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Physics Letters B

www.elsevier.com/locate/physletb

Viscous Ricci dark energy

Chao-Jun Fenga b *, Xin-Zhou Li

a Shanghai United Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China b Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

ARTICLE INFO

ABSTRACT

Article history:

Received 21 July 2009

Received in revised form 31 August 2009

Accepted 9 September 2009

Available online 11 September 2009

Editor: T. Yanagida

We investigate the viscous Ricci dark energy (RDE) model by assuming that there is bulk viscosity in the linear barotropic fluid and the RDE. In the RDE model without bulk viscosity, the universe is younger than some old objects at certain redshifts. Since the age of the universe should be longer than any objects living in the universe, the RDE model suffers the age problem, especially when we consider the object APM 08279 + 5255 at z = 3.91 with age t = 2.1 Gyr. In this Letter, we find that once the viscosity is taken into account, this age problem is alleviated.

© 2009 Elsevier B.V. All rights reserved.

Recent observations like CMB anisotropy, supernovae and galaxies clustering have strongly indicated that our universe is spatially flat and there exists an exotic cosmic fluid called dark energy with negative pressure, which constitutes about two thirds of the total energy of the universe. The dark energy is characterized by its equation of state w, which lies very close to -1, probably being below —1 indicated by the current data.

Many candidates including the cosmological constant, quintessence, phantom, quintom, holographic dark energy, etc., have been proposed to explain the acceleration. However, so far people still do not understand what's dark energy. Ricci dark energy, which can be regarded as a kind of holographic dark energy [1] with the square root of the inverse Ricci scalar as its infrared cutoff, has been proposed by Gao et al. [2]. This model is also phenomeno-logically viable. As we known, the energy density of holographic dark energy is inversely proportional to the square of an infrared cutoff L as p ~ L—2. Therefore, the energy density of RDE in a flat universe reads

pR = - R = 3a( H + 2 H2)

where we have set 8^ G = 1 and a is a dimensionless parameter which will determine the evolution behavior of RDE. Assuming the black hole is formed by gravitation collapsing of the perturbation in the universe, the maximal black hole can be formed is determined by the casual connection scale RCC given by the "Jeans" scale of the perturbations. For tensor perturbations, i.e. gravitational perturbations, R—C = Max(H + 2H2, —H) for a flat universe, where H = a/a is the Hubble parameter, and according to Ref. [3],

* Corresponding author at: Shanghai United Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China.

E-mail addresses: fengcj@shnu.edu.cn (C.-J. Feng), kychz@shnu.edu.cn (X.-Z. Li).

only in the case of R— C = H + 2H2, it could be consistent with the current cosmological observations when the vacuum density appears as an independently conserved energy component. As we known, in a flat FRW universe, the Ricci scalar is R = 6(H + 2H2), and it means the RCC a R. Therefore, if one chooses the casual connection scale as the IR cutoff, i.e. L = R—C, the Ricci dark energy model is obtained. For recent progress on Ricci dark energy and holographic dark energy, see Refs. [4-6].

Dissipative processes in the universe including bulk viscosity, shear viscosity and heat transport have been conscientiously studied [7]. The general theory of dissipation in relativistic imperfect fluid was put on a firm foundation by Eckart [8], and, in a somewhat different formulation, by Landau and Lifshitz [9]. This is only the first order deviation from equilibrium and may has a causality problem, the full causal theory was developed by Israel and Stewart [10,11], and has also been studied in the evolution of the early universe [12]. However, the character of the evolution equation is very complicated in the full causal theory. Fortunately, once the phenomena are quasi-stationary, namely slowly varying on space and time scale characterized by the mean free path and the mean collision time of the fluid particles, the conventional theory is still valid. In the case of isotropic and homogeneous universe, the dis-sipative process can be modeled as a bulk viscosity Z within a thermodynamical approach, while the shear viscosity n can be neglected, which is consistent with the usual practice [13]. For works on viscous dark energy models, see Ref. [14].

The bulk viscosity introduces dissipation by only redefining the effective pressure, peff, according to peff = p — 3f H where Z is the bulk viscosity coefficient and H is the Hubble parameter. The condition Z > 0 guaranties a positive entropy production, consequently, no violation of the second law of the thermodynamics [15]. In this Letter, we are interested when the universe is dominated by a usual fluid and the RDE, and both of them have a bulk

0370-2693/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.09.013

viscosity. The case Z = implying the bulk viscosity is pro-

portional to the fluid's velocity vector, is physically natural, and has been considered earlier in a astrophysical context, see the review article of Gr0n [16].

