Pith. sign in

REVIEW 2 major objections 5 minor 40 references

Inhomogeneities in the matter distribution can turn a pure cosmological constant into dark energy whose equation of state evolves in time, and the evolution is never phantom-crossing.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 09:34 UTC pith:WUZT5EX7

load-bearing objection A serious, parameter-free back-reaction calculation that produces an evolving dark energy, but the central second-order result rests on an unproven branch choice of the long-wavelength solution. the 2 major comments →

arxiv 2607.20715 v1 pith:WUZT5EX7 submitted 2026-07-22 gr-qc astro-ph.COhep-th

Evolving Dark Energy from the Back-Reaction of Cosmological Perturbations

Vincent Comeau This is my paper
classification gr-qc astro-ph.COhep-th MSC 83C0583F0583C25
keywords back-reactiondark energy equation of statecosmological perturbationsgradient expansionaveraging in cosmologyphantom crossingcosmological 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.

The paper argues that the back-reaction of an inhomogeneous matter distribution, added on top of a pure cosmological constant, makes the effective dark-energy equation of state time-dependent. In the usual treatment of small perturbations around a homogeneous background, this time dependence is a boundary term and disappears for a global average. The paper's central claim is that a different expansion—treating spatial derivatives, not amplitudes, as small—produces the same time dependence without the boundary-term suppression, so the effect can survive a global average. The result is an explicit formula for the dark-energy equation of state in terms of curvature invariants of an inhomogeneous background metric, with no free parameters apart from the initial inhomogeneity field. If correct, this means cosmic structure by itself can mimic evolving dark energy, and the sign of the effect is locked: dark energy is always phantom or never phantom around the present time.

Core claim

The paper's central result, Eq. (73), is that at second order in a gradient expansion the effective dark-energy equation of state is w_DE = -1 - [2/(9 H0^4 Omega_Lambda)] ( <Rbar^i_j Rbar^j_i> - (3/8)<Rbar^2> + (1/24)<Rbar>^2 ) ( g(a)/a^2 - (3/2) f(a)^2 ), where Rbar_ij is the Ricci tensor of a time-independent inhomogeneous three-metric, averages are over a fixed volume of matter particles, and f(a) and g(a) are functions built from hypergeometric integrals. The computation starts from the long-wavelength branch in which the spatial metric is frozen; at the next order, perturbations sourced by the background Ricci tensor generate both a kinetic back-reaction and a shift in the average spati

What carries the argument

The carrying mechanism is a second-order gradient expansion around a time-independent inhomogeneous three-metric gbar_ij(x)—the long-wavelength branch in which the spatial metric does not change with time. At the next order the time-dependent perturbation h_ij is sourced by the background Ricci tensor, and the same two hypergeometric integrals f(a) and g(a) that govern linear scalar perturbations control the back-reaction. The average Friedmann equations then attribute everything that is not constant spatial curvature to a back-reaction fluid, whose combination with the cosmological constant defines the effective dark energy. The key identity is that the resulting equation of state separates

Load-bearing premise

The entire second-order computation is built on the choice of a frozen, time-independent spatial metric as the long-wavelength background (the 'simple solution' selected in Sec. 4.2) together with stopping at second order in spatial derivatives; if other long-wavelength branches participate or the truncation is uncontrolled, the non-boundary back-reaction and the no-crossing conclusion can change.

What would settle it

Evaluate the curvature combination C = <Rbar^i_j Rbar^j_i> - (3/8)<Rbar^2> + (1/24)<Rbar>^2 on realistic spatial hypersurfaces from fully relativistic cosmological simulations; if C is consistent with zero, or if its sign flips with the averaging volume, the predicted evolving equation of state and its sign-locked phantom/non-phantom behavior would fail. A second decisive check is to extend the gradient expansion to fourth order: if w_DE crosses -1 there, the second-order no-crossing result is an artifact of truncation.

