The effects of anisotropy and non-adiabaticity on the evolution of comoving curvature perturbation

We derive the equation for the evolution of the curvature perturbation on the comoving time slice, $\mathcal{R}_c$, in the presence of anisotropic and non-adiabatic terms in the energy-momentum tensor of matter fields. The equation is obtained by manipulating the perturbed Einstein's equations in the comoving time slice. It could be used to study the evolution of the comoving curvature perturbations for systems with an anisotropic energy-momentum tensor, such as in the presence of vector fields, in the presence of entropy, such as in a multi-field system, or in modified gravity theories. As a simple application, after checking that the comoving time slice for a multi-field system does not coincide with the uniform field time slice in general, we use the equation in the case of two minimally coupled scalar fields and derive a closed set of equations for the curvature and entropy perturbations on the comoving time slice.