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REVIEW 4 major objections 5 minor 93 references

Constraining inflation with nonminimal derivative coupling with the Parkes Pulsar Timing Array third data release

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Using PPTA DR3, this paper constrains the position, height, and width of the nonminimal derivative coupling that spikes small-scale curvature perturbations in inflation, reporting $\phi_c = 3.7^{+0.3}_{-0.5}M_{\mathrm{P}}$…

desk verdict Useful but contained model-application paper: first PPTA DR3 constraints on nonminimal derivative coupling inflation, built on an analytic phi-k mapping validated at only one point; deserves review but needs a validation scan. read the letter →

arxiv 2412.09755 v2 pith:KLCVV5YG submitted 2024-12-12 gr-qc astro-ph.CO

Chang Han , Li-Yang Chen , Zu-Cheng Chen , Chengjie Fu , Puxun Wu , Hongwei Yu , N. D. Ramesh Bhat , Xiaojin Liu
show 8 more authors
Valentina Di Marco Saurav Mishra Daniel J. Reardon Christopher J. Russell Ryan M. Shannon Lei Zhang Xingjiang Zhu Andrew Zic
This is my paper · ORCID
classification gr-qcastro-ph.CO MSC 83F0583C35 PACS 98.80.Cq04.30.-w
keywords nonminimalderivativecouplinginflationprimordialblackholesscalar-inducedgravitationalwavespulsartimingarraycurvaturepowerspectrumultra-slow-rollPPTADR3
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 the Parkes Pulsar Timing Array's third data release can constrain the shape of a nonminimal derivative coupling that briefly drives inflation into ultra-slow-roll, creating a sharp spike in small-scale curvature perturbations. The authors replace expensive numerical Mukhanov-Sasaki integration with an analytic power spectrum built from a piecewise field-to-wavenumber mapping, and they show that at a representative parameter point the analytic spectrum agrees with the full numerical result to about 10% inside the PTA band. Fitting the model to PPTA DR3 yields $\phi_c = 3.7^{+0.3}_{-0.5} M_{\mathrm{P}}$, $\log_{10} \omega_L = 7.1^{+0.6}_{-0.3}$, and $\log_{10} \sigma = -8.3^{+0.3}_{-0.6}$ at 90% credibility, with a lower bound $\phi_c \gtrsim 3.2 M_{\mathrm{P}}$ at 95%. If correct, pulsar timing arrays can probe the position, height, and width of the friction-enhancement region in an inflation model, complementing cosmic microwave background observations at large scales.

What carries the argument

The load-bearing object is the piecewise analytic mapping from inflaton field value $\phi$ to comoving wavenumber $k$, derived in the Supplementary Material. The authors divide inflation into slow-roll, transition, and ultra-slow-roll stages, use a per-stage approximation for $H/\dot{\phi}$ from the background equations, integrate to get $N(\phi)$, and invert the horizon-crossing condition $c_s k = aH$ to get $\phi(k)$. Substituting the approximated coupling function into the analytic power spectrum gives $P_{\mathcal{R}}(k)$, which is then fed into the standard scalar-induced gravitational wave integral to produce $\Omega_{\mathrm{GW}}(f)$. The approximation is validated at one representative point ($\phi_c/M_{\mathrm{P}}=3.9$, $\omega\lambda=1.53\times10^7$, $\sigma=3\times10^{-9}$), where it agrees with the numerical Mukhanov-Sasaki result to better than about 10% in $\Omega_{\mathrm{GW}}$ in the PTA band.

What would settle it

Evaluate the full numerical Mukhanov-Sasaki power spectrum and the resulting $\Omega_{\mathrm{GW}}(f)$ for a sample of high-posterior-weight parameter points; if the relative error of the analytic approximation in the PTA band ($10^{-9}$–$10^{-7}$ Hz) exceeds about 10% for a substantial fraction of those points, the quoted credible intervals are not reliable.

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Extended reading notes

Core claim

This paper claims that the nonminimal derivative coupling inflation model — in which the inflaton's derivative couples to the Einstein tensor and the coupling function $\theta(\phi)$ is sharply peaked near a field value $\phi_c$ — naturally produces a small-scale enhancement of curvature perturbations through gravitationally enhanced friction, and that PPTA DR3 data can measure the parameters that shape that enhancement. Using an analytically derived $\phi(k)$ mapping, the authors find $\phi_c = 3.7^{+0.3}_{-0.5}M_{\mathrm{P}}$, $\log_{10}\omega_L = 7.1^{+0.6}_{-0.3}$, and $\log_{10}\sigma = -8.3^{+0.3}_{-0.6}$ at 90% credibility, with a 95% lower bound $\phi_c \gtrsim 3.2M_{\mathrm{P}}$. The predicted scalar-induced gravitational wave spectrum is consistent with the free spectra estimated from the data, while a Bayes factor of 0.9 relative to the supermassive-black-hole-binary power-law model says the current data cannot yet decide between the two sources.

