The Action Principle and Tensorial Perturbations in the $\Psi$-Field Cosmology: Theoretical Stability and Gravitational Wave Amplification

We present the fundamental action formulation for the damped harmonic oscillator scalarfield cosmology (Ψ-field). By embedding the cosmic damper mechanism within a modifiedscalar-tensor gravity framework, we derive the exact field equations from the principle of leastaction under fixed late-time structural constraints (A ≈ 0.25, m ≈ 3.0). Numerical analysisdemonstrates that the derived Energy-Momentum tensor strictly satisfies the Weak, Null, andDominant Energy Conditions, confirming the theoretical stability of the vacuum and preservation of causality (cgw = c). Crucially, evaluation of the tensorial perturbations reveals a novelphenomenon of cosmological gravitational wave amplification. Due to the dynamic decay ofthe field’s coupling function f(Ψ) in the late-time universe, the effective friction term becomesnegative, operating as a natural cosmic amplifier. Our simulation establishes a systematic6.55% increase in the gravitational wave amplitude at z = 0 for deep-space sources comparedto standard General Relativity (ΛCDM). This distinct, measurable signature provides a robustobservational test for the Ψ-field theory using next-generation gravitational wave detectors.