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Stellar feedback strength alone shifts supermassive black hole merger delays from ~30 to ~500 Myr in sub-Milky Way galaxies, a tenfold-plus spread driven by central stellar density.

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2026-08-02 03:42 UTC pith:NWCA2BOF

load-bearing objection New and credible evidence that stellar feedback strength alone can shift LISA-relevant SMBH merger delays by an order of magnitude in low-mass galaxies; the causal chain has one loose knot at t_hard, but the paper deserves a real referee. the 3 major comments →

arxiv 2607.13786 v1 pith:NWCA2BOF submitted 2026-07-15 astro-ph.GA

RABBITS IV: Stellar feedback and SMBH merging time-scales in the sub-Milky Way mass regime

Ruby J. Wright , Roosa Heiskanen , Shihong Liao , Alexander Rawlings , Peter H. Johansson , Max Mattero , Fiona H. Panther , Atte Keitaanranta This is my paper
classification astro-ph.GA
keywords supermassive black hole binariesstellar feedbacksupernova feedbackgalaxy mergersmerger time-scalesLISAstellar hardeninggravitational waves
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that in low-mass galaxy mergers, plausible variation in stellar feedback—here parameterized by the supernova outflow velocity, changed by a factor of about four—translates into an order-of-magnitude uncertainty in how quickly supermassive black holes merge after their galaxies collide. Sixteen equal-mass merger simulations, which follow the black hole binary through the gravitational-wave inspiral, yield post-hardening merger delays of roughly 30 to 500 million years, with the strongest feedback producing the longest waits. The channel is that vigorous feedback suppresses nuclear star formation, so the remnant has a lower central stellar density, and the binary loses orbital energy to stars more slowly. The same simulations show that the classic stellar-dynamical hardening formula, evaluated with the density and velocity dispersion at the binary's sphere of influence, reproduces the measured merger times, and that these nuclear properties can be extrapolated from scales up to about one hundred influence radii with useful accuracy. If this holds, cosmological simulations that cannot resolve parsec-scale dynamics could still estimate merger delays, and stellar feedback becomes a leading uncertainty for predicting LISA event rates.

Core claim

Using the KETJU code, the authors run sixteen idealized equal-mass mergers of gas-rich disc galaxies below 10^10 M_sun, each hosting a non-accreting ~7.5x10^6 M_sun black hole. They vary only the supernova outflow velocity (2828, 4000, 5657, 8000 km/s), spanning a factor of four in energy while keeping galaxies within observed scaling relations. The central finding is a systematic progression: average post-hardening merger time-scales grow from ~31 Myr in the weakest-feedback runs to ~357 Myr in the strongest, with individual realizations from ~20 to ~530 Myr. The physical chain is that stronger feedback vents merger-driven gas from the nucleus and suppresses central star formation, lowering

What carries the argument

The load-bearing object is the KETJU software extension of the GADGET-3 code: it replaces softened SMBH dynamics with the algorithmically regularized MSTAR integrator, so the binary interacts with individual stars without softening and includes post-Newtonian corrections to 3.5PN order. The analytic machinery is the stellar-dynamical hardening law, which states that the binary's hardening rate s = d/dt(1/a) equals H G rho_star / sigma_star, with H ~ 14.55 in an efficiently replenished loss cone. Combined with Peters' gravitational-wave inspiral term, this yields an explicit merger time T_m that scales as rho(R_infl)^(-4/5) sigma(R_infl)^(4/5) M_bin^(-3/5) F(e)^(-1/5), where R_infl is the inf

Load-bearing premise

The trend and the extrapolation accuracy both rest on the assumption that the four chosen supernova outflow velocities faithfully bracket the true range of stellar feedback, and that the unresolved inner density profile is adequately represented by a softened power law with a fixed core radius of 0.25 times the influence radius.

