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REVIEW 4 major objections 3 minor 73 references

From Quantum Correlations to Inflationary Tracking Scalar Field Evolution

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The inflationary tracking condition $\dot{\phi}^2 = \gamma H^{-m}$, previously a phenomenological ansatz, is argued to follow from critical-correlation physics of the pre-inflationary quantum state, with $m = z(2-\eta)$.

desk verdict A clearly written speculative bridge from critical phenomena to the tracking condition, but the bridge rests on an assumed power-law scaling that already contains the result. read the letter →

arxiv 2608.01165 v1 pith:PDUUOLSJ submitted 2026-08-02 gr-qc hep-th

V.K. Oikonomou This is my paper
classification gr-qchep-th MSC 83F0581T1782B27 PACS 98.80.Cq05.70.Jk11.10.Hi
keywords trackingconditioninflationscalarfieldcondensateWilsonianrenormalizationcriticalphenomenasusceptibilitycorrelationlengthCMBobservables
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 tries to give a theoretical basis for the tracking condition $\dot{\phi}^2 = \gamma H^{-m}$ for single-field inflation, the condition behind the only known analytic inflationary solutions compatible with current CMB data. The proposal is that the inflaton is not a fundamental particle but a collective condensate of pre-inflationary quantum degrees of freedom, formed as the Universe crosses from its quantum to its classical regime. Modeling that crossing with Wilsonian critical phenomena, the field's kinetic energy is tied to the susceptibility of the coarse-grained theory, and a power-law link between correlation length and Hubble radius converts this into $\dot{\phi}^2 = \gamma H^{-z(2-\eta)}$, so $m = z(2-\eta)$. The payoff would be that CMB observables stop being fitted numbers and become measurements of critical exponents characterizing the quantum-to-classical transition.

What carries the argument

The load-bearing object is the susceptibility $\chi = \int d^d r\, G(r)$ of the Wilsonian two-point function at criticality, where $G(r) = r^{-(d-2+\eta)}f(r/\xi)$ and near the fixed point $\chi \sim \xi^{2-\eta}$. The step that turns this into cosmology is the identification of the correlation length with a power of the Hubble radius, $\xi = \xi_0 H^{-z}$; combining the two scalings yields the tracking condition with $m = z(2-\eta)$. The same machinery would give the opposite scaling under equilibrium Landau-Khalatnikov dynamics, so the non-equilibrium identification $I \propto \chi$ is the physical hinge of the argument.

What would settle it

Measure the scalar spectral index and tensor-to-scalar ratio at the sensitivity of next-generation CMB experiments and check whether they lie on the tracking surface $n_S = 1 - \frac{m+4}{(m+2)N} - \frac{m+4}{(m+2)^2 N^2}$, $r = \frac{16}{(m+2)N} - \frac{16}{(m+2)^2 N^2}$ for any $m, N$; a decisive miss would rule out the tracking origin. Independently, a concrete microphysical model of the quantum cells whose correlation length has a different $H$-dependence than $\xi \propto H^{-z}$ would collapse the derivation.

Watch

Extended reading notes

Core claim

The central claim is that the tracking condition has a microscopic reading. The paper treats the pre-inflationary Universe as a quantum statistical system and identifies the scalar field with the condensate $\phi(x) = \langle \hat{O}(x) \rangle$ of its microscopic degrees of freedom, in the same way magnetization is a spin average. The Wilsonian free energy for this condensate has a critical two-point function $G(r) \sim r^{-(d-2+\eta)} f(r/\xi)$, whose zero-momentum value is the susceptibility $\chi \sim \xi^{2-\eta}$. Because the quantum-to-classical transition is far from equilibrium, the information-theoretic rate $I = \|\dot{\rho}\|^2$ driving the field's evolution is taken proportional

Load-bearing premise

The argument's load-bearing premise is that the correlation length of the pre-inflationary quantum degrees of freedom is exactly a power of the Hubble radius, $\xi = \xi_0 H^{-z}$; if that scaling fails, the tracking condition is not derived from microphysics but merely renamed.

