pith. machine review for the scientific record. sign in

arxiv: 2604.08317 · v1 · submitted 2026-04-09 · 🌌 astro-ph.CO · gr-qc

Recognition: 2 theorem links

· Lean Theorem

A Dynamical Equilibrium Linking Nanohertz Stochastic Gravitational Wave Background to Cosmic Structure Formation

Manjia Liang , Peng Xu , Ruijun Shi , Zhoujian Cao , Ziren Luo , Minghui Du , Qiong Deng , Bo Liang
show 1 more author
Jiaxiang Liang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:20 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords stochastic gravitational wave backgroundNANOGravdynamical equilibriumcosmic structure formationfluctuation-dissipation theoremgravitational screeningpulsar timing arrays
0
0 comments X

The pith

Treating the nanohertz gravitational wave background and cosmic matter as a coupled non-equilibrium system produces a dynamical equilibrium spectrum that fits NANOGrav data and derives its cutoff from structure formation scales.

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

The paper shows that the stochastic gravitational wave background cannot be treated as a passive relic because it couples dynamically to matter. Using linearized general relativity and the fluctuation-dissipation theorem, the authors derive a generalized Langevin equation whose stable solution is an equilibrium strain spectrum with a high-frequency cutoff and scale-dependent screening of gravity. This spectrum fits the NANOGrav 15-year dataset with a Bayes factor of 48 over the standard supermassive black hole binary model, all within general relativity and the Standard Model. The fitted screening mass scale coincides with the linear-to-nonlinear transition mass of cosmic structure, allowing the cutoff frequency to be expressed directly in terms of that mass without additional parameters.

Core claim

Combining linearized general relativity with the fluctuation-dissipation theorem yields a generalized Langevin framework that drives the SGWB-matter system to a dynamical equilibrium. The resulting strain spectrum is characterized by a high-frequency cutoff W and a scale-dependent coupling that screens gravity for structures above a characteristic mass m_c. When this spectrum is fitted to NANOGrav 15-year data, it is strongly preferred over the supermassive black hole binary model. The calibrated m_c range overlaps the Lambda-CDM linear-to-nonlinear transition mass M_NL at approximately 8 h^{-1} Mpc, so that W can be written entirely as a function of M_NL and its inverse interpreted as a SGW

What carries the argument

The generalized Langevin framework derived from linearized general relativity plus the fluctuation-dissipation theorem, which enforces dynamical equilibrium between the stochastic gravitational wave background and matter, producing a strain spectrum with high-frequency cutoff W and scale-dependent gravitational screening.

If this is right

  • The equilibrium spectrum fits the NANOGrav 15-year dataset with a Bayes factor of 48 over the supermassive black hole binary model.
  • The screening mass scale m_c calibrated by pulsar timing arrays overlaps the Lambda-CDM linear-to-nonlinear transition mass M_NL with no free cosmological parameters.
  • The high-frequency cutoff W is expressed entirely in terms of M_NL, linking nanohertz gravitational-wave observables directly to late-time cosmic structure.
  • The framework predicts distinctive scale-dependent modifications to gravity that can be tested by forthcoming large-scale structure surveys and space-borne gravitational-wave observatories.

Where Pith is reading between the lines

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

  • If the equilibrium holds, the same screening mechanism should appear as a suppression of power in the matter power spectrum above the M_NL scale, providing an independent test in galaxy surveys.
  • The cutoff W derived from M_NL could be checked against future space-based detectors such as LISA by searching for a corresponding spectral feature at higher frequencies.
  • The model implies that the coherence threshold for SGWB-matter coupling sets a fundamental limit on how far gravitational waves can propagate without being screened by large-scale structure.

Load-bearing premise

The stochastic gravitational wave background and matter can be treated as a dynamically coupled non-equilibrium system to which the fluctuation-dissipation theorem applies directly.

What would settle it

Observation of a sharp high-frequency cutoff in the SGWB spectrum at the frequency predicted by the measured M_NL, or the absence of the predicted scale-dependent screening of gravity in the clustering of the most massive halos in large-scale structure surveys.

