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Probing the Merger Rates of Supermassive Black Holes and Galaxies with Gravitational Waves

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Future LISA detections can recover galaxy and supermassive-black-hole merger rates, but only when the catalog reaches roughly forty events; with only a dozen events, high-redshift rates come out biased.

desk verdict Solid LISA-era mock-forecast paper; the unity detection fraction and a thin event-count basis are the main caveats, but the core results are credible. read the letter →

arxiv 2501.02748 v2 pith:EHK6JNUC submitted 2025-01-06 astro-ph.GA

Yun Fang , Rong-Gen Cai This is my paper · ORCID
classification astro-ph.GA
keywords gravitationalwavessupermassiveblackholesgalaxymergerratepulsartimingarraysLISAstochasticwavebackgroundhierarchicalBayesianinferenceholescalingrelations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that gravitational-wave detections of supermassive-black-hole mergers can be turned into direct measurements of how often galaxies and their central black holes merge, complementing surveys of galaxy pairs. The authors build a mock LISA catalog, add a PTA stochastic-background constraint, and fit a parameterized galaxy merger rate through the $M_\bullet$-$M_*$ relation; they find that the number of detected events and the joint distribution of binary mass and redshift decide how well the rates are recovered. A catalog of roughly forty events reproduces the galaxy merger rate from galaxy-pair observations, while a catalog of twelve events biases the high-redshift behavior. The same data constrain the delay between galaxy and black-hole mergers, and they show that for delays above about 0.5–0.8 Gyr, accretion rather than mergers becomes the dominant channel of black-hole mass growth at high redshift.

What carries the argument

The load-bearing machinery is a convolution chain from galaxy mergers to black-hole mergers: the galaxy stellar mass function $\Phi_{\mathrm{GSMF}}(M_*, z)$ times a parameterized merger rate per galaxy $R_{\mathrm{Gal}}(M_*, z)$ gives the galaxy merger rate, and a log-normal $M_\bullet$-$M_*$ relation with redshift-dependent parameters maps it to a SMBH mass function. A delay-time kernel $P_{\mathrm{delay}}(\tau)$ shifts the SMBH merger events to later lookback times, and an occupation fraction $f_{\mathrm{occ}}$ scales the number of host galaxies that actually contain a black hole. These rates enter an inhomogeneous Poisson hierarchical-Bayesian likelihood that compares predicted counts and the joint mass–redshift distribution to mock LISA events, while the same population feeds a power-law SGWB integral constrained by PTA strain. The paper sets LISA's detection fraction to unity, justified by the high signal-to-noise ratios of SMBH mergers in the considered range.

What would settle it

Once real LISA data exist, compare the galaxy merger rate recovered from a catalog of roughly forty events against independent galaxy-pair measurements at $z>3$; if the forty-event catalog disagrees with galaxy-pair rates while a smaller catalog agrees, the central claim is falsified. A simulation with a realistic signal-to-noise-dependent LISA selection function would settle the same point before launch.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the constraining power of future LISA observations resides in the number of detected SMBH mergers and in their joint distribution in binary mass and redshift. Using two mock realizations of 36 and 12 events, the hierarchical Bayesian recovery shows that the 36-event catalog reproduces the galaxy merger rate inferred from galaxy-pair observations, while the 12-event catalog misses the high-redshift behavior and biases parameters such as $n_0$, $\alpha_1$, and $\beta$ outside their $1\sigma$ credible regions. Adding a PTA stochastic-background amplitude tightens the posteriors, especially for the normalization $n_0$ and the delay time. For a fixed constant delay, the reconstructed SMBH mass assembly from mergers is suppressed at high redshift as $\tau$ grows, with accretion becoming the dominant growth channel beyond $z\sim 6$ for $\tau\gtrsim 0.5$ Gyr and beyond $z\sim 4$ for $\tau\gtrsim 0.8$ Gyr. The occupation fraction of SMBHs at $z>3$ is not recoverable, because it is degenerate with the delay time and the merger-rate parameters.

