escut: Numerical Screening in Horndeski Gravity

• escut is a Python module that reduces complex luminal Horndeski models to a static nonlinear scalar screening equation for precise numerical analysis.
• It distinguishes between Vainshtein, Chameleon, and Phaedrus screening regimes by computing scalar profiles and diagnostic quantities such as screening radii and slopes.
• The solver employs adaptive collocation and homotopy continuation methods to robustly solve the boundary-value problem and achieve high-accuracy convergence.

escut is a Python module introduced as part of the screening program developed in "A Master Equation for Screening in Luminal Horndeski Gravity" (Sirera et al., 5 May 2026). Its function is narrowly defined: once a luminal Horndeski model has been reduced, in the weak-field and quasi-static regime, to a nonlinear scalar screening equation, escut numerically solves that equation for static, spherically symmetric sources and thereby determines which screening behavior is active. In that sense, it is neither a general cosmology code nor a symbolic perturbation package. It is a dedicated numerical solver for the nonlinear scalar equation that controls screened fifth forces in luminal Horndeski gravity, and it serves as the step that turns the paper’s master screening equation into a practical diagnostic of Vainshtein, Chameleon, and Phaedrus screening (Sirera et al., 5 May 2026).

1. Definition and scientific role

escut operates within luminal Horndeski gravity with $c_{\rm GW}=c$, for theories of the form

$$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$

Its scientific purpose is to determine local screened behavior for static overdensities once the covariant theory has been reduced to a radial nonlinear scalar equation. The central problem addressed by the underlying paper is that the active screening mechanism is not generally obvious from the scalar-tensor Lagrangian alone. Even after deriving perturbation equations, the answer remains encoded in a nonlinear differential equation with competing operators. escut is the numerical tool introduced for that last step (Sirera et al., 5 May 2026).

The module sits at the end of a larger pipeline. A luminal Horndeski Lagrangian $K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$ is first translated into background and perturbation coefficients, including $\alpha_K,\alpha_B,\alpha_M$, the $\gamma$-functions, and then $M^2$, $M_{\rm nl}^2$, $\kappa_\pm$, and $\Gamma$. The weak-field and quasi-static limits are then imposed, the static spherical master equation is obtained, and escut takes the resulting dimensionless coefficients and source model as numerical input. Its outputs include the scalar profile $Q(r)$, its derivative $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$0, the scalar flux $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$1, and diagnostic quantities such as the screening-efficiency slope and screening radius. A plausible implication is that escut is best understood not as an isolated code, but as the phenomenological endpoint of a theory-to-screening workflow.

2. Perturbative reduction and the master equation

The module is built on a specific structural result of the perturbative analysis. Starting from the Newtonian-gauge metric

$$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$2

the paper derives the full second-order perturbation equations for the field multiplet

$$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$3

After imposing the weak-field limit and the quasi-static approximation, the metric equations remain effectively linear, while second-order nonlinearities survive only in the scalar field equation (Sirera et al., 5 May 2026).

The reduced metric sector is

$$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$4

By eliminating $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$5 and $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$6, the paper obtains a master screening equation whose relevant operator content can be organized into a linear kinetic term, linear and nonlinear mass terms, quadratic kinetic terms, and a Vainshtein-type second-derivative operator. The effective linear spatial kinetic coefficient is

$$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$7

with $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$8 the no-ghost parameter and $$S=\int d^4x\sqrt{-g}\Big[G_4(\phi)R+K(\phi,X)-G_3(\phi,X)\Box\phi+\mathcal L_m\Big].$$9 the scalar sound speed.

For the static, spherically symmetric configurations that escut solves, the physical radius is

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$0

and under $K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$1 the local operators reduce to radial form. The resulting screening equation is

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$2

with

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$3

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$4

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$5

This radial equation is the equation escut integrates.

The matter source used in the paper is a smoothed top hat,

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$6

with enclosed mass variable

$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$7

These definitions are numerically consequential because the source regularization parameter $K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$8 appears directly in the solver setup.

