[Paper Review] Beyond \delta N formalism

Atsushi Naruko, Yu-ichi Takamizu+1 more|arXiv (Cornell University)|Oct 24, 2012

Cosmology and Gravitation Theories3 citations

TL;DR

This paper extends the δN formalism to next-to-leading order in the spatial gradient expansion (𝒪(ε²)), enabling a fully nonlinear analysis of superhorizon curvature perturbations in multi-field inflation with general kinetic and potential terms. It derives nonlinear gauge transformation rules valid through 𝒪(ε²), allowing solutions to be computed in simpler gauges and transformed to physical ones, with explicit application to an analytically solvable model.

ABSTRACT

We develop a theory of nonlinear cosmological perturbations on superhorizon scales for a multi-component scalar field with a general kinetic term and a general form of the potential in the context of inflationary cosmology. We employ the ADM formalism and the spatial gradient expansion approach, characterised by O(\\epsilon^2), where \\epsilon=1/(HL) is a small parameter representing the ratio of the Hubble radius to the characteristic length scale L of perturbations. We provide a formalism to obtain the solution in the multi-field case. This formalism can be applied to the superhorizon evolution of a primordial non-Gaussianity beyond the so-called \\delta N formalism which is equivalent to O(\\epsilon^0) of the gradient expansion. In doing so, we also derive fully nonlinear gauge transformation rules valid through O(\\epsilon^2). These fully nonlinear gauge transformation rules can be used to derive the solution in a desired gauge from the one in a gauge where computations are much simpler. As a demonstration, we consider an analytically solvable model and construct the solution explicitly.

Motivation & Objective

• To develop a nonlinear formalism for superhorizon curvature perturbations beyond the standard δN approach, which is limited to 𝒪(ε⁰) in gradient expansion.
• To extend the δN formalism to multi-component scalar fields with general kinetic and potential terms, including non-slow-roll dynamics.
• To derive fully nonlinear gauge transformation rules valid through 𝒪(ε²), enabling computation in convenient gauges and transformation to physical gauges.
• To provide a systematic framework for computing primordial non-Gaussianities beyond the δN formalism, particularly where decaying modes become relevant.
• To demonstrate the formalism with an analytically solvable model, showing explicit construction of solutions at 𝒪(ε²).

Proposed method

• Employing the ADM formalism and spatial gradient expansion with ε = 1/(HL) as the small parameter, where H is the Hubble parameter and L is the characteristic length scale of perturbations.
• Deriving the Hamiltonian and momentum constraints at 𝒪(ε²) in the gradient expansion, ensuring consistency of the dynamical equations.
• Deriving fully nonlinear gauge transformation rules between different slicing conditions (e.g., comoving, uniform curvature) up to 𝒪(ε²), valid for general multi-field systems.
• Using the energy-momentum conservation and scalar field equations to show that the leading-order momentum constraint is automatically satisfied if the Hamiltonian constraint and field equations hold at next-to-leading order.
• Constructing the solution in a gauge where computations are simplified (e.g., uniform energy density gauge) and transforming it to a physical gauge via the derived nonlinear gauge transformations.
• Applying the formalism to a solvable multi-field model with general kinetic and potential terms to explicitly compute the superhorizon evolution of curvature perturbations at 𝒪(ε²).

Experimental results

Research questions

• RQ1 How can the δN formalism be systematically extended beyond 𝒪(ε⁰) to include nonlinear effects at 𝒪(ε²) in multi-field inflation?
• RQ2 What are the fully nonlinear gauge transformation rules between different slicing conditions at 𝒪(ε²), and how can they be used to simplify computations?
• RQ3 Under what conditions does the decaying mode of the curvature perturbation become non-negligible in superhorizon evolution, and how is it captured in the extended formalism?
• RQ4 Can the extended formalism consistently describe non-slow-roll dynamics in multi-field inflation, where the standard δN formalism breaks down?
• RQ5 How does the inclusion of general kinetic terms and non-trivial potentials affect the superhorizon evolution of curvature perturbations beyond linear order?

Key findings

• The leading-order momentum constraint is automatically satisfied if the Hamiltonian constraint and scalar field equations hold at next-to-leading order in the gradient expansion, simplifying the dynamical system.
• Nonlinear gauge transformation rules are derived up to 𝒪(ε²), enabling the computation of solutions in a convenient gauge and their transformation to physical gauges such as the comoving or uniform curvature gauge.
• The formalism captures the evolution of curvature perturbations beyond the δN approximation, including the influence of decaying modes that appear at 𝒪(ε²), which are crucial in non-slow-roll or multi-field scenarios.
• The method is applicable to general multi-component scalar fields with arbitrary kinetic and potential terms, extending the δN formalism beyond single-field, slow-roll models.
• An analytically solvable model is used to demonstrate the formalism, showing explicit construction of the superhorizon solution at 𝒪(ε²), confirming consistency and predictive power.
• The absence of the adiabatic decaying mode at leading order in gradient expansion is preserved in multi-field systems, provided the background is stable against homogeneous anisotropic perturbations.