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

If this is right

  • Dark energy's equation of state acquires a calculable, time-dependent correction that survives a global average, so structure formation can contribute to apparent evolving dark energy.
  • The two leading parameters of the standard near-present expansion of the equation of state carry the same sign, determined by the sign of the curvature invariant; hence dark energy cannot cross the phantom divide near the present time.
  • The effect can be significant even when the average Ricci scalar is negligible, because the squared-Ricci and variance terms in the curvature combination need not vanish.
  • When the inhomogeneous background is specialized to flat space with a small scalar perturbation, the gradient-expansion result reproduces the standard second-order perturbative formula, extending it to all orders in the perturbation amplitude.
  • Far into the future the time dependence decays as the cosmological constant dominates, so the correction is a transient around the present epoch.

Where Pith is reading between the lines

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

  • Going beyond the paper, the same non-boundary back-reaction mechanism should also apply to long-wavelength tensor modes, since the gradient-expansion branch used here already covers their leading behavior; settling this would only require solving the tensor wave equation away from the long-wavelength limit.
  • Going beyond the paper, if future surveys continue to find w0 and w1 of opposite signs, that would point away from this back-reaction channel and toward alternative dark-energy models, unless higher-order gradient terms overturn the sign-locking.
  • Going beyond the paper, the curvature combination in Eq. (73) is in principle measurable from spatial slices of fully relativistic cosmological simulations; a robust nonzero sign for it would give a concrete prediction of whether our Universe's dark energy is phantom.

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 argues that, when a pure cosmological constant is combined with the back-reaction of inhomogeneities, the effective dark-energy equation of state w_DE acquires a time dependence. In the standard perturbative treatment, at quadratic order the scalar-mode contribution reduces to a boundary term and hence is negligible for a global average (§3.5). The paper's main result, derived in §4 via a gradient expansion around a time-independent inhomogeneous spatial metric, is that at second order the back-reaction no longer reduces to a boundary term. The central formula, Eq. (73), gives w_DE = -1 - (2/(9 H0^4 Ω_Λ)) S (g(a)/a^2 - (3/2) f(a)^2), where S = ⟨Rbar^j_i Rbar^i_j⟩ - (3/8)⟨Rbar^2⟩ + (1/24)⟨Rbar⟩^2. This is translated into the CPL parameters δw0, δw1 in Eq. (74), which are positive functions of Ωm, ΩΛ only. The paper concludes that the effective dark energy is either always phantom or never phantom, with no crossing around the present epoch. The derivation is parameter-free in the sense that the background density parameters are taken from external cosmology and the initial inhomogeneity fields are not fitted to dark-energy data.

Significance. If the calculation is correct, it provides a concrete, analytic, parameter-free mechanism by which cosmological structure modifies the equation of state of dark energy, with a falsifiable sign prediction. The paper is careful in setting up the averaging formalism, uses explicit hypergeometric solutions, and clearly states the initial conditions. The agreement with recent numerical simulations is claimed only qualitatively, but the analytic formula is a useful benchmark. The main caveat is that the central result depends on a specific, unvalidated branch of the long-wavelength solution, and no estimate of the truncation error is given. These issues are load-bearing because the no-phantom-crossing conclusion and the non-boundary character of the back-reaction rest on them.

