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.

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 and by the evolution equations of the energy-momentum tensors of each fluid of the mixture, i.e.
where u β 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.
is covariantly conserved, i.e. ∇ µ T 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 by a viscous contribution, denoted byT µν a, b , i.e.
where the subscript in the various fluid quantities simply means that Eqs. (4), (5) and (6) hold, independently, for the a-and b-fluids. So, for instance, in Eqs. (5) and (6), u µ a and u µ b denote the peculiar velocities of each fluid of the mixture. In a spatially flat metric of Friedmann-Robertson-Walker (FRW) type charaterized by a background line element Eqs. (2) and (3) imply where the prime denotes a derivation with respect to the conformal time coordinate τ and H = a ′ /a. In Eqs. (8) and (9), P a, b denote the total effective pressure of each species, i.e.
while p a, b denote the inviscid pressures of each species. In Eq. (10), ξ a, b denote the bulk viscosity coefficient evaluated on the background geometry. As will be discussed later, the bulk viscosity coefficient may depend on both ρ a and ρ b . Equations (8) and (9) lead to the evolution of the total energy and pressure densities where ρ = ρ a + ρ b and P = P a + P b . Equations (8) and (9) must be supplemented by the explicit background form of Eq. (1), i.e.
where, again, (ρ + P) is the total effective enthalpy that contains the background viscosity coefficient of the mixture ξ = ξ a + ξ b .
We are now interested in deriving the evolution of the entropy and total-curvature fluctuations of the system. Both the entropy perturbations and the perturbations in the total spatial curvature can be written in terms of ζ a and ζ b which are related in the off-diagonal gauge [9] (see also [10,11]) to the density contrasts of the individual fluids of the mixture [12]: In the following we are going to exploit the off-diagonal gauge [9] (also called uniformcurvature gauge [10]), which is particularly convenient for the problem at hand. The results will be exactly the same of those obtainable in the framework of gauge-independent descriptions (see [13]). In fact, the quantities S and ζ, defined in terms of ζ a and ζ b , are invariant under infinitesimal coordinate transformations. Consequently they can be computed in any suitable (non-singular) coordinate system. Concerning Eq. (15) it could be noticed that the gauge-invariant definition of the spatial curvature perturbation can be slightly different for wavelengths smaller than the Hubble radius. However, since in the problem at hand we are mainly interested in super-Hubble fluctuations, Eq. (15) is numerically equivalent to the curvature inhomogeneities defined, more conventionally, from the curvature fluctuations on comoving orthogonal hypersurfaces [2,14,15] (see also [4]). The physical interpretation of the entropy fluctuations defined in Eq. (14) can be understood, for instance, in the case of an inviscid mixture of cold dark matter (CDM) and radiation. In this situation the a-fluid is given by pressureless matter and the b-fluid by radiation. Applying the covariant conservation equations for two (inviscid and non-interacting) species, it is clear that S is nothing but the fractional fluctuation in the specific entropy ς = T 3 /n CDM (where T is the radiation temperature and n CDM is the CDM number density), i.e. the entropy density per CDM particle: where δ CDM and δ r are the density contrasts for the CDM and for the radiation fluids.
In the off-diagonal gauge the spatial components of the perturbed metric vanish and, hence, the only components of the perturbed line element are: Since, in the long-wavelength limit, (H ′ − H 2 )φ = H 2 ζ, it turns out that in the off-diagonal gauge, δg 00 is connected to ζ. As anticipated, ζ a and ζ b can be expressed in terms of the fluctuations of the density contrasts of the individual fluids, i.e.
The evolution equations obeyed by the density contrasts δ a = δρ a /ρ a and δ b = δρ b /ρ b are derived by perturbing Eqs. (2) and (3) to first order in the amplitude of the metric and hydrodynamical fluctuations: Concerning Eqs. (19) and (20) a few comments are in order: • for notational convenience the barotropic indices (i.e. w a , w b ) and the sound speeds (i.e. • δ Γ = δΓ/Γ is the fractional fluctuation of the decay rate computed in the off-diagonal gauge; • δξ a and δξ b denote the fluctuations of the bulk viscosity coefficients; later on it will also be convenient to introduce the fluctuation in the total viscosity, i.e. δξ = δξ a + δξ b ; are the divergences of the peculiar velocity field of each species; note that the global velocity θ = ∂ i v i field (with δT i 0 = (ρ+P)v i ) is recovered from θ a and θ b by recalling that (p+ρ)θ = (p a +ρ a )θ a +(p b +ρ b )θ b .
Equations (19) and (20) must be supplemented by the perturbed components of Eq. (1); in particular by the Hamiltonian and momentum constraints: and by the other two equations stemming from the spatial components (i.e. (i = j) and (i = j)) of Eq. (1): In Eq. (21) the global energy and pressure density fluctuations (i.e. δρ and δp) have been introduced. As is clear from Eqs. (21)-(24), one of the advantages of the off-diagonal formulation is the absence of second time derivatives of the metric fluctuations. Strictly speaking the evolution equations of θ a and θ b should be added to the system. However, they are only relevant for typical length scales smaller than the Hubble radius at a given time.