Figures

Figures reproduced from arXiv: 2604.08317 by Bo Liang, Jiaxiang Liang, Manjia Liang, Minghui Du, Peng Xu, Qiong Deng, Ruijun Shi, Zhoujian Cao, Ziren Luo.

Figure 1
Figure 1. Figure 1: FIG. 1: Bayes factors for the model comparisons between the DEGWB, DEGWB with [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The two-dimensional marginalized posterior distribution of log [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The grey violin plot is the Bayesian free spectrum under Hellings-Downs [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

The stochastic gravitational wave background (SGWB) is conventionally treated as a passive relic of its astrophysical and cosmological sources, with negligible back-reaction on the matter content of the Universe. Here we show that this assumption needs to be modified once the SGWB and matter are treated as a dynamically coupled non-equilibrium system. Combining linearized general relativity with the fluctuation-dissipation theorem, we derive a generalized Langevin framework that drives the coupled system toward a dynamical equilibrium, which is characterized by a distinctive strain spectrum with a high-frequency cutoff $\mathcal{W}$, and a scale-dependent coupling parameter that screens gravity progressively for the most massive structures. Three findings support this framework. Fitting the equilibrium spectrum to the NANOGrav 15-year dataset yields a Bayes factor of $48\pm 3.8$ over the supermassive black hole binary baseline, achieved entirely within general relativity and the Standard Model. The PTA-calibrated screening mass scale $m_{c}\sim 10^{12}\text{--}10^{14}\,M_{\odot}$ overlaps, with no free cosmological parameter, the $\Lambda$CDM-derived linear-to-nonlinear transition mass $M_{\rm NL}$ of cosmic structure at $\sim 8\,h^{-1}\,\mathrm{Mpc}$. Most strikingly, promoting this concordance to a structural identification expresses $\mathcal{W}$ entirely in terms of $M_{\rm NL}$, and its inverse acquires a transparent physical reading as a coherence threshold for SGWB-matter coupling. $\mathcal{W}$ is thereby a derived quantity linking nanohertz gravitational-wave observables to the late-time cosmological sector. The framework makes distinctive scale-dependent predictions testable by forthcoming large-scale structure surveys and space-borne gravitational-wave observatories.

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

2 major / 2 minor

Summary. The manuscript proposes treating the stochastic gravitational wave background (SGWB) and cosmic matter as a dynamically coupled non-equilibrium system. Combining linearized general relativity with the fluctuation-dissipation theorem yields a generalized Langevin framework whose equilibrium is characterized by a distinctive strain spectrum possessing a high-frequency cutoff W and a scale-dependent screening parameter. Fitting this spectrum to the NANOGrav 15-year dataset produces a Bayes factor of 48±3.8 relative to the supermassive black hole binary baseline. The PTA-calibrated screening mass m_c is reported to overlap the ΛCDM linear-to-nonlinear transition mass M_NL, allowing W to be rewritten entirely in terms of M_NL with no additional free cosmological parameters.

Significance. If the central derivation is sound, the work would establish a direct, parameter-free link between nanohertz gravitational-wave observables and late-time cosmic structure formation entirely within general relativity and the Standard Model. The reported Bayes factor and the structural identification of m_c with M_NL would constitute notable strengths, together with the framework’s distinctive scale-dependent predictions for forthcoming large-scale structure surveys and space-borne gravitational-wave detectors.

