De Sitter diagrammar and the resummation of time

Published for SISSA by Springer Received: January 14, 2020 Accepted: June 18, 2020 Published: July 17, 2020 De Sitter diagrammar and the resummation of time Department of Physics, Arizona State University, Tempe, AZ 85287, U.S.A. Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, U.S.A. E-mail: , 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 momentum 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. Keywords: Cosmology of Theories beyond the SM, Effective Field Theories, Renormalization Group ArXiv ePrint: 1912.09502 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2020)119 JHEP07(2020)119 Matthew Baumgart and Raman Sundrum Contents 1 2 Massless scalar in De Sitter spacetime 4 3 Leading-logs to all orders 3.1 In-in perturbation theory 3.2 Causality constraints 3.3 The fast track to leading-log 7 8 10 12 4 Semiclassicality and first-orderness 15 5 Log resummation as Fokker-Planck evolution 17 6 Conclusion & discussion 22 A The nested commutator in-in formalism 26 B Closing loopholes in the leading-log argument 27 C Restoring acceleration 30 1 Introduction De Sitter may be thought of as the spacetime with the best sense of humor. A beautifully, indeed maximally, symmetric solution to Einstein’s equations, it nonetheless gives an excellent approximate description to both our cosmological past, through inflation, and our future, by the coming era of dark energy domination. However, at the quantum level, it poses a challenging set of questions, even in the infrared, where a UV-complete theory of quantum gravity seems unnecessary. All observers see a horizon, which leads to questions about the precise nature of De Sitter (DS) temperature and the appropriate microstate description. These horizons, along with the spacelike boundaries of the global spacetime, have made the proper holographic description elusive. Even at the level of perturbatively computing correlation functions, certain quantum field theories in DS face large infrared sensitivities that grow with time. The purpose of this work is to understand how one properly computes in one class of these theories, very light, non-derivatively-interacting scalars on a fixed De Sitter background.1 The main results are technical, and yet provide 1 As shown in [1–9], single-field inflation does not suffer from large infrared sensitivities in perturbation theory because the inflaton is determining the geometry. However, this perturbative breakdown would arise in an inflationary theory with a light spectator scalar for a sufficiently long period of inflation. –1– JHEP07(2020)119 1 Introduction –2– JHEP07(2020)119 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]. For an overview of more recent literature, see [13]. What is novel in our work is a rigorous, all-orders, diagrammatic derivation of the evolution equation for light-scalar correlation functions in De Sitter. A key simplifying feature in our presentation comes from the constraints of manifest causality [14], following from the reorganization of in-in perturbation theory given by Weinberg [15]. Another key feature is identifying the simple structure of propagators and vertices in the soft limit, with a careful accounting of where the soft approximation breaks down within hard loops (cf. appendix B). Recast as the evolution of a generating function, at leading-order in controlled approximations that we make explicit, we recover Starobinsky’s Fokker-Planck equation for stochastic inflation. An earlier diagrammatic approach to deriving stochastic inflation can be found in [16, 17], but without manifest causality it has a different character. As we will show, the leading infrared contributions to correlation functions are given by the convolution of causal, classical perturbative evolution with quantum noise. Using an a non-manifestly-causal basis, this simple property is obscured in these earlier diagrammatic papers. Furthermore, the graphical analysis in them maximizes the IR enhancement at each vertex, whereas in our formulation, we find in section 3 that one must analyze a diagram globally to capture the dominant soft physics. The key physical insight of stochastic inflation is that superhorizon modes in De Sitter follow a first-order, inhomogeneous classical equation of motion. The inhomogeneity is given by a stochastic source with a known distribution. Its intrinsic randomness is the remaining quantum feature in the problem. It reflects the fact that all comoving modes in De Sitter redshift, and even those in the UV that have heretofore admitted a healthy perturbative description will eventually “fall” into the nonperturbative regime. This entry into the strongly-interacting superhorizon sector can still be described by perturbation theory though. By power counting, we will derive the structure of De Sitter Feynman diagrams that has the leading sensitivity to the infrared breakdown of perturbation theory (or equivalently, the leading secular growth). The ingredients for this are nothing other than causality and the momentum scaling of two different types of propagators that arise in the in-in formalism for correlators. These leading diagrams then make sharp the sense in which the infrared of De Sitter field the (...truncated)