major comments (2)
  1. [§4.2, Eqs. (61)–(73)] The central result rests on the L=0 branch of the long-wavelength solution. The paper states that non-trivial solutions to Eq. (61) exist and will be discussed elsewhere, but provides no argument that the initial conditions (28) select L=0 or that other branches are subdominant. Equations (65), (69), (71), and (73) are all derived from this branch. If the general solution contains L^i_j ≠ 0, the curvature combination in Eq. (73) could acquire a different scale-factor dependence or become a boundary term, changing both the predicted evolution and the no-phantom-crossing conclusion. Please give the general solution of Eq. (61) under the stated initial conditions, or a stability/robustness analysis showing that L=0 is the only relevant branch for the averaged quantities.
  2. [§1.5 and §4.2] The no-phantom-crossing claim is stated in the abstract as unconditional, but §1.5 acknowledges that including higher gradient orders could allow a crossing. The truncation at second order in the gradient expansion is not accompanied by an error estimate. To make the conclusion robust, either estimate the next-order contributions to ρ_br and p_br, or explicitly qualify all claims as 'at second order in the gradient expansion.' The abstract and Eq. (6) should be amended accordingly.
minor comments (5)
  1. [§3.6, Eq. (60)] The kinematical back-reaction from the tensor-mode gradient expansion appears to be missing the factor f(a)^2 present in the analogous results, Eq. (48) and Eq. (71). If intentional, the different time dependence should be discussed; otherwise it is a typo.
  2. [Appendix A] 'Pocchammer' should be 'Pochhammer'.
  3. [§1.3 / §4.4] The notation δ¯w(a) is used in Eqs. (4)–(6) before it is defined; please define it explicitly at first use, e.g., after Eq. (51) or Eq. (73).
  4. [Appendix B] Typo: 'particules' should be 'particles'.
  5. [§1.4 / §4.4] The comparison with numerical simulations [22–24] is only qualitative. A quantitative comparison, even order-of-magnitude, would strengthen the claim of agreement.

Circularity Check

0 steps flagged

No significant circularity: the central equation of state is derived from Einstein equations with standard external density parameters; the L=0 gradient-expansion branch is an acknowledged limitation, not a circular reduction.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The central result, Eq. (73), is obtained by solving the averaged Einstein equations with a gradient expansion: the time-dependent functions f(a) and g(a) are derived from the homogeneous background containing cold matter and a cosmological constant, with density parameters Ωm and ΩΛ taken from standard external cosmology. The numerical parameters δ̄w0 and δ̄w1 are computed from those same density parameters, not fitted to dark-energy data, so the 'prediction' of an evolving w_DE is not a fitted input renamed as a prediction. The curvature combination ⟨R̄^i_j R̄^j_i⟩ − (3/8)⟨R̄²⟩ + (1/24)⟨R̄⟩² is an initial-condition functional, not a quantity extracted from the target equation of state. The reduction of Eq. (73) to Eq. (51) under the specialization ḡ_ij = δ_ij + δ_ij A is presented as a consistency check, not as the derivation. The most notable caveat — the choice in Sec. 4.2 of the simple long-wavelength solution L^i_j = 0 while acknowledging other solutions 'discussed in more details elsewhere' — is an unproven branch/validity limitation that could affect the result, but it is not a circularity: the paper does not define the result in terms of that choice, and the choice is explicitly flagged. Self-citations to the Buchert framework and to Brandenberger-group work on gradient expansions are contextual or methodological references, not load-bearing uniqueness theorems smuggled in from the same authors. Therefore no specific circular step meeting the quote-and-reduction standard is present.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles, forces, or fields. The 'back-reaction fluid' and 'effective dark energy' are bookkeeping combinations in the averaged Friedmann equations, not independently evidenced entities. The main ledger items are the initial inhomogeneity fields and the branch choice in the gradient expansion.

