their hierarchical assembly, these galaxies have experienced only a few major mergers at low redshift, but have accreted many low-mass galaxies across cosmic time, possibly hosting intermediate mass black holes (IMBHs). If some of these IMBHs migrate to the galactic center, they may form compact subsystems around the central SMBH. We investigate the evolution of such subsystems, consisting of ten IMBHs at three different concentrations around a SMBH. We evolve these systems both in isolation and in the presence of a companion SMBH, using \texttt{MSTAR}, a regularized integration method including relativistic effects up to post-Newtonian order 3.5PN. Our analysis focuses on gravitational--wave--driven intermediate--mass--ratio inspirals (heavy IMRIs) and direct plunges. We show that perturbations from a secondary SMBH enhance the number of IMBH direct plunges by more than a factor of two, making them the dominant merger channel. These plunges and IMRIs with a central SMBH will contribute to SMBH growth but will likely evade detection with future gravitational-wave interferometers and pulsar timing arrays (PTAs). However, for galaxies with lower--mass SMBHs (), heavy IMRIs will be detectable with the Laser Interferometer Space Antenna (LISA) and can provide direct observational constraints on the existence of IMBHs, while the more numerous plunges will still remain hidden.

Paper Structure

This paper contains 26 sections, 43 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical Background - Motion around Supermassive Black Holes
  3. Mass Precession
  4. Relativistic Precession
  5. Resonant Relaxation
  6. Non-resonant (two-body) relaxation
  7. Gravitational Wave Inspiral
  8. Eccentric von Zeipel-Kozai-Lidov Mechanism
  9. Methods
  10. $N$-body Code
  11. Relativistic dynamics at 3.5PN order
  12. Post-Newtonian Orbital Elements
  13. Geometric Eccentricity
  14. Adaptive Output Timestep
  15. Merger Conditions and Classification: Inspiral Vs Plunge
  16. ...and 11 more sections

Figures (11)

  • Figure 1: Orbital evolution of a merging SMBH binary with $M_1=10^{10} \: \mathrm{M_{\odot}}$, $M_2=10^{9} \: \mathrm{M_{\odot}}$. The initial eccentricity and semi-major axis are $e=0.9.$ and $a=0.5 \: \mathrm{pc}$ respectively. The inset panel in the top panel highlights the well behaved geometric eccentricity close to merger. The increase of the 3PN radial component $e_R$ does not correspond to a rise of the orbital eccentricity in the geometric sense.
  • Figure 2: Final orbital eccentricity as a function of normalized angular momentum for all the merger events in our simulations. Different marker colors indicate the two definitions of orbital elements. For orbits with $e < 0.6$, the 3PN-corrected elements overestimate the eccentricity, as the inspirals do not reach $e=0$. Conversely, for highly eccentric orbits ($e \geq 0.6$), the geometric elements fail to accurately represent the orbital properties due to rapid evolution and strong perturbations.
  • Figure 3: Initial configuration of the systems with wide semi-major axis distribution: $0.05\:\mathrm{pc} < a \leq 0.5\:\mathrm{pc}$. Left panel: single SMBH setup. Right panel: system with a companion SMBH on a fixed orbit with $a_{\mathrm{b}}=1 \: \mathrm{pc}$ and $e_{\mathrm{b}}=0.5$ corresponding to an orbital period of $P_{\mathrm{b}}=2000 \: \mathrm{yr}$. Both configurations are shown at $t = 1320 \: \mathrm{yr}$ after the start of the simulations, with trajectory tails tracing the preceding $t = 750 \: \mathrm{yr}$. Notice how the majority of orbits are being disrupted, already after the first pericentre passage of the companion SMBH. The animated versions of these runs are available here: https://doi.org/10.5281/zenodo.17237044.
  • Figure 4: Examples of DP orbits. The colormap traces the time evolution until merger, marked with a red cross. Left panels: single-SMBH run, where a lower-mass MBH precesses due to GR and receives a strong kick from another MBH $\sim 0.5 \: \mathrm{kyr}$ before merger, leading to a rapid plunge. Right panels: binary-SMBH run, where a wider orbit precesses under GR but is perturbed by the companion SMBH $\sim 2 \: \mathrm{kyr}$ before merger, driving it to high eccentricity and eventual plunge. Upper panels show the $x$–$y$ projection, while lower panels display the $x$–$z$ view.
  • Figure 5: Inspiral example orbits. Left panels: single-SMBH run ($\sim6 \: \mathrm{yr}$ to merger), showing an eccentric orbit that precesses due to relativistic effects while simultaneously shrinking and circularizing via GW emission until $d \leq r_{\mathrm{merge}} = 6.5 \: R_{\mathrm{s}}$. Right panels: binary-SMBH run ($\sim1 : \mathrm{kyr}$ to merger), where the inspiral is driven solely by GW emission. The color map indicates the time to merger (red cross), the dashed circle marks $R_{\mathrm{isco}} = 3 R_{\mathrm{s}}$, and the primary SMBH is shown as a black dot. Upper panels display the $x$–$y$ projection, while lower panels show the $x$–$z$ view.
  • ...and 6 more figures