Little ado about everything II: an `emergent' dark energy from structure formation to rule cosmic tensions

Andrea Lapi; Balakrishna S. Haridasu; Carlo Baccigalupi; Luigi Danese; Lumen Boco; Marcos M. Cueli

REVIEW 1 major objections 5 minor 1 cited by

Little ado about everything II: an `emergent' dark energy from structure formation to rule cosmic tensions

T0 review · 1 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper argues that the ensemble average of a stochastic, structure-formation-driven cosmology is equivalent to a flat universe with an emergent dark energy, and that this mechanism alone can ease both the $H_0$ and $f\sigma_8$ tensions.

desk verdict A serious but conditional alternative to LambdaCDM: the emergent dark energy is a closure residual, and the tension claims hinge on an unvalidated mean-field approximation. read the letter →

arxiv 2502.05823 v2 pith:EMNTG65Y submitted 2025-02-09 astro-ph.CO

Andrea Lapi , Balakrishna S. Haridasu , Lumen Boco , Marcos M. Cueli , Carlo Baccigalupi , Luigi Danese This is my paper · ORCID

classification astro-ph.CO

keywords emergentdarkenergystochasticbackreactioncosmicaccelerationH0tensionfsigma8etaCDMmodelstructureformationbaryonacousticoscillations

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 claims that the $\eta$CDM model, a stochastic description of how cosmic-web structure formation proceeds in $10$\,--\,$50\,h^{-1}\,\mathrm{Mpc}$ patches, has an ensemble-averaged evolution that can be recast as a spatially flat Friedmann cosmology with a time-dependent 'emergent' dark energy and no cosmological constant. Because the emergent component's equation of state runs from $-1/3$ in the early universe to below $-1$ today and settles at $-1$ in the future, late-time acceleration starts at the right redshift and the coincidence problem is avoided: the present densities sit near a future attractor rather than at a temporary accident. The authors fit the model to BAO, supernova, growth-rate, and cosmic-chronometer data and find that one parameter set, $h_0\approx0.7$, $\Omega_{m,0}\approx0.4$, $\sigma_8\approx0.78$, simultaneously eases the $H_0$ tension and removes the $f\sigma_8$ tension while remaining consistent with CMB pre-recombination constraints. If right, this would mean dark energy is an emergent statistical effect of structure formation rather than a fundamental ingredient, which is why a general reader should care.

What carries the argument

The load-bearing object is the stochastic continuity equation $\dot{\rho}_m = -3H\rho_m + \zeta\rho_m (H/H_\star)^\alpha H_\star^{1/2}\eta(t)$, with $\eta$ a Gaussian white noise and $\zeta,\alpha$ controlling strength and redshift evolution; the noise is a stand-in for the complex gravitational processes forming the cosmic web. The mean dynamics come from the Kramers-Moyal drift of this system, which converts the multiplicative noise into a deterministic correction. Integrating these mean equations yields the emergent dark-energy density $\Omega_\eta = -\kappa/(h^2 a^2)\exp\big[\tfrac{3}{2}\zeta^2\int_0^\tau d\tau'\, h^{2\alpha}(\Omega_m+\Omega_\gamma)^2\big]$, the central identity that turns a small early curvature-like term into late-time acceleration. The companion object is the growth equation $\delta''+(2+H'/H)\delta' = \tfrac{3}{2}\Omega_m\delta$ solved on this mean background; it produces an effective growth index $\gamma(a)\approx0.55+0.15a$, which is how the model reconciles growth data with a CMB-level $\sigma_8$.

What would settle it

Compute the actual ensemble-averaged backreaction term $Q=\frac{2}{3}\langle\theta^2\rangle+2(\langle\omega^2\rangle-\langle\sigma^2\rangle)$ on $10$\,--\,$50\,h^{-1}\,\mathrm{Mpc}$ scales from numerical-relativity or high-resolution N-body simulations and compare its distribution and mean with the assumed white noise and Kramers-Moyal drift; a material mismatch would remove the late-time acceleration. A separate, cheaper check is to measure the growth index at $z\lesssim1$ precisely: if it stays at the $\Lambda$CDM value $0.55$ rather than rising toward $0.6$\,--\,$0.65$, the growth side of the model is falsified.

