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The frozen vanilla model: Exploring dark sector interactions with delta effective theories

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A perturbation-only dark-sector interaction can shift structure growth while leaving the vanilla background untouched.

desk verdict A clean perturbation-only dark-sector interaction, but the headline ISW and fσ8 numbers need re-baselining against the wCDM model they claim to preserve. read the letter →

arxiv 2504.21110 v2 pith:7CNOVBEF submitted 2025-04-29 gr-qc astro-ph.COhep-th

Martín G. Richarte , Luiz F. Guimarães , Susana J. Landau , Júlio C. Fabris This is my paper · ORCID
classification gr-qcastro-ph.COhep-th MSC 83F0585A40
keywords darksectorinteractionsperturbation-levelcouplingfrozenvanillamodelintegratedSachs-Wolfeeffectfσ8growthestimatorS8tensionmatterpowerspectrumwCDMbackground
topics Dark Energy
open problems Dark MatterDark 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 paper proposes a new way to couple dark matter and dark energy: the energy exchange is switched off in the smooth background and activated only in the perturbations, so the expansion history stays identical to the vanilla wCDM model. The authors argue that this frozen mechanism still leaves measurable imprints: it can enhance or suppress the late integrated Sachs-Wolfe effect, alter the nonlinear matter power spectrum, and shift the growth estimator $f\sigma_8(z)$ by up to roughly 15 percent at low redshift, which in some branches lowers $S_8$ and eases the reported $S_8$ tension. A sympathetic reader would care because the model offers a concrete route by which dark-sector interactions could hide from geometric probes while revealing themselves in structure formation, without triggering the large-scale instabilities that plague many interacting dark-energy models.

What carries the argument

The load-bearing object is the perturbative delta-effective interaction of Eq. (36): $\delta Q_x = -\Sigma(3H \bar{\rho}_x)(K + \delta_x)$ with $K = h'/2 + \theta_T/(3H)$, together with the compensating $\delta Q_{\rm dm} = -\delta Q_x$. It acts only in the continuity equations for dark matter and dark energy, while the Euler equations remain unchanged and the dark-matter frame can still have zero initial velocity. Scale dependence enters through $\delta_x$, $h'$, and $\theta_x$, making the coupling efficient at small scales during matter domination and at intermediate scales only near dark-energy domination. The paper pairs this with $c^2_{sx}=1$ and $c^2_{ax}=w_x$, classifies models into four branches by the signs of $(1+w_x)$ and $\Sigma$, and evolves the system with a modified Boltzmann code.

What would settle it

Run N-body simulations with the frozen model's modified continuity equations, keeping the same background and scale-dependent $\delta Q$, and compare the $z=0$ nonlinear matter power spectrum at $0.1 < k < 10\ h/{\rm Mpc}$ with the HALOFIT-based predictions; discrepancies beyond the expected percent-level would invalidate the small-scale claims.

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Extended reading notes

Core claim

The central discovery is a covariant interaction vector confined to the perturbative sector, specified in the synchronous gauge by $\delta Q_x = -\Sigma(3H \bar{\rho}_x)(K + \delta_x)$, with $K = h'/2 + \theta_T/(3H)$, and the compensating source $\delta Q_{\rm dm} = -\delta Q_x$. Because the background energy-momentum conservation is untouched, the Friedmann expansion is fixed to the vanilla/CPL model with a constant equation of state $w_x$. The paper claims this mechanism is stable, with dark-energy pressure perturbations that remain bounded, a curvature perturbation $\zeta$ conserved on super-Hubble scales, and the usual background-level "doom factor" instability absent, while still generating distinctive late-time signatures: the ISW amplitude can change by tens of percent, the matter power spectrum is suppressed or enhanced depending on the sign of $\Sigma$, and $f\sigma_8(z)$ changes by up to 15 percent at low redshift, with $\sigma_8(z=0)$ in branch I falling in $[0.771, 0.800]$ and thereby alleviating the $S_8$ tension.

Load-bearing premise

The nonlinear matter power spectrum is computed with HALOFIT, a fitting formula calibrated to standard $\Lambda$CDM simulations, but the frozen model's growth and dark-sector interaction are scale-dependent, so if that calibration does not transfer, the small-scale $P(k)$ deviations and conclusions drawn from them are unsupported.

