The Spectrum of Density Perturbations produced during Inflation to Leading Order in a General Slow-Roll Approximation

TL;DR

This work addresses how the density-perturbation spectrum from inflation can be computed without assuming the running of the spectral index is small, using a general slow-roll framework. It develops both series and integral formulations that express the spectrum $\mathcal{P}_{\mathcal{R}_c}$, the spectral index, and its running in terms of slow-roll parameters $\epsilon,\delta_n$ and, alternatively, in terms of the inflaton potential derivatives via $\mathcal{U},\mathcal{V}_n$ and their associated generating functions. The author introduces new coefficients $d_n$ and $q_n$ and constructs a versatile toolkit (including optimized slow-roll) to capture features such as steps and bumps in the potential, or rapid variations in $f(\ln x)$, which can produce ringing and localized spectral features not captured by standard slow-roll. The results have significant implications for model-independent inflationary constraints, enabling precise matching of observed spectra to a broader class of inflationary scenarios with robust, feature-sensitive predictions.

Abstract

The standard calculation of the spectrum of density perturbations produced during inflation assumes (i) |n-1| << 1 and (ii) |dn/dlnk| << |n-1|. Slow-roll predicts and observations require (i), but neither slow-roll nor observations require (ii). In this paper I derive formulae for the spectrum, spectral index, and running of the spectral index, assuming only (i) and not (ii). I give a large class of observationally viable examples in which these general slow-roll formulae are accurate but the standard slow-roll formulae for the spectral index and running are incorrect.

Figures (3)

• Figure 1: The window function $-\frac{k}{aH}\,W'\left(\frac{k}{aH}\right)$ as a function of $\ln\left(\frac{aH}{k}\right)$.
• Figure 2: Spectrum for the arctangent step of Eq. (\ref{['astep']}) using standard slow-roll (long dash), standard slow-roll with 1st order corrections foc (dash) and standard slow-roll with 2nd order corrections Jinook (short dash), all evaluated at $aH=k$, and optimized standard slow-roll (dot dash) and general slow-roll (solid). The parameters are $\lambda=0.1$, $A=\lambda^3/\nu$ and $\lambda\nu=\{0.5,1,2,4\}$, top left to bottom right. Standard slow-roll fails to converge for $\lambda\nu\gtrsim 2$, while optimized standard slow-roll completely misses the ringing that occurs for $\lambda\nu\gtrsim 4$.
• Figure 3: Spectrum for steepening arctangent steps, with $\lambda=0.1$, $A=10^{-5}$ and $\nu=\{10,20,40,80,160\}$, enclosed by the sharp step limit of Eq. (\ref{['sharplimit']}).