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Extended reading notes

Core claim

In $\eta$CDM, each large-scale patch evolves under a Friedmann equation with a multiplicative Stratonovich white-noise term in the continuity equation, so different patches follow different histories; the Universe's background is defined as the average over the patch ensemble. This paper shows that the Kramers-Moyal mean of that ensemble obeys an integrated constraint $\Omega_m+\Omega_\gamma+\Omega_\eta=1$, where $\Omega_\eta$ starts as a small negative-curvature term in the early universe and is amplified by the noise into a dominant component at late times. The corresponding equation of state, $w_\eta = -1/3 - (\zeta^2/2) h^{2\alpha-1}(1-\Omega_\eta)^2$, satisfies $w_\eta\le -1/3$ at all times, so it always drives acceleration, and the matter density effectively dilutes more slowly than $a^{-3}$, which the model interprets as apparent rather than intrinsic. In the infinite future the system reaches an attractor with $\Omega_{m,\infty}\simeq\Omega_{m,0}$ and $w_m\simeq w_\eta\simeq -1$, so the observed coincidence between matter and dark energy densities is not special. Fitting DESI BAO, Pantheon+ supernovae, growth-rate, and chronometer data yields $h_0\approx0.69$, $\Omega_{m,0}\approx0.41$, $\zeta\approx1.46$, $\alpha\approx-0.77$, $\sigma_8\approx0.78$; the paper argues these values relieve the $H_0$ tension because distances and BAO agree, and solve the $f\sigma_8$ tension because the predicted growth is slower, letting $\sigma_8$ stay CMB-consistent.

Load-bearing premise

The central claim rests on the assumption that the ensemble effect of structure formation on the expansion can be captured by a multiplicative Gaussian white noise plus a first-order drift correction, an assumption the paper admits was partly postulated in the companion paper; if the true ensemble-averaged backreaction is not of this form, the emergent dark energy and both tension claims change.

Editorial extensions

If this is right

Late-time acceleration would follow from structure formation itself, so no cosmological constant, scalar field, or modified gravity is required.
The $H_0$ tension would be alleviated: the same $h_0\approx0.7$ that fits BAO and supernova distances needs no distance-ladder recalibration.
The $f\sigma_8$ tension would be gone: growth data fit with $\sigma_8\approx0.78$, consistent with CMB expectations, because the growth index rises at low redshift.
The dark-energy equation of state would be time-dependent and can cross $w=-1$ without violating energy conditions, because it is an emergent description rather than a field.
The Linder diagnostic, the joint plot of $H(z)$ versus $f\sigma_8(z)$, would separate $\eta$CDM from $\Lambda$CDM most clearly at $z\lesssim1$, where future galaxy surveys could decide between them.

Reading between the lines

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

The paper does not say this, but its predicted late-time growth index $\gamma(a)\approx0.55+0.15a$ means $\eta$CDM would look like modified gravity in low-redshift growth data; a survey that pinned $\gamma$ at $z\lesssim1$ could tell the two explanations apart.
The paper leaves implicit that $\zeta$, the noise strength, is set by the variance of the peculiar-velocity divergence; direct velocity-field reconstructions on the relevant smoothing scale could therefore predict $\zeta$ instead of fitting it.
A natural stress test the paper does not perform is to replace the truncated Kramers-Moyal mean with the exact ensemble average of the stochastic system, or with the backreaction measured in numerical relativity simulations; if the exact drift differs materially, the quantitative claims about tensions would need revision.

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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, here is my read on arXiv:2502.05823. The paper reformulates the etaCDM model as a flat cosmology with an emergent dark energy component, adds cosmographic expressions, and fits the model to Pantheon+, DESI BAO, growth-rate, and cosmic chronometer data. The fits are good and the claimed simultaneous relief of the H0 and f_sigma8 tensions is worth taking seriously. The authors are also honest about what is deferred: full CMB and weak-lensing analyses are not done, and the comparisons in Sections 4.3 and 4.4 rely on a LambdaCDM rescaling of high-redshift quantities.

The soft spots are concentrated in the step from the stochastic equations to the deterministic ensemble mean. The derivation uses a first-order Kramers-Moyal truncation, and no variance estimate or convergence test is given. At the best-fit parameters the noise term in the Omega_m equation is comparable to the drift, so the fluctuations are not obviously small. The noise form itself (Eq. A.14) is still postulated; the paper admits the stochasticity was 'somewhat postulated' in the companion paper, and Appendix A gives a plausibility argument rather than a derivation. The emergent dark energy Omega_eta is defined as the closure residual 1 - Omega_m - Omega_gamma, so its existence is formally an identity; the physical content is carried by the fitted noise parameters zeta and alpha. The requirement of a negative early-time curvature seed is an extra assumption, though it can be arbitrarily small.

