Friedmann Cosmology with a Generalized Equation of State and Bulk Viscosity

Commun. Theor. Phys. Beijing, China 47 2007 pp. 379 384 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Friedmann Cosmology with a Generalized Equation of State and Bulk Viscosity MENG Xin-He, REN Jie, and HU Ming-Guang Department of Physics, Nankai University, Tianjin 300071, China Received March 29, 2006 Abstract The universe content is considered as a non-perfect fluid with bulk viscosity and is described by a more general equation of state endowed some deviation from the conventionally assumed cosmic perfect fluid model. We assume the bulk viscosity is a linear combination of two terms: one is constant, and the other is proportional to the scalar expansion θ = 3/a. The equation of state is described as p = γ 1ρ + p 0, where p 0 is a parameter. In this framework we demonstrate that this model can be used to explain the dark energy dominated universe, and different proper choices of the parameters may lead to three kinds of fates of the cosmological evolution: no future singularity, big rip, or Type-III singularity as presented in [S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71 2005 063004]. PACS numbers: 98.80.Cq, 98.80.-k Key words: Friedmann cosmology, equation of state, bulk viscosity, dark energy 1 Introduction The cosmological observations indicate that the expansion of our universe accelerates. [1] Recently lots of research work on its possible mechanism, such as extended gravity, [2] modifying equation of state hereafter EOS or by introducing kinds of the so-called dark energy models, are to explain the cosmic acceleration expansion observed. To overcome the drawback of hydrodynamical instability, a linear EOS of a more general form, p = αρ p 0, was proposed, [3] and this form incorporated into cosmological model can describe the hydrodynamically stable dark energy behaviors. The astrophysical observations also indicate that the universe media is not a perfect fluid [4] and the viscosity is concerned in the evolution of the universe. [5 7] On the other hand, in the standard cosmological model, if the EOS parameter ω is less than 1, the universe shows the future finite singularity called Big Rip. [8,9] Several ideas were proposed to prevent the big rip singularity as thought it un-physical, like by introducing quantum effects terms in the action. [10] So an interesting question naturally arises: what kind of role the cosmic viscosity element can play for helping the above two facets, dark energy and cosmic dark energy model singularity? Considering some deviation from the ideal fluid model is also very helpful to nowadays cosmology studies. In Refs. [11] [17], the bulk viscosity in cosmology has been studied in various aspects. Dissipative processes are thought to be present in any realistic theory of the evolution of the universe. In the early universe, the thermodynamics is far from equilibrium, the viscosity should be concerned in the studies of the cosmological evolution. But even in the later cosmic evolution stage, for example, the temperature for the intergalactic medium IGM, baryonic gas, generally is about 10 4 K to 10 6 K and the complicated IGM is rather non-trivial. The sound speed c s in the baryonic gas is only a few km s 1 to a few tens km s 1 and the Jeans length λ yields a term as an effective viscosity c s λ. On the other hand, the bulk velocity of the baryonic gas is of the order of hundreds km s 1. [18] So it is helpful to consider the viscosity element in the later cosmic evolution. It is well known that in the framework of Friedmann Robertson Walker FRW metric, the shear viscosity has no contribution in the energy