Scalar-tensor theories at different scales: averaging the scalar sector

Hugo L\'evy; Jean-Philippe Uzan

Scalar-tensor theories at different scales: averaging the scalar sector

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that smoothing the matter distribution before solving the Klein–Gordon equation mis-estimates the coarse-grained scalar field's energy density, pressure, and equation of state, with errors exceeding five orders of…

desk verdict A solid toy-model proof that averaging matter before solving the scalar field equation is wrong by orders of magnitude for screened chameleons; the central claim is secure, the quantitative numbers are model-dependent, and the paper is honest about that. read the letter →

arxiv 2505.03909 v1 pith:ATPQG76J submitted 2025-05-06 gr-qc astro-ph.CO

Jean-Philippe Uzan , Hugo Lévy This is my paper · ORCID

classification gr-qcastro-ph.CO

keywords scalar-tensortheoryaveragingproblemKlein-GordonequationchameleonscreeningYukawafieldbackreactionofstateextendedquintessence

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 asks whether the standard cosmological shortcut—smooth the matter distribution first, then solve the scalar field's Klein–Gordon equation—gives the same answer as solving the true small-scale field and then averaging. For a linear (Yukawa) theory it does for the mean field value, but not for the field's energy density, pressure, or equation of state. For a non-linear (chameleon) theory even the mean field value is wrong whenever the matter source is screened, with $\delta\varphi$ exceeding $10^5$ in the computed parameter space. These results are established on a toy model in which matter is a regular lattice of identical homogeneous spheres, and they imply that homogeneous-fluid treatments of coupled dark energy and of laboratory fifth-force searches need to be revisited.

What carries the argument

The machinery is a two-scale toy model plus two averaging identities. Matter is described both as a homogeneous fluid of density $\rho_{\rm macro}$ and as a regular lattice of identical homogeneous spheres of radius $R$ and density $\rho_0$ in vacuum, with lattice spacing $L_0$ identified with the smoothing scale and $\rho_{\rm macro}=(4\pi/3)(R/L_0)^3\rho_0$. For each configuration one defines $\delta_\varphi$, $\delta_\rho$, $\delta_P$ as the ratios of coarse-grained field value, energy density, and pressure to their homogeneous-fluid counterparts. The argument runs on identities derived from the divergence theorem: for the Yukawa model $\int_\Omega \Delta\varphi\,d^3x=0$ under periodic boundary conditions gives $\langle\varphi\rangle=\varphi_{\rm macro}$, while for the chameleon model the same manipulation gives $\langle\varphi^{-(n+1)}\rangle=\varphi_{\rm macro}^{-(n+1)}$. The single dimensionless parameter $\tilde\lambda=\lambda/L_0$, the ratio of the field's Compton wavelength to the smoothing scale, controls the transition between unscreened ($\tilde\lambda\gtrsim 1$, commutation restored) and screened behavior.

What would settle it

Solve the chameleon Klein–Gordon equation on a realistic small-scale density field with the same mean density as the lattice (for example, a cosmological simulation box with halos and filaments, or a laboratory gas with measured particle positions) and compare $\langle\varphi\rangle$ with $\varphi(\langle\rho\rangle)$: if $\delta_\varphi$ remains of order one in the screened regime for these geometries, the toy model's claim of $\delta_\varphi\gg 1$ is not generic.

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Extended reading notes

Core claim

The central claim is that averaging and field solving do not generally commute in scalar-tensor theories: linearity of the Klein–Gordon equation guarantees commutation for the mean field, but non-linear theories can break it, and the paper shows this happens for chameleon models precisely in the screened regime. The argument compares two descriptions of the same mean density: a macroscopic homogeneous fluid whose field value is $\varphi_{\rm macro}$, and a microscopic lattice of identical spheres whose true field distribution is obtained numerically. For a Yukawa field, linearity plus periodic boundary conditions forces $\langle \varphi\rangle_{L_0}=\varphi_{\rm macro}$, so $\delta_\varphi=1$ always, but the energy density and pressure, being quadratic in $\varphi$, depend on how the mass is arranged and give an effective equation of state $W=-\delta_\rho/\delta_P$ that deviates from $-1$ when the Compton wavelength is below the smoothing scale. For a chameleon model the analogous identity is $\langle \varphi^{-(n+1)}\rangle_{L_0}=\varphi_{\rm macro}^{-(n+1)}$, which leaves $\delta_\varphi\neq 1$; in the screened regime the paper finds $\delta_\varphi$ above $10^5$ and derives the analytic approximation (Eqs. 21–22). In the unscreened regime, $\delta_\varphi=\delta_\rho=\delta_P=1$ even for the non-linear theory, and deep in either regime the equation of state returns to $-1$, with the largest deviations at intermediate screening.