Editorial extensions

If this is right

  • If the tracking condition has this origin, the scalar spectral index and tensor-to-scalar ratio depend only on $m$ and the e-foldings number $N$, and a measurement of $m$ from CMB data reads off the critical-exponent combination $z(2-\eta)$.
  • The single-field inflaton is reinterpreted as an emergent condensate rather than a fundamental particle, so inflationary phenomenology becomes a window on the pre-inflationary quantum-to-classical transition.
  • The analytic inflationary solutions built on the tracking condition retain their compatibility with current CMB data, but the exponent $m$ is no longer free: it is fixed by the critical behavior of the microscopic quantum degrees of freedom.
  • Because classicalization is assumed to be far from equilibrium, the tracking condition would not survive if the transition were governed by equilibrium critical dynamics, which would instead yield $\dot{\phi}^2 \sim \chi^{-1}$.

Reading between the lines

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

  • The argument is made at the scaling level; constructing a concrete microphysical model of the pre-inflationary quantum cells would be the step that turns $m = z(2-\eta)$ into a first-principles prediction.
  • A precise future CMB measurement of $m$ would fix only the product $z(2-\eta)$; separating the critical exponent $z$ from the anomalous dimension $\eta$ would need additional constraints or observables.
  • The same susceptibility logic should apply to any light spectator field born in the same transition, predicting analogous tracking laws with exponents built from the same $z$ and $\eta$; multi-field observables such as isocurvature or non-Gaussianity could test that.
  • If quantum-to-classical critical dynamics were shown to be equilibrium-like, the Landau-Khalatnikov route gives the opposite sign for the exponent, providing an internally accessible falsification of the paper's identification.
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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 / 3 minor

Summary. The paper aims to provide a theoretical origin for the phenomenological tracking condition dot_phi^2 = gamma H^{-m}, previously used in Ref. [66] to construct analytic inflationary solutions. The scalar field is modeled as a collective condensate of pre-inflationary quantum degrees of freedom. Using a Wilsonian two-point function, the paper derives the susceptibility scaling chi ~ xi^{2-eta} (Eq. 3.15), assumes the correlation length scales with the Hubble radius as xi = xi_0 H^{-z} (Eq. 3.17), connects the scalar kinetic term to an information measure I via (1/2) dot_phi^2 = alpha I (Eq. 3.22), assumes I proportional to chi (Eq. 3.23), and obtains dot_phi^2 = gamma H^{-z(2-eta)} (Eq. 3.24), defining m = z(2-eta) (Eq. 3.25). The conclusion acknowledges that the analysis is 'merely qualitative' and that a more detailed microscopic theory is required.

Significance. If the derivation were valid, it would connect a viable inflationary parametrization to critical exponents of a microscopic quantum theory, giving a microphysical rationale for the tracking condition and potentially linking CMB observables to critical phenomena. The conceptual picture of the inflaton as an information-theoretic condensate is interesting, and the paper clearly states its assumptions. However, the derivation is not established: the key H-dependence is inserted by hand, the information-theoretic link is posited rather than derived, and no predictive constraint on m is produced. The manuscript contains no machine-checkable proofs or numerical tests; its strength lies only in a qualitative analogy.

major comments (4)
  1. [III.B, Eq. (3.17)] The central step is the assertion xi = xi_0 H^{-z} 'from dimensional arguments.' Dimensional analysis only fixes the engineering dimension of xi; it does not select a pure power of H with arbitrary exponent z. With z and xi_0 free, Eq. (3.18) then yields chi ~ H^{-z(2-eta)}, and Eq. (3.24) yields dot_phi^2 ~ H^{-z(2-eta)}. The H^{-m} dependence of the tracking condition is therefore inserted at this step, not derived from the microscopic Hamiltonian or from the Wilsonian fixed point. The exponent m = z(2-eta) is a relabeling of the free exponents, so the central claim of emergence is not supported.
  2. [III.B, Eqs. (3.20)-(3.23)] The bridge from the density matrix to the scalar kinetic term is a chain of unproven identifications. Eq. (3.22) hides the entire coarse-graining factor in an unspecified constant alpha, and Eq. (3.23) states I proportional to chi merely because both 'measure the exact same thing.' Neither relation is derived from the Hamiltonian (3.1) or from any explicit information metric. Even if the critical scaling chi ~ xi^{2-eta} were accepted, the step to dot_phi^2 is assumed. Thus the inflationary kinetic dynamics is not obtained from microphysics.
  3. [III.B, Eq. (3.25); IV] The paper defines m = z(2-eta) but does not compute z or eta from any specific microscopic theory; the Hamiltonian (3.1) is left completely unspecified. As a result the tracking exponent m carries no new physical information: the observables (1.1)-(1.2) still depend on an unconstrained parameter. The conclusion's admission that the analysis is 'merely qualitative' and 'a more detailed microscopic theory is required' is in direct tension with the abstract's claim to 'demonstrate' the emergence of the tracking condition. This limitation is not merely cosmetic; it removes the predictive content of the proposal.
  4. [III.B, Eqs. (3.27)-(3.28)] The paper considers the equilibrium Landau-Khalatnikov route, which would give dot_phi^2 ~ chi^{-1} (Eq. 3.28), and discards it by declaring the quantum-to-classical transition to be far from equilibrium. However, no out-of-equilibrium Wilsonian framework is provided; the actual derivation (3.13)-(3.15) uses equilibrium-style critical scaling. The choice between the two scaling forms is therefore not justified within the manuscript.
minor comments (3)
  1. [III.B, Eqs. (3.4)-(3.6)] The functional F is presented both as a coarse-grained action and as the exponent in the partition function Z = integral Dphi e^{-F}; the 'maximization' condition delta F / delta phi = 0 is inconsistent with the usual sign convention for a probability weight e^{-F}. Please clarify whether F is to be minimized as a free energy or treated as a Euclidean action.
  2. [III.B, Eq. (3.19)] The expression dot_phi = (delta phi / delta rho) dot_rho is not defined in the text; a functional derivative with respect to a density matrix requires an information-geometric definition. Without such a definition, Eq. (3.20) and the passage to Eq. (3.22) are difficult to evaluate.
  3. [General; Eq. (1.1)] Minor typos and notation issues: the sentence after Eq. (3.15), 'which quite different from what Eq. (3.15)' should read 'which is quite different from Eq. (3.15)'. Also, Eq. (1.1) would be clearer with parentheses: n_S ~ 1 - (m+4)/[(m+2)N] - (m+4)/[(m+2)^2 N^2].

