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REVIEW 5 major objections 6 minor 93 references

A $\Lambda$CDM Extension Explaining the Hubble Tension and the Spatial Curvature $\Omega_{k,0} = -0.012 \pm 0.010$ Measured by the Final PR4 of the Planck Mission

T0 review · 5 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proposes that dark energy is a kinematic effect of the initial expansion rate, with an equation of state evolving from about -0.8 to -0.9 under void backreaction, raising the Hubble constant to 72.82 km/s/Mpc and explaining…

desk verdict The paper's central derivation of w_de is algebraically wrong; the owCDM results are fits, not predictions, so the model is not ready for publication. read the letter →

arxiv 2412.04126 v1 pith:QQDITXSC submitted 2024-12-05 astro-ph.CO gr-qc

Horst Foidl , Tanja Rindler-Daller This is my paper · ORCID
classification astro-ph.COgr-qc PACS 95.36.+x98.80.-k
keywords HubbletensiondarkenergyequationofstatevoidbackreactionspatialcurvaturecosmicwebCMBkinematic
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 proposes that dark energy is not a cosmological constant but a kinematic effect inherited from the initial expansion rate of the universe just after the big bang. With the same present-day matter density as LambdaCDM, the effective dark energy has equation of state $w_{\mathrm{de}} \simeq -0.8$, and once cosmic voids dominate the volume, their backreaction drives it to about $-0.9$ today. The resulting Hubble constant is $H_0 = 72.82$ km/s/Mpc, close to local distance-ladder measurements and larger than the CMB-inferred value, which the paper presents as a resolution of the Hubble tension. An extension that also accounts for our peculiar motion relative to a perfect comoving observer produces $\Omega_{k,0} = -0.0197$, matching the Planck PR4 value of $\Omega_{k,0} = -0.012 \pm 0.010$. If correct, the model would turn dark energy into a testable kinematic quantity and explain two observed anomalies without introducing new physics.

What carries the argument

The central object is the effective dark-energy density $\rho_{\mathrm{de}}$ with equation-of-state parameter $w_{\mathrm{de}}$, given by Eq. (24) as $w_{\mathrm{de}} = \frac{2}{3}\Omega_{\mathrm{phys},0} - 1$. This replaces the cosmological constant while keeping $\Omega_{k,0} = 0$ for the perceived flatness of comoving observers. The void-backreaction step makes $w_{\mathrm{de}}$ a piecewise-linear function of scale factor, constant until $a = 1/6$ and then evolving to about $-0.9$ today (Eqs. 30 and 31), using the volume evolution of the cosmic web extracted from LambdaCDM simulations. The machinery translates the initial conditions of the background universe into a time-dependent dark energy that raises $H_0$ and, with the local dipole included, produces the small spatial curvature $\Omega_{k,0} = -0.0197$.

What would settle it

A redshift-resolved measurement of the dark-energy equation of state, for example from BAO and supernova data across $0 \lesssim z \lesssim 2$, that returns $w = -1$ at all epochs with uncertainties below the predicted shift from about $-0.8$ to about $-0.9$ would exclude the model. A second test is the local expansion history: the predicted peak in the deviation of $H(z)$ from LambdaCDM near $a \sim 0.8$ could be sought directly in standard-siren or BAO data.

Watch

Extended reading notes

Core claim

The central claim is that the curvature term in the Friedmann equation does not measure global spatial curvature; it is a kinematic dark energy fixed by the ratio of the initial energy density to the initial expansion rate. Because freely falling comoving observers perceive flat space under the equivalence principle, a subcritical (open) universe is observationally consistent with the measured flatness of the CMB. The paper derives $w_{\mathrm{de}} = \frac{2}{3}\Omega_{\mathrm{phys},0} - 1$, which for $\Omega_{\mathrm{phys},0} \simeq 0.3$ gives $w_{\mathrm{de}} \simeq -0.8$. In the late universe, void backreaction makes $w_{\mathrm{de}}$ time-dependent, evolving to roughly $-0.9$ today, and raises $H_0$ to $72.82$ km/s/Mpc. The wCDM model reproduces the shape of the LambdaCDM CMB temperature spectrum with deviations at the 15% level, while the owCDM fit including curvature and the local dipole matches the spectrum at the 0.01% level and yields $\Omega_{k,0} = -0.0197$.

Load-bearing premise

The load-bearing premise is that freely falling comoving observers always perceive flat space regardless of the universe's energy density, so the curvature term in the Friedmann equation can be reinterpreted as a kinematic dark energy; if that premise is wrong, the derivation of $w_{\mathrm{de}}$ has no foundation.

