pith. sign in

arxiv: 2512.11665 · v1 · pith:YSXW47E6new · submitted 2025-12-12 · 🌌 astro-ph.GA

Rapid sinking and efficient mergers of supermassive black holes in compact high-redshift galaxies

Atte Keitaanranta , Peter H. Johansson , Alexander Rawlings , Toni Tuominen , Antti Rantala , Thorsten Naab , Shihong Liao , Basti\'an Reinoso This is my paper

Pith reviewed 2026-05-21 18:10 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords supermassive black holeshigh-redshift galaxiesdynamical frictiongravitational wavescosmological simulationscompact galaxiesLittle Red Dotsblack hole mergers
0
0 comments X

The pith

Supermassive black holes merge in 4 to 35 million years after becoming bound in dense early galaxies

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper runs cosmological zoom-in simulations of massive galaxies at high redshifts, with the largest reaching 8.5 times 10 to the 10 solar masses by redshift 5. It follows supermassive black hole dynamics from seeding all the way to coalescence at sub-parsec scales using an updated version of the KETJU code. The galaxies pass through a gas-rich compaction phase between redshifts 7 and 9 that builds extremely high central stellar densities above 10 to the 13 solar masses per cubic kiloparsec. These densities drive the central result that black hole binaries merge on short timescales of 4 to 35 million years. The same runs also reproduce the sizes and fluxes of JWST-observed compact systems known as Little Red Dots and track the full gravitational wave emission from pulsar timing array frequencies down to the final orbits visible to LISA.

Core claim

Due to the very high central stellar densities of 10 to the 13 solar masses per cubic kiloparsec or greater that form during the early compaction phase, supermassive black hole binaries in these compact high-redshift galaxies merge rapidly, typically only 4 to 35 million years after the binaries become bound.

What carries the argument

The KETJU code, which uses regularised integration for massive black holes combined with a dynamical friction subgrid model for lower-mass black holes to follow orbital decay and coalescence down to gravitational wave emission.

If this is right

  • The complete gravitational wave signal from each merger can be followed continuously from pulsar timing array frequencies through to the final orbits detectable by LISA.
  • The simulated compact galaxies at redshifts 5 to 9 match the observed sizes, masses, and fluxes of JWST Little Red Dots.
  • Central gas fractions drop sharply once the compaction phase ends while outer regions stay gas-rich and drive later size growth.
  • Black hole coalescence occurs early enough to shape the assembly and mass growth of the host galaxies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Rapid high-redshift mergers may add a larger share to the nanohertz gravitational wave background than models with longer binary lifetimes predict.
  • The short timescales imply that direct black hole coalescence, rather than prolonged binary hardening, dominates early black hole growth in dense environments.
  • Future LISA detections could include a population of these quick high-redshift events if the simulated densities are typical.

Load-bearing premise

The subgrid dynamical friction model for lower-mass black holes combined with regularized integration accurately captures the orbital decay and merger timescales at sub-parsec scales in these high-density environments.

What would settle it

Finding a supermassive black hole binary in a comparable high-redshift compact galaxy that remains unbound or unmerged for longer than 100 million years after the two black holes become bound would contradict the short merger timescale.

Figures

Figures reproduced from arXiv: 2512.11665 by Alexander Rawlings, Antti Rantala, Atte Keitaanranta, Basti\'an Reinoso, Peter H. Johansson, Shihong Liao, Thorsten Naab, Toni Tuominen.

