A causal viscous cosmology without singularities
aa r X i v : . [ g r- q c ] A p r A causal viscous cosmology withoutsingularities
Carlos E. LacianaDepartamento de Hidr´aulica, Facultad de Ingenier´ıa,Universidad de Buenos Aires,Av. Las Heras 2214, Ciudad Aut´onoma de Buenos Aires,C1127AAR, Argentina.email: clacian@fi.uba.arKeywords:dark energy, accelerated universe, relativistic fluid, causal correction.
Abstract
An isotropic and homogeneous cosmological model with a source ofdark energy is studied. That source is simulated with a viscous relativisticfluid with minimal causal correction. In this model the restrictions onthe parameters coming from the following conditions are analized: a)energy density without singularities along time, b) scale factor increasingwith time, c) universe accelerated at present time, d) state equation fordark energy with “ w ” bounded and close to -1. It is found that thoseconditions are satified for the following two cases. i) When the transportcoefficient ( τ Π ), associated to the causal correction, is negative, with theaditional restriction ζ | τ Π | > /
3, where ζ is the relativistic bulk viscositycoefficient. The state equation is in the “phantom” energy sector. ii) For τ Π positive, in the “k-essence” sector. It is performed an exact calculationfor the case where the equation of state is constant, finding that option (ii)is favored in relation to (i), because in (ii) the entropy is always increasing,while this does no happen in (i). As stressed in ref. [1], there is relevant experimental evidence about the ac-celeration of the universe. That effect can be attributed to the so-called darkenergy, which can be interpreted as a term of ”negative pressure” in the Einsteinequations, as remarked in ref. [2]. 1he most cited experimental observations related to universe accelerationare due to Perlmutter et al. [3] and Riess et al. [4], which include measuresof the redshift of supernovas. The results are compatible with the addition ofthe cosmological constant to the Einstein equations. This cosmological con-stant would be analogous to a negative pressure that leads the universe to anaccelerated expansion.The dark energy can be described by the state equation p = w ρ , with w < − /
3, in order to have an accelerated universe [5]. In agreement with experimen-tal observations [6], the value of w would be very close to − w = − . +0 . − . ).From a theoretical point of view, the different values of w considered in the lit-erature come from some Lagrangian formulations of field theories (for a review,see ref. [7]). It is so for a field with minimal coupling, i.e., − ≤ w ≤
1, which isknown as ”quintessence”. Also for potentials coming from string theory as, forexample, the formulation known as ”k-essence”, with − ≤ w ≤ − /
3. Froms-brane, in superstring theory, it is obtained for the ”phantom” field w < − Let us consider an isotropic and homogeneous universe with matter modelled asa relativistic fluid. Then, a spatially flat Friedmann-Robertson-Walker (FRW)universe is considered. From here on, the units for which c = 8 πG = k = 1 andthe signature for the metrics (+ , − , − , − ) are used, as in ref. [26]. The dynamics will be described by the Einstein equations with the energy-momentum tensor (EMT) as a source of viscous fluid (see ref. [26] or [27]),i.e.: R µν − g µν R = T µν , (1)where de EMT can be factorized as [24]: T µν = e T µν + Π µν . (2) e T µν is the ideal fluid part, given by e T µν = ( p + ρ ) u µ u ν − pg µν , (3)where ρ is the density of energy (dark energy in our case), p is the pressure, u µ the four-velocity, and g µν is the space-time metrics. Given the convention usedhere, we have the normalization equation u µ u µ = 1 . µν in Eq.