This paper studies a long-standing problem in quantum field theory on an expanding spacetime called de Sitter space. When a massless scalar field interacts with itself, the standard perturbative expansion of correlation functions produces a divergent series at late times. The authors test tools that turn those divergent series into well-defined, time-dependent answers and they compare the results with a separate “stochastic” picture that already resums the most dangerous infrared effects.

First, the authors apply a method they call autonomous equations. These are simple differential equations constructed so that their iterative solutions reproduce perturbative coefficients up to a chosen order. Solving those equations gives explicit, finite functions of cosmic time for two-point and higher correlation functions. The paper compares these analytic autonomous solutions with numerical solutions of the Fokker–Planck equation from the stochastic approach. The autonomous solutions track the stochastic time evolution reasonably well, and their late-time limits are close to the stochastic results. In one approximation the autonomous solution reduces to what is known as the Hartree–Fock result.

Second, the paper combines autonomous equations with a resummation technique called Borel–LeRoy (a generalization of Borel) transform. The authors apply autonomous equations to the Borel transforms of the correlation functions and then perform Borel resummation. These resummed answers match the stochastic time evolution substantially better than the raw autonomous solutions. The authors also investigate the singularity structure of the Borel transforms and propose an alternative method to extract perturbative coefficients.

The work offers a new derivation of the autonomous equation by building and truncating a system of Schwinger–Dyson–type differential equations. Schwinger–Dyson equations are an infinite set of relations that connect all n-point functions. By truncating that system — for example by setting higher n-point functions to zero or by using a Gaussian approximation for the six-point function — the authors obtain closed systems of first-order differential equations. Solving those truncated systems reproduces the autonomous equation for the two-point function and yields perturbative coefficients that agree with other methods.