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The decay rate of the cosmological gravitational potential, measured from CMB-galaxy cross-correlations, is consistent with general relativity at redshifts 0.2 to 1.4.

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2026-08-03 23:41 UTC pith:FVKETKQA

load-bearing objection A clean first application of the measured gravitational-potential decay rate to modified-gravity constraints; the result is consistent with GR and the analysis is honest, but the quoted errors depend on six DR points whose covariance is not shown. the 2 major comments →

arxiv 2511.04279 v2 pith:FVKETKQA submitted 2025-11-06 astro-ph.CO

Constraining gravity with the decay rate of cosmological gravitational potential

Xinyi Zhao , Pengjie Zhang , Fuyu Dong This is my paper
classification astro-ph.CO
keywords decay rate of gravitational potentialmodified gravityintegrated Sachs-Wolfe effectCMB lensingstructure growthgeneral relativity testcosmological gravitational potentialeffective gravitational constant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the decay rate (DR) of the cosmological gravitational potential — the quantity behind the integrated Sachs-Wolfe effect and CMB lensing — can serve as a clean, bias-free probe of the nature of gravity. Using a recent DR measurement over 0.2 ≤ z ≤ 1.4, the authors test four one-parameter modified-gravity extensions to flat ΛCDM. They find γ = 0.47^{+0.22}_{-0.15}, consistent with the general-relativistic value γ ≃ 0.55, and values of the effective gravitational-constant parameters Σ_Λ, Σ_1, Σ_2 all consistent with zero within about 0.5σ. The reason a reader should care is that even a modest 3.1σ measurement yields constraints on modified gravity as tight as those from combinations of several major surveys, suggesting DR is an efficient and largely model-independent route to testing GR at cosmological scales.

Core claim

The central claim is that the existing DR measurement, which combines the integrated Sachs-Wolfe effect and CMB lensing through galaxy cross-correlations, already delivers competitive one-parameter modified-gravity constraints. The authors find that the growth-index parameter γ is 0.47 with a 1σ range enclosing the GR prediction 0.55, and that the effective gravitational constant G_eff/G = 1 + Σ(a) shows no significant deviation from unity under three different redshift parameterizations (Σ ∝ Ω_Λ(a), Σ ∝ a, Σ ∝ a²). All three Σ parameters are consistent with zero within about half a sigma. The paper interprets this as support for general relativity and demonstrates that the DR method, despit

What carries the argument

The decay rate DR is the combination of the ISW-galaxy and CMB lensing-galaxy cross-correlations. It equals [−d ln D_{ψ+ϕ}/d ln a] × [aH(z)/c W_L(z)], where D_{ψ+ϕ} is the linear growth factor of the Weyl potential ψ+ϕ and W_L is the lensing kernel. Its power comes from the fact that the galaxy bias cancels in the ratio, so DR depends essentially only on the matter density Ω_m and on gravity, not on H_0, σ_8, n_s, or the sound horizon. Under sub-horizon linear perturbation theory, DR is sensitive to the growth index f = Ω_m^γ and to any departure of the effective Newton constant from unity, encoded in Σ(a). The paper fixes Ω_m = 0.3169 from CMB constraints and varies one gravity parameter at

Load-bearing premise

All constraints rest on six DR measurements (Table 1) being unbiased, Gaussian, and effectively independent; the paper does not propagate the imaging and magnification systematics mitigated in the companion paper, nor does it provide the covariance, so a shared systematic shift or bin correlation could move γ and Σ_X by more than the quoted errors.

What would settle it

Recompute the DR fit using the full covariance matrix for the six bins (e.g., from the companion paper's maps). If the off-diagonal correlations inflate the reported 1σ errors on γ and Σ_X by more than a factor of 1.5, the current GR-consistency loses statistical significance. Alternatively, a future DR measurement at z > 1.4 with σ_DR ≈ 0.03 that deviates from the ΛCDM prediction by more than 2σ would directly challenge the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The DR probe provides a nearly bias-free, independent confirmation that structure growth at z ≲ 1.4 is consistent with GR, strengthening the ΛCDM+GR picture.
  • Constraints on Σ (the effective gravitational constant modification) at the σ ~ 0.05 level are achievable without external datasets, and can be used to cross-check more complex modified-gravity analyses.
  • The measured γ = 0.47 is in tension with the higher γ ≈ 1.2 values inferred from some cluster counts and CMB-lensing S8 measurements, indicating that systematics or new physics may affect one of the probes.
  • Upcoming full-sky galaxy surveys are expected to halve DR uncertainties, making DR a standard tool in gravity tests.
  • The DR dependency on Ω_m is mild, so the current BAO/CMB/SNe Ω_m tension does not affect the conclusions.

