Abstract

This technical note derives the full set of linear scalar perturbation equations for a minimally coupled canonical scalar field I(x) intended to model a phenomenological late-time dark-energy–like component. Building on the background evolution introduced in Relational Gravity v1.2.1 (Technical Addendum: Background Scalar-Field Sector), we obtain the perturbation equations in both conformal Newtonian and synchronous gauges, compute the associated energy-density, pressure, and momentum perturbations, and construct a quasi-static, scale-dependent effective Poisson equation. The scope of this v1.3 note is strictly technical: to provide a clean, background-consistent perturbation framework suitable for numerical implementation in Boltzmann codes (e.g. CLASS and hi_class), and to make explicit the form of the effective gravitational coupling G_{\rm eff}(a,k) in the sub-horizon regime. No global parameter estimation is attempted. This document is part of the Relational Gravity technical series and complements the background analysis presented in v1.2.1.