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REVIEW 3 major objections 5 minor 58 references

Can quasars, triggered by mergers, account for NANOGrav's stochastic gravitational wave background?

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A quasar-driven scenario, with nearly every bright quasar tied to a merging supermassive black hole binary, can reproduce the measured pulsar-timing gravitational wave background.

desk verdict A transparent, well-executed consistency check between quasars and the NANOGrav GWB; the amplitude match is partly fitted, and the contemporaneity assumption is the load-bearing caveat. read the letter →

arxiv 2412.12726 v1 pith:LTDHGWOL submitted 2024-12-17 astro-ph.CO astro-ph.HEgr-qc

Ágnes Kis-Tóth , Zoltán Haiman , Zsolt Frei This is my paper · ORCID
classification astro-ph.COastro-ph.HEgr-qc
keywords stochasticgravitationalwavebackgroundpulsartimingarrayssupermassiveblackholebinariesquasarluminosityfunctionEddingtonratiolifetimegalaxymergersnanohertzwaves
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 tests whether the recently detected nanohertz gravitational wave background can be produced by the same galaxy mergers that are thought to switch on bright quasars. It takes the observed quasar luminosity function as the only empirical input, converts quasars into supermassive black hole binaries through a quasar lifetime and an Eddington-ratio assumption, and integrates their gravitational-wave emission over cosmic time. With nearly all quasars associated with merging binaries, a log-normal Eddington ratio peaking near 0.25, mass ratios near one, and a quasar lifetime around 2.7e7 years, the predicted spectrum reproduces the measured amplitude and slope. If correct, this removes the need for merger simulations in predicting the background and points to billion-solar-mass black holes at redshifts 2 to 3 as the dominant sources. It also implies that the background should be smoother and less anisotropic than in models dominated by nearby binaries.

What carries the argument

The machinery is the quasar luminosity function used as a stand-in for the supermassive black hole binary merger rate. A mass-luminosity relation with Eddington ratio $f_{\rm Edd}$ converts the observed quasar number density $\Phi(M,z)$ into a black hole number density $\Psi(M,z)$; dividing by the quasar lifetime $t_Q$ turns it into a merger rate. The chirp mass $\mathcal{M}=(m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$ then fixes both the emitted gravitational-wave energy and the residence time per logarithmic frequency interval, so the characteristic strain $h_c(f)$ follows from an integral over mass, mass ratio, and redshift. A second route, integrating the gravitational-wave luminosity density over cosmic time, is shown to be mathematically identical, and environmental hardening appears only as a low-frequency correction.

What would settle it

Measure the Eddington-ratio distribution of the redshifts 2 to 3 quasars that dominate this model directly from virial black hole masses: if the typical value is well above the assumed log-normal peak of 0.25, the inferred black hole masses, and therefore the predicted strain, drop by a factor of several and miss the measured pulsar-timing amplitude.

Watch

Extended reading notes

Core claim

The central claim is that a scenario with binary fraction $f_{\rm bin} \sim 1$ — essentially every bright quasar marking a galaxy merger whose supermassive black holes are already coalescing — is quantitatively consistent with the pulsar-timing background. Starting from the empirical quasar luminosity function, the authors derive a merger rate density by dividing the quasar number density by the quasar lifetime, then compute the characteristic strain both by summing individual inspiraling binaries and by integrating their gravitational-wave energy output, showing the two routes are identical. Their fiducial model yields $h_c(f)=2.4\times10^{-15}(f/\mathrm{yr}^{-1})^{-2/3}$, matching the measured amplitude, and the match is robust to reasonable parameter choices because lower Eddington ratios, higher mass ratios, and shorter quasar lifetimes all raise the predicted strain. The implied sources are distant, roughly $10^9\,M_\odot$ binaries at $z\approx 2{-}3$, near the peak of quasar activity. The authors emphasize the main physical caveat: the calculation assumes quasar activity and the binary's passage through the pulsar-timing frequency band happen at the same time.

Load-bearing premise

The load-bearing premise is that every bright quasar is caught during the same short episode in which its supermassive black hole binary is also emitting gravitational waves in the pulsar-timing band, so the observed quasar count directly equals the binary merger rate.

