Dynamical Dark Energy at Late Time $\Lambda$CDM

E. J. Thompson; J. W. Moffat

REVIEW 3 major objections 4 minor 4 cited by

Dynamical Dark Energy at Late Time $\Lambda$CDM

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Using the FLRW equations and recent baryon-acoustic-oscillation measurements, the paper derives a present-day dark energy equation of state w ≈ −0.7, a statistically significant departure from the cosmological constant value −1, with no…

desk verdict The claimed dynamical dark energy signal is an artifact of dropping the matter term from the exact equation of state, and the paper's own caveat at Eq. (30) gives away the error. read the letter →

arxiv 2505.18900 v2 pith:QIADUHEF submitted 2025-05-24 astro-ph.CO gr-qchep-thphysics.data-an

J. W. Moffat , E. J. Thompson This is my paper

classification astro-ph.COgr-qchep-thphysics.data-an PACS 95.36.+x98.80.-k

keywords darkenergyequationofstatedynamicalFLRWcosmologyHubbleparameterbaryonacousticoscillationsCPLparametrizationnullcondition

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 tries to establish that the cosmic expansion history alone fixes the dark energy equation of state at late times, without assuming a microphysical model for dark energy. From the FLRW equations it derives $w_{\mathrm{DE}}(z) = -1 + \frac{2(1+z)}{3H(z)}\frac{dH}{dz}$, then evaluates that expression on the flat $\Lambda$CDM Hubble law using matter-density priors from recent DESI DR2 measurements. The result is a present-day value $w_{\mathrm{DE}}(0)\approx -1 + \Omega_m \approx -0.7$, which the authors read as a statistically significant deviation from the cosmological constant value $w=-1$. Because the derived equation of state stays above $-1$ at all redshifts and approaches $-1$ in the future, the model avoids phantom energy and a Big Rip while still making dark energy dynamical today.

What carries the argument

The load-bearing object is the identity $w_{\mathrm{DE}}(z) = -1 + \frac{2(1+z)}{3H(z)}\frac{dH}{dz}$, obtained from the Friedmann equations after dropping the matter density from the denominator, combined with the flat $\Lambda$CDM Hubble law. Inserting that Hubble law turns the identity into the closed form $w_{\mathrm{DE}}(z) = -1 + \frac{\Omega_m(1+z)^3}{\Omega_m(1+z)^3 + 1 - \Omega_m}$, which the paper uses to map observed expansion rates into an equation of state, to fit the CPL parameters $w_0 = -1 + \Omega_m$ and $w_a = 3\Omega_m(1-\Omega_m)$, and to compare against DESI DR2 parametrized fits.

What would settle it

Take the flat $\Lambda$CDM Hubble law used in the paper, compute the exact dark energy equation of state from Eq. (18) without dropping $8\pi G\rho_m$, and check whether it equals $-1$ at all redshifts; if it does, the claimed $w_{\mathrm{DE}}(0)\approx -0.7$ is an artifact of the approximation rather than a property of the data.

Watch

Extended reading notes

Core claim

The central claim is that, in a flat universe whose expansion follows the standard $\Lambda$CDM Hubble law $H(z) = H_0\sqrt{\Omega_m(1+z)^3 + 1 - \Omega_m}$, the dark energy equation of state inferred from $w_{\mathrm{DE}}(z) = -1 + \frac{2(1+z)}{3H(z)}\frac{dH}{dz}$ is $w_{\mathrm{DE}}(z) = -1 + \frac{\Omega_m(1+z)^3}{\Omega_m(1+z)^3 + 1 - \Omega_m}$. Evaluated at $z=0$ this gives $w_{\mathrm{DE}}(0) = -1 + \Omega_m$, which with the DESI DR2 matter-density priors is about $-0.7$ with small quoted uncertainties; the paper interprets this as a statistically significant departure from a pure cosmological constant. The same formula makes $w_{\mathrm{DE}}(z)$ monotonic, always greater than $-1$, and asymptotic to $-1$ at both high redshift and future times, so the model keeps the $\Lambda$CDM trajectory while making dark energy dynamical at late times.

Load-bearing premise

The central deviation from $w = -1$ rests on the approximation in Eq. (22), which drops the matter density $8\pi G\rho_m$ from the denominator of $w_{\mathrm{DE}}$; at redshifts $z \lesssim 1$ matter and dark energy are still comparable, and if the matter term is retained the same $\Lambda$CDM Hubble law gives $w_{\mathrm{DE}} = -1$ exactly.

