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arxiv: 2605.05512 · v1 · submitted 2026-05-06 · 🌀 gr-qc · astro-ph.HE

Recognition: unknown

Testing General Relativity with Individual Supermassive Black Hole Binaries

Qinyuan Zheng , Bjorn Larsen , Ellis Eisenberg , Chiara M. F. Mingarelli
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:41 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords continuous gravitational wavessupermassive black hole binariespulsar timing arraystests of general relativitygravitational wave polarizationsmodified dispersion relationsnanohertz frequencies
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The pith

Continuous waves from individual supermassive black hole binaries enable tests of gravity beyond general relativity using pulsar timing arrays.

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

The paper develops a framework for using continuous gravitational waves from single supermassive black hole binaries to search for departures from general relativity. These long-lived nanohertz signals preserve phase coherence across cosmological distances and produce inter-pulsar correlations that carry information about gravity's properties. The authors derive the cross-correlations, antenna responses, and propagation phase delays for additional polarization modes, modified dispersion relations, and parity-violating birefringence. Simulations show that injected breathing-mode and massive-graviton signals can be recovered at current limits and separated from background models, while a pure general relativity template still recovers source parameters without significant bias.

Core claim

The paper develops a unified framework for testing gravity beyond general relativity with continuous gravitational waves from individual supermassive black hole binaries. For non-tensorial polarizations the cross correlation scales linearly in the alternative amplitude when the modes are sub-dominant. Modified dispersion relations enhance both antenna-pattern modifications and pulsar-term phase delays at low frequencies, while birefringence remains suppressed at nanohertz frequencies. Injection-and-recovery tests recover the injected beyond-general-relativity parameters and distinguish the continuous wave signal from both correlated and uncorrelated background models.

What carries the argument

The inter-pulsar cross-correlation function for continuous waves, modified for beyond-general-relativity effects and scaling linearly with sub-dominant alternative polarization amplitude.

If this is right

  • Pulsar timing arrays are competitive for testing modified dispersion relations because low frequencies amplify both the antenna-pattern modification and the pulsar-term phase delay.
  • Parity-violating birefringence produces negligible effects at nanohertz frequencies for most theories.
  • Individual continuous wave signals can be distinguished from both correlated and uncorrelated gravitational wave background models.
  • A two-stage analysis strategy works: first identify candidates under general relativity, then test for deviations without introducing significant bias in source parameters.

Where Pith is reading between the lines

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

  • Future higher-sensitivity pulsar timing arrays could detect the first individual sources and apply these beyond-general-relativity tests immediately.
  • The linear scaling for polarization modes means small deviations could appear in continuous waves even when undetectable in the stochastic background.
  • The framework could be extended to other continuous wave sources or combined with electromagnetic observations of the same binaries.

Load-bearing premise

Beyond-general-relativity polarization modes remain much weaker than the standard tensor modes so that cross-correlations scale linearly, and the signals remain nearly monochromatic with coherent phase evolution over cosmological distances.

What would settle it

An injection-and-recovery simulation in which the beyond-general-relativity parameters for breathing-mode or massive-graviton signals are not recovered from simulated pulsar timing array data at current observational limits.

Figures

Figures reproduced from arXiv: 2605.05512 by Bjorn Larsen, Chiara M. F. Mingarelli, Ellis Eisenberg, Qinyuan Zheng.

Figure 1
Figure 1. Figure 1: FIG. 1. The inter-pulsar correlations induced by the alter view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The inter-pulsar correlations induced by massive gravity are distinct from GR, and GW frequency dependent. The view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Most birefringent theories are suppressed at ultra-low frequencies, making high frequency detectors better detection view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The beyond-GR cross-correlation matrix (the off-diagonal terms) offers the numerical basis for discerning the modifi view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The cross-correlated CW model is favored by the data over both the uncorrelated CW model (top) and the GWB view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The cross-correlated CW model is preferred by the view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. On top of the cross correlation, the pulsar term view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. An improving timing precision results in better recovery of the injected beyond-GR parameters with higher Bayes view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Joint posterior distributions for data containing a breathing mode in addition to the transverse tensor modes, with view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Joint posterior distributions for data containing a massive graviton signal, with the injected values indicated by black view at source ↗
read the original abstract