Editorial extensions

If this is right

  • The gravitational-wave background can be predicted directly from the observed quasar census, with no need for galaxy merger simulations or semi-analytic models.
  • The dominant sources are distant, roughly $10^9\,M_\odot$ binaries at $z\approx 2{-}3$, so the background is built from a larger and more distant population than in most previous models.
  • Because the sources are numerous and distant, the background should be smoother in frequency and lower in angular anisotropy, and individual binaries should be harder for pulsar timing arrays to resolve.
  • The fit constrains only a degenerate combination of quasar lifetime, Eddington ratio, and mass ratio; many combinations of these three parameters match the data.
  • Discreteness of the brightest sources produces only a mild high-frequency steepening, leaving the spectrum compatible with current errors.

Reading between the lines

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

  • Beyond the paper: if quasar activity and pulsar-timing-band inspiral are separated by even a few quasar lifetimes, the predicted strain would drop substantially, so the match found here is best read as an upper limit on how many quasars can be tied to merging binaries.
  • Beyond the paper: the same construction could be applied to deeper future quasar luminosity functions; a measured change in the bright-end slope would shift the predicted gravitational-wave background in a knowable direction.
  • Beyond the paper: the model implies a quantitative variability rate for wide-field time-domain surveys, roughly $t_Q/t_{\rm GW}\sim10^{-3}$ of quasars should show year-scale periodic modulation if the one-to-one quasar-binary correspondence is real.
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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 / 5 minor

Summary. The paper proposes that the stochastic gravitational wave background (GWB) detected by NANOGrav and other PTAs can be explained by supermassive black hole binaries associated with quasars, under the assumptions that all luminous quasars are activated by galaxy mergers and that each such merger produces a promptly coalescing SMBH binary. Starting from the observed quasar luminosity function (QLF) of Kulkarni et al. 2019, the authors convert the quasar space density into a merger rate using a quasar lifetime t_Q (Eq. 6), then compute the GWB in two independent ways: by summing the strain of individual inspiraling binaries (Eq. 16) and by integrating the GW energy injected by past mergers (Eq. 24), showing the two methods are mathematically equivalent. The fiducial model adopts a log-normal Eddington ratio distribution with peak f_Edd = 0.25, a log-normal mass ratio distribution with peak q = 1, and t_Q = 2.7e7 yr, and reproduces the NANOGrav best-fit power-law h_c(f) = 2.4e-15 (f/yr^-1)^(-2/3). The paper explores parameter degeneracies using PTArcade, identifies the mass and redshift ranges that dominate the background, and examines the effects of environmental inspiral and source discreteness on the spectrum. The central claim is that the observed QLF, together with reasonable parameter choices, can make the quasar-merger scenario consistent with the measured GWB.

Significance. If the scenario holds, this paper provides a direct empirical link between the quasar population and the GWB, bypassing detailed galaxy-formation simulations. The derived predictions are falsifiable: the GWB would be dominated by ~1e9 Msun SMBHs at z~2-3, the spectrum would be smoother than in lower-redshift models, the angular anisotropy would be lower, and individual sources would be harder to resolve. The explicit proof that the two standard computational routes to h_c (source summation and energy injection) are algebraically equivalent is a useful pedagogical and cross-checking contribution. The treatment of discreteness effects and the use of a public MCMC package to map parameter degeneracies are also strengths, and the paper is transparent about the role of fitting in the fiducial choice of t_Q. The main weakness is that the consistency claim depends critically on an assumption—contemporaneity of the quasar phase and the PTA-band inspiral—that the authors themselves flag as unresolved, and whose violation by a time delay longer than t_Q would suppress the predicted amplitude below the observed value.

