A Master Equation for Screening in Luminal Horndeski Gravity

Determining the active screening mechanism from a general scalar-tensor Lagrangian remains a challenging problem. As a diagnostic tool, we present a systematic study of nonlinear cosmological perturbations in luminal Horndeski theories. Working in the $\alpha$-basis on a flat FLRW background, we derive and organise the full set of unapproximated second-order perturbation equations, and systematically apply the quasi-static and weak-field limits. We find that second-order effects modify only the scalar field equation. We derive, for static and spherically symmetric configurations, a master screening equation recovering the Vainshtein and Chameleon mechanisms. We also identify a novel regime, which we term Phaedrus screening, characterised by a screening radius that scales linearly with the source mass. For each mechanism, we derive analytical and numerical solutions and clarify the conditions under which they activate. Finally, we introduce two new publicly available software packages: (i) xAlpha, a Mathematica package to compute and organise perturbation equations in scalar-tensor theories, and (ii) escut, a Python module to solve the nonlinear scalar equation. In many cases, these tools enable the identification of the active screening type directly from a luminal Horndeski Lagrangian.