Load-bearing premise

All quantitative results rest on the toy description of matter as a regular lattice of identical homogeneous spheres in vacuum, with the lattice spacing set equal to the smoothing scale; if real small-scale structure has a different geometry or a range of scales, the specific numbers—though probably not the qualitative failure of commutation—would change.

Editorial extensions

If this is right

In Yukawa-type theories the mean scalar field is unbiased ($\delta_\varphi=1$), yet the coarse-grained energy density, pressure, and effective equation of state depend on the sub-grid arrangement of matter whenever the Compton wavelength is smaller than the smoothing scale.
In chameleon-type theories, once the matter source is screened, the mean field itself is biased, with $\delta_\varphi$ exceeding $10^5$ for the dilute-gas and cosmological parameters considered, so the homogeneous-fluid approximation fails in those regimes.
In the unscreened regime ($\tilde\lambda\gtrsim 1$), averaging commutes with the nonlinear field equation and $\delta_\varphi=\delta_\rho=\delta_P=1$, which justifies the standard treatment only for sufficiently light fields.
For extended quintessence and scalar fields coupled to dark matter, the equation of state inferred from a smoothed matter distribution can differ from $W=-1$, with the largest deviations at intermediate screening; late-time cosmological predictions are therefore sensitive to how the matter is clumped.
For laboratory searches, the common assumption that the field relaxes to $\varphi_{\min}(\rho_{\rm space})$ in the surrounding medium is only valid when $\delta_\varphi=1$, so vacuum-chamber and space-based fifth-force constraints may need to be rederived with proper small-scale boundary conditions.

Reading between the lines

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

If the lattice result carries over to realistic cosmic structure, standard coupled-dark-energy (extended quintessence) forecasts may miss order-of-magnitude corrections to the late-time field energy density; the natural next step is a chameleon Klein–Gordon solve on a realistic N-body density field with the same mean density, comparing $\langle\varphi\rangle$ with $\varphi(\langle\rho\rangle)$.
The relation $\langle\varphi^{-(n+1)}\rangle=\varphi_{\rm macro}^{-(n+1)}$ is a distinctive non-linear signature: measuring the distribution of $\varphi$ in a dilute gas while changing particle spacing at fixed mean density would test whether chameleon screening really produces the predicted $\delta_\varphi$ enhancement.
The static-lattice restriction suggests the effect could be time-dependent: in an expanding universe the screened-to-unscreened transition moves with the density, so the bias in $\delta_\varphi$ would be redshift-dependent; extending the calculation to a background FLRW spacetime would show whether the effect is strongest at late times, as the paper speculates.

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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 — read it. The one thing to know: the central claim is right and worth remembering. For nonlinear scalar-tensor fields, smoothing matter first and then solving the Klein–Gordon equation is not the same as solving the true field and then averaging. The exact identity (20), <φ^{-(n+1)}> = <ρ> for chameleons, is a clean formal result that makes the non-commutation obvious in a way I hadn't seen stated before. The analytic screened-regime approximation for δφ (Eqs. 21–22) is genuinely new, and it agrees with the FEM numerics remarkably well. The Yukawa half is less new — δφ = 1 from linearity, and the binding-energy interpretation goes back to Ref. [45] — but the paper is honest about that, and it usefully collects the δρ, δP, and W behavior in one place.

The soft spots are real but mostly mechanical. The chameleon exponent n is never stated for the figures; the FEM results have no convergence or error metrics; and the magnitudes in Figs. 6 and 8 depend on a lattice of identical homogeneous spheres with the smoothing scale set equal to the lattice spacing. That last point matters: for real small-scale structure — halos with density profiles, filaments, a spectrum of masses — the specific numbers will change. The authors concede this in Section V and repeatedly call it a toy model, so it is a scoping limitation rather than a hidden flaw. The extrapolation to cosmology is explicitly tentative, and the claim that the equation of state is untouched deep in both the screened and unscreened regimes is interesting but should be checked in a less symmetric setup. None of this undercuts the qualitative conclusion; it does mean the order-of-magnitude numbers should be cited as toy-model estimates, not predictions.