major comments (3)
  1. [Section 2.1 and Section 4] The conversion from quasar space density to merger rate (Eq. 6) and the residence-time weighting (Eq. 11) both assume that every quasar corresponds to a binary that passes through the nHz band within its t_Q-long active phase. As the authors note in Section 4, it is not clear whether the bright quasar phase and the PTA-band inspiral are contemporaneous, and either could precede the other. If the typical delay Delta_t between peak quasar activity and entry into the nHz band exceeds t_Q, the effective number of contributing binaries is suppressed by roughly t_Q/Delta_t, and h_c would fall below the NANOGrav best-fit value. Because t_Q = 2.7e7 yr is already at the short end of observationally inferred quasar lifetimes, absorbing such a delay by shortening t_Q further is not obviously possible. The paper should quantify this effect, even with a simple parameterization of the delay, and show how the inferred parameters (t_Q/f_bin, f_Edd, q) shift when contemporaneity is relaxed. This is the single most load-bearing assumption for the paper's central consistency claim.
  2. [Abstract and Section 3] The abstract states that the GWB is computed under the assumption t_Q ~ 10^8 yr, while the fiducial model in Section 3 uses t_Q = 2.7e7 yr, a factor of roughly four smaller. The amplitude match is achieved with this smaller value, which was explicitly 'chosen to fit the NANOGrav data' (Section 3). The abstract thus presents as an assumed parameter a value that is actually the fitted one, and it overstates the degree to which the match follows from the stated assumptions. Please reconcile the abstract and the fiducial model, either by quoting the fitted t_Q in the abstract or by presenting the match as a posterior constraint rather than an assumption.
  3. [Section 3.1 and Section 5] The GWB amplitude in the fiducial model is not an independent prediction: t_Q (and to some extent the peaks of the f_Edd and q distributions) are adjusted to match NANOGrav's A_yr. The paper is transparent about this in Section 3, but the abstract and conclusions say the model 'reproduces' and 'is consistent' without emphasizing the fitting step. Please add a sentence in the abstract and conclusions clarifying that the match is obtained by fitting these parameters, and that the falsifiable content lies in the shape of the spectrum, the source redshift/mass distribution, and the implied joint parameter constraints, not in the absolute normalization.
minor comments (5)
  1. [Section 2.1] The word 'actived' should be 'activated' in the sentence 'all quasars we see on the sky have been actived by a merger event.'
  2. [Section 3.4] Typo: 'Thesese panels' should be 'These panels', and 'lever of angular anisotropy' should be 'level of angular anisotropy'.
  3. [Section 4] Typo: 'correspondance' should be 'correspondence', and 'parameteric' should be 'parametric'.
  4. [Section 3.2] The uncertainties from the QLF itself (Kulkarni et al. 2019) are not propagated into the PTArcade posteriors. A sentence stating that only the model parameters were varied, and that QLF errors were neglected, would set expectations for the reported confidence regions.
  5. [Figure 3 caption] The notation 'log(f_Edd) ~ N(-0.6, 0.3)' is concise but could be clarified as a log-normal distribution with mean -0.6 in log10 and dispersion 0.3 dex, to match the text in Section 3.

Circularity Check

1 steps flagged · score 6.0 of 10

The GWB amplitude is fitted, not predicted: the quasar lifetime t_Q (or equivalently t_Q/f_bin) is chosen to match the NANOGrav amplitude, so the central claim of reproducing the measured GWB is partly by construction.

  1. fitted input called prediction [Section 3, opening of Results (fiducial-model paragraph); Eq. (16) and Eq. (24)]
    "The quasar lifetime in the fiducial model is a constant tQ ≃ 2.7 × 10^7 yr, consistent with expectations from observations [24], and chosen to fit the NANOGrav data (see below)."

    In both equivalent expressions for the GWB, Eq. (16) and Eq. (24), the characteristic strain satisfies h_c^2(f) ∝ (1/t_Q) ∫ Ψ_q M^{5/3} ... / [E(z)(1+z)^{4/3}]. The QLF and the mass-magnitude mapping fix everything except t_Q (and f_bin, which enters linearly and is set to 1). Because t_Q is explicitly 'chosen to fit the NANOGrav data', the subsequent statement that the fiducial model 'reproduces this best-fit power-law GWB' is a normalization fit rather than an independent prediction. The paper itself notes the degeneracy: 'we constrain only their ratio tQ/fbin', confirming that the amplitude match is absorbed by a free parameter.

