Addendum: Fitting the DESI BAO Data with Dark Energy Driven by the Cohen-Kaplan-Nelson Bound

Heinrich P\"as; Martin Hirsch; Mustafa Tabet; Patrick Adolf; Sara Krieg

REVIEW 2 major objections 4 minor 1 cited by

Addendum: Fitting the DESI BAO Data with Dark Energy Driven by the Cohen-Kaplan-Nelson Bound

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A dark-energy term tied to the CKN bound beats Lambda-CDM by 2.6 sigma in a new DESI fit.

desk verdict The CKN model in Eq. (1) is degenerate with ΛCDM, so the reported preference is an artifact of the fitting implementation rather than a real dark-energy signal. read the letter →

arxiv 2504.15332 v2 pith:QVUMQ5CX submitted 2025-04-21 astro-ph.CO gr-qchep-phhep-th

Patrick Adolf , Martin Hirsch , Sara Krieg , Heinrich Päs , Mustafa Tabet This is my paper · ORCID

classification astro-ph.COgr-qchep-phhep-th PACS 95.36.+x98.80.-k

keywords darkenergytime-varyingCKNboundbaryonacousticoscillationsDESIDR2cosmologicalconstantHubbleparameterLambda-CDM

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

This addendum asks whether the preference for time-varying dark energy seen in the first DESI BAO release survives the higher-statistics Year-2 release. The authors update their fits of a model in which the dark-energy density contains a piece proportional to the squared Hubble rate, motivated by the Cohen-Kaplan-Nelson bound on vacuum energy. Combining the DESI DR2 BAO data with supernova samples and model-independent Hubble measurements, they find that this model and its one-parameter extension are preferred over the cosmological-constant model, with the nuCKN variant reaching about 2.6 standard deviations with one supernova sample and about 1.8 with the other. The Year-2 data tighten the parameter regions and strengthen the time-varying preference relative to Year 1, while the paper also notes that the commonly used wCDM and w0waCDM models fit the same data even better in most comparisons.

What carries the argument

The load-bearing object is the modified dark-energy density $\rho_{\mathrm{DE}}(z)=\Lambda_0+\nu M_{\mathrm{Pl}}^2 H^2(z)/(16\pi^2)$, in which the cosmological constant is augmented by a term proportional to the squared Hubble rate. This term is motivated by the CKN bound, which limits vacuum-energy contributions by the horizon size, and it is what changes the background expansion at the redshifts probed by baryon acoustic oscillations. The argument is carried by chi-squared-minimum comparisons, with the Akaike information criterion used to account for the extra parameter nu, and by the change in chi-squared between the DESI DR1 and DR2 fits, which isolates how much the new data sharpen the preference.

What would settle it

A direct check: take the best-fit nuCKN and wCDM parameters, predict the BAO distance ratio $D_V/r_d$ at redshifts $2<z<3$, and compare against future DESI BAO measurements; a deviation toward wCDM and away from nuCKN beyond $2\sigma$ would rule out the CKN-specific scaling. A shorter-term check is to refit the Pantheon+ sample with a re-calibrated absolute magnitude and see whether the 1.75-standard-deviation preference remains.

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Extended reading notes

Core claim

The central claim is that the CKN-motivated scaling $\rho_{\mathrm{DE}}(z)=\Lambda_0+\nu M_{\mathrm{Pl}}^2 H^2(z)/(16\pi^2)$ gives a better simultaneous description of the DESI DR2 BAO, supernova, and cosmic-chronometer data than Lambda-CDM. For the nuCKN model the chi-squared difference relative to Lambda-CDM is -6.94 with the DESY5 supernova sample and -3.07 with Pantheon+, which the authors translate into 2.63 and 1.75 standard-deviation preferences; the CKN model with nu=1 gives -6.90 and -2.05. The same comparison against wCDM and w0waCDM shows that those flexible equation-of-state models produce even lower chi-squared values for both supernova datasets, so the paper's claim is that a time-varying dark energy is preferred, with the CKN form being one competitive realization.

Load-bearing premise

The whole result rests on assuming that the CKN bound is realized specifically as a term proportional to the squared Hubble rate; if the true time-varying dark energy has a different functional form, the model's reported preference over Lambda-CDM would not be the meaningful conclusion.

Editorial extensions

If this is right

DESI Year-2 BAO data continue to favor a dark-energy density that changes with redshift over a constant cosmological term, narrowing the parameter contours of the nuCKN and CKN models.
If the CKN-motivated H-squared term is real, the dark-energy sector consists of a small constant plus a term tied to the cosmic expansion, with nu consistent with unity within about one standard deviation in both supernova samples.
The larger improvement between DR1 and DR2 for time-varying models than for Lambda-CDM means future BAO releases should separate the models more cleanly.
Because wCDM and w0waCDM fit the same data even better, the current data cannot single out the CKN functional form; they support the broader conclusion of a time-varying dark energy.