major comments (2)
  1. [Derivation of the generalized Langevin framework and equilibrium spectrum] The derivation of the equilibrium spectrum (abstract and the section presenting the generalized Langevin framework): the manuscript invokes the fluctuation-dissipation theorem to obtain a stable equilibrium spectrum with cutoff W and scale-dependent screening, yet provides no explicit construction of the requisite noise kernel or two-point correlators from the linearized Einstein equations on an FLRW background with clustering matter. Standard FDT assumptions (near-equilibrium conditions, time-translation invariance, and detailed balance) are not automatically satisfied for a propagating tensor background; without this derivation the reported Bayes factor and the subsequent m_c–M_NL identification rest on an unverified premise.
  2. [Concordance between m_c and M_NL and structural identification] Identification of m_c with M_NL (abstract, paragraph on the three findings): m_c is calibrated by fitting the model to the NANOGrav dataset; the same m_c is then declared to match the independent ΛCDM value of M_NL, after which W is rewritten as a function of M_NL. This procedure makes the claimed “parameter-free” concordance dependent on the data fit rather than an a priori prediction, undermining the assertion that W acquires a transparent physical reading solely from M_NL.
minor comments (2)
  1. [Abstract] The abstract states that “three findings support this framework” but explicitly details only the Bayes factor, the m_c–M_NL overlap, and the structural identification of W; the third finding should be stated clearly.
  2. [Fitting procedure and results] The explicit functional form of the equilibrium strain spectrum used in the NANOGrav fit, together with the precise definition of the scale-dependent coupling parameter and the error-propagation procedure for the Bayes factor, should be provided in the main text rather than referenced only by result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments raise important points about the explicitness of our derivation and the interpretation of the m_c–M_NL concordance. We address each below and have revised the manuscript to improve clarity and detail where appropriate.

read point-by-point responses

  1. Referee: [Derivation of the generalized Langevin framework and equilibrium spectrum] The derivation of the equilibrium spectrum (abstract and the section presenting the generalized Langevin framework): the manuscript invokes the fluctuation-dissipation theorem to obtain a stable equilibrium spectrum with cutoff W and scale-dependent screening, yet provides no explicit construction of the requisite noise kernel or two-point correlators from the linearized Einstein equations on an FLRW background with clustering matter. Standard FDT assumptions (near-equilibrium conditions, time-translation invariance, and detailed balance) are not automatically satisfied for a propagating tensor background; without this derivation the reported Bayes factor and the subsequent m_c–M_NL identification rest on an unverified premise.

    Authors: We thank the referee for identifying the need for a more explicit derivation. The manuscript combines linearized general relativity with the fluctuation-dissipation theorem to obtain the generalized Langevin equation and equilibrium spectrum, but we acknowledge that the step-by-step construction of the noise kernel from the two-point correlators of tensor perturbations on an FLRW background with clustering matter was not presented in full detail. In the revised manuscript we have added an expanded subsection that derives the noise kernel explicitly from the linearized Einstein equations, incorporating the matter density fluctuations as the source of the stochastic forcing. We discuss the applicability of the FDT by noting that, for the nanohertz frequencies and cosmological timescales involved, the system can be treated in a quasi-stationary regime where time-translation invariance holds approximately and the back-reaction drives the spectrum toward equilibrium. This addition supplies the missing foundation and supports the subsequent Bayesian analysis. revision: yes

  2. Referee: [Concordance between m_c and M_NL and structural identification] Identification of m_c with M_NL (abstract, paragraph on the three findings): m_c is calibrated by fitting the model to the NANOGrav dataset; the same m_c is then declared to match the independent ΛCDM value of M_NL, after which W is rewritten as a function of M_NL. This procedure makes the claimed “parameter-free” concordance dependent on the data fit rather than an a priori prediction, undermining the assertion that W acquires a transparent physical reading solely from M_NL.

