action (\[einstein\]). In spatial-flat gauge which will be applied in this paper, one chooses $\psi=0$, and thus $Q^i$ identified with $\delta\phi^i$.

What we want to calculate is the comoving curvature perturbation $\cal R$, because it can be directly connected with the observables to test our model. However, what we can handle from the action (\[einstein\]) is the field perturbations $Q^i$. Therefore, we need to connect between these two variables. One convenient technique is the so-called “(instantaneous) Adiabatic-Entropy decomposition" [@Gordon:2000hv; @Lalak:2007vi], where one can decompose the field perturbations in the field-space $\{\dot\phi,\dot\chi\}$ into adiabatic and entropy directions, which traces along with/orthogonal to the field trajectory. Perturbations along the two directions are often called adiabatic and entropy perturbations of field respectively, which is proportional to the curvature perturbation $\cal R$ and the isocurvature perturbation, $\cal S$. Note that in single-field limit, there is only one field perturbation and the field-space is also one-dimensional, so the field perturbation can only goes along the field trajectory producing adiabatic perturbation, thus ${\cal S}\rightarrow 0$. In multi-field case however, isocurvature perturbations orthogonal to the trajectory can also exist, and as will be seen later, can act as a source of the adiabatic ones on large scales.

