Pith. sign in

REVIEW 3 major objections 4 minor 43 references

de Sitter duality and logarithmic decay of dark energy

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that infrared fluctuations of the conformal mode make the de Sitter coupling $g = G_N H^2/\pi$ asymptotically free toward the future, so dark energy decays logarithmically.

desk verdict The paper has a real one-loop computation and a plausible resummation, but its headline prediction rests on a sign choice the authors select by hand, so the logical status is conditional, not established. read the letter →

arxiv 1908.02534 v3 pith:DGGEDOJT submitted 2019-08-07 hep-th astro-ph.COgr-qc

Hiroyuki Kitamoto , Yoshihisa Kitazawa , Takahiko Matsubara This is my paper · ORCID
classification hep-thastro-ph.COgr-qc
keywords deSitterspaceinfraredquantumgravityconformalmodeFokker-Planckequationasymptoticfreedomentropydarkenergydecay/inflationduality
topics Dark Energy
open problems Dark Energy
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

The paper sets out to show that the infrared quantum fluctuations of Einstein gravity in de Sitter space are not harmless: the conformal mode of the metric, a scalar with a negative kinetic term, diffuses at the cosmological horizon, and this diffusion screens the only dimensionless coupling $g = G_N H^2/\pi$. The proposed one-loop $\beta$ function is $\beta(g) = -\tfrac{1}{2} g^2$ in the cosmic-time variable $\log(1+6Ht)$, so $g$ runs to zero toward the future. Since $g$ is the inverse of the de Sitter entropy, the same process makes entropy grow and dark energy decay logarithmically rather than stay constant. The paper also postulates a quantum-gravity/inflation duality in which these quantum effects are reproduced classically by an inflaton with a uniquely fixed linear potential. If the claim is right, the late-time state of an accelerating universe is flat space, and the dark-energy equation of state is slightly but observably different from a cosmological constant.

What carries the argument

The load-bearing object is the conformal zero mode $\omega$, the spatially constant part of the metric conformal factor, whose kinetic term has the wrong sign. The paper treats $\omega$ stochastically: its probability distribution $\rho(\xi,\omega)$ obeys a Fokker-Planck diffusion equation, and the parameter $\xi$ is found to evolve as $\xi = 1/(1+6Ht)$, spreading the distribution and increasing its von Neumann entropy $S = -\operatorname{tr}(\rho\log\rho)$. The identity that carries the argument is the correspondence between the de Sitter entropy $S = 1/g = \pi/(G_N H^2)$ and this von Neumann entropy; demanding that the bare action $S_B = 1/g + \tfrac{1}{2}\log\xi$ be time independent then yields $\beta(g) = -\tfrac{1}{2}g^2$. A second, auxiliary machinery is the quantum/classical duality: the same screening is represented classically by an inflaton with an exponential, and at one loop linear, potential, which is introduced as a covariant counterterm to restore general covariance.

What would settle it

A next-generation dark-energy survey measuring the equation-of-state parameters could settle the claim: the paper predicts $w_0 = -1 + 1/(3e) \approx -0.877$ and $w_a = -2/(3e^2) \approx -0.090$, whereas a cosmological constant has $w_0 = -1$, $w_a = 0$. If those parameters are measured at the cosmological-constant values with errors much smaller than the gap, the logarithmic-decay claim is ruled out.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dimensionless combination $g = G_N H^2/\pi$, the only dimensionless coupling in Einstein gravity in de Sitter space, is dynamically screened by infrared fluctuations of the conformal mode. With cosmic time measured by $T = 1+6Ht$, the one-loop $\beta$ function is exact within the Gaussian approximation: $\beta(g) = dg/d\log T = -g^2/2$. The negative sign makes the coupling asymptotically free toward the future, so $H^2(t)$, and the dark energy density it represents, falls as $1/\log(1+6Ht)$ after the recent accelerated expansion begins. The paper identifies the de Sitter entropy $S = 1/g$ with the von Neumann entropy of the conformal zero mode; solving the Fokker-Planck diffusion equation for that mode gives $\xi = 1/(1+6Ht)$ and an entropy increase at rate $\dot S = 3H$, matching the semiclassical horizon-entropy result. In the past direction the Gaussian $\beta$ function has an ultraviolet fixed point at $g = 1/2$, which the paper reads as the de Sitter expansion starting at the Planck scale with minimal entropy $S = 2$.