The citation pattern is sound. The paper points clearly to earlier work by Briddon, Clifton, and Fleury and to Fleury's Gauss's-law paper, and it acknowledges the prior macroscopic Compton-wavelength results of Mota and Shaw. There are no fitted parameters; the analytic approximations are derived, not tuned to match the numerics.

Verdict: seriously engage. This is a paper I would send to a referee. The formal core is exact, the numerics support the analytic work, and the implications for extended quintessence and laboratory fifth-force searches are concrete. I would want the missing n and the FEM convergence metrics fixed before publication, but the central message stands. For a reading group on screening mechanisms or backreaction, it is worth an hour, and I would cite it. Send it to peer review with minor-to-moderate revision expected.

Referee Report

0 major / 6 minor

Summary. This paper addresses the averaging problem in the scalar sector of scalar-tensor theories. It asks whether solving the Klein-Gordon equation with the smoothed matter source ρ̄ reproduces the coarse-grained field obtained by averaging the field computed from the microscopic source ρ. The setup is a periodic lattice of identical homogeneous spheres of radius R and density ρ0, with lattice spacing L0 taken as the smoothing scale (Section II.B). For Yukawa theories, linearity of the Klein-Gordon equation implies δφ=1 exactly, but the averaged energy density and pressure differ from their macroscopic values; the paper derives an analytic approximation (Eqs. 16-17) for δρ that reproduces the numerical plateau values 2.4×10^5, 2.4×10^2, and 8.8 for R̃=0.01, 0.1, and 0.3. For chameleon theories, the nonlinear Klein-Gordon equation yields the exact invariant ⟨φ^{-(n+1)}⟩=φ_macro^{-(n+1)} (Eq. 20), while δφ departs from unity in the screened regime, reaching values beyond 10^5 in Fig. 6; a thin-shell analytic approximation (Eqs. 21-22) agrees with the numerics. The paper concludes that smoothing matter before solving the field equation can mis-estimate the scalar field's averaged energy density, pressure, and equation of state, with consequences for extended quintessence and for fifth-force or atomic-transition experiments in low-density media. The authors explicitly restrict attention to static configurations and describe the lattice as a toy model (Section V).

Significance. If correct, the paper establishes a concrete and quantitatively large failure of the standard practice of replacing the matter distribution by its smoothed value before solving the scalar field equation. The central formal claims are robust: the Yukawa identity δφ=1 follows from the divergence theorem and periodic boundary conditions, and the chameleon invariant (20) is an exact consequence of the rescaled Klein-Gordon equation. The paper's strengths include closed-form Green's function solutions (Appendix D), analytic approximations that are checked rather than fitted against the finite-element numerics, and the explicit statement of the model's limitations. The static, regular, single-species lattice is an acknowledged idealization; it affects the quantitative cosmological extrapolation in Fig. 8 but not the non-commutation principle, which would survive in any screened configuration. The main open questions are the time-dependent generalization and the sensitivity of the quantitative values to realistic density profiles, both of which the paper identifies as future work.

minor comments (6)

[§IV.A, Figs. 5-8] The chameleon exponent n used in the numerical computations is not stated, although Eqs. (21)-(22) and the values of δφ in Fig. 6 depend on n. Please specify the value(s) of n and, if practical, the corresponding (Λ, β) mapping for each panel.
[Appendix C] No convergence or mesh-resolution tests are reported for the finite-element solutions; a brief statement of the mesh size and convergence criterion would allow the claimed agreement with the analytic plateaus to be assessed.
[Fig. 7 caption] The caption of Fig. 7 says 'The parameters are varied as in Fig. 3,' but Fig. 3 is the Yukawa case; the reference should be to the chameleon case, presumably Fig. 5.
[§IV.B] The sentence 'Still the equation of state is reduced but less sharply than for the Yukawa case' is unclear, since the surrounding discussion concerns the chameleon model and the comparison is not quantified; please rephrase or support it with a quantitative statement.
[Abstract and §III.A] The statement that in Yukawa theories 'all small-scale distributions of matter lead to field distributions with the same mean' is derived under periodic boundary conditions and equal mean density; this qualification should appear in the abstract or at the first use of the statement.
[§III.C] In the first bullet of Section III.C, the sentence 'This is indeed not the case otherwise' would be clearer as 'This is not the case when λ̃≲1.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central non-commutation result is derived from the field equations, and the quantitative estimates are closed-form or standard approximations checked against independent numerics.

