Tensor induced gravitational waves

Di Wu; Fei-Yu Chen; Jing-Zhi Zhou; Peng-Yu Wu; Zhi-Chao Li

REVIEW 4 major objections 5 minor 147 references

Tensor induced gravitational waves

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

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Pith's one-line read Small-scale primordial gravitational waves generate their own second-order signal, and one inflationary model with that signal fits pulsar timing array data better than black-hole binaries alone.

desk verdict The NANOGrav application is new and worth refereeing, but the headline Model 2 preference rests on an unchecked assumption—small scalar perturbations—that the paper itself flags. read the letter →

arxiv 2507.22688 v1 pith:BGFXBW3Z submitted 2025-07-30 astro-ph.CO gr-qchep-phhep-th

Fei-Yu Chen , Jing-Zhi Zhou , Di Wu , Zhi-Chao Li , Peng-Yu Wu This is my paper

classification astro-ph.COgr-qchep-phhep-th PACS 04.30.-w98.80.Cq

keywords tensor-inducedgravitationalwavesprimordialsecond-ordercosmologicalperturbationspulsartimingarraysstochasticwavebackgroundenergydensityspectrumblackholesinflation

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

Primordial gravitational waves on small scales are only weakly constrained, so in principle they can be much larger than the almost-scale-invariant tensor modes seen at CMB scales. This paper computes the gravitational-wave signal those small-scale primordial tensor modes generate at second order in cosmological perturbation theory, and gives a closed double-integral formula for its contribution to the energy-density spectrum during radiation domination. Applying the formula to four models that amplify small-scale tensor modes, it finds that second-order tensor induced gravitational waves become pronounced in particular frequency bands once the primordial amplitude is large. When the small-scale primordial waves dominate the nHz background seen by pulsar timing arrays, the correction matters for interpreting the signal; in one of the four models the combined primordial-plus-induced spectrum is preferred over the standard supermassive-black-hole-binary explanation, with a Bayes factor of 16.83 (63.94 when a black-hole component is included).

What carries the argument

The machinery is the second-order tensor perturbation equation with a source built from products of first-order tensor modes, Eq. (13), solved by Green's function methods. In radiation domination the first-order transfer function is $T_h(x)=\sin x/x$, the Green's function is $G_h(x,\bar{x})=(\bar{x}/x)\sin(x-\bar{x})$, and the five second-order kernel functions $I_i^{(2)}(u,v,x)$ reduce to compact expressions in terms of sine and cosine integrals, Eqs. (27)-(31). Contracting the four-point primordial tensor correlator with Wick's theorem and averaging over oscillations converts these kernels into the closed double integral in Eq. (36), with the integration domain $|1-v|\le u\le 1+v$. A property stressed throughout is that this second-order TIGW spectrum is independent of the sound speed and of gauge choice, unlike the scalar-induced gravitational wave spectrum.

What would settle it

Compute the small-scale curvature power spectrum $P_\zeta(k)$ for the four models: if $P_\zeta$ at the relevant scales is not far below the tensor power spectrum, scalar-induced gravitational waves enter at the same order and the claimed Model 2 preference would not survive unchanged. A complementary observational check is to derive the primordial black hole abundance from the second-order density perturbations induced by the same tensor modes for Model 2's best-fit parameters; if that abundance already exceeds observational upper limits, the scenario is excluded even before considering PTA data.

Watch

Extended reading notes

Core claim

The central claim is that second-order tensor induced gravitational waves are not negligible when primordial tensor modes on small scales are large, and that their spectrum is computable from the primordial tensor power spectrum alone during radiation domination. The calculational result is Eq. (36), which expresses $\Omega_{\rm GW}^{(2)}(k)$ as a double integral over $u$ and $v$ of $P_h(uk)P_h(vk)$ times an algebraic kernel built from logarithms; because the two-point function of second-order TIGWs is quartic in the primordial tensor amplitude, the correction scales as the square of $P_h$. Feeding in four inflationary models with enhanced small-scale PGWs, the paper finds that the correction concentrates in different frequency bands depending on the model, and that for its Model 2 the PGW+TIGW spectrum fits the PTA free-spectrum data with a Bayes factor of 16.83 relative to SMBHBs alone, rising to 63.94 when an SMBHB component is included; in that fit the inferred coupling $\alpha$ is slightly lower than in the PGW-only fit.

Load-bearing premise

The calculation assumes the only significant first-order perturbations are the primordial tensor modes; if ordinary density fluctuations at small scales are also large, their induced gravitational waves and mixed scalar-tensor effects appear at the same order and could dominate or alter the correction.

Editorial extensions

If this is right

Eq. (36) turns the TIGW computation into a direct integral: given any primordial tensor power spectrum $P_h(k)$, the second-order correction to $\Omega_{\rm GW}(k)$ follows without re-solving the perturbation equations.
Because the correction scales as $A_h^2$ while the linear spectrum scales as $A_h$, there is a threshold amplitude below which TIGWs are irrelevant and above which they dominate in specific frequency bands; this threshold is crossed by Model 2, and partly by Model 4.
If small-scale PGWs are the source of the pulsar timing array background, the observed spectrum is not a pure power law; the TIGW correction changes its shape at the high-frequency end of the PTA band, which is why the Bayesian comparison can distinguish scenarios.
Models 1 and 3 cannot satisfy large-scale cosmological bounds together with PTA data when TIGWs are included, while Model 2 is the preferred explanation and Model 4 is a weaker alternative.
Under a monochromatic primordial tensor spectrum, combining PTA, CMB+BAO, $\Delta N_{\rm eff}$, and PBH-abundance constraints leaves no region where PGWs+TIGWs alone dominate the PTA signal.

