Inflating in a Trough: Single-Field Effective Theory from Multiple-Field Curved Valleys

TL;DR

The paper addresses how multiple scalar fields with a trough-like potential can be effectively described by a single light field, deriving a covariant low-energy EFT that remains valid as long as three geometric scales—$m$, $\kappa$, and $\rho$—are large enough to suppress heavy-field dynamics.By integrating out the heavy mode and working in a covariant field-space framework, it produces a $P(\ell,X)$-type action for the light field with explicit expressions for $V_{\rm eff}$, $G_{\rm eff}$, and $\mathcal{H}_{\rm eff}$ in terms of trough data, and demonstrates how curved troughs generate a centrifugal $\mathcal{H}_{\rm eff}(\partial\ell)^4$ term.The EFT is tested on flat examples (Mexican and Cowboy hats) and then applied to inflationary cosmology, where it yields corrected slow-roll parameters, a reduced speed of sound $c_s$, and standard single-field predictions for power spectra and non-Gaussianity, while providing a mapping to Cheung et al.'s EFT of inflation.Overall, the work offers a covariant criterion for when a single-field effective description is justified, clarifies the role of trough curvature in decoupling, and provides practical tools to extract cosmological predictions from multi-field theories.

Abstract

We examine the motion of light fields near the bottom of a potential valley in a multi-dimensional field space. In the case of two fields we identify three general scales, all of which must be large in order to justify an effective low-energy approximation involving only the light field, $\ell$. (Typically only one of these -- the mass of the heavy field transverse to the trough -- is used in the literature when justifying the truncation of heavy fields.) We explicitly compute the resulting effective field theory, which has the form of a $P(\ell,X)$ model, with $X = - 1/2(\partial \ell)^2$, as a function of these scales. This gives the leading ways each scale contributes to any low-energy dynamics, including (but not restricted to) those relevant for cosmology. We check our results with the special case of a homogeneous roll near the valley floor, placing into a broader context recent cosmological calculations that show how the truncation approximation can fail. By casting our results covariantly in field space, we provide a geometrical criterion for model-builders to decide whether or not the single-field and/or the truncation approximation is justified, identify its leading deviations, and to efficiently extract cosmological predictions.