Differential Reductions and Cosmological Correlations

TL;DR

The work develops a general framework for simplifying cosmological correlators by interpreting integrals as GKZ systems and exploiting reducibility via reduction operators. It unifies D-module theory, Euler-Koszul homologies, and GKZ data to produce partial solution bases, then leverages algebraic and permutative identities to compress the full solution landscape. A key contribution is the recursive reduction algorithm, which expresses tree-level cosmological correlators in terms of a small set of minimal functions, while establishing diagrammatic relations such as cuts and contractions through first- and higher-order reduction operators. The Pfaffian approach to complexity is used to quantify and bound the complexity of correlators and to illuminate limitations of purely analytic representations. Overall, the thesis provides a scalable, structurally transparent method for solving GKZ systems in cosmology and outlines pathways to extending to loop-level calculations and broader quantum-field contexts.

Abstract

The study of cosmological correlators, and more generally Feynman integrals, is greatly aided by considering them as solutions to differential equations. Often, such systems of differential equations are reducible, which, broadly speaking, implies that the differential system is composed of various subsystems. Studying such decompositions and subsystems can greatly aid in solving the full differential system, as well as bring to light a substantial amount of structure. In this PhD thesis, we study reducibility for a particular system of differential equations known as GKZ (Gelfand, Kapranov and Zelevinsky) systems. We show how reducibility manifests itself in the differential equations through the existence of certain special operators, reduction operators, and explain their properties. Furthermore, we apply this framework to cosmological correlators, exemplifying how these reduction operators can be used to obtain and understand the structure within the system. Here the amount of structure seems remarkably large, and we leverage this structure to obtain many algebraic and permutative identities within the space of solutions to the differential equations. Interestingly, these include various cut and contraction relations between diagrams. We show how to obtain all such relations and how they reduce the full solution set to a certain, remarkably small, subset. Finally, we explain how such simplifications can be understood through the lens of o-minimality and Pfaffian complexity, as well as some of the limitations of this perspective.

Figures (5)

• Figure 1: Imprints of the initial conditions of the universe are still visible in the CMB and LSS. Pictures of the CMB and LSS are adapted from esa_planck_2013 and nasa_hubble_2014 respectively.
• Figure 2: The general in-in diagram corresponding to $n$ external particles.
• Figure 3: The 4-point single-exchange diagram. Here $X_1=\vert \vec{k}_1\vert+\vert\vec{k}_2\vert$, $X_2=\vert \vec{k}_3\vert+\vert\vec{k}_4\vert$ and $Y=\vert \vec{k}_1+\vec{k}_2\vert=\vert \vec{k}_3+\vec{k}_4\vert$.
• Figure 4: The diagrammatical interpretation of the integrals in equation \ref{['eq:Qactions']}. In particular, we have that (a) corresponds to the left-ordered part of the propagator, (b) to the right-ordered part and (c) to the non-time-ordered part. Finally, (d) corresponds to the collapsed propagator.
• Figure 5: Acting with the reduction operators makes it possible to either contract or remove an edge from the diagram.