Figure 1
Figure 1. Figure 1: A dichotomous key describing whether ketju or the dynamical friction subgrid model is used for the BH dynamics. The dynamical friction subgrid model is used when a SMBH has a mass of m• < mKetju and the separation from a ketju integrated SMBH is larger than 10 × rKetju. calculation. Finally, assuming the limit m• ≫ mi, i.e. that the BH is much more massive than the background field par￾ticles causing the d… view at source ↗
Figure 2
Figure 2. Figure 2: The sinking of a SMBH into the centre of a Plummer sphere. The mass ratio between the SMBH and a stellar particle is m•/m⋆ = 100 in each simulation. For each SMBH dynamics model, the separation from the centre (main panels) and the residual (smaller panels) from the analytical estimate is shown. The lines are averaged over 25 Myr. A shaded gray area is the region where ketju begins to deviate from the anal… view at source ↗
Figure 3
Figure 3. Figure 3: The sinking of a SMBH into the centre of a Plummer sphere, i.e. the same as [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The median distance of the SMBH from the centre for various SMBH masses, with errorbars covering the range from 25th percentile to 75th percentile. The results are shown for a set of simulations run with ketju and for a set where the dynamical friction subgrid model is used. For ketju integrated SMBHs, the Brownian motion starts to increase as the mass ratio falls below m•/m⋆ ≲ 100. With the subgrid model,… view at source ↗
Figure 5
Figure 5. Figure 5: The evolution of the SMBH mass as a function of the stellar mass for each system that reaches m• ⩾ mKetju. The diamonds and dashed lines show the evolution from the simulation zoom-G, while the circles and solid lines represent the simulation zoom-K+DF. The mass limit of switching ketju integration on for a SMBH is shown as a horizontal dotted line. Markers are shown for each system at intervals of ∼ 105 M… view at source ↗
Figure 6
Figure 6. Figure 6: Left: The evolution of the stellar mass and the effective radius of the stellar component of each system which reaches m• ⩾ mKetju in the simulation zoom-K+DF. The black dots show the estimated sizes and masses of LRDs observed with JWST (Baggen et al. 2023; Kokorev et al. 2024; Wang et al. 2025; Akins et al. 2025a; Ma et al. 2025; Baggen et al. 2024; Akins et al. 2023). For observations where only a upper… view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of eight properties for each system which reaches m• ⩾ mKetju in the zoom-K+DF simulation. The left column, from top to bottom, shows the stellar mass of the galaxy, the star formation rate, the effective radius of the stellar component and the gas fraction fgas = Mgas/(Mgas + M⋆) calculated within the three-dimensional stellar half-mass radius, respectively. The right column, from top to bo… view at source ↗
Figure 8
Figure 8. Figure 8: The evolution of the stellar mass and the stellar metallicity, as measured by 12+O/H of each system which reaches m• ⩾ mKetju in the zoom-K+DF simulation. The red markers show JWST observations in the redshift range 4 < z < 10 (Nakajima et al. 2023). The evolution of metallicity matches observations throughout the mass range 107 − 1010 M⊙. to each galaxy keeps the gas content from depletion. The left panel… view at source ↗
Figure 9
Figure 9. Figure 9: Mock SKIRT images and SED of galaxy A at a redshift of z = 5. The upper panels show broadband images in six different JWST bands, NIRCam F115W, F150W, F200W, F277W and F444W, and MIRI F770W, as well as an RGB image of three bands, F115W/F200W/F444W. The red circle in the MIRI F770W image indicates the aperture size Raperture = 0.3 ′′ used for the photometry, while the white bars in the F770W and RGB colour… view at source ↗
Figure 10
Figure 10. Figure 10: Mock SKIRT images and SED of galaxy B at redshift z = 7.986. As in [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The evolution of mass and separation from the cen￾tre of the host galaxy from the simulation zoom-K+DF for each SMBH which reaches the ketju integration mass limit, shown as a horizontal dashed line. The vertical dashed line shows the physical stellar softening length ϵ⋆. The results are from snapshots with an interval of ∼ 60 Myr. Three distinct stages of evolution are visible: seed mass SMBHs orbit arou… view at source ↗
Figure 12
Figure 12. Figure 12: Snapshot from redshift z = 5.33. Left: surface density profile of dark matter in a part of the zoom-in region. The black circles represent the positions of SMBHs. Middle: surface density of the gas particles from a region including an ongoing galaxy merger. Right: surface density of the stellar particles of the ongoing galaxy merger. The galaxy merger will result a SMBH binary (merger J→G), with the coale… view at source ↗
Figure 13
Figure 13. Figure 13: Binary parameters (top: semimajor axis and bottom: eccentricity) of the ketju integrated SMBH binaries in the cosmological zoom-in simulation. Zoomed panels are included for the system B→A, which reaches coalescence just ∼ 3.5 Myr after becoming bound. have merged. As a comparison, we also include densities of galaxies from the ketju integrated mergers from a cosmo￾logical zoom-in simulation presented in … view at source ↗
Figure 14
Figure 14. Figure 14: The three-dimensional stellar densities of galaxies, measured from the first snapshot after a ketju integrated SMBH merger has occurred in the galaxy. D→A, B→A and J→G show galaxies from the zoom-K+DF simulation performed in this study, the dashed line is the used stellar softening length ϵ⋆ and the grey lines are galaxies from Mannerkoski et al. (2022). In the central regions, the differences in stellar … view at source ↗
Figure 15
Figure 15. Figure 15: The evolution of the separation between the SMBHs (top row), the SMBH mass accretion rate (averaged over 1 Myr, middle row) and the masses of the SMBHs (bottom row) for binaries D→A (left column), B→A (middle column) and J→G (right column) as a function of time. The solid lines show results from the simulation zoom-K+DF, while the dashed line shows results from the run zoom-DF and the dotted-dashed line i… view at source ↗
Figure 16
Figure 16. Figure 16: The characteristic strain as a function of the observed frequency for the three binaries, assuming an observation time of 30 yr. The dashed lines show the characteristic strain calculated with PhenomD assuming a dimensionless spin parameter of χ = 0.8 (the spin is set by hand to a value equal to the choice in Katz et al. 2020) and the solid lines show the evolution calculated from ketju integrated binarie… view at source ↗
read the original abstract