(2) is the viscosity term. This term can also be fac-torized into two tensors, one of them the traceless part ( π µν ), related with theshear viscosity, and the other part with non-vanishing trace ( Π), representingthe bulk viscosity. Then, we have (as in [22])Π µν = π µν + ∆ µν Π , (4)with ∆ µν = g µν − u µ u ν . (5)It is convenient to re-write e T µν in the form: e T µν = ρu µ u ν − p ∆ µν . (6)Then, e T µµ = ρ − p. (7)By replacing Eq.(4) into Eq.(2), we obtain T µν = ρu µ u ν − ( p − Π) ∆ µν + π µν . (8)Therefore, now we have T µµ = ρ − p − Π) , (9)because π µµ = 0 . (10)By comparing Eq. (7) with Eq. (9), we can see that the quantity p − Π isequivalent to a corrected pressure. Then, we can say that − Π is a kind of“negative pressure” that represents the viscosity effect. On this basis, it isconvenient to define p † ≡ p − Π (11)By taking the trace in Eq. (2), and by using Eq. (9) with the definition (11)and the Hubble coefficient H = · a/a , with a ( t ) the scale factor, one finds · H + 2 H = 16 (cid:0) ρ − p † (cid:1) . (12)If the 00 component is taken from Eq. (1), and the term π µν is neglected in Eq.(8) because at lower order it goes as ∂ <α u β> (see ref. [22]), then one obtains H = 13 ρ. (13)For the closed universe, a term of the form a − must be added in Eqs. (12) and(13). Then, when a >>
1, the formulation is approximately that corresponding4o a flat universe. By derivation of Eq. (13), and by replacing the result in Eq.(12), the following well-known and useful expression can be obtained: · ρ + 3 H (cid:0) p † + ρ (cid:1) = 0 . (14)It is easy to prove that the last equation is valid for both flat and closed universein exact way. The functional form of the quantity Π was developed in the references [28], [21],[29], [25], and the generalization to second order in velocity gradients is alsogiven by [24]. Hereafter, the approach by Koide et al. [23] will be used. Suchproposal corresponds to the lowest order, in which the violation of causalitydoes not occur. The quantity Π, in that reference, is approximated by:Π ≃ ζ ▽ µ u µ − τ Π u µ ▽ µ Π , (15)where ζ is the bulk viscosity coefficient and τ Π , sometimes called ”second vis-cosity coefficient”, comes from the causal correction (see [24]). The product ζτ Π ≡ τ R is called “relaxation time” (see [23]).As we can see, Eq. (15) is an implicit equation of Π; therefore, to calculatethe source of the dynamical equation it is useless. However, we can deduce anapproximate, but explicit, expression to determine Π. In order to do this, wecan substitute Eq. (15) into itself and neglect the terms higher than the firstorder in τ Π . Then, one obtainsΠ ≃ ζ ▽ µ u µ − ζτ Π u µ ▽ µ ▽ α u α . (16)The first term of Eq. (16) corresponds to the Landau-Liftshitz theory [13], whilethe second term is the minimun necessary in order to avoid causality violation.Now we will express Π as a function of the scale factor a ( t ) and its derivatives.So, it is convenient to employ the continuity equation ( ▽ µ ( nu µ ) = 0 with n thedensity number) and to express the equations by means of the proper time. So,we finally get Π = 3 ζH − ζτ Π · H. (17)This functional form for Π is particularly convenient to solve the dynamicalequation. The equation of state (EoS), which relates the pressure and the density of darkenergy, is added to the equations given above. Following ref. [30], we can usethe following expression for the dark energy w = pρ = − − λρ α − , (18)5ith α an arbitrary parameter.On the other hand, by replacing Eq. (14) into Eqs. (13) and (18) we obtain (cid:18) ζτ Π (cid:19) · H − α λH α − ζH = 0 . (19)As we will see, by the resolution of Eq. (19), in some cases it is possible to obtainan exact expression for the dark energy density as a time dependent function.The questions that we want to answer in this paper are referred to theconditions that the parameters must satisfy for the following requirements befulfilled: i) energy density without singularities at finite time, ii) scale factor a ( t ) as an increasing function of the time, iii) accelerated universe at presenttime, and iv) w close to − t = 