    Authors: We agree that m_c is determined from the fit to the NANOGrav 15-year data and that the numerical overlap with the independently computed ΛCDM M_NL is therefore a consistency check rather than an a priori prediction. We have revised the relevant paragraphs to state this distinction explicitly: the agreement between the fitted screening mass and the known nonlinear transition mass supports promoting the concordance to a structural identification, which then allows W to be expressed solely in terms of M_NL without introducing additional free cosmological parameters. The physical interpretation of W as a coherence threshold for SGWB-matter coupling follows directly from this identification. The Bayes factor and model fits remain unchanged; the revision affects only the clarity of the discussion. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation from GR+FDT is independent of the data fit and identification step

full rationale

The paper's core derivation combines linearized general relativity with the fluctuation-dissipation theorem to obtain a generalized Langevin framework, equilibrium strain spectrum, high-frequency cutoff W, and scale-dependent screening mass m_c. This step is presented as first-principles within GR and the Standard Model and does not reduce to any fitted input or self-citation by the provided text. Fitting the resulting spectrum to NANOGrav 15-year data to obtain a Bayes factor and calibrate m_c is a standard model-validation procedure, not a circular redefinition. Noting the numerical overlap between the fitted m_c range and the independent LambdaCDM M_NL, then promoting the overlap to an identification so that W can be rewritten in terms of M_NL, is an interpretive post-processing step rather than a reduction of the claimed result to its inputs by construction. No quoted equation or derivation is shown to be equivalent to a fitted parameter or prior self-citation; the central claims retain independent content from the initial GR+FDT framework. The paper is therefore self-contained against external benchmarks (NANOGrav data and LambdaCDM M_NL) with no load-bearing circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on applying the fluctuation-dissipation theorem to a previously untreated SGWB-matter system and on introducing a new scale-dependent coupling whose only independent support is the data fit itself.

free parameters (1)
  • screening mass scale m_c = 10^{12}--10^{14} M_odot
    Calibrated directly to the NANOGrav 15-year dataset to produce the reported Bayes factor; sets the strength of the scale-dependent gravity screening.
axioms (2)
  • standard math Linearized general relativity governs the coupled dynamics
    Invoked to derive the generalized Langevin equation for the SGWB-matter system.
  • domain assumption Fluctuation-dissipation theorem applies to the SGWB-matter non-equilibrium system
    Used to obtain the equilibrium strain spectrum with high-frequency cutoff W.
invented entities (1)
  • scale-dependent coupling parameter no independent evidence
    purpose: Screens gravity progressively for the most massive structures, producing the equilibrium spectrum
    Newly introduced to characterize the dynamical equilibrium; no independent evidence outside the NANOGrav fit is provided.

pith-pipeline@v0.9.0 · 5639 in / 1881 out tokens · 74280 ms · 2026-05-10T17:20:05.332927+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

56 extracted references · 2 canonical work pages

  1. [1]

    Baumann, Cosmology (Cambridge University Press, 2022)

    D. Baumann, Cosmology (Cambridge University Press, 2022)

    2022

  2. [2]

    Aghanim et al

    N. Aghanim et al. (Planck), Astronomy & Astrophysics 641, A6 (2020)

    2020

  3. [3]

    P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, 1980)

    1980

  4. [4]

    R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh, Reviews of Modern Physics 88, 015004 (2016)

    2016

  5. [5]

    P. J. E. Peebles and B. Ratra, Reviews of modern physics 75, 559 (2003)

    2003

  6. [6]

    A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Mu- rakami, G. S. Anand, L. Breuval, et al. , The Astrophysical Journal Letters 934, L7 (2022)

    2022

  7. [7]

    W. L. Freedman, The Astrophysical Journal 919, 16 (2021)

    2021

  8. [8]

    Asgari, C.-A

    M. Asgari, C.-A. Lin, B. Joachimi, B. Giblin, C. Heymans, H. Hildebrandt, A. Kannawadi, B. St¨ olzner, T. Tr¨ oster, J. L. van den Busch, A. H. Wright,et al. , Astronomy & Astrophysics 645, A104 (2021)

    2021

  9. [9]

    L. F. Secco, S. Samuroff, E. Krause, B. Jain, J. Blazek, M. Raveri, J. Rollins, A. Amon, A. Choi, C. Chang, et al. , Physical Review D 105, 023515 (2022)

    2022

  10. [10]

    Christensen, Reports on Progress in Physics 82, 016903 (2018)

    N. Christensen, Reports on Progress in Physics 82, 016903 (2018)

    2018

  11. [11]

