Speaker
Description
The quasi-static (QSA) and sub-horizon (SHA) approximations are standard tools used to simplify the complex dynamics of linear cosmological perturbations in modified gravity. However, their conventional implementation relies on an ad-hoc truncation of time derivatives, which obscures theoretical errors and inadvertently eliminates fundamental background dynamics. In this work, we propose a mathematically rigorous framework that elevates the QSA and SHA from heuristic truncations to controlled asymptotic expansions within the broad class of Horndeski scalar-tensor theories. By introducing a set of dimensionless tracking parameters driven by the spatial ratio $\varepsilon = aH/k$, we formulate an exact rewriting of the perturbation equations where the standard QSA/SHA emerges purely as a well-defined zeroth-order asymptotic limit. Our systematic approach explicitly tracks higher-order dynamical corrections and reveals that the standard approach spuriously intertwines spatial scales with unexpanded temporal derivatives. To determine the physical regime where this expansion is strictly valid, we introduce a deep-matter tracking parameter, defining a clear validity window for the formalism. We evaluate the dynamical convergence of our $\mathcal{O}(\varepsilon^2)$ expansion by comparing it against the standard approach and the full unapproximated system using the Designer Horndeski (HDES) model as a case study. We find that our systematic expansion significantly improves accuracy over the standard approach, robustly tracking the exact evolution of the gravitational potentials and the effective dark energy fluid variables.