Load-bearing premise

The load-bearing premise is that the analytic shortcut that maps the inflaton field to the wavenumber of perturbations is verified at only one choice of parameters; if it becomes inaccurate elsewhere in the allowed parameter range, the quoted constraints would shift.

Editorial extensions

If this is right

  • The data place a 95% lower bound $\phi_c \gtrsim 3.2 M_{\mathrm{P}}$, so the coupling peak cannot sit too deep in the field range allowed by theory.
  • The model's median predicted spectrum is compatible with the PPTA DR3 free spectra, keeping this inflation mechanism a candidate explanation of the common-spectrum process.
  • A Bayes factor of 0.9 against the power-law supermassive-black-hole-binary model means present PTA sensitivity cannot distinguish the two explanations.
  • The analytic spectrum cuts the computational cost from minutes per parameter set to near-instant, enabling full posteriors that would be impractical with numerical Mukhanov-Sasaki integration.

Reading between the lines

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

  • The analytic mapping is checked at only one point; a systematic error analysis across the actual posterior would determine whether the 10% accuracy holds where the constraints concentrate, and could widen the credible intervals if it does not.
  • Because the same three parameters fix the height and location of the curvature spike, these PTA constraints translate directly into a predicted primordial-black-hole mass window under this model, a connection the paper motivates but does not develop.
  • Joint fits to independent pulsar-timing arrays, made cheap by the analytic spectrum, would sharpen these constraints and could begin to distinguish the scalar-induced gravitational wave peak shape from a power law.
  • The normalization $\lambda$ is fixed by CMB-scale observations rather than sampled; treating it as a free parameter in the fit could shift the posterior, especially where the coupling enhances CMB-scale power.
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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 paper derives analytic approximations for the primordial curvature power spectrum in a nonminimal derivative coupling inflation model, including a piecewise mapping from the inflaton field value to wavenumber, and uses the resulting scalar-induced gravitational wave spectrum to analyze the PPTA DR3 data. The authors report posterior constraints on the coupling parameters, φc = 3.7^{+0.3}_{-0.5} M_P, log10 ωL = 7.1^{+0.6}_{-0.3}, and log10 σ = -8.3^{+0.3}_{-0.6} at 90% credibility, and a Bayes factor of 0.9 relative to a power-law SMBHB model. The paper claims the analytic power spectrum reproduces the numerical Mukhanov-Sasaki result within about 10% in ΩGW and about 1% in its spectral index across the parameter space.

Significance. If the analytic mapping is accurate over the relevant parameter space, the paper provides a computationally efficient method for exploring this inflation model and demonstrates that PTA data can, in principle, constrain the shape parameters of the coupling function. The PTA likelihood and noise model follow standard PPTA pipelines, and the analytic expressions are benchmarked against the Mukhanov-Sasaki solver at one parameter point, showing good agreement. However, the central accuracy claims (10% in ΩGW, 1% in ns) are asserted across the whole prior without a supporting scan, and the reported credible intervals are computed with this unvalidated mapping. This is the main caveat that needs to be addressed.