Reading between the lines

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

Because Eq. (36) is model-agnostic, the same integral can be applied to other proposed PGW spectra, such as broken power laws or spectra from other particle-production mechanisms, to test whether second-order corrections will contaminate future high-frequency gravitational-wave searches.
The paper neglects first-order scalar perturbations, but all four models contain scalar fields; if any model predicts a non-negligible curvature power spectrum at the same scales, scalar-induced and mixed scalar-tensor gravitational waves enter at the same order and could either enhance or reduce the claimed preference for Model 2.
A concrete cross-check would be to compute the second-order density perturbations induced by the same tensor modes in Model 2 and compare the implied primordial black hole abundance with observational upper limits; the paper performs this comparison only for a monochromatic spectrum.
Re-running the Bayesian analysis with the full noise-marginalized likelihood underlying the PTA free-spectrum representation, instead of the KDE approximation, would test whether the reported Bayes factors of 16.83 and 63.94 are robust to likelihood details.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one for the PTA application, not for Eq. (36). The second-order TIGW spectrum is a review result from the group's earlier paper [76]; Sec. II is explicitly a review. The genuinely new work is the NANOGrav 15-year likelihood analysis: model-by-model Bayes factors, posterior constraints on the Nieh-Yan coupling alpha, and the observation that including TIGWs shifts the inferred alpha. That part is a legitimate extension, and the conclusion that a monochromatic PGW+TIGW spectrum cannot dominate the PTA signal is clean and useful. The KDE data are public, which helps reproducibility even though no analysis code is shipped.

The central phenomenological claim—that TIGW corrections become pronounced and that Model 2 is preferred (BF 16.83 without SMBHB, 63.94 with)—is conditional on neglecting first-order scalar perturbations. All four models contain scalar fields, and those fields generate curvature perturbations. The paper never computes P_zeta for the fitted parameters, and it concedes in Sec. IV C that it "neglects potentially significant primordial curvature perturbations." That is an acknowledged limitation, but it is load-bearing: at second order, scalar-induced GWs and mixed scalar-tensor contributions enter at the same order as the TIGW term. If P_zeta is not tiny on PTA scales, the Model 2 preference and the inferred alpha could change. A referee should ask for either a computation of P_zeta for the parameters that fit NANOGrav or an explicit presentation of the results as conditional on P_zeta being subdominant.

Two smaller issues. Eq. (35) uses Wick's theorem to reduce the four-point function of PGWs; the paper does not check Gaussianity for the amplification mechanisms, which is an assumption worth stating. Eq. (36) is quoted as a closed-form result without showing the momentum-polynomial contractions, which is acceptable for a review section but should be cross-checked against [76] during refereeing.

I disagree with the reader's circularity concern. The fact that Omega^(2) scales as Ph^2 and the amplitude is fitted to the same signal is not circular; it is the expected consistency condition. The conclusion that corrections become pronounced when PGWs dominate is physical, not tautological.

This paper is for people working on induced GWs, PTA background interpretations, and small-scale PGW constraints. It deserves a serious referee, not a desk rejection. My recommendation: send it to review, with instructions to focus on the scalar-perturbation assumption and, secondarily, the Gaussianity assumption.

Referee Report

4 major / 5 minor

Summary. The paper derives the second-order tensor-induced gravitational wave (TIGW) energy density spectrum sourced by primordial tensor perturbations, giving the closed-form RD-era result in Eq. (36). It applies this formula to four models that generate large-amplitude small-scale primordial gravitational waves (PGWs), fits the models to the NANOGrav 15-year data both with and without a SMBHB component, and reports that Model 2 yields Bayes factors 16.83 (PGW+TIGW vs SMBHB alone) and 63.94 (PGW+TIGW+SMBHB vs SMBHB alone). The paper also places constraints on a monochromatic PGW spectrum using PBH abundance and large-scale cosmological bounds. The central claim is that when small-scale PGWs dominate the PTA signal, second-order TIGW corrections can become pronounced in certain frequency bands.

Significance. If Eq. (36) is correct, the paper provides a new closed-form observable for second-order tensor-tensor induced GWs and a potentially important correction to the interpretation of PTA signals in terms of primordial tensor backgrounds. The Bayesian analysis is concrete, uses public NANOGrav data, and carefully distinguishes which models pass large-scale cosmological constraints. The derivation follows standard second-order perturbation theory, and the paper is explicit about several of its limitations. However, the significance of the PTA claim is conditional on the neglect of first-order scalar perturbations, a condition that is asserted but not verified for the four models considered.

major comments (4)

[Sec. II A, Eq. (1) and Sec. IV C] The calculation sets first-order scalar and vector perturbations to zero, yet all four models in Sec. III contain dynamical scalar fields whose fluctuations can source curvature perturbations. The paper never computes or bounds P_zeta(k) for the fitted parameter values, e.g., Model 2 at alpha ~ 1.24e5. At second order, scalar-induced gravitational waves and mixed scalar-tensor contributions enter at the same order as the TIGW term in Eq. (36); if P_zeta is not tiny on the relevant scales, the PTA-band spectrum and the reported Bayes factors (16.83 and 63.94) can be dominated or reshaped by these additional contributions. Because the manuscript itself acknowledges in Sec. IV C that it neglects potentially significant primordial curvature perturbations, the authors should either compute P_zeta for each model and demonstrate that the neglect is valid, or substantially weaken the PTA-preference claim to an explicitly conditional statement.
[Sec. II C, Eqs. (34)-(36)] The derivation of the central energy-density formula jumps from the two-point correlation function in Eq. (34) to the closed-form double integral in Eq. (36). The contraction of the polarization tensors, the momentum-space algebra, and the integration leading to the large polynomial and logarithmic kernel are not shown. This is the main new result of the paper, and the reported corrections and Bayesian results cannot be independently audited from the text. Please include the intermediate algebra in an appendix or as supplementary material, and verify the numerical coefficients against at least one known limiting case, such as a monochromatic or log-normal power spectrum.
[Sec. II C, Eq. (35)] The four-point function of PGWs is evaluated with Wick's theorem, which assumes Gaussian statistics for the primordial tensor perturbations. The paper does not discuss the conditions under which this holds for the models of Sec. III, where the tensor mode equation is modified by a dynamical scalar field. If the scalar field contributes stochastic fluctuations or non-Gaussianity, Eq. (35) misses connected four-point contributions and Eq. (36) would need correction terms. Please state the Gaussianity assumption explicitly and estimate the magnitude of non-Gaussian corrections for the models studied, or cite prior work establishing that the models are Gaussian at the relevant order.
[Sec. IV C and Table I] The PBH constraints that close the parameter space are computed only for a monochromatic PGW power spectrum. For the four models of Sec. III, the paper states that the corresponding second-order density perturbations and PBH abundance must be computed in future work. Since Question 5 of Sec. I and the summary in Table I are framed as constraints on small-scale PGWs, the model-specific PBH constraints are not actually provided for the models that are the focus of the paper. Please either extend the PBH calculation to the model spectra or restrict the constraints section to the monochromatic case with an explicit caveat in the abstract and conclusions.