Load-bearing premise

The conformal mode has a kinetic term with the wrong sign, so the diffusion equation can in principle run either forward or backward in time; the paper chooses the forward direction by requiring entropy to increase, and the entire screening and decay picture depends on that choice.

Editorial extensions

If this is right

  • The coupling runs as $1/g(t) = 1/g_i + \tfrac{1}{2}\log(1+6Ht)$; because $S = 1/g$, the horizon entropy grows logarithmically at rate $dS/d\log(1+6Ht) = 1/2$.
  • Dark energy, instead of being constant, decays logarithmically with cosmic time, and the expansion asymptotically approaches flat spacetime rather than de Sitter space.
  • The equation of state of dark energy is predicted to be $w_0 = -1 + 1/(3e)$ and $w_a = -2/(3e^2) \approx -0.877, -0.090$, close enough to $-1$ to fit current data but distinguishable with future surveys.
  • The Gaussian beta function has a past ultraviolet fixed point at $g = 1/2$, implying that the de Sitter phase starts at the Planck scale with minimal entropy $S = 2$.
  • In the dual picture the otherwise arbitrary inflaton potential is fixed: at one loop it is linear, with slow-roll parameters $\epsilon = \gamma$ and $\eta = 0$, so quantum gravity supplies a concrete quintessence model.

Reading between the lines

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

  • The sign of the Fokker-Planck diffusion is chosen by the paper to point in the direction of increasing entropy; if that sign were fixed independently from first principles, the entire run-to-flat-space scenario would follow without a free choice. A derivation of that sign is the key next step.
  • Taking the ultraviolet fixed point at $g = 1/2$ seriously suggests that de Sitter-like phases have a bounded past controlled by a conformal fixed point; a non-Gaussian calculation of the fixed point would show whether the Gaussian result is more than an artifact.
  • Because the entropy formula is tied to the Gaussian distribution of the conformal zero mode, non-Gaussian corrections would appear as deviations from the $(1/2)\log(1+6Ht)$ entropy law; computing those corrections would test whether the von Neumann-entropy identification survives beyond the Gaussian approximation.
  • The mechanism depends only on the conformal mode, so the same infrared beta function should apply to any nearly de Sitter epoch, including early-universe inflation; translating the late-time equation-of-state prediction into an inflationary-spectrum prediction could give a CMB test of the duality.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies infrared effects in four-dimensional Einstein gravity on de Sitter space. It parametrizes the metric by the conformal mode, computes one-loop IR logarithms in a background gauge, introduces an inflaton field as a covariant counterterm, and postulates a quantum-gravity/inflation duality. In Sec. 4 the authors set up a Fokker-Planck equation for the conformal zero mode, identify the de Sitter entropy with the von Neumann entropy of that mode, and derive the beta function β(g) = −(1/2)g² for g = G_N H²/π. This leads to the paper's main physical prediction: dark energy decays logarithmically and Einstein gravity is asymptotically free toward the future. The paper also derives an 'exact' Gaussian beta function with a UV fixed point and compares the model with H(z) data and with the standard ΛCDM model.