3. Screening regimes identified by escut

The scientific value of escut follows from the fact that one radial equation contains the three screening regimes analyzed in the paper. The code is designed to determine which nonlinear operator dominates for a given model and source, and thus which mechanism is realized in practice (Sirera et al., 5 May 2026).

Regime	Dominant structure	Characteristic signature
Vainshtein	$K(\phi,X),\,G_3(\phi,X),\,G_4(\phi)$9	$\alpha_K,\alpha_B,\alpha_M$0, $\alpha_K,\alpha_B,\alpha_M$1 for $\alpha_K,\alpha_B,\alpha_M$2
Chameleon	$\alpha_K,\alpha_B,\alpha_M$3 in $\alpha_K,\alpha_B,\alpha_M$4	field-dependent $\alpha_K,\alpha_B,\alpha_M$5, thin-shell behavior
Phaedrus	$\alpha_K,\alpha_B,\alpha_M$6	$\alpha_K,\alpha_B,\alpha_M$7, $\alpha_K,\alpha_B,\alpha_M$8

For Vainshtein screening, the relevant operator is $\alpha_K,\alpha_B,\alpha_M$9. Integrating once yields an algebraic relation for $\gamma$0, and the Vainshtein radius satisfies

$\gamma$1

The activation condition for a real screened solution around an overdensity is

$\gamma$2

Deep inside the screened region one obtains $\gamma$3, corresponding to screening efficiency $\gamma$4, where

$\gamma$5

For Chameleon screening, the dominant effect is the nonlinear mass term. The equation reduces to a radial Klein-Gordon-type form with field-dependent effective mass

$\gamma$6

The effective potential is defined by $\gamma$7, and deep inside the source the field sits near the minimum of that effective potential. Outside the source the profile is Yukawa-like, and in the thin-shell approximation the paper gives an exterior solution with effective shell mass $\gamma$8. escut is used to compute the full numerical solutions that interpolate across the shell.

For Phaedrus screening, which is introduced in the paper as a new regime, the source operators are the quadratic kinetic terms $\gamma$9. Outside the source, a power-law ansatz gives the nontrivial screening efficiency

$M^2$0

Its defining property is the screening-radius scaling

$M^2$1

that is, linear in source mass rather than proportional to $M^2$2 as in the Vainshtein case. In the special case $M^2$3, the paper gives the exact solution

$M^2$4

with

$M^2$5

The corresponding screening radius is

$M^2$6

The paper further states that Phaedrus can govern the exterior only if the standard kinetic term is heavily suppressed, roughly $M^2$7. It also states that this raises concerns about the quasi-static approximation and stability, since $M^2$8 and suppressing it tends to push $M^2$9. escut can still solve the static equation in that regime, but the physical viability of that limit is left to future work.

4. Numerical formulation and solver architecture

escut solves a dimensionless version of the radial master equation using

$M_{\rm nl}^2$0

The equation implemented numerically is

$M_{\rm nl}^2$1

The paper states that the coefficients

$M_{\rm nl}^2$2

correspond directly to the theoretical coefficients from the master equation, while $M_{\rm nl}^2$3 is the regularized matter source (Sirera et al., 5 May 2026). The exact coefficient-by-coefficient dictionary is not printed. This suggests that the mapping is structurally direct but not fully documented in the paper’s main text.

To pass the problem to a boundary-value solver, escut rewrites the equation as a first-order system for $M_{\rm nl}^2$4, with

$M_{\rm nl}^2$5

where

$M_{\rm nl}^2$6

$M_{\rm nl}^2$7

The domain is

$M_{\rm nl}^2$8

with regularity at the origin,

$M_{\rm nl}^2$9

and a mixed asymptotic Robin condition at $\kappa_\pm$0 enforcing decay toward a prescribed $\kappa_\pm$1, typically $\kappa_\pm$2. The far-field decay scale is estimated from the linearized outer effective mass.

