Cosmological perturbations for imperfect fluids

Interacting fluids, endowed with bulk viscous stresses, are discussed in a unified perspective with the aim of generalizing the treatment of cosmological perturbation theory to the case where both fluctuating decay rates and fluctuating bulk viscosity coefficients are simultaneously present in the relativistic plasma. A gauge-invariant treatment of the qualitatively new phenomena arising in this context is provided. In a complementary approach, faithful gauge-fixed descriptions of the gravitational and hydrodynamical fluctuations are developed and exploited. To deepen the interplay between bulk viscous stresses and fluctuating decay rates, illustrative examples are proposed and discussed both analytically and numerically. Particular attention is paid to the coupled evolution of curvature and entropy fluctuations when, in the relativistic plasma, at least one of the interacting fluids possesses a fluctuating bulk viscosity coefficient. It is argued that this class of models may be usefully employed as an effective description of the decay of the inflaton as well as of other phenomena involving imperfect relativistic fluids. e-mail address: [email protected] 1 Formulation of the problem It is plausible, as observations suggest, that the large-scale temperature fluctuations detected in the microwave sky may arise, via the Sachs–Wolfe effect, from primordial curvature perturbations amplified during the early stages of the life of the Universe. It is therefore mandatory for those interested in this possibility to scrutinize in detail the evolution of curvature fluctuations when the composition of the primeval plasma cannot be parametrized in terms of a single, perfect, relativistic fluid. A class of dissipative effects that can enter the energy-momentum tensor without spoiling the isotropy of the background geometry can be parametrized in terms of a bulk viscosity coefficient [1] (see also [2]). In fact, in the relativistic theory of imperfect fluids, the dissipative effects can be parametrized in terms of a shear coefficient, a heat flow coefficient and a bulk viscosity coefficient. When written in general coordinates [1] (see also [3]), the contributions of the shear tensor and of the heat flow (unlike the bulk viscosity) do not contribute to a homogeneous and isotropic background geometry of Friedmann-Robertson-Walker type [1]. Following then Ref. [1], for a single fluid, the total energy-momentum T ν μ tensor can then be split into a perfect contribution, denoted in the following by T ν μ , and into an imperfect contribution, denoted by ∆T ν μ , i.e. T ν μ = T ν μ +∆T ν μ , (1.1) In general coordinates, and within our set of conventions, the contribution of bulk viscous stresses can be written, in turn, as