Emergent Time, Nonlocal Gravity, and Gravitational Wave Constraints from Spectral Functional Renormalization Group

We derive a closed integro-differential evolution equation for the graviton spectral density ρk(ω,p) from the functional renormalization group (FRG), showing that the flow takes the form of a quantum Boltzmann equation with a positive-definite collision operator B[ρ] ∼ ρ ⋆ ρ. This structure implies an H-theorem for the spectral entropy, ∂tS[ρ] ≥ 0, establishing an intrinsic arrow of time generated by RG coarse-graining. The spectral convolution induces a causal memory kernel K(t), which we express covariantly as a nonlocal form factor F(□) in the effective gravitational action. Introducing a scalar field τ(x) representing the local RG entropy current—analogous to standard Hubbard Stratonovich transformations used to localize nonlocal effective actions—we obtain modified Einstein equations with nonlocal curvature terms. For tensor modes, this leads to a modified gravitational wave equation with damping rate Γ ∼ G2H3. Combining constraints from gravitational wave propagation (|cT − 1| < 10−15) and the tensor-to-scalar ratio (r < 0.036), we derive the bound Hinf ≲ 107 GeV, providing a sharp and falsifiable prediction of the framework. This connects RG irreversibility, emergent time, and observable signatures in gravitational wave physics.