Dynamical analysis of a first order theory of bulk viscosity
DDynamical analysis of a first order theory of bulk viscosity
Giovanni Acquaviva ∗ Institute of Theoretical Physics, Faculty of Mathematics and Physics,Charles University in Prague, 18000 Prague, Czech Republic
Aroonkumar Beesham † Department of Mathematical Sciences, University of Zululand,Private Bag X1001, Kwa-Dlangezwa 3886, South Africa
We perform a global analysis of curved Friedmann-Robertson-Walker cosmologies in thepresence of a viscous fluid. The fluid’s bulk viscosity is governed by a first order theoryrecently proposed in [18], and the analysis is carried out in a compactified parameter spacewith dimensionless coordinates. We provide stability properties, cosmological interpretationand thermodynamic features of the critical points.
PACS numbers: 98.80.-k, 95.36.+x
I. INTRODUCTION
Most studies in cosmology are based on the assumption that the matter content of the Universe iswell approximated by a perfect fluid description, i.e. , one without viscosity nor heat conduction.However there are stages in the evolution of the universe when viscosity and entropy-producing pro-cesses are expected to be important, especially during the early Universe: the reheating at the endof inflation, the decoupling of neutrinos from the primordial plasma, the nucleosynthesis and thedecoupling of photons from matter during the recombination era. At the same time, the analysis ofdata from the recent Planck survey [1] confirms a background geometry which, at very large scales,is isotropic and homogeneous (see also [2] for an independent analysis on the same dataset). Whileshear viscosity and heat fluxes are related to the presence of anisotropies and inhomogeneities,bulk viscosity is related just to the kinematical expansion of the fluid’s flow: the observational ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] A ug evidence hence justifies the choice of considering the cosmological effect of bulk viscosity aloneamong the possible dissipative processes, as long as the description is restricted to very large scales.The role of bulk viscosity in cosmology has been explored from many points of view. It has beenproposed long ago as a way to avoid the initial Big Bang singularity [3] and to obtain a betterunderstanding of the singularity itself [4]. As a phenomenological model of dissipative processes,bulk viscosity should describe the effect of isotropic expansion on the thermodynamic properties offluids: for instance, one could interpret the dissipation as the result of a friction between differentmatter species undergoing a common Hubble expansion. At the same time, bulk viscosity canprovide a phenomenological description of particle creation in strong gravitational fields [5]. Oneof the main features of viscosity is the possibility to lower the total effective pressure of the fluid tonegative values: given that the Universe is currently undergoing accelerated expansion, and giventhat such effect in GR can be obtained by a sufficiently negative pressure of the matter source, bulkviscous fluids have been proposed as possible candidates for dark energy. In this respect, it is worthmentioning that the traditional first-order model of bulk viscosity due to Eckart [6] is formallyequivalent to a generalized Chaplygin gas for some ranges of the free parameters involved. Apartfrom cosmology, bulk viscosity could play a substantial role as well in astrophysical scenarios, suchas in the growth of inhomogeneities that seed large-scale structures [7, 8] and in the gravitationalcollapse of compact objects [9, 10]. However, in these cases the other dissipative processes areexpected to contribute as well.The formulation of a relativistic theory of dissipative processes dates back to the studies ofEckart [6], who introduced viscous contributions to the stress-energy tensor as functions of thefour-velocity and the thermodynamical variables of the fluid. In [11] Hiscock and Lindblomprovided general arguments showing that a wide class of first-order theories are unstable:Eckart’s approach, which considers first-order deviations from thermodynamic equilibrium, fallsinto such a class, and hence presents several shortcomings in terms of stability and