aa r X i v : . [ a s t r o - ph ] S e p Bulk Viscous Cosmology
R. Colistete Jr. ∗ , J.C. Fabris † , J. Tossa ‡ , and W. Zimdahl § Universidade Federal do Esp´ırito Santo, Departamento de F´ısicaAv. Fernando Ferrari, 514, Campus de Goiabeiras,CEP 29075-910, Vit´oria, Esp´ırito Santo, Brazil Institut de Math´ematiques et de Sciences Physiques - IMSPUniversit´e d’Abomey-Calavi, BP613, Porto Novo, B´enin
Abstract
We propose a scenario in which the dark components of the Universe are manifestations of a singlebulk viscous fluid. Using dynamical system methods, a qualitative study of the homogeneous, isotropicbackground scenario is performed in order to determine the phase space of all possible solutions. Thespecific model which we investigate shares similarities with a generalized Chaplygin gas in the backgroundbut is characterized by non-adiabatic pressure perturbations. This model is tested against supernova typeIa and matter power spectrum data. Different from other unified descriptions of dark matter and darkenergy, the matter power spectrum is well behaved, i.e., there are no instabilities or oscillations on smallperturbation scales. The model is competitive in comparison with the currently most popular proposals forthe description of the cosmological dark sector. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail: [email protected] . INTRODUCTION The crossing of observational data from high redshift supernovae of type Ia (SNe Ia) [1, 2, 3, 4],cosmic microwave background (CMB) radiation [5], matter power spectra [6], X-rays from clustersof galaxies [7, 8], and weak gravitational lensing [9] strongly suggests that the present Universeis dynamically dominated by a dark sector which is responsible for about 96% of its total energycontent. It is usually assumed that this dark sector has two different components: (i) dark matter,which is supposed to consist of weakly interacting massive particles (WIMPS) with zero effectivepressure and (ii) dark energy, a mysterious entity which is equipped with a negative pressure. Darkmatter candidates include axions (a particle present in the multiplet of grand unified theories) andneutralinos (light particles present in broken supersymmetric models), but none of these particlescould be detected until now. The most natural dark energy candidate is a cosmological constantwhich arises as the result of a combination of quantum field theory and general relativity. However,its theoretical value is between 60-120 orders of magnitude greater than the observed value for thedark energy. An alternative to the cosmological constant is a self-interacting scalar field, knownas quintesssence. For a brief but enlightening review of these and other proposals, see [10] andreferences therein.There exists another route of investigations in which dark matter and dark energy are describedwithin a one-component model. According to this idea, dark matter and dark energy are just“different faces” of a single, exotic fluid. To the best of our knowledge, the first proposal alongthis line was the Chaplygin gas in its original and modified forms [11, 12, 13]. However, thisunified description of dark energy and dark matter, in spite of many attractive features, seemedto suffer from a major drawback: it predicted strong small scale oscillations or instabilities in thematter power spectrum, in complete disagreement with the observational data [14]. (On the otherhand, the observed matter power spectrum corresponds to the baryonic matter distribution whichdoes not exhibit strong oscillations [15], so that this point is still controversial.) The apparentlyunrealistic predictions of the unified Chaplygin gas type models are the result of an adiabaticperturbation analysis. It has been suggested that non-adiabatic perturbations may alleviate oreven avoid this problem[16, 17, 18].This paper explores to what extent a viscous fluid can provide a unified description of the darksector of the cosmic medium. The general influence of shear and bulk viscosity on the characterof cosmological evolution has been studied, e.g. in [19], in the context of Bianchi type I models.Under the conditions of spatial homogeneity and isotropy, a scalar bulk viscous pressure is the2nly admissible dissipative phenomenon. The cosmological relevance of bulk viscous media hasbeen investigated in some detail for an inflationary phase in the early universe (see [20, 21, 22, 23]and references therein). However, as was argued in [16, 24], an effective bulk viscous pressurecan also play the role of an agent that drives the present acceleration of the Universe. (Noticethat the possibility of a viscosity dominated late epoch of the Universe with accelerated expansionwas already mentioned in [25]). For a homogeneous and isotropic universe, the ΛCDM model andthe (generalized) Chaplygin gas models can be reproduced as special cases of this imperfect fluiddescription [16]. Moreover, in a gas dynamical model the existence of an effective bulk pressurecan be traced back to a non-standard self-interacting force on the particles of the gas. While theseinvestigations were performed for the homogeneous and isotropic background dynamics (a study ofthe background dynamics which is similar to the setup of the present paper was recently performedin [26]), a first perturbation theoretical analysis for a unifying viscous fluid description of the darksector was performed in [27].The bulk viscous pressure p visc will be described by Eckart’s expression [28] p visc = − ξu µ ; µ ,where the (non-negative) quantity ξ is the (generally not constant) bulk viscosity coefficient and u µ ; µ is the fluid expansion scalar which in the homogeneous and isotropic background reduces to3 H , where H = ˙ aa is the Hubble parameter and a is the scale factor of the Robertson-Walkermetric. By this assumption we ignore all the problems inherent in Eckart’s approach which havebeen discussed and resolved within the Israel-Stewart theory [29, 30] (see also [20, 21, 22, 23] andreferences therein). We expect that for the applications we have in mind here, the differences areof minor importance.It is obvious that the bulk viscosity contributes with a negative term to the total pressure andhence a dissipative fluid seems to be a potential dark energy candidate. However, a cautionaryremark is necessary here. In traditional non-equilibrium thermodynamics the viscous pressure rep-resents a (small) correction to the (positive) equilibrium pressure. This is true both for the Eckartand for the Israel-Stewart theories. Here we shall admit the viscous pressure to be the dominat-ing part of the pressure. This is clearly beyond the established range of validity of conventionalnon-equilibrium thermodynamics. As already mentioned, non-standard