We develop a unified framework for testing gravity beyond General Relativity (GR) with continuous gravitational waves (CWs) from individual supermassive black hole binaries (SMBHBs). These long-lived, nearly monochromatic nanohertz signals offer unique strengths for precision tests of gravity, since their coherent phase evolution and inter-pulsar correlations in pulsar timing arrays (PTAs) retain detailed information about departures from GR over cosmological propagation distances. We consider three representative classes of deviations from GR: additional polarization states, modified dispersion relations, and parity-violating birefringence. For each, we derive the inter-pulsar cross correlation, the modified antenna response, and the propagation-induced pulsar-term phase delay. For non-tensorial polarizations, the CW cross correlation scales linearly in the alternative-polarization amplitude, compared to the quadratic scaling of the gravitational-wave background (GWB), provided the beyond-GR modes are sub-dominant. PTAs are also competitive for modified dispersion relations, where low frequencies enhance both the antenna-pattern modification and the pulsar-term phase delay. Birefringence, by contrast, is suppressed at nanohertz frequencies for most parity-violating theories. We validate the framework with injection-and-recovery simulations for breathing-mode and massive-graviton signals at current observational limits, recovering the injected beyond-GR parameters and distinguishing the CW signal from both correlated and uncorrelated background models. We further show that a pure-GR CW template recovers source parameters without significant bias when beyond-GR physics is present in the data, supporting a two-stage analysis strategy: identify candidates under GR, then test for deviations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper develops a unified framework for testing gravity beyond GR with continuous gravitational waves from individual SMBHBs in PTAs. It derives the inter-pulsar cross-correlations, modified antenna responses, and propagation-induced pulsar-term phase delays for three classes of deviations: additional polarization states, modified dispersion relations, and parity-violating birefringence. For non-tensorial polarizations the CW cross-correlation is shown to scale linearly with alternative-polarization amplitude (versus quadratic for the GWB) provided the beyond-GR modes are sub-dominant. The framework is validated with injection-and-recovery simulations for breathing-mode and massive-graviton signals at current observational limits, recovering the injected beyond-GR parameters, distinguishing the CW signal from correlated and uncorrelated backgrounds, and showing that a pure-GR template recovers source parameters without significant bias.

Significance. If the central derivations hold, the work supplies a concrete, unified toolkit for precision tests of gravity with nanohertz CWs that exploits their coherent phase evolution and inter-pulsar correlations over cosmological distances. The analytic treatment across deviation classes and the successful injection-recovery tests (including the two-stage analysis strategy) are clear strengths. The approach is particularly competitive for modified dispersion relations at low frequencies and offers a practical way to separate individual-source signals from the GWB. These elements would meaningfully advance PTA-based gravity tests if the sub-dominance assumption and associated scaling are placed on firmer footing.

major comments (1)
  1. [framework for non-tensorial polarizations (abstract and associated derivations)] The central claim that the CW cross-correlation scales linearly with alternative-polarization amplitude (versus quadratic for the GWB) is conditioned on the beyond-GR modes being sub-dominant. The injection-recovery tests are performed only at current observational limits, which implicitly keep these amplitudes small. An explicit test or analytic discussion of the regime in which alternative amplitudes become comparable to the tensor mode is required, because higher-order interference could modify the correlation structure and affect the claimed ability to distinguish the CW signal from background models.
minor comments (2)
  1. The manuscript would benefit from additional intermediate steps or explicit definitions when presenting the modified antenna response and pulsar-term phase delay for the three deviation classes.
  2. Specific numerical values or references for the PTA sensitivity limits used in the injection-recovery tests would help readers assess how close the simulated signals are to current observational reach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review, which highlights a key assumption in our framework. We address the major comment below and will revise the manuscript to strengthen the discussion of the sub-dominance regime.

read point-by-point responses

  1. Referee: The central claim that the CW cross-correlation scales linearly with alternative-polarization amplitude (versus quadratic for the GWB) is conditioned on the beyond-GR modes being sub-dominant. The injection-recovery tests are performed only at current observational limits, which implicitly keep these amplitudes small. An explicit test or analytic discussion of the regime in which alternative amplitudes become comparable to the tensor mode is required, because higher-order interference could modify the correlation structure and affect the claimed ability to distinguish the CW signal from background models.

    Authors: We thank the referee for this insightful comment. Our derivation of the linear scaling for the CW cross-correlation indeed relies on the beyond-GR polarization amplitudes being sub-dominant, as explicitly stated in the manuscript; this yields a cross term linear in the alternative amplitude when computing the expected inter-pulsar correlations for a deterministic signal. For the stochastic GWB the scaling is quadratic by construction. We agree that the comparable-amplitude regime merits explicit treatment, as interference terms (tensor-alternative cross terms and alternative self-terms) would then appear and could alter the correlation structure. In the revised manuscript we will add an analytic discussion (in the section on non-tensorial polarizations) that expands the full correlation function to include these higher-order terms and delineates the regime of validity of the linear approximation. We will further note that when alternative amplitudes become comparable, the signal deviates strongly from a pure-GR template, which is consistent with our two-stage analysis strategy of first identifying candidates under GR and then testing for deviations. While we do not add new injection-recovery simulations at high amplitudes (our focus remains on signals near current observational limits), the analytic treatment will place the assumption on firmer footing without requiring additional computational work. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives inter-pulsar cross-correlations, modified antenna responses, and pulsar-term phase delays directly from standard modified-gravity Lagrangians for additional polarizations, dispersion relations, and birefringence. These steps compute observables from first principles without reducing predictions to fitted parameters or self-definitions. Injection-and-recovery simulations serve as an independent check, recovering injected parameters without bias in pure-GR templates. No load-bearing self-citations, uniqueness theorems from prior author work, or ansatzes smuggled via citation appear in the provided derivation chain. The sub-dominance condition for linear scaling is an explicit stated assumption, not a derived loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard domain assumptions from gravitational-wave physics and modified-gravity models; no new free parameters or invented entities are introduced in the provided description.