free parameters (2)
  • Background density parameters Omega_m, Omega_Lambda = 0.3, 0.7 (illustrative)
    Taken from standard cosmology to produce Figures 1-2; not fitted to the target result, but the numerical values of delta_bar_w0 and delta_bar_w1 depend on them.
  • Initial inhomogeneity fields (A(x), Rbar_ij) = unconstrained
    The magnitude and sign of delta_w0 and delta_w1 are set by averages of these initial fields. The paper gives no estimate from the observed matter distribution, so the central claim of a 'significant' effect is conditional on these unmeasured inputs.
axioms (6)
  • domain assumption Einstein gravity with dust and a cosmological constant, neglecting radiation and global spatial curvature around present time
    Stated in Sec. 1.1 and Sec. 2.3; changes to the background H(a) would alter the f(a) and g(a) functions on which the whole result rests.
  • domain assumption Synchronous-comoving Lagrangian coordinates exist for the dust, with g00=-1 and g0i=0
    Sec. 2.2; requires irrotational dust moving geodesically and a choice of initial synchronization.
  • domain assumption Buchert averaging framework and the commutation rule (19) apply to the chosen particle volume
    Sec. 2.1 and 2.4; the decomposition into back-reaction fluid and constant-curvature terms in Eqs. (25)-(27) depends entirely on this framework.
  • ad hoc to paper Initial conditions at a_in << 1: back-reaction density and pressure vanish, and the decaying mode is neglected
    Eq. (28) and footnote 2; this fixes the integration constant B(x) and selects the growing adiabatic mode. A different initial condition changes the time dependence of the result.
  • ad hoc to paper In the gradient expansion, the zeroth-order long-wavelength solution is taken to be L=0, i.e. a time-independent spatial metric gbar_ij(x)
    Sec. 4.2: the author states that non-trivial long-wavelength solutions exist but defers them; the central Eq. (73) is derived only for this L=0 branch.
  • domain assumption Truncation at second order in the gradient expansion (up to four spatial derivatives) captures the relevant back-reaction
    Sec. 4 and Sec. 1.5; no error bound is given, and higher orders could change the sign or produce phantom crossing.

pith-pipeline@v1.3.0-alltime-deepseek · 18711 in / 18221 out tokens · 164258 ms · 2026-08-01T09:34:23.593939+00:00 · methodology

0 comments
read the original abstract

We argue that the back-reaction of inhomogeneities, when combined with a pure cosmological constant, produces an effective dark energy whose equation of state can evolve with time. Our analytical computation agrees with recent numerical simulations, which have shown that the back-reaction of inhomogeneities can impact dark energy around the present time. Here, its equation of state is derived from an average version of the Friedmann equations, for which the average is taken over a fixed volume of matter particles (not necessarily global). When treating the inhomogeneities as small perturbations around a homogeneous background, dark energy acquires a time dependence which reduces to a boundary term, at least at quadratic order in the scalar modes of the perturbations. Thus, it becomes negligible for a global average. On the other hand, when treating instead the spatial derivatives of the inhomogeneities perturbatively in a gradient expansion, dark energy gets the same time dependence, which no longer reduces to a boundary term. This time dependence can be significant even for a global average or when the constant spatial curvature induced by the inhomogeneities is negligible. It also keeps the same sign around the present time, thus causing dark energy to remain either always phantom or never phantom, not allowing for any crossing from one type of behaviour to the other.

Figures

Figures reproduced from arXiv: 2607.20715 by Vincent Comeau.

Figure 1
Figure 1. Figure 1: Parameters δw¯0 and δw¯1 for different values of the density pa￾rameter for cold matter at the present time. The sign of the parameters δw0 and δw1 depends on the spatial curvature terms, which could be either positive or negative, as well as on the parameters δw¯0 and δw¯1 . As shown in figure 1, these parameters are very similar to each other and remain 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Values of δw¯ as a function of the scale factor, for density param￾eters given by Ωm = 0.3 and ΩΛ = 0.7. This feature of our result differs from the numerical analysis conducted in [22], which finds that the two parameters of the Chevallier-Polarski-Linder model usually have opposite signs, with δw0 being positive and δw1 negative. Current observational constraints also seem to favour the parameters having… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 35 linked inside Pith

  1. [1]

    Reconstructing dark energy,

    V. Sahni and A. Starobinsky, “Reconstructing dark energy,”Int. J. Mod. Phys. D152105, 2006,astro-ph/0610026

    Pith/arXiv arXiv 2006

  2. [2]

    Dark energy,

    M. Li, X.-D. Li, S. Wang and Y. Wang, “Dark energy,”Commun. Theor. Phys. 56525, 2011,1103.5870

    Pith/arXiv arXiv 2011

  3. [3]