momentum tensor, and the bulk viscosity behaves like an effective pressure. At the later time, since we do not know the nature of the universe content very clearly, concerning the bulk viscosity is reasonable and practical. Moreover, the cosmic viscosity here can also be regarded as an effective quantity as caused by other not clearly known mechanisms and may play a role as a dark energy candidate. [19 21] The EOS p = γ 1ρ and the bulk viscosity ζ = αρ s is studied in the full causal theory of bulk viscosity, and the case s = 1/2 has possessed exact solutions as shown in Refs. [13] and [14]. However, both the pressure and the bulk viscosity coefficient may have constant components. We argue that the non-causal approximation is reasonable in the late times of the universe. In this paper, we show that the Friedmann equations can be solved with both a more general EOS and bulk viscosity detailed as follows. The EOS is p = γ 1ρ + p 0, 1 where p 0 and γ are two parameters. The bulk viscosity is expressed as ζ = ζ 0 + ζ 1 a, 2 The project partly supported by National Natural Science Foundation of China under Grant No. 10675062 and the Doctoral Foundation of China E-mail: xhm@nankai.edu.cn E-Mail: jrenphysics@hotmail.com E-mail: huphys@hotmail.com

380 MENG Xin-He, REN Jie, and HU Ming-Guang Vol. 47 where ζ 0 and ζ 1 are two constants conventionally. The motivation of considering this bulk viscosity is that by fluid mechanics we know the transport/viscosity phenomenon is related to the velocity, which is related to the scalar expansion θ = 3/a. Both ζ = ζ 0 constant and ζ θ are considered in the previous papers, [5,22] so a linear combination of the two is more physical. The ω = p/ρ is constrained as 1.38 < ω < 0.82 [23] by present observation data, so the inequality in our case should be 1.38 < γ 1 + p 0 ρ < 0.82. 3 The parameter p 0 can be positive attractive force or negative repulsive force, and conventionally ζ 0 and ζ 1 are regarded as positive. To choose the parameters properly, it can prevent the Big Rip problem or some kind of singularity for the cosmology model, like in the phantom energy phase, as shown below. Additionally, the sound speed in this model can also keep the casuality. This paper is organized as follows. In Sec. 2 we describe our model and give out the exact solution. In Sec. 3 we consider some special cases of the solution. In Sec. 4 we discuss the acceleration phase and the future singularities in this model. In Sec. 5 we study the sound speed constraints, and in the last section we summarize our conclusions. 2 Model and Calculations We consider the Friedamnn Roberson Walker metric in the flat space geometry k = 0 as the case favored by WMAP satellite mission on cosmic background radiation CMB data ds 2 = dt 2 + at 2 dr 2 + r 2 dω 2, 4 and assume that the cosmic fluid possesses a bulk viscosity ζ. The energy-momentum tensor can be written as T µν = ρu µ U ν + p + ΠH µν, 5 where in comoving coordinates U µ = 1, 0, and H µν = g µν + U µ U ν. [22] By defining the effective pressure as p = p + Π and from the Einstein equation R µν 1/2g µν R = 8πGT µν, we obtain the Friedmann equations, 2 a 2 = 8πG 3 ρ, 6a a = 4πG ρ + 3 p. 3 6b The conservation equation for energy T;ν 0ν yields ρ + ρ + pθ = 0, 7 where θ = U ;µ µ = 3/a. According to the framework of the causal theory of bulk viscosity, [11] the entropy production density is S ;µ µ = Π [ θ + τ T ζ Π + 1 τ 2 T Π ζt U µ ]. 