[Paper Review] Second-Order Cosmological Perturbations from Inflation

Viviana Acquaviva, N. Bartolo+2 more|CERN Bulletin|Sep 9, 2002

Cosmology and Gravitation TheoriesPhysics and Astronomy340 citations

TL;DR

This paper presents the first gauge-invariant computation of cosmological perturbations during single-field slow-roll inflation up to second order in perturbation theory. It derives the exact expression for the curvature perturbation bispectrum in terms of slow-roll parameters or scalar spectral index $n_S$ and tensor-to-scalar ratio $r$, showing that non-Gaussianity from second-order effects is small and challenging to detect with upcoming CMB experiments like Planck.

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.

Motivation & Objective

• To compute cosmological perturbations during inflation up to second order in deviations from homogeneity.
• To provide a gauge-invariant definition of the comoving curvature perturbation $\mathcal{R}$ at second order.
• To derive the primordial bispectrum of scalar perturbations as a probe of non-Gaussianity.
• To assess the detectability of second-order non-Gaussianity in CMB anisotropies from upcoming satellite missions.

Proposed method

• Uses a spatially flat Robertson-Walker metric and expands the metric and energy-momentum tensor up to second-order perturbations.
• Applies the formalism of Refs. [6,7] to derive second-order perturbed Einstein equations and Klein-Gordon equations for the inflaton field.
• Implements a gauge-invariant definition of the comoving curvature perturbation $\mathcal{R}$ at second order, ensuring physical consistency.
• Performs a slow-roll expansion to lowest order in slow-roll parameters to study super-horizon evolution.
• Computes the bispectrum of the gauge-invariant gravitational potential using slow-roll parameters or $n_S$ and $r$.
• Validates the conservation of the second-order $\mathcal{R}$ on super-horizon scales, analogous to first-order behavior.

Experimental results

Research questions

• RQ1 What is the exact form of the gauge-invariant curvature perturbation bispectrum generated during single-field slow-roll inflation at second order?
• RQ2 How do inflaton self-interactions and metric fluctuations contribute to non-Gaussianity at second order?
• RQ3 Is the second-order curvature perturbation conserved on super-horizon scales, as in the first-order case?
• RQ4 Can the non-Gaussianity from second-order effects be detected by upcoming CMB experiments like Planck and MAP?
• RQ5 How does the primordial bispectrum depend on observable cosmological parameters such as $n_S$ and $r$?

Key findings

• The paper derives the first exact, gauge-invariant expression for the curvature perturbation bispectrum during single-field slow-roll inflation at second order.
• The second-order comoving curvature perturbation $\mathcal{R}$ is conserved on super-horizon scales, extending the first-order result.
• The primordial bispectrum is expressed in terms of slow-roll parameters or directly in terms of the scalar spectral index $n_S$ and tensor-to-scalar ratio $r$, enabling observational comparison.
• The amplitude of non-Gaussianity from second-order effects is found to be too small to be detected by upcoming satellite experiments such as MAP and Planck.
• The dominant source of non-Gaussianity at second order arises from non-linear gravitational perturbations, not inflaton self-interactions, confirming earlier stochastic approach results.