These are real limitations, but the paper does not hide them, and the internal math is consistent. The growth-rate equation is derived in Appendix B, and the model is tested with modern data. This is not a mature theory yet, but it is a serious phenomenological proposal. The claims about solving cosmic tensions, however, should not be accepted until two things happen: a direct simulation of the stochastic differential equations to check whether Eq. (2.3) is a good approximation to the true ensemble mean, and a full etaCDM likelihood analysis of CMB and weak-lensing data. Without those, the tension claims are conditional on an unvalidated closure.

I would send this to peer review rather than desk-reject it. The referee should ask for the SDE simulation and the full CMB analysis as conditions for acceptance. If the closure fails, the model collapses; if it holds, this is a useful alternative to LambdaCDM that deserves attention.

Referee Report

1 major / 5 minor

Summary. The paper presents the ηCDM model, in which the continuity equations for matter and radiation are promoted to stochastic differential equations with a multiplicative Stratonovich white-noise term (Eq. 2.1). The authors argue that the ensemble-averaged evolution can be approximated by the first-order Kramers-Moyal drift equations (Eq. 2.3), and that this average can be repackaged as a spatially flat cosmology with an 'emergent' dark energy component Ωη whose equation of state wη(z) evolves from -1/3 at early times to phantom-like values at z≈0 (Eqs. 2.5 and 2.7). They further claim that the model solves the coincidence problem via a future attractor, and they fit the parameters {h0, Ωm,0, ζ, α, rd, Mb, σ8} to DESI BAO, Pantheon+ supernovae, 18 growth-rate measurements, and 33 cosmic chronometers. The best fit (h0≈0.69, Ωm,0≈0.41, ζ≈1.46, α≈-0.77, σ8≈0.78) is reported as simultaneously alleviating the H0 and fσ8 tensions, and the paper provides cosmographic expansions and a Linder diagnostic as future discriminators.

Significance. If the stochastic closure and the noise model were validated, the paper would offer an interesting alternative to dark energy as an emergent, backreaction-driven phenomenon. The manuscript is transparent about its data analysis, provides analytic cosmographic expressions, and explicitly acknowledges several limitations, including the phenomenological origin of the noise and the deferred full CMB and weak-lensing analyses. However, the central physical claim is not yet established: the 'emergent' dark energy is defined as a closure residual of a fitted background, the noise term is a postulated phenomenological input, and the ensemble-average equations are obtained through a first-order Kramers-Moyal approximation whose accuracy is not demonstrated. The claimed resolution of the Hubble and fσ8 tensions consequently rests on an unvalidated mean-field closure and on consistency checks rather than on a full joint analysis with CMB data. The significance is therefore conditional on validation of the statistical model and on a more direct confrontation with high-redshift data.

major comments (1)

[Section 4.2, Figure 5] The Linder diagnostic is proposed as a decisive future test, but the current data shown in Figure 5 are binned with error bars that the authors acknowledge do not include systematics, especially for cosmic chronometers. The statement that ηCDM 'reproduces very well the binned data' while ΛCDM requires a lower σ8 is not backed by a quantitative model-comparison statistic on the binned points. If this diagnostic is to be a central claim, the authors should provide a quantitative significance for the difference between the two models, including systematic error bars. As it stands, the discussion is suggestive rather than demonstrative.

minor comments (5)

[Section 2.1] The text says 'at the hearth of the ηCDM model'; 'hearth' should be 'heart'.
[Appendix B] The phrase 'Finally, the redsfhit dependence of the growth index' contains a typo: 'redsfhit' should be 'redshift'.
[Abstract and Section 4.2] The diagnostic is attributed to 'Linders' in the abstract and Section 4.2, but the reference is to Linder (2017); the possessive should be 'Linder's diagnostic'.
[Section 3.1] The text introduces 'SGR' as an abbreviation for 'structure growth rate', but later uses 'growth rate data' and 'fσ8 data' interchangeably; please define the acronym at first use and use it consistently.
[Table 1] The fourth data row (SN only) has a very wide posterior for α (-0.90⁺⁰·⁷⁷₋₀·₃₉), and the fifth row reports σ8 with a 0.05 uncertainty; adding the derived S8 ≡ σ8(Ωm,0/0.3)^{1/2} in the table would facilitate comparison with the cosmic-shear literature.