physicalviability. The most evident of such problems is the superluminal propagation of signals. Inorder to address the ensuing issue of causality, Mueller, Israel and Stewart [12, 13] considered aformulation based on second-order deviations from equilibrium, in which the dissipative variablesare treated as independent quantities satisfying their own evolution equations. During the1980s, the formulation of such extended irreversible thermodynamics has been deeply refined[14, 15] and several applications in cosmology have been presented [16, 17]. However, it hasbeen argued [18] that the MIS theory is not uniquely defined, as the equations of motion thatsuch variables satisfy rely on a certain degree of arbitrariness, and that its causal character isnot yet fully understood. Nonetheless, both Eckart’s and MIS approaches (including possiblemodifications, e.g. , [19]) have been applied extensively to cosmology and their impact on the ac-celerated expansion of universe [20–25] and on the formation of structures [7, 26] has been analysed.We consider here a recent first-order fomulation of relativistic dissipative processes [27], basedon a previous approach due to Lichnerowicz [28] and which does not introduce new, independentvariables for the viscous quantities. Although first-order in nature, such an approach does not fallinto the class of theories that were proven unstable by Hiscock and Lindblom. We present the basicstructure of the theory in Sec. II, specializing to a curved Friedmann-Robertson-Walker (FRW)background metric and considering a single viscous fluid. In Sec. III we recast the equations in theform of an autonomous dynamical system and analyse the general features of its critical elementsfrom the cosmological point of view. In Sec. IV, we further specialize to the case of viscous radiationand show the behaviour of the trajectories in the compactified parameter space. Finally, in Sec.Vwe discuss and comment on the results. II. EQUATIONS
The viscous energy-momentum tensor proposed in [18] has the form T µν = ρ u µ u ν + (cid:16) p − ζ ∇ α C α (cid:17) ( g µν + u µ u ν ) , (2.1)where ρ and p are, respectively, the energy density and equilibrium pressure of the fluid. Theequilibrium pressure is modified by the bulk viscous term, where ζ ≥ C α := F u α is the dynamical velocity (or canonical momentum) of a fluid element and F is thespecific enthalpy of the fluid, F = ρ + pµ , (2.2)with µ its rest mass density. The adoption of the dissipative source in the form of eq.(2.1) canbe supported by the natural requirement that all the information about the properties of mattershould be conveyed only by the structure of the energy-momentum tensor, without the need ofintroducing additional assumption on the dynamics of the variables involved – which is instead thecase in MIS formulation. The dynamical velocity C α appearing in the energy-momentum tensorhas been considered as a suitable relativistic generalization of the concept of fluid’s velocity inpresence of dissipation (see [18] for a discussion). It is worth noticing that C α plays the role of canonical momentum in a Hamiltonian formulation of relativistic nonisentropic flows [29]. The bulkviscosity of the fluid is measured by ∇ α C α , that is the kinematical expansion of such vector field;the definition is compatible with the notion that bulk viscosity vanishes for an incompressible fluid,for which ∇ α C α = 0. A more general dissipative source would include also shear viscosity in termsof the spatial projection of the symmetrized quantity ∇ ( α C β ) : however, in an isotropic backgroundsuch contribution plays no role and for the purpose of the present analysis we will discard it. In aFRW background we have ∇ α C α = ˙ F + 3 H F , (2.3)where the dot is the derivative with respect to cosmic time. Moreover, the rest mass is conservedalong the flow lines ∇ α ( µu α ) = 0, so that µ = µ a − . The field and continuity equations obtainedwith the viscous energy-momentum tensor (2.1) are the following: H + ka = 13 ρ , (2.4)2 ˙ H + 3 H + ka = − ω ρ + ζ ∇ α C α , (2.5)˙ ρ + 3 H (1 + ω ) ρ − H ζ ∇ α C α = 0 , (2.6)where we have already implemented the barotropic equation of state p = ω ρ for the equilibriumpressure. Moreover, it is usual to assume a generic power-law dependence of the bulk viscosity onthe energy density of the fluid: ζ = ζ ρ α with ζ ≥ . (2.7)Note that, in an expanding universe ( H > T ∇ β S β = 3 ζ H ∇ α C α , (2.8)where T >