Reading between the lines

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

One could convert the fitted dark-energy coupling together with the Hubble constant and matter density into an effective present-day equation-of-state parameter, placing the CKN model directly on the w0-wa plane used by DESI comparisons; the paper does not do that, but it would make the comparison to wCDM and w0waCDM apples-to-apples.
If the H-squared scaling is taken at face value, it predicts a specific relation between the dark-energy density and the horizon size that could be tested with future high-redshift BAO and cosmic-chronometer data before the next DESI release.
The difference in significance between the two supernova samples suggests that the supernova absolute-magnitude calibration, rather than the BAO data themselves, is the main lever on the preference; a reanalysis with a unified calibration could shift the significance in either direction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline: this addendum updates the CKN-motivated dark energy fit to DESI DR2 and claims the preference over ΛCDM grows to ~2.6σ. But the model as written in Eq. (1) cannot produce that preference. Substituting ρDE = Λ0 + ν M_Pl² H²/(16π²) into the flat Friedmann equation and solving for H² gives an expansion history identical to ΛCDM, with the matter density shifted by a factor 1/(1−ν/48π²). Since ν, H0, Ωm, and rd are all free, every νCKN background is exactly a ΛCDM background. The χ² landscape should be the same and ν should be completely unconstrained. The reported Δχ² of −6.9 (2.6σ) is therefore impossible for the model as stated. Something in the numerical implementation is solving a different system—likely evaluating the H² term from a fixed reference background rather than self-consistently.

What the paper does well: it is clearly written, the fits use public DESI DR2, Pantheon+ and DESY5 data, the comparison with ωCDM and ω0ωaCDM is honest, and the DR1-to-DR2 comparison is useful. Those parts are fine. The issue is not lack of derivation of the CKN bound; it is internal consistency between Eq. (1) and the reported numbers. The paper explicitly says that ωCDM and ω0ωaCDM fit even better, so the qualitative "dark energy is time-varying" conclusion might survive, but the CKN-specific claim is unsupported.

I checked the algebra twice; the cancellation is exact. Unless the authors clarify that they are using an approximate equation—e.g., treating the H² term as a perturbation evaluated on a ΛCDM background—the central numbers in Tables 1 and 3 are not trustworthy. They need to redo the fit with the correct self-consistent equation or explicitly state the approximation if that was intended.

Who this is for: readers monitoring DESI-driven dark energy model comparisons. The paper deserves a referee because the flaw is subtle enough to need expert confirmation, but my expectation is rejection or major revision. The reader's report over-scored the soundness; the stress-test note is on target.

Referee Report

2 major / 4 minor

Summary. This addendum updates fits of the CKN and nuCKN dark-energy models to DESI DR2 BAO data combined with the DESY5 or Pantheon+ supernova samples and model-independent Hubble measurements. The authors report that the nuCKN model is preferred over Lambda-CDM at 2.63 sigma (DESY5) and 1.75 sigma (Pantheon+), while Omega-CDM and Omega0-Omega-a-CDM provide even better fits, and they show updated parameter correlations relative to DESI DR1.

Significance. The paper is a concise, clearly presented update with standard statistical methods: chi-square minimization, AIC comparisons, and delta-chi^2 significance estimates. The tables and figures are informative and the reported numbers appear internally consistent on their face. If the reported preference were correct, it would provide mild evidence for time-varying dark energy. However, the central claim is undermined by an algebraic degeneracy: the model as defined by Eq. (1) is exactly equivalent to flat Lambda-CDM after a redefinition of the matter density, so the reported delta-chi^2 values are impossible for the equations as written. This is a load-bearing internal inconsistency, not merely a missing derivation.

major comments (2)

