Consistent perturbations in an imperfect fluid

TL;DR

This work develops a fluid-based framework to analyze linear cosmological perturbations in broad scalar-field dark-energy theories whose energy-momentum tensors are imperfect fluids due to second-derivative terms. By decomposing the EMT covariantly into fluid variables and deriving two closure relations from the underlying degrees of freedom, the authors show that hydrodynamics does not generally apply and introduce a general parameterisation of closures that captures k-essence and higher-derivative effects. A key result is the identification of a new transition scale $k_T$ that separates perfect- and imperfect-fluid regimes, with the Jeans scale governed by the physical sound speed $c_s^2$ and the anisotropic stress playing a crucial role in small-scale dynamics. The worked example of non-minimally coupled k-essence demonstrates how second-derivative terms modify perturbation evolution, alter the lensing potential, and produce scale-dependent growth, providing a framework applicable to broader Horndeski-type theories and guiding observational tests of dark-energy models.

Abstract

We present a new prescription for analysing cosmological perturbations in a more-general class of scalar-field dark-energy models where the energy-momentum tensor has an imperfect-fluid form. This class includes Brans-Dicke models, f(R) gravity, theories with kinetic gravity braiding and generalised galileons. We employ the intuitive language of fluids, allowing us to explicitly maintain a dependence on physical and potentially measurable properties. We demonstrate that hydrodynamics is not always a valid description for describing cosmological perturbations in general scalar-field theories and present a consistent alternative that nonetheless utilises the fluid language. We apply this approach explicitly to a worked example: k-essence non-minimally coupled to gravity. This is the simplest case which captures the essential new features of these imperfect-fluid models. We demonstrate the generic existence of a new scale separating regimes where the fluid is perfect and imperfect. We obtain the equations for the evolution of dark-energy density perturbations in both these regimes. The model also features two other known scales: the Compton scale related to the breaking of shift symmetry and the Jeans scale which we show is determined by the speed of propagation of small scalar-field perturbations, i.e. causality, as opposed to the frequently used definition of the ratio of the pressure and energy-density perturbations.