Circularity Check

2 steps flagged · score 8.0 of 10

Tracking condition is not derived: Eq. (3.17) assumes the same power-law H-dependence that is then relabeled as m=z(2-η).

  1. self definitional [Sec. III.B, Eq. (3.17)]
    "From dimensional arguments, the correlation length must be related to a fundamental length of the newly emerged spacetime. The only such length is the Hubble radius, RH = 1/H, thus we generally assume that the correlation length is related to the Hubble radius as follows, ξ=ξ0 Rz H =ξ0 H−z, (3.17) with z>0 some arbitrary critical exponent"

    This assumption inserts the H-dependence of the final tracking condition at the outset. Via the standard scaling χ∼ξ^{2−η} (3.15) and the subsequent identifications (3.22)–(3.23), Eq. (3.24) gives dotϕ^2 = γ H^{−z(2−η)}, and Eq. (3.25) defines m=z(2−η). Thus m is a relabeling of the arbitrary exponents z and η. The functional form—a pure power of H—is not derived from the microscopic Hamiltonian (3.1) or from a Wilsonian fixed point; it is postulated in (3.17). Without (3.17), no H-dependence would follow, so the central claim reduces to its own input.

  2. self definitional [Sec. III.B, Eq. (3.23)]
    "Equation (3.20) therefore becomes, 1/2 dotϕ^2 = αI, (3.22) ... The simplest realization is I ∝ χ, (3.23) which is another quantity that measures the exact same thing, the microscopic coherence."

    The link between the scalar-field kinetic energy and the susceptibility is asserted, not derived. The paper needs dotϕ^2 ∝ χ to obtain H^{−z(2−η)}; this proportionality is a second assumption that essentially builds the tracking condition's dependence into the framework. Even if one granted the standard scaling χ∼ξ^{2−η}, the claimed 'emergence' of the tracking condition from correlations is a consistency check of an ansatz rather than a derivation from independent microphysics.

full rationale

The derivation chain (3.13)→(3.15)→(3.17)→(3.18)→(3.22)–(3.23)→(3.24)–(3.26) is algebraically self-contained, but the decisive physics is inserted at Eq. (3.17): the correlation length is assumed to be a pure power of the Hubble radius, ξ∼H^{−z}. Since the tracking condition is exactly a pure power of H, the output has the same functional form as the input. The exponents z and η are free parameters, and m=z(2−η) is a redefinition, not a prediction. The additional identification I∝χ (3.23) is likewise assumed rather than derived. The paper's own conclusion concedes the analysis is 'merely qualitative' and requires 'a more detailed microscopic theory,' which is consistent with this being an ansatz-consistency check rather than an emergent derivation. No independent external benchmark fixes the critical exponents; the CMB compatibility is inherited from the earlier analytic-solution paper [66], which is a same-author citation, but the present derivation itself does not rely on a self-citation chain. The central claim therefore reduces by construction: Eq. (3.17) is the tracking condition in disguise, and the rest renames it.

