A derived evolution of dark energy from cosmic inhomogeneity confronted with baryon acoustic oscillations and type Ia supernovae

Hans-Otto Carmesin; Jacky David Yang

REVIEW

A derived evolution of dark energy from cosmic inhomogeneity confronted with baryon acoustic oscillations and type Ia supernovae

Jacky David Yang, Hans-Otto Carmesin

Cosmology and Nongalactic Astrophysics · Submitted 25 Sep 2026

Pith's verdict: The headline numbers all flow from one formula, Eq. (E33), cited to the authors' own self-published work rather than derived here — yet the low-redshift predictions are genuinely non-tuned and the authors are unusually candid about what they did and did not test.

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arXiv ID2609.31543 (version 5)

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CitationHans-Otto Carmesin, Jacky David Yang. "A derived evolution of dark energy from cosmic inhomogeneity confronted with baryon acoustic oscillations and type Ia supernovae." arXiv:2609.31543 (2026). https://pith.science/paper/DEJMFIVG

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Two things up front. The headline results — H0,het(z) rising to 73.6, Omega_Lambda rising from 0.679 to 0.735, and the 0.306 Gyr age shift — all collapse onto a single auxiliary function, Eq. (E33): kappa ∝ Omega_m·sigma_8/(2·Omega_vol)·(1+z)^{-2.5}. The exponent and prefactor, including the claim that quantization leaves a non-vanishing mean-square field driving locally formed volume at roughly 30% of Omega_vol today, are cited to the authors' reference [2], not derived in this text. I checked the arithmetic: plugging the Planck inputs into (E32)–(E34) reproduces H0,het(0)=73.62 and Omega_Lambda,het(0)=0.7351 exactly. And the validation curve substitutes the measured 0.679 for the framework's own derived 2/3, so the Planck point anchors the curve rather than testing it — the authors say this openly (b75). What is actually new: redshift-binned H0 and time-varying dark energy are old news (Krishnan, Dainotti, DESI all cited), but deriving those trends from one mechanism, with no new fitted parameter, is the contribution, and the specific numbers are new. Credit where due: at low redshift the predictions are honest in the strongest sense. Nothing is tuned to SH0ES or the independent Omega_Lambda probes — given only Planck inputs, 73.6 against SH0ES's 73.04 and 0.735 against the late-Universe values are non-tuned hits. The Zenodo workbooks are shipped and the sums reproduce. The authors also state their limitations plainly: no likelihood analysis was run (b91), the literature values are effective parameters under homogeneous assumptions (b69–b70), and the SnIa entries are flagged as non-independent (b111). That candor is real and unusual. The soft spots are proportionate. The load-bearing concern is E33 itself: one asserted formula, cited to a self-published framework, carrying the entire redshift evolution, with an amplitude orders of magnitude above the perturbative backreaction bounds (Green & Wald) the authors themselves cite. The appendix derivations are identification-level — the Schrodinger equation is obtained by equating a normalized rate with the wave function (E6), and rho_vol = H0^2/(4πG) rests on the identification in (E19). These may be defensible inside the framework, but the framework lives in Verlag Dr. Koester books, so a referee cannot audit it without buying the series. There is also a visible strain: at z≈1.31 the prediction runs about 1.4σ below DESI's 68.51. Bottom line: this is not ready to stand as a derivation, and I would not publish it as is. But it is coherent rather than incoherent, honestly presented, with reproducible numerics and falsifiable predictions. I would send it to a referee with a brief concentrated on E33 and on enforcing the 2/3 prediction in the validation. Expect rejection on the derivation gap, but referee time is warranted. I would not cite it in my own work; I might bring it to a reading group as a clean case study of where identification ends and derivation begins.

Referee report

Significance

If established, the claims would constitute a parameter-free derivation of three cosmological parameters, a resolution of the Hubble tension, and a concrete prediction of time-evolving dark energy, with the admitted virtue that the low-z values (H0,het(0) ≈ 73.6 against SH0ES 73.04; Ω_Λ,het(0) = 0.735 against independent probes) are not tuned to those points. The authors are also commendably explicit about what the comparison does not establish: no likelihood analysis is performed, literature values are effective parameters under homogeneous assumptions, and the Planck point anchors rather than tests the curve. These strengths are, however, entirely conditional on a small set of asserted relations — chiefly κ(z) in Eq. (E33) and ρ_vol = H0²/(4πG) in Eq. (E19) — cited to the authors' own prior work rather than derived in this manuscript, together with a quantum mean-square mechanism that is not made checkable here. As it stands, the paper's predictive content reduces to a single unverified functional form, so the significance of the empirical agreement cannot yet be assessed.