Circularity Check

2 steps flagged · score 4.0 of 10

The 'emergent dark energy' is defined as the closure residual by Eq. (2.5), and the 'coincidence solution' is imposed as a prior via Omega_m0 < 2/(4-alpha); the remaining claims depend on a postulated noise form, though the fits to external data are genuine.

fitted input called prediction [Section 3.1 (Data and Analysis) and Section 2.4 (Fate of the Universe and coincidence problem)]
"The bound Omega_m,0 < 2/(4 - alpha) is also set to ensure late-time physical solutions solving the cosmic coincidence problem (see Section 2.4). ... it is easy to check that for alpha >= -1 and zeta ~ 1 the cosmological parameters will hover around values similar to the current ones for an infinite amount of time in the future ... the cosmic coincidence is solved without recurring to anthropic considerations."

The attractor that 'solves' the coincidence problem exists only under the condition Omega_m < 2/(4-alpha). Rather than being derived from the data or from an independent ab initio calculation, this condition is imposed as a hard bound in the parameter fit. The posterior therefore automatically satisfies the condition, and the paper presents this guaranteed constraint as a solved physical problem. The solution is an input enforced by the prior, not a predicted output of the model.
self definitional [Section 2.2, Eq. (2.5)]
"Omega_eta = -kappa/h^2 a^2 e^{3/2 zeta^2 int dtau h^{2alpha} (Omega_m+Omega_gamma)^2} = 1 - Omega_m - Omega_gamma ... In terms of the latter, the ensemble-average Universe is formally equivalent to a flat cosmology with Omega_m + Omega_gamma + Omega_eta = 1."

The 'emergent dark energy' Omega_eta is defined as the closure residual 1 - Omega_m - Omega_gamma. Therefore its existence as a component in a flat decomposition is an identity, true for any background, rather than an empirical detection. The nontrivial content is the noise-induced equation of state, but that content depends on the free parameters zeta and alpha, which are fitted to the same datasets used to claim success. The naming of the residual as emergent dark energy is thus a re-description of the fitted dynamics rather than an independent prediction.

full rationale

The paper is not blatantly circular overall: the background evolution, the derived equation of state, and the growth-rate predictions are genuine outputs of the assumed stochastic system, and they are tested against external Pantheon+, DESI, growth-rate, and cosmic-chronometer data. However, two central claims have a definitional or fitted character. First, Eq. (2.5) defines Omega_eta as exactly 1 - Omega_m - Omega_gamma, so the existence of an 'emergent dark energy' component is an identity; the physical content is carried by the fitted noise parameters. Second, the claimed solution to the coincidence problem is effectively guaranteed by the prior Omega_m,0 < 2/(4-alpha), which is imposed to ensure the attractor, rather than being a derived consequence. The noise form itself, Eq. (A.14), is admitted to have been 'somewhat postulated' in the companion paper and is only motivated, not derived, in Appendix A; this is a correctness and validation risk rather than a circular reduction. The H0 and f_sigma8 alleviation claims are model-dependent fits, not out-of-sample predictions, but they are not forced by construction because the growth equation and background are nontrivial functions of the model. Score 4 reflects partial circularity in the 'emergent dark energy' and 'coincidence' claims, while the overall framework retains independent empirical content.

Assumptions & free parameters 7 free parameters · 6 assumptions · 2 invented entities

The central claim rests on a phenomenological stochastic noise with fitted parameters, a first-order ensemble-averaging approximation, and a negative early curvature seed. The emergent dark energy is a derived quantity rather than an independent component. No machine-checked proof or independent benchmark is provided, so the ledger documents the unverified assumptions and fitted inputs the reader must accept.

free parameters (7)

h0 = 0.69 (joint fit)
Normalized Hubble constant, fitted to BAO+SN+growth+CC data.
Omega_m0 = 0.41 (joint fit)
Present matter density parameter, fitted.
zeta = 1.46 (joint fit)
Noise strength, fitted; controls the amplitude of the stochastic term and hence the emergent dark energy.
alpha = -0.77 (joint fit)
Noise redshift evolution index, fitted; controls when the stochasticity becomes important.
rd = 147.1 Mpc (with prior)
Comoving sound horizon at drag epoch, fitted with a Gaussian prior from Planck LambdaCDM.
Mb = -19.38 mag (with prior)
Type-Ia supernova absolute magnitude, fitted with a prior from the distance ladder.
sigma8 = 0.78 (joint fit)
Linear matter power spectrum amplitude, fitted to growth rate data.

