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What solves the Hubble tension in phenomenological dark energy models at background level?

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Pith's one-line read Minimal dark energy models resolve the Hubble tension only through a future sign-switch in the dark-sector interaction strength, equivalent to a future violation of the null energy condition.

desk verdict Worth reading for the dataset forensics; the headline claim about inevitable future NEC violation is a parametrization artifact. read the letter →

arxiv 2505.24743 v2 pith:CC4GLG43 submitted 2025-05-30 astro-ph.CO gr-qc

Manosh T. Manoharan This is my paper
classification astro-ph.COgr-qc
keywords HubbletensionphenomenologicalemergentdarkenergyholographicinteractingsectorsnullconditionBAOLyman-alphacosmicchronometersPantheon+
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 asks what, at the level of the expansion history alone, actually lets minimal dark energy models raise the Hubble constant enough to match SH0ES measurements. Using two models with no extra parameters once their internal constants are fixed—PEDE and GOHDE—it shows that the apparent resolution of the Hubble tension is driven by specific data choices and priors: BAO Lyman-α points at z≈2.3, derived under a fiducial low matter density, push H0 up, while cosmic chronometer, Pantheon+, and CMB shift-parameter data pull it back down. When the models' free parameters are allowed to vary, they fall back to low H0 values; only hand-fixed variants (ν=1/ln10, β=1/3) resolve the tension. The paper's central conclusion is that the common feature behind these successes is a sign-switching interaction strength γ_Λ with a pole, which corresponds to a future violation of the null energy condition and of the second law of horizon thermodynamics. In other words, a future NEC violation is claimed to be inevitable for any minimal background model that solves the Hubble tension.

What carries the argument

The central object is the interaction strength γ_Λ(z) defined by Q = 3H γ_Λ ρ̃_m, obtained by rewriting the non-interacting PEDE and GOHDE densities in an interacting dark-sector picture via the mapping ρ̃_Λ = -w_Λ ρ_Λ, ρ̃_m = ρ_m + (1+w_Λ)ρ_Λ (Eq. 20). The paper computes explicit expressions for γ_Λ(z) for both models and identifies the parameter values that resolve the Hubble tension with the ones where γ_Λ diverges at a pole and changes sign from negative to positive. That sign-switch is the mechanism: it marks the onset of a violation of the null energy condition, equivalently a future turning point in the Hubble flow and a future violation of the second law of horizon thermodynamics. The alternative interaction strength γ_m shows no such distinctive behaviour, which is why the γ_Λ formulation is the one that carries the argument.

What would settle it

Look for a marked past turning point in the Hubble flow: if H(z) is observed to rise toward high redshift at any z > 0 while a minimal PEDE or GOHDE fit still yields H0 ≈ 73 km/s/Mpc, the claim that the required sign-switch lies in the future is refuted. Alternatively, a full CMB power-spectrum analysis that lets ν and β vary freely and still recovers H0 ≈ 73 km/s/Mpc would break the paper's conclusion that only the hand-fixed, future-NEC-violating branches resolve the tension.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that phenomenological dark energy models which reduce to ΛCDM with no extra free parameters—PEDE and GOHDE—do not by themselves favour the high Hubble constant. The preference for H0 ≈ 73 km/s/Mpc in earlier literature comes from the BAO Lyman-α data at z≈2.3 together with the fiducial Ωm=0.27 cosmology used to derive those points, and from fixing the models' internal parameters to specific values. Reinterpreting both models as interacting dark sectors through the dictionary Q = 3Hγ_Λ ρ̃_m, the paper shows that the parameter choices which resolve the tension are exactly those in which γ_Λ diverges and switches sign, flipping the direction of energy transfer between dark energy and dark matter. This sign-switch is tied to a turning point in the Hubble flow and to a maximum of the Hubble-horizon entropy followed by a decrease, i.e. a future violation of the second law of horizon thermodynamics and of the null energy condition. The paper therefore claims that the timing of the NEC violation—future, not past—is the decisive feature, and that a future violation is inevitable in any minimal background-level solution of the Hubble tension.

