Scattering Amplitudes via Intersection Theory: Algebraic-Geometric Techniques for Gravitational Waveforms and Cosmological Correlators

Particle–physics colliders rely on precise predictions of differential cross sections to test the Standard Model and to search for new physics. Gravitational–wave interferometers require equally precise predictions of the waveforms emitted by coalescing compact objects to detect and characterize astrophysical events. In cosmology, correlation functions at the spacelike boundary of a nearly de Sitter spacetime illuminate the physics of inflation and the large–scale structure of the Universe we observe today. Despite their differences, these topics share one common physical language: scattering amplitudes in perturbative Quantum Field Theory. Computations at higher orders in perturbation theory are needed to increase the precision of theoretical predictions. These become increasingly complex due to the proliferation of multi–loop, multi–leg Feynman integrals and generalizations. Across particle physics, gravity, and cosmology, such integrals share a common mathematical framework: twisted period integrals. This thesis develops and exploits this setup in which integrals inhabit finite–dimensional cohomology spaces, obey linear (integration–by–parts) and quadratic relations, and satisfy differential and difference equations. Within twisted de Rham cohomology, intersection theory provides a natural bilinear pairing, called intersection number, to express these relations. A central contribution of this thesis is the development of an efficient computational framework for intersection numbers. Building on tools from computational algebraic geometry—polynomial ideals, global residues, and companion matrices—together with the existence of boundary-supported dual integrals, I introduce a companion–tensor approach that exposes the underlying tensor structure of intersection numbers. This reformulates intersection problems as matrix operations that couple naturally to finite–field reconstruction techniques. This greatly improves stability and performance while clarifying the mathematical patterns behind the results. I apply these methods to the full decomposition of two–loop four– and five–point integral families by projection through intersection numbers. Beyond standard Feynman Integrals, Fourier integrals are twisted period integrals with the exponential as a twist. This leads to Fourier–space integral relations and first–order differential systems directly in the frequency domain. Leveraging this property, we can study gravitational waveforms generated by the scattering of two compact bodies as twisted period integrals. Within the observable-based formalism, waveforms can be expressed as Fourier transforms of five-point amplitudes in impact-parameter space. I construct the integrand via generalized unitarity and perform integration-by-parts with an exponential twist. The combined Fourier–loop integrals then reduce to a compact basis of master integrals. This yields the first fully analytic, fully relativistic two–body waveform at second Post-Minkowskian order, and it extends naturally to including spin and higher orders. Finally, I show that cosmological wave functions in nearly de Sitter space for conformally-coupled scalars admit a twisted period integral representation. I analyze the combinatorial and vector space structure, and I study the differential equations that govern the two- and three-point one-loop corrections, as functions of external kinematic variables. This provides new insight into quantum corrections in cosmology and a pathway to higher–point and higher–loop correlators. Twisted period integrals and intersection theory furnish a unified language across particle physics, gravitational physics, and cosmology, delivering concrete advances: a fast companion–tensor framework, two–loop Feynman integral decompositions, an extension of integration-by-parts to Fourier integrals, analytic gravitational waveforms, and analytic studies of cosmological correlators.