interactions are requiredto support such type of approach [16, 24]. Of course, this reflects the circumstance that darkenergy is anything but a “standard” fluid. (We mention that viscosity has also been suggestedto have its origin in string landscape [31]). To successfully describe the transition to a phase ofaccelerated expansion, preceded by a phase of decelerated expansion in which structures can form,it is necessary that the viscous pressure is negligible at high redshifts but becomes dominant later3n.In ref. [27] a one-component bulk viscous model (BV model) of the cosmic medium was inves-tigated in which baryons were not taken into account. As far as the SNe Ia data are concerned,the results of this model were similar to those obtained for a generalized Chaplygin gas (GCG)model. But while GCG models predict small scale instabilities and oscillations at the perturbativelevel, the corresponding matter power spectrum of the BV model turned out to be well behaved.Of course, the results of a model that does not include baryons cannot be seen as conclusive; afterall, the observed power spectrum describes the distribution of baryonic matter. In the presentpaper, we perform an advanced analysis which properly includes a separately conserved baryoncomponent. Thus we establish a more realistic viscous fluid scenario of the cosmic substratum.For the quantitative calculations will use a constant coefficient of bulk viscosity. A qualitativeanalysis with the help of dynamical system methods is applied to visualize the space of moregeneral cosmological background scenarios. The corresponding phase space reveals interestingnew features compared with those generally found in similar models (see, for example, [33] andreferences therein). In the present approach there is an entire singular axis. Solutions of thedesired type are generated, i.e., solutions for which an initial subluminal expansion is followed bysuperluminal expansion. These solutions are characterized by a bulk viscosity coefficient with apower law dependence on the energy density, ξ = ξ ρ ν , where ξ =const and ν < / χ value for the SNe Ia data is similar tothe χ values of those models. Furthermore, the matter power spectrum represents a good fit to thecorresponding observational data. We argue that the absence of oscillations and instabilities is aconsequence of the fact that the pressure perturbations in our model are intrinsically non-adiabatic.The paper is organized as follows. In Section II we define our model and specify the homogeneousand isotropic background dynamics. Section III performs a qualitative analysis using dynamicalsystem methods. In section IV, the SNe Ia data are used to restrict the values of the physicallyrelevant free parameters. In section V the matter power spectrum is determined and comparedwith large scale structure observations. In section VI we present our conclusions.4 I. BACKGROUND RELATIONS
A bulk viscous fluid is characterized by an energy density ρ and a pressure p which has aconventional component p β = βρ and a bulk viscosity component p visc = − ξ ( ρ ) u µ ; µ , such that p = βρ − ξ ( ρ ) u µ ; µ . (1)On thermodynamical grounds the bulk viscosity coefficient ξ ( ρ ) is positive, assuring that the vis-cosity pushes the effective pressure towards negative values. In fact, the expression (1) is theoriginal proposition for a relativistic dissipative process [28]. As already mentioned, it follows fromthe more general Israel-Stewart theory [29, 30] in the limit of a vanishing relaxation time. We shallassume this approximation to be valid throughout the paper.Let us consider the cosmic medium to consist of a viscous fluid of the type (1), which is supposedto characterize the dark sector, and of a pressureless fluid that describes the baryon component.Hence, the relevant set of equations is R µν − g µν R = 8 πG (cid:26) T vµν + T bµν (cid:27) , (2) T µνv ; µ = 0 , T µνv = ( ρ v + p v ) u µ u ν − p v g µν , (3) p = p v = βρ v − ξ ( ρ v ) u µ ; µ , (4) T µνb ; µ = 0 , T µνb = ρ m u µ u ν . (5)The (super)subscripts v and b indicate the viscous and the (baryonic) matter components, respec-tively. Since the matter is pressureless, the total pressure p of the cosmic medium coincides withthe pressure p v of the viscous fluid. For the homogeneous and isotropic background dynamics weshall restrict ourselves to the flat Friedmann-Lemaˆıtre-Robertson-Walker (FRLW) metric, ds = dt − a ( t )( dx + dy + dz ) , (6)favored by the CMB anisotropy spectrum [5]. The dynamic equations then are: (cid:18) ˙ aa (cid:19) = 8 πG ρ b + ρ v ) , (7)2 ¨ aa + ˙ a a = − πGp v (8)˙ ρ v + 3 ˙ aa (cid:18) ρ v + p v (cid:19) = 0 , (9)˙ ρ b + 3 ˙ aa ρ b = 0 , (10) p v = βρ v − aa ξ ρ νv , (11)5here the bulk viscosity coefficient was assumed to have a power law dependence on the energydensity ρ v according to ξ ( ρ v ) = ξ ρ νv with ξ = const. The dot denotes differentiation with respectto the cosmic time. Of course, not all these equations are independent. Equation (10) is decoupledand leads to ρ b = ρ b a . (12)Here and in the following, quantities with a subscript 0 refer to the present epoch and we haveused a = 1. From now on we will set β = 0, thus assuming the dissipative pressure to be thedominating contribution.With a change of variables, ˙ ρ v = dρ v dt = dρ v da dadt = ρ ′ v ˙ a , (13)where the prime means derivative with respect to the scale factor a , and using relation (7), theconservation equation for the viscous component becomes ρ ′ v + 3 a (cid:18) ρ v − aa ξ ρ νv (cid:19) = 0 . (14)This equation can be recast in an integral form: Z dεε ν ( ε + 1) = 2 k − ν ) a (1 − ν ) , ν = 12 ; (15) Z dεε + 1 = k ln a , ν = 12 ; (16) ε = ρ v ρ b , k = 9 r πG ξ ρ ν − b . (17)We will be mainly interested in the case ν = .There is a simple, direct relation between ρ v and the scale factor a for ν = 0. In this particularcase, we find, ρ v = ρ b (cid:26)(cid:20) B + k a (cid:21) − (cid:27) , ( ν = 0) , (18)where B is an integration constant. It can easily be verified that ρ v → a − when a → ρ v → cte (cosmological constant) when a → ∞ . It is expedient to notice that for ν = 0the total energy density ρ = ρ v + ρ b coincides with the energy density of a specific GCG [11, 12, 13].Generally, a GCG is characterized by an equation of state ( E = const > p GCG = − Eρ α GCG , (19)6hich corresponds to an energy density ρ GCG = (cid:20) E + Fa α ) (cid:21) / (1+ α ) , (20)where F is another (non-negative) constant. With the identifications E = k ρ / b and F = Bρ / b itbecomes obvious that for α = − / ρ = ρ v + ρ b from (18) and ρ GCG in (20)coincide. The analogy between dissipative and GCG models was also pointed out in [26].The structure of the total energy density ρ = ρ v + ρ b allows us to perform a decomposition