Load-bearing premise

The forecast assumes LISA detects every SMBH merger in the mass and redshift range considered, so if real selection effects remove a significant fraction of low-mass or high-redshift events, the claimed constraining power of roughly forty detections would be optimistic.

Editorial extensions

If this is right

  • A real LISA catalog of roughly forty SMBH mergers should reproduce the galaxy merger rate inferred from galaxy-pair observations, including the high-redshift end.
  • A catalog of twelve or fewer events will not resolve the high-redshift merger rate; using it as a definitive measurement would inject systematic bias.
  • PTA measurements of the stochastic gravitational-wave background will tighten the parameters inferred from LISA alone, even though PTAs sample a different mass and redshift window.
  • If the true delay between galaxy and black-hole merger is longer than roughly 0.5–0.8 Gyr, merger-driven mass growth of SMBHs falls below accretion beyond $z\sim 6$ to $z\sim 4$.
  • The SMBH occupation fraction at $z>3$ cannot be pinned down by this data combination on its own; it stays degenerate with delay time and the galaxy merger rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A straightforward extension would rerun the recovery with a mass- and redshift-dependent LISA selection function; the paper's unity-detection assumption means the effective event count, not the raw detection count, is what should be compared with the roughly forty-event threshold.
  • If the local $M_\bullet$-$M_*$ relation holds at all redshifts rather than the near-infrared-motivated relation used here, the recovered high-redshift merger rates shift systematically upward, so independent constraints on that relation at $z>4$ will directly set the normalization of GW-inferred merger rates.
  • Relaxing the constant-delay assumption to a distribution of delay times, which the paper notes is computationally expensive, may widen the delay posteriors and make the occupation-fraction degeneracy even harder to break; testing this before LISA flies would clarify how much of the reported $f_3$ degeneracy is due to the constant-delay simplification.
  • The same hierarchical Poisson framework could be applied to a future catalog of individually resolved PTA binaries, giving a lower-redshift cross-check of the LISA-based rates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents a hierarchical Bayesian framework for inferring galaxy and SMBH merger rates from future LISA detections combined with current PTA constraints on the stochastic gravitational-wave background. The SMBH merger rate is constructed from the galaxy merger rate per galaxy, the galaxy stellar mass function, a redshift-dependent M_BH–M_* relation, and a delay time between galaxy and SMBH mergers. The authors generate mock LISA event catalogs with 36 and 12 events, and recover the hyperparameters of two galaxy-merger-rate parameterizations, with and without PTA data. They report that the event count and the joint mass–redshift distribution drive constraining power, that a 36-event catalog recovers merger rates consistent with galaxy-pair observations while a 12-event catalog produces high-redshift biases, that PTA data sharpen the constraints, that a constant delay time can be recovered, and that the SMBH occupation fraction at z>3 is poorly constrained.

Significance. If the result holds, the paper provides a useful forecast for LISA planning: roughly forty well-characterized SMBH merger events, combined with PTA data, may be sufficient to cross-check galaxy merger rates out to high redshift. The framework is standard and the injection–recovery tests are internally consistent; the authors explicitly report the imperfect recovery in the 12-event realization and in the tau=0.8 Gyr case, which is a sign of honesty. The paper also clearly labels its analysis as a mock-data self-test, so I see no circularity problem. The main significance is quantitative guidance on event counts and on the combination of LISA and PTA data, not a new astrophysical measurement.