Major objections (4)

[Appendix E, Eqs. (E32)–(E34)] All headline numbers — H0,het(0) = 73.62, Ω_Λ,het(0) = 0.7351, and Δt_U = 0.306 Gyr — follow from the single auxiliary function κ(z) = Ω_m σ_8/[2 Ω_vol (1+z)^{5/2}], which is cited to ref. [2] (Eqs. 223, 244, 237–240) but is neither derived nor testable within this manuscript. The exponent 5/2 and the prefactor 1/(2 Ω_vol) carry the entire redshift dependence of both H0,het(z) and Ω_Λ,het(z). The central claim of the paper — that cosmic heterogeneity produces a redshift-dependent dark energy and expansion rate — therefore stands or falls on an external, non-standard source rather than on anything the reader can verify here.
[Appendix E, step (10), Eq. (E19)] The identification ˙ε_L,rr = H0, and hence ρ_vol = H0²/(4πG) and Ω_vol = 2/3, is asserted rather than derived: the integration over shells with light-travel times up to the Hubble time is cut off by a dimensional argument, and the 'completeness' check in Eq. (E20) merely restates that the formed volume equals the probe volume. Since Ω_vol = 2/3 is pure algebra once ρ_vol = H0²/(4πG) is accepted, the abstract's claim that the framework derives the dark-energy density from first principles rests entirely on this unverified identification.
[Appendix E, steps (4), (5), (7), (11)] Three load-bearing physical ingredients are asserted rather than derived: (i) the indivisibility of volume portions (step 5), which is what 'quantizes' the model; (ii) the non-vanishing mean square ⟨(G*)²⟩ of the stochastic field, invoked in step (11) and the main text as the mechanism that makes heterogeneity drive backreaction several orders of magnitude above the Green–Wald bound the authors themselves cite as ref. [7]; and (iii) the derivation of the Schrödinger equation in step (7), which proceeds by 'identifying the normalized rate t_n ˙ε_L with the wave function Ψ' — a definitional stipulation, not a derivation. The mean-square mechanism is essential: without it, the perturbative backreaction amplitude would be tiny and κ(z) would collapse, taking the headline redshift evolution with it.
[Section IV and Eq. (E36)] The empirical validation is partially circular in a way the authors acknowledge but do not resolve: the measured CMB value Ω_vol = 0.679 is substituted for the derived 2/3 in the numerical run (Eq. (E36)), while Ω_m = 0.321 and σ_8 = 0.8118 are also measured inputs, so the Planck point anchors the curve rather than tests it. The derived-parameter variant (Table III) changes the headline rise from 8.3% to 9.0% with Ω_Λ,het(0) = 0.7265, so the quoted numbers are not robust to the measured-versus-derived input choice. The comparison in Figs. 1–2 is therefore only a test of the (unverified) shape of κ(z) against quantities the authors themselves call effective parameters, and no statistical evaluation is provided for Tables I–II; as a concrete strain, the DESI H0 point at z ≈ 1.31 lies about 1.4σ below the predicted H0,het ≈ 67.7 km/s/Mpc.

Minor objections (5)

[Sections II and V, Eqs. (B3) and (6)] The paper quotes three different homogeneous reference ages (13.775 Gyr via the unexplained correction factor 0.95, 13.827 Gyr from direct integration in Eq. (6), and the familiar ~13.8 Gyr from ΛCDM) without reconciling them; the 0.95 factor in Eq. (B3) appears with no derivation.
[Table I and Fig. 1] The inclusion rule for the comparison points mixes criteria: the Pantheon+ value (ref. [33]) is excluded for not presupposing flatness while other entries are converted to flat-space values; please state a single explicit selection rule for the points included in Fig. 1.
[Equations (3), (E32) and (B2)] The notation is confusing: H0,het(z) is a redshift-dependent function despite the subscript 0, and the same symbol H0,ΛCDM is used for 66.88 km/s/Mpc in some equations and for the parameter entering different numerical runs in others; clarify which inputs each numerical line uses.
[Throughout] The manuscript shows numerous rendering artifacts from the source (for example 'Schr¨odinger', broken math symbols in Appendix E, 'u vol,obs', and misplaced sub/superscripts in Table II); please ensure the text compiles cleanly before resubmission.
[Appendix B and reference list] The derivations of Eqs. (E16)–(E34) are carried by refs. [1], [2], [20]–[22] and the authors' own book series (Verlag Dr. Köster); an independent reader cannot verify the central functional form from the present text, so at minimum the explicit derivation of Eq. (E33) should be reproduced in an appendix.

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The paper

The paper claims

The paper derives from first principles a redshift-dependent dark energy and Hubble constant, claiming that cosmic unevenness alone resolves the Hubble tension, with no new free parameter.

