Cosmological perturbations with multiple scalar fields

We consider the evolution of perturbed cosmological spacetime with multiple scalar fields in Einstein gravity. A complete set of scalar-type perturbation equations is presented in a gauge-ready form, and the closed sets of second-order differential equations in several useful forms are derived. Conserved behaviors of the perturbed three-space curvature in the comoving gauge, ϕv, under several conditions are clarified. Under the slow-roll conditions, the adiabatic and isocurvature modes decouple from each other, and in the large-scale limit we have (i) the adiabatic mode is generally conserved, (ii) for a couple of special potential the isocurvature modes decouple from each other and are described by conserved quantities, (iii) in the two field system, the isocurvature mode is described by a conserved quantity for the general potential.