major comments (4)
  1. [II.B and Supplementary Eqs. (S30)-(S50)] The analytic φ-to-k mapping is validated against the numerical Mukhanov-Sasaki solution at a single parameter point (φc=3.9 M_P, ωλ=1.53×10^7, σ=3×10^-9; Figs. 1-3). The paper then claims that the relative error of ΩGW h² is ≲10% and the error in the spectral index ns is ≲1% 'across all model parameter choices' (Sec. II.B). This claim is not supported by any parameter scan or error surface. The posterior reported in Fig. 4 spans φc∈[3.2,4.0] M_P, log10 ωL∈[6.8,7.7], and log10 σ∈[-8.9,-8.0], which includes regions where the transition-region width κ²ωλσφc^p (appearing in Eqs. S31-S34) differs from the validation point by a large factor. The credible intervals in Fig. 4 are computed with this mapping but without propagating its error; if the 10%/1% errors are parameter-dependent or larger than stated, the constraints would be biased. Please run the numerical solver at a grid of points spanning the posterior (or at least the prior extremes), report the error surface, and either incorporate the systematic error into the credible intervals or demonstrate that it is smaller than the statistical uncertainty.
  2. [II.B] The statement that the relative error in ns remains ≲ O(1%) across all model parameter choices is an empirical claim with no supporting calculation in the manuscript or supplementary material. The single-point comparison in Fig. 3 shows agreement in the PTA band, but ns is a local derivative of ΩGW and can be more sensitive to mapping errors, particularly near the peak where the spectrum bends. A derivative-based error estimate or a multi-point scan is needed before this claim can be used to support the published posterior.
  3. [IV and Fig. 5] The posterior predictive spectrum in Fig. 5 is constructed from parameters inferred from the same PPTA DR3 data, so the agreement with the free spectrum is an in-sample consistency check rather than an independent validation of the model. This is not a fatal flaw, but the paper should state this clearly. The Bayes factor of 0.9 (Sec. IV) additionally shows that the data do not prefer this model over a power-law SMBHB background; the quoted parameter constraints are therefore conditional on the model being correct, and the text should emphasize this dependence.
  4. [IV, Fig. 4 and priors] The 90% credible interval for φc is reported as 3.2-4.0 M_P, and the upper endpoint coincides exactly with the prior boundary (φc/M_P ∈ [3,4] set by 'theoretical considerations'). Consequently, the quoted interval is not a fully data-driven 90% range; the upper side is truncated by the prior. The paper should either show a robustness check with a wider prior or explicitly state that only the lower bound is constrained by the data.
minor comments (5)
  1. [Abstract and Sec. IV] The abstract uses '90% confidence level' while the body (Sec. IV) correctly says '90% credible intervals'; please use one terminology consistently throughout the paper.
  2. [Eq. (6), Sec. II.B, and Fig. 4] The notation for the coupling height is inconsistent: Eq. (6) uses ωλ, while Sec. II.B introduces ωL ≡ ωλ and the posterior plots label the parameter as log10 L (presumably ωL). Please define and use a single symbol, e.g., ωL, consistently to avoid confusion.
  3. [Eq. (22) and Supplementary Eqs. (S31)-(S34)] Please explicitly state the normalization convention for aend. The text says a = aend e^N with N=0 at the end of inflation, but it is not stated whether aend is set to 1; an explicit convention is needed to reproduce the peak frequency and the mapping formulas.
  4. [Sec. II.B, λ normalization] The adopted value λ = 8.0×10^-10 is stated with little explanation (Sec. II.B). Please provide the CMB-scale power spectrum value and the procedure used to derive this number, since the combination ωL = ωλ is the effective parameter constrained by the data and a different CMB-consistent normalization could shift the inferred coupling height.
  5. [Supplementary Eqs. (S31)-(S34)] The expressions for k1-k4 contain products of aend and exponentials without consistent bracketing; please add parentheses to make the order of operations unambiguous.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity: the PPTA constraints come from a genuine likelihood fit with an independently benchmarked analytic spectrum; only the Fig. 5 posterior-predictive 'prediction' is an in-sample check presented as viability support.

  1. fitted input called prediction [Section IV, Results and Discussion, paragraph discussing Fig. 5]
    "Fig. 5 presents the posterior predictive distribution for the SIGW energy density spectrum from our model. The median prediction (blue line) and 90% credible region (shaded area) are consistent with the free spectrum estimates from PPTA DR3 (orange violins), supporting the viability of our model."

    The parameters φc, ωL, and σ are inferred by evaluating the likelihood against the same PPTA DR3 dataset whose free-spectrum estimates are shown as orange violins in Fig. 5. The plotted median and 90% credible band are therefore a posterior predictive check of the fit, not an independent prediction: agreement with the free spectra is an in-sample consistency statement, so calling it a 'prediction' that 'support[s] the viability of our model' overstates its evidential weight. This is a presentational circularity rather than a load-bearing one, because the quoted parameter constraints come from the likelihood computation itself and do not reduce to this plot.

full rationale

I walked the derivation chain from the action Eq. (1), through the background equations, the approximate piecewise H/φ̇ solution Eq. (10) and Supplementary Eqs. (S23)-(S50), to PR(k), ΩGW, and the PPTA DR3 likelihood. The only genuinely circular presentation is the Fig. 5 posterior-predictive comparison: the parameters were fitted to PPTA DR3 and the same data are then shown to be consistent with the model's median prediction, which is an in-sample check rather than independent confirmation. The central constraints are not defined into existence by construction: the analytic φ-to-k mapping is benchmarked against the numerical Mukhanov-Sasaki solution at a representative point (Figs. 1-3), the starting power-spectrum form Eq. (6) and spectral indices Eq. (8) rest on prior derivations [64,65] that are themselves compared with the numerical solver and are not fitted to the PTA data, and λ is anchored by CMB-scale normalization to PR(k*) ≃ 2.10×10^-9. The self-citations to [64,65] are therefore supported by independent numerical evidence in this paper, not used as an unreviewed uniqueness constraint. The Bayes factor of 0.9 against the power-law SMBHB model is reported transparently and shows the data do not strongly prefer this model, which is a modeling limitation rather than a circular step. The paper's untested generalization that the ≲10% ΩGW error and ≲1% spectral-index error hold 'across all model parameter choices' is a robustness/correctness concern because the validation is shown at one parameter point only, but that is not circularity. Overall, the core inference chain is self-contained, so the circularity score is low.

