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Optimal Detection Bands for Intermediate-Mass Black Hole Binaries: Prospects for LISA and AMIGO in General Relativity and $f(R, T)$ Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper forecasts that LISA will detect about $10^4$ intermediate-mass black hole binary mergers in four years and AMIGO hundreds to thousands in three, with minimal f(R,T) gravity changing those totals by under five percent.

desk verdict A readable side-by-side LISA/AMIGO IMBH forecast whose headline event counts rest on an assumed rate and mock detection functions, not on the Section 2 population model; the f(R,T) conclusion is plausible but the quantitative claims are not yet supported. read the letter →

arxiv 2507.04392 v1 pith:ZBG5BYP2 submitted 2025-07-06 gr-qc astro-ph.CO

Marcelo M. Lapola , Oswaldo D. Miranda This is my paper · ORCID
classification gr-qcastro-ph.CO MSC 83C3583D0583C57
keywords intermediate-massblackholesgravitationalwaveastronomyLISAAMIGOdetectionbandforecastsf(RT)gravitymodifiedeventrates
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

The paper tries to establish that intermediate-mass black hole binaries — pairs of black holes between roughly $10^2$ and $10^5$ solar masses — will be a high-yield target for the space-based gravitational wave observatories LISA and AMIGO. It builds a redshift-dependent population model from extrapolated black hole mass functions, flat mass-ratio pairing, and a distribution of orbital separations, then folds in the standard gravitational-wave merger timescale and each detector's sensitivity curve to map the detectable mass–redshift bands. The central numbers are about $10^4$ LISA mergers in four observing years and between $10^2$ and $10^3$ AMIGO mergers in three, with LISA seeing heavier systems at higher redshift and AMIGO lighter systems nearby. The paper then asks whether the minimal matter-coupled modified gravity model $f(R,T) = R + 2\lambda T$ changes this picture; it finds that for couplings compatible with Solar System tests, $|\lambda| \le 2\times10^{-3}$, the peak detection redshift shifts by less than one percent and the total event count by less than five percent.

What carries the argument

The carrying machinery is the semi-analytic redshift-dependent binary mass function of Section 2 together with the analytic rescaling identity of Section 4. The binary mass function $\Phi(M_1,z)\,f_{\rm bin}(M_1,M_2,z)\,p(q)$ starts from an extrapolated IMBH mass function, a flat mass-ratio distribution $p(q)\propto q^0$, and a logarithmically flat distribution of initial orbital separations integrated against the standard gravitational-wave inspiral timescale; this produces the intrinsic coalescence rate. For forecasting, the detector response is compressed into Gaussian redshift profiles $P_{\rm det}(z)$, and the modified-gravity part is carried entirely by the factor $\alpha(\lambda)=(1-\lambda/8\pi)^{5/6}$, which rescales the Gaussian centers and widths. That identity converts the f(R,T) strain change into a transparent shift of the event-rate forecast without recomputing signal-to-noise ratios, and is stated to reproduce the full numerical SNR rescaling to better than $2\%$.

What would settle it

A redshift-resolved catalog from LISA's first four years would settle it: the model predicts a peak in $dN/dz$ near $z \approx 4$--$5$ with about $10^4$ total events, so an observed peak shifted by more than roughly $\Delta z \approx 1$, a total count differing from $10^4$ by more than about a factor of two or three, or a LISA/AMIGO count ratio that deviates by more than $5\%$ from the predicted ratio would contradict the fiducial model and the f(R,T) claim.

Watch

Extended reading notes

Core claim

The central claim is that intermediate-mass black hole binaries are a rich multi-detector population. Starting from a constant comoving merger rate density $R_0 = 100\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ and Gaussian detection probabilities $P^{\rm LISA}_{\rm det}(z) = \exp[-(z-5)^2/(2.5)^2]$ and $P^{\rm AMIGO}_{\rm det}(z) = \exp[-(z-1.5)^2/(0.75)^2]$, multiplied by the $\Lambda$CDM comoving volume and a $(1+z)^{-1}$ time-dilation factor, the model yields about $10^4$ events for LISA in four years and $10^2$–$10^3$ for AMIGO in three years. In the linear model $f(R,T) = R + 2\lambda T$, the gravitational-wave strain is rescaled by $\alpha(\lambda) = (1 - \lambda/8\pi)^{5/6}$ while the phase is unchanged at leading order, so the detection-probability Gaussians are shifted from $(z_0,\sigma)$ to $(\alpha z_0,\alpha\sigma)$. For $|\lambda| \le 2\times10^{-3}$, this rescaling moves the peak detection redshift by less than $1\%$ and the integrated event count by less than $5\%$; the paper concludes that, at population level, this minimal matter-coupled modified gravity is observationally indistinguishable from general relativity and can serve mainly as a consistency check.

