The effective speed of sound in cosmological perturbation theory

TL;DR

This paper introduces a new, gauge-invariant definition of the effective speed of sound in complex multi-field and multi-fluid cosmological systems, which governs the propagation of perturbations and generalizes previous single-field concepts.

Contribution

It defines a background-dependent effective speed of sound for multi-field/fluid systems, extending Garriga and Mukhanov's single-field formulation, and derives evolution equations for perturbations in such systems.

Findings

Effective speed of sound governs perturbation propagation at small scales.

In large-scale limit, adiabatic initial conditions remain adiabatic throughout evolution.

Multi-field/fluid universe behaves as a single barotropic fluid at large scales.

Abstract

In a multi-field/fluid cosmological system consisting of a number of minimally coupled canonical scalar fields, non-canonical scalar fields and barotropic perfect fluids, we introduce a new definition of effective speed of sound of the entire system for describing the evolution of cosmological perturbations. This effective speed of sound is not just gauge invariant but also a background dependent quantity and therefore can 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. Further, the effective speed of sound defined in this paper generalizes the one defined by Garriga and Mukhanov for a single non-canonical scalar field to a system consisting of many minimally coupled barotropic perfect fluids, canonical and non-canonical scalar fields. Moreover, just like in the case of a single pure-kinetic non-canonical scalar field, this effective speed of the sound of the total system turns out to be identically equal to the total adiabatic speed of sound when the dynamic of the universe is driven by a number of pure kinetic non-canonical scalar fields making such a system tantamount to a system of equivalent multi barotropic perfect fluids. We also derive a set of equations which governs the evolution of perturbations in a general multi-field/fluid universe containing barotropic perfect fluids, canonical and non-canonical scalar fields. Using these equations we show that, in the large scale limit ($k = 0$ scales), if the perturbations are initially adiabatic, then it continues to remain so at that scales throughout the evolution of the universe. Consequently, at that scales, such multi-field/fluid universe dynamically behaves as if it contains only a single barotropic perfect fluid.