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

The model contributes three physically motivated coupling parameters (the fit targets), one hand-set potential normalization λ, and a sequence of analytic approximations. No new entities are postulated. The CMB-scale normalization anchors the amplitude externally, but the analytical mapping carries unquantified systematic risk.

free parameters (4)
  • λ (potential normalization) = 8.0 × 10^-10 (adopted)
    Set by requiring the CMB-scale curvature power spectrum to match observations after accounting for the coupling-induced enhancement. The correction is estimated by hand and not derived from first principles.
  • φc (coupling position) = 3.7 (+0.3/-0.5) M_P
    Target parameter inferred from PPTA DR3. Prior range [3,4] M_P is informed by CMB consistency and PTA detectability.
  • ωL ≡ ωλ (coupling peak height) = log10 = 7.1 (+0.6/-0.3)
    Target parameter inferred from PPTA DR3. Prior range log10 ∈ [4,8].
  • σ (coupling width) = log10 = -8.3 (+0.3/-0.6)
    Target parameter inferred from PPTA DR3. Prior range log10 ∈ [-10,-6].
assumptions (4)
  • domain assumption The inflaton potential is a fractional power law V = λ M_P^{4-p}|φ|^p with p = 2/5.
    Chosen because it satisfies CMB spectral index constraints; no first-principles derivation. Invoked in Sec II A.
  • ad hoc to paper The coupling function θ has the localized functional form of Eq (2).
    Introduced in [64] to create ultra-slow-roll via enhanced friction; its shape is not fixed by an independent symmetry or measurement.
  • domain assumption Instantaneous reheating and radiation-dominated generation of SIGWs with standard present-day propagation factors.
    Used in Eqs (19)-(21) to convert the spectrum to Ω_GW h^2 and frequency; no reheating uncertainty is modeled.
  • domain assumption The piecewise approximations of H/φdot and the φ-k mapping (Eq 10 and Supplementary Eqs S23-S50) remain accurate over the full prior volume.
    Validation is shown for a single parameter set in Figs 1-3; errors are quoted as at most O(10%) in the PTA band but are not propagated into the posteriors.

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Cite this review

Pith. "Pith review of Constraining inflation with nonminimal derivative coupling with the Parkes Pulsar Timing Array third data release." pith.science (2026). https://pith.science/paper/KLCVV5YG

@misc{pith2026241209755,
  author       = {Pith},
  title        = {Pith review of: Constraining inflation with nonminimal derivative coupling with the Parkes Pulsar Timing Array third data release},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLCVV5YG}},
  note         = {Machine review of arXiv:2412.09755}
}
abstract

We study an inflation model with nonminimal derivative coupling that features a coupling between the derivative of the inflaton field and the Einstein tensor. This model naturally amplifies curvature perturbations at small scales via gravitationally enhanced friction, a mechanism critical for the formation of primordial black holes and the associated production of potentially detectable scalar-induced gravitational waves. We derive analytical expressions for the primordial power spectrum, enabling efficient exploration of the model parameter space without requiring computationally intensive numerical solutions of the Mukhanov-Sasaki equation. Using the third data release of the Parkes Pulsar Timing Array (PPTA DR3), we constrain the model parameters characterizing the coupling function: $\phi_c = 3.7^{+0.3}_{-0.5} M_\mathrm{P}$, $\log_{10} \omega_L = 7.1^{+0.6}_{-0.3}$, and $\log_{10} \sigma = -8.3^{+0.3}_{-0.6}$ at 90\% confidence level. Our results demonstrate the growing capability of pulsar timing arrays to probe early Universe physics, complementing traditional cosmic microwave background observations by providing unique constraints on inflationary dynamics at small scales.

Figures

Figures reproduced from arXiv: 2412.09755 by the authors.