Since we are interested in the opposite regime, these equations will be omitted, but they will be discussed elsewhere in their full generality (see [13]). By combining Eqs. (21) and (23) the evolution equation for ζ can be easily obtained; it is given byζ where we passed, for later convenience, from the conformal time coordinate τ to the cosmic time coordinate t (i.e. dt = a(τ )dτ ). Equations (19) and (20) lead to the evolution equations of ζ a and ζ b whose explicit form is given bẏ and byζ b + where q a =ρ a H , Various identities can be used to bring Eqs. (25), (26) and (27) to slightly different (but equivalent) forms. In particular: • using Eq. (15), we can always trade the combinations (ζ a −ζ) and • according to Eq. • finally the background evolution of each fluid, i.e. Eqs. (8) and (9), may always be employed to obtain equivalent forms of the above equations.
Specific limits of Eqs. (25)-(27) will now be reproduced. In the limit ξ a = ξ b = 0, with δ Γ = δξ a = δξ b = 0 andΓ = 0, Eqs. (25)-(27) reaḋ In the case w a = 0 and w b = 1/3, Eqs. (29)-(31) coincide with the set of equations used in Ref. [16] to describe the radiative decay of a massive curvaton whose effective pressure, at the oscillatory stage, reproduces that of dusty matter, i.e. w a = 0. It is then clear, taking the difference between Eqs. (31) and (30), that the evolution equation of entropy perturbationṡ is homogeneous and does not contain any ζ-dependent source term.
Sticking to the case of the radiative decay of a dusty fluid, but including the fluctuations of the decay rate, the following system of evolution equationṡ can be derived from Eqs. (25)-(27) when w b = 1/3 and w a = 0. In Eqs. (33) and (34), g a = −Hρ a /ρ a . Equations (33) and (34) describe the situation discussed in Ref. [17], where the dynamics of the inflaton with inhomogeneous decay rate has been discussed (see, for instance, also [18,19,21,20]). If the spatial fluctuations of the decay rate are not a function of the local energy density of the mixture, curvature fluctuations may be generated for length scales larger than the Hubble radius. Consider now the case where the Γ is constant, the decay is homogeneous (i.e. δ Γ = 0), but ξ a = ξ a (ρ a ) and ξ b = ξ b (ρ b ). This occurrence implies that Hence, from Eqs. (25)-(27), we obtain, respectivelẏ Again, in this case, it can be easily argued that the evolution of entropy fluctuations obeys a homogeneous equation in S. In fact, combining Eqs. (37) and (38) it is possible to obtain: Equations (42) and (43) describe the evolution of ζ and δ a for typical wavelengths larger than the Hubble radius. Initial conditions of the system are then set by requiring ζ(t 0 ) = 0 and δ a (t 0 ) = δ b = 0, where t 0 is the initial integration time. From Fig. 1 (plot at the right-hand side) the evolution is such that curvature fluctuations grow from 0 to a value proportional to δ Γ , i.e. proportional to the fluctuations of the decay rate over length scales larger than the Hubble radius. The final asymptotic value of ζ can be determined analytically and it turns out to be In the limit ǫ → 0, the results reproduce the findings of Ref. [17] leading to a Bardeen potential |Ψ final | ≃ δ Γ /9, which implies |ζ final | ≃ δ Γ /6 by using the well-known relation of ζ and Ψ in a radiation-dominated phase (see, for instance, [4]). In the example discussed so far the values of δ Γ have been taken in the ranges 10 −6 -10 −9 . The class of examples reported so far can be generalized in various ways. Different barotropic indices for the fluids of the mixture can be studied. Equation (44) can then be generalized to the cases of generic w a and w b . Furthermore, the functional dependence of the viscosity coefficients can be chosen to be different. Possible generalizations will be present elsewhere [13]. We would like to point out that the simple examples presented here may be made more realistic by thinking that a dust fluid is an effective description of a scalar field oscillating in a quadratic potential [22]. Thus, the simple fluid model of a dust fluid decaying into radiation has been used [17] (with some caveats [20]) to infer some properties of the inflaton decay when the inflaton decay rate is not homogeneous. If the inhomogeneous decay occurs after an inflationary phase at low curvature (i.e. H inf ≪ 10 −6 M P ), it is plausible to argue that the spectrum of δ Γ may be converted into the spectrum of ζ for typical frequencies smaller than the Hubble rate. We are not interested here in supporting a specific model of inhomogeneous reheating. The purpose of the examples discussed so far is purely illustrative. However, the lesson to be drawn is that bulk viscous stresses may play a relevant role.
In the present paper, various results have been achieved. First of all, the concept of interacting viscous mixtures has been introduced, i.e. a mixture of interacting fluids with viscous corrections. In this framework, the coupled evolution of curvature and entropy fluctuations has been derived in the case where both the decay rate and the bulk viscosity coefficients are allowed to fluctuate over typical length scales larger than the Hubble radius. Different situations have been systematically discussed. If the decay rate is constant and homogeneous, with bulk viscosities that depend separately on the energy density of each fluid of the mixture, the evolution of entropy fluctuations obeys a source-free evolution equation. If, on the contrary, the bulk viscosity has a more general dependence on the energy densities of the fluids composing the mixture, the evolution equations of the entropy perturbations may inherit a source term that involves, in one way or another, curvature fluctuations. In similar terms, if the decay rate is allowed to fluctuate without being a function of the local density of the fluid, entropy fluctuations will not obey a source-free equation.