III. DYNAMICAL SYSTEM
Dealing with a curved FRW background, we know that positive spatial curvature can lead to bounc-ing/recollapsing scenarios, whereas a negative or vanishing curvature is either always expanding oralways collapsing. The usual definition of expansion-normalized variables breaks down in the formercase whenever H = 0 at finite times. For this reason we divide the analysis in two parts, the k ≤ k > ω ∈ ( − ,
1) and we will specialize toviscous radiation ω = 1 / A. Non-positive spatial curvature
Imposing k ≤ k/a in the field equations is non-positive. Hence we cansafely define the new variables in the following way:Ω ρ = ρ H , (3.9)Ω k = | k | H a , (3.10)Ω C = ζ ∇ α C α ρ . (3.11)In terms of such variables, the Friedmann and Raychaudhuri equations respectively take theform 1 = Ω ρ + Ω k , (3.12)˙ HH = 32 Ω ρ (cid:104) Ω C − (1 + ω ) (cid:105) . (3.13)The first equation is a constraint that allows us to disregard the evolution of, e.g. , Ω k . Sucha constraint tells us that Ω ρ ∈ [0 , C instead is unbounded both from aboveand from below, which means that some trajectories of the system might escape to infinity: inorder to capture such asymptotic behaviour, we define the new variable X = arctan Ω C , such that X ∈ [ π/ , π/ τ = log a ( t ). The derivative withrespect to τ will be denoted by a prime and its relation with the cosmic time derivative is suchthat X (cid:48) = H − ˙ X . Taking the prime derivative of the definitions of the relevant variables Ω ρ and X , and making use of eqs.(2.6) and (3.13), we arrive at the following autonomous system:Ω (cid:48) ρ = − ρ (1 − Ω ρ ) (cid:104) (1 + ω ) − tan X (cid:105) , (3.14) X (cid:48) = − − ω ) cos X sin X (cid:16) − ω + tan X (cid:17)(cid:104) (2 α + Ω ρ ) (1 + ω − tan X ) − (cid:105) . (3.15)The entropy production is related to the compact variables by T ∇ β S β = 9 H Ω ρ tan X . (3.16)This means that for expanding models (
H > X ∈ (0 , π/
2) correspond to dynamics with positive entropy production; conversely,in collapsing models (
H < X ∈ ( − π/ , X = 0 is an invariant subset of the system, so we cannot expect positiveentropy-producing initial conditions to evolve into negative entropy-producing states.The critical points P = { X ∗ , Ω ∗ ρ } of the system are easily found by solving the system X (cid:48) (cid:0) X ∗ , Ω ∗ ρ (cid:1) = 0 (3.17)Ω (cid:48) ρ (cid:0) X ∗ , Ω ∗ ρ (cid:1) = 0 , (3.18)and they are given by P = { , } , P = { , } P = {− arctan(1 − ω ) , } , P = {− arctan(1 − ω ) , } P = { arctan(1 + ω − /α ) , } , P = { arctan(1 + ω − / (1 + 2 α )) , } , For purposes of clarity, their stability properties for the case of expanding models (
H >
0) arelisted in Table I for − < ω < < ω <
1. The case of collapsing modelscan be obtained by swapping the roles of sinks and sources. The deceleration parameter q and the effective EoS parameter ω E are given by q ≡ − − ˙ HH = 32 Ω ρ (cid:104) (1 + ω ) − tan X (cid:105) − , (3.19) ω E ≡ p − ζ ∇ α C α ρ = ω − tan X , (3.20) P P P P P P α < − / − / < α < < α < saddle source sink source saddle saddle < α < − ω ω ) saddle source saddle source sink saddle − ω ω ) < α < ω saddle saddle saddle source sink source α > ω sink saddle saddle source saddle source TABLE I: Stability of the critical points of the system with k ≤ − < ω < P P P P P P α < − / − / < α < < α < − ω ω ) saddle source sink source saddle saddle − ω ω ) < α < saddle saddle sink source saddle source < α < ω saddle saddle saddle source sink source α > ω sink saddle saddle source saddle source TABLE II: Stability of the critical points of the system with k ≤ < ω < P , P and P are vacuum models with exponential expansion of the scale factor ( q = − P is an inviscid fluid solution with ω E = ω and q = (1 + 3 ω ) / P is a stiff matter model with q = 2 and ω E = 1, withpossible character of future attractor. Finally, point P has an effective equation of state whichdepends explicitly on the viscosity parameter as ω E = − α α ; it can represent a phantom modelwhen α < − /
2, in which case is a future attractor for the system.