assumptions (6)

ad hoc to paper The stochastic evolution of patches is described by Stratonovich Gaussian white noise with the multiplicative form in Eq. (2.1).
This is the core of the model. Appendix A gives only a heuristic justification relating Q to <theta^2> and assuming Q ~ <theta^2>_* H^{alpha+1} eta(t), so the exact noise scaling is an assumption.
ad hoc to paper The ensemble average can be approximated by the first-order Kramers-Moyal drift equations (2.3).
This truncation is not rigorously justified in the paper and underlies all subsequent results.
ad hoc to paper The early-time spatial curvature constant kappa is negative, though it can be arbitrarily small.
Section 2.2 states that if kappa = 0, then Omega_eta = 0 and no emergent dark energy is produced; the negative sign and smallness are required to match data.
domain assumption Pre-recombination physics is identical to LambdaCDM, so Planck-derived values of rd, theta*, and r*/rd can be imposed as priors.
Section 3.1 uses these priors to anchor BAO and the CMB first-peak scale; if etaCDM modified early physics, these priors would be invalid.
domain assumption The ensemble-averaged Universe is described by a single FLRW metric with the standard redshift-distance relation.
Section 2.3 footnote 4 adopts this as a 'reasonable approximation'; the stochastic patch structure could affect observations in ways not captured by a single metric.
ad hoc to paper The noise terms for matter and radiation share the same parameters zeta and alpha.
Section 2.1 states this is a simplifying assumption; the paper argues it is irrelevant because radiation is negligible when the noise is important.

invented entities (2)

Multiplicative noise term (the 'little ado')
purpose: To represent stochastic patch-to-patch variations in the expansion due to structure formation and to drive the ensemble average toward acceleration.
The noise is not derived from first principles; it is a phenomenological term chosen to reproduce the observed lognormal density distribution, and its parameters are fitted.
Emergent dark energy Omega_eta
purpose: To reinterpret the ensemble-averaged acceleration as a time-dependent dark energy component in a flat FLRW model.
Omega_eta is constructed as the closure residual 1 - Omega_m - Omega_gamma (Eq. 2.5); it is an effective quantity with no independent observable handle.

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Cite this review

Pith. "Pith review of Little ado about everything II: an `emergent' dark energy from structure formation to rule cosmic tensions." pith.science (2026). https://pith.science/paper/EMNTG65Y

@misc{pith2026250205823,
  author       = {Pith},
  title        = {Pith review of: Little ado about everything II: an `emergent' dark energy from structure formation to rule cosmic tensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMNTG65Y}},
  note         = {Machine review of arXiv:2502.05823}
}

abstract

[abridged] The $\eta$CDM framework is a new cosmological model aimed to cure some drawbacks of the standard $\Lambda$CDM scenario, such as the origin of the accelerated expansion at late times, the cosmic tensions, and the violation of the cosmological principle due to the progressive development of inhomogeneous/anisotropic conditions in the Universe during structure formation. To this purpose, the model adopts a statistical perspective envisaging a stochastic evolution of large-scale patches in the Universe with typical sizes $10-50\, h^{-1}$ Mpc, which is meant to describe the complex gravitational processes leading to the formation of the cosmic web. The stochasticity among different patches is technically rendered via the diverse realizations of a multiplicative noise term (`a little ado') in the cosmological equations, and the overall background evolution of the Universe is then operationally defined as an average over the patch ensemble. In this paper we show that such an ensemble-averaged evolution in $\eta$CDM can be described in terms of a spatially flat cosmology and of an `emergent' dark energy with a time-dependent equation of state, able to originate the cosmic acceleration with the right timing and to solve the coincidence problem. Then we test the $\eta$CDM model against the most recent supernova type-I$a$, baryon acoustic oscillations and structure growth rate datasets, finding an excellent agreement. Remarkably, we demonstrate that $\eta$CDM is able to alleviate simultaneously both the $H_0$ and the $f\sigma_8$ tensions. Finally, we discuss that the Linders' diagnostic test could be helpful to better distinguish $\eta$CDM from the standard scenario in the near future via upcoming galaxy redshift surveys at intermediate redshifts such as those being conducted by the Euclid mission.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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