The Geometry of Cosmological Correlators

TL;DR

This work introduces weighted cosmological polytopes as a first-principles geometric framework for cosmological correlators in generic FRW backgrounds, unifying their computation with the wavefunction of the universe. It develops contour-integral representations that relate higher-loop correlators to lower-loop building blocks and proves factorisation and Steinmann-like relations through the boundary structure of these geometries. A central achievement is the explicit mapping between correlator integrands and the canonical forms of weighted polytopes, including a detailed treatment of their adjoint surfaces and the resulting vanishing conditions. The approach offers a robust, triangulation-based toolkit to compute and bootstrap cosmological correlators, elucidating the inheritance of correlator properties from the wavefunction and providing new mathematical structures within weighted positive geometries.

Abstract

We provide a first principle definition of cosmological correlation functions for a large class of scalar toy models in arbitrary FRW cosmologies, in terms of novel geometries we name {\it weighted cosmological polytopes}. Each of these geometries encodes a universal rational integrand associated to a given Feynman graph. In this picture, all the possible ways of organising, and computing, cosmological correlators correspond to triangulations and subdivisions of the geometry, containing the in-in representation, the one in terms of wavefunction coefficients and many others. We also provide two novel contour integral representations, one connecting higher and lower loop correlators and the other one expressing any of them in terms of a building block. We study the boundary structure of these geometries allowing us to prove factorisation properties and Steinmann-like relations when single and sequential discontinuities are approached. We also show that correlators must satisfy novel vanishing conditions. As the weighted cosmological polytopes can be obtained as an orientation-changing operation onto a certain subdivision of the cosmological polytopes encoding the wavefunction of the universe, this picture allows us to sharpen how the properties of cosmological correlators are inherited from the ones of the wavefunction. From a mathematical perspective, we also provide an in-depth characterisation of their adjoint surface.

Figures (11)

• Figure 1: Reduced graphs. A Feynman graph (on the left) can be represented as a reduced graph (on the right) by suppressing the lines associated to the external states as well as the line representing the late-time boundary. It is a weighted graph whose site weights are the sums of the energies of the external states on the given site, while the edge weights are the energies of the internal states.
• Figure 2: Poles location in the $z$-plane for a generic one-parameter family of correlators $\mathcal{C}_{\hbox{\tiny $\mathcal{G}$}}(x,y;\,z)$. The poles $z_{\mathfrak{g}_j}$ belong to the set $\mathcal{P}_{-}$, while $z_{\mathfrak{g'}_j}$ belong to $\mathcal{P}_{+}$. The contour can be closed either in the negative half-plane, enclosing with counter-clockwise orientation the poles which are function of $x_i$; or in the negative half-plane, enclosing the poles which are function of $x_j$ with clockwise orientation.
• Figure 3: From a collection of $2$-site line graphs to connected graphs. Given a collection of $2$-site line graphs, it is possible to generate different topologies of connected graphs by merging them in their sites. Here we depict a collection of $3$$2$-site line graphs (on the left) and the possible connected topologies that can be generated from them (on the right).
• Figure 4: Cosmological and weighted cosmological polytopes. A cosmological polytope is endowed with an orientation determined by positivity conditions among the vertices, in the case above by $\langle 123 \rangle >0$ (on the left). When considering a polytope subdivision, its elements inherit the orientation from the full polytope in such a way that the codimension-$1$ boundaries belonging to the elements of the subdivision but not to the polytope are un-oriented and hence spurious (center). Reversing the mutual orientation between the elements of the subdivision provides an orientation to these boundaries turning them into internal boundaries (on the right).
• Figure 5: Adjoint surface for a weighted cosmological polytope. For the weighted cosmological polytope associated to a $2$-site line graph, it is given by the green line $(AB_1B_2)$ with just one of its points, $\mathcal{Z}_{\hbox{\tiny $A$}}^{\hbox{\tiny $I$}}$, reflecting a straightforward generalisation of the usual definition of the adjoint for polytopes. Further conditions are needed, and they fix the points $\mathcal{Z}_{\hbox{\tiny $B_1$}}^{\hbox{\tiny $I$}}$ and $\mathcal{Z}_{\hbox{\tiny $B_2$}}^{\hbox{\tiny $I$}}$.