Editorial extensions

If this is right

  • Geometric and expansion probes remain consistent with vanilla wCDM by construction, so the model does not address the Hubble tension and directs attention to growth-based observables.
  • The late ISW effect can be enhanced by more than 10 percent for strong couplings in branch I, or suppressed in the reversed-sign branches, making CMB temperature cross-correlations with galaxy or quasar catalogues a direct test.
  • $f\sigma_8(z)$ is the most sensitive observable: branch I produces lower clustering than $\Lambda$CDM with relative differences up to roughly 15 percent at low redshift.
  • In branch I, $\sigma_8(z=0)$ falls in the range $[0.771, 0.800]$ as $\Sigma$ varies, which shifts the model toward the lower $S_8$ values preferred by cosmic-shear surveys and alleviates the $S_8$ tension.
  • No large-scale instabilities appear in the numerical results: dark-energy pressure and curvature perturbations stay bounded, so the model is a viable minimal extension for future Bayesian comparison against $\Lambda$CDM.

Reading between the lines

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

  • If the frozen mechanism is correct, the $S_8$ tension can be addressed without changing the expansion history or early-universe parameters, isolating the discrepancy to late-time growth and giving surveys a sharper target than models that alter both.
  • The scale dependence of the interaction implies a weakly scale-dependent growth rate in the observable window $0<z<2$; detecting such scale dependence in redshift-space-distortion or weak-lensing statistics would be a distinguishing signature.
  • A natural extension would be to replace the sharp activation of the interaction with a smooth time-localized coupling and check whether super-Hubble conservation of $\zeta$ survives; the paper's Lagrangian sketch points in that direction but does not develop it.
  • The HALOFIT-based nonlinear predictions should be checked against dedicated N-body simulations before small-scale power-spectrum deviations are used as evidence, because the fitting formula is calibrated to standard $\Lambda$CDM simulations rather than to scale-dependent growth.
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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

3 major / 5 minor

Summary. The paper proposes a 'frozen vanilla model' in which dark matter and dark energy exchange energy only at the level of cosmological perturbations, through δQ_x = -Σ(3Hρ̄_x)(K+δ_x) with K = (h'/2 + θ_T)/(3H), while the wCDM background is left unchanged. The authors derive the coupled perturbation equations and radiation-era initial conditions, implement them in a modified CLASS code, and explore four sign branches of (1+w_x) and Σ. They report that the interaction modifies the late ISW amplitude, the nonlinear matter power spectrum, fσ8(z), and S8(z) by up to about 15% relative to ΛCDM, and that curvature and dark-energy pressure perturbations remain bounded, suggesting the absence of large-scale instabilities. The central claim is that the interaction is hidden from geometric and background probes while leaving observable growth and ISW signatures.

Significance. If the findings hold, the model provides a concrete way for dark-sector energy exchange to escape background and geometric constraints while producing late-time ISW, P(k), fσ8, and S8 signatures, and it offers a possible route to alleviating the S8 tension. The linear perturbation derivation in Secs. III-V is internally consistent, the parameter choices are explicit, and the numerical results are forward predictions rather than fits to target observables. The use of a modified public CLASS code aids reproducibility, and the distinction from momentum-transfer models is clearly stated: here the continuity equations are modified while the Euler equations remain unchanged. The main quantitative claims, however, rest on two load-bearing assumptions: the choice of ΛCDM as the reference baseline and the transfer of HALOFIT calibration to a model with scale-dependent growth. Both need to be addressed before the central claim can be regarded as established.