major comments (3)
  1. [Section 4.1, Eq. (13) and Fig. 2] The detection fraction xi(Lambda) is set to unity in Eq. (13) on the grounds that SMBH mergers in the considered mass and redshift ranges have large signal-to-noise ratios. However, the mock catalogs in Fig. 2 extend down to M_bullet about 1e5 M_sun and out to z about 10, and the paper provides no SNR distribution or detection-threshold calculation for these events. If a realistic LISA selection function removes a non-negligible fraction of low-mass or high-redshift events, the likelihood should use N_exp(Lambda) in both the Poisson count factor and the per-event normalization, and the effective event count would be smaller than the injected value. Because the paper's central claim is the contrast between 36 and 12 detected events, an unquantified selection function is load-bearing; the authors should add a recovery test with a fiducial LISA sensitivity and an explicit SNR threshold (for example SNR>8) and show how the posterior widths and the high-redshift biases change.
  2. [Section 5.1.3 and Abstract] The headline statement that datasets with around forty events yield results consistent with galaxy-pair observations is based on a single 36-event realization and a single 12-event realization. Two realizations do not establish a threshold; the Poisson scatter in the 36-event result is not quantified, and the claimed contrast could be driven by the particular random draws. The repeated tests mentioned in Section 5.1.4 vary merger-rate parameters but not the event count. The authors should include several realizations at intermediate event counts (for example 20, 30, and 40) and report the distribution of recovery biases and credible-interval coverage.
  3. [Section 5.2] The delay-time inference assumes a delta-function delay distribution, and the recovery shown in Fig. 7 is a test of recovering a point delay under that same assumption. The abstract's statement that the method 'effectively constrains the delay time' is therefore stronger than what is demonstrated: the method has not been tested on, for example, a power-law or Gaussian delay distribution, which the authors themselves note in Section 5.2. The abstract and conclusion should be reworded to say that a constant delay time is assumed in the analysis.
minor comments (4)
  1. [Abstract] The sentence 'According to our mock analysis, the models with delay times longer than 0.5Gyr (0.8Gyr), accretion becomes the primary driver of SMBH mass growth beyond z~6 (4)' is grammatically incomplete; it should read 'for models with delay times longer than 0.5 Gyr (0.8 Gyr), accretion becomes...'.
  2. [Section 5.1.2] The expression 'a= log10 kappa + 9 - 11 * b' uses a nonstandard asterisk and should be written as a = log10 kappa + 9 - 11 b. Also, the numerical values in Table A2 for the z<4 relation (for example a=-5.36, b=1.28 for GW eg1) differ from the central values quoted in the text (b=1.17 +/- 0.08); please clarify that the table entries are random draws from the observational priors rather than the central values.
  3. [Section 4.1, Eq. (14)] The default prior P_emptyset(theta) in Eq. (14) is described only as 'usually set to a uniform distribution'; please specify the explicit ranges used for the Monte Carlo averages over total mass and redshift so that the analysis is reproducible.
  4. [Section 5.3] In Section 5.3, the text says 'accretion generally dominants the mass assembly'; this should be 'accretion generally dominates the mass assembly'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the forecasts are explicitly mock-data injection–recovery exercises with an external PTA constraint; headline 'predictions' are labeled as mock analysis and follow from injected parameters rather than from circular definitions.

full rationale

The derivation chain is self-contained: Eq. (4) maps galaxy merger rates, delay-time distributions, occupation fraction, GSMFs and M•–M* relations into an SMBH merger rate; Eqs. (7)–(12) convert that rate into a PTA SGWB strain; Eqs. (13)–(17) define a hierarchical Bayesian likelihood for LISA events. Mock LISA catalogs are generated from specified injected parameter values (n0 = 0.03, α0 = 0.2, α1 = −0.01, β = 2.4 and chosen delay times) and then fed through the same likelihood, so the recovered posteriors and the statements about 'around forty events' are injection-recovery calibration results, not empirical measurements. This is a legitimate way to forecast constraining power, and the paper is explicit that these are mock data. The PTA branch uses the actual NANOGrav amplitude as an external constraint. The high-redshift accretion-versus-merger statements are conditional consequences of the injected delay times, explicitly framed as 'according to our mock analysis.' The self-citation to Fang & Yang (2023) for Eq. (4) and for earlier delay-time work is not load-bearing: Eq. (4) is a standard convolution formula and the cited prior work is used as a modeling reference, not as a uniqueness theorem that forces the conclusion. The unity detection fraction in §4.1 is a simplifying assumption with no fitted input renamed as a prediction; it is a robustness/correctness concern, not a circular step. No equation is defined in terms of the quantity it is claimed to predict, and no fitted parameter is relabeled as an independent prediction. Hence the paper shows no significant circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The framework rests on parameterized galaxy merger rate models, externally measured GSMFs and M_bullet-M_star relations, and simplifying assumptions (unit detection fraction, constant delay, q=0.6). None of these are derived in the paper; the inferred hyperparameters are targets of the mock analysis rather than discoveries.