1 √ Z ρρ P M ( ρ ) ρ / dρ, (20)ln (cid:18) aa P (cid:19) = Z ρρ P M ( ρ ) dρ, (21) ·· aa = 13 ρ (cid:18) ρM ( ρ ) (cid:19) , (22)with ∆ t ≡ t − t P (subindex “ P ” indicate Planck era), and M ( ρ ) ≡ (cid:18) ζτ Π (cid:19) / (cid:16) λρ α + √ ζρ / (cid:17) . (23)Eqs. (20) - (23), plus Eq. (18), are the set of equations that, in the nextsection, will be used to analyze the restrictions in the parameters necessary toavoid singularities in the physical quantities. In particular, the main interest isto determine the influence of the causal correction on the results. In this section, the conditions that the quantities on the left side of Eqs. (20) -(23) and the state equation (18) must satisfy will be formulated, i.e.:i) The condition on ∆ t = ∆ t ( ρ ). In this case, two possibilities are considered:I) The dark energy density as a decreasing function of time, i.e.lim ρ → ρ f ∆ t = ∞ , for ρ f < ρ P , (in particular, ρ f can be zero).II) The dark energy density as an increasing function of time, i.e.6im ρ → ρ f ∆ t = ∞ , for ρ f > ρ P . A particular case is ρ f = ∞ , which isknown in the literature as ”Little Rip” [30].ii) The scale factor a ( t ) as an increasing function of time, i.e.lim ρ → ρ f a ( ρ ) = a f , such that a f > a o , where a o is the scale factor observedat present .iii) Accelerated universe (at least for t ∼ t o ), i.e. ·· a > w ∼ − ± . In this subsection, the restrictions on the parameters due to the above condi-tions are analyzed in general. In particular, the two possibilities I and II areconsidered: ρ as a decreasing function Condition (i) tells us that ∆ t is a decreasing function of ρ , because ρ decreaseswith t . Then, the following inequality must be satisfied: d ∆ t/dρ <
0. As aconsequence, from Eq. (20) the implication is M ( ρ ) < . (24)From condition (ii), ln (cid:16) aa P (cid:17) is a decreasing function of ρ . Therefore, theinequality (24) is implied again, i.e., condition (ii) does not introduce a newrestriction. It is noteworthy that the condition (24) may be satisfied by negativenumerator or denominator of M ( ρ ). In the first case, it should be τ Π <
0, withthe additional condition ζ | τ Π | > / . (25)In the second case, we should make λ < | λ | > √ ζρ − α . (26)Since in model I ρ ( t ) decreases with time, the above condition requires α ≤ / | λ | , or that ρ ( t )tends to a finite value, as we shall see in the next subsection.From condition (iii), the following inequality (besides that of (24)) must besatisfied: ρ > | M ( ρ ) | − , ∀ ρ ≤ ρ P . (27)7his implies that there is a density value, let’s call it ρ c , below which thecondition (iii) is not met. This value will depend on how close to 1 the value of ζ | τ Π | is, and how small λ is. This will be clear when, in the next subsection,a specific case will be analyzed.Condition (iv) implies that | λ | ρ α − . . . (28)Then, for w limited, it must be α >
1, which is in contradiction with thecondition coming from (24). ρ as an increasing function Condition (i) in this case tells us that ∆ t is an increasing function of ρ , i.e.: d ∆ t/dρ >
0. Therefore, M ( ρ ) > . (29)Condition (ii) ⇒ d ln( a/a p ) /dρ > ⇒ (29).Condition (iii) is directly satisfied.Condition (iv) is the same as in the case of (28), but with λ = | λ | (phantomenergy sector). But, since ρ can grow indefinitely, in order to keep w bounded,in this case it should hold that α ≤ α = 1. Hence,it is interesting to analyze this case in more detail, a task that we will undertakein the next subsection. α = 1 Why to study the detail of a particular case, if the general conditions were justgiven? The reason is that the general conditions are valid for functions withmonotonic behavior, i.e., functions that increase or decrease along all the timeinterval. However, this analysis is beyond that case. For example, · a ( t ) is anincreasing function of time in a range of time, but in another range it is decreas-ing