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Accelerating cosmological expansion from shear and bulk viscosity

The dissipation of energy from local velocity perturbations in the cosmological fluid affects the time evolution of spatially averaged fluid dynamic fields and the cosmological solution of Einstein's field equations. We show how this backreaction effect depends on shear and bulk viscosity and other material properties of the dark sector, as well as the spectrum of perturbations. If sufficiently large, this effect could account for the acceleration of the cosmological expansion.

The dissipation of energy from local velocity perturbations in the cosmological fluid affects the time evolution of spatially averaged fluid dynamic fields and the cosmological solution of Einstein's field equations. We show how this backreaction effect depends on shear viscosity and other material properties of the dark sector, as well as the spectrum of perturbations. If sufficiently large, this effect could account for the acceleration of the cosmological expansion.
The possibility that dissipative phenomena in the form of bulk viscosity might affect the expansion history of the Universe has been discussed repeatedly. Indeed, for a homogeneous and isotropic expansion, the bulk viscous pressure is negative and could account for effects usually attributed to dark energy and held responsible for the current accelerated expansion of the Universe [1][2][3]. However, models that aim at replacing the need for separate dark matter and dark energy components in the cosmological concordance model by invoking bulk viscosity of dark matter, are strongly challenged by cosmological precision data [4][5][6][7].
In this letter, we point out that dissipative effects relevant for the expansion history of the Universe could arise also from the shear viscous properties of the cosmological fluid and, at least in principle, one could have a similar effect from the gain in internal energy due to fluid motion against local pressure gradients. If the Universe were exactly homogeneous and isotropic, such effects would vanish for symmetry reasons. However, homogeneity and isotropy are preserved only statistically and they are broken locally by perturbations in the cosmological fluid. Here we discuss how in such a case the viscous damping of velocity gradients feeds (at non-linear order in the fluid perturbations) into the evolution of the spatially averaged fluid fields.
In general relativity, the matter fields that enter the energy-momentum tensor T µν and the metric g µν that enters T µν as well as the Einstein tensor G µν , are both dynamical variables. The time evolution of the latter is determined by Einstein's field equations If the universe were completely homogeneous and isotropic, these symmetries would lead to strong constraints. The metric would have to be of the form where τ is the conformal time and where we have neglected spatial curvature for simplicity. The energymomentum tensor could deviate from its ideal form at most by a bulk viscous term, T 0 0 = , T i j = (p+π bulk )δ i j , and the Einstein equations would reduce to the standard Friedmann equations that express the time evolution of the scale factor a(τ ) in terms of the energy density and the effective pressure p eff = p + π bulk .
For the more realistic case of a universe that is homogeneous and isotropic only in a statistical sense, a Friedmann-type solution acts as a background. The evolution of this background is not affected by the perturbations if they are small enough for only linear terms to be kept. However, the fluctuations can backreact on the background at non-linear order. On the gravity side (left hand side of eq. (1), broadly speaking), these backreaction effects have come under scrutiny and are likely to be small [8,9]. Here, we discuss backreaction effects on the matter side of eq. (1).
For matter described as a relativistic viscous fluid, the energy-momentum tensor in the Landau frame (where the fluid velocity is defined by the condition that there is no energy current in the fluid rest frame, −u µ T µν = u ν ) reads Here, ∆ µν = u µ u ν + g µν is a projector orthogonal to the fluid velocity, and π µν is the shear stress, satisfying u µ π µν = π µ µ = 0. To first order in the gradient expansion of hydrodynamics, one has the following constitutive relations where η and ζ denote the shear and bulk viscosity, respectively. In addition, there can be conserved charges.
