Nonlinear Scales and Screening

Screening allows an extra gravitational degree of freedom to affect diffuse cosmological structure while becoming weak in dense, deep-potential, or large-gradient environments. Verifying screening requires more than fitting a suppressed matter spectrum: the same theory and parameters must reproduce an unscreened limit, a spatially resolved transition tied to environment, and a screened local limit. Numerical resolution, baryonic feedback, halo finding, and the effective-theory cutoff can otherwise imitate the transition.

Required background. Probability and Statistical Inference supplies emulator uncertainty, simulation covariance, selection functions, and hierarchical comparison of environments.

Helpful background. Weak-Field Gravity supplies Newtonian potentials and force laws; Relativistic Cosmology supplies initial conditions and structure formation; and Tests of Gravity supplies the cross-regime requirement that one theory map both cosmological and local observables.

Screening is a field solution, not a switch

Consider first a canonical scalar with conformal matter coupling $A(\phi)=e^{\beta\phi/M_{\rm Pl}}$, where $M_{\rm Pl}=(8\pi G)^{-1/2}$ is the reduced Planck mass and $\beta$ is dimensionless. In a local weak-field region small compared with the Hubble scale, use physical coordinates $\mathbf r$ and neglect time derivatives. The field and the acceleration of an unscreened test particle obey

$$\nabla_{\mathbf r}^2\phi=V_{{\rm eff},\phi}, \qquad V_{\rm eff}(\phi)=V(\phi)+\rho A(\phi),$$

where $\rho$ is the conserved nonrelativistic density used in the Einstein-frame effective potential. To leading weak-field order,

$$\mathbf a=-\nabla_{\mathbf r}\Psi_N -\frac{\beta}{M_{\rm Pl}}\nabla_{\mathbf r}\phi.$$

Here $\Psi_N$ is the Newtonian potential, and the factor $\beta/M_{\rm Pl}$ ensures that both acceleration terms have the same dimensions. In comoving simulation coordinates, $\nabla_{\mathbf x}^2\phi=a^2V_{{\rm eff},\phi}$ instead; the particle evolution then also carries expansion factors. Mixing the comoving field equation with a physical-coordinate force law would produce an erroneous scale-factor dependence.

Chameleon-like screening makes the scalar massive or gives an object a thin shell in a dense environment. Symmetron-like models suppress coupling through a density-dependent field value. Vainshtein-like models require additional nonlinear derivative terms in the field equation; they are not described by $V_{\rm eff}$ alone. These mechanisms depend on different environmental variables, so a universal density threshold cannot represent all of them.

In chameleon models, an extended object’s scalar charge depends on its internal potential and the exterior field, so two halos of equal mass can screen differently in different environments. Vainshtein screening is sensitive to source geometry and derivatives, and its superposition properties differ from a simple Yukawa force. Burrage and Sakstein derive the effective potential, thin-shell behavior, and laboratory-to-astrophysical tests Burrage and Sakstein 2018, §§2–5.

A spherical thin-shell example

For an approximately uniform sphere of radius $R$ and mass $M$, let the scalar sit near its dense interior minimum until a thin outer layer of thickness $\Delta R$. Only that layer efficiently sources the exterior scalar. Its mass fraction is

$$Q=1-\left(1-\frac{\Delta R}{R}\right)^3 \simeq3\frac{\Delta R}{R}\quad(\Delta R/R\ll1).$$

For a light exterior scalar, in the interval $R\lt r\ll m_{\rm env}^{-1}$, the exterior solution is approximately $\phi(r)=\phi_{\rm env}-\beta QM/(4\pi M_{\rm Pl}r)$. Differentiation and $G=1/(8\pi M_{\rm Pl}^2)$ give

$$\frac{F_\phi}{F_N}=2\beta^2Q$$

for an unscreened test body. Thus $Q\ll1$ suppresses the force without requiring a small microscopic coupling. A screened test body contributes its own charge factor. The usual thin-shell matching estimates $\Delta R/R\simeq(\phi_{\rm env}-\phi_{\rm in})/[6\beta M_{\rm Pl}(GM/R)]$; outside the thin-shell regime one must solve or match to the unscreened limit rather than extrapolate to $Q\gt1$ Khoury and Weltman 2004, Eqs. (5)–(8).

As an illustration, $\beta=1/\sqrt6$ gives an unscreened force ratio $1/3$, whereas $Q=0.03$ reduces it to $0.01$. This is a controlled spherical example, not a prediction for every halo. Its environmental boundary value explains why mass alone is not a universal screening classifier.

A transition requires several observables

No single power-spectrum ratio demonstrates screening. A persuasive transition has linked signatures:

• the large-scale linear growth approaches the theory’s unscreened prediction;
• halo abundance and clustering respond across the characteristic mass and environment;
• dynamical mass inferred from motions differs from lensing mass where the fifth force acts, then converges in screened objects;
• density and velocity profiles change with radius near the screening boundary;
• voids and low-potential environments show enhanced sensitivity;
• local or Solar-System observables recover their screened limit using the same field boundary condition.