minor comments (5)

[Sec. II B, Eqs. (30)-(31)] The indexing of the asymptotic kernels is inconsistent: Eq. (30) labels the sin x/(4x) term as I_3, while Eq. (31) also includes i = 3. From Eq. (27), the sin x/(4x) form is the large-x limit of I_1, not I_3. The labels should be corrected so that the decomposition in Eq. (15) matches the formula used to derive Eq. (36).
[Sec. II A, Eq. (8)] The text says 'The first-order transfer function Th(x) in Eq. (9)' but Eq. (9) appears later and Eq. (8) is the general expression. Please fix the cross-reference.
[Sec. II, Eq. (12)] The transverse projector is written with 'delta_i^i' in T^i_j; this should be delta^i_j. The same typo appears in the surrounding discussion of the decomposed operator.
[Introduction] The acronym FLR W is typeset with an artificial space; it should be FLRW. There are also several similar spacing issues in the equations and headings that should be corrected in a final proof.
[Sec. IV B, Fig. 15] The Bayes factor labels in Fig. 15 are not legible in the text version, and the relationship between the plotted numbers and the values quoted in the text (e.g., 16.83 and 63.94 for Model 2) is not immediately clear. Please enlarge the labels or add a table with the numerical values.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Eq. (36) is a genuine convolution of the input PGW spectrum, and the PTA conclusions are conditional model fits rather than re-labeled inputs.

full rationale

The derivation chain is self-contained: starting from the perturbed FLRW line element (Eq. (1)), the paper derives the first-order tensor equation (Eq. (5)), the second-order tensor equation (Eq. (10)) with source (Eq. (13)), and the kernel functions (Eqs. (22)-(28)) via a Green's function integral. The energy density spectrum (Eq. (36)) is then obtained by applying Wick's theorem to the four-point function of the PGWs, leaving a double integral over Ph(uk)Ph(vk) with a fixed u,v kernel. This is a convolution, not an identity; the output Omega^(2) is not equal to any input by construction. The conditional claim that corrections become pronounced when PGWs dominate PTA observations is a quantitative consequence of the fitted amplitude and the kernel, and the paper explicitly shows that some models (e.g., Model 4) yield negligible corrections, so the statement is not forced. The neglect of first-order scalar and vector perturbations is an explicit assumption (Sec. II A and Sec. IV C: 'we focus solely on PGWs with large amplitudes on small scales while neglecting potentially significant primordial curvature perturbations'), not a circular step; it is a validity condition of the calculation and is acknowledged. Minor self-citations occur (e.g., Ref. [76] for the standard density-perturbation expression, Eq. (14), and Refs. [90], [100], [137] for peripheral constraints), but they are not load-bearing: the cited formulas are standard, externally checkable perturbation-theory results. No fitted parameter is renamed as a prediction; the Bayes factors and posterior alpha values are standard model-comparison outputs from the same data. Therefore the paper exhibits no significant circularity; the score reflects only the presence of minor non-load-bearing self-citations.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central derivation introduces no new free parameters beyond the model couplings alpha that are fitted to PTA data; the model Lagrangians and their parameter choices are imported from Refs [82-84]. The main unstated inputs are Gaussianity of the tensor fluctuations, neglect of scalar perturbations, and correctness of the cited primordial gravitational wave spectra.

free parameters (6)

alpha, Model 1 (Nieh-Yan coupling) = 23.4 (posterior median, PGW-only); with TIGW the posterior peak violates CMB+BAO bound alpha < 22.76
Fitted to NANOGrav 15-year KDE with uniform prior [0,26]; sets the primordial gravitational wave amplitude and therefore the size of the TIGW correction.
alpha, Model 2 (Nieh-Yan coupling) = 1.24^{+0.01}_{-0.02} (PGW+TIGW posterior median)
Fitted to PTA data; the shift from 1.25 (PGW-only) to 1.24 shows that TIGWs supply part of the signal and drives Model 2's high Bayes factor.
alpha, Model 3 (Nieh-Yan coupling) = 1.508 +/- 0.008 (posterior median); model inconsistent with large-scale constraints
Fitted to PTA data, but all alpha values fail CMB+BAO and Neff bounds and the Bayes factor is about 10^-30; included for completeness.
alpha, Model 4 (Nieh-Yan coupling) = 26.9^{+0.2}_{-0.2} (PGW+TIGW posterior median)
Fitted to PTA data; the TIGW correction appears mainly above 10^-7.5 Hz, so it does not affect the PTA band strongly.
SMBHB amplitude A_BHB and spectral index gamma_BHB = Model 2 PGW+TIGW+SMBHB: log10 A_BHB = -15.4^{+0.3}_{-0.4}, gamma_BHB = 4.7^{+0.3}_{-0.3}
Nuisance parameters in the hybrid fits; priors from the NANOGrav multivariate Gaussian in Eq. (62).
Monochromatic PGW spectrum parameters A_h and f_* = posterior support in log10 A_h in [-4,0] and log10(f_*/Hz) in [-10,-5]; excluded by PBH and CMB+BAO constraints
Used in Sec. IV C to show that a monochromatic PGW+TIGW spectrum cannot dominate PTA while obeying other cosmological bounds.

assumptions (5)

standard math Standard FLRW background and second-order cosmological perturbation theory with transverse-traceless projection are valid.
Sec. II A, Eqs. (1)-(13); the derivation of Eq. (10) relies on the Einstein field equations expanded to second order in the FLRW metric.
domain assumption Primordial tensor perturbations are Gaussian, so Wick's theorem factorizes the four-point correlator.
Eq. (35); no non-Gaussianity estimate is given for the amplification mechanisms in Models 1-4, which involve coupled scalar fields and tachyonic or sound-speed-resonance growth.
domain assumption First-order scalar and vector perturbations are negligible for the TIGW calculation.
Sec. II A, text near Eq. (1) and the paragraph on rho^(1) approximately 0; not checked for the model scalar fields in Sec. III.
domain assumption The present-day spectrum is obtained by the standard radiation-era transfer and thermal history mapping in Eq. (38).
Eq. (38); assumes radiation domination at production and standard g* evolution; no modified thermal history is considered.
domain assumption The primordial gravitational wave power spectra for Models 1-4 are correctly computed by the cited model papers.
Sec. III uses spectra from Refs [82-84] as inputs; the paper does not re-derive them from the Lagrangians.

