Inflating in a trough: single-field effective theory from multiple-field curved valleys

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, ℓ. (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(ℓ, X) model, with $$ X=-\frac{1}{2}{{\left( {\partial \ell } \right)}^2} $$, 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.