The effective speed of sound in cosmological perturbation theory

Abstract

In a multi-field/fluid cosmological system consisting of minimally coupled canonical scalar fields, non-canonical scalar fields, and barotropic perfect fluids, we introduce a new definition for the effective speed of sound of the entire system to describe the evolution of cosmological perturbations. This effective speed of sound is not only gauge-invariant but also a background-dependent quantity; it can, therefore, be treated as a parameter to quantify perturbations in such multi-field/fluid systems. It is with this effective speed that the gauge-invariant Bardeen potential and the curvature perturbation propagate at scales much smaller than the sound horizon. Furthermore, the effective speed of sound defined in this paper generalizes the one provided by Garriga and Mukhanov for a single non-canonical scalar field to a system consisting of multiple minimally coupled barotropic perfect fluids, canonical scalar fields, and non-canonical scalar fields. Moreover, as in the single pure-kinetic non-canonical scalar field case, this effective speed of sound for the total system equals the total adiabatic speed of sound when the dynamics are driven by multiple pure-kinetic non-canonical scalar fields. This makes such a system tantamount to a system of equivalent multi-barotropic perfect fluids. We also derive a set of equations governing the evolution of perturbations in a general multi-field/fluid universe. Using these equations, we demonstrate that in the large-scale limit ($k \to 0$), initially adiabatic perturbations remain adiabatic at those scales throughout the evolution, extending this well-known result to a general multi-field/fluid system containing non-canonical scalar fields. Consequently, at those scales, such a multi-field/fluid universe dynamically behaves as if it contains only a single barotropic perfect fluid.