ofthe viscous-baryon system into three non-interacting components plus the baryonic fluid: ρ = ρ + ρ + ρ + ρ b , (21)with ρ = E , ρ = 2 EFa , ρ = F − ρ b a , ρ b = ρ b a , (22)and E = k √ ρ b , F = √ ρ b B . (23)The individual equations of state of the different components are: p = − ρ , p = − ρ , p = 0 , p b = 0 . (24)Consequently, our system is equivalent to a mixture of two dark energy type fluids, one darkmatter fluid and the baryon component. While the BV model is a unified description of darkmatter/energy, the option of a decomposition may be useful in comparing the scenario with someobservational data. The X-ray measurement of galactic clusters [34] is an example which requiresa separation of a matter component of the cosmic medium. For a Chaplygin gas, different decom-positions have been proposed in the literature [35, 36]. Moreover, the decomposition (21) with (22)- (24) reveals, that the background dynamics of our model differs from that of the ΛCDM model(including a baryon component) by the existence of the second dark energy component with energydensity ρ .The relevant cosmological quantities are the present density parameters Ω v and Ω b whichrepresent the fractions of the viscous fluid and of the pressureless matter, respectively, with respectto the total density (remember that we restrict ourselves to the flat case), the Hubble parametertoday H , the present value of the deceleration parameter q , and the age of the universe t . Theapparently new parameter of the present model, the viscosity coefficient, can be expressed in terms7f the other quantities. In fact, the relation (18) can be rewritten in a more convenient way. UsingFriedmann’s equation today, we have H = 8 πG ρ b + ρ v ) ⇒ Ω v + Ω b = 1 , (25)Ω b = 8 πG H ρ b , Ω v = 8 πG H ρ v . (26)Combining (18) with (25) and (26) we obtain the relation B + k √ Ω b . (27)Now, if we use equation (8) with p = p v = − ξ aa and the definition of the deceleration parameter q = − ¨ aa ˙ a , we find − q + 1 = 24 πGξ H . (28)From the definition of k in equation (17) we have k = 24 πGξ H √ Ω b . (29)Using equation (8), evaluated today, it follows that k = 1 − q √ Ω b , (30)which leads to B = 23 1 + q √ Ω b . (31)Hence, the viscous fluid energy density becomes, ρ v = ρ b (cid:26) f ( a ) − (cid:27) , f ( a ) = 13 √ Ω b (cid:20) q ) + (1 − q ) a / (cid:21) . (32)All these relations will be useful for defining the observables that will be constrained later on bythe SNe Ia and matter power spectrum data. III. A DYNAMICAL SYSTEM ANALYSIS
The relations of the previous section that will be used to compare the theoretical predictions ofour model with observational data, are valid under the restriction ν = 0. It is important to havean idea of how serious this restriction is. If the case ν = 0 is very particular, our results will notbe conclusive. Therefore it is desirable to have at least a qualitative analysis of the general case8 xy y S x xy y S FIG. 1:
Phase diagrams for ν < / / < ν < / ν > / (7)-(11) which, because of its complexity, cannot be solved analytically. Such an analysis couldreveal whether or not the case ν = 0 is, in a sense, typical. To this purpose we shall use dynamicalsystem techniques for a system of non-linear differential equations [32].The system of equation (7)-(11) can be recast in a more convenient form by using geometricunities 8 πG = 1 and fixing the scale so that ξ = 1. Defining x = ˙ aa and y = ρ v , the followingsystem of equations is obtained:˙ x = − x ( x − y ν ) = P ( x, y ) (33)˙ y = − xy (cid:0) − xy ν − (cid:1) = Q ( x, y ) . (34)Since the density ρ b = 3 x − y of the baryonic matter must be non-negative, the solutions thathave physical meaning are those which satisfy y ≤ x .In a first step we have to identify the critical points ( x , y ) for which P ( x , y ) = Q ( x , y ) = 0.These points are either attractors or repellers or saddle points. The nature of a critical point isdetermined by the eigenvalues of the matrix P x ( x , y ) P y ( x , y ) Q x ( x , y ) Q y ( x , y ) ,where P x i = ∂P∂x i , x i = x, y and Q x i = ∂Q∂x i . In general, there are two eigenvalues λ , for thecharacteristic equation of this matrix. If both eigenvalues are positive, than the critical point is9epeller; if both are negative, the critical point is an attractor; if one is positive and the other oneis negative, than the critical point is a saddle. With the help of a suitable transformation [32], thecritical points at infinity can be studied. A projection onto the Poincar´e sphere [32] will then allowus to represent the entire phase space in a finite region in the x − y plane.It turns out that the entire Oy axis is a stationary solution for the system, representing astatic space-time. A singular point, denoted by Σ, is ( x , y ) = (cid:16) ν − ν , − ν (cid:17) where ν = . Thecharacteristic equation for this critical point is given by λ + α (cid:0) − ν (cid:1) λ + α (cid:0) − ν (cid:1) = 0 where α = 3 − ν − ν . The eigenvalues are λ = (cid:0) ν − (cid:1) α and λ = − α . The phase diagram depends onthe value of the parameter ν . If ν < /
2, the critical point Σ is an attractor: all solutions convergeto it. This point represents the de Sitter phase. Hence, all solutions approach a de Sitter phaseasymptotically.If 1 / < ν < /
2, the point Σ becomes a saddle point and all solutions approach a Minkowskispace-time in the origin. If ν > /
2, Σ is still a saddle point, but the directions of the curves change:all solutions leave the Minkowski space-time. These different behaviors are shown in figure (1).For the point Σ we have 3 x = y , which corresponds to a vanishing baryon density. (Recall thatthe range y > x is unphysical.) The deceleration parameter q can be written as q = − − ˙ HH ,equivalent to q = − − ˙ xx . Combination with Eq. (33) shows that there is accelerated expansion q < x < y ν . The special case ν = 0 has ( x , y ) = (1 ,
3) and accelerated expansion for x < x axis. The curve that points from the origin towards Σ inthe left part of Fig. 1 corresponds to a phantom scenario with ρ b = 0 which separates the physicaland unphysical regions. We are not interested here in this type of models. IV. CONSTRAINING THE MODEL USING SNE IA DATA
The type Ia supernovae allow us to obtain informations about the universe at high redshifts.Today, observers have detected about 300 of these objects with redshifts up to almost 2. In whatfollows, we will use the more restricted sample of 182 SNe Ia of the Gold06 data set which consistsof objects for which the observational data are of very high quality [37]. The relevant quantity forour analysis is the moduli distance µ , which is obtained from the luminosity distance D L by the10 ulk Generalized Traditional ΛCDMViscosity Chaplygin gas Chaplygin gas χ .