What would settle it

Run the same merger suite with an independent stellar feedback implementation and check whether the central stellar density at the influence radius—and the resulting 30–500 Myr spread—is reproduced; or, observationally, if high-resolution nuclear density measurements of post-merger galaxies at M_star ~ 10^10 M_sun show no scatter correlated with feedback proxies, the proposed mechanism would be in doubt.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • SMBH merger-delay distributions inherit a >10x spread purely from plausible stellar-feedback uncertainty; LISA population forecasts that fix one feedback model are likely overconfident.
  • In these low-mass, gas-rich remnants, the post-hardening environment is stellar-dominated, so stellar-only hardening prescriptions are adequate once the nuclear stellar density is known.
  • Cosmological simulations with ~100 pc resolution, where the sphere of influence is unresolved, can recover SMBH merger times to ~0.2 dex by extrapolating fitted outer density profiles inward.
  • The quantity that sets the delay is the central stellar density at hardening, which is imprinted by feedback during the merger itself—so galaxy-formation physics and SMBH binary dynamics cannot be treated independently.

Where Pith is reading between the lines

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

  • One testable extension is to repeat the extrapolation exercise with a physically motivated core radius calibrated to observed nuclear profiles, rather than the fixed 0.25 R_infl—this would show how much of the quoted ~0.2 dex accuracy is assumption.
  • If the feedback-delay trend persists for unequal-mass mergers, the LISA-detectable population could skew to even longer delays, pushing more events to lower redshift than current catalogs assume.
  • Feedback-regulated delays should also shape the nanohertz stochastic gravitational-wave background: longer delays mean fewer actively inspiralling binaries, raising the expected sky anisotropy—a signature the next round of pulsar-timing-array anisotropy searches could confront.
  • The strong correlation between merger time and extrapolated central density suggests a practical subgrid calibration recipe: train delay prescriptions on resolved KETJU mergers across feedback strengths instead of adopting a constant post-coalescence delay.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents 16 idealised equal-mass galaxy merger simulations with the KETJU code, varying supernova feedback strength via the SN outflow velocity (2828–8000 km/s, four values × four realisations), for low-mass galaxies (M* < 10^10 Msun). The authors track SMBH binary evolution from kpc scales through dynamical friction, stellar hardening, and GW-driven inspiral with post-Newtonian terms. They report post-hardening merger time-scales of ~30–500 Myr, increasing systematically with feedback strength, and attribute this to feedback suppressing nuclear star formation and lowering central stellar densities. They further test the Sesana & Khan (2015) analytic merger-time prescription against their simulations, finding good agreement with no adjusted parameters, and show that merger times can be recovered to ~0.2 dex accuracy when central stellar properties are extrapolated from scales up to ~100 R_infl. The paper concludes that stellar feedback uncertainty alone can cause order-of-magnitude variations in SMBH merger delays, with implications for LISA and PTA predictions.

Significance. If the central causal claim holds, this is an important result: it demonstrates that a plausible variation in sub-grid stellar feedback—within a range that still yields galaxies consistent with observed scaling relations—changes post-hardening SMBH merger delays by more than an order of magnitude in the low-mass regime most relevant to LISA. The strength of the paper is its systematic approach: four feedback strengths, four realisations each, a direct KETJU treatment of the binary, and a parameter-free comparison to the Sesana & Khan (2015) and Sesana et al. (2006) analytic frameworks. The quantitative agreement with those frameworks over two orders of magnitude in T_m is a strong, falsifiable test. The extrapolation experiment in Section 4.4 is also a constructive step toward sub-grid prescriptions in cosmological simulations. However, the causal interpretation (feedback → lower density → longer T_m) is the load-bearing part of the abstract, and it is currently supported by density profiles measured only at feedback-dependent epochs; this needs a direct fixed-epoch test to rule out a partly reversed causal arrow.