Assumptions & free parameters 5 free parameters · 6 assumptions · 1 invented entities

The central claim rests on a chain of domain assumptions and ad hoc identifications. The most important is the power-law relation between correlation length and Hubble radius, which already contains the functional form of the result. The microscopic Hamiltonian is never specified, and the free parameters z, \eta, \xi_0, and \alpha are unconstrained. No invented entities carry independent evidence.

free parameters (5)
  • z = unknown
    Critical exponent in the assumed power law \xi = \xi_0 H^{-z} (Eq. 3.17). Introduced ad hoc; m = z(2-\eta) inherits its freedom.
  • eta = unknown
    Anomalous dimension in the two-point function scaling (Eq. 3.13). Not computed from microphysics; enters m through \chi ~ \xi^{2-\eta}.
  • xi_0 = unknown
    Proportionality constant connecting correlation length to Hubble radius (Eq. 3.17). Dimensionful and unspecified.
  • alpha = unknown
    Coefficient in (1/2)\dot{\phi}^2 = \alpha I, containing (\delta\phi/\delta\rho)^2/2 (Eq. 3.22). Never evaluated.
  • proportionality constant in I ~ chi = unknown
    The relation I \propto \chi (Eq. 3.23) is posited; the proportionality constant is free and can be absorbed into \gamma.
assumptions (6)
  • domain assumption The scalar field is a collective condensate of quantum degrees of freedom: \phi(x) = <O(x)>.
    Eq. (3.3). Central conceptual assumption; not derived from any microscopic theory.
  • domain assumption The quantum-to-classical transition of the Universe is an ordering transition governed by critical phenomena.
    Section II: 'we shall assume that the quantum-to-classical transition of the Universe is a sort of ordering transition.'
  • standard math Near criticality, the two-point function has the universal scaling form G(r) = r^{-(d-2+\eta)} f(r/\xi).
    Eq. (3.13). Standard result from critical phenomena, used to derive the susceptibility scaling.
  • standard math The susceptibility scales as \chi ~ \xi^{2-\eta}.
    Eq. (3.15), obtained by integrating Eq. (3.13). Standard for d > 2-\eta.
  • ad hoc to paper The correlation length is a power of the Hubble radius: \xi = \xi_0 H^{-z}, with z > 0.
    Eq. (3.17). Called a 'critical assumption' by the author; it inserts the H-dependence that directly produces the tracking condition.
  • ad hoc to paper The scalar kinetic term is related to information and susceptibility: (1/2)\dot{\phi}^2 = \alpha I and I \propto \chi.
    Eqs. (3.20)-(3.23). The connection between the rate of change of the density matrix and the susceptibility is posited, not derived.
invented entities (1)
  • Primordial quantum information network (Planck cells of information)
    purpose: Pre-geometric substrate from which continuous spacetime and the scalar field emerge as collective condensates.
    Section II and Fig. 1. Entirely speculative; no falsifiable handle provided outside the paper.

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Cite this review

Pith. "Pith review of From Quantum Correlations to Inflationary Tracking Scalar Field Evolution." pith.science (2026). https://pith.science/paper/PDUUOLSJ

@misc{pith2026260801165,
  author       = {Pith},
  title        = {Pith review of: From Quantum Correlations to Inflationary Tracking Scalar Field Evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDUUOLSJ}},
  note         = {Machine review of arXiv:2608.01165}
}
abstract

The tracking condition $\dot{\phi}^2=\gamma H^{-m}$ for single scalar field theory leads to analytic inflationary solutions, which are compatible with the current cosmic microwave background radiation experiments. To our knowledge this is the only analytic solution of inflation which is compatible with the data, to date. In this work we seek for some theoretical basis that can lead to the tracking condition $\dot{\phi}^2=\gamma H^{-m}$. As we show, if the Universe is seen pre-inflationary as a quantum statistical system, the scalar field may emerge as a collective condensate of the quantum degrees of freedom. In the quantum-to-classical transition of the Universe, the scalar field two point function yields the susceptibility of the theory with an inherent correlation length. Using information theoretic motivation and Wilsonian quantum field theoretic arguments, we demonstrate that the tracking conditions $\dot{\phi}^2=\gamma H^{-m}$ emerge from this framework.

Figures

Figures reproduced from arXiv: 2608.01165 by the authors.