Editorial extensions

If this is right

Today's dark energy equation of state is pinned to the matter density by $w_0 \approx -1 + \Omega_m$, so any independent measurement of $\Omega_m$ fixes the present deviation from $-1$.
Cosmic acceleration is slower at late times than in a pure cosmological-constant model, consistent with DESI DR2 hints of weaker late-time acceleration.
The equation of state never crosses the phantom divide: $w_{\mathrm{DE}}(z) > -1$ for all redshifts and approaches $-1$ in the future, so no Big Rip occurs.
The deceleration-to-acceleration transition lands near $z \approx 0.7$, matching the standard $\Lambda$CDM timeline.
Within the approximation, the CPL form with $w_0 = -1 + \Omega_m$ and $w_a = 3\Omega_m(1-\Omega_m)$ reproduces the model's evolution for $0 < z < 1$ and can be compared directly with DESI fits.

Reading between the lines

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

If the exact Eq. (18) is used instead of the matter-neglected approximation, a flat $\Lambda$CDM $H(z)$ yields $w_{\mathrm{DE}} = -1$ exactly, so the paper's deviation may be an artifact of dropping $8\pi G\rho_m$ below $z \approx 1$; testing that drop is the natural next step.
The same machinery could be applied to non-flat models, interacting dark sectors, or evolving matter densities; each choice of $H(z)$ produces a different inferred $w_{\mathrm{DE}}(z)$, so the method is more a mapping from expansion history to equation of state than a prediction of a specific dark energy model.
A direct observational falsifier would be to reconstruct $w_{\mathrm{DE}}(z)$ from combined geometric probes at $z < 1$ without assuming $\Lambda$CDM $H(z)$; if the reconstructed value at $z=0$ is compatible with $-1$ at the few-percent level, the claimed deviation disappears.

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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 short version: the paper's central claim—that DESI data imply w_DE(0) ≈ -0.7 and dynamical dark energy—does not survive contact with its own equations. Eq. (22) is the standard total equation-of-state reconstruction, valid only when matter is negligible. The paper derives the exact Eq. (18) with the matter term in the denominator, then drops that term in Eq. (19) to get Eq. (22). At z ≲ 1 matter is not negligible (Ωm ≈ 0.3), so Eq. (30) is the total effective w of flat ΛCDM, not the dark-energy EOS. Plug the same ΛCDM H(z) back into the exact Eq. (18) and you get w_DE = -1 exactly. The claimed w0 = -1 + Ωm and wa = 3Ωm(1-Ωm) are therefore algebraic functions of the DESI-fitted Ωm priors—comparing them to DESI CPL fits is comparing a reparametrization of the prior to the same data. The paper even admits this at Eq. (30): 'This formula does not represent the dark energy equation of state in a general cosmology.' Then it uses it anyway. That is a load-bearing internal contradiction.

What the paper does well: it lays out the FLRW derivation cleanly, the DESI DR2 parameter table (Table 1) is handy, and the figures are readable. The CPL fitting arithmetic is straightforward. But none of this is new—Eq. (22) is textbook reconstruction, and Eq. (30) is the known total EOS for flat ΛCDM.

Soft spots: the central comparison to DESI is circular; the transition redshift z ≈ 0.7 is just standard ΛCDM; the 'prediction' avoids phantom crossing by construction, so the NEC discussion adds nothing. The prose sometimes overclaims, e.g. 'statistically significant deviation' that is an artifact of dropped terms.

Who is this for? Possibly a reader wanting a quick introduction to reconstruction formulas, but not someone looking for a valid dynamical DE result. It deserves a serious referee? No—the main result is demonstrably wrong, and the paper itself contains the caveat that refutes it. I'd desk-reject.

Referee Report

3 major / 4 minor

Summary. The paper claims to derive the dark-energy equation of state w_DE(z) from the Friedmann equations and, using the flat ΛCDM Hubble rate H(z), obtains w_DE(z) ≈ -1 + Ωm(1+z)^3/[Ωm(1+z)^3 + 1 - Ωm]. From this it derives w0 = -1 + Ωm and wa = 3Ωm(1-Ωm), compares these with DESI DR2 CPL constraints, and concludes that w_DE(0) ≈ -0.7 is a statistically significant deviation from ΛCDM. The manuscript asserts that dark energy is dynamical today while asymptotically approaching w=-1, avoiding phantom behaviour.