full rationale

The paper's central claim is that averaging the matter distribution before solving a nonlinear Klein-Gordon equation does not commute with solving the equation, with chameleon models giving order-of-magnitude differences in the coarse-grained field and its energy density. This claim is established by the paper's own exact identity, Eq. (20), obtained by integrating the dimensionless KG equation over the period cell and using the periodic boundary conditions; no parameter is fitted to the target averaged field. The Yukawa results follow from the explicit Green's function solution, Eq. (14), and the analytic approximation, Eq. (17), which is then compared with, not fitted to, the numerical integrations shown in Fig. 4. For the chameleon model, the analytical approximation for delta-phi, Eqs. (21)-(22), is derived from a thin-shell ansatz attributed to Khoury-Weltman (Ref. [37]), an external source, and is validated against the FEM numerics in Fig. 6; the agreement is reported as confirmation rather than used as the definition of the result. The exact invariant <phi^{-(n+1)}> = <rho> is also independently checked in the numerics. The self-citations present, notably the femtoscope solver (Refs. [34-36]) and earlier applications such as the binding-energy interpretation (Ref. [45]), are infrastructural or contextual; the paper does not invoke a self-cited uniqueness theorem or require that an alternative be excluded on authority. The acknowledged simplifications, such as the static regular lattice of identical spheres and the choice L0 equal to the lattice spacing, are limitations on the quantitative extrapolation to realistic cosmologies, not circular reductions. No fitted parameter is renamed as a prediction, and no derived quantity is identical to an input by construction. The derivation chain is internally consistent and self-contained against the stated model assumptions.

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

No target quantities are fitted. The central outputs δφ, δρ, δP, W are derived from closed-form Green's-function integrals and an algebraic thin-shell model; the theory parameters (β, m, Λ, n) and geometric scales (R, L0, ρ0) are inputs scanned over. The FEM solver settings (relaxation parameter 0.4) are implementation details. The analytic formulas contain no constants adjusted to match the numerics.

assumptions (7)

domain assumption Static configuration and Minkowski background: the KG equation reduces to Δφ = V' + α(φ)ρ with time derivatives dropped (Eq. A8).
All numerical solutions and averaged quantities (Figs. 3-8) are static; cosmological implications are extrapolated with an explicit caveat that field dynamics are not solved.
domain assumption Non-relativistic matter: the source term in the KG equation is the matter density ρ (T ≈ -ρ), a standard weak-field approximation.
Used in Eq. (A8) and throughout; invalid for radiation-dominated epochs, but the paper targets late-time dust-like clustering.
domain assumption Lattice model: matter consists of identical homogeneous spheres in vacuum centered on a regular lattice of size L0, with mean density matching ρmacro (Section II.B, Eq. 10).
This is the defining toy model. All numbers, including the δρ plateau 3/(4πR̃³), depend on it. The paper acknowledges it is a simplifying description.
domain assumption Periodic boundary conditions on the unit cube (Eq. C1).
Used in the divergence-theorem argument for δφ=1 (Section III.A) and in the FEM implementation. It permits the conclusion that the mean of the Yukawa field equals ϕmacro.
domain assumption Potential and coupling restricted to V=Cφ^p and lnA=βφ/M_P (Eq. 3), covering Yukawa (p=2) and chameleon (p=-n).
The whole analysis is for this two-parameter family; results do not generalize to arbitrary scalar-tensor potentials without further work.
ad hoc to paper Thin-shell approximation: in the screened regime the chameleon field is constant in the sphere and in the inter-particle space except a shell of width εR̃, with ε given by the Khoury-Weltman formula (Eq. E3).
Used to derive the analytic δφ (Eqs. 21-22). It is an approximation, not a solution of the full equation; validated against numerics in Fig. 6 for a subset of parameters.
ad hoc to paper Bi-valued field approximation (E4): φ̃ = 1 in the screened region, φ̃ = φ̃vac elsewhere.
This collapses the actual continuous field profile to a step function to compute δφ; the paper notes φ̃vac is a lower bound for the maximum field.