Figure 1
Figure 1. FIG. 1. The Universe depicted as information framework. Primordially, quantum degrees of freedom are loosely [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Works this paper leans on

73 extracted references · 20 canonical work pages

  1. [66]

    V. K. Oikonomou, [arXiv:2603.11794 [gr-qc]]

    arXiv

  2. [1]

    A. D. Linde, Lect. Notes Phys.738(2008) 1 [arXiv:0705.0164 [hep-th]]

    arXiv 2008

  3. [2]

    A. D. Linde, Phys. Lett. B108(1982), 389-393 doi:10.1016/0370-2693(82)91219-9

    doi:10.1016/0370-2693(82)91219-9 1982

  4. [3]

    Introduction to the theory of the early universe: Cosmological pertur- bations and inflationary theory,

    D. S. Gorbunov and V. A. Rubakov, “Introduction to the theory of the early universe: Cosmological pertur- bations and inflationary theory,” Hackensack, USA: World Scientific (2011) 489 p

    2011

  5. [4]

    Linde, arXiv:1402.0526 [hep-th]

    A. Linde, arXiv:1402.0526 [hep-th]

    arXiv

  6. [5]

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

    arXiv 1999

  7. [6]

    J. c. Hwang and H. Noh, Phys. Rev. D71(2005) 063536 doi:10.1103/PhysRevD.71.063536 [gr-qc/0412126]

    arXiv 2005

  8. [7]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and M. Sami, Phys. Rev. D74(2006) 046004 doi:10.1103/PhysRevD.74.046004 [hep-th/0605039]

    arXiv 2006

Show all 73 references
  1. [8]

    Cognola, E

    G. Cognola, E. Elizalde, S. Nojiri, S. Odintsov and S. Zerbini, Phys. Rev. D75(2007) 086002 doi:10.1103/PhysRevD.75.086002 [hep-th/0611198]

    2007 arXiv

  2. [9]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D71(2005) 123509 doi:10.1103/PhysRevD.71.123509 [hep-th/0504052]. 8

    2005 arXiv

  3. [10]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Lett. B631(2005) 1 doi:10.1016/j.physletb.2005.10.010 [hep-th/0508049]

    2005 arXiv

  4. [11]

    Satoh, S

    M. Satoh, S. Kanno and J. Soda, Phys. Rev. D77(2008) 023526 doi:10.1103/PhysRevD.77.023526 [arXiv:0706.3585 [astro-ph]]

    2008 arXiv

  5. [12]

    Bamba, A

    K. Bamba, A. N. Makarenko, A. N. Myagky and S. D. Odintsov, JCAP1504(2015) 001 doi:10.1088/1475- 7516/2015/04/001 [arXiv:1411.3852 [hep-th]]

    2015 arXiv

  6. [13]

    Z. Yi, Y. Gong and M. Sabir, Phys. Rev. D98(2018) no.8, 083521 doi:10.1103/PhysRevD.98.083521 [arXiv:1804.09116 [gr-qc]]

    2018 arXiv

  7. [14]

    Z. K. Guo and D. J. Schwarz, Phys. Rev. D80(2009) 063523 doi:10.1103/PhysRevD.80.063523 [arXiv:0907.0427 [hep-th]]

    2009 arXiv

  8. [15]

    Z. K. Guo and D. J. Schwarz, Phys. Rev. D81(2010) 123520 doi:10.1103/PhysRevD.81.123520 [arXiv:1001.1897 [hep-th]]

    2010 arXiv

  9. [16]

    P. X. Jiang, J. W. Hu and Z. K. Guo, Phys. Rev. D88(2013) 123508 doi:10.1103/PhysRevD.88.123508 [arXiv:1310.5579 [hep-th]]

    2013 arXiv

  10. [17]

    van de Bruck, K

    C. van de Bruck, K. Dimopoulos, C. Longden and C. Owen, arXiv:1707.06839 [astro-ph.CO]

    arXiv

  11. [18]

    E. O. Pozdeeva, M. R. Gangopadhyay, M. Sami, A. V. Toporensky and S. Y. Vernov, Phys. Rev. D102 (2020) no.4, 043525 doi:10.1103/PhysRevD.102.043525 [arXiv:2006.08027 [gr-qc]]

    2020 arXiv

  12. [19]

    Vernov and E

    S. Vernov and E. Pozdeeva, Universe7(2021) no.5, 149 doi:10.3390/universe7050149 [arXiv:2104.11111 [gr- qc]]

    2021 arXiv

  13. [20]