Significance. If the central claim were correct, a model with only Ωm as input would predict a present-day dark-energy equation of state w0 ≈ -0.7 and a specific CPL slope wa ≈ 0.6, which would be a striking and falsifiable result. However, the central claim is internally inconsistent with the paper's own exact equations: Eq. (30) is not the dark-energy equation of state but the total effective equation of state of flat ΛCDM, and the exact Eq. (18) yields w_DE = -1 when the same ΛCDM H(z) is inserted. The 'predictions' in Eqs. (35) and (38) are deterministic reparametrizations of the Ωm priors taken from DESI, so the comparison to DESI is circular rather than an independent test. The paper is transparent about the limitation immediately after Eq. (30), but then proceeds to use the mislabelled quantity as its main prediction. No new data, code, or machine-checked derivation is provided. The paper therefore does not establish dynamical dark energy.

major comments (3)

[§2, Eqs. (18), (22), (30)] The central claim that Eq. (30) gives the dark-energy equation of state is not supported. Eq. (18) is exact for a flat universe with pressureless matter plus dark energy. Inserting the flat ΛCDM H(z) of Eq. (25), one has Hdot = -4πG ρm(z) and 3H^2/(8πG) = ρm(z) + ρΛ, so the numerator of Eq. (18) equals -ρΛ and the denominator equals ρΛ; hence w_DE = -1 exactly at every redshift. Eq. (22) follows from Eq. (18) only after dropping the 8πGρm term in the denominator, which requires ρDE >> ρm. At z ≲ 1, matter is not negligible (Ωm ≈ 0.3), so this step is invalid. The manuscript itself states after Eq. (30) that the formula 'does not represent the dark energy equation of state in a general cosmology' and instead represents what one would infer for w_DE in a universe with a cosmological constant. Indeed, Eq. (30) is exactly the total effective equation of state w_tot = -ρΛ/(ρm+ρΛ) = -1 + Ωm(z) for flat ΛCDM. The claimed deviation w0 = -1 + Ωm ≈ -0.7 is therefore an artifact of the dropped denominator term, not a prediction about dark energy.
[§3, Eqs. (35), (38), Table 3] The quantities presented as predictions, w0 = -1 + Ωm and wa = 3Ωm(1-Ωm), are deterministic functions of the Ωm priors taken from DESI in Table 1. Comparing these functions with DESI's CPL constraints is comparing a reparametrization of the input prior with the same dataset, not an independent model test. The claim in Section 3 that 'the majority of best-fit curves across all models and datasets converge around wDE(0) ≈ -0.7' is also not supported by Table 3, whose entries span roughly -0.648 to -0.797 depending on the model and dataset. For example, the DESI-only wCDM row gives w0 = -0.916 ± 0.078 in Table 1, while Eq. (35) with the same Ωm prior gives -0.703 ± 0.009; this difference is not a small scatter but a dataset/model dependence that the paper does not address.
[§3, Eq. (39), Figs. 3–4] The comparison with DESI CPL constraints is not quantitatively supported. For Ωm ≈ 0.3, Eq. (39) gives wa ≈ +0.6, whereas the DESI w0waCDM constraints in Table 1 have wa negative for most combinations, e.g., -0.62 ± 0.22 for DESI+CMB+Pantheon+. The paper acknowledges a mismatch Δw ≈ 0.1-0.2 at z ≈ 0.5 but provides no likelihood or significance calculation demonstrating agreement; a model whose wa has the opposite sign from the best-fit data cannot be described as consistent with those fits. The 'statistically significant deviation' claim in Section 3 is likewise unsupported because the error bars quoted in Table 3 merely propagate the Ωm prior uncertainties and do not represent a test against the ΛCDM hypothesis.

minor comments (4)

[§2, Eq. (7)] Equation (7) is algebraically inconsistent with Eq. (8): solving Eq. (7) as written gives w = -1 - 2Hdot/H^2, not w = -1 - 2Hdot/(3H^2). The correct intermediate relation is Hdot = -(3H^2/2)(1+w) for the total fluid, which does lead to Eq. (8). Please correct the factor in Eq. (7).
[§2, Eq. (13)] The signs of the pm and pr terms in Eq. (13) appear incorrect. From the acceleration equation, p_DE = -(2Hdot + 3H^2)/(8πG) - pm - pr. The plus signs shown in Eq. (13) give the wrong result for a mixture of radiation and a cosmological constant, although the error does not affect the later analysis because radiation is neglected and pm = 0 for pressureless matter.
[Abstract and Eq. (33)] The CPL parametrization is written inconsistently: the abstract has w(z) = -1 + wa/(1+z), while the standard form used in Eq. (33) is w(z) = w0 + wa z/(1+z). Please ensure the same definition is used throughout; the abstract formula is missing the w0 term and the redshift factor.
[References] Reference [4] is cited for the CPL parametrization, but [4] is a later test of CPL by Linden and Virey. The original parametrization should be credited to Chevallier and Polarski (2001) and Linder (2003), which are already listed as references [6] and [7].