71 157 .
52 157 .
88 160 . B . − − − α − .
90 1 0¯ E − .
897 0 .
825 1Ω b . . . . dm − − − . H .
38 63 .
97 63 .
56 62 . t .
92 13 .
83 14 .
09 14 . q − . − . − . − . χ is minimum, for the BV model, the GCG model, thetraditional Chaplygin gas model ( α = 1) and the ΛCDM model ( α = 0) for a flat Universe (Ω k = 0). H is given in km/Mpc.s , ¯ E ≡ EE + F in units of c ( c - speed of light) and t in Gy . relation µ = 5 log (cid:18) D L M pc (cid:19) +25 . (35)The computation of the luminosity distance follows the standard procedure [1, 38]. First, werecall its definition [39, 40], D L = ra = (1 + z ) r , (36)where r is the comoving coordinate of the source and z is the redshift, z = − a . The coordinate r can be obtained by considering the propagation of light ds = 0. This implies, ds = c dt − a dr = 0 ⇒ r = − c Z a daa ˙ a . (37)From Friedmann’s equation one obtains,˙ a = r Ω m a H f ( a ) . (38)11 ulk Generalized Traditional ΛCDMViscosity Chaplygin gas Chaplygin gas B . +0 . − . − − − α − . +4 . − . E − . +0 . − . . +0 . − . b . +0 . − . . +0 . − . . +0 . − . . +0 . − . Ω dm − − − . +0 . − . H . +1 . − . . +1 . − . . +1 . − . . +1 . − . t . +1 . − . . +1 . − . . +0 . − . . +0 . − . q − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . p ( q <
0) 6 . σ
100 % 100 % 100 %TABLE II: The estimated parameters for the BV model, the GCG model, the traditional Chaplygin gasmodel and the ΛCDM model for a flat Universe (Ω k = 0). We use the Bayesian analysis to obtain the peakof the one-dimensional marginal probability and the 2 σ confidence region for each parameter. H is givenin km/s/Mpc, ¯ E ≡ EE + F in units of c ( c - speed of light) and t in Gy. In terms of the redshift z , the luminosity distance may be expressed as D L = 3(1 + z ) cH Z z dz ′ q )(1 + z ′ ) / + (1 − q ) , (39)or, using B and Ω m , D L = (1 + z ) cH Z z d z ′ √ Ω m B (( z ′ + 1) / − . (40)In order to compare the theoretical results with the observational data, the first step is tocompute the quality of the fitting through the least square fitting quantity χ . We adopt the HST(Hubble Space Telescope) prior [41] for H , as well as the cosmic nucleosynthesis prior for thebaryonic density parameter Ω b h (where h is H divided by 100 km/s/Mpc), which leads to χ = X i (cid:16) µ o ,i − µ t ,i (cid:17) σ µ ,i + ( h − . . + (Ω b h − . . . (41)12he quantities µ o ,i are the distance moduli, observationally measured for each supernova of the182 Gold06 SNe Ia dataset [37] and the µ t ,i are the corresponding theoretical values. The σ µ ,i represent the measurement errors and include the dispersion in the distance moduli, resulting fromthe galaxy redshift dispersion which is due to the peculiar velocities of the objects (cf. [3, 37]).The smaller the χ , the better the agreement between the theoretical model and the observa-tional data. Table I shows that the χ value for BV model is competitive with the GCG model (cf.Eq. (20)), the traditional Chaplygin gas model (cf. Eq. (20) with α = 1) and the ΛCDM model.Compared with the ΛCDM and the (generalized) Chaplygin gas models, the age of the Universe isgreater for the bulk viscous model and its deceleration parameter is less negative, i.e., the Universeis less accelerated.Using Bayesian statistics, a probability distribution can be constructed from the χ parameter[38]. In the present case, there are three free parameters: the Hubble parameter today H , thepressureless baryonic density parameter Ω b and the auxiliary quantity B which is connected tothe present value of the deceleration parameter (cf. (31)). Marginalization over one or two ofthese parameters will lead to corresponding two- or one-dimensional representations. The detailsof the Bayesian statistics and computational analysis, relying on BETOCS ( B ay E sian T ools for O bservational C osmology using S Ne Ia), are given in ref. [38].The main feature of the parameter estimation for the viscous fluid model is the Gaussian-likedistribution around the preferred value.This can easily be seen both in the table II (almost symmetric error bars) and in the two andone-dimensional diagrams of figures 2 and 3. In figure 2 we show the two-dimensional probabilitydistribution for the parameters B , Ω b and H at one, two and three sigma, corresponding, re-spectively, to 68%, 95% and 99% confidence levels. The corresponding one-dimensional probabilitydistributions are shown in figure 3. The main features of the model are: the age of the universeis considerably larger than for the ΛCDM and GCG models, around t ∼