Load-bearing premise

The argument treats the mapping in Eq. (20) as a physical equivalence between the non-interacting densities and the interacting-sector densities, so that the divergence and sign switch of γ_Λ is a real event rather than a bookkeeping artifact; if that dictionary is only a re-labeling, the inevitability of a future NEC violation does not follow.

Editorial extensions

If this is right

  • If the claim is right, any minimal background-level model that resolves the Hubble tension must predict a future epoch in which the null energy condition is violated, meaning the Hubble parameter eventually starts increasing again.
  • The apparent success of PEDE and GOHDE in earlier work is not a generic property of the models: with the free parameters left free, both models prefer H0 closer to the Planck value, so claims of resolution depend on fixing ν ≈ 1/ln10 and β ≈ 1/3 and on including the z≈2.3 BAO Lyα points.
  • Because the BAO Lyα-derived H(z) points assume a fiducial Ωm = 0.27 cosmology, the high H0 they produce is partly a prior effect; including Pantheon+ or the CMB shift parameter, which pin down matter density, re-introduces the tension unless the shift parameter is also included.
  • The sign-switch in γ_Λ is observable in principle as a future turning point in the Hubble flow; observations of a past turning point would disqualify the models discussed.

Reading between the lines

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

  • A testable extension: the same interaction-picture analysis could be applied to wCDM with a phantom w; one would predict that its γ_Λ also shows a pole, which would unify the 'phantom crossing' and 'future NEC violation' descriptions of Hubble-tension resolutions.
  • The claim that a future second-law violation is 'inevitable' may be an artifact of restricting to two-parameter phenomenological families; models with early dark energy or a time-varying sound speed that alter the background differently might evade the no-go, so the inevitability should be read as applying within the minimal class considered.
  • If the sign-switch is the real effector, low-redshift probes that measure H(z) or the dark-energy equation of state near z≈0.5–2, or gravitational-wave standard sirens, could detect the precursor behaviour (steepening of w_Λ) before the pole, providing an observational foretaste of the future violation.
  • The paper's reanalysis suggests that past claims of PEDE/GOHDE resolving the tension should be re-audited with the SH0ES prior removed and the Lyα points' fiducial cosmology accounted for; the same audit could be applied to DESI DR1/DR2 BAO points, which the paper finds mildly tension with SDSS.
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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 / 6 minor

Summary. The paper performs a background-level analysis of two single-parameter dark-energy extensions of LCDM, PEDE and GOHDE, using cosmic-chronometer, BAO (including SDSS and DESI Ly-alpha), Pantheon+ supernova, and CMB shift-parameter data. It finds that the apparent ability of these models to resolve the Hubble tension depends strongly on (i) fixing the extra model parameters to particular values, (ii) the fiducial cosmology assumed in the Ly-alpha H(z) measurements around z ~ 2.3, and (iii) the correlation between H0 and Omega_m. When the extra parameters are left free, the data prefer values that yield H0 below the SH0ES value. The paper then reinterprets PEDE and GOHDE as interacting dark-sector models and claims that the parameter choices that raise H0 produce a sign-switching, singular interaction strength gamma_Lambda, concluding that a future violation of the null energy condition and of the horizon second law is 'inevitable' for models that resolve the Hubble tension at the background level.

Significance. If the paper's central interpretive claim were correct, it would provide a general statement that background-level Hubble-tension resolutions require future NEC violation. However, the empirical dataset-dependence analysis is the more robust part of the paper and has independent value: it clearly shows the role of the BAO Ly-alpha fiducial cosmology, the anti-correlation between H0 and Omega_m, and the difference between SDSS and DESI Ly-alpha estimates. The paper also reports tension statistics and AIC/BIC/DIC values carefully, and it correctly distinguishes the fitted matter density from the effective matter density in GOHDE when interpreting the CMB shift parameter. The main weakness is that the 'inevitability' conclusion is not supported by the analysis, because the sign-switch in gamma_Lambda is a parametrization-dependent feature. The paper would be suitable for publication after a substantial revision that removes or carefully reframes the necessity claim.