B. Positive spatial curvature If k > i.e. when H = 0. However the quantity D = (cid:114) H + ka (3.21)is always positive definite, so we can define the following normalized variables:Ω H = HD , (3.22)Ω ρ = ρ D , (3.23)Ω C = ζ ∇ α C α H D . (3.24)Again we define the compactified variable X = arctan Ω C , so that the Friedmann constraint andRaychaudhuri equation readΩ ρ = 1 , (3.25)˙ HH = −
12 Ω H (cid:2) H + 3 ( ω − Ω H tan X ) (cid:3) . (3.26)We see immediately that Ω ρ is not a dynamical degree of freedom. The variable Ω H is definedin the interval [ − ,
1] and its sign is positive/negative iff the metric is expanding/contracting; theboundary values Ω H = ± D :˙ DD = −
32 Ω H (cid:104) ω − Ω H tan X (cid:105) . (3.27) Q +0 Q − Q +1 Q − Q +2 Q − α < − / − / < α < < α < < α <
12 1 − ω ω saddle saddle source sink sink source α >
12 1 − ω ω (cid:26) sink X → − saddle X → + (cid:26) saddle X → − source X → + source sink saddle saddle TABLE III: Stability of the finite critical points of the system with k > − < ω < − / X (cid:48) = D − ˙ X , so that the primederivatives of the dynamical variables give us the following system:Ω (cid:48) H = − (cid:0) − Ω H (cid:1) (cid:104) ω − Ω H tan X ) (cid:105) , (3.28) X (cid:48) = − sin X − ω ) (cid:104) − ω ) ( ω + 2 α (1 + ω ) −
1) Ω H cos X + (3.29)+ sin X (cid:16) ω + Ω H (3 (1 + 4 α ) ω − − H tan X (cid:0) α Ω H (cid:1) (cid:17)(cid:105) . (3.30)In this case, the entropy production is given by T ∇ α S α = 9 H tan X Ω H . (3.31)Hence, in the parameter space spanned by ( X, Ω H ), entropy production is positive iff X ∈ (0 , π/ X = 0 is an invariant subset of the system, so trajectories startingwith positive entropy production cannot cross to the negative entropy production part.The critical points are calculated as before and they are given by Q +0 = { , } , Q − = { , − } Q +1 = {− arctan(1 − ω ) , } , Q − = {− arctan(1 − ω ) , − } Q +2 = { arctan(1 + ω − /α ) , } , Q − = { arctan(1 + ω − / (1 + 2 α )) , − } , Q +0 Q − Q +1 Q − Q +2 Q − α < − / − / < α < < α <
12 1 − ω ω source sink source sink saddle saddle
12 1 − ω ω < α < α > TABLE IV: Stability of the finite critical points of the system with k > − / < ω < − < ω < − / − / < ω <
1, which are shown respectively in Tables III and IV. The cosmologicalparameters that characterize the critical points in this case are q = 12 1 + 3 ( ω − Ω H tan X )Ω H , (3.32) ω E = ω − Ω H tan X . (3.33)The critical points Q ± represent inviscid fluid models with ω E = ω . Points Q ± are stiff mattersolutions with ω E = 1. Points Q ± have ω E = − α α and q = − α )1+2 α , so that they can representphantom models for α < − / IV. VISCOUS RADIATION
In order to visualize a physically meaningful case, we now specialize the analysis to the case ofviscous radiation ( ω = 1 / k ≤
0, and in Fig.3 and Fig.4 for k >
0. We choose representative values of theparameter α corresponding to the ranges specified in the Tables of stability given in the previoussection. The dots in the plots identify sinks (green), saddles (blue) and sources (red). The greenshaded region in the plots corresponds to positivity of entropy production, according to eqs. (3.16)and (3.31).1We stress that in the case k ≤ H >
0) portion of the system:the collapsing part can be obtained by inverting the direction of the flows; moreover, in this casethe positive entropy production regions will be
X <
0. As noted before, X = 0 is an invariantsubset, so the sign of entropy production is preserved during the dynamics. The same happens inthe k > X = ± π/
2, identified inthe figures by a dashed line. In both curvature cases, the trajectories reach such boundaries witha specific constant values of the bounded coordinate (Ω ρ for k ≤ H for k > ρ ∈ (0 ,
1) on theboundary allows us to calculate the scale factor by integrating the Friedmann constraint: a ( t ) = (cid:115) | k | (cid:0) − Ω ρ (cid:1) ( t − t BB ) . (4.34)This corresponds to an asymptotically Milne-like model, with Big Bang time t BB = − a (cid:113) (cid:0) − Ω ρ (cid:1) / | k | and a = a ( t = 0). Consequently, the Hubble expansion and the energydensity scale like H = (cid:115) | k | (cid:0) − Ω ρ (cid:1) a − , (4.35) ρ = | k | Ω ρ − Ω ρ a − . (4.36)By considering a usual functional dependence of the temperature of the fluid on its energy density, i.e. T = T ρ ω/ (1+ ω ) with ω = 1 /
3, eventually one can evaluate the scaling of the entropy productionin such asymptotic states: ∇ a S a = c ± · a − α +1 / , (4.37)where c + (resp. c − ) is a positive (resp. negative) constant on the boundary X C = π/ X C = − π/ • if α + 1 / > – ∇ a S a → a → ∞ (stable boundary) – ∇ a S a → ∞ for a → k ≤ ω = 1 / H > α = − α = − / α = 1 /
10. Dots identify sources (red),saddles (blue) and sinks (green). The green shaded area is the positive entropy production region.The dashed line is the compactified boundary of the system.3FIG. 2: Trajectories in the parameter space for the system with k ≤ ω = 1 / H > α = 1 / α = 2 / α = 2. Dots identify sources (red), saddles(blue) and sinks (green). The green shaded area is the positive entropy production region. Thedashed line is the compactified boundary of the system.4FIG. 3: Trajectories in the parameter space for the system with k > ω = 1 /