Circularity Check

3 steps flagged · score 8.0 of 10

Claimed w_DE deviation is the total effective EOS of ΛCDM relabeled as dark energy, with w0 and wa deterministic functions of the DESI Ωm priors.

self definitional [Section 2, Eqs. (19), (22), (30)]
"For the dark energy dominated and matter negligible case ρDE > ρm, we have at late time for z <1: wDE = −1 − 2 ˙H/3H^2 . ... wDE(z) = −1 + 2(1+z)/3H(z) dH/dz . ... We can now approximate wDE for z ≲ 1: wDE(z) ≈ −1 + Ωm(1+z)^3/(Ωm(1+z)^3 + 1 − Ωm)."

Eq. (22) is Eq. (8), the equation of state of the total fluid, not of dark energy; it equals w_DE only if the matter term in the denominator of Eq. (18) is dropped. Substituting flat-ΛCDM H(z) into Eq. (22) gives Eq. (30), which is precisely the total effective equation of state of ΛCDM, w_eff = −Ω_Λ/[Ω_m(1+z)^3 + Ω_Λ]. Putting the same H(z) into the exact Eq. (18) returns w_DE = −1 identically. The claimed deviation of w_DE from −1 is therefore generated by the dropped matter term, i.e. by construction, not by the DESI data.
fitted input called prediction [Section 3, Eqs. (34)–(35) and Table 3]
"We use (22) taking the Ωm priors provided by the DESI measurements and compute for various redshifts how our model deviates from ΛCDM. ... This gives us w0 = −1 + Ωm."

The 'prediction' w0 is simply the DESI Ωm prior shifted by −1; the claimed convergence around w0 ≈ −0.7 is just the input Ωm ≈ 0.3 rewritten. Comparing these w0 values to DESI CPL constraints on (w0, wa) is not an independent test but a re-parametrization of the same fitted parameter.

1 more flagged steps

fitted input called prediction [Section 3, Eqs. (37)–(39), Fig. 4]
"Evaluating this expression at z = 0 yields dwDE/dZ |_{z=0} = 3Ωm(1 − Ωm) = wa."

wa is also a deterministic function of the DESI Ωm prior. The reconstructed CPL model, Eq. (39), has no free parameter other than Ωm; the plotted 'theoretical CPL' curves are a relabeling of the ΛCDM total-EOS formula evaluated at the DESI Ωm values. Agreement or disagreement with DESI w0–wa fits is therefore forced algebraically rather than constituting a falsifiable prediction of dynamical dark energy.

full rationale

The paper contains no load-bearing self-citation; the circularity is algebraic and definitional. The central derivation takes the total-fluid equation of state from Eq. (8), relabels it as the dark-energy equation of state in Eq. (22), and substitutes the flat-ΛCDM Hubble history. The resulting Eq. (30) is the total effective w of ΛCDM, not w_DE: for a cosmological constant, the exact dark-energy EOS from Eq. (18) remains −1. The paper even concedes this at Eq. (30): 'This formula does not represent the dark energy equation of state in a general cosmology. It represents what you would infer for w_DE(z), if you apply the formula derived for dark energy to a universe that has a cosmological constant w = −1,' before proceeding to use it as a prediction. The subsequent w0 = −1 + Ωm and wa = 3Ωm(1 − Ωm) are one-to-one functions of the DESI Ωm priors, so comparing them with DESI CPL constraints is comparing a re-parametrization of the input prior to the same data. The claimed statistically significant deviation of w0 from −1 is thus forced by the choice of formula and priors rather than derived from independent physics. Score 8 reflects that the central claim reduces by definition.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on standard FLRW assumptions plus the specific unjustified assumption that the matter term can be dropped for z<1. No new entities are introduced. The only fitted input is Omega_m, taken from DESI; w0 and wa are deterministic functions of Omega_m.

free parameters (1)

Omega_m (matter density parameter prior) = 0.203 to 0.357 across DESI DR2 model and dataset combinations (Table 1)
The paper's predicted w0 = -1 + Omega_m and wa = 3 Omega_m (1 - Omega_m) are direct functions of this input. No new fit is performed; the values are taken from DESI DR2 constraints, so the 'prediction' is a re-labeling of the fitted parameter.