major comments (4)
  1. [Sec. III, Eq. (20) and Sec. V.C] The central conclusion that resolving the Hubble tension 'necessitates' a sign-switching gamma_Lambda is not established, because the sign-switch is an artifact of the chosen interaction parametrization. Equation (20) defines tilde_rho_Lambda = -w_Lambda rho_Lambda and tilde_rho_m = rho_m + (1+w_Lambda) rho_Lambda; these auxiliary densities can be negative (e.g., tilde_rho_Lambda < 0 when w>0, and tilde_rho_m < 0 when w<-1), and the pole in gamma_Lambda arises when these bookkeeping densities cross zero. The physically equivalent parametrization Q = 3H gamma_m tilde_rho_Lambda (Eq. 19) produces no divergent sign-switch: gamma_m is constant for GOHDE (Eq. 24) and shows no singular behavior for PEDE (Eq. 22). A feature that disappears under an equivalent reparametrization cannot be used to infer an inevitable future NEC violation.
  2. [Sec. V.B] The sign-switch and pole in gamma_Lambda are realized only for parameter values that the authors' own free fits disfavor. The broad-prior fits reported in Sec. V.B give nu ~ -1.12 with H0 ~ 62.95 km/s/Mpc for PEDE and beta ~ 1.45 with H0 ~ 64.9 km/s/Mpc for GOHDE, whereas the pole and sign-switch shown in Figs. 11-12 occur for nu > 0 and beta < 2/3, including the hand-fixed values nu = 1/ln 10 and beta = 1/3 used in the earlier sections. Thus the 'future NEC violation' is not a generic property of models that resolve the Hubble tension; it is a property of parameter choices that the data reject. The abstract's language of 'necessitates' and the conclusion's 'inevitable' should be removed or rederived under the best-fit parameter values.
  3. [Sec. V.C, entropy discussion] The thermodynamic interpretation of the sign flip is not compelling as stated. For PEDE with nu = 1/ln 10, the late-time asymptote is de Sitter (H tends to a constant), so the horizon entropy tends to a constant rather than decreasing in the future; for GOHDE with beta = 1/3, the decreasing horizon entropy is simply the well-known phantom future, not an independent diagnostic of a second-law violation. The claim that the condition in Eqs. (35)-(37) marks the onset of a second-law violation requires a physical, coordinate-invariant diagnostic that the manuscript does not provide.
  4. [Sec. V.B, Fig. 10] The statement that PEDE and GOHDE 'perform significantly better' at higher H0 is based on Fig. 10, which fixes H0 = 73 km/s/Mpc and keeps all other cosmological parameters at their Planck 2018 LambdaCDM best-fit values. This is not a likelihood fit and does not account for parameter correlations or data constraints. Without a full exploration of the parameter space and a comparison of chi^2 or information criteria, Fig. 10 cannot support a comparative goodness-of-fit claim.
minor comments (6)
  1. [Abstract and keywords] The keyword 'DegenerDynamical Dark Energy' appears to contain a typo and should be corrected, for example to 'Degenerate Dynamical Dark Energy' or a similar intended phrase.
  2. [Eqs. (21)-(23)] The typesetting of the gamma_Lambda and gamma_m expressions is garbled in places (for example, '2/3 v tanh' and missing parentheses), which makes the formulas difficult to check; these equations should be carefully typeset.
  3. [Table I] The four Ly-alpha H(z) points in Table I are treated as independent data points, but they are derived from overlapping SDSS/BOSS Ly-alpha analyses; the paper notes they are 'technically related' but does not account for possible covariance, which may underestimate the uncertainties in the combined likelihood.
  4. [Data and code availability] The paper states that codes and datasets are available only upon reasonable request and that no code is deposited; given the emphasis on dataset-dependent conclusions, providing public access to the MCMC chains and analysis scripts would strengthen reproducibility.
  5. [Eq. (28) and surrounding text] The shift parameter is written as R in Eq. (27) and then as ar{R} in Eq. (28) with the value ar{R} = 1.7502; please use a single notation consistently and clarify that the quantity used is the Planck 2018 value.
  6. [Figure 3 and Tables IV-VII] The text says tables report the effective matter density for GOHDE, but some figure captions (e.g., Fig. 3) state that the plotted quantity is Omega_m rather than tilde_Omega_m; please make the distinction consistent in all captions and in the main text.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'necessity' of sign-switching gamma_Lambda and of future NEC/second-law violation is a reparametrization artifact of Eq. (20) and restates the phantom ansatz; the data analysis itself is independent.