Significance. If established, the result would be significant: it offers a concrete IR mechanism for the smallness of g = G_N H²/π, connects the Gibbons-Hawking entropy to the von Neumann entropy of a conformal zero mode, and makes a falsifiable prediction for the time dependence of dark energy. The manuscript contains useful explicit material, including the background-gauge one-loop computation in Sec. 3, the propagator collection in Appendix A, and a transparent resummation in Sec. 4. The data comparison is honest in stating that the difference from ΛCDM is not statistically significant. The central weakness is that the sign of the Fokker-Planck diffusion term — the decisive input for the sign of β(g) — is chosen by hand rather than derived; until that sign is fixed by a first-principles argument, the headline prediction remains conditional.

major comments (3)
  1. [Sec. 4, Eq. (4.9)] The sign of the diffusion term in the Fokker-Planck equation is chosen, not derived. Because the conformal mode has a negative kinetic term (Appendix A, Eq. (A.11) and the discussion around it), the quadratic action does not fix whether ∂²ρ/∂ω² appears with a plus or a minus sign. The text immediately after Eq. (4.9) acknowledges this and selects the sign that makes the entropy increase: 'We might imagine that the sign of the right-hand side is flipped into the negative... The sensible choice is to let it coincide with that of entropy.' With the selected sign, Eq. (4.15) gives ξ = 1/(1+6Ht) and Eq. (4.24) gives β(g) = −(1/2)g². With the opposite sign, the same Gaussian ansatz gives ξ = 1/(1−6Ht), a decreasing von Neumann entropy, and, at leading order in Ht, β(g) = +g²/2. Since the headline prediction of logarithmic decay of dark energy follows only from the selected sign, an independent derivation of the sign is required before the central claim can be accepted.
  2. [Sec. 4, Eqs. (4.18) and (4.23)] The consistency checks cited for the sign choice are not independent. The positive entropy-production rate ˙S = 3Hξ in Eq. (4.18) and the agreement with the Gibbons-Hawking increase in Eq. (4.2) are consequences of the already-inserted positive sign in Eq. (4.9), not verifications of it. Similarly, the bare action in Eq. (4.23) is constructed so that S_B is time-independent, and this yields the beta function only after ξ(t) has been fixed by the signed equation. The argument therefore does not provide a separate test of the central sign; it builds the desired answer into the input.
  3. [Sec. 3 vs. Sec. 4, Eqs. (3.30)-(3.31) and (4.29)-(4.30)] The one-loop local estimate in Sec. 3 shows screening at leading order in log a_c, but it does not determine the resummed global form. The exponential local running in Eq. (4.29) and the logarithmic running in Eq. (4.30) agree only to first order. The logarithmic decay is a property of the Fokker-Planck resummation and therefore inherits the undetermined sign of Eq. (4.9). The text should not present the Sec. 3 computation as independent support for the logarithmic prediction; at most it supports the weaker statement that the dimensionless coupling is screened at one loop.
minor comments (4)
  1. [Abstract] The abstract contains a typo: 'stared' should be 'started'.
  2. [Sec. 5, Eq. (5.39)] The replacement 1+Ht → e + log(1+z) introduces a time-translation freedom and an e-shift normalization; the statement in Sec. 5 that 'there is no free parameter here' is too strong, since the e-shift is a convention that affects the normalization of log(1+Ht0)=1.
  3. [Table 1] The χ²/dof values 0.623 and 0.739 are close, and the paper itself notes the difference is not statistically significant. The wording that the model 'fares well' and is 'promising' should be restricted to consistency with current data, not presented as evidence in favor of the model.
  4. [Abstract and Sec. 5, Eq. (5.30)] The word 'exact' for the β function with backreaction is misleading: Eq. (5.30) is derived within the Gaussian ansatz, and the text later acknowledges this is not a proof. The abstract should say 'exact within the Gaussian approximation'.

Circularity Check

2 steps flagged · score 5.0 of 10

The central β(g)=−(1/2)g² prediction is not fitted to external data, but its sign is imposed by the Fokker-Planck sign chosen to match entropy increase; with the sign the paper itself admits could be flipped, the same Gaussian ansatz gives β(g)=+g²/2.