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Cite this review

Pith. "Pith review of Tensor induced gravitational waves." pith.science (2026). https://pith.science/paper/BGFXBW3Z

@misc{pith2026250722688,
  author       = {Pith},
  title        = {Pith review of: Tensor induced gravitational waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BGFXBW3Z}},
  note         = {Machine review of arXiv:2507.22688}
}

read the original abstract

Primordial gravitational waves on small scales are not tightly constrained by current cosmological observations, which allows for the possibility of large amplitudes at small scales. We investigate second-order tensor induced gravitational waves (TIGWs) sourced by primordial gravitational waves and present the corresponding corrections to the total energy density spectrum of gravitational wave. We analyze primordial gravitational waves with large amplitudes generated by various models at small scales. Our results indicate that when primordial gravitational waves on small scales sufficiently dominate the current PTA observations, corrections to the total energy density spectrum from second-order TIGWs may become pronounced in certain frequency bands.

Figures

Figures reproduced from arXiv: 2507.22688 by the authors.

**Figure 3.** FIG. 3: Energy density spectra in Model 3. Dashed [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 6.** FIG. 6: The corner plot of the posterior [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

Figures from the paper (7 more)

**Figure 7.** Figure 7: FIG. 7: The corner plot of the posterior [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: FIG. 9: The energy density spectra of PGWs and [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: FIG. 10: The corner plot of the posterior [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

**Figure 12.** Figure 12: FIG. 12: The corner plot of the posterior [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 14.** Figure 14: FIG. 14: The energy density spectra of PGWs and [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 15.** Figure 15: FIG. 15: The Bayes factors between different models. The vertical axis represents the Bayes factor of [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 16.** Figure 16: FIG. 16: The blue shaded region represents the [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

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Reference graph

Works this paper leans on

147 extracted references · 4 canonical work pages

[76]

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

arXiv 2024
[1]

5 presents the posterior distributions of PGW and PGW+SMBHB, along with constraints from large-scale cosmological observations

Model 1 Fig. 5 presents the posterior distributions of PGW and PGW+SMBHB, along with constraints from large-scale cosmological observations. The prior dis- tribution of α is set as a uniform distribution over the interval [0 , 26]. When only PGWs are consid- ered, PTA data yield the median value of the pos- terior distribution of the parameter α = 23 .4. ...
[2]

7 and Fig

Model 2 For Model 2, the posterior distributions derived from current PTA observations are shown in Fig. 7 and Fig. 8. The prior distribution of α is set as a uniform distribution over the interval [0, 1.56 × 105]. Unlike Model 1, the posterior distribution of param- eter α in Model 2 lies entirely to the left of the black solid line, indicating that Mode...
[3]

10 and Fig

Model 3 Based on current PTA observations, Fig. 10 and Fig. 11 present the posterior distribution of the pa- rameter space for Model 3. The prior distribution of α is set as a uniform distribution over the inter- val [1 .5, 2.4]. In contrast to Models 1 and Model 2, the constraints from large-scale cosmological ob- servations—represented by the black soli...
[4]

12 and Fig

Model 4 As shown in Fig. 12 and Fig. 13, similar to Model 2, Model 4 is capable of dominating the current PTA observations while still satisfying large-scale cosmo- logical constraints. And the prior distribution of α is set as a uniform distribution over the inter- val [24, 30]. The corresponding energy density spec- trum for Model 4 is presented in Fig....
[5]

Baumann, Cosmology (Cambridge University Press, 2022)

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

2022
[6]

J. M. Bardeen, Phys. Rev. D 22, 1882 (1980)

1980
[7]

Kodama and M

H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984)

1984

Show all 147 references

[8]

K. A. Malik and D. Wands, Phys. Rept. 475, 1 (2009), arXiv:0809.4944 [astro-ph]

2009 arXiv
[9]

D. H. Lyth and Y. Rodriguez, Phys. Rev. Lett. 95, 121302 (2005), arXiv:astro-ph/0504045

2005 arXiv
[10]

Weinberg, Phys

S. Weinberg, Phys. Rev. D 72, 043514 (2005), arXiv:hep-th/0506236

2005 arXiv
[11]

B. A. Bassett, S. Tsujikawa, and D. Wands, Rev. Mod. Phys. 78, 537 (2006), arXiv:astro- ph/0507632

2006
[12]

Arkani-Hamed, P

N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga, JCAP 04, 001 (2004), arXiv:hep-th/0312100

2004 arXiv
[13]

Kachru, R

S. Kachru, R. Kallosh, A. D. Linde, J. M. Malda- cena, L. P. McAllister, and S. P. Trivedi, JCAP 10, 013 (2003), arXiv:hep-th/0308055

2003 arXiv
[14]

Armendariz-Picon, T

C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, Phys. Lett. B 458, 209 (1999), arXiv:hep-th/9904075

1999 arXiv
[15]

P. J. E. Peebles and A. Vilenkin, Phys. Rev. D 59, 063505 (1999), arXiv:astro-ph/9810509

1999 arXiv
[16]

Kobayashi, M

T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev. Lett. 105, 231302 (2010), arXiv:1008.0603 [hep-th]

2010 arXiv
[17]

Bezrukov, A

F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. Sibiryakov, JHEP 01, 016 (2011), arXiv:1008.5157 [hep-ph]

2011 arXiv
[18]

Arkani-Hamed, D

N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, JHEP 04, 105 (2020), arXiv:1811.00024 [hep-th]

2020 arXiv
[19]

Dom` enech, Universe 7, 398 (2021), arXiv:2109.01398 [gr-qc]

G. Dom` enech, Universe 7, 398 (2021), arXiv:2109.01398 [gr-qc]

2021 arXiv
[20]

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

2007 arXiv
[21]