We present a cosmological zoom-in simulation targeting the high redshift compact progenitor phase of massive galaxies, with the most massive galaxy reaching a stellar mass of $M_{\star}=8.5\times 10^{10} \ M_{\odot}$ at $z=5$. The dynamics of supermassive black holes (SMBHs) is modelled from seeding down to their coalescence at sub-parsec scales due to gravitational wave (GW) emission by utilising a new version of the KETJU code, which combines regularised integration of sufficiently massive SMBHs with a dynamical friction subgrid model for lower-mass SMBHs. All nine massive galaxies included in this study go through a gas-dominated phase of early compaction in the redshift range of $z\sim 7-9$, starting at stellar masses of $M_\star\gtrsim 10^8\ \mathrm{M}_\odot$ and ending at a few times $M_{\star}\sim 10^9\ \mathrm{M}_\odot$. The sizes, masses and broad band fluxes of these compact systems are in general agreement with the population of systems observed with JWST known as `Little Red Dots'. In the compact phase, the stellar and SMBH masses grow rapidly, leading to a sharp decline in the central gas fractions. The outer regions, however, remain relatively gas-rich, leading to subsequent off-centre star formation and size growth. Due to the very high central stellar densities ($\rho_{\star}\gtrsim 10^{13}\,\mathrm{M_\odot/kpc^3}$), the SMBHs merge rapidly, typically just $\sim 4-35\ \mathrm{Myr}$ after the SMBH binaries have become bound. Combining KETJU with the phenomenological PhenomD model resolves the complete evolution of the GW emission from SMBH binaries through the Pulsar Timing Array frequency waveband up to the final few orbits that produce GWs observable with the future LISA mission.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper presents cosmological zoom-in simulations of nine high-redshift galaxies reaching stellar masses up to 8.5e10 M⊙ at z=5, using an updated KETJU code that combines regularized N-body integration for massive SMBHs with a subgrid dynamical friction model for lower-mass SMBHs. The galaxies undergo gas-dominated compaction at z~7-9, forming compact systems whose sizes, masses, and fluxes align with JWST-observed 'Little Red Dots'. Due to central stellar densities ρ⋆ ≳ 10^13 M⊙/kpc³, SMBH binaries are reported to merge in 4-35 Myr after becoming bound. The work extends GW modeling from PTA frequencies to LISA using the PhenomD prescription.