function, in such a way that in the present time the universe is accelerated,consistent with observations, but in a far later time, slowdown occurs. Then,this example is not covered by the criterion given above. Moreover, this casesatisfies automatically one of the required conditions: the EoS remains boundedduring the whole evolution of the universe.Then we start with Eq. (19), which is solved exactly by means of the method-ology of ref. [16], we obtain the following solution: H ( t ) = − ζ λ [1 + coth ( γt )] , (30)with γ ≡ (3 ζ/ / (1 + 3 ζτ Π / . n , d n H ( t ) /dt n = ∞ ∀ t = 0 then, the studied case, does not present the singularity,identified as Type IV in the literature [31], [32].Taking into account that H = d ln a/dt , we can integrate in time to obtain a = a ( t ), which results: a ( t ) = a P (cid:20) − e γt − e γt P (cid:21) − ζ/ λγ . (31)From Eq. (31) we can see that, for a ( t ) be an increasing function of t , as ζ > λ > . a) γ < τ Π < γ .Then, we can write a ( t ) = a P (cid:20) − e − | γ | t − e − | γ | t P (cid:21) ζ/ λ | γ | , (32)whereupon lim t → ∞ a ( t ) = a P h − e − | γ | t P i − ζ/ λ | γ | , (33)As we can see from Eq. (33), a ( ∞ ) = ∞ , but it can be as large as we want. Infact, if for example | γ | ∼
1, as t P ≪
1, developing the exponential up to linearterm, we obtain a ( ∞ ) ≃ (2 t P ) − ζ/ λ : also the exponent can be in absolute valueas large as we want, if λ ≪ . The last requirement is consistent with the factthat w ∼ − ρ ( t ) = 34 (cid:18) ζλ (cid:19) [1 − coth ( | γ | t )] , (34)Clearly lim t → ∞ ρ ( t ) = 0 (is included in the family of models labeled by I). Thisresult creates some conceptual conflict in relation with the above result: if thescale factor reaches a finite value in infinite time when the density is null, whathappened with dark energy? Did it disappear? Nevertheless, as we saw above,this would not have a “noticeable” effect because a ( ∞ ) would be a power of theinverse of Planck time as large as we would like.Now we can analyze if this model gives us an accelerated universe. If thetwo derivatives of Eq. (32) are performed, we obtain: ·· a ( t ) = 2 ζ | γ | λ Q ( t P ) (cid:16) − e − | γ | t (cid:17) ζ λ | γ | − e − | γ | t (cid:18) ζ λ | γ | − e | γ | t (cid:19) , (35)9ith Q ( t P ) ≡ a P (cid:0) − e − | γ | t P (cid:1) − ζ λ | γ | . So, for ·· a ( t ) >
0, the following inequalitymust hold: ζ λ | γ | > e | γ | t . (36)The two quantities of the above inequality would be equal for a “change time” t c given by t c = ln (cid:18) ζ | τ Π | − / λ (cid:19) ζ | τ Π | − / ζ . (37)Then, the value of λ can be set so the universe will slow at t c > t a , where t a isthe current observation time. Again, we obtain λ <<
1, whereby the value of w should be very close to −
1, in agreement with the observation. b) γ > λ < a) γ > a ( t ) = a P (cid:20) e γt − e γt P − (cid:21) ζ/ | λ | γ . (38)Then lim t → ∞ a ( t ) = ∞ . The energy density is in this case is ρ ( t ) = 34 (cid:18) ζλ (cid:19) [1 + coth ( γt )] . (39)As we see from Eq.(39), a function that converges to a finite value is obtained,i.e.: lim t → ∞ ρ ( t ) = 3 ( ζ/λ ) . It is worth to notice that, for a very small valueof λ , i.e., | λ | < √ ζ/ρ / P , the density could grow beyond the density at thePlanck time; in that case, the model would be included in the family that wecalled II. We can say that, density of dark energy will grow up to a finite valueand therefore does not become a ”Little Rip” type singularity [30],[33] for anytime, provided λ = 0.Now, from Eq. (38) the acceleration can be computed, and the followingexpression is obtained: ·· a ( t ) = f ( t ) (cid:18) ζ | λ | γ − e − γt (cid:19) , (40)where f ( t ) > ∀ t . Therefore, the condition for accelerated expansion isobtained from the condition that the quantity into the parenthesis in Eq. (40)be positive. As a consequence, the following condition on time results:10 > ζ (cid:18)