16 Gy, with a smalldispersion; the baryon density is around 0 .
05, a little larger than predicted by nucleosynthesis, butthere is a marginal agreement; the Hubble constant is around 62 km/s/Mpc, i.e. much smallerthan the one predicted by the CMB, which is around 72 km/s/Mpc, but in agreement with theΛCDM and GCG models when only the data from SNe Ia are used; the deceleration parameter isbigger (smaller absolute value) than in those models. For the parameter estimations in the ΛCDMand in the GCG models, see ref. [38] and references therein.The Gaussian nature of the distribution is responsible for the fact that the best fitting set ofvalues is very close to the marginalized parameter estimations, with a small dispersion. Generally,13 .04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 W b0 B PDF for BVM with W b0 h , HST prior, W k0 = W dm0 = W b0 H PDF for BVM with W b0 h , HST prior, W k0 = W dm0 = FIG. 2:
The two-dimensional plots of the Probability Density Function (PDF) as function for the parameter B , thebaryon density and the Hubble constant. The joint PDF peak is shown by the large dot, the confidence regions of 1 σ (68 , σ (95 , σ (99 , this is neither true for the ΛCDM model nor for the GCG model. Again: the most remarkabledifferences between the viscous model and the ΛCDM model are the larger age of the universe andthe smaller absolute value of the deceleration parameter. V. THE MATTER POWER SPECTRUMA. The perturbed equations
In order to write the first order perturbed equations, we re-express the field equations as R µν = 8 πG (cid:26) T vµν − g µν T v (cid:27) +8 πG (cid:26) T bµν − g µν T b (cid:27) , (42) T µνv ; µ = 0 , T µνv = ( ρ v + p v ) u µ u ν − p v g µν , (43) p v = − ξ u µ ; µ , (44) T µνb ; µ = 0 , T µνb = ρ b u µ u ν . (45)We introduce the quantities,˜ g µν = g µν + h µν , ˜ ρ = ρ + δρ , ˜ u µ = u µ + δu µ , (46)14 .2 1.4 1.6 1.8 2 2.2 2.4 B p PDF for BVM with W b0 h , HST prior, W k0 = W dm0 = W b0 p PDF for BVM with W b0 h , HST prior, W k0 = W dm0 =
59 60 61 62 63 64 65 H p PDF for BVM with W b0 h , HST prior, W k0 = W dm0 = - - - - q p PDF for BVM with W b0 h , HST prior, W k0 = W dm0 = FIG. 3:
The one-dimensional plots of the PDF as function for the parameter B , the baryonic density, the Hubbleconstant and the deceleration parameter. The joint PDF peak is shown by the large dot, the confidence regions of 1 σ (68 , σ (95 , σ (99 , where g µν , ρ and u µ are the known background solutions for the metric, the energy density andthe four velocity, respectively, and h µν , δρ and δu µ are the corresponding perturbations. We willperform the calculations using the synchronous coordinate condition, h µ = 0 . (47)Since we will be interested mainly in perturbations that are inside the horizon, the choice of thegauge is not really essential.The perturbed Ricci tensor takes the form, δR µν = χ ρµν ; ρ − χ ρµρ ν (48)where χ ρµν = g ρσ (cid:20) h σµ ; ν + h σν ; µ − h µν ; σ (cid:21) , (49)15re the perturbations of the Christoffel symbols. The relevant non-vanishing components are: χ i j = − (cid:18) h ij a (cid:19) · , (50) χ ij = − ˙ h ij , (51) χ kij = − a (cid:26) h ik,j + h ij,k − h ij,k (cid:27) . (52)For the components of the perturbed energy-momentum tensor we have δT = δρ , (53) δT i = ( ρ + p ) δu i , (54) δT ij = h ij p − g ij δp . (55)Using the definitions, h = h kk a , Θ = δ i,i , δ b = δρ b ρ b , ∆ p v = δp v ξ , (56)and performing a plane wave decomposition of all perturbed functions according to δf ( ~x, t ) = δf ( t ) e i~k · ~x , (57)where ~k is the wave vector, we find, after a long but straightforward calculation, the following setof first order perturbed equations:¨ h + 2 ˙ aa ˙ h = 8 πG ( δρ v + 3 δp v ) + 8 πGρ b δ b , (58) δ ˙ ρ v + 3 ˙ aa ( δρ v + δp v ) + ( ρ v + p v ) (cid:18) Θ − ˙ h (cid:19) = 0 , (59)( ˙ ρ v + ˙ p v )Θ + ( ρ v + p v ) ˙Θ + 5 ˙ aa ( ρ v + p v )Θ − k a δp v = 0 , (60)˙ δ b = ˙ h . (61)Replacing ˙ h by using the last relation, we obtain¨ δ b + 2 ˙ aa ˙ δ b − πGρ b δ b − πGδρ v − πGξ ∆ p v = 0 , (62) δ ˙ ρ v + 3 ˙ aa ( δρ v + ξ ∆ p v ) − ( ρ v + p v )∆ p v = 0 , (63)( ˙ ρ v + ˙ p v )Θ + ( ρ v + p v ) ˙Θ + 5 ˙ aa ( ρ v + p v )Θ − k ξ a ∆ p v = 0 , (64)Θ − ˙ δ b + ∆ p v = 0 . (65)16ith the help of the constraint (65), this system can be further reduced to a set of three coupleddifferential equations: ¨ δ b + 2 ˙ aa ˙ δ b − πGρ b δ b − πGδρ v − πGξ ∆ p v = 0 ; (66) δ ˙ ρ v + 3 ˙ aa ( δρ v + ξ ∆ p v ) − ( ρ v + p v )∆ p v = 0 ; (67)( ˙ ρ v + ˙ p v ) ˙ δ b + ( ρ v + p v )¨ δ b + 5 ˙ aa ( ρ v + p v ) ˙ δ b =( ˙ ρ v + ˙ p v )∆ p v + ( ρ v + p v )∆ ˙ p v + 5 ˙ aa ( ρ v + p v )∆ p v + k ξ a ∆ p v . (68)Now it is convenient to introduce the background equations and the quantities in Eq. (32) fromwhich we find 4 πGρ b H = 32 Ω b a , (69)4 πGρ v H = 32 Ω b a [ f ( a ) − , (70)12 πGξ H = 12 (1 − q ) . (71)The following relations will be useful later on: p v ρ v ≡ g ( a ) = − f ( a ) f ( a ) − − q √ Ω b a / , (72)˙ ρ v = − aa ( ρ v + p v ) , (73)˙ p v = 1 − q ρ v + ρ b + p v ) . (74)In a next step we divide the set of equations (66) - (68) by H and redefine H − ∆ p v → ∆ p v .Again, after a fairly long calculation, the perturbed equations reduce to the following system ofcoupled linear differential equations:¨ δ b + 2 ˙ aa ˙ δ b −
