Nonminimally Coupled Massive Scalar Field in QuadraticGravity: Effective Gravitational Response and Matter Growth

We examine a massive scalar field with a nonminimal coupling −βRϕ2/2 in an R + λR2 gravitational theory. The model belongs to the established mixed scalar–scalaron class; λ, β, and the scalar mass are independent parameters. Starting from an action with explicit conventions, we derive the metric and scalar equations, distinguish the separately conserved geometric and matter sources from the jointly conserved scalar and interaction10 contributions, and obtain the spatially flat cosmological equations. Retaining both scalar responses and metric slip gives a scale-dependent quasi-static coupling Geff = Gμ(k, a) for linear matter growth. We provide the limiting response plateaus, their signed difference, and a half-transition scale, stating the conditions under which these limits lie in the approximation’s domain. Illustrative background and growth integrations compare different15 values of β with matched-background force-law controls. The results specify an effective dark-sector application of known gravitational operators; they do not establish observational viability, a complete dark-matter replacement, or the accuracy of the quasi-static approximation for arbitrary initial conditions.