Model-independent approach to effective sound speed in multi-field inflation

For any physical system satisfying the Einstein's equations, the comoving curvature perturbations satisfy an equation involving the momentum-dependent effective sound speed, valid for any system with a well defined energy-stress tensor, including multi-fields models of inflation. We derive a general model-independent formula for the effective sound speed of comoving adiabatic perturbations, valid for a generic field-space metric, without assuming any approximation to integrate out entropy perturbations, but expressing the momentum-dependent effective sound speed in terms of the components of the total energy-stress tensor. As an application, we study a number of two-field models with a kinetic coupling between the fields, identifying the single curvature mode of the effective theory and showing that momentum-dependent effective sound speed fully accounts for the predictions for the power spectrum of curvature perturbations. Our results show that the momentum-dependent effective sound speed is a convenient scheme for describing all inflationary models that admit a single-field effective theory, including the effects of entropy perturbations present in multi-fields systems.