[Paper Review] Inflationary perturbations with multiple scalar fields

B. Van Tent, Stefan Groot Nibbelink|ArXiv.org|Nov 28, 2001

Cosmology and Gravitation Theories6 references3 citations

TL;DR

This paper develops a first-order slow-roll formalism for computing scalar gravitational and matter perturbations during multi-field inflation with non-minimal kinetic terms. It introduces a dynamically induced orthonormal basis in field space and shows that perpendicular field perturbations—via the slow-roll function $\tilde{\eta}^\perp$—can contribute at leading order to the gravitational potential correlator and spectral index, even when suppressed by slow-roll factors, as demonstrated numerically in a quadratic potential model.

ABSTRACT

The calculation of scalar gravitational and matter perturbations during multiple-field inflation valid to first order in slow roll is discussed. These fields may be the coordinates of a non-trivial field manifold and hence have non-minimal kinetic terms. A basis for these perturbations determined by the background dynamics is introduced, and the slow-roll functions are generalized to the multiple-field case. Solutions for a perturbation mode in its three different behavioural regimes are combined, leading to an analytic expression for the correlator of the gravitational potential. Multiple-field effects caused by the coupling to the field perturbation perpendicular to the field velocity can even contribute at leading order. This is illustrated numerically with an example of a quadratic potential. (The material here is based on previous work by the authors presented in hep-ph/0107272.)

Motivation & Objective

• To generalize scalar perturbation theory to multiple scalar fields with non-minimal kinetic terms in a curved field manifold.
• To identify and quantify multiple-field effects in the gravitational potential correlator during slow-roll inflation.
• To develop a formalism that distinguishes between adiabatic and entropy-type contributions using a dynamically induced field-space basis.
• To compute the vacuum correlator of the gravitational potential at recombination, including first-order slow-roll corrections.
• To determine the spectral index $n-1$ in the multi-field context, accounting for non-adiabatic contributions.

Proposed method

• Introduce a field-space orthonormal basis $\{\mathbf{e}_1, \mathbf{e}_2, \ldots\}$ induced by the background field dynamics to separate adiabatic and entropy modes.
• Generalize slow-roll parameters into vector-valued functions, particularly $\tilde{\eta}^\perp$, which measures field acceleration perpendicular to the background velocity.
• Define Mukhanov-Sasaki variables as a vector on the field manifold to unify gravitational and matter perturbations in a covariant way.
• Solve the coupled system of perturbation equations in three asymptotic regimes (sub-Hubble, super-Hubble, and transition) using analytic matching in $k\eta$.
• Construct the full first-order solution for the gravitational potential by combining mode solutions across regimes, with explicit treatment of particular solutions.
• Evolve the perturbations post-inflation under adiabaticity to compute the correlator at recombination, including the spectral index $n-1$.

Experimental results

Research questions

• RQ1 How do multiple scalar fields with non-minimal kinetic terms affect the gravitational potential correlator during inflation?
• RQ2 Can entropy perturbations—specifically those perpendicular to the field velocity—contribute at leading order in the gravitational potential, despite slow-roll suppression?
• RQ3 What is the role of the generalized slow-roll function $\tilde{\eta}^\perp$ in determining the amplitude and spectral index of the primordial power spectrum?
• RQ4 How does the dynamically induced field-space basis improve the separation of adiabatic and non-adiabatic contributions in multi-field inflation?
• RQ5 To what extent do particular solution terms in the gravitational potential equation dominate the final spectrum, even when they appear with slow-roll factors?

Key findings

• The gravitational potential correlator at recombination is computed analytically to first order in slow roll, with relative errors significantly smaller than $\mathcal{O}(\tilde{\epsilon}_\mathcal{H})$, validating the approximation.
• The particular solution terms, driven by perpendicular field perturbations, contribute nearly half of the total amplitude and spectral index in the numerical example, despite being suppressed by $\tilde{\eta}^\perp$.
• Multiple-field effects from $\tilde{\eta}^\perp$ can enter at leading order in the gravitational potential, challenging the assumption that such terms are always subleading.
• The spectral index $n-1$ is derived in the multi-field context, with contributions from both homogeneous and particular solution terms.
• The slow-roll approximation for $U_P$ remains accurate even near the end of inflation, as confirmed by numerical comparison with exact results.
• The formalism successfully separates adiabatic and non-adiabatic contributions via the dynamically induced orthonormal basis, enabling precise quantization and perturbation analysis.