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, $\ell$. (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.