[Section 1, Eq. (1); Section 2, Tables 1 and 3] The model defined by Eq. (1) is degenerate with flat Lambda-CDM, so the reported chi-square differences in Table 3 cannot arise from the equations as written. Substituting rho_DE = Lambda_0 + nu M_Pl^2 H^2/(16 pi^2) into the flat Friedmann equation and imposing flatness gives E(z)^2 = [Omega_m/(1-beta)](1+z)^3 + 1 - Omega_m/(1-beta), where beta = nu/(48 pi^2) for the reduced Planck mass; this is exactly the flat Lambda-CDM background with a redefined matter density Omega_m^eff = Omega_m/(1-beta). Since Omega_m is a free fitted parameter, every nuCKN background is identical to some flat Lambda-CDM background, and the likelihood is flat along the direction that keeps Omega_m/(1-beta) fixed. Therefore the minimum chi^2 of nuCKN must equal that of Lambda-CDM, in direct contradiction with the Table 3 values of Delta-chi^2 = -6.94 (DESY5) and -3.07 (Pantheon+). Either the numerical fit solved a different system (for example, by evaluating the H^2 term on a fixed Lambda-CDM background without self-consistency) or the equations in the manuscript are incomplete. The authors must state the exact equations actually solved and reconcile them with Eq. (1); as it stands, the headline preference over Lambda-CDM is unsupported.
[Section 1, Eq. (1)] The paper describes this as a dark-energy density 'driven by the Cohen-Kaplan-Nelson bound,' but no derivation is given connecting the CKN bound to the specific scaling rho_DE proportional to H^2. As a purely phenomenological ansatz the model could stand, but then the title and abstract overstate the CKN connection. The authors should either provide a derivation or explicitly frame Eq. (1) as an ad hoc parametrization whose physical motivation is yet to be established.

minor comments (4)

[Abstract] The word 'strengthend' is a typo; it should be 'strengthened'.
[Section 2, Table 3] The caption does not restate the sign convention for Delta-chi^2 and Delta-AIC; the text defines it, but a brief parenthetical in the caption would aid the reader.
[Figure 1] The caption does not explicitly label which panel corresponds to DESY5 and which to Pantheon+, although the text mentions the left/right split; adding '(left: DESY5, right: Pantheon+)' would improve clarity.
[Section 2] The sentence stating that 'only according to the Delta-AIC values for the combination with the Pantheon+ dataset, both the CKN and nuCKN models are slightly preferred with respect to the Omega0-Omega-a-CDM model' is grammatically dense and could be clarified by specifying the actual Delta-AIC numbers.

Circularity Check

1 steps flagged · score 8.0 of 10

νCKN model of Eq. (1) is algebraically identical to flat ΛCDM, so the claimed 2.6σ preference cannot come from the model as written.

renaming known result [Section 1, Eq. (1); Section 2, Table 3]
"We consider a dark energy scaling proportional to the squared Hubble parameter H(z) as ρDE(z) = Λ0 + ν M_Pl^2 H^2(z)/(16π^2). ... In case of the νCKN model, the χ2 difference can be translated into a significance of 2.63σ and 1.75σ for the DESY5 and the Pantheon+ data, respectively."

Substituting Eq. (1) into the flat Friedmann equation gives H² = (8πG/3)(ρ_m + Λ0 + ν M_Pl² H²/16π²), so (1−ν/48π²)H² = (8πG/3)(ρ_m + Λ0). With ρ_m = Ω_m ρ_crit(1+z)³ and E(0)=1, this becomes E²(z) = Ω_eff(1+z)³ + 1 − Ω_eff, where Ω_eff = Ω_m/(1−ν/48π²). That is exactly the flat ΛCDM background. Since H0, Ω_m, and r_d are free parameters, every νCKN model is a flat ΛCDM model, and χ²_min for νCKN must equal χ²_min for ΛCDM. The reported Δχ² = −6.94 (DESY5) and −3.07 (Pantheon+) therefore cannot be produced by the model defined by Eq. (1); the claimed 'preference' is a reparameterization artifact, i.e., the νCKN prediction reduces to ΛCDM by construction.

full rationale

The central claim is that the CKN/νCKN time-varying dark energy is preferred over ΛCDM when DESI DR2 data are used. However, the model defined by Eq. (1) is degenerate with flat ΛCDM: the νH² term can be absorbed into the left-hand side of the Friedmann equation, rescaling the effective matter density and the effective cosmological constant. The resulting expansion history is a subset of flat ΛCDM, so the model cannot improve the fit over ΛCDM at all; Δχ² must be identically zero. The numerical Δχ² values and significances in Table 3 are therefore not predictions of the model as written, but artifacts of an internal inconsistency or of a different, unspecified implementation. This is a constructional reduction, not merely a missing derivation of the CKN bound. The self-citation to the authors' previous work [1] for the model and statistical procedure is not the load-bearing problem here; the degeneracy is. Thus the highest-scoring circularity pattern is 'renaming known result': the paper presents a reparameterized ΛCDM as a distinct model and reports a preference that cannot follow from its own equations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends mainly on the assumed dark-energy form in Eq. (1) and on the fitted parameter nu. The other parameters are standard cosmological fit parameters. No new particles, forces, or entities are introduced.

