A Differential Representation of Cosmological Wavefunctions

TL;DR

The paper introduces a differential representation for tree-level cosmological wavefunction coefficients in de Sitter space, recasting the traditional nested time-integral calculation into a three-step procedure that starts from flat-space seeds, passes through a differential-intermediate stage, and ends with a final de Sitter wavefunction obtained by vertex-specific differential operators. This method applies to scale-invariant, parity-invariant theories of massless scalars and gravitons with boost-breaking interactions, including EFT of inflation and solid inflation, and extends to minimal coupling to gravity. By trading time integrals for derivatives with respect to boundary kinematics and judicious edge-collapsing, the approach yields compact, rational expressions whose complexity grows much more slowly with diagram size, enabling efficient calculations even for multi-vertex processes. The framework naturally encodes key physical properties such as scale invariance, pole structure, and manifest locality, and it provides explicit, compact formulae for a wide range of diagrams, including scalar and graviton exchanges. The work opens avenues for holographic interpretations and non-perturbative bootstrap ideas in cosmology, while offering practical tools for EFT-based cosmological predictions.

Abstract

Our understanding of quantum field theory rests largely on explicit and controlled calculations in perturbation theory. Because of this, much recent effort has been devoted to improve our grasp of perturbative techniques on cosmological spacetimes. While scattering amplitudes in flat space at tree level are obtained from simple algebraic operations, things are harder for cosmological observables. Indeed, computing cosmological correlation functions or the associated wavefunction coefficients requires evaluating a growing number of nested time integrals already at tree level, which is computationally challenging. Here, we present a new "differential" representation of the cosmological wavefunction in de Sitter spacetime that obviates this problem for a large class of phenomenologically relevant theories. Given any tree-level Feynman-Witten diagram, we give simple algebraic rules to write down a seed function and a differential operator that transforms it into the desired wavefunction coefficient for any scale-invariant, parity-invariant theory of massless scalars and gravitons with general boost-breaking interactions. In particular, this applies to large classes of phenomenologically relevant theories such as those described by the effective field theory of inflation or solid inflation. Trading nested bulk time integrals for derivatives on boundary kinematical data provides a great computational advantage, especially for processes involving many vertices.

Figures (11)

• Figure 1: Depiction of the sequence of differential operators acting on the flat space seed function $\psi^{\text{flat}}$ to obtain the function $\psi$ specified by the edges and vertices.
• Figure 2: For a half-edge $\phi$ the graphic defines flowing vertex side or opposite-vertex side. A choice of flow determines a differential operator summing over energies in $G_L$ including the edge itself or in $G_R$ including the vertex.
• Figure 3: A particular contribution to the quintic wavefunction coefficient from the indicated cubic interactions.
• Figure 4: The two topologies for the four-site skeleton diagrams, namely the flux capacitor (left) and the four-site chain (right). The diagram also indicates the total external energies $q_{A}$ associated with each vertex $A=1,\dots,4$ and the energy $p_{m}$ of each internal line (edge) $m=1,2,3$.
• Figure 5: Edge-labeled diagram with a choice of routing for the two $\phi$-type half-edges attached to the central vertex.