Numerically, the module builds an initial logarithmic mesh of $\kappa_\pm$3 points, adaptively refines it up to a maximum number of nodes, and uses adaptive collocation to solve the nonlinear boundary-value problem to tolerance $\kappa_\pm$4. It also supports homotopy continuation, namely a sequence of gradually more nonlinear problems that is used to reach the target equation robustly. The explicitly named solver settings are

$\kappa_\pm$5

where $\kappa_\pm$6 is the source-edge smoothing parameter inherited from the density profile.

5. Inputs, outputs, and diagnostics

The physical inputs required by escut are the coefficients of the screening equation, which are ultimately functions of the background Horndeski model and the $\kappa_\pm$7-basis variables, together with the matter source profile and asymptotic boundary data. In practical terms, the workflow described in the paper is: choose a luminal Horndeski Lagrangian, compute the background and perturbation coefficients, reduce the system to the static spherical master equation, pass the resulting dimensionless coefficients and source model to escut, and obtain radial solutions and screening diagnostics (Sirera et al., 5 May 2026).

The outputs are not presented as a formal application programming interface, but the paper makes clear what the code computes in practice. These include the radial scalar profile $\kappa_\pm$8, the scalar flux $\kappa_\pm$9, the screening-efficiency slope

$\Gamma$0

and the screening radius $\Gamma$1, defined as the location where nonlinear and linear terms become comparable. In the numerical examples discussed in the paper, escut reproduces the expected Vainshtein plateau $\Gamma$2, the Chameleon thin-shell behavior and effective-mass growth, and the Phaedrus plateau, for example $\Gamma$3 when $\Gamma$4.

A practical output is therefore classification. By examining which nonlinear operator dominates, together with the asymptotic slope or mass-scaling of the screening radius, the code diagnoses whether the active mechanism is Vainshtein, Chameleon, Phaedrus, or absent. This suggests that escut functions not merely as an ODE solver, but as a model-discrimination instrument for the local phenomenology of luminal Horndeski theories.

6. Validation, scope, and limitations

The paper reports that the current implementation was used to compute numerical solutions for Vainshtein, Chameleon, and Phaedrus cases, and that it is robustly validated for isolated screening regimes, while mixed-mechanism solutions remain more challenging (Sirera et al., 5 May 2026). Since exact solutions are not generally available, validation is carried out against high-accuracy reference solutions $\Gamma$5 using

$\Gamma$6

with $\Gamma$7 and $\Gamma$8 norms. For derivative-driven mechanisms the comparison is performed on the flux $\Gamma$9, while for Chameleon it is performed on the amplitude $Q(r)$0. The reported result is stable convergence to sub-percent or better precision in the relevant interior region.

The scope of escut is sharply delimited by the derivation of the master equation. It applies to luminal Horndeski theories, not to general Horndeski or beyond-Horndeski/DHOST in the paper’s implementation. It assumes scalar perturbations only, a flat FLRW background, truncation at second perturbative order, the weak-field limit, the quasi-static approximation, and static, spherically symmetric local sources. These assumptions are not incidental; they are the conditions under which the equation solved by escut is obtained.

The limitations are correspondingly clear. The paper explicitly notes that third order is needed to systematically capture $Q(r)$1-mouflage and the Symmetron as full screening mechanisms, although some leading effects can already appear at second order. It also notes that the framework is not yet the final answer for non-spherical sources, time-dependent stability, or the simultaneous robust resolution of multiple screening operators. For Phaedrus screening in particular, the static equation can be solved, but physical viability remains uncertain because the required $Q(r)$2 regime may invalidate the quasi-static approximation and trigger instabilities. On the numerical side, Chameleon solutions are reported to be especially sensitive to the source-edge smoothing parameter $Q(r)$3, and very large domains can be difficult because the thin-shell transition is localized while the asymptotic region is large.

escut is paired with xAlpha, the symbolic Mathematica package used in the same work to compute and organize the perturbation equations and coefficients. xAlpha handles the symbolic reduction from a luminal Horndeski Lagrangian to the weak-field quasi-static coefficients, while escut solves the nonlinear boundary-value problem that determines actual local screened behavior. In many cases, the combined pipeline enables identification of the active screening type directly from a luminal Horndeski Lagrangian.