free parameters (4)

nu = 0.92 +/- 0.35 (DESY5); 0.64 +/- 0.36 (Pantheon+)
Free parameter controlling the H^2 contribution in Eq. (1); fitted to DESI DR2, supernova, and Hubble data.
H0 = 68.83 to 69.46 km/s/Mpc depending on dataset
Standard Hubble constant parameter fitted in all cosmological models compared in the paper.
Omega_M = 0.297 to 0.352 depending on model and dataset
Standard matter density parameter fitted in all models.
rd = 143.6 to 144.3 Mpc depending on dataset
Baryon drag epoch fitted as part of the BAO distance calibration.

assumptions (5)

domain assumption The Friedmann-Lemaitre-Robertson-Walker background and the standard expansion history with the modified dark-energy density of Eq. (1) are assumed.
All fits use the cosmological background from the authors' model; see Section 1 and Eq. (1).
ad hoc to paper The dark-energy density scaling in Eq. (1), rho_DE = Lambda_0 + nu M_Pl^2 H^2/(16 pi^2), is assumed as the realization of the CKN bound without a derivation.
Section 1 states the scaling is motivated by the CKN bound, but the functional form is not derived from it.
domain assumption The DESI DR2 BAO, supernova, and Hubble datasets are independent and their published likelihoods can be combined by summing chi-square contributions.
Section 2 describes a combined chi-square fit over these datasets, which assumes the standard combination procedure.
standard math The significance conversion from delta chi-square to sigma assumes Wilks' theorem for nested models.
Section 2 converts the nuCKN versus Lambda-CDM delta chi-square into Gaussian significances, requiring asymptotic chi-square behavior.
domain assumption Flatness (Omega_k = 0) is implicitly assumed in the distance fits.
The fits report only Omega_M with no curvature parameter; Tables 1 and 2.

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Cite this review

Pith. "Pith review of Addendum: Fitting the DESI BAO Data with Dark Energy Driven by the Cohen-Kaplan-Nelson Bound." pith.science (2026). https://pith.science/paper/QVUMQ5CX

@misc{pith2026250415332,
  author       = {Pith},
  title        = {Pith review of: Addendum: Fitting the DESI BAO Data with Dark Energy Driven by the Cohen-Kaplan-Nelson Bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVUMQ5CX}},
  note         = {Machine review of arXiv:2504.15332}
}

abstract

Motivated by the recent Year-2 data release of the DESI collaboration, we update our results on time-varying dark energy models driven by the Cohen-Kaplan-Nelson bound. The previously found preference of time-dependent dark energy models compared to $\Lambda$CDM is further strengthend by the new data release. For our particular models, we find that this preference increases up to $\approx 2.6\,\sigma$ depending on the used supernova dataset.

Figures

Figures reproduced from arXiv: 2504.15332 by the authors.

**Figure 1.** Shown is the angle-averaged distance quantity DV/(rdz 2/3 ) normalised to the ΛCDM value at our best-fit point together with the DR2 DESI measurements [2] for all considered dark energy models. In the left plot, the best-fit points of the combination with the DESY5 dataset are used and in the right plot with the Pantheon+ dataset. Note that in the left plot the best-fit lines for CKN and νCKN overlap. separately. Th… view at source ↗

**Figure 2.** Correlations of H0–Ω0 M (top left), H0–rd (top right), rd–Ω0 M (bottom left) in the CKN model for the DESI BAO+Hubble+DESY5 DR1 (black dashed lines) and DESI BAO+Hubble+DESY5 DR2 (blue area) dataset at the 95 % and 68 % CL. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Correlations of H0–Ω0 M (top left), H0–rd (top right), rd–Ω0 M (bottom left) in the CKN model for the DESI BAO+Hubble+Pantheon+ DR1 (black dashed lines) and DESI BAO+Hubble+Pantheon+ DR2 (blue area) dataset at the 95 % and 68 % CL. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Correlations of ν–Ω0 M (top left), ν–H0 (top right), ν–rd (bottom left) in the νCKN model for the DESI BAO+Hubble+DESY5 DR1 (black dashed lines) and DESI BAO+Hubble+DESY5 DR2 (blue area) dataset at the 95 % and 68 % CL. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Correlations of ν–Ω0 M (top left), ν–H0 (top right), ν–rd (bottom left) in the νCKN model for the DESI BAO+Hubble+Pantheon+ DR1 (black dashed lines) and DESI BAO+Hubble+Pantheon+ DR2 (blue area) dataset at the 95 % and 68 % CL. 7 [PITH_FULL_IMAGE:figures/full_fig_p007…

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