8 ;µ The simplest way to guarantee S µ ;µ 0 implies the evolution equation, [11] Π + τ Π = ζθ 1 [ 2 τπ θ + τ τ ζ ζ T T ], 9 where T is the temperature and τ is the relaxation time. In Eq. 9, τ = 0 gives the non-causal Eckart theory. Taking the relation τ = ζ ρ 10 is a simple procedure to ensure that the speed of viscous pulses does not exceed the speed of light. In the early times of the universe, especially the inflation epoch, the full causal theory of bulk viscosity should be considered in the study of the cosmic evolution. However, in the late times, we argue that the approximation of τ 0 is reasonable, because the density ρ approaches to a constant related to Λ while the bulk viscosity coefficient ζ 0 as the asymptotic behavior. Therefore, we study the noncausal approximation of the bulk viscosity in the following context. Using the EOS to eliminate ρ and p, we obtain the equation which determines the scale factor at 2 a = 3 γ 2 2 a 2 + 12πGζ 0 a 4πGp 0, 11 where the effective EOS parameter is shifted from the original one as γ = γ 8πGζ 1. 12 So we can see that the equivalent effect of the second term in ζ is to change the parameter γ to γ in the EOS. As shown in Ref. [5], the barrier ω = 1 between the quintessence region ω > 1 and the phantom region ω < 1 can be crossed, as a consequence of the bulk viscosity available. Since the dimension of the two terms 12πGζ 0 and 4πGp 0 is [time] 1 and [time] 2, respectively, we define then equation 11 becomes 12πGζ 0 = 1, 13 4πGp 0 = 1 2, 14 a = 3 γ 2 2 a 2 + 1 a + 1. 15 Here and T 2 are criteria to determine whether we should concern ζ and p 0. If t, the cosmic time scale, the effect of ζ can be neglected, and if T 2 t, the effect of p 0 can be neglected likewise. Concerning the initial conditions of at 0 = a 0 and θt 0 = θ 0, if γ 0, the solution can be obtained as { 1 at = a 0 1 + γθ 0 T T [ t t0 1 exp 2 2 T + 1 ] + 1 1 γθ 0 T + T [ exp t t 0 1 2 2 T 1 ] } 2/3 γ. 16

No. 2 Friedmann Cosmology with a Generalized Equation of State and Bulk Viscosity 381 And we obtain directly a = 1 1 + γθ 0 T T 1 T + 1 exp t t0 T 1 γθ 0T + T 1 T 1 3 γ 1 + γθ 0 T T exp t t0 T + 1 γθ 0T + T. 17 Here we define T = 1 + 6 γt1 /T 2 2. 18 We can see that when T 2, T = ; when, T = T 2 / 6 γ. 3 γ = 0 and Special Cases For γ = 0, we should use the mathematical L Hospital s rule to calculate the limit of Eq. 16 rigourously and note dt 3 lim γ 0 d γ = 3 T2 2. 19 The limit of solution at when γ 0 is [ 1 at = a 0 exp 3 θ 0 + 2 e t t0/t1 1 Directly solving Eq. 15 t t 0 ] T2 2. 20 a = 2 a 2 + 1 a + 1 T 2 21 gives the same result. So equation 16 is consistent for γ crossing zero. The T 2 limit of Eq. 20 is [ 1 ] at = a 0 exp 3 θ 0 e t t0/t1 1. 22 and the limit of Eq. 20 is [ 1 at = a 0 exp 3 θ 0t t 0 + t t 0 2 ]. 23 2 These two special limits are also consistent with directly solving Eq. 15, by checking. For and T 2, the limit case is at = a 0 e θ0t t0/3, 24 which corresponds to the de Sitter universe with constant scalar expansion. Let us discuss two special cases in the following. When the constant term in the EOS is not concerned, i.e. T 2, at = a 0 [1 + 1 2/3 γ 2 γθ 0T e t t0/t 1], 25 and when the constant term in the bulk viscosity is not concerned, i.e., at = a 0 cosh t t 0 2T + γθ 0T sinh t t 2/3 γ 0. 26 2T When T, the above two cases become at = a 0 [1 + 1 2 γθ 0t t 0 ] 2/3 γ, 27 which corresponds to the case of p = γ 1ρ and ζ θ. 4 Acceleration and Big Rip If the universe accelerates, then mathematically a > 0. 28 From Eq. 11, we can qualitatively see that the bulk viscosity and a negative pressure p 0 can cause the universe to accelerate. Since the expression of /a is too complicated in this situation, now we only discuss a special case, with p 0 = 0. Here /a > 0 yields ω = γ 1 < 2 3 et t0/t 1 + 2 θ 0 T + 8πGζ 1. 