Load-bearing premise

The headline numbers assume a constant comoving merger rate of 100 intermediate-mass black hole binaries per cubic gigaparsec per year at every redshift, and model each detector's sensitivity as a fixed bell-shaped curve in redshift; if the true rate is ten times smaller or larger, or the sensitivity shape differs, every headline count changes by that factor.

Editorial extensions

If this is right

  • If LISA runs for four years as assumed, it should see on the order of $10^4$ IMBH binary mergers, making intermediate-mass black holes a statistically rich population rather than a handful of special events.
  • AMIGO would open a complementary decihertz window on lighter binaries at $z \sim 1.5$--$2$, and together with LISA the two observatories would cover total masses from about $10^2$ to $10^6\,M_\odot$ continuously.
  • Within current Solar System bounds, $f(R,T)=R+2\lambda T$ cannot be distinguished from general relativity by IMBH event counts; the GR forecasts therefore remain the reference numbers for both detectors.
  • The amplitude rescaling $\alpha(\lambda)$ provides a consistency test: after marginalizing astrophysical uncertainties, a systematic few-percent offset between the predicted and observed merger rates would point to a varying effective gravitational constant or a similar uniform rescaling.
  • The estimate of about $6\times10^6$ IMBH binaries in the observable universe sets an upper bound on the total merger yield available to any future detector.

Reading between the lines

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

  • A natural extension is to replace the constant $R_0 = 100\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ with the redshift- and mass-dependent merger rate derived from the paper's own mass-function model, which would give the event counts a fully internal normalization rather than an external input.
  • Because the f(R,T) forecast is made by rescaling the Gaussian detection functions rather than recomputing SNR maps, a full numerical rerun with realistic noise curves could find larger differences near the detection threshold, particularly for AMIGO's narrower band.
  • If both detectors fly, the ratio of LISA to AMIGO event counts becomes a useful test that partially cancels population uncertainties, since many astrophysical factors affect both instruments the same way.
  • The same rescaling technique could be extended to nonlinear f(R,T) variants that are equipped with screening, where population-level shifts of order ten percent are plausible even while Solar System bounds are satisfied.
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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

4 major / 5 minor

Summary. The manuscript presents a semi-analytic forecast for the detection of intermediate-mass black hole binaries with the space-based detectors LISA and AMIGO, based on an extrapolated IMBH mass function, and extends the analysis to the minimal matter-coupled modified gravity model f(R,T)=R+2λT. The abstract reports event rates of about 10^4 mergers for LISA in four years and 10^2–10^3 for AMIGO in three years, and claims that for |λ|≤2×10^-3 the peak detection redshift shifts by less than one percent and the total event number changes by less than five percent. The paper identifies optimal detection bands in mass and redshift and discusses complementarity between the two detectors.

Significance. If the quantitative forecasts were reliable, the paper would provide useful mission-planning information for LISA and AMIGO and a simple population-level test of f(R,T) gravity. The authors are candid that the redshift-dependent detection probabilities are 'mock functions' intended as illustrative tools rather than derived detector responses. However, the central event-rate numbers and the modified-gravity percentages are not computed from the mass-function model advertised in Section 2, and the f(R,T) amplitude rescaling contains algebraic and dimensional errors. The qualitative conclusion that LISA and AMIGO are complementary is plausible, but the specific quantitative claims require a substantial reworking.