For a single conserved current N α (corresponding e.g. to conserved baryon number or a conserved number of dark matter particles), one has to first order in hydrodynamical gradients a particle diffusion current ν α that points along chemical potential gradients orthogonal to the fluid velocity and whose strength is set by the thermal conductivity κ, While keeping the next (second) order in the gradient expansion is important for maintaining causal dynamics arXiv:1411.3280v1 [gr-qc] 11 Nov 2014 and linear stability [10,11], first-order viscous hydrodynamics is usually sufficient for practical calculations, and we restrict the following discussion to it. From the covariant conservation of energy, momentum and particle number, one finds the fluid dynamic equations of motion for the energy density the fluid velocity and the particle number density In eqs. (9), (10) and (11), only energy density = u µ u ν T µν and particle number density n = u µ N µ are independent thermodynamic variables. Pressure p as well as other thermodynamic quantities such as µ and T are related to these via the thermodynamic equation of state (e.o.s.) in terms of the standard thermodynamic relations. Also the transport coefficients η, ζ and κ are functions of the independent thermodynamic variables.
Einstein's field equations (1) imply the conservation laws (8) and thus the equations of motion (9), (10) and (11). For our purpose we find it more useful to work with the latter instead of analyzing directly the former. Once supplemented by an e.o.s., the equations (9), (10) and (11) form a closed set for the evolution of fluid dynamic fields. At least locally, these equations provide sufficient information about the time evolution of the thermodynamic variables and the fluid velocity. Also, eqs. (9), (10) and (11) are valid for arbitrary gravitational fields, on which they depend via the dependence of the fluid velocity u µ , the projector ∆ µν and the covariant derivatives ∇ µ on the metric g µν . To analyze this dependence in more detail, let us now consider a perturbative ansatz for the metric where Φ and Ψ denote potentials (in conformal Newtonian gauge). We follow here the general expectation that, at least at late times, the main modification of a simple homogeneous and isotropic expansion is mediated by scalar fluctuations around the background metric (2), and that the influence of vector and tensor excitations is negligible.
With the metric of eq. (12), the fluid velocity can be written as We specialize now to the cosmologically relevant case of small fluid velocity ( v 2 1, γ → 1). The different terms entering eqs. (9), (10) and (11) can now be computed, and to linear order in Φ, Ψ one finds for instance In the regime of structure formation at late times one expects that the Newton potentials are small (Φ , Ψ 1), that they vary slowly in time (typically with the Hubble rate,Φ ∼ȧ a Φ, and similarly for Ψ) and that they vary in space on similar scales as the fluid dynamic fields. In this case, only the first two terms on the right hand side of eq. (13), i. e. the ones that are independent of Φ and Ψ, must be kept. We analyze other terms in eq. (9) in a similar way and find that it becomeṡ where sub-leading Φ-and Ψ-dependent terms have now been suppressed. An analogous argument 1 applies also to the time evolution (11) of the particle number density that reads in the same limiṫ (15) We turn next to the expectation values or spatial av-erages¯ = andn = n . From eq. (15), one finds (neglecting surface terms as usual) with Hubble parameter H =ȧ/a 2 . This shows simply that the standard dilution of particle number due to the expansion is not modified by dissipative effects. On the other hand, we find from eq. (14) for the cosmological evolution of the average energy density where we have introduced the shorthand The term D enters eq. (17) as a backreaction of fluid fluctuations onto the time evolution of the background field . Interestingly, while the assumption of isotropy and homogeneity precludes a shear stressπ µν in the background, eq. (18) demonstrates that the time-evolution of background fields like¯ depends on the shear viscosity of the cosmological fluid. The shear and bulk viscous contributions of the first and second term on the r.h.s. of eq. (18) are positive semi-definite, since they are expectation values of squares. They describe the gain in internal energy due to the dissipation of local gradients in the fluid velocity. The third term accounts for the work done by the fluid expanding out of high-pressure regions (or a corresponding gain in internal energy due to contraction against a pressure gradient). When structures form due to local gravitational collapse, one expects that the shear and bulk viscous contributions to D increase the internal energy. Depending on the equation of state and the dissipative properties of the fluid, this effect might be small or sizable. In the following we simply assume that it is non-negligible and discuss possible consequences for the cosmic expansion.