Where Pith is reading between the lines

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

  • If DR is combined with independent growth-rate probes such as redshift-space distortions, the Σ–η degeneracy could be broken, yielding a two-dimensional map of both the gravitational slip parameter and the effective Newton constant.
  • The method could be extended to higher redshifts and to galaxy clusters as tracers, and to higher-resolution CMB lensing maps, which would sharpen the test and probe gravity in a regime where GR has not been directly checked.
  • The discrepancy between the DR-based γ and the cluster-based γ may reflect scale dependence in modified gravity or in the bias treatment of cluster samples; a scale-dependent joint analysis would be a natural next step.
  • Publication of the full covariance between the six DR bins would allow future analyses to propagate systematic errors robustly and could widen or narrow the quoted uncertainties.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper uses the recently measured decay rate (DR) of the cosmological gravitational potential at 0.2 ≤ z ≤ 1.4 (Dong et al. 2025) to constrain one-parameter modified-gravity extensions of flat ΛCDM. The DR combines ISW and CMB lensing cross-correlations, cancelling galaxy bias and clustering amplitude. The authors fit the growth index γ in the f = Ω_m^γ parametrization, finding γ = 0.47^{+0.22}_{-0.15}, consistent with the GR value ~0.55. They also fit three parametrizations of the effective gravitational constant, Σ(a) = Σ_Λ Ω_Λ(a)/Ω_Λ, Σ_1 a, and Σ_2 a^2, finding Σ_Λ = 0.018^{+0.052}_{-0.053}, Σ_1 = 0.020^{+0.065}_{-0.062}, Σ_2 = 0.027^{+0.067}_{-0.069}, all consistent with zero. The paper argues the DR constraints are competitive with multi-probe analyses and checks the sensitivity to Ω_m, showing the results are robust within current Ω_m uncertainties.

Significance. If the input DR errors are reliable, this paper offers a qualitatively new and independent probe of gravity at late times, with distinctive advantages: DR is insensitive to galaxy bias, σ8, H0, and several other nuisance parameters that affect standard structure-growth probes. The derived constraint on Σ_Λ (σ ≃ 0.05) is claimed to be comparable to state-of-the-art combined DESI/Planck/DES analyses, which would be notable for a single 3.1σ measurement. The paper is transparent about its modeling assumptions (fixing the gravitational slip η = 1, adopting Ω_m from CMB) and explicitly tests the dependence on Ω_m. The main risk is the statistical treatment of the six DR data points, which is not fully presented; the central claim of compatibility with GR is likely robust, but the quoted precision and the 'competitive' claim rest on that treatment.

major comments (2)
  1. [§2, Table 1] The six DR measurements in Table 1 are treated as independent, unbiased Gaussian points, but the paper does not provide a covariance matrix or a systematic-error budget for them. These measurements share the same Planck CMB maps, the same DESI DR9 galaxy catalog, and adjacent bins have overlapping ISW and lensing kernels; residual systematics such as imperfect magnification correction are not propagated from Dong et al. (2025). If the bins are positively correlated, the effective number of independent measurements is less than six and the quoted 1σ errors on γ and Σ in Table 2 are underestimated. This is load-bearing for the claimed precision and for the comparison to other probes. The authors should either incorporate the full covariance from the companion paper or explicitly justify the independence assumption (e.g., by showing the correlation matrix) and propagate the systematic uncer
  2. [§3] The fitting procedure is underspecified. The paper does not state the likelihood function, whether the asymmetric 1σ errors in Table 1 are symmetrized or handled with a Gaussian approximation, what priors (if any) are imposed on γ and Σ, or which redshift bin centers are used in the model prediction (the listed z_m or the bin edges). Without this information the analysis is not reproducible. The authors should provide the χ² definition, the priors, and a table of per-bin model predictions and residuals for each of the four fits.
minor comments (5)
  1. [Table 1 caption] Typo: "T able" should be "Table".
  2. [References] The DOI for Abdul Karim et al. (2025) appears as "10.1103/tr6y-kpc6", which looks invalid. Please verify and correct.
  3. [§3.1, Fig. 2] The comparison with the eROSITA and CMB-lensing γ constraints is qualitative. The statement "The data prefer γ~1.2 or even larger, unless the data point in the lowest redshift bin is excluded" would be more informative if quantified, e.g., by reporting the Δχ² between fits with and without that point. As written, it risks being interpreted as cherry-picking without a statistical justification.
  4. [§3.2, Eq. (5)] The symbol f in Eq. (5) is not explicitly defined in that section; it is defined earlier in §3.1 as the matter growth rate, but for the Σ parameterizations the reader must infer that f = dln D/dln a. Please restate the definition for clarity.
  5. [References] The text cites "P. Zhang (2006)" in two places, but there are two 2006 papers by this author in the bibliography (ApJ 647, 55 and PRD 73, 123504). Please disambiguate, for example by citing the ApJ paper for the DR method and the PRD paper for the f(R) simplification, or citing both where appropriate.