Baumann, P

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

2007 arXiv
[22]

Chang, Y.-T

Z. Chang, Y.-T. Kuang, D. Wu, and J.-Z. Zhou, JCAP 2024, 044 (2024), arXiv:2312.14409 [astro- ph.CO]

2024 arXiv
[23]

P. Bari, A. Ricciardone, N. Bartolo, D. Bertacca, and S. Matarrese, Phys. Rev. Lett. 129, 091301 (2022), arXiv:2111.06884 [astro-ph.CO]

2022 arXiv
[24]

P. Bari, D. Bertacca, N. Bartolo, A. Ricciardone, S. Giardiello, and S. Matarrese, JCAP 07, 034 (2023), arXiv:2209.05329 [astro-ph.CO]

2023 arXiv
[25]

Abdelaziz, P

M. Abdelaziz, P. Bari, S. Matarrese, and A. Ric- ciardone, Phys. Rev. D 112, 023505 (2025), arXiv:2504.07063 [astro-ph.CO]

2025 arXiv
[26]

Chang, Y.-T

Z. Chang, Y.-T. Kuang, X. Zhang, and J.-Z. Zhou, Universe 10, 39 (2024), arXiv:2211.11948 [astro- ph.CO]

2024 arXiv
[27]

Zhou, Y.-T

J.-Z. Zhou, Y.-T. Kuang, Z. Chang, X. Zhang, and Q.-H. Zhu, (2023), arXiv:2307.02067 [astro- ph.CO]

2023 arXiv
[28]

Nakama and T

T. Nakama and T. Suyama, Phys. Rev. D 94, 043507 (2016), arXiv:1605.04482 [gr-qc]

2016 arXiv
[29]

Nakama and T

T. Nakama and T. Suyama, Phys. Rev. D 92, 121304 (2015), arXiv:1506.05228 [gr-qc]. 17

2015 arXiv
[30]

De Luca, A

V. De Luca, A. Kehagias, and A. Riotto, Phys. Rev. D 108, 063531 (2023), arXiv:2307.13633 [astro-ph.CO]

2023 arXiv
[31]

Mollerach, D

S. Mollerach, D. Harari, and S. Matarrese, Phys. Rev. D 69, 063002 (2004), arXiv:astro-ph/0310711

2004 arXiv
[32]

B. Cyr, T. Kite, J. Chluba, J. C. Hill, D. Jeong, S. K. Acharya, B. Bolliet, and S. P. Patil, Mon. Not. Roy. Astron. Soc. 528, 883 (2024), arXiv:2309.02366 [astro-ph.CO]

2024 arXiv
[33]

Gurian, D

J. Gurian, D. Jeong, J.-c. Hwang, and H. Noh, Phys. Rev. D104, 083534 (2021), arXiv:2104.03330 [astro-ph.CO]

2021 arXiv
[34]

D’Amico, N

G. D’Amico, N. Bartolo, S. Matarrese, and A. Ri- otto, JCAP 01, 005 (2008), arXiv:0707.2894 [astro- ph]

2008 arXiv
[35]

Yamauchi, T

D. Yamauchi, T. Namikawa, and A. Taruya, JCAP 10, 030 (2012), arXiv:1205.2139 [astro-ph.CO]

2012 arXiv
[36]

S. Saga, D. Yamauchi, and K. Ichiki, Phys. Rev. D 92, 063533 (2015), arXiv:1505.02774 [astro- ph.CO]

2015 arXiv
[37]

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
[38]

Afzal et al

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

2023 arXiv
[39]

Agazie et al

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

2023 arXiv
[40]

Antoniadis et al

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

2023 arXiv
[41]

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

2023 arXiv
[42]

Xu et al., Res

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

2023 arXiv
[43]

Assadullahi and D

H. Assadullahi and D. Wands, Phys. Rev. D 79, 083511 (2009), arXiv:0901.0989 [astro-ph.CO]

2009 arXiv
[44]

Alabidi, K

L. Alabidi, K. Kohri, M. Sasaki, and Y. Sendouda, JCAP 05, 033 (2013), arXiv:1303.4519 [astro- ph.CO]

2013 arXiv
[45]

Dom` enech, C

G. Dom` enech, C. Lin, and M. Sasaki, JCAP 04, 062 (2021), [Erratum: JCAP 11, E01 (2021)], arXiv:2012.08151 [gr-qc]

2021 arXiv
[46]

Dom` enech, S

G. Dom` enech, S. Pi, and M. Sasaki, JCAP08, 017 (2020), arXiv:2005.12314 [gr-qc]

2020 arXiv
[47]

Inomata, K

K. Inomata, K. Kohri, T. Nakama, and T. Terada, JCAP 10, 071 (2019), [Erratum: JCAP 08, E01 (2023)], arXiv:1904.12878 [astro-ph.CO]

2019 arXiv
[48]

S. Wang, T. Terada, and K. Kohri, Phys. Rev. D 99, 103531 (2019), [Erratum: Phys.Rev.D 101, 069901 (2020)], arXiv:1903.05924 [astro-ph.CO]

2019 arXiv
[49]

Liu, Z.-C

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

2023 arXiv
[50]

Dom` enech, S

G. Dom` enech, S. Pi, A. Wang, and J. Wang, JCAP 08, 054 (2024), arXiv:2402.18965 [astro-ph.CO]

2024 arXiv
[51]

Harigaya, K

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

2023 arXiv
[52]

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
[53]

Pearce, L

M. Pearce, L. Pearce, G. White, and C. Bal- azs, JCAP 06, 021 (2024), arXiv:2311.12340 [astro- ph.CO]

2024 arXiv
[54]

Inomata, K

K. Inomata, K. Kohri, T. Nakama, and T. Ter- ada, Phys. Rev. D 100, 043532 (2019), [Erratum: Phys.Rev.D 108, 049901 (2023)], arXiv:1904.12879 [astro-ph.CO]

2019 arXiv
[55]

Zhang, J.-Z

X. Zhang, J.-Z. Zhou, and Z. Chang, Eur. Phys. J. C 82, 781 (2022), arXiv:2208.12948 [astro-ph.CO]

2022 arXiv
[56]

