Abstract
Light scalars in inflationary spacetimes suffer from logarithmic infrared divergences at every order in perturbation theory. This corresponds to the scalar field values in different Hubble patches undergoing a random walk of quantum fluctuations, leading to a simple toy “landscape” on superhorizon scales, in which we can explore questions relevant to eternal inflation. However, for a sufficiently long period of inflation, the infrared divergences appear to spoil computability. Some form of renormalization group approach is thus motivated to resum the log divergences of conformal time. Such a resummation may provide insight into De Sitter holography. We present here a novel diagrammatic analysis of these infrared divergences and their resummation. Basic graph theory observations and momen- tum power counting for the in-in propagators allow a simple and insightful determination of the leading-log contributions. One thus sees diagrammatically how the superhorizon sector consists of a semiclassical theory with quantum noise evolved by a first-order, interacting classical equation of motion. This rigorously leads to the “Stochastic Inflation” ansatz developed by Starobinsky to cure the scalar infrared pathology nonperturbatively. Our approach is a controlled approximation of the underlying quantum field theory and is systematically improvable.
Highlights
Suggestive hints for some of the deeper conceptual issues in quantum De Sitter correlators and cosmology
Before describing the connections to important topics such as eternal inflation, the measure problem, and holography, we state that our resolution to the infrared pathologies of certain De Sitter theories follows the familiar formulation of “stochastic inflation”, developed originally by Starobinsky in the mid-1980s [10], and further elucidated by [11, 12]
We will see that an arbitrary diagram with an arbitrary number of loops contains within it a substructure of tree diagrams with all non-tree features corresponding to the inhomogeneous stochastic noise sources that arise from the redshifting of UV modes
Summary
This Universe could have been in a radiation-FRW phase early on (ρtot = ρrad + ρΛ), but after sufficient redshifting of the radiation, the cosmic evolution would be dominated by the cosmological constant, and would enter a DS expansion.5 Since this theory (like pure radiation FRW) is IR-safe, we can find some kIR such that cutting off modes with k < kIR introduces only small power corrections, kInR, relative to a complete matching procedure between pre-inflationary and inflation phases. Since we are tracking and resumming logarithmic dependence on kIR in this paper, such power corrections are negligible This equation defines the modified state in which we will compute expectation values, which we denote |BD. We will demonstrate that in the late-time limit, the leading-log contributions are resummable and one obtains a finite result for η → 0.6 This is identical mathematically to taking kIR → 0, removing the cutoff, pushing the pre-DS cosmology infinitely far into the past. Whether we are interested in a pure, infinite-time DS, or an inflation-like scenario with a finite duration, we can adopt the same approach so long as inflation lasts for sufficient time to make the leading-log analysis relevant
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