Editorial extensions

If this is right

  • The model raises the Hubble constant to $H_0 = 72.82$ km/s/Mpc, matching local distance-ladder measurements that report $73.04$ km/s/Mpc.
  • The owCDM model predicts a spatial curvature of $\Omega_{k,0} = -0.0197$, compatible with the Planck PR4 value of $\Omega_{k,0} = -0.012 \pm 0.010$.
  • The effective dark-energy equation of state evolves from about $-0.8$ to about $-0.9$, a concrete deviation from a cosmological constant that future surveys can test.
  • Both wCDM and owCDM keep the early-time expansion history of LambdaCDM unchanged, so big bang nucleosynthesis and the CMB peak structure are preserved.
  • The model yields $S_8 = 0.784$ (wCDM) and $0.798$ (owCDM), within $1\sigma$ of current weak-lensing survey values, mitigating the $\sigma_8$ tension.

Reading between the lines

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

  • Beyond the paper, if the mechanism is right, the dark-energy density is not a new substance but a boundary condition of the early universe, reframing the cosmological constant problem as a question about initial conditions.
  • Beyond the paper, the linear-in-redshift ansatz for $w_{\mathrm{de}}$ is an approximation; a calibration of the void volume fraction in the model's own cosmology would sharpen the prediction and test the 8% backreaction.
  • Beyond the paper, the same machinery should predict a specific local bulk flow that produces the CMB dipole; measuring that dipole with independent kinematic tracers could isolate the claimed local-curvature effect.
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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

5 major / 6 minor

Summary. The paper proposes two extensions of ΛCDM, called wCDM and owCDM, in which the cosmological constant is replaced by an effective dark-energy component whose equation of state is purportedly derived from the initial conditions of the background universe and from a reinterpretation of the Friedmann curvature term. The model adds a void-backreaction-induced time dependence to the dark-energy equation of state, claims to raise H0 to 72.82 km/s/Mpc and thus resolve the Hubble tension, and claims that the fitted curvature Ωk,0 = -0.0197 explains the Planck PR4 value Ωk,0 = -0.012 ± 0.010. The model is implemented in a modified version of CLASS and compared to ΛCDM and to observations.

Significance. If the central derivation were sound, the paper would address a topical problem with a novel, physically motivated alternative to ΛCDM, and its explicit implementation in CLASS would be a useful starting point for further tests. However, the derivation of the key parameter w_de is internally inconsistent, the claimed agreements with the CMB and with H0 are obtained by fitting free parameters rather than by prediction, and the reported CMB spectrum of the baseline wCDM model deviates from ΛCDM at the ~15% level, which is far larger than the observational uncertainties. The paper is therefore not suitable for publication in its present form.

