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 definition 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 case of a single pure-kinetic non-canonical scalar field, this effective speed of sound for the total system turns out to be identically equal to the total adiabatic speed of sound when the dynamics of the universe 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 (\(k \rightarrow 0\)), if the perturbations are initially adiabatic, they remain so at those scales throughout the evolution of the universe, thus extending this well-known result to a general multi-field/fluid system consisting of 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.