Theoretical framework for gravitational wave memory in scaler-tensor and f(R) gravity

We investigate gravitational wave memory effects in modified gravity theories using the Bondi-Metzner-Sachs formalism to derive exact expressions for memory tensors incorporating additional gravitational degrees of freedom. Our analysis demonstrates that scalar-tensor theories with Brans-Dicke parameter ω = 1000 produce memory amplitude enhancements of (8.2 ± 1.1)% for binary black hole systems with M = 60M _⊙ at D = 400 Mpc, while f(R) theories with Starobinsky coupling λ = 10 ¹² m ² yield frequency-dependent corrections reach- ing (12.6±2.3)% during merger phases. These predictions emerge from detailed post-Newtonian source modeling with comprehensive error analysis accounting for systematic uncertainties. Space-based gravitational wave detectors operat- ing in the millihertz regime can distinguish these modified gravity signatures with signal-to-noise ratios exceeding ρ = 8 for massive binary systems (M > 10 ⁵ M _⊙ ) at distances up to z = 3. The theoretical framework enables precision tests of fundamental gravitational physics through next-generation observations, with projected constraints suffcient to constrain the Brans-Dicke parameter to ∆ω/ω ∼ 0.08 and Starobinsky coupling to |∆λ| < 10 ¹¹ m ² , representing significant improvements over existing bounds.