major comments (5)
  1. [§4, Eqs. (21)–(24)] The derivation of the dark-energy equation of state is mathematically inconsistent. From Eq. (21), with κ constant, differentiating the second term gives d/da(GΩ_phys/a) ∝ -Ω_phys/a^2. Inserting ρ_de ∝ a^{-2} into Eq. (13) yields 3(1+w_de) = 2, hence w_de = -1/3, independent of Ω_phys,0. Equation (24), w_de = 2Ω_phys,0/3 - 1, would require ρ_de ∝ a^{-2Ω_phys,0}, but that scaling is never derived; Eq. (22) provides only a single power of a^{-2}. Moreover, a constant κ corresponds exactly to the standard curvature component ρ_k ∝ a^{-2} with w = -1/3, so a component with w ≈ -0.8 is not described by Eq. (21). Since this w_de is the input to all subsequent CLASS computations (H0 = 72.82, Ωk,0 = -0.0197), the central parameter of the model is unestablished by the paper's own equations.
  2. [§7, Ωk,0 result] The claimed explanation of the Planck PR4 curvature is circular. The text states that the authors fit the wCDM model to the ΛCDM CMB spectrum and to H0 = 73.04 km/s/Mpc (Riess et al. 2022), and then report Ωk,0 = -0.0197 as the outcome. Because Ωk,0 is a free parameter of that fit, and because the endpoint w_de(1) is likewise adjusted via Eq. (31), the agreement with Ωk,0 = -0.012 ± 0.010 is a post-hoc match rather than a prediction. No likelihood or model-comparison statistic is provided to substantiate that the model actually explains the PR4 measurement.
  3. [§6.2, Fig. 7] The CMB temperature power spectrum of the wCDM model is stated to deviate from ΛCDM at the ~15% level (figure caption and main text). This is orders of magnitude larger than the measurement uncertainties in Planck PR4 data, yet the text claims the model 'agrees well with current data' and that there are 'no significant differences in the structure of the peaks.' Without a full likelihood analysis and a quantification of goodness of fit, the claim of agreement with observations is unsupported; a 15% deviation in the TT spectrum is a serious discrepancy, not a minor one.
  4. [§5.4, Eq. (30)] The void-backreaction parameterization is not derived from the cited works. The text invokes Cautun et al. (2014) and Icke (2001), but the specific linear interpolation in Eq. (30), the threshold a = 1/6, and the endpoint w_de(1) ≈ -0.9 are introduced without a quantitative derivation or a clear mapping to the simulation products. Consequently, the ~8% rise in H0 is imposed by construction, and the 'solution to the Hubble tension' is not a robust prediction but a consequence of the chosen interpolation. The same applies to Eq. (31), where the endpoint is adjusted to 0.91 in the fit.
  5. [§3.1, flat-space premise] The foundational premise that comoving FLRW observers perceive flat space irrespective of the universe's energy density is asserted, not derived. The equivalence principle guarantees a local inertial frame, but the Friedmann curvature term κ/a^2 is a global quantity that is not a local coordinate artifact; for open or closed FLRW geometries the spatial curvature scalar does not vanish in the comoving frame. Because this premise is the basis for rewriting the Friedmann equation as Eq. (21) and for setting Ωk = 0 in the wCDM background, it is a correctness-risk concern that should be substantiated with a concrete calculation (for example, the Riemann tensor in comoving coordinates) rather than asserted.
minor comments (6)
  1. [§4, Eq. (25)] Equation (25) is typeset in a malformed way: 'wde,early = -1/3 - Θ(Ωde,0) 2/3 Ωde,0' lacks a clear second term and is not a well-defined expression as printed; it needs to be rewritten with proper parentheses and the correct dependence on Ωde,0.
  2. [Fig. 1 caption] The figure caption lists 'wde = 0.33' and 'wde = 0.80' for supercritical and subcritical models, respectively; these values should presumably be negative (e.g., wde = -1/3 for the critical-density case), and the sign convention should be clarified.
  3. [§4, Eq. (23)] The notation ρ_de ∝ a^{-2}Ω_phys,0 is ambiguous: it should be stated explicitly whether Ω_phys,0 is in the exponent or a prefactor, since the two readings lead to different equations of state and the ambiguity contributes to the derivation problem.
  4. [§6.3] The paper reports S8 = 0.784 for wCDM and S8 = 0.798 for owCDM and claims consistency with DES-Y3's S8 = 0.782 ± 0.019, but no uncertainty is quoted and no description is given of how S8 is computed in the modified CLASS implementation; a reference to the output and a propagation of the model parameters would make the comparison meaningful.
  5. [§5.4 and §7] The paper repeatedly refers to Foidl & Rindler-Daller (2024) for the fitting procedure and for the forward-in-time integration of the dark-energy density; the present paper should at least summarize the key steps so that Eqs. (30) and (31) can be reproduced without requiring the companion paper.
  6. [Throughout] There are numerous typographical and formatting issues, including 'di fferent' for 'different', 'T ension' in the Section 6.3 header, and inconsistent use of subscripts (e.g., 'wde,early' vs 'w_de,early'); a careful proofreading pass is needed.

Circularity Check

3 steps flagged · score 8.0 of 10

The headline results (w_de ≈ −0.8, H0 ≈ 72.82, and the PR4 curvature match) reduce to input definitions or to a fit to the very quantities they are said to explain.

  1. self definitional [Sec. 4, Eqs. (21)–(24)]
    "Since κ is a constant, the evolution of the first term is determined by the evolution of the second term, given by d/da(GΩ_phys/a) ∝ −Ω_phys/a^2. ... We use relation (22) in Eq. (13) which yields ρ_de ∝ a^{−2Ω_phys,0}, and equate the exponents in (23) and ρ_k ∝ a^{−3(1+w)} [see Eq. (13)], rearranged to express w_de it reads as w_de = 2/3 Ω_phys,0 − 1, where w_de is constant."

    Equation (22) gives at most a derivative whose integrated scaling is a single power a^{−2}, corresponding to w = −1/3. The exponent −2Ω_phys,0 in Eq. (23) is inserted by hand, not derived from Eq. (22). Equation (24) is then just the algebraic relabeling of the assumed exponent in terms of Ω_phys,0; it is true by definition, not by derivation. The advertised value w_de ≈ −0.8 is therefore a restatement of the chosen input Ω_m,0 ≈ 0.3, and every later quantity that depends on this w_de inherits that input rather than being independently predicted.

