The adiabatic/entropy decomposition in $P(\phi^I,X^{IJ})$ theories with
  multiple sound speeds
  We consider $P(\phi^I,X^{IJ})$ theories of multi-field inflation and ask the
question of how to define the adiabatic and entropy perturbations, widely used
in calculating the curvature and isocurvature power spectra, in this general
context. It is found that when the field perturbations propagate with different
speeds, these adiabatic and entropy modes are not generally the fundamental
(most natural to canonically quantise) degrees of freedom that propagate with a
single speed. The alternative fields which do propagate with a single speed are
found to be a rotation in field space of the adiabatic and entropy
perturbations. We show how this affects the form of the horizon-crossing power
spectrum, when there is not a single "adiabatic sound speed" sourcing the
curvature perturbation. Special cases of our results are discussed, including
$P(X)$ theories where the adiabatic and entropy perturbations are fundamental.
We finally look at physical motivations for considering multi-speed models of
inflation, particularly showing that disformal couplings can naturally lead to
the kind of kinetic interactions which cause fields to have different sound
speeds.