  1. self definitional [Sec. 4, Eqs. (4.8)-(4.9), (4.15)-(4.24)]
    "To the leading order in the log ac = Ht expansion, its growing speed is expected as follows ˙S = 1/2 (− ˙ξ/ξ) = 3H, to be consistent with semiclassical result (4.2). ... We might imagine that the sign of the right-hand side is flipped into the negative. However, the direction of time flow is not prefixed in quantum gravity. The sensible choice is to let it coincide with that of entropy."

    The Fokker-Planck diffusion sign in (4.9) is expressly not fixed by the calculation; the paper selects the sign so that entropy increases. With the chosen sign, ξ=1/(1+6Ht) and ˙S=3Hξ>0. With the opposite sign, which the paper says one 'might imagine,' the same Gaussian reduction gives ξ=1/(1−6Ht), a decreasing von Neumann entropy, and the same bare-action construction yields β(g)=+g²/2. The sign of the headline β function is therefore put in by the requirement that entropy increase, not derived from it; the later entropy-increase checks (4.18)-(4.19) are consequences of that imposed sign.

  2. fitted input called prediction [Sec. 4, Eqs. (4.23)-(4.24)]
    "Since SB is the bare action, we derive the β functions in a standard way, i.e., by requiring SB to be time independent, β(g) = −1/2 g2, β(g) ≡ ∂/∂ log(1 + 6Ht) g."

    With the chosen ξ(t)=1/(1+6Ht), the bare action is SB=1/g(t)−(1/2)log(1+6Ht) up to constants. The condition ∂SB/∂log(1+6Ht)=0 is algebraically d(1/g)/dlog(1+6Ht)=1/2, i.e., exactly β=−g²/2. Thus (4.24) restates the ξ(t) already fixed by matching the Gibbons-Hawking entropy growth ˙S=3H; the advertised 'logarithmic decay of dark energy' is the differential form of that input, not an independent consequence. The coefficient −1/2 is not fitted to external data, but neither is it a prediction independent of the sign-entropy input.

full rationale

The derivation is largely self-contained and is tested against external H(z) and dark-energy observations in Sec. 5; the self-citations to the authors' earlier IR-logarithm results are supported by the one-loop computation in Sec. 3 and Appendix A, so they are not load-bearing by citation alone. The circularity is concentrated in Sec. 4: the Fokker-Planck equation's diffusion sign is not determined by the negative kinetic term, and the paper chooses it by demanding entropy increase. That choice alone fixes the sign of β(g), and the subsequent RG step (4.23)-(4.24) converts the already imposed time dependence ξ=1/(1+6Ht) into the claim β(g)=−(1/2)g². Flipping the admitted sign reverses the central prediction, so the headline logarithmic decay is partly circular: its direction is an input rather than a consequence. The magnitude and mechanism, and the local one-loop screening, retain independent content, which keeps the overall circularity score moderate rather than maximal.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central construction rests on a duality postulate, a chosen sign in the Fokker-Planck equation, a Gaussian ansatz, and the identification of de Sitter entropy with conformal-mode entropy. The free parameters in the observational comparison are the e-shift normalization and H0. The inflaton counterterm is the main invented entity, with no independent evidence outside the paper.