assumptions (7)

standard math FLRW metric and Friedmann equations describe the background cosmology
Used throughout Section 2, Eqs. (1)-(9).
domain assumption Spatial flatness (k=0, Omega_K=0)
Invoked in Eq. (9) and in the H(z) formula Eq. (25); the paper uses flat priors for its central curves.
domain assumption Radiation is negligible for z < 1
Stated in Section 2 before Eq. (15); reasonable for late times but not exact at z near 1.
domain assumption Perfect fluid with p = w rho
Eq. (5), used to derive the continuity-equation result.
domain assumption Energy conditions (NEC, WEC, SEC) are physical requirements
Section 4, Eqs. (40)-(42); used to argue phantom crossing is unphysical.
domain assumption CPL parametrization is a faithful approximation for z < 1
The paper fits its own formula to CPL and compares with DESI CPL constraints; no validation of CPL adequacy at the claimed precision is given.
ad hoc to paper The matter term in Eq. (18) can be dropped for z < 1
This step makes Eq. (22) follow from Eq. (18), but at z<1 Omega_m is about 0.3, so the drop is not justified. It is the load-bearing assumption behind the predicted w_DE(z).

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Pith. "Pith review of Dynamical Dark Energy at Late Time $\Lambda$CDM." pith.science (2026). https://pith.science/paper/QIADUHEF

@misc{pith2026250518900,
  author       = {Pith},
  title        = {Pith review of: Dynamical Dark Energy at Late Time $\Lambda$CDM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIADUHEF}},
  note         = {Machine review of arXiv:2505.18900}
}

abstract

We investigate the dynamical properties of dark energy through a detailed analysis of its equation of state parameter $w(z)$ as a function of redshift. We derive a general expression for $w(z)$ from the Friedmann-Lema\^itre-Robertson-Walker (FLRW) equations, establishing a direct relationship between the dark energy equation of state and the observable Hubble parameter $H(z)$ and its derivative. Using the relation $w(z) = -1 + \frac{2(1+z)}{3H(z)} \frac{dH}{dz}$, we develop an approximation method valid for $z \lesssim 1$ that accounts for the changing balance between matter and dark energy contributions to cosmic expansion. We compare our theoretical framework with recent observational data from the Dark Energy Spectroscopic Instrument (DESI) DR2, analysing how well the commonly used Chevallier-Polarski-Linder (CPL) parametrization $w(z) = -1 + w_a \frac{z}{1+z}$ captures the evolution of dark energy. Our results indicate that the dark energy equation of state exhibits a monotonic evolution with redshift, transitioning from deceleration to acceleration around $z \approx 0.7$. Notably, our predicted $w_{\mathrm{DE}}$ remains greater than $-1$ across all redshifts, avoiding phantom energy scenarios that would violate the null energy condition. This work demonstrates how precise measurements of the cosmic expansion history can constrain the nature of dark energy and provides a framework for testing dynamical dark energy models against current and future cosmological observations.

Figures

Figures reproduced from arXiv: 2505.18900 by the authors.

**Figure 2.** Dark energy equation of state wDE(z) plotted for the Ωm priors with the tightest errors. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Evolution of the theoretical CPL parametrized dark energy equation of state, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Comparison between the theoretical CPL dark energy equation of state model and DESI DR2 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Evolving dark energy equation of state wDE(z) plotted for various Ωm priors. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Evolving dark energy equation of state wDE(z) plotted for various Ωm priors with the tightest constraints. From our model, we can see that we do not violate the NEC as even at z ≲ 0, we are always greater than −1 and we recover the standard ΛCDM predictions. While seve…

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Simple quintessence models in light of DESI-BAO observations
astro-ph.CO 2025-06 conditional novelty 6.0 of 10

Thawing quintessence with linear or quadratic potentials is favored over LambdaCDM only when the DESY5 supernova catalog is used; with Pantheon+ or Union3 the preference is mild.
Dynamical dark energy in the Bianchi Type-V Universe with DESI DR2 BAO, SNIa compilation and RSD measurements
physics.gen-ph 2026-06 conditional novelty 5.0 of 10

Bianchi-V models with constant-w or CPL dynamical dark energy accommodate DESI DR2 + SNIa + RSD data and reduce H0/S8 tensions relative to flat ΛCDM, though BIC still prefers ΛCDM.
Topological defects as effective dynamical dark energy
hep-ph 2025-06 conditional novelty 4.0 of 10

A few percent domain-wall component gives a mild (Δχ² = -1.72) improvement over ΛCDM when fitting DESI DR2 BAO and DESY5 supernovae, but the evidence is inconclusive.
Comment on "Dynamical Dark Energy at Late Time $\Lambda$CDM"
astro-ph.CO 2025-07 conditional novelty 2.0 of 10

Cline shows that Moffat and Thompson's apparent dynamical dark energy in Lambda-CDM comes from neglecting the matter density term in the equation of state formula.

Reference graph

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