Significance. If the merger timescales prove robust, the results would be significant for models of early SMBH assembly, expected LISA event rates, and the role of dense stellar environments in driving rapid coalescence. The hybrid KETJU scheme to bridge parsec to sub-parsec scales, the sample of nine galaxies, and the direct link to observed compact high-z systems are clear strengths that enhance the work's impact.

major comments (2)
  1. [Abstract and Methods] Abstract and Methods (KETJU implementation): The central claim of 4-35 Myr merger times after binary binding is load-bearing and emerges from the subgrid dynamical friction model translating the reported ρ⋆ ≳ 10^13 M⊙/kpc³ into rapid orbital decay. No resolution convergence tests, parameter sensitivity studies for the friction coefficients, or validation against the extreme densities and velocity dispersions are mentioned, leaving the quantitative result sensitive to the Chandrasekhar-like assumptions in the subgrid prescription.
  2. [Results] Results (merger timescale reporting): The 4-35 Myr range is presented without per-galaxy values, medians, or scatter across the nine systems, which weakens the ability to assess whether the rapid-merger conclusion holds uniformly or depends on specific realizations of the compaction phase.
minor comments (3)
  1. [Abstract] Abstract: The description of the transition from compact phase to subsequent size growth via off-centre star formation would benefit from a short physical explanation of the driving mechanism.
  2. [Abstract] Abstract: Include explicit citations to the original KETJU papers and the PhenomD model to provide immediate context for the numerical and GW modeling choices.
  3. The reported decline in central gas fraction during the compact phase should be accompanied by a brief note on whether this is measured within a fixed aperture or adaptive radius for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address each major comment below and have revised the paper accordingly to improve the robustness and clarity of our presentation.

read point-by-point responses

  1. Referee: [Abstract and Methods] Abstract and Methods (KETJU implementation): The central claim of 4-35 Myr merger times after binary binding is load-bearing and emerges from the subgrid dynamical friction model translating the reported ρ⋆ ≳ 10^13 M⊙/kpc³ into rapid orbital decay. No resolution convergence tests, parameter sensitivity studies for the friction coefficients, or validation against the extreme densities and velocity dispersions are mentioned, leaving the quantitative result sensitive to the Chandrasekhar-like assumptions in the subgrid prescription.

    Authors: We acknowledge that the current manuscript does not present new resolution convergence tests or dedicated parameter sensitivity studies for the subgrid dynamical friction coefficients at the extreme densities encountered here. The KETJU hybrid scheme and the underlying Chandrasekhar-type dynamical friction prescription have been tested and validated against direct N-body integrations in prior work at somewhat lower (but still high) densities. At the central stellar densities ρ⋆ ≳ 10^13 M⊙/kpc³ reported in our galaxies, the dynamical friction timescale remains short even when the Coulomb logarithm is varied by factors of a few or when modest changes are made to the velocity dispersion scaling. Nevertheless, we agree that an explicit discussion of these assumptions is warranted. In the revised manuscript we have added a dedicated paragraph in the Methods section that (i) recalls the calibration of the subgrid model, (ii) estimates the sensitivity of the merger time to plausible variations in the friction coefficient, and (iii) notes that full convergence at these densities would require substantially higher resolution than is computationally feasible in the present cosmological zoom-in runs. We believe this addition addresses the referee’s concern without altering the central conclusion. revision: partial

  2. Referee: [Results] Results (merger timescale reporting): The 4-35 Myr range is presented without per-galaxy values, medians, or scatter across the nine systems, which weakens the ability to assess whether the rapid-merger conclusion holds uniformly or depends on specific realizations of the compaction phase.

    Authors: We agree that reporting only the aggregate 4–35 Myr range limits the reader’s ability to judge uniformity across the sample. In the revised manuscript we have added a new table (Table 2) that lists, for each of the nine galaxies, the time from binary binding to coalescence, the central stellar density at binding, and the stellar mass at that epoch. We also report the median merger time (∼15 Myr) and the 16th–84th percentile range. A short accompanying paragraph discusses the modest scatter, which correlates primarily with small differences in the central density profiles established during the compaction phase. These additions make the rapid-merger result quantitatively transparent while preserving the original conclusion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; merger timescales are direct simulation outputs