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Cite this review

Pith. "Pith review of Scalar-tensor theories at different scales: averaging the scalar sector." pith.science (2026). https://pith.science/paper/ATPQG76J

@misc{pith2026250503909,
  author       = {Pith},
  title        = {Pith review of: Scalar-tensor theories at different scales: averaging the scalar sector},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATPQG76J}},
  note         = {Machine review of arXiv:2505.03909}
}

read the original abstract

This article investigates the averaging of a scalar degree of freedom that couples universally to matter. It quantifies the approximation of smoothing the matter distribution before solving the Klein--Gordon equation. In the case of Yukawa theories, which enjoy a linear Klein--Gordon equation, the averaging commutes with the field equation as one might expect. While all small-scale distributions of matter lead to field distributions with the same mean, the latter can have different energy densities and pressures when the Compton wavelength of the field is smaller than the smoothing scale. In the non-linear case, such as chameleon theories, this study quantifies the error made by averaging the matter distribution before solving the Klein--Gordon equation. While field fluctuations can become arbitrarily large when the matter source is screened, the commutativity property of linear theories is recovered in the unscreened regime. Implications for cosmology -- and in particular the equation of state in extended quintessence models -- and for laboratory experiments in low density medium are discussed. This analysis, although based on a simplifying description, sheds light on the effects of the small-scale distribution of matter as well as on the care required to define their equation of state and their cosmological signature.

Figures

Figures reproduced from arXiv: 2505.03909 by the authors.

**Figure 2.** FIG. 2. Definition and relation of the microscopic and macro [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Yukawa field profile [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Dependency of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Chameleon field profiles [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Due to the non-linearities of the KG equations, chameleon models generically predict that [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Dependency of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Reviewed August 15, 2026 · model on record in the stance chip above.

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Chameleon cosmology

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On a large scales, the medium is ho- mogeneous and continuous so that it can be de- scribed as a fluid with constant density ρmacro

Macroscopic. On a large scales, the medium is ho- mogeneous and continuous so that it can be de- scribed as a fluid with constant density ρmacro. For a static configuration the scalar field has thus the constant value ϕmacro≡ϕmin(ρmacro), (9) that is indeed not necessarily strictly positive; see Eq. (6). TABLE I. Orders of magnitude of the physical featur...

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1− e− ˜R ˜λ ˜R ˜λ + 1 ! sinhu u # ≃ 3 4π ˜R3

Microscopic. On the local scale, the fluid is de- scribed by spherical particles of radius R with con- stant mass densityρ0 in vacuum. For simplicity, we assume that these particles occupy the centers of a regular lattice of size L0. For each representative elementary volume, the local matter distribution is thus given by ρ(x)≡ 0 if x /∈B (R) and ρ(x) = ρ...

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post-Newtonian approach to cosmological model- ing

In the cosmological context, the energy density and pressure of the scalar field smoothed on cosmologi- cal scales shall enter the Friedmann equation. This effect has been investigated in Ref. [44] in their “post-Newtonian approach to cosmological model- ing”, with which we share the same description of the microscopic distribution of matter. They con- cl...

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So far, we have used the Einstein-frame description but observations shall be analyzed in the Jordan frame [50, 51]

An additional point is worth emphasizing. So far, we have used the Einstein-frame description but observations shall be analyzed in the Jordan frame [50, 51]. It follows that aJF =A(ϕ)aEF and ρJF =A4(ϕ)ρEF, PJF =A4(ϕ)PEF respectively for the scale factor, energy density and pressure in both frames. Since the coupling A(ϕ) is not linear, it is then obvious...

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4, the equation of state remains larger than −1, W ≥− 1, and is still driving an acceleration since W <−1/3

As seen on Fig. 4, the equation of state remains larger than −1, W ≥− 1, and is still driving an acceleration since W <−1/3. Hence, one shall ex- pect the effect of the clustering to lead to a milder acceleration. Even in the case of a Yukawa model with a large mass compared to the Hubble param- eter, so that the macroscopic dynamics leads to a pure cosmo...

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This also the case in models [48, 49] in which the scalar field coupled simply to a dark matter component to avoid varia- tions of fundamental constants [52]

It follows that extra-care is needed for the inter- pretation of extended quintessence models even if the KG equation is linear and ¯ϕ = ϕmacro since the clumpiness of matter will be more important at small redshifts and hence modify the late time prediction of the observables. This also the case in models [48, 49] in which the scalar field coupled simply...

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When the system is unscreened on the smoothing scale L0, which is typically the case when ˜λ ≳ 1, one recovers δϕ = 1 despite the non-linerarities

work page