Linear Perturbations in a Minimally Coupled Scalar-Field Cosmology: Technical Addendum v1.3

This technical note presents the full linear perturbation framework for a minimally coupled canonical scalar-field cosmology, building on the background scalar-field evolution developed in Relational Gravity: Technical Addendum v1.2.1. The scalar field I(x) is treated purely phenomenologically as a canonical field with potential V(I), without assuming any microscopic interpretation.

The note derives the complete set of linear scalar perturbation equations in both conformal Newtonian and synchronous gauges, including the evolution of \delta I, the density and pressure perturbations \delta\rho_I, \delta p_I, and the velocity divergence \theta_I. Einstein equations are presented consistently in cosmic time, and care is taken to provide expressions directly implementable in Boltzmann solvers such as CLASS or hi_class.

A quasi-static, scale-dependent effective gravitational coupling G_{\rm eff}(a,k) is introduced as a phenomenological rewriting of the scalar-field contribution to the Poisson equation. This does not imply any modification of general relativity; rather, it serves as a compact parametrization of the scalar-field clustering behaviour.

The scope of this note is deliberately narrow and technical: to provide a self-contained perturbation sector that complements the background analysis of v1.2.1 and allows full numerical testing of the scalar-field cosmology against observational datasets (Planck, DESI, Euclid, LSST, etc.). No global likelihood analysis is performed here; those steps are deferred until the sector is implemented and validated within a Boltzmann code.

This document therefore constitutes Technical Addendum v1.3 in the Relational Gravity research series, focusing exclusively on the linear perturbation sector.