The reading

This paper argues that the standard model's treatment of dark energy and cosmic time is an idealization that fails because it ignores the unevenness of matter. From a 'volume dynamics' of space — space made of indivisible, quantized volume portions — the author derives that the dark-energy density parameter is exactly 2/3 in the homogeneous early universe, and that as structure grows, heterogeneity drives extra volume formation that makes both dark energy and the Hubble constant rise toward the present. The predicted curve takes dark energy from 0.679 at high redshift to 0.735 today and the expansion rate from 66.88 to about 73.6 km/s/Mpc, which the paper reads as closing the Hubble tension with one mechanism and no adjustable parameter. Integrating the resulting expansion history gives a cosmic age of 13.521 Gyr, 0.306 Gyr shorter than the homogeneous value, interpreted as a global time dilation. If right, the early-versus-late expansion discrepancy and the reported drift of dark energy are two faces of one fact: the universe is not homogeneous.

What carries the argument

The volume dynamics (VD) equation (2), ∂ε_L/∂τ + ∂ε_L/∂L⃗ · c e_v = 0, governs indivisible, massless volume portions of space and is claimed to imply the Schrödinger equation, gravity, and spacetime curvature. Its consequence, the law of locally formed volume (LFV), ε̇_L,rr = G*/c (E16), states that volume formation is set by the gravitational field strength; squaring it makes the non-vanishing mean-square field ⟨(G⃗*)²⟩ of a heterogeneous universe produce extra volume even when the mean field vanishes. That extra volume, growing as κ(z) = Ωm σ8 / [2Ωvol (1+z)^5/2], is what turns the constant dark-energy density into ρvol(1+κ)^ξ and the constant Hubble rate into H0,het(z). The value Ωvol = 2/3 comes from identifying the local volume-formation rate with the Hubble constant, which yields ρvol = H0²/(4πG); dividing by the critical density 3H0²/(8πG) leaves the pure ratio 2/3.

Extended reading notes

Core claim

Once cosmic unevenness is promoted from a nuisance to a dynamical component, both the dark-energy density parameter and the expansion rate become functions of redshift. The central derived numbers are Ωvol = 2/3 for the early, homogeneous universe; ΩΛ,het(z) rising from 0.679 to 0.735±0.010 today; H0,het(z) rising from the Planck value 66.88 toward roughly 73.6 km/s/Mpc; and an integrated cosmic age of 13.521 Gyr, which is 0.306 Gyr shorter than the homogeneous age, stated as a global time dilation. Because the redshift dependence emerges from the growth of heterogeneity — entering through σ8 — and uses no newly fitted parameter, the author presents the mechanism as resolving the Hubble tension and as predicting, rather than accommodating, the time-evolving dark energy favored by recent combined analyses.

Load-bearing premise

The whole edifice rests on the claim that the rate at which space manufactures volume at any point equals the Hubble constant, so that the dark-energy density is H0²/(4πG), together with the companion claim that because volume portions are indivisible the mean-square gravitational field of a heterogeneous universe does not vanish — if the first identification is wrong, or if the non-vanishing mean-square effect washes out under realistic averaging, both the 2/3 value and the entire redshift evolution of dark energy collapse.

Editorial extensions

If this is right

The Hubble tension between early- and late-universe measurements is explained by a single mechanism — the growth of heterogeneity — with no new free parameter.
Dark energy becomes intrinsically time-varying: ΩΛ rises by roughly eight percent from the microwave background to the present, matching the direction favored by recent combined BAO, supernova, and CMB analyses.
The cosmic age becomes model-dependent: the heterogeneous universe is 0.306 Gyr younger than the homogeneous one, a difference read as global time dilation that the author argues remains compatible with stellar cosmic-clock lower bounds.
Three of the six standard-model parameters (ΩΛ = 2/3, Ωk = 0, and Ωm by subtraction) are derived rather than fitted, making the dark-energy budget a prediction testable at any redshift.

Reading between the lines

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

Because heterogeneity enters through σ8, the framework implies a direct coupling between the amplitude of matter fluctuations and the present-day dark-energy density: a sharper measurement of σ8 would shift the predicted ΩΛ,het(0) and its uncertainty — a cross-check the paper does not spell out.
The 2/3 value arises as the pure geometric ratio of 1/4π to 3/8π, suggesting the same ratio might reappear in any scale-free self-gravitating system, which could be probed in simulations of volume formation rather than in cosmology alone.
The global time-dilation prediction — that events separated by the same redshift lapse a shorter cosmic time in the heterogeneous model — could be tested independently by comparing ages of old stellar populations against model-independent distance measures, giving a falsifiable age difference.