23 + ζτ Π (cid:19) ln | λ | + ζτ Π . (41)This means that the condition | λ | < + ζτ Π , is sufficient for an acceleratedregimen during all the time interval. b) γ < λ and γ ( τ Π < In this section, the following question will be addressed: which is the contribu-tion to the entropy due to the causal corrective term in the viscosity?From Eq. (12) we can see that the quantity p † is equivalent to an “effectivepressure”. On the other hand, we can write the Gibbs equation in the form dE = − (cid:18) p − T dSdV (cid:19) dV. (42)The second term in the parenthesis can be interpreted as the heat per unit ofvolume absorbed by the system as the result of viscosity. It can be conceived asa negative pressure, analogous to the correction to the pressure introduced inEq. (11), which we identified with − Π, associated to the bulk viscosity. Then,it is natural to propose the identification
T dSdV ≡ Π , (43)with Π given by Eq. (17). Then, there are two viscosity contributions to thechange of entropy, indicated as dS = dS + dS , (44)with dS = 3 T ζHdV, (45) dS = − T ζτ Π · HdV, (46)Eq. (45) gives the bulk viscosity contribution, and Eq. (46) supplies the secondbulk viscosity contribution, related with the causal correction.The first term of Eq. (44) is always positive, since we assume that, as inmost physical systems,
T >
0. According to the theory of fluids [13], it shouldhold that ζ >
0, since an expansion stage with dV > · a > HdV remains positive. Therefore,the first term of Eq. (44) does not break time invariance (see also ref. [34] forthe case where particle creation is considered).The analysis of the influence due to the second term of Eq. (44) is easierwhen the equation is rewritten in the following convenient form: dS = dS " τ Π · aa − ·· a · a ! . (47)A first remark is that the term into the parentheses that multiplies τ Π becomeszero for an evolution of the form a ( t ) ∝ exp ( αt ). As we did above, we willseparate the analysis of the contribution to dS , according τ Π is greater or smallerthan zero.We see that, when τ Π >
0, the suficient condition for the term of entropyassociated with the coefficient τ Π gives a positive contribution is · aa > ·· a · a . (48)It has to be noticed that a solution of the form a ( t ) ∝ t β satisfies the abovecondition for any β . Another possibility would be ·· a/ · a <
0, but this does notagree with observations since it would lead to a deceleration of the universe.When τ Π <
0, the sufficient condition is · aa < ·· a · a . (49)Since · a and a are positive, this condition also implies that ·· a > α = 1 To analyze the details of an exact calculation example, we can see again the casein which the state equation is constant (i.e. with α = 1). in order to simplifynotation, it is convenient to define δ ≡ · aa − ·· a · a . (50)Two subclasses are considered: λ < and τ Π > . By performing the derivatives of Eq. (38), we can calculate δ : δ = 2 γ/ (cid:0) e γt − (cid:1) , (51)with γ defined as in Eq. (30). Then, δ > ∀ t and, therefore, dS > ∀ t . Wecan see also that lim t → ∞ dS = dS . This means that, when high values of t ,12nd hence of a ( t ), are reached, the effect due to the causal correction becomesnegligible. λ > and τ Π < . Now we derive Eq. (32) to calculate δ . In this case we obtain δ = 2 | γ | / (cid:16) − e − | γ | t (cid:17) . (52)Therefore δ > ∀ t . However now dS is dS = dS (1 − | τ Π | δ ) . (53)If we also consider that ζ | τ Π | > / t > t → ∞ dS = − dS / (cid:0) ζ | τ Π | − (cid:1) . These are then good arguments againstthis case. A cosmological model as an isotropic and homogeneous universe, with a sourceof matter that simulates dark energy, was proposed. The source consists of arelativistic viscous fluid with minimal causal correction. Due to the symmetryof the model, the only viscous contribution is the bulk viscosity, which providesthe negative pressure necessary to maintain an accelerated expansion, consistentwith the