On the time evolution of cosmological correlators

TL;DR

This work develops a robust, model-independent framework for the time evolution of cosmological correlators during inflation by working in the bulk Schrödinger picture on de Sitter space. It derives simple, general equations of motion for the wavefunction coefficients c_n, identifies unitarity-induced constants of motion β_n, and shows how de Sitter isometries constrain both bulk coefficients and boundary data, reducing to conformal Ward identities at late times. The authors introduce transfer functions linking conformal-boundary data to bulk evolution, emphasize locality leading to analytic transfer functions outside the horizon, and address boundary divergences via boundary-wavefunction renormalisation and Boundary Operator Expansion. They illustrate the formalism with explicit examples (conformally coupled and massless scalars) and discuss the EFT of inflation, revealing a cohesive picture that connects bulk dynamics with boundary bootstrap ideas and paves the way for new diagnostics of the inflationary era. Overall, the paper provides a coherent, symmetry- and locality-driven methodology to analyze cosmological correlators beyond specific Lagrangians, with implications for interpreting primordial non-Gaussianity in upcoming observations.

Abstract

Developing our understanding of how correlations evolve during inflation is crucial if we are to extract information about the early Universe from our late-time observables. To that end, we revisit the time evolution of scalar field correlators on de Sitter spacetime in the Schrodinger picture. By direct manipulation of the Schrodinger equation, we write down simple "equations of motion" for the coefficients which determine the wavefunction. Rather than specify a particular interaction Hamiltonian, we assume only very basic properties (unitarity, de Sitter invariance and locality) to derive general consequences for the wavefunction's evolution. In particular, we identify a number of "constants of motion": properties of the initial state which are conserved by any unitary dynamics. We further constrain the time evolution by deriving constraints from the de Sitter isometries and show that these reduce to the familiar conformal Ward identities at late times. Finally, we show how the evolution of a state from the conformal boundary into the bulk can be described via a number of "transfer functions" which are analytic outside the horizon for any local interaction. These objects exhibit divergences for particular values of the scalar mass, and we show how such divergences can be removed by a renormalisation of the boundary wavefunction - this is equivalent to performing a "Boundary Operator Expansion" which expresses the bulk operators in terms of regulated boundary operators. Altogether, this improved understanding of the wavefunction in the bulk of de Sitter complements recent advances from a purely boundary perspective, and reveals new structure in cosmological correlators.

Figures (5)

• Figure 1: A cartoon of the expanding de Sitter spacetime. We refer to late times, $\eta \to 0$, as the conformal boundary, and denote the wavefunction coefficients there by $\alpha_n$. This boundary condition can be translated into a bulk wavefunction, with coefficients $c_n (\eta)$, by means of various "transfer functions" $\mathcal{I}_{{\bf k}_1 ... {\bf k}_n}^{\nu_1 ... \nu_n} (\eta)$ which we define in Section \ref{['sec:superhorizon']}---these objects are analytic (for any local Hamiltonian) until horizon crossing at $| k \eta | \sim 1$ where they develop non-analyticities (such as the $1/k_T$ pole). In the far past, $\eta \to -\infty$, we denote the wavefunction coefficients as $\alpha_n^{\rm in}$. Imposing the Bunch-Davies vacuum state in the far past corresponds simply to $\alpha_n^{\rm in} = 0$.
• Figure 2: The Hamilton-Jacobi equations for the wavefunction phase, $\Gamma$. The time derivative is given by all possible ways of splitting the interaction into two pieces (exchange contribution), plus all possible ways of contracting a single higher-point coefficient into a loop (loop contribution), plus any interactions in the Hamiltonian (contact contribution).
• Figure 3: Adding a quadratic $\alpha_2 \phi^2$ to the boundary wavefunction (at time $\eta = 0$) affects the two-point coefficient $c_2 (\eta) \phi^2$ at a bulk time $\eta$ via the resummation shown above. This is analogous to a mass insertion, $m^2 \phi^2$, shifting the propagator from $1/p^2$ to $1/(p^2-m^2)$ in a standard Lorentzian QFT.
• Figure 4: Diagrammatic representation of the wavefunction coefficient $\tilde{c}_{{\bf k}_1 {\bf k}_2 {\bf k}_3 {\bf k}_4} (\eta)$ and its dependence on the boundary wavefunction coefficients $\alpha_{{\bf k}_1 ... {\bf k}_n}$ at $\eta = 0$.
• Figure :