  1. self definitional [Section III, Eqs. (17)-(20)]
    "Following the approach outlined in [22], the PEDE and GOHDE models can be reformulated as interacting models with specific expressions for the interaction term ˜Q. ... This transformation is given by, ˜ρΛ=−wΛ(a)ρΛ and ˜ρm=ρm+[1+wΛ(a)]ρm (20)"

    γΛ is read off from Q̃=3HγΛρ̃m after substituting Eq. (20), so it is an algebraic function of the same wΛ(a), ρΛ, and ρm used to define the model. The pole and sign switch occur where this bookkeeping density crosses zero, and they are not invariant: Eqs. (18)-(19) define two equivalent interaction strengths for the same Q̃, and the paper's own Eqs. (22) and (24) show γm is constant (GOHDE) or featureless (PEDE). Thus the abstract's claim that solving the tension 'necessitates a varying γΛ characterised by a singular sign-switch behaviour' is a property of the chosen parametrization, not an independent prediction.

  2. renaming known result [Section V.C (summary) and Section VI]
    "From this, we conclude that a future violation of the second law is an inevitable feature of models capable of resolving the Hubble tension. This can also be interpreted as a turning point in the Hubble flow."

    The future violation is inserted by the model ansatz before any data are used: for ν>0 the PEDE equation of state (3) is everywhere <−1, and Section II.B states that β<2/3 for GOHDE predicts 'a phantom-like future'. The paper then labels the γΛ pole as the onset of NEC violation and plots the horizon entropy of these already-phantom models (Fig. 14). Calling the resulting entropy decrease an 'inevitable feature' is therefore a restatement of the phantom assumption in new vocabulary, not a new implication of the data; the admitted absence of the feature in γm confirms its parametrization dependence.

full rationale

Most of the paper's statistical work (Sections IV-V.B) is not circular: the fits to CC, BAO, Pantheon+, and the CMB shift parameter are anchored to external datasets, and the finding that the apparent H0 preference is prior- and dataset-dependent is an independent contribution. The circularity is concentrated in the 'interaction picture' conclusion. The singular γΛ is constructed from Eq. (20) out of the same wΛ(a) that defines the model, and the two formulations of Q in Eqs. (18)-(19) give γm no such pole, so the sign-switch is not a physical necessity. The 'future violation of the second law / NEC' is likewise just the phantom future already built into PEDE (ν=1/ln10) and GOHDE (β=1/3); Section V.B shows that when the parameters are free, the data prefer ν<0 or β>2/3 and no resolution. No author self-citation is load-bearing here: [22] is external and [20],[21] do not carry the inevitability claim. Score 6 reflects a partially circular central claim with otherwise independent data analysis.

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

The paper introduces no new particles or forces; its new quantities (gamma_Lambda, gamma_m) are derived functions. The central conclusion leans on hand-fixed parameters nu and beta and on the Eq. (20) mapping from [22], which the paper does not independently verify.