    E. O. Pozdeeva and S. Y. Vernov, Eur. Phys. J. C81(2021) no.7, 633 doi:10.1140/epjc/s10052-021-09435-8 [arXiv:2104.04995 [gr-qc]]

    2021 arXiv

  14. [21]

    Fomin, Eur

    I. Fomin, Eur. Phys. J. C80(2020) no.12, 1145 doi:10.1140/epjc/s10052-020-08718-w [arXiv:2004.08065 [gr-qc]]

    2020 arXiv

  15. [22]

    De Laurentis, M

    M. De Laurentis, M. Paolella and S. Capozziello, Phys. Rev. D91(2015) no.8, 083531 doi:10.1103/PhysRevD.91.083531 [arXiv:1503.04659 [gr-qc]]

    2015 arXiv

  16. [23]

    Chervon, I

    Scalar Field Cosmology, S. Chervon, I. Fomin, V. Yurov and A. Yurov, World Scientific 2019, doi:10.1142/11405

    2019 doi

  17. [24]

    Nozari and N

    K. Nozari and N. Rashidi, Phys. Rev. D95(2017) no.12, 123518 doi:10.1103/PhysRevD.95.123518 [arXiv:1705.02617 [astro-ph.CO]]

    2017 arXiv

  18. [25]

    S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D98(2018) no.4, 044039 doi:10.1103/PhysRevD.98.044039 [arXiv:1808.05045 [gr-qc]]

    2018 arXiv

  19. [26]

    Kawai, M

    S. Kawai, M. a. Sakagami and J. Soda, Phys. Lett. B437, 284 (1998) doi:10.1016/S0370-2693(98)00925-3 [gr-qc/9802033]

    1998 arXiv

  20. [27]

    Yi and Y

    Z. Yi and Y. Gong, Universe5(2019) no.9, 200 doi:10.3390/universe5090200 [arXiv:1811.01625 [gr-qc]]

    2019 arXiv

  21. [28]

    van de Bruck, K

    C. van de Bruck, K. Dimopoulos and C. Longden, Phys. Rev. D94(2016) no.2, 023506 doi:10.1103/PhysRevD.94.023506 [arXiv:1605.06350 [astro-ph.CO]]

    2016 arXiv

  22. [29]

    K. i. Maeda, N. Ohta and R. Wakebe, Eur. Phys. J. C72(2012) 1949 doi:10.1140/epjc/s10052-012-1949-6 [arXiv:1111.3251 [hep-th]]

    2012 arXiv

  23. [30]

    W. Y. Ai, Commun. Theor. Phys.72(2020) no.9, 095402 doi:10.1088/1572-9494/aba242 [arXiv:2004.02858 [gr-qc]]

    2020 arXiv

  24. [31]

    Easther and K

    R. Easther and K. i. Maeda, Phys. Rev. D54(1996) 7252 doi:10.1103/PhysRevD.54.7252 [hep-th/9605173]

    1996 arXiv

  25. [32]

    Codello and R

    A. Codello and R. K. Jain, Class. Quant. Grav.33(2016) no.22, 225006 doi:10.1088/0264-9381/33/22/225006 [arXiv:1507.06308 [gr-qc]]

    2016 arXiv

  26. [33]

    V. K. Oikonomou, Class. Quant. Grav.38(2021) no.19, 195025 doi:10.1088/1361-6382/ac2168 [arXiv:2108.10460 [gr-qc]]

    2021 arXiv

  27. [34]

    V. K. Oikonomou, Astropart. Phys.141(2022), 102718 doi:10.1016/j.astropartphys.2022.102718 [arXiv:2204.06304 [gr-qc]]

    2022

  28. [35]

    S. D. Odintsov, V. K. Oikonomou and F. P. Fronimos, Nucl. Phys. B958(2020), 115135 doi:10.1016/j.nuclphysb.2020.115135 [arXiv:2003.13724 [gr-qc]]

    2020

  29. [36]

    V. K. Oikonomou, Phys. Lett. B856(2024), 138890 doi:10.1016/j.physletb.2024.138890 [arXiv:2407.12155 [gr-qc]]

    2024

  30. [37]

    J. Fier, H. Han, B. Li, K. Lin, S. Mukohyama and A. Wang, [arXiv:2503.01975 [gr-qc]]

    arXiv

  31. [38]