free parameters (2)
  • e-shift normalization = e (the constant e in x -> x + e)
    In Eq. (5.39) the time translation x -> x + e is inserted so that log(1 + H t0) = 1 at z = 0; this choice tunes the dark-energy normalization and is not derived from the theory.
  • Hubble constant H0 = 67.2 +/- 2.5 km/s/Mpc in this work
    H0 is fitted to 51 H(z) data points in Table 1 with Npar = 1, so the model comparison has one fitted parameter.
assumptions (6)
  • ad hoc to paper There is a duality between quantum effects in Einstein gravity and classical evolutions in an inflation or quintessence model.
    Stated as a working assumption in Sec. 2: 'Our working assumption is that there is a duality between a quantum gravity and an inflation theory.' The paper postulates this rather than deriving it.
  • ad hoc to paper The sign of the Fokker-Planck diffusion term is chosen so that the von Neumann entropy increases.
    Sec. 4, near Eq. (4.9): the direction of time flow is said to be not prefixed, and the 'sensible choice' is to let it coincide with that of entropy. This sign choice is load-bearing for the screening result.
  • domain assumption Only the massless minimally coupled conformal mode contributes the infrared logarithms; the conformally coupled modes are set to zero.
    Sec. 3 and Appendix A restrict the field space to h00 about 2 omega and b0 about 0, neglecting h0i, Y, and massive ghost modes. This is a standard infrared approximation but is not fully proven.
  • domain assumption The distribution of the conformal zero mode stays Gaussian at all times.
    Sec. 4 assumes a Gaussian ansatz for rho(xi, omega). The authors say the Gaussian approximation is excellent because g is small, but they later apply it at the strong-coupling fixed point g = 1/2, where the approximation is uncontrolled.
  • ad hoc to paper The de Sitter entropy is identified with the von Neumann entropy of the conformal zero mode.
    Sec. 4 states: 'Our hypothesis is that the von Neumann entropy accounts for the time dependent part of the de Sitter entropy.' This identification is a conjecture that the paper tests for consistency but does not prove.
  • domain assumption The Gibbons-Hawking formula S = pi/(G_N H^2) remains valid as H evolves.
    The paper repeatedly uses S = 1/g = pi/(G_N H^2) to translate entropy growth into coupling decay, assuming the semiclassical area formula holds throughout the evolution.
invented entities (1)
  • Inflaton counterterm field f
    purpose: A covariant counterterm to cancel the noncovariant infrared logarithm and to satisfy the h00 equation of motion in the dual inflation picture.
    The inflaton field appears in Eq. (2.22) and is identified on-shell with the conformal mode via a = e^f. It has no independent observational signature; it is introduced to make the effective action manifestly covariant.

how reviews work

0 comments
Cite this review

Pith. "Pith review of de Sitter duality and logarithmic decay of dark energy." pith.science (2026). https://pith.science/paper/DGGEDOJT

@misc{pith2026190802534,
  author       = {Pith},
  title        = {Pith review of: de Sitter duality and logarithmic decay of dark energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGGEDOJT}},
  note         = {Machine review of arXiv:1908.02534}
}
abstract

We investigate infrared dynamics of four-dimensional Einstein gravity in de Sitter space. We set up a general framework to investigate dynamical scaling relations in quantum/classical gravitational theories. The conformal mode dependence of Einstein gravity is renormalized to the extent that general covariance is not manifest. We point out that the introduction of an inflaton is necessary as a counterterm. We observe and postulate a duality between quantum effects in Einstein gravity and classical evolutions in an inflation (or quintessence) model. The effective action of Einstein gravity can be constructed as an inflation model with manifest general covariance. We show that $g=G_N H^2/\pi$: the only dimensionless coupling of the Hubble parameter $H^2$ and the Newton's coupling $G_N$ in Einstein gravity is screened by the infrared fluctuations of the conformal mode. We evaluate the one-loop $\beta$ function of $g$ with respect to the cosmic time $\log Ht$ as $\beta(g)=-(1/2)g^2$, i.e., $g$ is asymptotically free toward the future. The exact $\beta$ function with the backreaction of $g$ reveals the existence of the ultraviolet fixed point. It indicates that the de Sitter expansion stared at the Planck scale with a minimal entropy $S=2$. We have identified the de Sitter entropy $1/g$ with the von Neumann entropy of the conformal zero mode. The former evolves according to the screening of $g$ and the Gibbons-Hawking formula. The latter is found to increase by diffusion in the stochastic process at the horizon in a consistent way. Our Universe is located very close to the fixed point $g=0$ with a large entropy. We discuss possible physical implications of our results such as logarithmic decay of dark energy.