Figure 1
Figure 1. FIG. 1. Relationship between the e-folding number [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Primordial power spectrum [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Present-day energy density spectrum of SIGWs, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Posterior distributions for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison between the predicted SIGW energy den [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Works this paper leans on

93 extracted references · 5 canonical work pages

  1. [1]

    A. H. Guth, Phys. Rev. D 23, 347 (1981)

    1981

  2. [2]

    A. D. Linde, Phys. Lett. B 108, 389 (1982)

    1982

  3. [3]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982)

    1982

  4. [4]

    A. H. Guth and S. Y. Pi, Phys. Rev. Lett. 49, 1110 (1982)

    1982

  5. [5]

    D. H. Lyth and A. Riotto, Phys. Rept. 314, 1 (1999), arXiv:hep-ph/9807278

    arXiv 1999

  6. [6]

    Akrami et al

    Y. Akrami et al. (Planck), Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO]

    arXiv 2020

  7. [7]

    Hawking, Mon

    S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 (1971)

    1971

  8. [8]

    B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399 (1974)

    1974

Show all 93 references
  1. [9]

    Garcia-Bellido and E

    J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. 18, 47 (2017), arXiv:1702.03901 [astro-ph.CO]

    2017 arXiv

  2. [10]

    Sasaki, T

    M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Class. Quant. Grav. 35, 063001 (2018), arXiv:1801.05235 [astro-ph.CO]

    2018 arXiv

  3. [11]

    Braglia, D

    M. Braglia, D. K. Hazra, F. Finelli, G. F. Smoot, L. Sri- ramkumar, and A. A. Starobinsky, JCAP 08, 001 (2020), arXiv:2005.02895 [astro-ph.CO]

    2020 arXiv

  4. [12]

    S. Bird, I. Cholis, J. B. Mu˜ noz, Y. Ali-Ha ¨ ımoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, Phys. Rev. Lett. 116, 201301 (2016), arXiv:1603.00464 [astro-ph.CO]

    2016 arXiv

  5. [13]

    Sasaki, T

    M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Phys. Rev. Lett. 117, 061101 (2016), [Erratum: Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 [astro-ph.CO]

    2016 arXiv

  6. [14]

    Chen and A

    Z.-C. Chen and A. Hall, (2024), arXiv:2402.03934 [astro- ph.CO]

    2024 arXiv

  7. [15]

    Seto and A

    N. Seto and A. Cooray, Phys. Rev. D 70, 063512 (2004), arXiv:astro-ph/0405216

    2004 arXiv

  8. [16]

    Saito and J

    R. Saito and J. Yokoyama, Phys. Rev. Lett. 102, 161101 (2009), [Erratum: Phys.Rev.Lett. 107, 069901 (2011)], arXiv:0812.4339 [astro-ph]

    2009 arXiv

  9. [17]

    Antoniadis et al

    J. Antoniadis et al. (EPTA), Astron. Astrophys. 678, A48 (2023), arXiv:2306.16224 [astro-ph.HE]

    2023 arXiv

  10. [18]

    Antoniadis et al

    J. Antoniadis et al. (EPTA, InPTA:), Astron. Astrophys. 678, A50 (2023), arXiv:2306.16214 [astro-ph.HE]

    2023 arXiv

  11. [19]

    Agazie et al

    G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L9 (2023), arXiv:2306.16217 [astro-ph.HE]

    2023 arXiv

  12. [20]

    Agazie et al

    G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L8 (2023), arXiv:2306.16213 [astro-ph.HE]

    2023 arXiv

  13. [21]

    Zic et al., Publ

    A. Zic et al., Publ. Astron. Soc. Austral. 40, e049 (2023), arXiv:2306.16230 [astro-ph.HE]

    2023 arXiv

  14. [22]

    D. J. Reardon et al., Astrophys. J. Lett. 951, L6 (2023), arXiv:2306.16215 [astro-ph.HE]

    2023 arXiv

  15. [23]

    Xu et al., Res

    H. Xu et al., Res. Astron. Astrophys. 23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE]

    2023 arXiv

  16. [24]

    R. w. Hellings and G. s. Downs, Astrophys. J. Lett. 265, L39 (1983)

    1983

  17. [25]

    M. T. Miles et al., Mon. Not. Roy. Astron. Soc.536, 1467 (2025), arXiv:2412.01148 [astro-ph.HE]

    2025 arXiv

  18. [26]

    M. T. Miles et al., Mon. Not. Roy. Astron. Soc.536, 1489 (2025), arXiv:2412.01153 [astro-ph.HE]

    2025 arXiv

  19. [27]

    Agazie et al

    G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 952, L37 (2023), arXiv:2306.16220 [astro-ph.HE]

    2023 arXiv

  20. [28]

    Ellis, M

    J. Ellis, M. Fairbairn, G. H¨ utsi, J. Raidal, J. Urru- tia, V. Vaskonen, and H. Veerm¨ ae, Phys. Rev. D 109, L021302 (2024), arXiv:2306.17021 [astro-ph.CO]

    2024 arXiv

  21. [29]

    Bi, Y.-M

    Y.-C. Bi, Y.-M. Wu, Z.-C. Chen, and Q.-G. Huang, Sci. China Phys. Mech. Astron. 66, 120402 (2023), arXiv:2307.00722 [astro-ph.CO]