major comments (4)
  1. [Section 3.6, Eq. (3.4)] The differential event rate dNbin/dz is computed from an assumed constant comoving rate density R0 = 100 Gpc^-3 yr^-1 and Gaussian detection probabilities Pdet(z) that the text explicitly calls 'mock functions' and 'illustrative tools rather than physically rigorous detection probabilities.' No derivation connects R0 to the mass-function model of Section 2; the scenario parameters fbin, eta_merge, and <a0> defined in Section 2.4 do not appear in Eq. (3.4), and the paper states that the Gaussians were 'chosen to reflect' the full SNR maps rather than derived from them. Since Ndet is linear in R0, the headline event counts in the abstract and Section 3.5 inherit this external normalization, and the f(R,T) percentages computed in Section 4.3 rescale these same Gaussians. The authors should either compute R0 from the binary mass function or reframe the forecasts as explicitly conditional on an assumed rate density.
  2. [Section 2.2-2.3 and 3.7, Eqs. (2.3), (2.6), (3.8)] The binary mass function and coalescence rate are defined in three mutually inconsistent forms. Eq. (2.3) gives dNbin/(dM1 dM2 dz dV) = Φ(M1) fbin p(q) with no 1/M1 Jacobian, Eq. (2.6) instead uses a product Φ(M1)Φ(M2) and an integral over p(a0)/τmerge, and Eq. (3.8) reintroduces the 1/M1 factor. The merger efficiency ηmerge introduced in Section 2.3 and used in the scenario definitions of Section 2.4 does not appear in Eq. (2.6). The paper does not state which expression is inserted into the numerical integration that produces the event counts, so the quantitative results are not reproducible.
  3. [Section 4.1, Eqs. (4.3)-(4.5)] The working field equation (4.4), Gµν = 8πG Tµν + 2λ Tµν + λ T gµν, does not follow from Eq. (4.3) when Θµν = -2Tµν for dust; the substitution gives Gµν = (8πG + 2λ)Tµν with no λTgµν term. In addition, the Newtonian limit Eq. (4.5), Geff = G(1 - λ/8π), is dimensionally inconsistent unless G is set to unity and has a sign opposite to what follows from Eq. (4.4) with a (-,+,+,+) signature. Because the f(R,T) strain rescaling α(λ) = (1 - λ/8π)^(5/6) used throughout Section 4 is derived from this Geff, the modified-gravity conclusions are founded on an incorrect derivation.
  4. [Section 4.3, Eqs. (4.9)-(4.11)] The transformation (z0, σ) → (α z0, α σ) is not the correct transformation of a detection probability that was defined by an SNR threshold. The SNR is not proportional to 1/DL(z) ∝ 1/z at all redshifts, since the redshifted chirp mass and the detector noise enter the SNR integral nontrivially, and rescaling a Gaussian's center and width does not reproduce the effect of multiplying the SNR by α. The paper asserts that these closed forms 'reproduce the full numerical SNR rescaling to better than 2%' but gives no numerical comparison or derivation; this claim needs to be substantiated or removed.
minor comments (5)
  1. [Section 3.5] The event-rate prediction is given only as 'preliminary results suggest', with no indication of which mass-function version from Section 2 was integrated or over what domain; please specify the exact computation.
  2. [Section 3.7, Eq. (3.10)] The text writes ln(10^3) ≈ 6.9 but the displayed expression uses log(10^5/10^2); clarify the base or use a single notation.
  3. [Section 4.4] The statement that GW observatories could verify 'a difference in the propagation speed of waves with different frequencies' is not supported by the model, which has no massive mode at linear order; either connect this to a concrete observable or remove it.
  4. [References [41] and [42]] These references are cited to support the use of Gaussian mock detectability functions, but those references concern population synthesis and hierarchical mergers, not this specific approximation; please provide a more direct citation or note that the Gaussian is a simplifying assumption.
  5. [Section 1] The paper claims to be the 'first side-by-side forecast' of IMBH mergers for LISA and AMIGO, but does not compare with existing IMBH rate estimates (e.g., Sesana 2007; Fragione et al. 2018); a brief quantitative comparison would help the reader calibrate the results.

Circularity Check

2 steps flagged · score 7.0 of 10

Headline event counts are the assumed R0=100 normalization times a geometric factor, and the f(R,T) "sub-1% / sub-5%" conclusions are baked into the rescaled Gaussian definition.

  1. fitted input called prediction [Section 3.6, Eq. (3.4); Fig. 2 caption; Section 5.1 item 4]
    "R0 is the assumed constant comoving merger rate density (e.g.,100 events Gpc−3 yr−1) ... A constant comoving merger rate R0 = 100 Gpc−3 yr−1 is assumed. ... absolute numbers remain high: ∼ 104 IMBH mergers for LISA over a four-year mission and hundreds–to–thousands for AMIGO in three years (fiducial scenario)."