Circularity Check

0 steps flagged

No significant circularity: the MG parameters are fitted to an external DR measurement, not recovered from the parameters themselves.

full rationale

The derivation chain is linear and non-circular. DR is defined in Eq. (2) from ISW and CMB-lensing cross-correlations, and the six measured DR values are taken from the companion paper (F. Dong et al. 2025) and listed in Table 1. The modified-gravity models are then forward-modeled: Eqs. (4)-(6) predict DR given γ or Σ_X, and a likelihood fit compares these predictions to the external Table 1 values, yielding the results in Table 2. No step defines the fitted MG parameters in terms of the measured DR values, and no prediction is obtained by renaming the input. The self-citations to P. Zhang (2006) and F. Dong et al. (2025) supply the method and the input measurement, respectively, but neither is used as a substitute for the fitting derivation, and the measurement itself is based on independent DESI imaging and Planck data. The reviewer-identified concern about covariance or systematics among the six DR bins is a data-quality and statistical-robustness issue, not circularity: it does not make the fitted parameters equivalent to the inputs by construction. Under the stated rules, no circular step is present.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The analysis is deliberately frugal: each MG model has exactly one fitted parameter plus a fixed Ωm. The major external inputs are the six DR measurements from the authors' companion paper and the stated assumption that the Weyl potential follows GR under these parameterizations. No new particles or forces are introduced.

free parameters (4)
  • γ (growth index) = 0.47^{+0.22}_{-0.15}
    Fitted to the six DR measurements in the f(a)=Ω_m^γ(a) parameterization.
  • Σ_Λ = 0.018^{+0.052}_{-0.053}
    Fitted amplitude for Σ(a)=Σ_Λ Ω_Λ(a)/Ω_Λ.
  • Σ_1 = 0.020^{+0.065}_{-0.062}
    Fitted amplitude for the linear redshift dependence Σ(a)=Σ_1 a.
  • Σ_2 = 0.027^{+0.067}_{-0.069}
    Fitted amplitude for the quadratic redshift dependence Σ(a)=Σ_2 a^2.
axioms (5)
  • domain assumption The background expansion is fixed to flat ΛCDM with Ωm=0.3169.
    Section 3 states: 'we adopt Ωm = 0.3169 ± 0.0086 ... fix Ωm = 0.3169'. The DR model predictions depend on H(z) through Eq. (5) and Eq. (6).
  • domain assumption The (ψ+ϕ)–δ relation remains the same as in GR for the γ model.
    Section 3.1: 'we assume that the (ψ+ϕ)-δ relation remains the same as in GR.' This enters the growth equation used to predict DR.
  • domain assumption The gravitational slip η is fixed to 1 for the G_eff parameterizations.
    Section 3.2: 'we will fix η=1 and only consider Σ≠0'. DR cannot break the η–Σ degeneracy, so the constraints are conditional on this choice.
  • domain assumption The six measured DR values in Table 1 are unbiased and their errors are Gaussian.
    The likelihood is built on these values; the companion paper's systematics mitigation is not reproduced or propagated here.
  • standard math Flat geometry and Newtonian gauge.
    Eq. (1) defines the metric in Newtonian gauge with flat spatial sections; this underlies ISW and lensing expressions.

pith-pipeline@v1.3.0-alltime-deepseek · 8058 in / 9832 out tokens · 94857 ms · 2026-08-03T23:41:45.298378+00:00 · methodology