free parameters (3)
  • PEDE log-base parameter nu = -1.12 (full OHD fit); fixed to 1/ln(10) ~ 0.43 in minimal version
    Controls the phantomness and determines whether the model predicts high H0; the 'minimal PEDE' value is hand-fixed, and when free, data prefer a negative value that lowers H0.
  • GOHDE tracker parameter beta = 1.45 (full OHD fit); fixed to 1/3 in minimal version
    Governs the effective matter density and dark energy equation of state; higher best-fit values lower H0, and the high-H0 minimal version requires beta = 1/3 by hand.
  • H0 and Omega_m (or Omega_tilde_m) = per dataset, e.g., PEDE H0=73.65, Omega_m=0.25 for BAO Gal+Ly-alpha
    Standard cosmological parameters fitted to data; they are anti-correlated and central to the tension analysis.
assumptions (5)
  • domain assumption FLRW background with flat geometry and negligible radiation is assumed for the reduced models
    Section II, Eqs. (1)-(14); curvature and radiation are dropped to obtain the minimal PEDE and GOHDE expressions.
  • domain assumption The mapping between interacting and non-interacting densities, Eq. (20), is taken from [22] and assumed valid
    Section III, Eq. (20); this mapping underlies the gamma_Lambda sign-switch diagnostic and is adopted without independent derivation.
  • domain assumption The CMB shift parameter R uses effective matter density Omega_tilde_m for GOHDE
    Section IV.A, discussion of Eq. (28); using Omega_m instead would bias the constraint.
  • domain assumption The BAO Ly-alpha H(z) data points at z ~ 2.3 are treated as independent direct measurements with the stated fiducial cosmology
    Section IV.A and Tables I-II; this assumption is the fulcrum of the dataset-dependence finding.
  • standard math Standard MCMC likelihood with Gaussian errors and covariance where available
    Section IV.B; non-Gaussianity is not modeled.

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Pith. "Pith review of What solves the Hubble tension in phenomenological dark energy models at background level?." pith.science (2026). https://pith.science/paper/CC4GLG43

@misc{pith2026250524743,
  author       = {Pith},
  title        = {Pith review of: What solves the Hubble tension in phenomenological dark energy models at background level?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CC4GLG43}},
  note         = {Machine review of arXiv:2505.24743}
}
abstract

Few phenomenological models tend to favour higher values of the Hubble parameter, often at the expense of invoking phantom transitions. These models achieve this without introducing additional parameters, akin to the simplicity of the concordance $\Lambda$CDM model. In this work, we investigate two such models -- Phenomenologically Emergent Dark Energy (PEDE) and Granda-Oliveros Holographic Dark Energy (GOHDE) -- to assess how correlations between $H_0$ and $\Omega_m$, as well as the choice of datasets, influence conclusions regarding their potential to address the Hubble tension at the background level. We find that minimally extended versions of these models favour notably low values for the Hubble parameter, with the perceived preference for higher values driven by the associated prior. Excluding BAO Ly$\alpha$-$H(z)$ data points at a redshift of $\sim 2.3$ results in a Hubble parameter that remains in significant tension with SH0ES measurements. In contrast, including these data points favours a higher $H_0$, as they suggest a relatively lower matter density within the framework of the assumed fiducial cosmology. Additionally, recent DESI DR1 and DR2 data exhibit mild tension with BAO-$H(z)$ estimates from SDSS. We demonstrate that the inclusion of stringent constraints, such as the CMB shift-parameter along with Pantheon$^+$, on the effective pressure less matter density significantly impacts the estimation of the Hubble parameter. Finally, reinterpreting these models in terms of interacting dark sectors with $Q = 3H\gamma_{\Lambda}\tilde{\rho}_{m}$ reveals that addressing the Hubble tension necessitates a varying $\gamma_{\Lambda}$ characterised by a singular sign-switch behaviour. This phantom behaviour, or equivalently, the onset of violation of the null energy condition in the future, is crucial for minimal models to solve the Hubble tension.

Figures

Figures reproduced from arXiv: 2505.24743 by the authors.

Figure 1
Figure 1. FIG. 1. Comoving Hubble parameter as a function of redshift for the [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of Hubble parameter estimates for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Corner plots for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of Hubble parameter estimates for [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of matter density estimates for the PEDE and [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of matter density estimates for the GOHDE [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Posterior distributions of the additional free parameters [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Corner plots of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Impact of varying the parameter [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Deviation of the CMB TT power spectrum from the Planck [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Interaction strength [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Interaction strengths [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Evolution of the horizon entropy normalised to unity at [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

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