observations.The constrains in the model parameters due to the following conditions, werestudied: i) energy density tending to a finite value along the time, ii) scale factor a ( t ) increasing function of time, iii) accelerated present universe, and iv) stateequation for dark energy p/ρ = w , with w close to − ρ is a decreasing function, for alltimes, when the second viscosity coefficient τ Π is negative. It is worth recallingthat the purpose of the term of TEM associated with this coefficient is to correctthe defects of the non causal theory of relativistic fluids [13], in which fluiddisturbances are propagated at superluminal speeds. This term (as it can easilybe verified from calculations ref. [22]) leads to a bound for the propagationvelocity. This behavior is not affected by the change of sign in τ Π .In particular, a detailed study of the case where the state equation is constant( w = − − λ ) was performed. It was found that, in order to meet the imposedconditions, in particular a ( t ) as an increasing function of time, assuming one ofthe two following restrictions was necessary: a) τ Π < λ > τ Π > λ < ζ | τ Π | > , which leadsto a ( t → ∞ ) ≃ (2 t P ) − ζ/ λ . So, the value of the scale factor is finite but aslarge as we want, provided that we make λ small enough. It is interesting to13ote that the latter requirement makes w to be very close to −
1. Moreover, ρ ( t → ∞ ) = 0 . Under these conditions the universe is accelerated until acertain time t c , which can be as large as we want, for a λ close enough to zero.After that time, the universe begins to slow the velocity of expansion.In the second case, a ( t → ∞ ) = ∞ was obtained, but with ρ ( t → ∞ ) =3( ζ/λ ) , which can give us a density that increases with time if | λ | < √ ζ/ρ / P .When it is taken into account that ρ P is very large ( ∼ g/m ), this boundis extremely small. Whenever we consider small λ , but not to the above value,the energy density is a decreasing function of time. Moreover, the universe isaccelerated for all times provided it is | λ | < / ζτ Π , which is not a strongconstraint because it leaves open an interval of physically reasonable values. Inaddition, w will be limited, since α = 1, and λ could be set to a value smallenough to make the deviation from − ρ and a EMT representing a viscous fluid with causal correction.However, this can also be seen as an ideal fluid source but with an effective EoSwith w † including the viscous correction ( w † = p † /ρ , with p † given by Eq. (11)),in this way it can be considered as a case particular of the inhomogeneous EoSproposed in ref. [35]. But this choice has not been arbitrary but comes from theviscous fluid model with causal correction first developed by Israel-Stewart [21]and later generalized, so as to consider non-linear terms in the velocity gradients[22], [24].This work opens different lines for future research. On the one hand, to con-sider the causal relativistic theory of fluid in a more complete version, includingnonlinear terms in the velocity gradients. This will require more computationaleffort, but it would be interesting to discover the new conditions on the morecomplete set of coefficients and to evaluate their influence on dynamics of theuniverse. On the other hand, it can be studied whether, at least at the levelof approximation used in this work, it is possible to establish a correspondencewith some formulation from an analysis at a more fundamental level, i.e. witha field theory formulation. It would also be interesting to consider the stateequations used in recent papers [36] to describe halos of dark matter, which areinferred from some experiments [37]. Acknowledgement 1
I would like to thank Olimpia Lombardi for her criticallyreading and her comments on the manuscript.This research was supported by theUniversity of Buenos Aires grant: UBACyT-01/Q710.
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