    2023 arXiv

  22. [30]

    Afzal et al

    A. Afzal et al. (NANOGrav), Astrophys. J. Lett. 951, L11 (2023), [Erratum: Astrophys.J.Lett. 971, 8 L27 (2024), Erratum: Astrophys.J. 971, L27 (2024)], arXiv:2306.16219 [astro-ph.HE]

    2023 arXiv

  23. [31]

    Antoniadis et al

    J. Antoniadis et al. (EPTA, InPTA), Astron. Astrophys. 685, A94 (2024), arXiv:2306.16227 [astro-ph.CO]

    2024 arXiv

  24. [32]

    Wu, Z.-C

    Y.-M. Wu, Z.-C. Chen, and Q.-G. Huang, Sci. China Phys. Mech. Astron.67, 240412 (2024), arXiv:2307.03141 [astro-ph.CO]

    2024 arXiv

  25. [33]

    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, Phys. Rev. D 109, 023522 (2024), arXiv:2308.08546 [astro-ph.CO]

    2024 arXiv

  26. [34]

    D. G. Figueroa, M. Pieroni, A. Ricciardone, and P. Simakachorn, Phys. Rev. Lett. 132, 171002 (2024), arXiv:2307.02399 [astro-ph.CO]

    2024 arXiv

  27. [35]

    K. N. Ananda, C. Clarkson, and D. Wands, Phys. Rev. D 75, 123518 (2007), arXiv:gr-qc/0612013

    2007 arXiv

  28. [36]

    Baumann, P

    D. Baumann, P. J. Steinhardt, K. Takahashi, and K. Ichiki, Phys. Rev. D 76, 084019 (2007), arXiv:hep- th/0703290

    2007

  29. [37]

    Kohri and T

    K. Kohri and T. Terada, Phys. Rev. D 97, 123532 (2018), arXiv:1804.08577 [gr-qc]

    2018 arXiv

  30. [38]

    Z.-C. Chen, C. Yuan, and Q.-G. Huang, Phys. Rev. Lett. 124, 251101 (2020), arXiv:1910.12239 [astro-ph.CO]

    2020 arXiv

  31. [39]

    Franciolini, A

    G. Franciolini, A. Iovino, Junior., V. Vaskonen, and H. Veermae, Phys. Rev. Lett. 131, 201401 (2023), arXiv:2306.17149 [astro-ph.CO]

    2023 arXiv

  32. [40]

    Liu, Z.-C

    L. Liu, Z.-C. Chen, and Q.-G. Huang, Phys. Rev. D 109, L061301 (2024), arXiv:2307.01102 [astro-ph.CO]

    2024 arXiv

  33. [41]

    Wang, Z.-C

    S. Wang, Z.-C. Zhao, J.-P. Li, and Q.-H. Zhu, Phys. Rev. Res. 6, L012060 (2024), arXiv:2307.00572 [astro-ph.CO]

    2024 arXiv

  34. [42]

    Z. Yi, Q. Gao, Y. Gong, Y. Wang, and F. Zhang, Sci. China Phys. Mech. Astron. 66, 120404 (2023), arXiv:2307.02467 [gr-qc]

    2023 arXiv

  35. [43]

    Tagliazucchi, M

    M. Tagliazucchi, M. Braglia, F. Finelli, and M. Pieroni, Phys. Rev. D 111, L021305 (2025), arXiv:2310.08527 [astro-ph.CO]

    2025 arXiv

  36. [44]

    Liu, Z.-C

    L. Liu, Z.-C. Chen, and Q.-G. Huang, JCAP 11, 071 (2023), arXiv:2307.14911 [astro-ph.CO]

    2023 arXiv

  37. [45]

    Balaji, G

    S. Balaji, G. Dom` enech, and G. Franciolini, JCAP 10, 041 (2023), arXiv:2307.08552 [gr-qc]

    2023 arXiv

  38. [46]

    L. Liu, Y. Wu, and Z.-C. Chen, JCAP 04, 011 (2024), arXiv:2310.16500 [astro-ph.CO]

    2024 arXiv

  39. [47]

    Wang, Z.-C

    S. Wang, Z.-C. Zhao, and Q.-H. Zhu, Phys. Rev. Res. 6, 013207 (2024), arXiv:2307.03095 [astro-ph.CO]

    2024 arXiv

  40. [48]

    Zhu, Z.-C

    Q.-H. Zhu, Z.-C. Zhao, S. Wang, and X. Zhang, Chin. Phys. C 48, 125105 (2024), arXiv:2307.13574 [astro- ph.CO]

    2024 arXiv

  41. [49]