32 Ω b a δ −
32 Ω b a ( f − δ v −
12 (1 − q )∆ p v = 0 , (75)˙ δ v − aa gδ v − (1 + 2 g )∆ p v = 0 , (76)(1 + g )¨ δ b + (cid:26) aa (1 + g ) + 1 − q (cid:20) f f − g (cid:21)(cid:27) ˙ δ b =(1 + g )∆ ˙ p v + (cid:26) aa (1 + g ) + 1 − q (cid:20) f f − g (cid:21) + (cid:18) k l H (cid:19) (1 − q ) a b ( f − (cid:27) ∆ p v , (77)17here l H = c/H .Now, we re-express the perturbed equations in terms of the variable a according to (13), whichimplies ˙ δ = δ ′ ˙ a , ¨ δ = ˙ a δ ′′ + ¨ aδ ′ . (78)We also need the following relations, which result from the background equations:˙ a = Ω b f a , ¨ a ˙ a = − a (cid:26) g f − f + 1 (cid:27) . (79)With the redefinition ∆ v = ∆ p v ˙ a , the final form for the set of perturbed equations is: δ ′′ b + (cid:26) − g f − f a + 32 a (cid:27) δ ′ b − δ b f a =32 f − f a δ v − ga f − f ∆ v ; (80) δ ′ v − ga δ v − (1 + 2 g )∆ v = 0 ; (81)(1 + g ) δ ′′ b + (cid:26) − ga (1 + 2 g ) f − f + 32 a (cid:27) δ ′ b =(1 + g )∆ ′ v + (cid:26) − ga (1 + 2 g ) f − f + 32 a − (cid:18) k l H (cid:19) g f Ω b (cid:27) ∆ v . (82) B. Numerical integration
Evidently, equations (80)-(82) are too complicated to admit analytical solutions. Hence, weproceed integrating these equations numerically. In order to do so, an important problem is tofix the initial conditions. A scale invariant primordial spectrum (as predicted by the inflationaryscenario) is assumed. But, since we are interested in the power spectrum today, we have to followthe evolution of this spectrum until the present phase.The corresponding procedure for the ΛCDM model has been carried out in terms of the BBKStransfer function [42, 43, 44]. This function is characterized by˜ q = ˜ q ( k ) = k ( h Γ) Mpc − , Γ = Ω M h e − Ω B − ( Ω B / Ω M ) , (83)where Ω M and Ω B are the density parameters of dark matter and baryonic matter of the ΛCDMmodel. The baryonic transfer function is approximated by by the numerical fit [42]: T ( k ) = ln (1 + 2 . q )2 .
34 ˜ q h . q + (16 . q ) + (5 . q ) + (6 . q ) i − / . (84)18he quantity of interest is the power spectrum for the baryonic matter today. This spectrum isgiven by P ( k ) = | δ b ( k ) | = A k T ( k ) g (Ω T ) g (Ω M ) , (85)where Ω T is the total density parameter of the ΛCDM model. The function g in (85) is the growthfunction g (Ω) = 5Ω2 (cid:20) Ω / − Ω Λ + (cid:16) (cid:17)(cid:16) Λ (cid:17)(cid:21) − (86)in which Ω Λ represents the fraction of the total energy, contributed by the cosmological constant.The normalization coefficient A in (85) can be fixed by using the COBE measurements of the CMBanisotropy spectrum. This coefficient is connected to the quadrupole momentum [43, 44] Q rms ofthis spectrum by the relation A = (2 l H ) π Q rms T , (87)where T = 2 . ± .
001 is the present CMB temperature. The quadrupole anisotropy is taken as Q rms = 18 µ K . (88)This value is obtained from the COBE normalization and is consistent with the more recent resultsof the WMAP measurements which use the prior of a scale invariant spectrum [44, 45]. Taking allthese estimates into account, one may fix A = 6 . × h − Mpc , (89)which corresponds to the initial vacuum state for the density perturbations used in the BBKStransfer function.We recall that all the relations (83)-(89) are valid for the ΛCDM model. Strictly speaking,we would have to redo all the calculations to obtain the analogue of the expression (85) for ourbulk viscous model. However, since we are interested in the shape of the spectrum, there existsa simpler way which relies on the circumstance that the growth function g in (86) is a purebackground quantity. Use of the ΛCDM growth function and the ΛCDM initial values also for ourmodel will result in an overall shift of the spectrum P without affecting the structure of the latter.The resulting data will then be a fit to a “wrong”, i.e. unphysical ΛCDM model, in which thecomposition of the dark sector differs from the standard one with roughly 30% of dark matter and70% of a cosmological constant. However, this procedure provides us with the correct structure19f the spectrum for our bulk viscous model. Keep in mind that the background dynamics of ourmodel differs from that of the ΛCDM model by the existence of a second dark energy component(component 2) in Eq. (22).Well inside the matter dominated phase (say, z ∼ P ( k ) = δ b ( k ) at z = 0, equivalent to a = 1. Finally, we compare the theoretical resultswith the power spectrum obtained by the 2dFGRS observational program. The spectrum dependsessentially on two parameters: Ω b and q . From now on, relying on the results of the type Ia SNeanalysis and the primordial nucleosynthesis constraint, we fix Ω b = 0 .