major comments (3)
  1. [§4.2, Fig. 9] The central claim—'stronger stellar feedback produces longer merger delays through its impact on the central stellar density'—is supported by Fig. 9, which compares density profiles at t≈t_hard. But t_hard increases systematically with feedback strength (Fig. 8, Table 4), so strong-feedback binaries have had substantially more time to scatter and eject stars before the density is measured. The influence radii in these remnants are only ~6–10 pc, comparable to the KETJU regularized region, so binary back-reaction could plausibly lower the density at exactly the radius used in the analysis. The statement in §4.3 that the divergence is 'imprinted prior to the SMBH binary phase' is not directly demonstrated; Fig. 7 shows gas/SFR evolution, not stellar density at a fixed epoch. Please provide a direct fixed-epoch stellar density comparison (e.g., at a common time after coalescence, or at t=2.
  2. [§4.4, Fig. 14, footnote 7] The extrapolation experiment assumes a universal core radius R_core = 0.25 R_infl for all simulated remnants. The claim that T_m is recovered to ~0.2 dex at ~100 R_infl (Fig. 14, top panel) may be sensitive to this ad hoc choice; no sensitivity test is shown. Since this section is presented as a practical route for cosmological simulations, please either (i) vary the core factor (e.g., 0.1–0.5 R_infl) and show the effect on the recovered T_m, or (ii) provide a physical justification for the universality of R_core = 0.25 R_infl based on the simulated profiles. Without this, the robustness of the extrapolation claim is unclear.
  3. [§4.3/§5.4] The analytic comparison in Figs. 12 and 13 uses stellar properties measured at the same time as the hardening rate. The good agreement with Sesana & Khan (2015) may be partly self-consistent: if the binary modifies the density at R_infl while hardening, then measuring both the hardening rate and the density at the same epoch could yield an apparent correlation that does not fully validate the predictive use of the framework. The authors note this concern indirectly in §5.4 (and the framework is not claimed universal), but a quantitative check—e.g., comparing hardening rates at early times against initial densities, or using densities measured before hardening—would strengthen the predictive claim.
minor comments (6)
  1. [§2.1.3] The phrase 'AGB feedback is implemented using the same temporal framework as SNIa feedback' may confuse AGB mass loss with supernova events; consider clarifying that only the temporal sampling is analogous.
  2. [§4.1, Table 3] Table 3 lists mean/min/max dynamical-friction delays, but the text says 'the duration of time spent in the dynamical friction phase is presented'; consider indicating that only summary statistics are shown.
  3. [§4.3, end of first paragraph] Typo: 'increases by approximately an of magnitude' should read 'an order of magnitude'.
  4. [§4.3, Eq. (17) region] 'dynamical ou bli ette' appears to be a typo for 'dynamical oubliette'.
  5. [§4.4, Fig. 14] The eccentricity at t_hard is taken directly from the simulation, which is not available in a purely cosmological context. This is acknowledged in the text, but it should be explicitly restated in the caption of Fig. 14 and in the conclusions, since it is a key input to the extrapolation framework.
  6. [§5.2] The discussion of PTA anisotropy is somewhat tangential; the connection between merger-delay distributions and SGWB anisotropy could be tightened, but it is not a blocking issue.

Circularity Check

1 steps flagged

Central analytic comparison is independent; mild oracle-input contamination in the Section 4.4 extrapolation experiment.

specific steps
  1. fitted input called prediction [Section 4.4 and footnote 7 (Fig. 14, Fig. A1)]
    "The choice to add a core avoids a slight overestimation of ρ(R_infl) from extrapolation of average 0.05−0.1 dex due to the inner flattening of the density profiles in our sample of remnants – this is shown in Fig. A1. ... The notable exception to this is the eccentricity of the binary, which we directly take from the simulation at t≈t_hard."