29 As we know, if the bulk viscosity is zero as in the standard Friedammn Robertson-Walker cosmology model, an accelerating expansion universe corresponds to ω < 1/3. Inequality Eq. 29 tells us that if the bulk viscosity is large enough, the universe expansion can accelerate even if ω > 1/3. According to Ref. [9], the future singularities can be classified in the following way: Type I Big Rip : For t t s, a, ρ and p ; Type II sudden : For t t s, a a s, ρ ρ s and p ; Type III: For t t s, a a s, ρ and p ; Type IV: For t t s, a a s, ρ 0, p 0 and higher derivatives of H diverge. In this paper, ρ means p we assume γ 1 generally. In the following we show that different choices of the parameters may lead to three fates of the universe evolution: no future singularity, big rip, or the Type-III singularity. i γ < 0 From Eq. 6a, we see ρ a. 30 If the denominator of Eq. 17 is zero at a finite time t, 1 + γθ 0 T T t t0 exp + 1 γθ 0 T + T = 0, 31 T then a, ρ, so the big rip occurs. The solution for t is t s = T ln 1 γθ 0T + T/T 1 + t 0. 32 1 + γθ 0 T T/

382 MENG Xin-He, REN Jie, and HU Ming-Guang Vol. 47 Note that T is a complex number, in the following context we assume T is real. If we want to prevent the big rip, there should be no real solution for t > t 0, so cosmological constant. Actually, the bulk viscosity model fits the SNe Ia data at an acceptable level. [20] The inequality is equivalent to 1 γθ 0T + T/ 1 + γθ 0 T T/ < 1. 33 1 + γθ 0 T T > 0. 34 The above inequality can be satisfied in some conditions for the phantom energy, so the big rip will not occur. Furthermore, we can see that even if the dark energy is in the quintessence region, there also can be future singularity. For example, if γ > 0 and p 0 < 0, it is possible that inequality 34 is not satisfied, so the future singularity may occur, which will be discussed in the next subsection below. The more explicit form of inequality 34 is 1 + γθ 0 T 1 1 + 6 γt1 /T 2 1 2 1 + 6 γt1 /T 2 > 0. 35 2 From this inequality, we obtain the following conclusions: a p 0 < 0, i.e. T2 2 > 0: inequality 34 is always unsatisfied, so there will be a big rip at time t s. b p 0 > 0, i.e. < 0: inequality 34 is not always unsatisfied. If it is unsatisfied, there will be a big rip at time t s. If it is satisfied, there is no future singularity. ii γ > 0 If the denominator of inequality 34 is zero, then a 0, ρ, so the Type III singularity occurs. Following the same steps as before, we obtain the following points. i p 0 < 0, i.e. T2 2 > 0: inequality 34 is always satisfied, so there is no future singularity. ii p 0 > 0, i.e. T2 2 < 0: inequality 34 is not always satisfied. If it is satisfied, there is no future singularity; while if it is unsatisfied, there will be Type-III singularity at time t s. iii γ = 0 For the case γ = 0, a = 1 3 θ 0 e t t0/t1 + e t t0/t1 1. 36 So there is no future singularity in this case. To illustrate the parameters in the general solution Eq. 16 more clearly, we draw some graphics in Figs. 1 3. The initial condition is [22] t 0 = 1000 s, θ 0 = 1.5 10 3 s 1. At this time, the bulk viscosity ζ = 7.0 10 3 g/cm s, and the corresponding = 5.1 10 28 s. We assume ζ 1 = 0 for simplicity. In Eq. 15, the term 1/ plays the role of the effective cosmological constant, so figure 2 corresponds to the ΛCDM model. Comparing Fig. 1 with Fig. 2, we can see that the constant bulk viscosity has possessed the similarly dynamical effects of the Fig. 1 The normalized scale factor evolution with cosmic time, p 0 = 0 T 2, γ = 1, the dashed line corresponds to = 5.1 10 17 s, and the solid line corresponds to = 5.1 10 28 s. Fig. 2 The normalized scale factor evolution with cosmic time, ζ 0 = 0, γ = 1, the dashed line corresponds to T 2 = 5.1 10 17 s, and the solid line corresponds to T 2 = 5.1 10 28 s. Additionally, if < 0, the scale factor may have the oscillatory behavior, i.e. the scale factor may become negative and positive alternatively with the cosmic time evolution, as the solution of Eq. 15. For example, equation 26 can be rewritten as at = a 0 cos t t 0 2 T + γθ 0 T sin t t 2/3 γ 0, 37 2 T when T 2 < 0. This solution may become imaginary, and the evolution of the real part in two cases is shown in Fig. 4. We will investigate whether the case is interesting to our physical universe in other studies, that is, if the real