full rationale

The paper's derivation chain from the observed quasar luminosity function to the GWB is otherwise self-contained: the two methods (Eqs. 14-16 and Eqs. 17-24) are shown to be mathematically identical, and the input QLF (Kulkarni et al. 2019 / Xin & Haiman 2021) is an external empirical dataset, not derived from the NANOGrav signal. The log-normal distributions for f_Edd and q are also taken from independent or only partially overlapping literature and are not shaped by the GWB data. The central circular element is the amplitude normalization: t_Q is a free parameter explicitly adjusted to match NANOGrav, and f_bin is degenerate with it, so the headline consistency of the amplitude is partly by construction. However, the paper is transparent about this and does make predictions that were not fitted: the f^{-2/3} spectral slope in the GW-driven case, the redshift/mass distribution peaking at z~2-3 and M~10^9 M_sun, the relative smoothness of the spectrum, reduced anisotropy, and reduced resolvability of individual sources. These give the model independent falsifiable content. The contemporaneity assumption (bright quasar phase overlapping the nHz-band inspiral) is explicitly flagged as uncertain in Section 4; it is a physical limitation affecting the predicted normalization, not an additional circular step. On balance, one central fitted parameter makes the amplitude claim partly circular, but the model retains independent content, so a score of 6 is appropriate.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model introduces no new particles or forces; it combines an existing QLF with standard SMBH binary gravitational wave formulas and a small set of free parameters (t_Q, f_Edd peak, q peak, f_bin).

free parameters (4)
  • quasar lifetime t_Q = 2.7e7 yr (fiducial); log(t_Q/1e8 yr) in [-2,1] in MCMC
    Chosen to fit NANOGrav GWB amplitude; degenerate with binary fraction f_bin.
  • Eddington ratio distribution peak f_Edd* = log f_Edd* = -0.6 (fiducial); [-2,0] in MCMC
    Fitted to NANOGrav in Section 3.2; dispersion fixed at 0.3 dex from Kollmeier+06.
  • Binary mass ratio distribution peak q* = log q* = 0 (fiducial); [-3,0] in MCMC
    Fitted to NANOGrav in Section 3.2; dispersion fixed at 1.2 dex from Sesana+18.
  • Binary fraction f_bin = 1 (assumed)
    All quasars associated with SMBHBs; amplitude scales linearly with f_bin, so constraints are on t_Q/f_bin.
assumptions (6)
  • domain assumption The Kulkarni+19 QLF, as parameterized by Xin & Haiman 21, is a complete census of luminous quasars.
    Adopted in Section 2.1; obscured quasars would raise the predicted amplitude, as the authors acknowledge.
  • domain assumption Quasar luminosity maps to SMBH mass through Eq (4) with constant bolometric correction and Eddington ratio.
    Section 2.1; a spread in bolometric correction would change the inferred mass function.
  • ad hoc to paper Every quasar is activated by a merger and hosts a coalescing SMBH binary contemporaneously with the quasar phase; f_bin = 1.
    Section 2.1 states all quasars have been activated by a merger; Section 4 flags the contemporaneity assumption as unclear.
  • domain assumption The quasar population is in steady state, so the merger rate density equals the quasar number density divided by t_Q.
    Equation (6); assumes merger rate is constant over t_Q, which is much shorter than the Hubble time.
  • domain assumption Binaries are circular and their orbital evolution is driven by GW emission, plus the parametric environmental model of Agazie+23.
    Sections 2.2 and 2.4; eccentricity would alter the spectrum shape.
  • standard math Standard flat LCDM cosmology is used for distances and volume elements.
    Equations (15) and (20); H0 and E(z) are taken from standard values.

