Interacting viscous mixtures

Gravitational and hydrodynamical perturbations are analysed in a relativistic plasma containing a mixture of interacting fluids characterized by a non-negligible bulk viscosity coefficient. The energy-momentum transfer between the cosmological fluids, as well as the fluctuations of the bulk viscosity coefficients, are analyzed simultaneously with the aim of deriving a generalized set of evolution equations for the entropy and curvature fluctuations. For typical length scales larger than the Hubble radius, the fluctuations of the bulk viscosity coefficients and of the decay rate provide source terms for the evolution of both the curvature and the entropy fluctuations. According to the functional dependence of the bulk viscosity coefficient on the energy densities of the fluids composing the system, the mixing of entropy and curvature perturbations is scrutinized both analytically and numerically. e-mail address: [email protected] If a relativistic plasma contains a mixture of inviscid fluids with negligible transfer of energy and momentum, the evolution of entropy fluctuations is characterized, in the longwavelength limit, by the absence of source terms containing curvature perturbations. This property has relevant consequences on the dynamics of the coupled system of gravitational and hydrodynamical perturbations. It implies, for instance, that curvature perturbations are conserved, in the long-wavelength limit, under rather general assumptions [1, 2, 3]. Longwavelength fluctuations in the spatial curvature determine, via the Sachs–Wolfe effect, the temperature inhomogeneities observed in the microwave sky (see, for instance, [4]). Mixtures of relativistic fluids are a useful toy model that can be investigated with the purpose of inferring some general properties of the evolution equations of curvature and entropy fluctuations. Moreover, multifluid systems are per se relevant to the model-independent discussion of initial conditions of CMB anisotropy [1, 4]: for instance, the five isocurvature modes supported by the predecoupling plasma may be discussed, in their simplest realization, by a truncated Einstein–Boltzmann system of equations whose lower multipole moments reproduce indeed a multifluid hydrodynamical description [5]. One of the assumptions often invoked in the analysis of multifluid systems is that the bulk viscosity coefficient and its possible spatial variation have a negligible impact on the dynamics. While this assumption may be justified in some specific system, it may not be true in the early stages of the life of the Universe (see, for instance, [6, 7, 8]). Unlike other dissipative effects, the presence of bulk viscosity does not spoil the isotropy of the background geometry. Therefore, consider a mixture of two relativistic fluids (the a-fluid and the b-fluid) obeying a set of generally covariant evolution equations formed by the Einstein equations 2 R μ − 1 2 δ μR = 1 2 T ν μ (1) and by the evolution equations of the energy-momentum tensors of each fluid of the mixture, i.e. ∇μT μν a = −Γguβ(pa + ρa), (2) ∇μT μν b = Γguβ(pa + ρa), (3) where uβ is the total velocity field of the mixture. Equations (2) and (3) describe the situation where the a-fluid decays into the b-fluid with decay rate Γ. It is evident from the form of Eqs. (2) and (3) that the total energy-momentum tensor of the mixture, i.e. T μν = T μν a + T μν b is covariantly conserved, i.e. ∇μT μν = 0. The total energy-momentum tensor of each species is given by the sum of an inviscid contribution, denoted by T μν a, b and Units of 8πG = 1 will be used throughout. Notice, to avoid confusions, that the Latin (lower-case roman) subscripts a, b, c d, ... will denote, in the present paper, different fluids present in the relativistic plasma. Greek (lower-case) subscripts will denote tensor indices. Latin (lower-case italic) subscripts i, j, k, ... will denote the spatial components of a tensor.