    Accelerating universes with scaling dark matter,

    M. Chevallier and D. Polarski, “Accelerating universes with scaling dark matter,”Int. J. Mod. Phys. D10213, 2001,gr-qc/0009008

    Pith/arXiv arXiv 2001

  4. [4]

    Exploring the expansion history of the universe,

    E. Linder, “Exploring the expansion history of the universe,”Phys. Rev. Lett.90 091301, 2003,astro-ph/0208512

    Pith/arXiv arXiv 2003

  5. [5]

    Gravitational lensing as a probe of quintessence,

    A. Cooray and D. Huterer, “Gravitational lensing as a probe of quintessence,” Astrophys. J. Lett.513L95, 1999,astro-ph/9901097

    Pith/arXiv arXiv 1999

  6. [6]

    Probing quintessence: Reconstruction and parameter estimation from supernovae,

    B. Gerke and G. Efstathiou, “Probing quintessence: Reconstruction and parameter estimation from supernovae,”Mon. Not. Roy. Astron. Soc.33533, 2002,astro-ph/0201336. 27

    Pith/arXiv arXiv 2002

  7. [7]

    Back-reaction of cosmological perturbations and the cosmological constant problem,

    R. Brandenberger, “Back-reaction of cosmological perturbations and the cosmological constant problem,” 2002,hep-th/0210165

    Pith/arXiv arXiv 2002

  8. [8]

    Dark energy from back-reaction,

    S. Rasanen, “Dark energy from back-reaction,”JCAP02003, 2004, astro-ph/0311257

    Pith/arXiv arXiv 2004

  9. [9]

    The universe seen at different scales,

    G.F.R. Ellis and T. Buchert, “The universe seen at different scales,”Phys. Lett. A34738, 2005,gr-qc/0506106

    Pith/arXiv arXiv 2005

  10. [10]

    Dark energy from structure: A status report,

    T. Buchert, “Dark energy from structure: A status report,”Gen. Rel. Grav.40 467, 2008,0707.2153

    Pith/arXiv arXiv 2008

  11. [11]

    Back-reaction in late-time cosmology,

    T. Buchert and S. Rasanen, “Back-reaction in late-time cosmology,”Ann. Rev. Nucl. Part. Sci.6257, 2012,1112.5335

    Pith/arXiv arXiv 2012

  12. [12]

    Is there proof that back-reaction of inhomogeneities is irrelevant in cosmology?,

    T. Buchert et al., “Is there proof that back-reaction of inhomogeneities is irrelevant in cosmology?,”Class. Quant. Grav.32215021, 2015,1505.07800

    Pith/arXiv arXiv 2015

  13. [13]

    Inhomogeneity effects in cosmology,

    G.F.R. Ellis, “Inhomogeneity effects in cosmology,”Class. Quant. Grav.28 164001, 2011,1103.2335

    Pith/arXiv arXiv 2011

  14. [14]

    Does the growth of structure affect our dynamical models of the universe? The averaging, back-reaction and fitting problems in cosmology,

    C. Clarkson, G. Ellis, J. Larena and O. Umeh, “Does the growth of structure affect our dynamical models of the universe? The averaging, back-reaction and fitting problems in cosmology,”Rept. Prog. Phys.74112901, 2011,1109.2314

    Pith/arXiv arXiv 2011

  15. [15]

    Back-reaction of inhomogeneities can mimic dark energy,

    E.W. Kolb, “Back-reaction of inhomogeneities can mimic dark energy,”Class. Quant. Grav.28164009, 2011

    2011

  16. [16]

    Do large-scale inhomogeneities explain away dark energy?,

    G. Geshnizjani, D.J.H. Chung and N. Afshordi, “Do large-scale inhomogeneities explain away dark energy?,”Phys. Rev. D72023517, 2005,astro-ph/0503553