    Z.-Q. You, Z. Yi, and Y. Wu, JCAP 11, 065 (2023), arXiv:2307.04419 [gr-qc]

    2023 arXiv

  42. [50]

    S. A. Hosseini Mansoori, F. Felegray, A. Talebian, and M. Sami, JCAP 08, 067 (2023), arXiv:2307.06757 [astro- ph.CO]

    2023 arXiv

  43. [51]

    Yi, Z.-Q

    Z. Yi, Z.-Q. You, and Y. Wu, JCAP 01, 066 (2024), arXiv:2308.05632 [astro-ph.CO]

    2024 arXiv

  44. [52]

    Yi, Z.-Q

    Z. Yi, Z.-Q. You, Y. Wu, Z.-C. Chen, and L. Liu, JCAP 06, 043 (2024), arXiv:2308.14688 [astro-ph.CO]

    2024 arXiv

  45. [53]

    Harigaya, K

    K. Harigaya, K. Inomata, and T. Terada, Phys. Rev. D 108, 123538 (2023), arXiv:2309.00228 [astro-ph.CO]

    2023 arXiv

  46. [54]

    Z.-C. Chen, J. Li, L. Liu, and Z. Yi, Phys. Rev. D 109, L101302 (2024), arXiv:2401.09818 [gr-qc]

    2024 arXiv

  47. [55]

    Chen and L

    Z.-C. Chen and L. Liu, (2024), arXiv:2402.16781 [astro- ph.CO]

    2024 arXiv

  48. [56]

    W. H. Kinney, Phys. Rev. D 56, 2002 (1997), arXiv:hep- ph/9702427

    1997

  49. [57]

    Inoue and J

    S. Inoue and J. Yokoyama, Phys. Lett. B 524, 15 (2002), arXiv:hep-ph/0104083

    2002 arXiv

  50. [58]

    W. H. Kinney, Phys. Rev. D 72, 023515 (2005), arXiv:gr- qc/0503017

    2005

  51. [59]

    A. Y. Kamenshchik, A. Tronconi, T. Vardanyan, and G. Venturi, Phys. Lett. B 791, 201 (2019), arXiv:1812.02547 [gr-qc]

    2019 arXiv

  52. [60]

    Ballesteros, J

    G. Ballesteros, J. Beltran Jimenez, and M. Pieroni, JCAP 06, 016 (2019), arXiv:1811.03065 [astro-ph.CO]

    2019 arXiv

  53. [61]

    Y.-F. Cai, X. Tong, D.-G. Wang, and S.-F. Yan, Phys. Rev. Lett. 121, 081306 (2018), arXiv:1805.03639 [astro- ph.CO]

    2018 arXiv

  54. [62]

    Chen and Y.-F

    C. Chen and Y.-F. Cai, JCAP 10, 068 (2019), arXiv:1908.03942 [astro-ph.CO]

    2019 arXiv

  55. [63]

    Z. Zhou, J. Jiang, Y.-F. Cai, M. Sasaki, and S. Pi, Phys. Rev. D 102, 103527 (2020), arXiv:2010.03537 [astro- ph.CO]

    2020 arXiv

  56. [64]

    C. Fu, P. Wu, and H. Yu, Phys. Rev. D 100, 063532 (2019), arXiv:1907.05042 [astro-ph.CO]

    2019 arXiv

  57. [65]

    C. Fu, P. Wu, and H. Yu, Phys. Rev. D 101, 023529 (2020), arXiv:1912.05927 [astro-ph.CO]

    2020 arXiv

  58. [66]

    Amendola, Phys

    L. Amendola, Phys. Lett. B 301, 175 (1993), arXiv:gr- qc/9302010

    1993

  59. [67]

    Kaloper, Phys

    N. Kaloper, Phys. Lett. B 583, 1 (2004), arXiv:hep- ph/0312002

    2004

  60. [68]

    S. V. Sushkov, Phys. Rev. D 80, 103505 (2009), arXiv:0910.0980 [gr-qc]

    2009 arXiv

  61. [69]

    Germani and A

    C. Germani and A. Kehagias, Phys. Rev. Lett. 105, 011302 (2010), arXiv:1003.2635 [hep-ph]

    2010 arXiv

  62. [70]

    Germani and Y

    C. Germani and Y. Watanabe, JCAP 07, 031 (2011), [Addendum: JCAP 07, A01 (2011)], arXiv:1106.0502 [astro-ph.CO]

    2011 arXiv

  63. [71]

    Tsujikawa, Phys

    S. Tsujikawa, Phys. Rev. D 85, 083518 (2012), arXiv:1201.5926 [astro-ph.CO]

    2012 arXiv

  64. [72]