Figures

Figures reproduced from arXiv: 1908.02534 by the authors.

Figure 1
Figure 1. The Hubble parameter measurements and their errors (in [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 11 canonical work pages

  1. [1]

    Entropy Generation at the Horizon Diffuses Cosmological Constant in 2D de Sitter Space

    H. Kitamoto and Y. Kitazawa, Phys. Rev. D 99, no. 8, 085015 (2019) [arXiv:1902.06571 [hep-th]]

    work page Pith review arXiv 2019

  2. [2]

    A. M. Polyakov, Sov. Phys. Usp. 25, 187 (1982) [Usp. Fiz. Nauk 136, 538 (1982)]. A. M. Polyakov, hep-th/9304146

    arXiv 1982

  3. [3]

    A. M. Polyakov, Nucl. Phys. B 797, 199 (2008) [arXiv:0709.2899 [hep-th]]

    arXiv 2008

  4. [4]

    Quantum Relaxation of the Cosmological Constant

    R. Jackiw, C. Nunez and S.-Y. Pi, Phys. Lett. A 347, 47 (2005) [hep-th/0502215]

    work page Pith review arXiv 2005

  5. [5]

    Soft Gravitons Screen Couplings in de Sitter Space

    H. Kitamoto and Y. Kitazawa, Phys. Rev. D 87, no. 12, 124007 (2013) [arXiv:1203.0391 [hep-th]]

    work page Pith review arXiv 2013

  6. [6]

    Time Dependent Couplings as Observables in de Sitter Space

    H. Kitamoto and Y. Kitazawa, Int. J. Mod. Phys. A 29, no. 8, 1430016 (2014) [arXiv:1402.2443 [hep-th]]. ¶ Recall that ac = 1/ (−HRτ), H 2 R = H 2(1 + 3 2 γ). 37

    work page Pith review arXiv 2014

  7. [7]

    N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 474, 235 (1996) [hep-ph/9602315]

    arXiv 1996

  8. [8]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. D 72, 043514 (2005) [hep-th/0506236]. S. Weinberg, Phys. Rev. D 74, 023508 (2006) [hep-th/0605244]

    arXiv 2005

Show all 43 references
  1. [9]

    D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973)

    1973

  2. [10]

    H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973)

    1973

  3. [11]

    A. A. Starobinsky, Lect. Notes Phys. 246, 107 (1986). A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357 (1994) [astro-ph/9407016]

    1986 arXiv

  4. [12]

    N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 724, 295 (2005) [gr-qc/0505115]

    2005 arXiv

  5. [13]

    A. V. Frolov and L. Kofman, JCAP 0305, 009 (2003) [hep-th/0212327]

    2003 arXiv

  6. [14]

    Kachru, R

    S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [hep-th/0301240]

    2003 arXiv

  7. [15]

    Obied, H

    G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, arXiv:1806.0836 2 [hep-th]

  8. [16]

    A. A. Starobinsky, JETP Lett. 30, 682 (1979) [Pisma Zh. Eksp. Teor. Fiz. 30, 719 (1979)]

    1979

  9. [17]

    A. H. Guth, The Inflationary Universe (Perseus Books, New York, 1997)

    1997

  10. [18]

    A. D. Linde, Phys. Lett. 108B, 389 (1982) [Adv. Ser. Astrophys. Cosmol. 3, 149 (1987)]

    1982

  11. [19]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982) [Adv. Ser. Astro- phys. Cosmol. 3, 158 (1987)]

    1982

  12. [20]

    Ratra and P

    B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988)

    1988

  13. [21]

    R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998) [astro-ph/9708069]

    1998 arXiv

  14. [22]

    Kitamoto and Y

    H. Kitamoto and Y. Kitazawa, Phys. Rev. D 83, 104043 (2011) [arXiv:1012.5930 [hep- th]]

    2011 arXiv

  15. [23]