Original abstract

The volume dynamics of space is a framework in which the possibility that matter can be unevenly distributed enters the expansion dynamics directly. It yields a redshift-dependent effective Hubble constant and dark-energy density parameter from the Planck parameters alone. We derive the expansion history $H(z)$ of the framework and show that it is flat $\Lambda$CDM with the cosmological constant replaced by a dark-energy density that grows by 31 percent toward the present, with an effective equation of state $w(0) = -1.21$ approaching $-1$ at high redshift and a present-day age of 13.52 Gyr, 0.3 Gyr younger than in the homogeneous limit. We confront it with the DESI DR2 baryon acoustic oscillations and the Pantheon+ type Ia supernovae, with the Hubble constant, the sound horizon and the supernova absolute magnitude free, so that only the shape of $H(z)$ is tested. Relative to $\Lambda$CDM the prediction is disfavored by $\Delta\chi^2 = 65$ with one parameter fewer. A free matter density reduces this to $\Delta\chi^2 = 42$, because each probe alone accommodates the model only with incompatible matter densities (0.35 from the oscillations, 0.54 from the supernovae). Their combination bounds the heterogeneity amplitude to $\sigma_8 < 0.14$ at 95 percent confidence, against the Planck input 0.81; the best fit of the heterogeneous model is its homogeneous limit, flat $\Lambda$CDM. At the leading order currently evaluated, the heterogeneous contribution cannot account for the Hubble tension, and it acts opposite to the evolution of dark energy that current distance data favor.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-28: 1 request, 7,059 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.61 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Early- and late-Universe measurements of the Hubble constant disagree, a puzzle known as the Hubble tension. We derive cosmological parameters from first principles, using only the dynamics of cosmic volume and the possibility that matter can be unevenly distributed. Within this framework, three of the six parameters of the standard model are obtained from the underlying dynamics rather than fitted to observations: volume formation determines the dark-energy density parameter to be two thirds, spatial curvature vanishes, and the matter density follows from the remaining cosmic energy budget. The Hubble constant remains an observational input because it represents a calendar date rather than an intrinsic property of the Universe, while the two remaining parameters are likewise taken from observation. Once cosmic unevenness is included as a dynamical component, both the expansion rate and the dark-energy density parameter become redshift dependent. The latter increases by eight percent toward the present, from 0.679 in the homogeneous early Universe to 0.735 today, consistent with independent probes spanning several orders of magnitude in redshift, including the microwave background, supernovae, baryon acoustic oscillations, X-ray clusters and gravitational waves. This same mechanism yields the time-evolving dark energy favored by recent combined analyses without introducing a new free parameter. Integrating the resulting expansion history gives a cosmic age of 13.521 Gyr, shorter than the homogeneous value by 0.306 Gyr. This difference is a global time dilation: dark energy and time evolve together, shaped by the unevenness of the cosmos.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Early- and late-Universe measurements of the Hubble constant disagree, a puzzle known as the Hubble tension.	background 1.00	0.04	0.03	asserted 1.00
s1	We derive cosmological parameters from first principles, using only the dynamics of cosmic volume and the possibility that matter can be unevenly distributed.	method 0.84	0.96	0.07	asserted 0.95
s2	Within this framework, three of the six parameters of the standard model are obtained from the underlying dynamics rather than fitted to observations: volume formation determines the dark-energy density parameter to be two thirds, spatial curvature vanishes, and the matter density follows from the remaining cosmic energy budget.	result 0.94	0.92	0.31	asserted 1.00
s3	The Hubble constant remains an observational input because it represents a calendar date rather than an intrinsic property of the Universe, while the two remaining parameters are likewise taken from observation.	method 0.72	0.51	0.08	asserted 1.00
s4	Once cosmic unevenness is included as a dynamical component, both the expansion rate and the dark-energy density parameter become redshift dependent.	result 0.93	0.84	0.26	asserted 1.00
s5	The latter increases by eight percent toward the present, from 0.679 in the homogeneous early Universe to 0.735 today, consistent with independent probes spanning several orders of magnitude in redshift, including the microwave background, supernovae, baryon acoustic oscillations, X-ray clusters and gravitational waves.	result 1.00	0.61	0.25	asserted 0.75
s6	This same mechanism yields the time-evolving dark energy favored by recent combined analyses without introducing a new free parameter.	result 0.67	0.81	0.22	asserted 1.00
s7	Integrating the resulting expansion history gives a cosmic age of 13.521 Gyr, shorter than the homogeneous value by 0.306 Gyr.	result 1.00	0.91	0.12	asserted 1.00
s8	This difference is a global time dilation: dark energy and time evolve together, shaped by the unevenness of the cosmos.	interpretation 1.00	0.80	0.56	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Comparison of the theoretical Ω

**Figure 2.** FIG. 2. Comparison of the theoretical

**Figure 3.** FIG. 3. The global time difference ∆

**Figure 4.** FIG. 4. Paths of derivation

**Figure 5.** FIG. 5. The global time difference ∆

Figures reproduced from arXiv:2609.31543 by the authors.

References

Discussion

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