Mangilli, N

A. Mangilli, N. Bartolo, S. Matarrese, and A. Riotto, Phys. Rev. D 78, 083517 (2008), arXiv:0805.3234 [astro-ph]

2008 arXiv
[57]

Yu and S

Y.-H. Yu and S. Wang, Sci. China Phys. Mech. As- tron. 68, 210412 (2025), arXiv:2405.02960 [astro- ph.CO]

2025 arXiv
[58]

Loverde and Z

M. Loverde and Z. J. Weiner, JCAP02, 064 (2023), arXiv:2208.11714 [astro-ph.CO]

2023 arXiv
[59]

S. Saga, K. Ichiki, and N. Sugiyama, Phys. Rev. D 91, 024030 (2015), arXiv:1412.1081 [astro-ph.CO]

2015 arXiv
[60]

X.-B. Sui, J. Liu, X.-Y. Yang, and R.-G. Cai, Phys. Rev. D110, 103541 (2024), arXiv:2407.04220 [astro-ph.CO]

2024 arXiv
[61]

Madge, E

E. Madge, E. Morgante, C. Puchades-Ib´ a˜ nez, N. Ramberg, W. Ratzinger, S. Schenk, and P. Schwaller, JHEP 10, 171 (2023), arXiv:2306.14856 [hep-ph]

2023 arXiv
[62]

Dom` enech and S

G. Dom` enech and S. Pi, Sci. China Phys. Mech. As- tron. 65, 230411 (2022), arXiv:2010.03976 [astro- ph.CO]

2022 arXiv
[63]

Hwang, D

J.-C. Hwang, D. Jeong, and H. Noh, Astrophys. J. 842, 46 (2017), arXiv:1704.03500 [astro-ph.CO]

2017 arXiv
[64]

De Luca, G

V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, JCAP 03, 014 (2020), arXiv:1911.09689 [gr-qc]

2020 arXiv
[65]

Y. Lu, A. Ali, Y. Gong, J. Lin, and F. Zhang, Phys. Rev. D 102, 083503 (2020), arXiv:2006.03450 [gr- qc]

2020 arXiv
[66]

Tomikawa and T

K. Tomikawa and T. Kobayashi, Phys. Rev. D101, 083529 (2020), arXiv:1910.01880 [gr-qc]

2020 arXiv
[67]

Dom` enech and M

G. Dom` enech and M. Sasaki, Phys. Rev. D 103, 063531 (2021), arXiv:2012.14016 [gr-qc]

2021 arXiv
[68]

Inomata and T

K. Inomata and T. Terada, Phys. Rev. D 101, 023523 (2020), arXiv:1912.00785 [gr-qc]

2020 arXiv
[69]

Yuan, Z.-C

C. Yuan, Z.-C. Chen, and Q.-G. Huang, Phys. Rev. D 101, 6 (2020), arXiv:1912.00885 [astro-ph.CO]

2020 arXiv
[70]

Yuan, Z.-C

C. Yuan, Z.-C. Chen, and L. Liu, Phys. Rev. D 111, 103528 (2025), arXiv:2410.18996 [gr-qc]

2025 arXiv
[71]

A. A. Kugarajh, (2025), 10.1088/1361- 6382/ade2b3, arXiv:2503.00083 [gr-qc]

2025 arXiv
[72]

Dom` enech, Int

G. Dom` enech, Int. J. Mod. Phys. D 29, 2050028 (2020), arXiv:1912.05583 [gr-qc]

2020 arXiv
[73]

Gong, Astrophys

J.-O. Gong, Astrophys. J. 925, 102 (2022), arXiv:1909.12708 [gr-qc]

2022 arXiv
[74]

Balaji, G

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

2023 arXiv
[75]

Jin, Z.-C

J.-H. Jin, Z.-C. Chen, Z. Yi, Z.-Q. You, L. Liu, and Y. Wu, JCAP 09, 016 (2023), arXiv:2307.08687 [astro-ph.CO]

2023 arXiv
[77]

Li, G.-H

J. Li, G.-H. Guo, and Y. Zu, Eur. Phys. J. C 84, 1083 (2024), arXiv:2312.04589 [gr-qc]. 18

2024 arXiv
[78]

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

2024 arXiv
[79]

Wang, L.-Y

L.-S. Wang, L.-Y. Chen, and Q.-T. Xie, (2025), arXiv:2503.01460 [astro-ph.CO]

2025
[80]

Zhou, Y.-T

J.-Z. Zhou, Y.-T. Kuang, D. Wu, F.-Y. Chen, H. L¨ u, and Z. Chang, JCAP 12, 021 (2024), arXiv:2409.07702 [gr-qc]

2024 arXiv
[81]

Maggiore, Phys

M. Maggiore, Phys. Rept. 331, 283 (2000), arXiv:gr-qc/9909001

2000 arXiv
[82]

Yuan and Q.-G

C. Yuan and Q.-G. Huang, iScience 24, 102860 (2021), arXiv:2103.04739 [astro-ph.GA]

2021 arXiv
[83]

M. A. Gorji and M. Sasaki, Phys. Lett. B 846, 138236 (2023), arXiv:2302.14080 [gr-qc]

2023 arXiv
[84]

M. A. Gorji, M. Sasaki, and T. Suyama, Phys. Lett. B 846, 138214 (2023), arXiv:2307.13109 [astro-ph.CO]

2023 arXiv
[85]

Biagetti, M

M. Biagetti, M. Fasiello, and A. Riotto, Phys. Rev. D 88, 103518 (2013), arXiv:1305.7241 [astro- ph.CO]

2013 arXiv
[86]

C. Fu, J. Liu, X.-Y. Yang, W.-W. Yu, and Y. Zhang, Phys. Rev. D 109, 063526 (2024), arXiv:2308.15329 [astro-ph.CO]

2024 arXiv
[87]

R.-G. Cai, C. Fu, and W.-W. Yu, Phys. Rev. D 105, 103520 (2022), arXiv:2112.04794 [astro- ph.CO]

2022 arXiv
[88]

Jiang, Y

Z.-W. Jiang, Y. Cai, F. Wang, and Y.-S. Piao, JHEP 09, 067 (2024), arXiv:2406.16549 [astro- ph.CO]

2024 arXiv
[89]