0 comments
read the original abstract

A key task in cosmology is to test the validity of general relativity (GR) at cosmological scales and, therefore, to distinguish between dark energy and modified gravity (MG) as the driver of the late-time cosmic acceleration. The decay rate ($DR$) of cosmological gravitational potential, being sensitive to gravity and being immune to various astrophysical uncertainties, enables GR tests independent to other structure growth probes. Recently we have measured $DR$ at $0.2\leq z\leq 1.4$, combining the DR9 galaxy catalog from the DESI imaging surveys and Planck cosmic microwave background maps \citep{arXiv:2411.12594}. Here we use this measurement to test gravity, and restrict the analysis to one-parameter extensions to the standard $\Lambda$CDM cosmology. We consider four one-parameter MG parameterizations. One is $f(a)=\Omega_m^\gamma(a)$. The other three adopt the gravitational slip parameter $\eta=1$ and consider variations in the effective gravitational constant $G_{\rm eff}/G$ with the parameterization $\Sigma(a)=\Sigma_\Lambda \Omega_\Lambda(a)/\Omega_\Lambda$, $\Sigma(a)=\Sigma_1 a$ or $\Sigma(a)=\Sigma_2 a^2$. We find $\gamma=0.47^{+0.22}_{-0.15}$, consistent with the GR prediction $\gamma\simeq 0.55$. We also find $\Sigma_\Lambda=0.018^{+0.052}_{-0.053}$, $\Sigma_1=0.020^{+0.065}_{-0.062}$, and $\Sigma_2=0.027^{+0.067}_{-0.069}$, fully consistent with the GR case of $\Sigma=0$, regardless of parameterizations of $\Sigma(a)$. The constraining power is already competitive, while a factor of 2 further improvement is expected for the upcoming full-sky galaxy surveys.

Figures

Figures reproduced from arXiv: 2511.04279 by Fuyu Dong, Pengjie Zhang, Xinyi Zhao.

Figure 1
Figure 1. Figure 1: Constraint on the γ parameter using DR. The shaded areas indicate the 68% confidence region. γ = 0.47+0.22 −0.15, consistent with the GR value of γ ≃ 0.55. (Weyl) potential. From the definition, DR is indepen￾dent of galaxy bias, the matter clustering amplitude and slope, and the Hubble constant H0. Hence DR is signif￾icantly more sensitive to the nature of dark energy and gravity than other probes, in cas… view at source ↗
Figure 2
Figure 2. Figure 2: γ constraints from various data sets. In the bot￾tom is γ constrained by DR. The constraint labeled with ”Artis et al” is from E. Artis et al. (2024) using cluster num￾ber counts. The one with label ”S8” is calculated by all the S8(z) measured by Fig.19 of F. J. Qu et al. (2024) (includes S8(z) measurements of F. J. Qu et al. (2024); N. Sailer et al. (2024); G. S. Farren et al. (2024) ). The other constrai… view at source ↗
Figure 3
Figure 3. Figure 3: Similar to Fig.1, but for the ΣΛ, Σ1 and Σ2 parameterizations. We numerically solve the following equation to deter￾mine f, δ ′′ + δ ′ ( H′ H + 3 a ) − 3 2 ΩmH2 0 H2a 3 δ a 2 (1 + Σ) = 0 . (6) Here ′ means d/da. We need to specify the redshift dependence of Σ. Since cosmic acceleration occurred at late times, we nat￾urally expect Σ → 0 when a → 0. Then we adopt the following three forms of Σ(a). • Σ(a) = Σ… view at source ↗
Figure 4
Figure 4. Figure 4: The dependence of MG parameter constraints on Ωm. Over the range of bestfit Ωm from BAO, CMB and SNe Ia, the induced variation in γ and ΣΛ is subdominant to the statistical error, and the agreement with GR remains unchanged. This work demonstrates the impressive power of DR to constrain MG. Therefore a major task in this direc￾tion is to improve the DR measurement precision and accuracy. Upcoming full sky … view at source ↗

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Works this paper leans on

34 extracted references · 10 canonical work pages

  1. [1]

    Abbott, T. M. C., Abdalla, F. B., Avila, S., et al. 2019, Phys. Rev. D, 99, 123505, doi: 10.1103/PhysRevD.99.123505 Abdul Karim, M., Aguilar, J., Ahlen, S., et al. 2025, PhRvD, 112, 083515, doi: 10.1103/tr6y-kpc6

    doi:10.1103/physrevd.99.123505 2019

  2. [2]

    2021, PhRvD, 103, 083533, doi: 10.1103/PhysRevD.103.083533

    Alam, S., Aubert, M., Avila, S., et al. 2021, PhRvD, 103, 083533, doi: 10.1103/PhysRevD.103.083533

    doi:10.1103/physrevd.103.083533 2021

  3. [3]

    2008, Journal of Cosmology and Astroparticle Physics, 2008, 013, doi: 10.1088/1475-7516/2008/04/013