free parameters (7)
  • n0 = 0.03 Gyr^-1 (injected)
    Normalization of the galaxy merger rate per galaxy, Eq. (18); inferred from mock LISA data and showing degeneracy in Case 1.
  • alpha0 = 0.2 (injected)
    Power-law index of the merger rate with stellar mass, Eq. (18); inferred.
  • alpha1 = -0.01 (injected)
    Redshift-dependent mass slope in Eq. (18); inferred.
  • beta = 2.4 (injected)
    Power-law index in (1+z) in Eq. (18); inferred.
  • tau = 0.2, 0.5, 0.8, 1.0 Gyr in different mock sets
    Constant delay time between galaxy merger and SMBH merger; inferred in Sec 5.2; an assumed delta-function distribution.
  • f3 = 0.46 (injected)
    Occupation fraction parameter at z>=3, Eq. (23); poorly constrained due to degeneracy.
  • q_bullet = 0.6
    Assumed average SMBH mass ratio in Eq. (22) for mass assembly; chosen for simplicity, not fitted.
assumptions (6)
  • standard math GW strain from SMBH binaries follows the standard quadrupole formula with circular orbits and pure GW-driven evolution (Eqs. 7-10).
    Standard general relativity background; not the paper's contribution.
  • domain assumption The merger rate of SMBH binaries equals the galaxy merger rate mapped through the M_bullet-M_star relation, occupation fraction, and a delay distribution (Eq. 4).
    Central mapping connecting galaxies to SMBH mergers; Section 2.
  • domain assumption The M_bullet-M_star relation of Kormendy & Ho (2013) at z<4 and Pacucci & Loeb (2024) at z>=4 is correct.
    Section 5.1.2; Appendix B shows that using a different (local-only) relation changes the recovered merger rates.
  • ad hoc to paper Detection fraction xi = 1 (Section 4.1).
    Assumed for simplicity; ignores LISA's actual selection function.
  • ad hoc to paper Delay time is a constant delta function at tau (Section 5.2).
    Authors acknowledge computational limitations; more realistic distributions are left for future work.
  • domain assumption Galaxy merger rate per galaxy follows Eq. (18) parameterization, with Case 2 approximating Illustris.
    Parameterized form; the true merger rate is assumed to be captured by this functional family.

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Pith. "Pith review of Probing the Merger Rates of Supermassive Black Holes and Galaxies with Gravitational Waves." pith.science (2026). https://pith.science/paper/EHK6JNUC

@misc{pith2026250102748,
  author       = {Pith},
  title        = {Pith review of: Probing the Merger Rates of Supermassive Black Holes and Galaxies with Gravitational Waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHK6JNUC}},
  note         = {Machine review of arXiv:2501.02748}
}
abstract