To do so we need to supplement the fluid dynamic evolution equations with an equation for the scale factor. A direct spatial average of Einstein's field equation (1) with the energy-momentum tensor (3) would involve on its right hand side unknown quantities such as ( + p + π bulk )u µ u ν . One could project to the different terms in eq. (3) by contracting with the fluid velocity, e. g. u µ u ν G µν = −8πG N but the space average of this equation would involve unknown averages of velocities on the left hand side. We therefore look in the Einstein equations for a suitable constraint that is independent of u µ and find it in the trace R = 8πG N T µ µ . The averaged part R = 8πG N T µ µ reads a a 3 = 1 aḢ For given e.o.s. and thermodynamic transport properties one can determine the time evolution of the Hubble parameter and the scale factor a(τ ) by solving eq. (19) together with eqs. (16) and (17). However, one also needs the parameter D in eq. (18) which depends on correlation functions of perturbations.
To illustrate the physics encoded in this set of equations, we assume first for the e.o.s. a simple relation p +π bulk =ŵ¯ , withŵ a numerical constant. A straightforward calculation gives for the acceleration parameter q = 1 +Ḣ/(aH 2 ) dq d ln a + 2(q + 1) q (20) For D = 0, eq. (20) has an attractive fixed point at the well known value q = −(1 + 3ŵ)/2. In particular for a matter dominated universe withŵ = 0 one finds deceleration, q = −1/2, which contradicts observations. Interestingly, if the right hand side of equation (20) is positive, the fixed point is shifted to larger values of q. More spe- cific, the fixed point is accelerating, i. e. q > 0, for As a result, a positive D can actually contribute to the acceleration of the expansion, similarly to an effective negative pressureŵ < 0 induced by bulk viscous pressurē π bulk or a positive cosmological constant or dark energy. Fig. 1 illustrates eq. (20) graphically and shows in particular the value the dissipative term must take in order to account for a given acceleration parameter q. We concentrate here on vanishing effective pressure, p eff = 0. For the experimentally favored value of q ≈ 0.6 [12], we conclude that the set of equations (17), (19) could account for the observed accelerated expansion of the universe if 4πG N D/(3H 3 ) ≈ 3.5 (assuming dq/d ln a 1). In general, whether dissipative effects encoded in D can affect the expansion history of the Universe significantly depends on material properties that enter the analysis of eqs. (16), (17) and (19) via the e.o.s. and via the numerical value of D. We comment now on both these dependencies: First, the above analysis of the acceleration parameter was done for a very simple e.o.s.,p +π bulk =ŵ¯ . For a universe filled with pure radiation,ŵ → 1/3, no finite dissipative term D can satisfy eq. (21). But for a more realistic e.o.s., one has p = p( , n), and eq. (20) for the acceleration parameter is replaced by a more complicated expression that depends on first and second derivatives of the thermodynamic potential p( , n). One has to check then for each e.o.s. whether the combination of eqs. (16), (17) and (19) contributes to deceleration or acceleration and how sizable the effect can be.
Second, remarkably, how the evolution of the Universe at the largest scales is affected by D depends mainly on the physics at the smallest scales that enter D. To see this more explicitly, it is useful to write eq. (18) in momentum space. Decomposing v into a sum of a gradient, characterized by θ = ∇ · v, and a rotation, characterized by the vorticity w = ∇ × v, and going to Fourier space θ(x) = d 3 qθ(q) e i q x , etc., we have Here, we assumed for simplicity that ζ =ζ and η =η are constant in space, and we defined the power spectra If the spectra P θp ( q), P θθ ( q) and (P w ) jj ( q) in eq. (22) do not die out faster than 1/q 3 , D is dominated by the UV, i. e. by the fine structures in position space. (The criterion for D to be UV-dominated is more complicated if one relaxes the assumption of constant ζ and η). Hence, one expects that the value of D will be set by the smallest relevant scale which is the dissipation or virialization scale below which a fluid dynamic description does not apply.