    Amendola, L., Kunz, M., & Sapone, D. 2008, Journal of Cosmology and Astroparticle Physics, 2008, 013, doi: 10.1088/1475-7516/2008/04/013

    doi:10.1088/1475-7516/2008/04/013 2008

  4. [4]

    2024, arXiv e-prints, arXiv:2410.09499, doi: 10.48550/arXiv.2410.09499

    Artis, E., Bulbul, E., Grandis, S., et al. 2024, arXiv e-prints, arXiv:2410.09499, doi: 10.48550/arXiv.2410.09499

    Pith/arXiv arXiv doi:10.48550/arxiv.2410.09499 2024

  5. [5]

    2010, Phys

    Bean, R., & Tangmatitham, M. 2010, Phys. Rev. D, 81, 083534, doi: 10.1103/PhysRevD.81.083534

    work page doi:10.1103/physrevd.81.083534 2010

  6. [6]

    2007, Phys

    Caldwell, R., Cooray, A., & Melchiorri, A. 2007, Phys. Rev. D, 76, 023507, doi: 10.1103/PhysRevD.76.023507

    doi:10.1103/physrevd.76.023507 2007

  7. [7]

    G., Padilla, A., & Skordis, C

    Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. 2012, Physics Reports, 513, 1–189, doi: 10.1016/j.physrep.2012.01.001

    doi:10.1016/j.physrep.2012.01.001 2012

  8. [8]

    G., & Turok, N

    Crittenden, R. G., & Turok, N. 1996, PhRvL, 76, 575, doi: 10.1103/PhysRevLett.76.575 DES Collaboration, Abbott, T. M. C., Acevedo, M., et al. 2024, ApJL, 973, L14, doi: 10.3847/2041-8213/ad6f9f

    doi:10.1103/physrevlett.76.575 1996

  9. [9]

    2022, ApJ, 938, 72, doi: 10.3847/1538-4357/ac905b

    Dong, F., Zhang, P., Sun, Z., & Park, C. 2022, ApJ, 938, 72, doi: 10.3847/1538-4357/ac905b

    doi:10.3847/1538-4357/ac905b 2022

  10. [10]

    2025, ApJ, 982, 99, doi: 10.3847/1538-4357/adb585 6

    Dong, F., Zhang, P., Xu, H., & Qin, J. 2025, ApJ, 982, 99, doi: 10.3847/1538-4357/adb585 6

    work page doi:10.3847/1538-4357/adb585 2025

  11. [11]

    S., Krolewski, A., MacCrann, N., et al

    Farren, G. S., Krolewski, A., MacCrann, N., et al. 2024, The Astrophysical Journal, 966, 157, doi: 10.3847/1538-4357/ad31a5

    doi:10.3847/1538-4357/ad31a5 2024

  12. [12]

    Ferreira, P. G. 2019, ARA&A, 57, 335, doi: 10.1146/annurev-astro-091918-104423

    doi:10.1146/annurev-astro-091918-104423 2019

  13. [13]

    Fry, J. N. 1985, Physics Letters B, 158, 211, doi: 10.1016/0370-2693(85)90957-8

    work page doi:10.1016/0370-2693(85)90957-8 1985

  14. [14]

    2018, Living Reviews in Relativity, 22, doi: 10.1007/s41114-018-0017-4

    Ishak, M. 2018, Living Reviews in Relativity, 22, doi: 10.1007/s41114-018-0017-4

    doi:10.1007/s41114-018-0017-4 2018

  15. [15]

    2025, JCAP, 2025, 053, doi: 10.1088/1475-7516/2025/09/053

    Ishak, M., Pan, J., Calderon, R., et al. 2025, JCAP, 2025, 053, doi: 10.1088/1475-7516/2025/09/053

    doi:10.1088/1475-7516/2025/09/053 2025

  16. [16]

    2008, Phys

    Jain, B., & Zhang, P. 2008, Phys. Rev. D, 78, 063503, doi: 10.1103/PhysRevD.78.063503

    doi:10.1103/physrevd.78.063503 2008

  17. [17]

    2015, Physics Reports, 568, 1–98, doi: 10.1016/j.physrep.2014.12.002

    Joyce, A., Jain, B., Khoury, J., & Trodden, M. 2015, Physics Reports, 568, 1–98, doi: 10.1016/j.physrep.2014.12.002

    doi:10.1016/j.physrep.2014.12.002 2015

  18. [18]