    I. D. Gialamas, A. Karam, A. Lykkas, and T. D. Pappas, Phys. Rev. D 102, 063522 (2020), arXiv:2008.06371 [gr- qc]

    2020 arXiv

  65. [73]

    I. D. Gialamas, T. Katsoulas, and K. Tamvakis, JCAP 06, 005 (2024), arXiv:2403.08530 [gr-qc]

    2024 arXiv

  66. [74]

    Silverstein and A

    E. Silverstein and A. Westphal, Phys. Rev. D 78, 106003 (2008), arXiv:0803.3085 [hep-th]

    2008 arXiv

  67. [75]

    Mukhanov, Physical Foundations of Cosmology (Cam- bridge University Press, Oxford, 2005)

    V. Mukhanov, Physical Foundations of Cosmology (Cam- bridge University Press, Oxford, 2005)

    2005

  68. [76]

    Inomata and T

    K. Inomata and T. Nakama, Phys. Rev. D 99, 043511 (2019), arXiv:1812.00674 [astro-ph.CO]

    2019 arXiv

  69. [77]

    Liu, Z.-K

    J. Liu, Z.-K. Guo, and R.-G. Cai, Phys. Rev. D 101, 083535 (2020), arXiv:2003.02075 [astro-ph.CO]

    2020 arXiv

  70. [78]

    Kerr et al

    M. Kerr et al. , Publ. Astron. Soc. Austral. 37, e020 (2020), arXiv:2003.09780 [astro-ph.IM]

    2020 arXiv

  71. [79]

    Goncharov et al

    B. Goncharov et al. , Mon. Not. Roy. Astron. Soc. 502, 478 (2021), arXiv:2010.06109 [astro-ph.HE]

    2021 arXiv

  72. [80]

    D. J. Reardon et al., Astrophys. J. Lett. 951, L7 (2023), arXiv:2306.16229 [astro-ph.HE]

    2023 arXiv

  73. [81]

    Arzoumanian et al

    Z. Arzoumanian et al. (NANOGrav), Astrophys. J. 821, 13 (2016), arXiv:1508.03024 [astro-ph.GA]

    2016 arXiv

  74. [82]

    Enterprise: Enhanced numerical toolbox enabling a ro- bust pulsar inference suite,

    J. A. Ellis, M. Vallisneri, S. R. Taylor, and P. T. Baker, “Enterprise: Enhanced numerical toolbox enabling a ro- bust pulsar inference suite,” Zenodo (2020)

    2020

  75. [83]

    Thrane and J

    E. Thrane and J. D. Romano, Phys. Rev. D 88, 124032 (2013), arXiv:1310.5300 [astro-ph.IM]

    2013 arXiv

  76. [84]

    Aghanim et al

    N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    2020 arXiv

  77. [85]

    R. E. Kass and A. E. Raftery, Journal of the American 9 Statistical Association 90, 773 (1995)

    1995

  78. [86]

    enterprise extensions,

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

    2021

  79. [87]

    Mitridate, D

    A. Mitridate, D. Wright, R. von Eckardstein, T. Schr¨ oder, J. Nay, K. Olum, K. Schmitz, and T. Trickle, (2023), arXiv:2306.16377 [hep-ph]

    2023 arXiv

  80. [88]

    Mitridate, (2023), 10.5281/zenodo.7876430

    A. Mitridate, (2023), 10.5281/zenodo.7876430

    2023 doi

  81. [89]

    B. P. Carlin and S. Chib, Journal of the Royal Statistical Society. Series B (Methodological) 57, 473 (1995)

    1995

  82. [90]

    S. J. Godsill, Journal of Computational and Graphical Statistics 10, 230 (2001)

    2001

  83. [91]

    S. Hee, W. Handley, M. P. Hobson, and A. N. Lasenby, Mon. Not. Roy. Astron. Soc. 455, 2461 (2016), arXiv:1506.09024 [astro-ph.CO]

    2016 arXiv

  84. [92]

    S. R. Taylor, R. van Haasteren, and A. Sesana, Phys. Rev. D 102, 084039 (2020), arXiv:2006.04810 [astro- ph.IM]

    2020 arXiv

  85. [93]

    √ 3kp λϕp c aend #) 1 2 , (S41) ϕ2(k) = ϕc + σ √ 3e−N1−N2−N3 kp λϕp c aend !−3+α , (S42) ϕ3(k) = (ϕc + σ)

    T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog. Theor. Phys. 126, 511 (2011), arXiv:1105.5723 [hep-th]. 1 Constraining inflation with nonminimal derivative coupling with the Parkes Pulsar Timing Array third data release Chang Han, Li-Yang Chen, Zu-Cheng Chen, Chengjie Fu, P...

    2011 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.