No. 2 Friedmann Cosmology with a Generalized Equation of State and Bulk Viscosity 383 part can be employed to mimic the current cosmic accelerating expansion as observed and earlier matter dominated evolution phase, for example. Fig. 3 The normalized scale factor evolution with cosmic time, ζ = 0, p 0 = 0, the dashed line corresponds to γ = 0.18, and the solid line corresponds to γ = 0.181 case. sensitivity to c s, however, there is a slight difference between models with no dark energy perturbations and models with dark energy behaving as a fluid. In general, the sound speed is defined as c 2 s = p ρ. 38 In an acceptable model, the sound speed should be bounded by a constant, like the upper bound of light speed or approach to a constant in the late times of the universe. From the EOS in the above section p = γ 1ρ + p 0, we obtain c 2 s = γ 1, 39 which is a constant. This result is the same as that of the ΛCDM model, so we seek some generalizations of the EOS. In Ref. [20], we show that this time-dependent bulk viscosity ζ = ζ 0 + ζ 1 a + ζ 2 40 is effectively equivalent to the form derived by using the EOS p = γ 1ρ + p 0 + w H H + w H2 H 2 + w dh Ḣ. 41 So here, we consider the EOS of the form in Eq. 41. In this case, the square of the sound speed is c 2 1 1 s = γ 1 ρ. 42 3κT1 When γ = 0, should be negative if the sound speed is a real number. Moreover, < 0 has possessed another important benefit that the density approaches to a constant when t, i.e. c 2 s approaches to a constant. For example, if γ = 0, when t, Fig. 4 The real part of normalized scale factor evolution with cosmic time, ζ = 0, γ = 1, p 0 > 0T2 2 < 0, the dashed line corresponds to T 2 = 1.0 10 17 s, and the solid line corresponds to T 2 = 1.5 10 17 s case. 5 Sound Speed Constraints To ensure the causality, the sound speed must not exceed the speed of light. The astrophysics observations related to the sound speed in the universe have provided some constraints to the related model parameters, and several research papers have studied this topic, [25 27] for which the CMB results imply a tentative 1σ detection of the sound speed, c 2 s < 0.04. [26] In Ref. [27], the authors find that the current observations have got no significant ρ = 3 κ 2 T2 2, 43 which corresponds to the de Sitter universe, just as the late-time evolution of the ΛCDM model. Detailed study of the case < 0 for the more complicated equation of state with observational data fitting will be published elsewhere. 6 Conclusions and Discussions In conclusion, we have solved the Friedmann equations with both a more general EOS and bulk viscosity, and discussed the acceleration expansion of the universe evolution and the future singularities for this model. Compared with the standard model of cosmology, this model has had three additional parameters, ζ 0, ζ 1 and p 0 : proper choices of ζ 0 and negative p 0 can cause the universe accelerate; ζ 1 can drive the cosmic fluid from the quintessence region to the phantom one, [5] and positive p 0 may both prevent the big rip for phantom phase and lead to the Type-III singularity of Ref. [9] for the quintessence phase. The relation between the choices of parameters and the future singularities of the cosmological evolution in this extended model is summarized in the following Table and we expect