    Kitamoto and Y

    H. Kitamoto and Y. Kitazawa, Phys. Rev. D 85, 044062 (2012) [arXiv:1109.4892 [hep- th]]

    2012 arXiv

  16. [24]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, Ann. Inst. H. Poincare Phys. T heor. A 20, 69 (1974)

    1974

  17. [25]

    Kawai, Y

    H. Kawai, Y. Kitazawa and M. Ninomiya, Nucl. Phys. B 404, 684 (1993) [hep-th/9303123]

    1993 arXiv

  18. [26]

    J. M. Maldacena, JHEP 0305, 013 (2003) [astro-ph/0210603]. 38

    2003 arXiv

  19. [27]

    Strominger, JHEP 0110, 034 (2001) [hep-th/0106113]

    A. Strominger, JHEP 0110, 034 (2001) [hep-th/0106113]

    2001 arXiv

  20. [28]

    Balasubramanian, J

    V. Balasubramanian, J. de Boer and D. Minic, Phys. Rev. D 65, 123508 (2002) [hep-th/0110108]

    2002 arXiv

  21. [29]

    Balasubramanian, J

    V. Balasubramanian, J. de Boer and D. Minic, Class. Quant. Grav . 19, 5655 (2002) [Annals Phys. 303, 59 (2003)] [hep-th/0207245]

    2002 arXiv

  22. [30]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977)

    1977

  23. [31]

    C. D. Froggatt and H. B. Nielsen, Phys. Lett. B 368, 96 (1996) [hep-ph/9511371]

    1996 arXiv

  24. [32]

    Hamada, H

    Y. Hamada, H. Kawai and K. Kawana, Int. J. Mod. Phys. A 29, 1450099 (2014) [arXiv:1405.1310 [hep-ph]]

    2014 arXiv

  25. [33]

    Maga˜ na, M

    J. Maga˜ na, M. H. Amante, M. A. Garcia-Aspeitia and V. Motta, Mon. Not. Roy. Astron. Soc. 476, no. 1, 1036 (2018) [arXiv:1706.09848 [astro-ph.CO]]

    2018 arXiv

  26. [34]

    Aghanim et al

    N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO]

    arXiv

  27. [35]

    A. G. Riess et al., Astrophys. J. 826, no. 1, 56 (2016) [arXiv:1604.01424 [astro-ph.CO]]

    2016 arXiv

  28. [36]

    Chevallier and D

    M. Chevallier and D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001) [gr-qc/0009008]

    2001 arXiv

  29. [37]

    E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003) [astro-ph/0208512]

    2003 arXiv

  30. [38]

    Alam et al

    S. Alam et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. 470 (2017) no.3, 2617 [arXiv:1607.03155 [astro-ph.CO]]

    2017 arXiv

  31. [39]

    Aghamousa et al

    A. Aghamousa et al. [DESI Collaboration], arXiv:1611.00036 [astro-ph.IM]

    arXiv

  32. [40]

    Laureijs et al

    R. Laureijs et al. [EUCLID Collaboration], arXiv:1110.3193 [astro-ph.CO]

    arXiv

  33. [41]

    Kitamoto, Y

    H. Kitamoto, Y. Kitazawa and R. Kojima, Phys. Rev. D 96, no. 2, 023535 (2017) [arXiv:1704.08845 [hep-th]]

    2017 arXiv

  34. [42]

    Nishimura and A

    J. Nishimura and A. Tsuchiya, JHEP 1906, 077 (2019) [arXiv:1904.05919 [hep-th]]

    2019 arXiv

  35. [43]

    Berglund, T

    P. Berglund, T. H¨ ubsch and D. Mini´ c, Phys. Lett. B 798, 134950 (2019) [arXiv:1905.08269 [hep-th]]. 39

    2019 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.