    Eq. (3.4) defines the detectable redshift distribution as R0 · Pdet(z) · dVc/dz/(1+z), with R0 an "assumed" constant and Pdet a hand-calibrated Gaussian ("mock functions"). The Section 2 mass function (Eq. 2.6) is never integrated to produce these counts; no derivation links R0=100 to Φ(M1,z)Φ(M2,z) fbin ∫ p(a0)/τ da0. Because Ndet is linear in R0, the claimed ~10^4 LISA and ~10^2–10^3 AMIGO events are exactly the assumed rate density multiplied by a geometric factor, i.e. the input parameter presented as a population-model forecast. The paper's own wording ("assumed", "mock functions") confirms this is an input, not an output.

  2. self definitional [Section 4.3, Eqs. (4.9)-(4.11); Section 5.1 item 3]
    "Because the inspiral SNR of a given system scales as 1/DL(z) ∝ 1/z in this toy description, the horizon redshift and therefore the effective (z0, σ) contract or dilate by the same factor α. Replacing (z0, σ) → (αz0, ασ) in the GR expressions of Sec. 3 yields the modified detection probabilities ... P LISA, f(R,T) det (z) = exp[-((z − α 5)/(α 2.5))^2], (4.10) ... (4.11)."

    By Eq. (4.10)-(4.11), P_f(R,T)(z) = exp[-((z−αz0)/(ασ))^2] = P_GR(z/α); the f(R,T) detection probability is defined as the GR Gaussian with center and width rescaled by α. Therefore the peak detection redshift is α z0 by definition, and the integrated event count changes by whatever ∫P_GR(z/α)dV/(1+z) gives. The abstract's claims that the peak shifts by <1% and the event count changes by <5% are algebraic consequences of this definition, not the result of a separate SNR recomputation. The assertion that the closed forms "reproduce the full numerical SNR rescaling to better than 2%" is a claim about the fit, not a derivation. The modified-gravity forecast thus reduces to the rescaled Gaussian ansatz.

full rationale

The paper is not circular via a self-citation chain: the GR detectability maps use published LISA/AMIGO sensitivity curves, and the α(λ)=(1−λ/8π)^(5/6) rescaling follows from the cited f(R,T) field equations. However, two load-bearing quantitative claims reduce by construction. First, the abstract's event rates are produced by Eq. (3.4) using R0=100 Gpc−3 yr−1, which the paper itself labels "assumed", and Pdet Gaussians that it calls "mock functions"; no calculation connects R0 to the Section 2 population model, so the ~10^4 and 10^2–10^3 forecasts are the input rate density times a fixed volume/detection integral. Second, in Section 4.3 the f(R,T) detection probability is defined as the GR Gaussian with (z0,σ)→(αz0,ασ), so the claimed sub-1% redshift shift and sub-5% event-count change are identities built into Eqs. (4.10)-(4.11) rather than independent predictions. Score 7 rather than 8 because the underlying GR SNR maps and the modified-gravity amplitude rescaling are not themselves circular; but the headline numbers presented as forecasts do reduce to assumed or definitional inputs.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The forecast depends on hand-picked inputs: the mass-function normalization Phi0 = 10^-4, scenario values for f_bin, eta_merge, and initial separation, the assumed constant merger rate R0 = 100 Gpc^-3 yr^-1, and the Gaussian P_det parameters. The f(R,T) results additionally depend on an asserted Geff rescaling. None of these are derived from independent data within the paper; the Gaussian curves are explicitly 'chosen to reflect' the intended result.

