Adiabatic and Isocurvature Perturbation Projections in Multi-Field
  Inflation
  Current data are in good agreement with the predictions of single field
inflation. However, the hemispherical asymmetry seen in the cosmic microwave
background data, may hint at a potential problem. Generalizing to multi-field
models may provide one possible explanation. A useful way of modeling
perturbations in multi-field inflation is to investigate the projection of the
perturbation along and perpendicular to the background fields' trajectory.
These correspond to the adiabatic and isocurvature perturbations. However, it
is important to note that in general there are no corresponding adiabatic and
isocurvature fields. The purpose of this article is to highlight the
distinction between a field redefinition and a perturbation projection. We
provide a detailed derivation of the evolution of the isocurvature perturbation
to show that no assumption of an adiabatic or isocurvature field is needed. We
also show how this evolution equation is consistent with the field covariant
evolution equations for the isocurvature perturbation in the flat field space
limit.