axioms (2)
  • domain assumption Continuous waves from individual SMBHBs are long-lived and nearly monochromatic with coherent phase evolution over cosmological distances.
    Explicitly stated as the basis for retaining detailed information about departures from GR.
  • domain assumption Beyond-GR polarization modes are sub-dominant, allowing linear scaling of the cross-correlation.
    Invoked to contrast with quadratic scaling of the gravitational-wave background.

pith-pipeline@v0.9.0 · 5603 in / 1508 out tokens · 55273 ms · 2026-05-08T15:41:11.932927+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

105 extracted references · 59 canonical work pages · 9 internal anchors

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    17 •GW propagation direction: ˆΩ = (sinθcosϕ,sinθsinϕ,cosθ)

    Geometry of the computational frame In practical computations, we may define a convenient coordinate system as follows: •Pulsars: ˆpa = (0,0,1),ˆp b = (sinζ,0,cosζ). 17 •GW propagation direction: ˆΩ = (sinθcosϕ,sinθsinϕ,cosθ). •Transverse basis: ˆm= (sinϕ,−cosϕ,0),(A4) ˆn= (cosθcosϕ,cosθsinϕ,−sinθ).(A5)

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    Polarization tensors e+ ij =m imj −n inj,(A6) e× ij =m inj +n imj,(A7) eb ij =m imj +n inj,(A8) eL ij = √ 2 ΩiΩj,(A9) ex ij =m iΩj + Ωimj,(A10) ey ij =n iΩj + Ωinj.(A11)

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    Breathing mode Using Eq. (A3), F b a (θ) = 1 2 sin2 θ 1 + cosθ , F b b (θ, ϕ, ζ) = 1 2 sin2 ϕsin 2 ζ+ (−sinθcosζ+ sinζcosϕcosθ) 2 1 + sinθsinζcosϕ+ cosθcosζ . Thus the breathing–mode correlation is Υ(b) ab (θ, ϕ, ζ) = sin2 θ 4(1 + cosθ) sin2 ϕsin 2 ζ+ (−sinθcosζ+ sinζcosϕcosθ) 2 1 + sinθsinζcosϕ+ cosθcosζ . (A12)

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    Therefore Υ(L) ab (θ, ϕ, ζ) = cos2 θ 2(1 + cosθ) (sinθsinζcosϕ+ cosθcosζ) 2 1 + sinθsinζcosϕ+ cosθcosζ

    Longitudinal scalar mode F L a (θ) = √ 2 2 cos2 θ 1 + cosθ , F L b (θ, ϕ, ζ) = √ 2 2 (sinθsinζcosϕ+ cosθcosζ) 2 1 + sinθsinζcosϕ+ cosθcosζ . Therefore Υ(L) ab (θ, ϕ, ζ) = cos2 θ 2(1 + cosθ) (sinθsinζcosϕ+ cosθcosζ) 2 1 + sinθsinζcosϕ+ cosθcosζ . (A13)

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    Vector-x mode For vector modes, the antenna pattern is transverse basis dependent. This means that F x a = 0, F x b (θ, ϕ, ζ) = (sinϕsinζ) (sinθsinζcosϕ+ cosθcosζ) 1 + sinθsinζcosϕ+ cosθcosζ . Thus in this computational frame Υ(x) ab (θ, ϕ, ζ) = 0.(A14)

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    Bayes factors The Bayes factors used in the model comparisons, which we denote withBModel 1 Model 2, are calculated with logB Model 1 Model 2 = logZ Model 1 −logZ Model 2,(E1) whereZ Model 1,Z Model 2 aretheevidenceforthetwomod- els. For the particular case where the two models are the cross-correlated CW models with and without the beyond-GR modification...

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    Beyond-GR parameter recovery with improving SNRs We show that the beyond-GR parameter recovery may be improved with higher SNRs. One way to achieve 20 5 4 3 2 1 0 log10 0.00 0.05 0.10 0.15 0.20 Injected breathing mode Corr. B ( TOA = 100 ns) Corr. B ( TOA = 10 ns) Corr. B( = 0) ( TOA = 100 ns) Corr. B( = 0) ( TOA = 10 ns) 26.0 25.5 25.0 24.5 24.0 23.5 23....

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