Second-order cosmological perturbations produced by scalar-scalar coupling during inflation stage

Abstract

We study the perturbations up to the 2nd-order for a power-law inflation driven by a scalar field in synchronous coordinates. We present the 1st-order solutions, and analytically solve the 2nd-order perturbed Einstein equation and scalar field equation, give the 2nd-order solutions for all the scalar, vector, and tensor metric perturbations, as well as the perturbed scalar field. During inflation, the 1st-order tensor perturbation is a wave and is decoupled from other perturbations, the scalar metric perturbation and the perturbed scalar field are coupled waves, propagating at the speed of light, differing from those in the dust and relativistic fluid models. The 1st-order vector perturbation is not wave and just decreases during inflation. The 2nd-order perturbed Einstein equation is similar in structure to the 1st-order one, but various products of the 1st-order perturbations occur as the effective source, among which the scalar-scalar coupling is considered in this paper. The solutions of all the 2nd-order perturbations consist of a homogeneous part similar to the 1st-order solutions, and an inhomogeneous part in a form of integrations of the effective source. The 2nd-order vector perturbation is also a wave since the effective source is composed of the 1st-order waves. We perform the residual gauge transformations between synchronous coordinates up to the 2nd-order, and identify the 1st-order and 2nd-order gauge modes.

54 pages, 2 figures

References in corpus (22)

• Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations
• The cosmological gravitational wave background from primordial density perturbations
• Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations
• Relic Gravitational Waves in the Accelerating Universe
• Imprints of Relic Gravitational Waves in Cosmic Microwave Background Radiation
• Gauge-invariant Formulation of the Second-order Cosmological Perturbations
• Analytic spectrum of relic gravitational waves modified by neutrino free streaming and dark energy
• Third-order Perturbation Theory With Non-linear Pressure
• An exact analytic spectrum of relic gravitational waves in an accelerating universe
• Vector modes generated by primordial density fluctuations
• Second-order perturbations of cosmological fluids: Relativistic effects of pressure, multi-component, curvature, and rotation
• Approximate Analytic Spectra of Reionized CMB Anisotropies and Polarization generated by Relic Gravitational Waves
• Modifications by QCD transition and $e^{+}{e}^{-}$ annihilation on analytic spectrum of relic gravitational waves in accelerating universe
• CMB Temperature Polarization Correlation and Primordial Gravitational Waves
• Exact non-linear equations for cosmological perturbations
• Second-order cosmological perturbations. IV. Produced by scalar-tensor and tensor-tensor couplings during the radiation dominated stage
• A Coarse-Grained Field Theory for Density Fluctuations and Correlation Functions of Galactic Objects
• Field Theory of the Correlation Function of Mass Density Fluctuations for Self-Gravitating Systems
• Fully nonlinear and exact perturbations of the Friedmann world model: Non-flat background
• Nonlinear Density Fluctuation Field Theory for Large Scale Structure
• Field Equation of Correlation Function of Mass Density Fluctuation for Self-Gravitating Systems
• Maxwell field with gauge fixing term in de Sitter space: exact solution and stress tensor

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