    The extrapolation experiment's 'recovery' of merger times from scales up to ~100 R_infl is partially aided by inputs taken directly from the same KETJU simulations: R_core is set to 0.25 R_infl based on the inner flattening measured in the same remnant sample, R_infl itself is a measured simulation quantity, and e_hard is taken directly from the simulated binary. The core prescription is thus calibrated to the very nuclear structure being predicted, so the quoted ~0.2 dex accuracy is a calibrated interpolation rather than a fully out-of-sample prediction. This does not affect the main analytic comparison (Figs. 12–13), where H=14.55 and the Sesana & Khan (2015) framework are external and no T_m value is fitted.

full rationale

The paper's principal derivation chain is not circular. The post-hardening merger times in Table 4 are measured directly from KETJU simulations, and the comparison with stellar-dynamical predictions uses the externally calibrated Sesana et al. (2006) hardening coefficient H=14.55 and the Sesana & Khan (2015) two-regime formula, with no parameter fit to the simulated T_m values. The stellar density and velocity dispersion are measured from the same simulations, but that is a consistency test, not a constructional reduction of the prediction to its inputs. The causal feedback-to-density-to-delay chain is supported by independent SFR and gas-evolution evidence (Fig. 7) and by direct density profiles (Fig. 9). The only mild contamination is the Section 4.4 extrapolation experiment, where the fixed core radius R_core=0.25 R_infl is chosen to correct for the known inner flattening of the same remnant sample, and R_infl plus e_hard are taken from the simulations; this makes the extrapolated 'recovery' partly calibrated to the target systems. However, T_m,KETJU is not used as a fit target, and the extrapolation result is clearly labeled as a benchmark test rather than a stand-alone prediction. The self-citations (Liao et al. 2023, 2024a,b; Keitaanranta et al. 2026) provide code and prior context but do not carry the central derivation. Overall circularity is low: score 2 reflects only the mild oracle-input aspect of the extrapolation experiment, not the main results.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central quantitative claims rest on well-established stellar-dynamical formulas and publicly documented simulation codes. The genuinely 'paid-for' inputs are the physically motivated but hand-chosen feedback parameters, the initial SMBH mass, and the ad hoc core prescription in the extrapolation experiment. No new entities are postulated.

free parameters (3)
  • Supernova outflow velocity v_SN = 2828, 4000, 5657, 8000 km/s
    Independent variable chosen by hand to span the physically plausible range; the resulting feedback energetics differ by factors of 0.5-4 relative to the fiducial model.
  • Extrapolation core radius factor = R_core = 0.25 R_infl
    Hand-chosen in Section 4.4/footnote 7 to avoid a known overestimation bias in the recovered central densities; the robustness of the 100 R_infl recovery statement to this choice is not systematically tested.
  • Initial SMBH seed mass = 7.53e6 M_sun
    Chosen in Section 2.3.1 so that merger remnants sit on the local M_BH-sigma relation, compensating for the lack of BH accretion; it sets R_infl and thereby affects both predicted and measured merger delays.
axioms (6)
  • standard math Binary hardening rate s = H G rho/sigma with H=14.55, and the Sesana & Khan (2015) two-phase inspiral formula
    Used in Section 4.3 to interpret and predict the simulated merger times; coefficients are adopted from the cited literature, not fit here.
  • domain assumption Equal-mass, gas-rich disc mergers on the G5 retrograde orbit with local-universe initial conditions and no cosmological gas replenishment
    Adopted in Section 2.3.2 and acknowledged in Section 5.4.1; restricts generalizability to other orbital configurations, mass ratios, and redshifts.
  • domain assumption SMBH accretion and AGN feedback are negligible in this mass regime
    Stated in Sections 2.2 and 5.4.2; if wrong, gas discs and AGN outflows could alter the nuclear stellar density and hardening evolution.
  • domain assumption The Nunez et al. (2017) subgrid stellar feedback implementation and the chosen v_SN variation faithfully represent the feedback physics relevant to nuclear star formation
    The whole feedback-strength experiment is built on this; the paper argues the values are physically plausible but cannot validate them against resolved feedback simulations or observations beyond broad consistency with scaling relations.
  • domain assumption KETJU's regularized integration and post-Newtonian terms resolve SMBH binary evolution without significant numerical artifacts from the 2.5 pc stellar softening
    Requires exact-solver treatment of SMBH-star scattering; star-star softening and particle masses set the effective resolving scale of the stellar background.
  • ad hoc to paper R_core = 0.25 R_infl is an acceptable universal core prescription for extrapolating unresolved density profiles
    Introduced in Section 4.4; without it the extrapolated densities are biased high, so the recovery claim depends on this choice.

