Induced gravitational waves for arbitrary higher orders: Vertex rules and loop diagrams in cosmological perturbation theory

Gravitational waves induced by primordial perturbations serve as crucial probes for studying the early Universe, providing a significant window into potential new physics during cosmic evolution. Due to the potentially large amplitudes of primordial perturbations on small scales, the contributions of high-order cosmological perturbations (CPs) are highly significant. We propose a vertex approach applicable to the study of induced gravitational waves for arbitrary higher orders. Using the vertex approach and tree diagrams, we can directly derive the explicit expressions of higher-order induced gravitational waves without involving the complex and lengthy calculations of higher-order CPs. Correlations between different tree diagrams correspond to the loop diagrams of two-point correlation functions of induced gravitational waves. Our investigation reveals that one-particle reducible diagrams impact tensor-scalar-induced gravitational waves while leaving scalar-induced gravitational waves unaffected.