Y.-F. Cai, C. Lin, B. Wang, and S.-F. Yan, Phys. Rev. Lett. 126, 071303 (2021), arXiv:2009.09833 [gr-qc]

2021 arXiv
[90]

Addazi, Y

A. Addazi, Y. Aldabergenov, and Y. Cai, Phys. Rev. D 110, 123530 (2024), arXiv:2408.05091 [gr- qc]

2024 arXiv
[91]

M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, Riv. Nuovo Cim. 39, 399 (2016), arXiv:1605.01615 [astro-ph.CO]

2016 arXiv
[92]

M. Li, H. Rao, and D. Zhao, JCAP 11, 023 (2020), arXiv:2007.08038 [gr-qc]

2020 arXiv
[93]

M. Li, H. Rao, and Y. Tong, Phys. Rev. D 104, 084077 (2021), arXiv:2104.05917 [gr-qc]

2021 arXiv
[94]

Zhou, Y.-T

J.-Z. Zhou, Y.-T. Kuang, Z. Chang, and H. L¨ u, Astrophys. J. 979, 178 (2025), arXiv:2410.10111 [astro-ph.CO]

2025 arXiv
[95]

Cang, Y.-Z

J. Cang, Y.-Z. Ma, and Y. Gao, Astrophys. J. 949, 64 (2023), arXiv:2210.03476 [astro-ph.CO]

2023 arXiv
[96]

Ben-Dayan, B

I. Ben-Dayan, B. Keating, D. Leon, and I. Wolf- son, JCAP 06, 007 (2019), arXiv:1903.11843 [astro- ph.CO]

2019 arXiv
[97]

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
[98]

Auclair et al

P. Auclair et al. (LISA Cosmology Work- ing Group), Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro-ph.CO]

2023 arXiv
[99]

Iacconi, M

L. Iacconi, M. Bacchi, L. F. Guimar˜ aes, and F. T. Falciano, (2024), arXiv:2412.02544 [astro-ph.CO]

2024 arXiv
[100]

Flauger, N

R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone, and J. Torrado, JCAP 01, 059 (2021), arXiv:2009.11845 [astro-ph.CO]

2021 arXiv
[101]

Wu et al

Y.-L. Wu et al. (Taiji Scientific), Commun. Phys. 4, 34 (2021)

2021
[102]

Chluba and D

J. Chluba and D. Grin, Mon. Not. Roy. Astron. Soc. 434, 1619 (2013), arXiv:1304.4596 [astro- ph.CO]

2013 arXiv
[103]

Bringmann, D

T. Bringmann, D. Croon, and S. Sevillano Mu˜ noz, (2025), arXiv:2506.20704 [astro-ph.CO]

2025 arXiv
[104]

Zhou, Z.-C

J.-Z. Zhou, Z.-C. Li, and D. Wu, (2025), arXiv:2505.22614 [astro-ph.CO]

2025 arXiv
[105]

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
[106]

Chluba, A

J. Chluba, A. L. Erickcek, and I. Ben-Dayan, As- trophys. J. 758, 76 (2012), arXiv:1203.2681 [astro- ph.CO]

2012 arXiv
[107]

Chluba, R

J. Chluba, R. Khatri, and R. A. Sunyaev, Mon. Not. Roy. Astron. Soc. 425, 1129 (2012), arXiv:1202.0057 [astro-ph.CO]

2012 arXiv
[108]

Sharma, J

D. Sharma, J. Lesgourgues, and C. T. Byrnes, JCAP 07, 090 (2024), arXiv:2404.18474 [astro- ph.CO]

2024 arXiv
[109]

C. T. Byrnes, J. Lesgourgues, and D. Sharma, JCAP 09, 012 (2024), arXiv:2404.18475 [astro- ph.CO]

2024 arXiv
[110]

T. J. Clarke, E. J. Copeland, and A. Moss, JCAP 10, 002 (2020), arXiv:2004.11396 [astro-ph.CO]

2020 arXiv
[111]

W. G. Lamb, S. R. Taylor, and R. van Haasteren, Phys. Rev. D108, 103019 (2023), arXiv:2303.15442 [astro-ph.HE]

2023 arXiv
[112]

C. J. Moore and A. Vecchio, Nature Astron.5, 1268 (2021), arXiv:2104.15130 [astro-ph.CO]

2021 arXiv
[113]

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
[114]

Kde representations of the gravitational wave background free spectra present in the nanograv 15-year dataset,

T. N. Collaboration, “Kde representations of the gravitational wave background free spectra present in the nanograv 15-year dataset,” (2023)

2023
[115]

Ashton et al., Astrophys

G. Ashton et al., Astrophys. J. Suppl. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM]

2019 arXiv
[116]

J. S. Speagle, Mon. Not. Roy. Astron. Soc. 493, 3132 (2020), arXiv:1904.02180 [astro-ph.IM]

2020 arXiv
[117]

joshspeagle/dynesty: v2.1.4,

S. Koposov, J. Speagle, K. Barbary, G. Ashton, E. Bennett, J. Buchner, C. Scheffler, B. Cook, C. Talbot, J. Guillochon, P. Cubillos, A. A. Ramos, M. Dartiailh, Ilya, E. Tollerud, D. Lang, B. John- son, jtmendel, E. Higson, T. Vandal, T. Day- lan, R. Angus, patelR, P. Cargile, ...

2024
[118]

B. J. Carr, Astrophys. J. 201, 1 (1975)

1975
[119]

B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Rept. Prog. Phys.84, 116902 (2021), arXiv:2002.12778 [astro-ph.CO]

2021 arXiv
[120]

Carr and F

B. Carr and F. Kuhnel, Ann. Rev. Nucl. Part. Sci. 70, 355 (2020), arXiv:2006.02838 [astro-ph.CO]

2020 arXiv
[121]

B. Carr, F. Kuhnel, and M. Sandstad, Phys. Rev. D 94, 083504 (2016), arXiv:1607.06077 [astro- ph.CO]

2016 arXiv
[122]

B. J. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Phys. Rev. D 81, 104019 (2010), arXiv:0912.5297 [astro-ph.CO]

2010 arXiv
[123]

Musco, Phys

I. Musco, Phys. Rev. D 100, 123524 (2019), arXiv:1809.02127 [gr-qc]. 19

2019 arXiv
[124]

