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Gauge-Invariant Gravitational Wave Polarization in Metric f(R) Gravity with Cosmological Implications

We develop a fully gauge invariant analysis of gravitational wave polarizations in metric f(R) gravity with a particular focus on the modified Starobinsky model, whose constant curvature solution provides a natural deSitter background for both early and late time cosmology. Linearizing the field equations around this background, we derive the Klein Gordon equation for the curvature perturbation and show that the scalar propagating mode acquires a mass, highlighting how the same scalar degree of freedom governs inflationary dynamics at high curvature and the propagation of gravitational waves in the current accelerating Universe. Using the scalar vector tensor (SVT) decomposition and a decomposition of the perturbed Ricci tensor, we obtain a set of fully gauge invariant propagation equations that isolate the contributions of the scalar, vector, and tensor modes in the presence of matter. We find that the tensor sector retains the two transverse traceless polarizations of General Relativity, while the scalar sector supports a massive breathing-longitudinal mode determined by the massive scalar propagating mode. Through the geodesic deviation equation, computed both in a local Minkowski patch and in fully covariant de Sitter form, we independently recover the same polarization content and identify its tidal signatures. The resulting framework connects the extra scalar polarization to cosmological observables, providing a unified, gauge invariant link between gravitational wave phenomenology and the cosmological implications of metric f(R) gravity.