major comments (3)
  1. [Sec. VI.C.2, Sec. VI.C.4, Figs. 5 and 7] All reported ISW enhancements and fσ8 deviations are quoted relative to ΛCDM, not relative to the wCDM model with the same w_x and Σ=0 that the frozen model is designed to preserve. The manuscript itself notes in Sec. VI.C.1 that approximately 20% of the CMB TT/EE difference from ΛCDM arises solely from the difference in w_x, independently of Σ. Since the frozen model's background is wCDM, the genuine interaction fingerprint is the difference between the frozen model and the w_x-identical, Σ=0 baseline. Without that decomposition, the reported 'more than 10%' ISW enhancement and 'up to 15%' fσ8 changes cannot be attributed to the dark-sector interaction. Please recompute the percentage differences against the Σ=0 wCDM baseline and state explicitly which part of each deviation is due to w_x and which to Σ.
  2. [Sec. VI.C.3, Fig. 6] The nonlinear matter power spectrum at z=0 is computed with the HALOFIT routine, which is calibrated to standard ΛCDM N-body simulations. The frozen model exhibits scale-dependent growth and a scale-dependent interaction in Eq. (36), and the paper provides no N-body or mock-calibration test showing that the halo-model parameters transfer to this scenario. Therefore the claimed small-scale P(k) deviations of about 10%, and any conclusions drawn from them about nonlinear structure formation, are unsupported. I recommend either restricting the P(k) claims to the linear regime or validating the nonlinear prescription with N-body simulations before presenting these deviations as model predictions.
  3. [Appendix A] Appendix A sketches a Lagrangian Lδ−EFT but does not derive δQ_x = -Σ(3Hρ̄_x)(K+δ_x); it states only that mixed terms involving u^μ and ∇_μ δφ_A should be included and that a full Lagrangian model will appear in future work. Thus Eq. (36) is currently a phenomenological ansatz rather than a derived effective-theory operator. This is a limitation rather than an internal inconsistency, but it should be acknowledged in the abstract or introduction, and the strength of the 'new mechanism' claim should be calibrated accordingly.
minor comments (5)
  1. [Eq. (46)] Equation (46) writes the ISW temperature shift as proportional to (2/c^2)∫dτ Ψ̇, but the text and figures refer to the derivative of (Ψ+Φ); the standard linear ISW contribution in the Newtonian gauge involves both metric potentials. Please correct the formula.
  2. [Sec. VI.C.5] The text labels the dark-energy pressure perturbation as 'δpx (24)', but Eq. (24) is the density-contrast equation; the pressure perturbation is defined in Eq. (56). Please correct the cross-reference.
  3. [Fig. 9 and Sec. VI.D] The caption of Fig. 9 lists repeated parameter combinations (for example, w_x=-1.06 with the same Σ value appears three times), and the text in Sec. VI.D states that Σ=0.1 is impossible for w_x=-1.06. The legend should be cleaned and made consistent with the stability statement.
  4. [Sec. VI.B] The phrase 'we have numerically proven' overstates what is a numerical observation; please replace it with 'we find' or 'we observe numerically'.
  5. [Throughout] The model naming is inconsistent across figures and text: 'frozen-branch-I', 'Frozen-branch-I', and 'Branch I' are all used. Please unify the terminology.

Circularity Check

1 steps flagged · score 1.0 of 10

The ISW/P(k)/fsigma8/S8 outputs are genuine forward predictions with parameters scanned by hand; only the background-preservation and doom-factor-stability claims reduce to the model's defining ansatz.

  1. self definitional [Sec. III (after Eq. 12), Sec. IV, and Sec. VI.C.5]
    "The main difference from other proposals in the literature is that the exchange of energy takes place at the perturbative level QA = δQA, so ¯QA = 0. ... We will adopt a framework where ¯Qdm = 0 is at the background level, ensuring that the background evolution equation for the density parameter aligns with that of the vanilla model."

    The advertised advantage that all geometric probes remain consistent with the vanilla model is not derived from the perturbation equations; it is imposed by setting the background transfer ¯QA = 0, which makes the background evolution identical to wCDM by construction. The same input underlies the doom-factor stability remark: with ¯Qx = 0, dx = ¯Qx/[3H¯ρx(1+wx)] vanishes identically, so the known large-scale nonadiabatic instability of background-coupled dark energy is excluded by the ansatz rather than by an independent stability proof. The ISW, P(k), fσ8, and S8 numbers are, however, forward outputs of a modified CLASS run with Σ and wx scanned by hand, so they are not fitted to the predicted observables and the central claim retains independent content.

full rationale

The model's defining property—an interaction active only at the perturbation level—is an input, and the paper is transparent that the background is the vanilla/wCDM model. The abstract's 'geometric probes consistent with vanilla' advantage follows directly from ¯QA=0 and is therefore self-definitional rather than a tested prediction; similarly, the doom-factor stability argument reduces to the same zero-background-coupling choice. However, none of the headline observables (ISW enhancement/suppression, matter power spectrum deviations, fσ8 and S8 changes) is used to fit Σ or wx: those parameters are scanned by hand and the spectra are computed with a modified CLASS code as forward predictions. The self-citations in the reference list are contextual and not load-bearing; the cited K-term perturbation is external ([113]). The HALOFIT calibration concern is an external-validity risk, not circularity. The comparison baseline for quoted percentages is ΛCDM rather than a wx-identical Σ=0 wCDM, which can make the interaction's marginal effect harder to isolate, but the paper does identify the wx-only contribution explicitly in Sec. VI.C.1. Overall, the central derivation is self-contained; the only definitional reductions are the background-preservation and doom-factor-stability statements, so the circularity score is 1.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central additions are one dimensionless coupling Σ, a fixed dark energy sound speed, and a perturbation-only interaction form. The most fragile postulate is that a ΛCDM-calibrated HALOFIT prescription remains valid for this scale-dependent growth model.