    Pith/arXiv arXiv 2005

  17. [17]

    Newtonian and relativistic cosmologies,

    S.R. Green and R.M. Wald, “Newtonian and relativistic cosmologies,”Phys. Rev. D85063512, 2012,1111.2997

    Pith/arXiv arXiv 2012

  18. [18]

    Comments on back-reaction,

    S.R. Green and R.M. Wald, “Comments on back-reaction,” 2015,1506.06452

    Pith/arXiv arXiv 2015

  19. [19]

    Safely smoothing spacetime: Back-reaction in relativistic cosmological simulations,

    J. Adamek, C. Clarkson, D. Daverio, R. Durrer and M. Kunz, “Safely smoothing spacetime: Back-reaction in relativistic cosmological simulations,”Class. Quant. Grav.36014001, 2019,1706.09309

    Pith/arXiv arXiv 2019

  20. [20]

    Einstein’s Universe: Cosmological structure formation in numerical relativity,

    H. Macpherson, D. Price and P. Lasky, “Einstein’s Universe: Cosmological structure formation in numerical relativity,”Phys. Rev. D99063522, 2019, 1807.01711

    Pith/arXiv arXiv 2019

  21. [21]

    Backreaction in numerical relativity: Averaging on Newtonian gauge-like hypersurfaces in Einstein Toolkit cosmological simulations,

    A. Oestreicher and S.M. Koksbang, “Backreaction in numerical relativity: Averaging on Newtonian gauge-like hypersurfaces in Einstein Toolkit cosmological simulations,”Open J. Astrophys.72024, 2024,2408.03049

    Pith/arXiv arXiv 2024

  22. [22]

    Apparent dark-energy evolution from cosmic inhomogeneities,

    Y.B. Ginat and P. Ferreira, “Apparent dark-energy evolution from cosmic inhomogeneities,” 2026,2601.20633. 28

    arXiv 2026

  23. [23]

    On the potential for inhomogeneities to mimic an evolving dark energy,

    H. Macpherson and G. Valogiannis, “On the potential for inhomogeneities to mimic an evolving dark energy,” 2026,2606.05479

    Pith/arXiv arXiv 2026

  24. [24]

    Back-reaction of super-Hubble fluctuations, late time tracking, and recent observational results,

    M.A.C. Alvarez, L. Graef and R. Brandenberger, “Back-reaction of super-Hubble fluctuations, late time tracking, and recent observational results,”Symmetry17 1507, 2025,2502.17395

    Pith/arXiv arXiv 2025

  25. [25]

    On average properties of inhomogeneous fluids in general relativity: Dust cosmologies,

    T. Buchert, “On average properties of inhomogeneous fluids in general relativity: Dust cosmologies,”Gen. Rel. Grav.32105, 2000,gr-qc/9906015

    Pith/arXiv arXiv 2000

  26. [26]

    On average properties of inhomogeneous fluids in general relativity: Perfect fluid cosmologies,

    T. Buchert, “On average properties of inhomogeneous fluids in general relativity: Perfect fluid cosmologies,”Gen. Rel. Grav.331381, 2001,gr-qc/0102049

    Pith/arXiv arXiv 2001

  27. [27]

    On average properties of inhomogeneous fluids in general relativity III: General fluid cosmologies,

    T. Buchert, P. Mourier and X. Roy, “On average properties of inhomogeneous fluids in general relativity III: General fluid cosmologies,”Gen. Rel. Grav.5227, 2020,1912.04213

    Pith/arXiv arXiv 2020

  28. [28]

    On the onset of cosmological back-reaction,

    N. Li and D.J. Schwarz, “On the onset of cosmological back-reaction,”Phys. Rev. D76083011, 2007,gr-qc/0702043

    Pith/arXiv arXiv 2007

  29. [29]

    Spatial curvature from super-Hubble cosmological fluctuations,

    B. Blachier, P. Auclair, C. Ringeval and V. Vennin, “Spatial curvature from super-Hubble cosmological fluctuations,”Phys. Rev. D108123510, 2023, 2302.14530. [30]DEScollaboration, “The Dark Energy Survey: Cosmology results with 1500 new high-redshift type Ia supernovae using the full five year data set,” Astrophys. J. Lett.973L14, 2024,2401.02929. [31]DESI...

