Infrared effects in inflationary correlation functions

TL;DR

Infrared effects in inflationary correlation functions are analyzed, focusing on three classes of divergences: time-dependent logs, box-cutoff logs, and logs from new physics. The paper reviews the origin of these logs, the calculations where they arise, and the methods proposed to control them, including δN formalism and dynamical renormalization group techniques. It also discusses spatial evolution via mosaicking and stochastic inflation, the possibility of dynamically generated masses, and implications for observable quantities such as the tensor-to-scalar ratio. The overall message is that, while infrared effects complicate predictions, these resummation frameworks provide a path to robust initial conditions and comprehension of UV-IR interplay in inflationary cosmology.

Abstract

In this article, I briefly review the status of infrared effects which occur when using inflationary models to calculate initial conditions for a subsequent hot, dense plasma phase. Three types of divergence have been identified in the literature: secular, "time-dependent" logarithms, which grow with time spent outside the horizon; "box-cutoff" logarithms, which encode a dependence on the infrared cutoff when calculating in a finite-sized box; and "quantum" logarithms, which depend on the ratio of a scale characterizing new physics to the scale of whatever process is under consideration, and whose interpretation is the same as conventional field theory. I review the calculations in which these divergences appear, and discuss the methods which have been developed to deal with them.

Figures (3)

• Figure 1: One-loop corrections to the power spectrum of an active scalar field. In the left-hand diagram a loop of spectator scalar fields, represented by dashed lines, corrects the two-point function of an active field, represented by solid lines. This loop was computed by Weinberg Weinberg:2005vyWeinberg:2006ac. An error in the numerical coefficient was corrected by Adshead, Easther & Lim Adshead:2008gk. In the right-hand diagram, a self-loop corrects the same two-point function of an active scalar, first computed in Ref. Seery:2007we and again in Ref. Adshead:2008gk. In Minkowski space this diagram would factorize, leaving a scale-free integral over the loop momentum $\bm{\mathrm{{q}}}$. The non-trivial time dependence of de Sitter endows the loop with a scale, of the form of Eq. eq:uvlog. This diagram is the leading correction when self-loops are included. It would be accompanied by self-loops of the same form as the left-hand diagram, which are suppressed by powers of the slow-roll parameter $\epsilon$. For spectator fields there is no contribution from the right-hand diagram, so the left-hand loop is the leading term.
• Figure 2: Graviton loop corrections to the power spectrum of a scalar field, calculated by Dimastrogiovanni and Bartolo Dimastrogiovanni:2008af. Unlike the case of scalar loops, the right-hand diagram is not slow-roll suppressed compared to the left-hand diagram. As before, the interior of these diagrams can be considered as a sort of instanton for the nucleation of gravitional (wavy lines) and scalar quanta (straight lines), which propagate to the time of observation $\tau_\ast$ on the external legs of the diagram.
• Figure 3: Loop in pure $\phi^3$ theory. A factor of the coupling $g$ is present at each vertex.