    P., & Schechter, P

    Lightman, A. P., & Schechter, P. L. 1990, ApJS, 74, 831, doi: 10.1086/191521

    work page doi:10.1086/191521 1990

  19. [19]

    2005, Physical Review D, 72, doi: 10.1103/PhysRevD.72.043529

    Linder, E. 2005, Physical Review D, 72, doi: 10.1103/PhysRevD.72.043529

    doi:10.1103/physrevd.72.043529 2005

  20. [20]

    Lue, A., Scoccimarro, R., & Starkman, G. D. 2004, Phys. Rev. D, 69, 124015, doi: 10.1103/PhysRevD.69.124015

    work page doi:10.1103/physrevd.69.124015 2004

  21. [21]

    Nishizawa, A. J. 2014, Progress of Theoretical and Experimental Physics, 2014, 6B110, doi: 10.1093/ptep/ptu062

    work page doi:10.1093/ptep/ptu062 2014

  22. [22]

    Peebles, P. J. E. 1980, The large-scale structure of the universe Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016a, A&A, 594, A21, doi: 10.1051/0004-6361/201525831 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016b, A&A, 594, A14, doi: 10.1051/0004-6361/201525814 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 64...

    doi:10.1051/0004-6361/201525831 1980

  23. [23]

    J., Hang, Q., Farren, G., et al

    Qu, F. J., Hang, Q., Farren, G., et al. 2024, arXiv e-prints, arXiv:2410.10808, doi: 10.48550/arXiv.2410.10808

    Pith/arXiv arXiv doi:10.48550/arxiv.2410.10808 2024

  24. [24]

    K., & Wolfe, A

    Sachs, R. K., & Wolfe, A. M. 1967, ApJ, 147, 73, doi: 10.1086/148982

    doi:10.1086/148982 1967

  25. [25]

    2024, arXiv e-prints, arXiv:2407.04607, doi: 10.48550/arXiv.2407.04607

    Sailer, N., Kim, J., Ferraro, S., et al. 2024, arXiv e-prints, arXiv:2407.04607, doi: 10.48550/arXiv.2407.04607

    Pith/arXiv arXiv doi:10.48550/arxiv.2407.04607 2024

  26. [26]

    2009, Journal of Cosmology and Astroparticle Physics, 2009, 048, doi: 10.1088/1475-7516/2009/01/048

    Song, Y.-S., & Koyama, K. 2009, Journal of Cosmology and Astroparticle Physics, 2009, 048, doi: 10.1088/1475-7516/2009/01/048

    work page doi:10.1088/1475-7516/2009/01/048 2009

  27. [27]

    2007, Phys

    Tsujikawa, S. 2007, Phys. Rev. D, 76, 023514, doi: 10.1103/PhysRevD.76.023514

    work page doi:10.1103/physrevd.76.023514 2007

  28. [28]

    Wang, L., & Steinhardt, P. J. 1998, ApJ, 508, 483, doi: 10.1086/306436

    doi:10.1086/306436 1998

  29. [29]

    H., Mortonson, M

    Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013, PhR, 530, 87, doi: 10.1016/j.physrep.2013.05.001

    doi:10.1016/j.physrep.2013.05.001 2013

  30. [30]

    2023, MNRAS, 520, 161, doi: 10.1093/mnras/stad136

    Xu, H., Zhang, P., Peng, H., et al. 2023, MNRAS, 520, 161, doi: 10.1093/mnras/stad136

    doi:10.1093/mnras/stad136 2023

  31. [31]

    2006, ApJ, 647, 55, doi: 10.1086/505297

    Zhang, P. 2006, ApJ, 647, 55, doi: 10.1086/505297

    work page doi:10.1086/505297 2006

  32. [32]

    2006, Phys

    Zhang, P. 2006, Phys. Rev. D, 73, 123504, doi: 10.1103/PhysRevD.73.123504

    work page doi:10.1103/physrevd.73.123504 2006

  33. [33]

    2007, PhRvL, 99, 141302, doi: 10.1103/PhysRevLett.99.141302

    Zhang, P., Liguori, M., Bean, R., & Dodelson, S. 2007, PhRvL, 99, 141302, doi: 10.1103/PhysRevLett.99.141302

    doi:10.1103/physrevlett.99.141302 2007

  34. [34]

    2009, Phys

    Zhao, G.-B., Pogosian, L., Silvestri, A., & Zylberberg, J. 2009, Phys. Rev. D, 79, 083513, doi: 10.1103/PhysRevD.79.083513

    doi:10.1103/physrevd.79.083513 2009