pith-pipeline@v1.3.0-alltime-deepseek · 36871 in / 14443 out tokens · 151017 ms · 2026-08-02T03:42:23.331563+00:00 · methodology

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read the original abstract

Merging supermassive black holes (SMBHs) in low- and intermediate-mass galaxies are important sources for future millihertz gravitational-wave observatories such as LISA. Predicting the delay between galaxy coalescence and SMBH merger is therefore critical for modelling the observable SMBH merger population. Using the KETJU code, we perform 16 equal-mass galaxy merger simulations as part of the Resolving supermAssive Black hole Binaries In galacTic hydrodynamical Simulations (RABBITS) series to investigate SMBH binary evolution in galaxies with stellar masses below $M_{\star}\lesssim10^{10}\,{\rm M}_{\odot}$. We systematically vary the strength of stellar feedback by altering the supernova outflow velocity by a factor of $\sim4$, while still producing galaxies consistent with observed scaling relations. We find post-hardening SMBH merger time-scales spanning $\sim30$-$500\,{\rm Myr}$, with stronger stellar feedback producing systematically longer merger delays through its impact on the central stellar density of the merger remnants. Across our suite, merging time-scales vary by more than an order of magnitude, demonstrating that uncertainties in stellar feedback alone can translate into large uncertainties in SMBH merger delays. At the onset of hardening, the binary evolution remains consistent with stellar-dynamical hardening models based on the local stellar density and velocity dispersion near the binary sphere of influence. Using KETJU as a benchmark, we show that merging time-scales can be recovered with useful accuracy when these nuclear stellar properties are extrapolated from scales up to $\sim 100\,R_{\rm infl}$. These results provide a promising route for modelling SMBH mergers in cosmological simulations.

Figures

Figures reproduced from arXiv: 2607.13786 by Alexander Rawlings, Atte Keitaanranta, Fiona H. Panther, Max Mattero, Peter H. Johansson, Roosa Heiskanen, Ruby J. Wright, Shihong Liao.