free parameters (4)
  • Σ (dark sector perturbation coupling) = ±10^-3, ±10^-2, ±10^-1 (scanned)
    Free coupling in δQ_x; scanned by hand over three orders of magnitude, not fitted to data.
  • w_x (dark energy equation of state) = -0.9, -0.8, -1.06, -1.1
    Chosen to represent non-phantom and phantom cases consistent with prior constraints, not fitted here.
  • c_sx^2 (dark energy sound speed) = 1
    Fixed by hand for all runs; a free parameter of the perturbation model.
  • Planck 2018 best-fit cosmological parameters = h=0.6736, τ_reio=0.0544, Ω_bh2=0.02237, Ω_cdmh2=0.1200, As=2.1e-9, ns=0.9649, r=0.055
    Used to set the frozen background; external inputs from Planck, not fitted in this work and not re-derived for w_x different from -1.
assumptions (6)
  • standard math Flat FLRW background with GR and standard components (radiation, baryons, CDM, DE).
    Basis of the Friedmann equations in Sec. II.
  • ad hoc to paper The dark sector interaction four-vector has zero background part and is Q_A = δQ_A u^mu with δQ_x = -Σ(3H ρbar_x)(K + δ_x).
    Defines the frozen model at Eq. (36); only a sketch of a Lagrangian is given in Appendix A.
  • domain assumption Dark energy is a fluid with constant equation of state w_x and sound speed c_sx = 1.
    Fixed in Sec. VI.A; c_sx=1 is a standard but not mandatory choice.
  • domain assumption No momentum transfer in the dark sector, so Euler equations for DM and DE are unchanged.
    Eqs. (33)-(35) set the momentum transfer f_A to zero, following the usual avoidance of fifth forces.
  • domain assumption Initial conditions are the radiation-era attractor solutions with δdm = h/2 and no growing interaction mode.
    Sec. V assumes Σ R_I << 1 in the radiation era to justify the attractor.
  • ad hoc to paper HALOFIT calibration from ΛCDM N-body simulations applies to the frozen model's scale-dependent growth.
    Used in Sec. VI.C.3 to compute nonlinear P(k); no N-body validation is provided.

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Cite this review

Pith. "Pith review of The frozen vanilla model: Exploring dark sector interactions with delta effective theories." pith.science (2026). https://pith.science/paper/7CNOVBEF

@misc{pith2026250421110,
  author       = {Pith},
  title        = {Pith review of: The frozen vanilla model: Exploring dark sector interactions with delta effective theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CNOVBEF}},
  note         = {Machine review of arXiv:2504.21110}
}
abstract

In this paper, we introduce a new interacting mechanism within the dark sector, encompassing both dark energy and dark matter, while grounding our analysis in the familiar framework of the $\mathrm{\Lambda CDM}$ model augmented by baryons and radiation components, including photons and neutrinos. The interaction between dark energy and dark matter is confined to the perturbative level. One significant advantage of this proposal is that all geometric probes yield constraints consistent with the so-called vanilla model or the extended vanilla model, where dark energy has a constant equation of state, $w_{x}$. However, the introduction of this new interacting mechanism affects several theoretical signatures, involving contrast dark matter and dark energy densities. We perform an exploratory analysis of those effects in the CMB power spectra, matter-power spectra, and redshift space distortions. For instance, it allows for a decrease/increment in the integrated Sachs-Wolfe (ISW) effect depending on the value taken by the interaction coupling. This effect could be observationally detected by looking for a cross-correlation between the ISW temperature fluctuations and the distribution of galaxies or quasars. At late times, the interaction in the dark sector becomes very effective, affecting the non-linear scale of structure formation. We discuss how the estimators $f\sigma_{8}(z)$ and $S_{8}(z)$ are affected by different interacting couplings; indicating that $f\sigma_{8}(z)$ can show a relative change of up to $15\%$ compared to the concordance model at low redshifts. Finally, we show how the various terms in the dark energy pressure perturbation (both adiabatic and non-adiabatic) are relevant for different scales, demonstrating the absence of large-scale instabilities.

Figures

Figures reproduced from arXiv: 2504.21110 by the authors.

Figure 1
Figure 1. FIG. 1. The different extensions of the vanilla model under the perturbative ET approach are shown upon the value of the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The classification of models based on their position [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We examine the cosmological evolution of the absolute value of the interaction strength within the dark energy [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The CMB angular power spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Cosmological evolution for models in the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The CMB angular power spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Cosmological evolution for models in the [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The density contrasts for dark matter, baryons, their relative contrast, and the density contrast for photons for [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The density contrasts for dark matter, baryons, their relative contrast, and the density contrast for photons for [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p030_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p031_18.png]

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Forward citations

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