A parton picture of de Sitter space during slow-roll inflation

TL;DR

This work reframes infrared issues in de Sitter space during slow-roll inflation as a DGLAP-like evolution of inflationary fluctuations, drawing a precise analogy between parton evolution in hadrons and superhorizon cosmological modes. By mapping the Hubble scale to a probing energy and long-wavelength scalar vevs to Bjorken variables, the author derives a de Sitter master equation that reproduces Starobinsky’s stochastic diffusion in the leading-log limit. The approach provides a concrete, renormalization-group–flavored picture of how large IR logs resummate into a diffusion process for the background fields, with a factorization scale $k_F$ separating hard subprocesses from nonperturbative, superhorizon evolution. The results unify stochastic inflation with the parton-evolution framework, clarifying when and how stochasticity affects observables like non-Gaussian correlators, and highlighting both the utility and limits of this leading-log resummation in inflationary predictions.

Abstract

It is well-known that expectation values in de Sitter space are afflicted by infra-red divergences. Long ago, Starobinsky proposed that infra-red effects in de Sitter space could be accommodated by evolving the long-wavelength part of the field according to the classical field equations plus a stochastic source term. I argue that--when quantum-mechanical loop corrections are taken into account--the separate-universe picture of superhorizon evolution in de Sitter space is equivalent, in a certain leading-logarithm approximation, to Starobinsky's stochastic approach. In particular, the time evolution of a box of de Sitter space can be understood in exact analogy with the DGLAP evolution of partons within a hadron, which describes a slow logarithmic evolution in the distribution of the hadron's constituent partons with the energy scale at which they are probed.

Figures (5)

• Figure 1: Kinematics of deep inelastic electron--hadron scattering. A high energy electron impinges on the target hadron and interacts electromagnetically with one of its constituent partons. After the interaction, the original hadron is disrupted and the ejected quark materializes as a jet of hadrons collinear with the motion of the initial electron.
• Figure 2: A parton brakes as it enters into a collision with some incoming particle $X$, before scattering into a final hadronic state $Y$. As it brakes, it radiates an arbitrary number of soft gluons $\{ \cdots, 3, 2, 1 \}$ and moves increasingly off-shell. The impinging $X$ projectile can resolve this emission cascade if it is sufficiently energetic.
• Figure 3: The analogue of Altarelli--Parisi splitting functions for de Sitter space. A de Sitter parton, represent by the hatched region, evolves in time from left to right and radiates soft quanta which materialize in the final state. These radiated quanta are fluctuations which are instantaneously drawn over the de Sitter horizon and classicalize. The solid lines represent scalar particles whereas the wavy line represents gravitons; the diagram with three scalar particles in the final state represents the leading non-Gaussian correction, although this turns out not to contribute in a leading-logarithm approximation. In principle, radiation into any light states is permitted.
• Figure 4: The spacetime interpretation of a leading-logarithm resummation in $H_\ast^2$.
• Figure 5: Parton evolution and recombination in de Sitter, with time evolving from left to right in the diagram. Hard subprocess quanta, represented by the solid lines, materialize above the horizon and form the initial condition for the subsequent "third phase" evolution, represented by the dashed lines. For modest evolution subsequent to horizon crossing, this "third phase" evolution corresponds to a mixture of time dependence and recombination. If the correlation functions are observed following only a modest number of e-folds subsequent to horizon crossing, these are the only effects which must be taken into account. In general the recombination process will be very strongly suppressed, together with any other quantum processes associated with the scale $H_\ast^2$. In this approximation, the non-perturbative de Sitter region, represented by the hatched region at the bottom of the diagram, undergoes only coherent time evolution from left to right.