04 (according to the type IaSNe analysis of the previous section, this is marginally satisfied at least at 2 − σ ).In figures 4 and 5 we present the best fitting for q = − . , − . , − . , − .
6. To have a goodagreement, the ratio between the quantity of dark matter/dark energy in the “wrong” ΛCDMreference model has to be in the range from 1 for q = − .
3, to about 2 / q = − .
6. It is seenthat the baryonic matter power spectrum of the viscous fluid model is in good agreement with theobservational data for values of q that are close to those predicted by the type Ia SNe analysis,that is, q ∼ − .
4. We remark en passant that a variation of the value of Ω b within the intervalpredicted by the type Ia SNe at 2 − σ does not change substantially the results presented in figures4 and 5. In any case, there is no instability in the computed power spectrum. This is true not onlyfor the baryonic fluid power spectrum but also for dark viscous fluid power spectrum. We note,however, that as q approaches zero, there is a power depression at large scales (small k ) comparedwith the ΛCDM reference model. C. A physical interpretation
In the homogeneous and isotropic background, the BV model is equivalent to a GCG with α = − /
2. The qualitative differences between both models at the perturbative level can be tracedback to a difference in the pressures that characterize the cosmic medium in each of these cases.In the background both pressures coincide. But while the perturbations for the GCG are entirelyadiabatic, this is not the case for our present model.In the background we have p = p v = − Hξ for the viscous fluid and p = − Eρ / for the GCGwith α = − / H = 8 πGρ (for the spatially flat case)reveals that in both cases we have an equation of state p ∝ − ρ / , where the constants are related20 Log k (cid:144) h @ Mpc - D Log P H k L (cid:144) h - @ M p c D -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 Log k (cid:144) h @ Mpc - D Log P H k L (cid:144) h - @ M p c D FIG. 4:
Predicted power spectrum for the dark viscous fluid compared with the 2dFGRS observational data for q = − . q = − . dm = 0 .
49 and Ω Λ0 = 0 .
51 (left) and Ω dm = 0 .
52 andΩ Λ0 = 0 .
48 (right). -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8
Log k (cid:144) h @ Mpc - D Log P H k L (cid:144) h - @ M p c D -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 Log k (cid:144) h @ Mpc - D Log P H k L (cid:144) h - @ M p c D FIG. 5:
Predicted power spectrum for the dark viscous fluid compared with the 2dFGRS observational data for q = − . q = − . dm = 0 .
44 and Ω Λ0 = 0 .
56 (left) and Ω dm = 0 .
52 and Ω Λ0 = 0 . E = ξ √ πG . The first order pressure perturbations for the GCG are δp = ˙ p ˙ ρ δρ (GCG model) , (90)where ˙ p ˙ ρ = − α pρ (GCG model) (91)is the adiabatic sound speed square. For α > α < δp − ˙ p ˙ ρ δρ = − ˙ p (cid:18) δρ ˙ ρ − δH ˙ H (cid:19) (BV model) , (92)where δH ≡ δ ( u µ ; µ ) /
3. In general, the right hand side of (92) does not vanish (For a similarsituation see [46]). It is this non-adiabatic character of the perturbations which makes the powerspectrum well behaved on small scales in contrast to the GCG case. A more detailed analysis ofthis feature will be given elsewhere.
VI. CONCLUSIONS
We presented a two-component cosmological model in which one component represents the darksector, the other one pressureless baryons. The dark sector is described by a single bulk viscousfluid with a constant bulk viscosity coefficient ξ = ξ =constant. A dynamical system analysis wasused to show that the assumption of a constant ξ does not seem to be too restrictive and that thismodel embodies quite general features and does not represent a very particular configuration. Inthe homogeneous and isotropic background the total energy density (i.e., including the baryons) isequivalent to that of a GCG with α = − /
2. But while the perturbation dynamics of a GCG isentirely adiabatic, the present model exhibits non-adiabatic features. As a consequence, it can avoidthe problems that usually plague such type of unified dark matter/dark energy models, namelythe appearance of (non-observed) small scale oscillations or instabilities. The predictions of ourmodel were compared with SNe Ia data and with the 2dFGRS results. For the comparison withSNe Ia data we employed the so-called gold sample of 182 good quality high redshift supernovae.22ur Bayesian statistics analysis leads to the following parameter evaluation at 2 σ level: q = − . +0 . − . , Ω b = 0 . +0 . − . , H = 62 . +1 . − . km/s/Mpc and t = 15 . +0 . − . Gy. The maindifferences in comparison with a similar analysis for the ΛCDM and the GCG models [38] are: theage of the universe is considerably larger; the deceleration parameter is larger (smaller in absolutevalue); the baryon density is only marginally compatible with the primordial nucleosynthesis result.A remarkable feature is the (almost) perfect Gaussian distribution for all parameters with a quitesmall dispersion.To compare the results of our perturbation analysis with the observed matter power spectrum wehave used the BBKS transfer function to fix the initial condition at a very high redshift ( z ∼ q which are in agreement with the type Ia SNe analysis. The best results arefound for − . & q & − .
6. Qualitatively, our analysis confirms previous results for a simplifiedmodel of the cosmic medium, in which the baryon component was neglected [27].