    Bajardi, Eur

    F. Bajardi, Eur. Phys. J. C84(2024) no.12, 1298 doi:10.1140/epjc/s10052-024-13673-x

    2024 doi

  32. [39]

    Bajardi, Phys

    F. Bajardi, Phys. Rev. D112(2025) no.10, 104028 doi:10.1103/pw9v-3ml2

    2025 doi

  33. [40]

    Bajardi, D

    F. Bajardi, D. Blixt and S. Capozziello, Int. J. Geom. Meth. Mod. Phys.20(2023) no.Supp01, 2450042 doi:10.1142/S0219887824500427 [arXiv:2308.03632 [gr-qc]]

    2023 arXiv

  34. [41]

    Bajardi, S

    F. Bajardi, S. Capozziello, T. Di Salvo and F. Spinnato, Symmetry15(2023) no.9, 1625 doi:10.3390/sym15091625 [arXiv:2308.00082 [gr-qc]]

    2023 arXiv

  35. [42]

    Bajardi, S

    F. Bajardi, S. Capozziello and D. Vernieri, Eur. Phys. J. Plus135(2020) no.12, 942 doi:10.1140/epjp/s13360- 020-00944-1 [arXiv:2011.01317 [gr-qc]]

    2020 arXiv

  36. [43]

    Bajardi and S

    F. Bajardi and S. Capozziello, Eur. Phys. J. C80(2020) no.8, 704 doi:10.1140/epjc/s10052-020-8258-2 [arXiv:2005.08313 [gr-qc]]

    2020 arXiv

  37. [44]

    Capozziello and G

    S. Capozziello and G. G. L. Nashed, Class. Quant. Grav.40(2023) no.20, 205023 doi:10.1088/1361- 6382/acfa5c [arXiv:2309.08894 [gr-qc]]

    2023 arXiv

  38. [45]

    de Martino, M

    I. de Martino, M. De Laurentis and S. Capozziello, Phys. Rev. D102(2020) no.6, 063508 doi:10.1103/PhysRevD.102.063508 [arXiv:2008.09856 [gr-qc]]

    2020 arXiv

  39. [46]

    Capozziello, C

    S. Capozziello, C. A. Mantica and L. G. Molinari, Int. J. Geom. Meth. Mod. Phys.16(2019) no.09, 1950133 doi:10.1142/S0219887819501330 [arXiv:1906.05693 [gr-qc]]

    2019 arXiv

  40. [47]

    Capozziello, M

    S. Capozziello, M. De Laurentis and K. F. Dialektopoulos, Eur. Phys. J. C76(2016) no.11, 629 doi:10.1140/epjc/s10052-016-4491-0 [arXiv:1609.09289 [gr-qc]]. 9

    2016 arXiv

  41. [48]

    Capozziello, M

    S. Capozziello, M. De Laurentis and S. D. Odintsov, Mod. Phys. Lett. A29(2014) no.30, 1450164 doi:10.1142/S0217732314501648 [arXiv:1406.5652 [gr-qc]]

    2014 arXiv

  42. [49]

    Capozziello, A

    S. Capozziello, A. N. Makarenko and S. D. Odintsov, Phys. Rev. D87(2013) no.8, 084037 doi:10.1103/PhysRevD.87.084037 [arXiv:1302.0093 [gr-qc]]

    2013 arXiv

  43. [50]

    Louiset al.[ACT], [arXiv:2503.14452 [astro-ph.CO]]

    T. Louiset al.[ACT], [arXiv:2503.14452 [astro-ph.CO]]

    arXiv

  44. [51]

    Calabreseet al.[ACT], [arXiv:2503.14454 [astro-ph.CO]]

    E. Calabreseet al.[ACT], [arXiv:2503.14454 [astro-ph.CO]]

    arXiv

  45. [52]

    M. H. Abitbolet al.[Simons Observatory], Bull. Am. Astron. Soc.51(2019), 147 [arXiv:1907.08284 [astro- ph.IM]]

    2019 arXiv

  46. [53]

    S. Hild, M. Abernathy, F. Acernese, P. Amaro-Seoane, N. Andersson, K. Arun, F. Barone, B. Barr, M. Bar- suglia and M. Beker,et al.Class. Quant. Grav.28(2011), 094013 doi:10.1088/0264-9381/28/9/094013 [arXiv:1012.0908 [gr-qc]]

    2011 arXiv

  47. [54]

    Baker, J

    J. Baker, J. Bellovary, P. L. Bender, E. Berti, R. Caldwell, J. Camp, J. W. Conklin, N. Cornish, C. Cutler and R. DeRosa,et al.[arXiv:1907.06482 [astro-ph.IM]]