    Kakkat, Physical Review D 109, 024013 (2024)

    V. Kakkat, Physical Review D 109, 024013 (2024)

    2024

  12. [12]

    Agazie et al

    G. Agazie et al. , The Astrophysical Journal Letters 951, L8 (2023)

    2023

  13. [13]

    Antoniadis, P

    J. Antoniadis, P. Arumugam, S. Arber, S. Babak, A.-S. Bak Nielsen, C. G. Bassa, 30 A. Berthereau, M. Bonber, E. Bortolas, P. R. Brook, et al. , Astronomy & Astrophysics 678, A50 (2023)

    2023

  14. [14]

    D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Kapur, A. F. Rogers, E. Thrane, J. Askew, et al. , The Astrophysical Journal Letters 951, L6 (2023)

    2023

  15. [15]

    H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu, Z. Xue, R. N. Caballero, J. Yuan, Y. Xu, et al. , Research in Astronomy and Astrophysics 23, 075024 (2023)

    2023

  16. [16]

    Agazie, A

    G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. B´ ecsy, L. Blecha, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., The Astrophysical Journal Letters 951, L8 (2023)

    2023

  17. [17]

    B. P. Abbott et al. , Physical Review Letters 116, 061102 (2016)

    2016

  18. [18]

    Allen, Physical Review D 37, 2078 (1988)

    B. Allen, Physical Review D 37, 2078 (1988)

    2078

  19. [19]

    Afzal et al

    A. Afzal et al. , The Astrophysical Journal Letters 951, L11 (2023)

    2023

  20. [20]

    Xue et al

    X. Xue et al. , Physical Review Letters 127, 251303 (2021)

    2021

  21. [21]

    D. G. Figueroa et al. , Physical Review Letters 132, 171002 (2024)

    2024

  22. [22]

    Cai, X.-C

    Y.-F. Cai, X.-C. He, X.-H. Ma, S.-F. Yan, and G.-W. Yuan, Science Bulletin 68, 2929 (2023)

    2023

  23. [23]

    Kubo, Reports on Progress in Physics 29, 255 (1966)

    R. Kubo, Reports on Progress in Physics 29, 255 (1966)

    1966

  24. [24]

    R. B. Tully, E. J. Shaya, I. D. Karachentsev, H. M. Courtois, D. D. Kocevski, L. Rizzi, and A. Peel, The Astrophysical Journal 676, 184 (2008)

    2008

  25. [25]

    M. I. Scrimgeour, T. M. Davis, C. Blake, L. Staveley-Smith, C. Magoulas, C. M. Springob, F. Beutler, M. Colless, A. Johnson, D. H. Jones, et al. , Monthly Notices of the Royal Astro- nomical Society 455, 386 (2016)

    2016

  26. [26]

    G. W. Ford, J. T. Lewis, and R. F. O’Connell, Physical Review Letters 55, 2273 (1985)

    1985

  27. [27]

    G. W. Ford, J. T. Lewis, and R. F. O’Connell, Physical Review A 37, 4419 (1988)

    1988

  28. [28]

    Earman, in PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Associ- ation, Vol

    J. Earman, in PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Associ- ation, Vol. 1978 (1978) pp. 156–185

    1978

  29. [29]

    Hakim, Journal of Mathematical Physics 8, 1315 (1967)

    R. Hakim, Journal of Mathematical Physics 8, 1315 (1967)

    1967

  30. [30]

    Dunkel and P

    J. Dunkel and P. H¨ anggi, Physics Reports471, 1 (2009)

    2009

  31. [31]

    Bao and Z.-W

    J.-D. Bao and Z.-W. Bai, Chinese Physics Letters 22, 1845 (2005)

    2005

  32. [32]

    Kubo, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1991)

    R. Kubo, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1991)

    1991

  33. [33]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges, Physics Reports 195, 127 (1990). 31

    1990

  34. [34]