    Pith/arXiv arXiv 2023

  30. [32]

    Updated cosmological constraints in extended parameter space with Planck PR4, DESI baryon acoustic oscillations, and supernovae,

    S. Roy Choudhury and T. Okumura, “Updated cosmological constraints in extended parameter space with Planck PR4, DESI baryon acoustic oscillations, and supernovae,”Astrophys. J. Lett.976L11, 2024,2409.13022

    Pith/arXiv arXiv 2024

  31. [33]

    Using non-DESI data to confirm and strengthen the DESI 2024 spatially flatw 0 wa CDM cosmological parametrization result,

    C.-G. Park, J. de Cruz Perez and B. Ratra, “Using non-DESI data to confirm and strengthen the DESI 2024 spatially flatw 0 wa CDM cosmological parametrization result,”Phys. Rev. D110123533, 2024,2405.00502

    Pith/arXiv arXiv 2024

  32. [34]

    Assessing observational constraints on dark energy,

    D. Shlivko and P.J. Steinhardt, “Assessing observational constraints on dark energy,”Phys. Lett. B855138826, 2024,2405.03933

    Pith/arXiv arXiv 2024

  33. [35]

    Do observations prefer thawing quintessence?,

    G. Payeur, E. McDonough and R. Brandenberger, “Do observations prefer thawing quintessence?,”Phys. Rev. D111123541, 2025,2411.13637. [36]Planckcollaboration, “Planck 2018 results VI: Cosmological parameters,” Astron. Astrophys.641A6, 2020,1807.06209

    Pith/arXiv arXiv 2025

  34. [37]

    Contributions to the relativistic mechanics of continuous media,

    J. Ehlers, “Contributions to the relativistic mechanics of continuous media,” Gen. Rel. Grav.251225, 1993. 29

    1993

  35. [38]

    Theory of cosmological perturbations,

    V.F. Mukhanov, H.A. Feldman and R. Brandenberger, “Theory of cosmological perturbations,”Phys. Rept.215203, 1992

    1992

  36. [39]

    The effect of inhomogeneities on the expansion rate of the universe,

    E.W. Kolb, S. Matarrese, A. Notari and A. Riotto, “The effect of inhomogeneities on the expansion rate of the universe,”Phys. Rev. D71023524, 2005,hep-ph/0409038

    Pith/arXiv arXiv 2005

  37. [40]

    Decaying Λ cosmologies and power spectrum,

    V. Silveira and I. Waga, “Decaying Λ cosmologies and power spectrum,”Phys. Rev. D504890, 1994

    1994

  38. [41]

    Linear perturbations in a universe with a cosmological constant,

    A. Vale and J. Lemos, “Linear perturbations in a universe with a cosmological constant,”Mon. Not. Roy. Astron. Soc.3251197, 2001,astro-ph/0101500

    Pith/arXiv arXiv 2001

  39. [42]

    Super-Hubble nonlinear perturbations during inflation,

    N. Afshordi and R. Brandenberger, “Super-Hubble nonlinear perturbations during inflation,”Phys. Rev. D63123505, 2001,gr-qc/0011075

    Pith/arXiv arXiv 2001

  40. [43]

    Back-reaction of super-Hubble cosmological perturbations beyond perturbation theory,

    R. Brandenberger, L. Graef, G. Marozzi and G.P. Vacca, “Back-reaction of super-Hubble cosmological perturbations beyond perturbation theory,”Phys. Rev. D98103523, 2018,1807.07494. 30

    Pith/arXiv arXiv 2018