    1907 arXiv

  48. [55]

    T. L. Smith and R. Caldwell, Phys. Rev. D100(2019) no.10, 104055 doi:10.1103/PhysRevD.100.104055 [arXiv:1908.00546 [astro-ph.CO]]

    2019 arXiv

  49. [56]

    Crowder and N

    J. Crowder and N. J. Cornish, Phys. Rev. D72(2005), 083005 doi:10.1103/PhysRevD.72.083005 [arXiv:gr- qc/0506015 [gr-qc]]

    2005

  50. [57]

    T. L. Smith and R. Caldwell, Phys. Rev. D95(2017) no.4, 044036 doi:10.1103/PhysRevD.95.044036 [arXiv:1609.05901 [gr-qc]]

    2017 arXiv

  51. [58]

    N. Seto, S. Kawamura and T. Nakamura, Phys. Rev. Lett.87(2001), 221103 doi:10.1103/PhysRevLett.87.221103 [arXiv:astro-ph/0108011 [astro-ph]]

    2001 arXiv

  52. [59]

    Kawamura, M

    S. Kawamura, M. Ando, N. Seto, S. Sato, M. Musha, I. Kawano, J. Yokoyama, T. Tanaka, K. Ioka and T. Akutsu,et al.[arXiv:2006.13545 [gr-qc]]

    2006 arXiv

  53. [60]

    Weltman, P

    A. Weltman, P. Bull, S. Camera, K. Kelley, H. Padmanabhan, J. Pritchard, A. Raccanelli, S. Riemer-Sørensen, L. Shao and S. Andrianomena,et al.Publ. Astron. Soc. Austral.37(2020), e002 doi:10.1017/pasa.2019.42 [arXiv:1810.02680 [astro-ph.CO]]

    2020 arXiv

  54. [61]

    Auclairet al.[LISA Cosmology Working Group], [arXiv:2204.05434 [astro-ph.CO]]

    P. Auclairet al.[LISA Cosmology Working Group], [arXiv:2204.05434 [astro-ph.CO]]

    arXiv

  55. [62]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept.692(2017) 1 [arXiv:1705.11098 [gr-qc]]

    2017 arXiv

  56. [63]

    Capozziello, M

    S. Capozziello, M. De Laurentis, Phys. Rept.509, 167 (2011); V. Faraoni and S. Capozziello, Fundam. Theor. Phys.170(2010)

    2011

  57. [64]

    Nojiri, S.D

    S. Nojiri, S.D. Odintsov, Phys. Rept.505, 59 (2011)

    2011

  58. [65]

    S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fronimos and E. C. Lymperiadou, Symmetry15 (2023) no.9, 1701 doi:10.3390/sym15091701 [arXiv:2307.16308 [gr-qc]]

    2023 arXiv

  59. [67]

    Brax and P

    P. Brax and P. Valageas, Phys. Rev. D90(2014) no.2, 023507 doi:10.1103/PhysRevD.90.023507 [arXiv:1403.5420 [astro-ph.CO]]

    2014 arXiv

  60. [68]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, [arXiv:2607.06582 [gr-qc]]

    arXiv

  61. [69]

    Cognola, R

    G. Cognola, R. Myrzakulov, L. Sebastiani and S. Zerbini, Phys. Rev. D88(2013) no.2, 024006 doi:10.1103/PhysRevD.88.024006 [arXiv:1304.1878 [gr-qc]]

    2013 arXiv

  62. [70]

    S. M. Carroll and G. N. Remmen, Phys. Rev. D93(2016) no.12, 124052 doi:10.1103/PhysRevD.93.124052 [arXiv:1601.07558 [hep-th]]

    2016 arXiv

  63. [71]

    D. A. Easson, P. H. Frampton and G. F. Smoot, Phys. Lett. B696(2011), 273-277 doi:10.1016/j.physletb.2010.12.025 [arXiv:1002.4278 [hep-th]]

    2011 arXiv

  64. [72]

    J. D. Bekenstein, Phys. Rev. D51(1995) no.12, R6608 doi:10.1103/PhysRevD.51.R6608

    1995 doi

  65. [73]

    J. D. Bekenstein, Phys. Rev. D49(1994), 1912-1921 doi:10.1103/PhysRevD.49.1912 [arXiv:gr-qc/9307035 [gr-qc]]

    1994 arXiv

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