This paper compares two ways of evolving small ripples produced during cosmic inflation: the standard quantum description and a classical one. The authors show that even if you force the two descriptions to agree at one moment during inflation, they will generally disagree later, as long as the fields interact. The size of the disagreement grows very fast with the amount of expansion that happens after the time of matching. In other words, when and how you match a classical model to the quantum one matters a great deal.

To make this comparison precise the authors write both quantum and classical time evolution in the same formal language. The quantum evolution uses operators and commutators (the usual quantum rules). The classical evolution uses ordinary numbers and Poisson brackets (the usual rules for classical fields), but starts from a probability distribution of initial conditions chosen so that classical statistics match quantum statistics at a chosen time t0. They then compute concrete examples. One example is the tree-level bispectrum of the curvature perturbation ζ, which is a three-point correlation function that measures non-Gaussianity. The other example mentioned is the one-loop power spectrum of tensor modes (these are gravitational-wave fluctuations).

At a high level the reason for the difference is that quantum and classical evolution are only formally similar. Quantum orderings and the choice of vacuum state introduce factors of Planck’s constant ℏ. For the bispectrum calculated in the paper the leading quantum contribution scales like ℏ^2. If one starts a purely classical evolution at a finite time t0 and forces the classical statistics to match the quantum ones there, extra time-dependent contributions appear during the subsequent classical evolution. These extra contributions vanish only if the matching is done after the perturbation ζ has already “frozen,” meaning its value no longer changes with time. The authors show the discrepancy between classical and quantum correlators is exponentially sensitive to the number of e-folds of expansion that happen between the matching time and horizon crossing. An e-fold is the standard way to measure how much the universe expands during inflation: one e-fold means the universe grew by a factor of e≈2.7.