Lensing–dynamical comparisons are especially useful because nonrelativistic motion and the Weyl potential can respond differently, but baryonic mass, velocity anisotropy, disequilibrium, line-of-sight structure, and selection remain competing explanations. An environmental trend should be predicted before examining the data, including which variable—density, potential, acceleration, curvature, or derivative invariant—controls it.

Ishak reviews mass-, coupling-, and kinetic-screening mechanisms and emphasizes the transition to nonlinear scales Ishak 2019, §5.7 and §§8.1–8.3. Hu and Sawicki’s PPF construction consistently joins superhorizon and quasistatic linear regimes in §§II–III; its §IV halo interpolation is a model-motivated nonlinear ansatz, not a general screening theorem Hu and Sawicki 2007, §§II–IV, especially Eqs. (16)–(21).

Matched simulation and inference design

The core numerical check uses matched random phases for GR and the modified theory. It pins the action and branch, background parameters, scalar initial and boundary data, box size, particle number, force and scalar meshes, refinement criteria, time steps, solver tolerance, residual norm, random seed, initial-condition generator, halo finder, environment classifier, and baryonic subgrid model. At least three resolutions and two box sizes test force convergence and missing long modes.

The solver must recover an analytic unscreened force around a low-mass isolated source and a screened solution around a deep-potential source. A code comparison should reproduce density, velocity, and force statistics from independently implemented solvers; Winther and collaborators demonstrate why such cross-code benchmarks are necessary Winther et al. 2015, §§3–5 and comparison figures.

An observational likelihood maps simulation outputs through galaxy or cluster selection, lensing and velocity calibration, redshift errors, masks, and covariance. At the evidence cutoff 2026-08-19, the Planck final legacy products, DES Year 3 3-by-2-point likelihood, and DESI DR2 cosmology products can constrain initial conditions, lensing/clustering, and background distances, respectively. DESI DR2 BAO products are not a released screening likelihood, and Euclid Q1 is explicitly a quick, non-cosmology release rather than a mature cosmological-gravity likelihood. Simulation emulators must state their training domain; extrapolated theory parameters or environments are a diagnosed failure.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or counterevidence
Analytic screened and unscreened solutions	Establish theory-specific limiting forces and environmental variables	Ideal geometry and quasistatic assumptions may not describe halos or time evolution
Matched-phase modified-gravity simulations	Predict transition signatures in power, halos, profiles, and velocities	Resolution, box size, scalar solver, baryons, halo finding, and emulator coverage matter
Lensing, clustering, and dynamical observations	Can compare light and matter response across selected environments	Mass calibration, anisotropy, disequilibrium, feedback, projection, and selection can imitate trends

The conservative position is that screening is viable in specified models and that cosmological data constrain their combined linear and nonlinear predictions; no model-independent screening transition has been established. A competing signal may come from baryonic feedback, halo assembly, mass calibration, or environment-dependent selection. Conversely, a null trend need not exclude a mechanism whose relevant systems are already screened. The conclusion remains unresolved and model-dependent.

An adversarial independent check runs matched-phase GR and modified simulations across resolution, box size, baryonic variants, and two scalar solvers. It must recover both analytic limits and the predicted radial and environmental transition in several observables, not just one fitted suppression factor. Blind injection into mock lensing and velocity catalogs tests whether the inference recovers the mechanism rather than the environment classifier.

What could change the conclusion. The interpretation could change when a new simulation comparison, baryonic emulator, lensing–dynamics dataset, environmental catalog, or local constraint changes the matched theory parameter space or invalidates the predicted screening boundary.

Exercises

Explain why observing the matter power spectrum approach its GR value at high wavenumber is insufficient to establish screening.

Solution

The approach could follow from finite force resolution, an emulator boundary, baryonic suppression, massive neutrinos, or a fitted interpolation rather than a nonlinear scalar solution. Screening predicts more: the solver should recover analytic screened and unscreened limits, and halos should show a radial and environmental transition in force, velocities, and lensing–dynamical mass. Therefore convergence across resolutions and agreement of several matched observables are required before the high-wavenumber trend has a gravitational interpretation.

References

• Clare Burrage and Jeremy Sakstein, “Tests of Chameleon Gravity,” Living Reviews in Relativity 21, article 1 (2018), DOI.
• DES Collaboration, “Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” Physical Review D 105, 023520 (2022), DOI.
• DESI Collaboration, “DESI DR2 Cosmology Results and Supporting Data Products,” official publication list and cosmology-chain release (2025–2026), DESI Data.
• Euclid Consortium, “Euclid Data Release: Q1,” official release overview (2025), Release page.
• Hans A. Winther et al., “Modified Gravity N-Body Code Comparison Project,” Monthly Notices of the Royal Astronomical Society 454, 4208–4234 (2015), DOI.
• Justin Khoury and Amanda Weltman, “Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space,” Physical Review Letters 93, 171104 (2004), DOI, Open PDF.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Planck Collaboration, “Planck 2018 Results. V, VI, and VIII,” Astronomy & Astrophysics 641, A5, A6, and A8 (2020), official publication list.
• Wayne Hu and Ignacy Sawicki, “Parameterized Post-Friedmann Framework for Modified Gravity,” Physical Review D 76, 104043 (2007), DOI.