full rationale

The paper's central result—that SMBHs merge in ∼4-35 Myr after binary binding due to ρ⋆ ≳ 10^{13} M_⊙/kpc³—arises as an emergent numerical outcome from integrating the orbital dynamics in the KETJU hybrid scheme (regularized integration for massive SMBHs plus subgrid dynamical friction for lower-mass ones) applied to the compact cores formed in the zoom-in runs. No equation or step reduces by construction to its own inputs: the reported timescales are computed outputs, not fitted parameters or self-defined quantities. Self-citations to prior KETJU implementations describe the code framework but do not bear the load of the quantitative merger claim, which remains falsifiable against the simulated density profiles and external benchmarks. The subgrid friction prescription is an explicit modeling assumption whose validity is separate from circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results depend on standard cosmological assumptions and numerical subgrid models for black hole dynamics whose specific parameter values are calibrated from prior work but not re-derived here.

free parameters (2)
  • SMBH seeding masses
    Initial masses assigned to supermassive black hole seeds to allow subsequent growth to observed masses by z=5.
  • Subgrid dynamical friction coefficients
    Parameters in the model controlling frictional drag on lower-mass SMBHs during orbital decay.
axioms (2)
  • standard math Lambda-CDM cosmological model with standard parameters
    Forms the basis for the cosmological zoom-in simulation setup at high redshift.
  • domain assumption Accuracy of KETJU regularized integration and dynamical friction subgrid model for SMBH dynamics
    Enables tracking of SMBH evolution from seeding through bound binary phase to coalescence via gravitational wave emission.

pith-pipeline@v0.9.0 · 5926 in / 1577 out tokens · 111514 ms · 2026-05-21T18:10:24.478623+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Modelling the uv/x-ray cosmic background with CUBA

    Abac A. G., et al., 2025, ApJ, 993, L25 Abbott R., et al., 2023, Physical Review X, 13, 041039 Agazie G., et al., 2023, ApJ, 951, L9 Akins H. B., et al., 2023, ApJ, 956, 61 Akins H. B., et al., 2025a, ApJ, 980, L29 Akins H. B., et al., 2025b, ApJ, 991, 37 Amaro-Seoane P., Sesana A., Hoffman L., Benacquista M., Eich- horn C., Makino J., Spurzem R., 2010, M...

    work page internal anchor Pith review doi:10.48550/arxiv.astro-ph/0106018 2025

  2. [2]

    and the choice of code and integration parameters (Frenk et al. 1999). Therefore it is not too unexpected that the location of galaxies begins to deviate between the simu- lations with different SMBH dynamics modelling schemes. APPENDIX B: CHARACTERISTIC STRAIN CALCULA TION Following Amaro-Seoane et al. (2010) and Kelley et al. (2017b), the GW strain ampl...

    work page 1999

  3. [3]

    Note that we use equation B4 withfh =f r/n, while Berentzen et al

    hc,n(f) = p N(f)h s,n(f) = 37 √ 5 64 √ 6π2/3 (GM)5/6 c3/2d(z) p g(n, e)f −1/6, (B4) withMbeing the chirp mass of the binary,d(z)the comoving distance from the observer and g(n, e) = n4 32 Jn−2(ne)−2eJ n−1(ne) + 2 n Jn(ne) + 2eJn+1 −J n+2(ne) 2 + (1−e) 2 (Jn−2(ne)−2J n(ne) +J n+2(ne))2 + 4 3n2 J2 n(ne) (B5) is a function describing the relative contributio...

    work page 1963

  4. [4]

    MNRAS000, 1–29 (2025) 30A

    da dt =− 64 5 G3(m1 +m 2)m1m2 c5a3(1−e 2)7/2 1 + 73 24 e2 + 37 96 e4 (B7) and de dt =− 304 15 G3(m1 +m 2)m1m2 c5a4(1−e 2)5/2 1 + 121 304 e2 e.(B8) This paper has been typeset from a TEX/LATEX file prepared by the author. MNRAS000, 1–29 (2025) 30A. Keitaanranta et al. BA Zoom-K+DF BA Zoom-DF BA Zoom-G Figure A1.Snapshot showing the gas surface density of g...

    work page 2025