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Pith. "Pith review of Can quasars, triggered by mergers, account for NANOGrav's stochastic gravitational wave background?." pith.science (2026). https://pith.science/paper/LTDHGWOL

@misc{pith2026241212726,
  author       = {Pith},
  title        = {Pith review of: Can quasars, triggered by mergers, account for NANOGrav's stochastic gravitational wave background?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LTDHGWOL}},
  note         = {Machine review of arXiv:2412.12726}
}
abstract

The stochastic gravitational wave background (GWB) recently discovered by several pulsar timing array (PTA) experiments is consistent with arising from a population of coalescing super-massive black hole binaries (SMBHBs). The amplitude of the background is somewhat higher than expected in most previous population models or from the local mass density of SMBHs. SMBHBs are expected to be produced in galaxy mergers, which are also thought to trigger bright quasar activity. Under the assumptions that (i) a fraction $f_{bin} \sim 1$ of all quasars are associated with SMBHB mergers, (ii) the typical quasar lifetime is $t_{Q} \sim 10^{8} yr$, and (iii) adopting Eddington ratios $f_{Edd} \sim 0.3$ for the luminosity of bright quasars, we compute the GWB associated directly with the empirically measured quasar luminosity function (QLF). This approach bypasses the need to model the cosmological evolution of SMBH or galaxy mergers from simulations or semi-analytical models. We find a GWB amplitude approximately matching the value measured by NANOGrav. Our results are consistent with most quasars being associated with SMBH binaries and being the sources of the GWB, and imply a joint constraint on $t_{Q}$, $f_{Edd}$ and the typical mass ratio $q \equiv M_{2}/M_{1}$. The GWB in this case would be dominated by relatively distant $\sim 10^{9} M_{\odot}$ SMBHs at $z \approx 2 - 3$, at the peak of quasar activity. Similarly to other population models, our results remain in tension with the local SMBH mass density.

Figures

Figures reproduced from arXiv: 2412.12726 by the authors.

Figure 1
Figure 1. The SMBHB merger rate per unit comoving volume as a function of the binary’s total mass for nine different redshifts. The merger rate was calculated based on the double power-law quasar luminosity function (QLF) with the assumption that all quasars have been activated by a merger event, and that quasars are accreting at the Eddington limit. In all cases, the number of merger events are dominated by the least massive… view at source ↗
Figure 2
Figure 2. The evolution of the SMBHBs’ separation for different parameter values of the environmentally-driven inspiral model. All four panels show the residence time (tres ≡ −a/(da/dt)) that the binary spends at orbital separation a as a function of the orbital time (torb) corresponding to that separation. The coloured curves correspond to SMBH binaries with total masses of M• = 106−11 M⊙ as labeled. The different panels sho… view at source ↗
Figure 3
Figure 3. In the top left panel, we show a comparison between the NANOGrav data [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 3
Figure 3. Figure 3: Predicted and observed GWB spectra. Panel (a) shows the NANOGrav 15-year results (datapoints with errors bars), and their best-fit power-law spectrum assuming canonical GW-driven SMBH binary inspirals (pink lane, showing the ±1σ range), compared to our fiducial model w…
Figure 4
Figure 4. Figure 4: Left panel: 68% and 95% confidence levels of the posterior in the three￾dimensional parameter space of our model, using PTArcade to fit NANOGrav’s 15- yr GWB data. Right panels: Two dimensional cross-sections of the posterior with the same confidence levels, and with t…
Figure 5
Figure 5. Figure 5: Contributions to the present-day GWB of SMBHBs with different masses and redshifts at the GW frequencies of f = 1, 3, 10 and 30 nHz. The bin-size of the two￾dimensional distributions in each panel is: ∆z × ∆ log(M•/M⊙) = 0.1 × 0.1. Binaries in the M• ∼ 108 − 1010 M⊙ an…
Figure 6
Figure 6. Figure 6: Top left: The GWB amplitude in the fiducial model compared to a model corrected for discreteness effects. In the high-frequency regime, the spectrum typically steepens due to the low number of sources contributing in higher-frequency bins. However, this steepening is n…

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    Introduction Several different pulsar timing array (PTA) campaigns have recently reported the discovery of a stochastic gravitational background (GWB) at nano-Hz frequencies, including the North American Nanohertz Observatory for Gravitational Waves (NANOGrav; [1]), the joint European PTA and Indian PTA (EPTA and InPTA; [2]), the Australian Parkes PTA (PP...

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    Methods In this section, we first describe the quasar luminosity function (QLF) we adopted, and then discuss how we use it to produce an estimate of the mass function of binary SMBHs and the stochastic GWB. For pedagogical reasons, we present two different ways to construct the latter estimate - versions of both of these appear in the literature, and here...

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