  2. fitted input called prediction [Sec. 7, first paragraph and closing paragraphs]
    "In the course of this section, we fit the wCDM model to the ΛCDM CMB spectrum as well as to the value H0 = 73.04 km/s/Mpc, derived by Riess et al. (2022), giving the owCDM model. ... We find a spatial curvature of Ω_k,0 = −0.0197 ... This value is compatible with the final PR4 of the Planck mission (Tristram et al. (2024)) with Ω_k,0 = −0.012 ± 0.010."

    The model is fitted to the ΛCDM CMB spectrum and to the local H0 = 73.04 km/s/Mpc, and the spatial curvature Ω_k,0 appears as part of the resulting fit parameters. Reporting that the fit gives Ω_k,0 = −0.0197 and that this is compatible with the PR4 value is not an independent explanation of the Planck curvature measurement; it is a description of the fitted parameter. Likewise, the claimed solution to the Hubble tension is built in by fitting to the Riess et al. H0 value, so the agreement is constructed rather than predicted.

1 more flagged steps
  1. self citation load bearing [Sec. 1 and Sec. 7]
    "In Foidl & Rindler-Daller (2024) we exemplify, based on empirical arguments, that a cosmological model including a CPL-based DE component with an EoS parameter w_de evolving from −0.8 to −0.9 describes the CMB temperature spectrum of ΛCDM and yields a Hubble constant H0 being compatible with direct measurements (e.g., Riess et al. (2022)) in the local Universe. ... To this end, we apply our proposed procedure by Foidl & Rindler-Daller (2024) to our model and fit it to the ΛCDM CMB spectrum and H0 = 73.04 km/s/Mpc as determined by Riess et al. (2022) in the local Universe."

    The central fitting procedure and the expected w_de range (−0.8 to −0.9) are imported from the authors' own prior paper rather than derived here. That prior work is explicitly empirical, and the procedure's target already includes the local H0 value that the present paper claims to explain. The self-citation thus carries the load of fixing the model's free behavior, while the confirmation consists of matching the same fitted target.

full rationale

The paper contains real computational work: the authors implement the modified background and perturbation equations in CLASS, compute CMB spectra, and compare S8 and ages with observations. Those comparisons are not themselves circular. However, the two headline successes reduce by construction. First, Eq. (24) is not a derivation of the dark-energy equation of state: the exponent in Eq. (23) is assumed to contain Ω_phys,0, and equating that assumed scaling with ρ ∝ a^{−3(1+w)} makes w_de a definitional function of Ω_phys,0. Second, the owCDM results in Sec. 7 are obtained by fitting to the ΛCDM CMB spectrum and to H0 = 73.04 km/s/Mpc, after which the paper reports H0 and Ω_k,0 as agreements with observations, including the PR4 curvature Ω_k,0 = −0.012 ± 0.010. Those agreements are properties of the fit, not independent predictions. The self-citation to Foidl & Rindler-Daller (2024) is load-bearing for the fitting procedure, but the main circularity is the fit-to-target disguised as validation. I therefore assign score 8: the headline results are forced by the input definitions and the fitting procedure, even though the CLASS implementation and spectrum comparisons retain independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model adds no new particles, forces, or dimensions; ρ_de is a relabeled curvature term and the void backreaction is a reparameterization of w_de. The unsupported inputs are the equivalence-principle reinterpretation, the significance of void backreaction, and the fitted values of w_de(1) and Ωk,0.

free parameters (3)
  • w_de at present, w_de(1) = -0.9 (Eq. 30); -0.91 (Eq. 31)
    The endpoint of the dark-energy equation of state is chosen so that the model yields H0 near 73 km/s/Mpc, matching Riess et al. (2022). The value is not derived from void dynamics; it is adjusted in the owCDM fit.
  • Ωk,0 (spatial curvature) = -0.0197
    Fitted in Sec. 7 so that the owCDM CMB spectrum matches ΛCDM and H0 matches Riess, then compared to Planck PR4.
  • H_ini (initial expansion rate) = not specified
    Introduced in Sec. 3.2 as the physical motivation for the model, but never quantified or used in the computations; effectively absorbed into Ωde,0.
assumptions (5)
  • ad hoc to paper Comoving FLRW observers perceive flat space in their local inertial frame regardless of the universe's global energy density.
    Used in Sec. 3.1 to justify setting Ωk = 0 in the Friedmann equation while reinterpreting the curvature term as ρ_de. This is an extrapolation of the equivalence principle beyond its standard local scope and is not a standard result.
  • ad hoc to paper The curvature term in the Friedmann equation can be identified with an effective dark-energy component ρ_de with EoS w_de given by Eq. (24).
    This identification is the core of the model in Sec. 4 and does not follow from the Einstein equations or the standard interpretation of the Friedmann curvature term.
  • ad hoc to paper Voids, once they dominate the volume of the universe, produce an 8 percent backreaction on the Hubble rate.
    Sec. 5.3 argues for this against a literature that generally finds negligible backreaction; the 8 percent magnitude comes from the chosen w_de(a) interpolation, not from an independent calculation.
  • domain assumption Birkhoff's theorem and the separate universe conjecture apply to voids as mini-universes.
    Invoked in Sec. 5.3 to argue that voids expand faster than the homogeneous background; standard in toy models but not rigorously applicable to the full cosmic web with interacting walls and filaments.
  • ad hoc to paper The spatial curvature measured by Planck can be interpreted as a local effect of our peculiar motion relative to the CMB.
    Secs. 7 and 8 attribute Ωk = -0.0197 to the CMB dipole decoupling, a non-standard reading of what CMB curvature constraints actually measure, since the CMB acoustic scale is a global geometric probe.

