Inflation: flow, fixed points and observables to arbitrary order in slow roll

TL;DR

The paper generalizes the inflationary flow formalism to arbitrary order in slow roll and analyzes predictions in the observable space of $r$, $n$, and $d n / d \ln k$ for single-field inflation. Using a Monte Carlo approach up to fifth order, it reveals strong clustering of models and identifies two primary attractor classes, with most trajectories ending at $r=0$ or approaching a nontrivial small-$r$ region; importantly, power-law inflation is not a universal attractor. The results highlight a generic, high-order structure of inflationary predictions and motivate a holographic interpretation via RG flow in a boundary CFT, while acknowledging limitations from truncation and single-field assumptions. These insights provide a more precise, model-agnostic picture of inflationary observables and their interrelations, guiding both data interpretation and theoretical exploration.

Abstract

I generalize the inflationary flow equations of Hoffman and Turner to arbitrary order in slow roll. This makes it possible to study the predictions of slow roll inflation in the full observable parameter space of tensor/scalar ratio $r$, spectral index $n$, and running $d n / d \ln k$. It also becomes possible to identify exact fixed points in the parameter flow. I numerically evaluate the flow equations to fifth order in slow roll for a set of randomly chosen initial conditions and find that the models cluster strongly in the observable parameter space, indicating a ``generic'' set of predictions for slow roll inflation. I comment briefly on the the interesting proposed correspondence between flow in inflationary parameter space and renormalization group flow in a boundary conformal field theory.

Figures (7)

• Figure 1: Models plotted in the $(n,r)$ plane for an $M=5$ Monte Carlo. The solid line is the power-law fixed point $n = 1 - 2 r / (1 - r)$. The error bar shows the size of the expected $2\ \sigma$ error from Planck. (See note in text regarding the somewhat unconventional normalization of $r$ used here.)
• Figure 2: Figure 1 zoomed in to the region preferred by observation.
• Figure 3: Spectral index vs. $\log(r)$, showing the behavior of the attractor region for small r.
• Figure 4: Models plotted in the $(n,dn/d\ln k)$ plane for an $M=5$ Monte Carlo.
• Figure 5: Models plotted in the $(r,dn/d\ln k)$ plane for an $M=5$ Monte Carlo.