We note that techniques for the calculation of D do exist. These follow the time evolution of the correlation functions as in eq. (23) and thus provide a dynamical understanding of their q dependence. 2 Leaving a detailed study of D to future work, and keeping in mind that any simple estimate of this UV-dominated quantity may be subject to significant uncertainties, we close this letter by exploring the possibility that D could be sizeable, in the sense that 4πG N D/(3H 3 ) 1 and eq. (20) allows for accelerated expansion. It is generally difficult to conceive that bulk viscosity is large enough to have a substantial effect, in particular because neither radiation (ultra-relativistic particles) nor simple non-relativistic gases can contribute to it [15]. We therefore focus on the shear viscous part of D, and we simply assume that typical gradients of the fluid velocity are of the same order as the Hubble rate H so that with σ of order one. This corresponds to realistic peculiar velocity variations of the order of 100 km/s on distances of 1 MPc, and it amounts to assuming that the smallest distances relevant for D are of this order.
It remains to estimate the shear viscosity. In general, this will depend on the unknown material properties of the dark sector, and thus involves inevitably modeldependent aspects. It is noteworthy, however, that a large shear viscosity arises for systems containing very 2 In a companion paper [13], we have calculated with one of these techniques [14] the time evolution of spectra in a simple fluid dynamic model of heavy ion collisions. A backreaction effect analogous to eqs. (17) and (18) has been characterized there in a technically more detailed way.
weakly interacting relativistic particles of long mean free paths (e.g. as one component in addition to cold dark matter with shorter range interactions). In this case, relativistic kinetic theory suggests that [16] where c η is a numerical prefactor of order one, R is the energy density carried by the weakly interacting particles and τ R is their mean free time. Accelerated expansion would result then if the e.o.s. of this system is not pure radiation and if is of order unity, where the critical energy density ρ c is defined by H 2 = 8πG N ρ c /3. On the other hand, for a description in terms of a single fluid to be applicable, the mean free times of the weakly interacting particles must be smaller than the expansion rate, τ R H < 1. Thus, the term in (25) can only become of order one if R is of the same order as ρ c and σ is somewhat larger than one. One may wonder whether there is any reasonable weakly coupled candidate particle that could satisfy these constraints. Photons or relativistic (massless) neutrinos can be excluded because of their too small interaction cross sections or, equivalently, too long mean free times. On the other hand, gravitons are expected to have a mean free time [17] One can solve eqs. (24) and (25) for η and τ G [18], and one finds For an accelerating expansion, one would have to require σ 10, c η ≈ 1, and most importantly, a fractional energy density of the gravitational radiation Ω G = G /ρ c not too far from one. This would also satisfy the consistency condition τ G H 1. The purpose of the above comment is not to argue whether a graviton gas of such high energy density, interacting with dark matter, can provide a phenomenologically viable component of the dark sector (in any case, this would only seem plausible if such a component plays essentially the role of Ω Λ in the standard concordance model). Rather, we sketch this scenario only to illustrate with an example how a specific particle content of the dark sector affects its material properties and how these properties may impact the large-scale dynamics of the Universe, or can be constrained by it.
In summary, the main result of this letter is the identification of a dissipative term D in the cosmological evolution (17) of the average energy density. This term arises from the backreaction of fluid velocity fluctuations, it depends on shear viscosity, and it may affect the expansion history of the Universe. If sufficiently large, it could lead to an accelerated cosmological expansion without assuming negative effective pressure. It would then provide a rather natural resolution of the coincidence problem and explain why accelerated expansion occurs at late times, in the same epoch as structure formation. Irrespectively of the size of D, we emphasize here that dissipative phenomena are ubiquitous in nature and that eqs. (18) and (22) for D provide a novel and more comprehensive framework to account for them in discussions of the cosmological expansion. At least in principle, eqs. (18) or (22) can be calculated also for non-equilibrium scenarios, which is of interest since it is a priori unclear to what extent the dark sector is equilibrated. Also, it is conceivable that contributions to D arise from sources not discussed so far. For instance, an effective viscosity may also arise on large length scales from a coarse-grained description of fluctuations in the cosmological fluid [19,20]. Or, at least in principle, a contribution to D could also arise from the contraction of the fluid against local pressure gradients (third term in eq. (18)), as they might be induced by gravitational collapse. In view of these many physics effects, we hope that the results derived in this letter will help to better constrain the role of dissipation in cosmology, and the material properties that may give rise to it.