free parameters (7)
  • IMBH mass-function normalization Phi0 = 10^-4 Mpc^-3 M_sun^-1
    Assumed constant in mass and redshift, extrapolated from SMBH mass functions; anchors the total IMBH population at about 6e6 binaries (Section 3.7).
  • Binary fraction f_bin = 0.001 / 0.01 / 0.05
    Pessimistic, fiducial, and optimistic scenarios; hand-picked to span plausible dense-stellar-environment values (Section 2.4).
  • Merger efficiency eta_merge = 0.01 / 0.1 / 0.5
    Listed in scenarios but not used in the a0-integrated rate formula (2.6), so its role in the numerics is unclear (Section 2.4).
  • Initial separation scale <a0> = 0.05 / 0.01 / 0.005 pc
    Scenario choices that conflict with the 10^-2 to 10 AU range used in Eq. (2.6); 0.05 pc is about 10^4 AU (Sections 2.3-2.4).
  • Comoving merger rate density R0 = 100 Gpc^-3 yr^-1
    Assumed constant and used for all headline event rates; not derived from the mass-function model (Section 3.6, Eq. 3.4).
  • Gaussian P_det parameters (z0, sigma) = LISA (5, 2.5), AMIGO (1.5, 0.75)
    Chosen by hand to reflect approximate sensitivity ranges; the f(R,T) forecasts are built by rescaling these fitted Gaussians (Section 3.6).
  • f(R,T) coupling lambda = ±2e-3
    Chosen at the largest value consistent with Solar System and pulsar bounds; drives the tiny alpha rescaling (Section 4.4).
assumptions (8)
  • domain assumption The IMBH mass function can be extrapolated from SMBH mass functions and is roughly constant over 10^2 to 10^5 solar masses and z from 0 to 10.
    Invoked in Section 2.1 and used in Eq. (3.10); no explicit Phi(M,z) is given.
  • domain assumption A constant binary fraction f_bin and a flat mass-ratio distribution p(q) describe binary formation in dense stellar environments.
    Section 2.2, Eqs. (2.2)-(2.3).
  • standard math Merger timescales follow the Peters formula for circular orbits with a log-flat separation distribution.
    Section 2.3, Eqs. (2.5)-(2.6).
  • domain assumption The inspiral-only characteristic strain and the SNR threshold of 8 are sufficient to define detectability.
    Section 3.1, Eqs. (3.2)-(3.3); no merger/ringdown or orientation averaging is included.
  • ad hoc to paper For f(R,T) = R + 2 lambda T with dust, the Newtonian limit gives Geff = G(1 - lambda/8pi).
    Eqs. (4.4)-(4.5); this is asserted and appears inconsistent with the 00 component of the paper's own Eq. (4.4), where the lambda terms cancel.
  • ad hoc to paper The Gaussian redshift detectability functions summarize the full SNR calculation, and rescaling them by alpha(lambda) reproduces the full f(R,T) SNR rescaling to better than 2%.
    Sections 3.6 and 4.3; no comparison to the full detectability maps is shown.
  • domain assumption The background expansion is LambdaCDM and is unchanged by the allowed f(R,T) coupling.
    Section 4.3, citing references [21,43,44].
  • standard math A flat LambdaCDM cosmology with H0 = 70 km/s/Mpc, Omega_m = 0.3, and Omega_Lambda = 0.7.
    Section 3.6, Eq. (3.5).

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Cite this review

Pith. "Pith review of Optimal Detection Bands for Intermediate-Mass Black Hole Binaries: Prospects for LISA and AMIGO in General Relativity and $f(R, T)$ Gravity." pith.science (2026). https://pith.science/paper/ZBG5BYP2

@misc{pith2026250704392,
  author       = {Pith},
  title        = {Pith review of: Optimal Detection Bands for Intermediate-Mass Black Hole Binaries: Prospects for LISA and AMIGO in General Relativity and $f(R, T)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBG5BYP2}},
  note         = {Machine review of arXiv:2507.04392}
}
read the original abstract

We present a semi analytic forecast for the detection of intermediate mass black hole (IMBH) binaries with the space based detectors LISA (millihertz band) and AMIGO (deci hertz band). A redshift dependent population model is built from extrapolated black hole mass functions and realistic pairing and merger time scales. Folding this population through the instrument sensitivities yields event rates of about 1e4 mergers for LISA in four observing years and between 1e2 and 1e3 mergers for AMIGO in three years, with complementary optimal detection bands in total mass and redshift. We then incorporate the minimal matter coupled modified gravity model f(R,T) = R + 2 lambda T. In this scenario the strain amplitude is rescaled by (1 - lambda / 8 pi)^(5/6) while the waveform phasing is unchanged at leading order. For Solar System compatible couplings abs(lambda) <= 2 x 10^-3, the peak detection redshift shifts by less than one percent and the total number of events changes by less than five percent. Combined statistics from LISA and AMIGO therefore provide a consistency check on f(R,T) gravity without spoiling standard IMBH forecasts.

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