The mergers of galaxies and supermassive black holes (SMBHs) are key drivers of galaxy evolution, contributing to the growth of both galaxies and their central black holes. Current and upcoming gravitational wave (GW) detectors -- Pulsar Timing Arrays (PTAs), LISA, Taiji, and Tianqin -- offer unique access to these processes by observing GW signals from SMBH binaries. We present a framework to infer galaxy and SMBH merger rates by combining mock LISA detections of SMBH mergers with PTA constraints on the stochastic GW background, while incorporating observational uncertainties in stellar mass functions and $M_\bullet$-$M_*$ relations. We find that the number of LISA-detected events and their joint distribution in mass and redshift are key to constraining merger rates -- datasets with around forty events yield results consistent with galaxy pair observations, whereas limited event counts lead to biases at high redshift. Including PTA data further reduces parameter uncertainties. Our method also effectively constrains the delay time between galaxy and SMBH mergers, with longer delays suppressing high-redshift SMBH merger rates and shifting mass growth from mergers to accretion. According to our mock analysis, the models with delay times longer than $0.5\text{Gyr}$ ($0.8\text{Gyr}$), accretion becomes the primary driver of SMBH mass growth beyond $z \sim 6$ ($4$). In contrast, the SMBH occupation fraction at $z>3$ remains poorly constrained due to its degeneracies with delay time and the galaxy merger rate. These findings highlight both the promise and limitations of using GW observations to probe the coevolution of galaxies and SMBHs.

Figures

Figures reproduced from arXiv: 2501.02748 by the authors.

Figure 1
Figure 1. Strain amplitude of SGWB detected by NANOGrav (Agazie et al. 2023). By substituting equation (2) into equation (9), and then into equa￾tion (7), and integrating over the mass ratio for major mergers, the strain of SGWB is finally given by: ℎ 2 c = 𝑓 ∫ ∫ ∫ ΦGSMF(𝑀∗, 𝑧′ ) 𝑑𝑁 𝑑𝑡 (𝑀∗, 𝑧′ ) 𝑑𝑉 𝑑𝑧  𝑑𝑡 𝑑𝑓 ℎ 2 𝑠  𝑑𝑧 𝑑𝑀∗ , (12) where 𝑧 ′ = 𝑧(𝑡𝐿 (𝑧) + 𝜏) represents the redshift at which the galaxy merger occurs, while 𝑧 den… view at source ↗
Figure 2
Figure 2. Left: the 36 mock LISA GW events from the first realization. Right: the 12 mock LISA GW events from the second realization [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Left: posteriors of the model parameters 𝚲dN/dt in Case 1 model of galaxy merger rate (discussed in subsection 5.1.1), estimated with mock LISA data from the first realization (left panel of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: The galaxy merger rate 𝑑𝑁 𝑑𝑡r (𝑧) for galaxy stellar mass in the range log10𝑀∗/𝑀⊙ = [8, 10]. Left panel: The pink (cyan) lines represent the recovered merger rate, 𝑑𝑁 𝑑𝑡r (𝑧), based on the posterior distributions of model parameters presented in the left (right) panel …
Figure 6
Figure 6. Figure 6: Mock LISA GW data generated assuming a delay time of 𝜏 = 0.2Gyr, 0.5Gyr, 0.8Gyr, and 1Gyr respectively. In this work, we only consider a constant delay model due to the computational challenges of performing high-dimensional integra￾tions as well as dealing with large …
Figure 7
Figure 7. Figure 7: Posterior distributions of the model parameters Λ = {𝜏, 𝛼0, 𝛼1, 𝛽}, inferred from the mock LISA data presented in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The comparison of the contributions from mergers ( 1 − 𝜎 credible region in the red shaded region, this work) and accretion (crossed green lines) to the mass assembly of SMBHs for different redshifts (𝑧 = 1 − 6) and SMBH mass bins (log10𝑀•/𝑀⊙ = [5, 10]). First column: …
Figure 9
Figure 9. Figure 9: Mock LISA data realized from merger rate model as￾suming {𝜏, 𝑓3, 𝛼0, 𝛼1, 𝛽} = {0.5Gyr, 0.46, 0.2, −0.01, 2.4} (navy) and {𝜏, 𝑓3, 𝛼0, 𝛼1, 𝛽} = {1Gyr, 0.46, 0.2, −0.01, 2.4} (gold) respectively, and the GSMF and 𝑀• − 𝑀∗ relationship are set to Tables A1-A2. In the previo…

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