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Cite this review

Pith. "Pith review of A $\Lambda$CDM Extension Explaining the Hubble Tension and the Spatial Curvature $\Omega_{k,0} = -0.012 \pm 0.010$ Measured by the Final PR4 of the Planck Mission." pith.science (2026). https://pith.science/paper/QQDITXSC

@misc{pith2026241204126,
  author       = {Pith},
  title        = {Pith review of: A $\Lambda$CDM Extension Explaining the Hubble Tension and the Spatial Curvature $\Omega_k,0 = -0.012 \pm 0.010$ Measured by the Final PR4 of the Planck Mission},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQDITXSC}},
  note         = {Machine review of arXiv:2412.04126}
}
abstract

The measurements of the CMB have determined the cosmological parameters with high accuracy, and the observation of the flatness of space have contributed to the status of the concordance $\Lambda$CDM model. However, the cosmological constant $\Lambda$, necessary to close the model to critical density, remains an open conundrum. We explore the observed late-time accelerated expansion of the Universe, where we consider that the Friedmann equation describes the expansion history of FLRW universes in the local reference frame of freely falling comoving observers, which perceive flat, homogeneous and isotropic space in their local inertial system, where, as a consequence of the equivalence principle, special relativity applies. We use this fact to propose an extension to $\Lambda$CDM, incorporating the initial conditions of the background universe, comprising the initial energy densities as well as the initial post big bang expansion rate. The observed late-time accelerated expansion is then attributed to a kinematic effect akin to a dark energy component. Choosing the same $\Omega_{m,0} \simeq 0.3$ as $\Lambda$CDM, its equation of state $w_{de} \simeq -0.8$. Furthermore, we include the impact on the expansion history caused by the cosmic web of the late Universe, once voids dominate its volume, and find that the initially constant $w_{de}$ becomes time-dependent, evolving to a value of $w_{de} \simeq -0.9$ at the present. While this impact by voids is minor, it is sufficient to provide a solution to the Hubble tension problem. We use CLASS to calculate the expansion history and power spectra of our extension and compare our results to concordance $\Lambda$CDM and to observations. We find that our model agrees well with current data, in particular with the final data release PR4 of the Planck mission, where it explains the reported spatial curvature of $\Omega_{k,0} = - 0.012 \pm 0.010$.

Figures

Figures reproduced from arXiv: 2412.04126 by the authors.

Figure 1
Figure 1. Expansion histories of model universes within the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Evolution of energy densities in ΛCDM (top panel) and wCDM (bottom panel) as a function of scale factor and proper time, respectively. The top panel displays the evolution of the individual con￾tributions to the energy density of the ΛCDM model according to (14). The red solid line indicates the evolution of the cosmological constant Λ. The bottom panel displays the evolution of the individual contri￾butions to the … view at source ↗
Figure 5
Figure 5. Evolution of scale factor and Hubble parameter in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Evolution of effective EoS parameter weff in extended and concordance ΛCDM. The top panel displays the evolution of weff in both models vs scale factor and proper time, respectively, with the same vertical lines as in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 6
Figure 6. Figure 6: Evolution of Hubble parameter vs scale factor [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The CMB temperature power spectrum in ΛCDM (blue) and in our wCDM model (magenta). The relative differences are shown at the bottom of the panel, and amount to ∼ 15 %. Although we see no significant differences in the structure of the peaks in the CMB temperature power…
Figure 8
Figure 8. Figure 8: Now, the deviation from the ΛCDM spectrum amounts to a ≲ 0.01% level (disregarding the higher deviations at small l, 15This acronym is customarily used for extensions to ΛCDM, when w(z) is different from −1 and additionally a spatial curvature is applied; see, for exam…

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