384 MENG Xin-He, REN Jie, and HU Ming-Guang Vol. 47 more detailed investigations on viscosity effects to be carried out, especially the connection with a more general EOS for describing dark fluid [20,24] and its possible role as a dark energy candidate to mimic the cosmologically later stage evolutions. Table 1 of parameters. The relation of the future singularities, choices Parameters Future singularity at t t s γ < 0 p 0 < 0 a, ρ p 0 > 0 1 + γθ 0 T T/ > 0 No 1 + γθ 0 T T/ < 0 a, ρ γ = 0 γ > 0 p 0 < 0 No p 0 > 0 1 + γθ 0 T T/ > 0 No 1 + γθ 0 T T/ < 0 a 0, ρ No Acknowledgments We thank Profs. I. Brevik, S.D. Odintsov, and Lewis H. Ryder for lots of interesting discussions. References [1] T. Totani, Y. Yoshii, and K. Sato, Astrophys. J. 483 1997 L75; S. Perlmutter et al., Nature London 391 1998 51; A.G. Riess et al., Astron. J. 116 1998 1009; N. Bahcall, J.P. Ostriker, S. Perlmutter, and P.J. Steinhardt, Science 284 1999 1481. [2] For example, X.H. Meng and P. Wang, Class. Quant. Grav. 20 2003 4949; 21 2004 951; 21 2004 2029; 22 2005 23; Gen. Rel. Gra. 36 2004 1947; Phys. Lett. B 584 2004 1; E. Flanagan, Class. Quant. Grav. 21 2003 417; S. Nojiri and S. Odintsov, Phys. Lett. B 576 2003 5; Phys. Rev. D 68 2003 123512; D. Vollick, D 68 2003 063510. [3] E. Babichev, V. Dokuchaev, and Y. Eroshenko, Class. Quantum Grav. 22 2005 143. [4] T.R. Jaffe, A.J. Banday, H.K. Eriksen, K.M. Górski, and F.K. Hansen, astro-ph/0503213. [5] I. Brevik and O. Gorbunova, gr-qc/0504001. [6] I. Brevik, O. Gorbunova, and Y.A. Shaido, grqc/0508038. [7] M. Cataldo, N. Cruz, and S. Lepe, Phys. Lett. B 619 2005 5. [8] R.R. Caldwell, M. Kamionkowski, and N.N. Weinberg, Phys. Rev. Lett. 91 2003 071301. [9] S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71 2005 063004. [10] S. Nojiri, and S.D. Odintsov, Phys. Lett. B 595 2004 1; E. Elizalde, S. Nojiri, and S.D. Odintsov, Phys. Rev. D 70 2004 0343539. [11] W. Zimdahl, Phys. Rev. D 53 1996 5483. [12] A.A. Coley, R.J. van den Hoogen, and R. Maartens, Phys. Rev. D 54 1996 1393. [13] L.P. Chimento, A.S. Jakubi, V. Méndez, and R. Maartens, Class. Quantum Grav. 14 1997 3363. [14] M.D. Mak and T. Harko, J. Math. Phys. 39 1998 5458. [15] W. Zimdahl, Phys. Rev. D 61 2000 083511. [16] A.D. Prisco, L. Herrera, and J. Ibáñez, Phys. Rev. D 63 2000 023501; T. Padmannabhan and S. Chitre, Phys. Lett. A 120 1987 433. [17] R. Maartens, astro-ph/9609119. [18] P. Peebles, Principles of Physical Cosmology, Princeton University Press, Princeton 1993; E. Kolb and M. Turner, The Early Universe, Addison-Wesley, New York 1990; J. Peacock, Cosmological Physics, Cambridge University Press, Cambridge 1999; A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, Cambridge 2000; S. Weinberg, Gravitation and Cosmology, John Wiley and Sons 1972; G. Borner, The Early Universe, Facts and Fiction, Springer-Verlag, New York 1988. [19] W. Zimdahl, D.J. Schwarz, A.B. Balakin, and D. Pavon, Phys. Rev. D 64 2001 063501, astro-ph/0009353. [20] J. Ren and X.H. Meng, Phys. Lett. B 633 2006 1; astroph/0511163; astro-ph/0602462. [21] M.G. Hu and X.H. Meng, Phys. Lett. B 635 2006 186; astro-ph/0511615. [22] I. Brevik, Phys. Rev. D 65 2002 127302. [23] A. Melchiorri, L. Mersini, C.J. Odman, and M. Trodden, Phys. Rev. D 68 2003 043509. [24] X.H. Meng, M.G. Hu, and J. Ren, astro-ph/0510357; astro-ph/0605694. [25] S. DeDeo, R.R. Caldwell, and P.J. Steinhardt, Phys. Rev. D 67 2003 103509, astro-ph/0301284; Erratum-ibid, Phys. Rev. D 69 2004 129902. [26] R. Bean and O. Dore, Phys. Rev. D 69 2004 083503, astro-ph/0307100. [27] S. Hannestad, Phys. Rev. D 71 2005 103519, astroph/0504017.