Second-Order Cosmological Perturbations from Inflation

TL;DR

This paper develops a complete second-order perturbation theory for single-field slow-roll inflation, providing a gauge-invariant definition of the second-order comoving curvature perturbation ${\cal R}$ that is conserved on super-horizon scales. By solving the second-order Einstein equations in a generalized longitudinal gauge, it derives the intrinsic second-order curvature perturbations and their sourcing by products of first-order modes, leading to a predicted curvature bispectrum characterized by a kernel $\mathcal{K}$ and an effective non-Gaussian parameter $f_{NL}$. The results express the bispectrum in terms of slow-roll parameters and the scalar spectral index $n_S$ (and ratio $r$), linking inflationary dynamics to a measurable CMB non-Gaussian signal, though the signal is expected to be small for slow-roll models. The work complements Maldacena's cubic-action approach and provides a framework for interpreting future observations (MAP/Planck) of primordial non-Gaussianity, highlighting the experimental challenge of detecting the second-order signature.

Abstract

We present the first computation of the cosmological perturbations generated during inflation up to second order in deviations from the homogeneous background solution. Our results, which fully account for the inflaton self-interactions as well as for the second-order fluctuations of the background metric, provide the exact expression for the gauge-invariant curvature perturbation bispectrum produced during inflation in terms of the slow-roll parameters or, alternatively, in terms of the scalar spectral $n_S$ and and the tensor to adiabatic scalar amplitude ratio $r$. The bispectrum represents a specific non-Gaussian signature of fluctuations generated by quantum oscillations during slow-roll inflation. However, our findings indicate that detecting the non-Gaussianity in the cosmic microwave background anisotropies emerging from the second-order calculation will be a challenge for the forthcoming satellite experiments.