    B. D. Hughes, Random Walks and Random Environments, Vol. 1: Random Walks (Oxford University Press, 1995)

    1995

  35. [35]

    Ashton et al

    G. Ashton et al. , Nature Reviews Methods Primers 2, 39 (2022)

    2022

  36. [36]

    Mitridate et al

    A. Mitridate et al. 10.48550/arXiv.2306.16377 (2023), arXiv:2306.16377 [astro-ph.CO]

    work page doi:10.48550/arxiv.2306.16377 2023

  37. [37]

    The NANOGrav Collaboration, KDE representations of the gravitational wave background free spectra present in the NANOGrav 15-year dataset (1.0.0) (2023)

    2023

  38. [38]

    J. A. Ellis, M. Vallisneri, S. R. Taylor, and P. T. Baker, ENTERPRISE: Enhanced numerical toolbox enabling a robust PulsaR inference SuitE (v3.0.0) (2020)

    2020

  39. [39]

    S. R. Taylor, P. T. Baker, J. S. Hazboun, J. Simon, and S. J. Vigeland, enterprise extensions v2.4.3 (2021)

    2021

  40. [40]

    J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Astrophys. J. 304, 15 (1986)

    1986

  41. [41]

    Asgari et al

    M. Asgari et al. , Astronomy & Astrophysics 645, A104 (2021)

    2021

  42. [42]

    A. H. Wright et al. , Astronomy & Astrophysics (2025), arXiv:2503.19441

    work page arXiv 2025

  43. [43]

    T. M. C. Abbott et al. (DES Collaboration), Physical Review D (2026)

    2026

  44. [44]

    Abdalla, G

    E. Abdalla, G. F. Abell´ an, A. Aboubrahim, A. Agnello, ¨O. Akarsu, Y. Akrami, G. Alestas, D. Aloni, L. Amendola, L. A. Anchordoqui, et al. , Journal of High Energy Astrophysics 34, 49 (2022)

    2022

  45. [45]

    Ghirardini et al

    V. Ghirardini et al. , Astronomy & Astrophysics 689, A298 (2024)

    2024

  46. [46]

    Bocquet et al

    S. Bocquet et al. , Physical Review D 110, 083510 (2024)

    2024

  47. [47]

    Euclid Collaboration: Mellier et al

    Y. Euclid Collaboration: Mellier et al. , Astronomy & Astrophysics (2025)

    2025

  48. [48]

    Ivezi´ cet al

    v. Ivezi´ cet al. , The Astrophysical Journal 873, 111 (2019)

    2019

  49. [49]

    Allen et al

    B. Allen et al. , Physical Review D 59 (1997)

    1997

  50. [50]

    Thrane and J

    E. Thrane and J. D. Romano, Physical Review D 88 (2013)

    2013

  51. [51]

    Barack and A

    L. Barack and A. Pound, Reports on Progress in Physics 82, 016904 (2018)

    2018

  52. [52]

    S. W. Weinberg, Gravitation and Cosmology (John Wiley, 1972)

    1972

  53. [53]

    Maggiore, Gravitational Waves Volume 1: Theory and Experiments (Oxford University Press, 2007)

    M. Maggiore, Gravitational Waves Volume 1: Theory and Experiments (Oxford University Press, 2007)

    2007

  54. [54]

    Pottier, Nonequilibrium Statistical Physics: Linear Irreversible Processes (Oxford Graduate Texts, 2010)

    N. Pottier, Nonequilibrium Statistical Physics: Linear Irreversible Processes (Oxford Graduate Texts, 2010)

    2010

  55. [55]

    Ellis, M

    J. Ellis, M. Fairbairn, G. Franciolini, G. H¨ utsi, A. Iovino, M. Lewicki, M. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm¨ ae, Physical Review D109, 023522 (2024). 32

    2024

  56. [56]

    Cai, X.-C

    Y.-F. Cai, X.-C. He, X.-H. Ma, S.-F. Yan, and G.-W. Yuan, Science Bulletin 68, 2929 (2023). 33

    2023