3. From topleft: α = − α = − / α = 1 /
10. Dots identify sources (red), saddles (blue) and sinks (green).The green shaded area is the positive entropy production region. The dashed line is thecompactified boundary of the system.5FIG. 4: Trajectories in the parameter space for the system with k > ω = 1 /
3. From left: α = 2 / α = 3 /
2. Dots identify sources (red), saddles (blue) and sinks (green). The greenshaded area is the positive entropy production region. The dashed line is the compactifiedboundary of the system. • if α + 1 / < – ∇ a S a → ∞ for a → ∞ (stable boundary) – ∇ a S a → a → k >
0, where a constant value Ω H on the compacti-fied boundaries allows to calculate the scale factor by integrating the definition of the variableitself: a ( t ) = (cid:114) k (Ω H ) − (Ω H ) ( t − t BB ) , for Ω H > (cid:114) k (Ω H ) − (Ω H ) ( t BC − t ) , for Ω H < , where t BB and t BC are the Big Bang and the Big Crunch times respectively. Having the sameasymptotic time-dependence of the scale factor as in the previous case, also the scaling propertiesof the entropy production are analogous.6 V. CONCLUSIONS
In the present work we have analysed the system of Einstein’s equations sourced by a single dissi-pative fluid in the context of a first-order theory of relativistic dissipation. The generically curvedFRW metric has been taken as cosmological background. The system has been recast in the formof a dimensionless, autonomous system of equations whose equilibrium points represent differentdynamics of the scale factor. The study of the stability of such critical points allowed us to assessthe past and future behaviour of the system, characterized unambiguously by the deceleration andeffective equation of state parameters. The results obtained here are in accord with those presentedin [27], for instance regarding the attractor behaviour of the phantom solution for α < − / P ), but as well highlight additional features: de Sitter-like future attractors exist fordifferent ranges of the parameter α in the non-positive curvature case (points P , P and P ); for α >
0, a stiff matter-dominated solution is found as a past attractor for k ≤ P ) and asboth a past and a future attractor for k > Q ± ); we notice further that in general theevolution preserves the sign of the entropy production, so that positive entropy-producing initialconditions cannot evolve into negative entropy-producing states.It seems natural to ask whether it is possible to obtain an evolution that could interpolate betweenan inflationary epoch due to viscosity up to the later radiation-dominated phase: the results pre-sented here indicate that it is not possible to have such a behaviour. Indeed, if we require positiveand non-divergent entropy production throughout the evolution, there is no trajectory connectingde Sitter-like solutions to radiation-dominated ones in an expanding dynamics. Instead, under thesame physical assumptions on entropic evolution, this model of viscous radiation seems to giverise quite easily to the opposite transition, from a radiation-dominated early epoch towards a late-time accelerated phantom behaviour. In the non-positive curvature scenario, this is obtained for α < − / e.g. trajectories connecting P and P in Fig.1) and α > / e.g. trajectories con-necting P and P in Fig.2), and the rate of acceleration is higher the closer α is to such bounds.Considering positive curvature instead and requiring only non-recollapsing scenarios, the possibilityto have such transition needs α < − /
2. These considerations suggest that the present model ofviscosity (at least in the single-fluid scenario) is not suitable for describing an effective inflationaryregime together with its exit mechanism towards reheating and subsequent radiation dominance.IBLIOGRAPHY 17Regarding instead the possibility of describing a late-time acceleration of the Universe – which arisesmore naturally in the global dynamics – the attractors discussed above have generically ω E < − α → ±∞ . On one hand, assuming a sufficiently big value for | α | ,one could think of including in the analysis an additional inviscid dust component which wouldlikely dominate right after the radiation: the expected dynamics could then interpolate betweenradiation, matter and accelerated expansion phases. On the other hand, a very big value of | α | thatwould ensure a late-time exponential expansion in accord with observations might instead have se-rious repercussions at perturbative level. In this regard, an analysis of the growth of perturbationsin this framework of dissipative processes would certainly be useful in assessing the viability of suchscenario. Acknowledgments
GA acknowledges financial support from the Grant No. 17-16260Y of the Czech Science Foundation(GA ˇCR). The authors thank the referees for their constructive comments, which lead to significantimprovements in the manuscript.
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