Acknowledgments
J.C.F. thanks the IMSP (B´enin) for the warm hospitality during part of the elaboration ofthis work. He thanks also CNPq (Brazil) and the CAPES/COFECUB French-Brazilian scientificcooperation for partial financial support. W.Z. acknowledges support by grants 308837/2005-3(CNPq) and 093/2007 (CNPq and FAPES). [1] A.G. Riess et al., Astron. J. , 1009(1998); S. Perlmutter et al., Astrophys. J. , 565(1999).[2] J.L. Tonry et al, Astrophys. J. , 1(2003).[3] A.G. Riess, Astrophys. J. , 665(2004).[4] P. Astier et al., Astron. Astrophys. , 31(2006).[5] D.N. Spergel et al. , ApJS, , 377 (2007) (arXiv:astro-ph/0603449v2).[6] M. Tegmark et al., Phys. Rev.
D69 , 103501 (2004).[7] S.W. Allen, R.W. Schimdt, H. Ebeling, A.C. Fabian and L. van Speybroeck, Mon. Not. Roy. Astron.Soc. , 457 (2004).[8] E.V. Linder,
Cosmology with X-ray cluster baryons , arXiv:astro-ph/0606602v2.
9] A. Lewis and A. Challinor, Phys. Rep. , 1 (2006).[10] S. Hannestad, Int. J. Mod. Phys.
A21 , 1938 (2006).[11] A. Yu. Kamenshchik, U. Moschella and V. Pasquier, Phys.Lett.
B511 , 265 (2001).[12] J.C. Fabris, S.V.B. Gon¸calves e P.E. de Souza, Gen. Rel. Grav. , 53 (2002).[13] M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev. D66 , 043507 (2002).[14] H. Sandvik, M. Tegmark, M. Zaldarriaga and I. Waga, Phys. Rev.
D69 , 123524 (2004).[15] J.C. Fabris and H. Velten, work in progress.[16] A.B. Balakin, D. Pav´on, D.J. Schwarz, and W. Zimdahl, NJP , 85.1 (2003).[17] R.R.R. Reis, I. Waga, M.O. Calv˜ao e S.E. Jor`as, Phys. Rev. D68 , 061302 (2004).[18] W. Zimdahl and J.C. Fabris, Class. Quant. Grav. , 4311( 2005).[19] V.A. Belinskii and I.M. Khalatnikov, Sov. Phys. JETP , 205 (1975).[20] R. Maartens, Class. Quantum Grav. , 1455 (1995).[21] W. Zimdahl, Phys. Rev. D , 5483 (1996).[22] R. Maartens 1997 Causal Thermodynamics in Relativity in Proceedings of the Hanno Rund Confer-ence on Relativity and Thermodynamics ed S D Maharaj, University of Natal, Durban pp 10 - 44.(astro-ph/9609119).[23] W. Zimdahl, Phys. Rev. D , 083511 (2000).[24] W. Zimdahl, D.J. Schwarz, A.B. Balakin, and D. Pav´on, Phys. Rev. D , 063501 (2001).[25] T. Padmanabhan and S. M. Chitre, Phys. Lett. A , 433 (1987).[26] M. Szyd lowski and O. Hrycyna, arXiv:astro-ph/0602118.[27] J.C. Fabris, S.V.B. Gon¸calves and R. de S´a Ribeiro, Gen. Rel. Grav. , 495 (2006).[28] C. Eckart, Phys. Rev. D58 , 919 (1940).[29] W. Israel and J.M. Stewart, Proc. R. Soc. Lond.
A365 , 43 (1979).[30] W. Israel and J.M. Stewart, Ann. Phys. , 341 (1979).[31] Jian-Huang She, JCAP , 021 (2007) (hep-th/0702006).[32] G. Sansone and R. Conti, EQUAZIONI DIFFERENZIALI NON LINEARI, Edizioni Cremonense,Roma, 1956.[33] A.A. Coley and R.J. van den Hoogen, Class. Quant. Grav. , 1977 (1995).[34] J.G. Bartlett and J. Silk, Astrophs. J. , 12 (1994); J.G. Bartlett, Astrophys. Sp. Scie. , 105(2004).[35] M.C. Bento, O. Bertolami and A.A. Sen, Phys. Rev. D70 , 083519 (2004).[36] W. Zimdahl and J.C. Fabris, Class. Quant. Grav. , 1731 (2005).[37] A.G. Riess et al. , New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ¿ 1: NarrowingConstraints on the Early Behavior of Dark Energy , arXiv:astro-ph/0611572.[38] R. Colistete Jr, J. C. Fabris, S.V.B. Gon¸calves and P.E. de Souza, Int. J. Mod. Phys.
D13 , 669 (2004);R. Colistete Jr., J. C. Fabris and S.V.B. Gon¸calves, Int. J. Mod. Phys.
D14 , 775 (2005); R. ColisteteJr. and J. C. Fabris, Class. Quant. Grav. , 2813 (2005); R. Colistete Jr. and R. Giostri, BETOCS sing the 157 gold SNe Ia Data : Hubble is not humble , arXiv:astro-ph/0610916.[39] S. Weinberg, Gravitation and cosmology , Wiley, New York (1972).[40] P. Coles and F. Lucchin,
Cosmology , Wiley, New York (1995).[41] W. Freedman, Astrophys. J. , 47 (2001).[42] J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, Astrophys. J. , 15 (1986).[43] N. Sugiyama, Astrophys. J. Suppl. , 281 (1995).[44] J. Martin, A. Riazuelo and M. Sakellariadou,
Phys. Rev.
D61 , 083518 (2000).[45] G. Hinshaw et al., ApJS, , 288 (2007) (arXiv:astro-ph/0603451v2).[46] W. Zimdahl, arXiv:0705.2131v1 [gr-qc]., 288 (2007) (arXiv:astro-ph/0603451v2).[46] W. Zimdahl, arXiv:0705.2131v1 [gr-qc].