Figure 1
Figure 1. Figure 1: Initial condition profiles for each progenitor galaxy. The top panel illustrates the 3D radial density profile, while the bottom panel shows the enclosed cumulative mass profile for each component. Dark matter is illus￾trated in grey, the stellar disc in yellow, and the gas in teal/green. The gaseous component is further broken down into contributions from the disc (dashed line) and the halo (dotted line).… view at source ↗
Figure 2
Figure 2. Figure 2: A visualisation of the orbit and merger configuration for one of the fiducial 4000 km s−1 realisations. This is an indicative visualisation of the merger that provides an overview of the merger geometry, with gas density indicated with the purple–orange colourmap, and stellar density is shown in pale yellow/cream. Descriptions of the merger phase, relative separation, and relative velocities of the SMBHs a… view at source ↗
Figure 3
Figure 3. Figure 3: Gas surface density (top) and star formation rate surface density (bottom) of simulations with varying supernova feedback strength at 𝑡 = 500 Myr. Columns show runs with supernova outflow velocities of 𝑣SN = 2828, 4000, 5657, 8000 km s−1 (left to right), corresponding to increasing energy injection. The top two rows show face-on and edge-on projections of the gas surface density Σgas, while the bottom two … view at source ↗
Figure 4
Figure 4. Figure 4: Gas phase diagrams stacked over all realisations for a given supernova feedback strength, increasing from left to right. Each panel shows the distribution of gas in temperature–density space after 500 Myr of simulation time. Each panel is colour-coded by the radial velocity of the gas, relative to the nearest SMBH. Columns correspond to supernova outflow velocities 𝑣SN = 2828, 4000, 5657, 8000 km s−1 (left… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the progenitor and remnant galaxies to various observations in the literature – namely (A) the stellar mass – halo mass relation, (B) the stellar mass – size relation, (C) the specific star formation rate main sequence, (D) the stellar mass – cold gas mass relation, (E) the stellar mass – stellar metallicity relation, (F) mass loadings as a function of stellar mass, (G) the stellar mass – bla… view at source ↗
Figure 6
Figure 6. Figure 6: Multi-scale evolution of the SMBH separation through 2.5 Gyr of simulation time. The separation of SMBHs is almost identical between runs (as expected) up until just prior to galaxy coalescence, where the differences in feedback strength and associated differences in dynamical friction influence the rate at which the SMBHs sink to become a hard binary. Times of binary coalescence vary from simulation times… view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the star formation rate (top panel), total gas mass (middle panel), and the fraction of gas located more than 5 kpc from either SMBH (bottom panel) for the merger simulations with different stellar feedback strengths, parameterised by the supernova wind velocity 𝑣SN. Coloured lines indicate the four feedback models, while transparent curves show different orbital realisations. Thin and thick l… view at source ↗
Figure 8
Figure 8. Figure 8: Stellar surface density projections at 𝑡 ≈ 𝑡hard (the time at which the semi-major axis drops below 𝑅hard) for individual realisations spanning a range of feedback strengths. Each panel shows a single merger remnant, with increasing 𝑣SN from left to right. All systems exhibit the characteristic features of a recent major merger, including a compact central core and extended tidal debris. However, with incr… view at source ↗
Figure 9
Figure 9. Figure 9: Stellar mass density profiles centered on the binary black hole center-of-mass at 𝑡 ≈ 𝑡hard for different feedback strengths. Profiles are con￾structed by depositing stellar mass into logarithmically spaced spherical shells using a three-dimensional top-hat kernel with radius equal to the stellar grav￾itational softening, 𝜖 = 2.5 pc. Thick lines show the mean profile across mul￾tiple realisations, with thi… view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of the SMBH binary semi-major axis (top panel) and eccentricity (bottom panel) following the onset of stellar hardening (𝑡hard) for merger remnants with different stellar feedback strengths. Thick coloured lines show individual orbital realisations, while horizontal markers indicate the mean merger time-scale ⟨𝑇m⟩ and the full range across realisations for each feedback model. Thin horizontal da… view at source ↗
Figure 11
Figure 11. Figure 11: Gas mass enclosed within the SMBH binary sphere of influence as a function of supernova feedback strength. Diamonds show the mean gas mass enclosed within 𝑅infl at the onset of the stellar hardening phase, averaged over the merger realisations for each feedback model, and circles show the mean gas mass enclosed within 𝑅infl after 𝑡 = 2.5 Gyr of simulation time. The enclosed gas mass is calculated using th… view at source ↗
Figure 12
Figure 12. Figure 12: Hardening rate, 𝑠 ≡ 𝑑/𝑑𝑡 (1/𝑎), measured near the onset of stellar hardening (𝑡 ≈ 𝑡hard), as a function of 𝐺𝜌★(𝑅infl )/𝜎★(𝑅infl ), where 𝜌★(𝑅infl ) is the stellar density at the binary influence radius and 𝜎★(𝑅infl ) is the one-dimensional stellar velocity dispersion measured within 𝑅infl. Colours indicate the stellar feedback strength, parameterised by the super￾nova wind injection velocity 𝑣SN, while ma… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison between the SMBH binary merger times measured in the KETJU simulations and the merger time-scales predicted using the Sesana & Khan (2015) prescription. Predictions are computed using the stellar density at the binary influence radius, 𝜌★(𝑅infl ), one-dimensional stellar velocity dispersion, 𝜎★(𝑅infl ), and the binary eccentricity at the onset of hardening, 𝑒hard. Colours indicate the stellar f… view at source ↗
Figure 14
Figure 14. Figure 14: Testing the recoverability of SMBH binary merger time-scales from progressively larger spatial scales. The top panel shows the median absolute offset between the merger times predicted using the Sesana & Khan (2015) prescription and the merger times measured directly in the KETJU simulations, while the remaining panels show the signed offset in merger time, the inferred central stellar density and velocit… view at source ↗

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