Germani and I

C. Germani and I. Musco, Phys. Rev. Lett. 122, 141302 (2019), arXiv:1805.04087 [astro-ph.CO]

2019 arXiv
[125]

Ferrante, G

G. Ferrante, G. Franciolini, A. Iovino, Junior., and A. Urbano, Phys. Rev. D 107, 043520 (2023), arXiv:2211.01728 [astro-ph.CO]

2023 arXiv
[126]

Shimada, A

M. Shimada, A. Escriv´ a, D. Saito, K. Uehara, and C.-M. Yoo, JCAP02, 018 (2025), arXiv:2411.07648 [gr-qc]

2025 arXiv
[127]

A. J. Iovino, G. Perna, A. Riotto, and H. Veerm¨ ae, JCAP 10, 050 (2024), arXiv:2406.20089 [astro- ph.CO]

2024 arXiv
[128]

Musco, V

I. Musco, V. De Luca, G. Franciolini, and A. Riotto, Phys. Rev. D 103, 063538 (2021), arXiv:2011.03014 [astro-ph.CO]

2021 arXiv
[129]

Young, C

S. Young, C. T. Byrnes, and M. Sasaki, JCAP 07, 045 (2014), arXiv:1405.7023 [gr-qc]

2014 arXiv
[130]

A. M. Green, A. R. Liddle, K. A. Malik, and M. Sasaki, Phys. Rev. D 70, 041502 (2004), arXiv:astro-ph/0403181

2004 arXiv
[131]

De Luca, G

V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto, and C. ¨Unal, JCAP 07, 048 (2019), arXiv:1904.00970 [astro-ph.CO]

2019 arXiv
[132]

Ballesteros and M

G. Ballesteros and M. Taoso, Phys. Rev. D 97, 023501 (2018), arXiv:1709.05565 [hep-ph]

2018 arXiv
[133]

Kannike, L

K. Kannike, L. Marzola, M. Raidal, and H. Veerm¨ ae, JCAP 09, 020 (2017), arXiv:1705.06225 [astro-ph.CO]

2017 arXiv
[134]

J. M. Ezquiaga, J. Garcia-Bellido, and E. Ruiz Morales, Phys. Lett. B 776, 345 (2018), arXiv:1705.04861 [astro-ph.CO]

2018 arXiv
[135]

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
[136]

Firouzjahi and A

H. Firouzjahi and A. Riotto, JCAP 02, 021 (2024), arXiv:2304.07801 [astro-ph.CO]

2024 arXiv
[137]

P. B. Ferraz and J. G. Rosa, JCAP 03, 040 (2025), arXiv:2410.10996 [hep-ph]

2025 arXiv
[138]

Kristiano and J

J. Kristiano and J. Yokoyama, Phys. Rev. Lett. 132, 221003 (2024), arXiv:2211.03395 [hep-th]

2024 arXiv
[139]

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
[140]

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
[141]

Wu, Z.-C

D. Wu, Z.-C. Li, P.-Y. Wu, F.-Y. Chen, and J.-Z. Zhou, (2025), arXiv:2507.07836 [astro-ph.CO]

2025
[142]

Choudhury, S

S. Choudhury, S. Panda, and M. Sami, Phys. Lett. B 845, 138123 (2023), arXiv:2302.05655 [astro- ph.CO]

2023 arXiv
[143]

Barker, B

W. Barker, B. Gladwyn, and S. Zell, Phys. Rev. D 111, 123033 (2025), arXiv:2410.11948 [astro- ph.CO]

2025 arXiv
[144]

B.-Y. Su, N. Li, and L. Feng, Eur. Phys. J. C 85, 197 (2025), arXiv:2502.14323 [astro-ph.CO]

2025 arXiv
[145]

Wang, X.-H

X. Wang, X.-H. Ma, and Y.-F. Cai, Int. J. Mod. Phys. D 34, 2550027 (2025), arXiv:2412.19631 [astro-ph.CO]

2025 arXiv
[146]

Inomata, JCAP 03, 013 (2021), arXiv:2008.12300 [gr-qc]

K. Inomata, JCAP 03, 013 (2021), arXiv:2008.12300 [gr-qc]

2021 arXiv
[147]

C. Chen, A. Ota, H.-Y. Zhu, and Y. Zhu, Phys. Rev. D107, 083518 (2023), arXiv:2210.17176 [astro-ph.CO]

2023 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [76]

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

arXiv 2024

[2] [1]

5 presents the posterior distributions of PGW and PGW+SMBHB, along with constraints from large-scale cosmological observations

Model 1 Fig. 5 presents the posterior distributions of PGW and PGW+SMBHB, along with constraints from large-scale cosmological observations. The prior dis- tribution of α is set as a uniform distribution over the interval [0 , 26]. When only PGWs are consid- ered, PTA data yield the median value of the pos- terior distribution of the parameter α = 23 .4. ...

[3] [2]

7 and Fig

Model 2 For Model 2, the posterior distributions derived from current PTA observations are shown in Fig. 7 and Fig. 8. The prior distribution of α is set as a uniform distribution over the interval [0, 1.56 × 105]. Unlike Model 1, the posterior distribution of param- eter α in Model 2 lies entirely to the left of the black solid line, indicating that Mode...

[4] [3]

10 and Fig

Model 3 Based on current PTA observations, Fig. 10 and Fig. 11 present the posterior distribution of the pa- rameter space for Model 3. The prior distribution of α is set as a uniform distribution over the inter- val [1 .5, 2.4]. In contrast to Models 1 and Model 2, the constraints from large-scale cosmological ob- servations—represented by the black soli...

[5] [4]

12 and Fig

Model 4 As shown in Fig. 12 and Fig. 13, similar to Model 2, Model 4 is capable of dominating the current PTA observations while still satisfying large-scale cosmo- logical constraints. And the prior distribution of α is set as a uniform distribution over the inter- val [24, 30]. The corresponding energy density spec- trum for Model 4 is presented in Fig....

[6] [5]

Baumann, Cosmology (Cambridge University Press, 2022)

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

2022

[7] [6]

J. M. Bardeen, Phys. Rev. D 22, 1882 (1980)

1980

[8] [7]

Kodama and M

H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984)

1984