First, let us consider a simple case when the universe is dominated by the RDE only, then the Friedmann equation reads:

H 2 = PR = «

( H 2)'

where prime denotes the derivative with respect to x = ln a and hereafter we set a0 = 1. The solution of the above equation is H2 = H^e—2(2—a)x and the energy density of the RDE is pR = 3H^e—2(2—a)x. By using the conservation equation

PR + 3(pr + pr - 3zh) = 0,

we obtain the pressure of the viscous RDE: PR = (1 — 2) x H2e—2(2— a)x + 3ZH and the equation of state:

w = -1 ( 2 -1) + L. 3 V « J h

If we choose Z = Tps, then 2

^2 - ij + 3srH20s-1e-(2s-1)(2-«)x.

In the special case of Z(t) = Zo = const, i.e. s = 0, w would be very large at very early time (a < 1/2) or in the later time (a > 1/2). In particular, for the case s = 1/2, it requires a < 2/(2 + \/3t) to accelerate the universe (w < —1/3).

If the universe also contains another component with the viscous RDE, the Friedmann equation and the corresponding equation of motion can be written as:

h2 = 3 (pr + PY),

p'y=-3(py + PY - 3ZyH), pr = -3(pr + Pr - 3ZrH),

where py is the density of fluid with a barotropic equation of state py = (y — \)py, and 0 < y < 2 to satisfy the dominant energy condition (DEC). So, the equation of state parameter wy = y — 1. Here we will choice Zr = TRyfp = V3rRH and Zy = xY\fP = V3ty H, where p = Py + pR. Then the Friedmann equation (6) becomes

where E± = 2 — 4a + 3ay ± 2aA. Since the definition of density parameter for the fluid is Qy0 = py0/(3H^), then

Ci = 1 - C2 =

a A 4aA

The energy density of viscous RDE is

pR = 3 H2e—^ x[C n — + C2n+e2Ax],

where n± = 2 + 4a — 3ay ± 2aA. By using Eq. (8), we obtain the equation of state for the viscous RDE:

= PR = C1(12V3rfi + x+n—) WR = pR = + 3(C 1n— + C2n+e2Ax)

C2 (12 V3rR + x—n+) 3(Ci n— + C 2n+e2Ax Obviously, the present value of wR is

WR0 = -1 +

„2Ax

1^V3 TR + C1À+n- + C2X-n+ 12(1 - Qy0)

where we have used Eq. (13) and if A is real, the past and future value of wR is

X+ 4V3TR WR (x ^ -to) = -1 + — + n- ,

, , 4V3tr

wr (x ^œ) = -1 + — + n+ ,

and one can see that the value of the equation of state parameter is determined by both the viscosity of the fluid and that of the RDE and does not blow up neither in the past nor in the future.

In the following, we will consider the case of y = 1, which corresponding the equation of state of dark matter wm = 0. By assuming Tm ^ (2 — a) 2/(2443a), we obtain

2-« A --Tm,

À+ ~ 3 - Tm,

' 4 — + Tm, «

E + « 4 — 2a — 2aTm

■■ 2a Tm

+ (4a + 3aY - 2)y' + 6(-y + 2aY + 43ty)y = 0, (9) n +- 4 - 2aTm,

where we have defined y = E2 = H2/H2. The general solution to Eq. (9) is

= 2a + 2aTm,

y = C1e-X+x + C 2e-X-x,

where X±= 2 + 3 y — a ± A and C1, C2 are integration constants. Here we have defined

C1 = 1 - C2 « 2Qm£ + - A,

2 - a 2 - a \2 - a /' where Tm = 623lm. And the present equation of state is

A = —J(2 - 4a + 3aY)2 - 24V3aTY. 2a

WR0 « -3

2 1 + 3V3(Tr + Tm)

1 - Qm0

(20) (21) (22)

From Eq. (10), one can see that the background evolution only depends on ty and does not depend on the viscosity of the RDE. By the definition of y, we have C2 = 1 — C1, and the energy density of the fluid is

so it requires

2(1 - Qm0)

2 - Qm0 + 3V3(Tr + Tm)

Py = - H 2e-X+x(C1E + + C 2E-e2Ax),

to accelerate the universe at present. The past and future the equation of state parameter are:

(28) values of

Fig. 1. The total age of the universe in RDE model with respect to parameter a. Here h = 0.72 and Qm0 = 0.20 (solid), 0.27 (dashed), 0.32 (dotted) are used. The horizontal shadow corresponds to the stellar age bound, namely t0 = 10-12 Gyr.

0 1 2 3 4 5

Fig. 2. The evolution of age of the universe in RDE model with a = 0.46 (solid), 0.76 (dashed), 1.0 (dot-dashed). Here h = 0.72 and Qm0 = 0.27 are used. a < 1/3 will make the perturbation unstable, see [5].

WR (X ^ — C)

TR —

2 — a

wr (x-

1/2 c) - — 1

+ 2V3( +

tr V 2

(31) t(z)

where we have assumed tr is the same order as ry and kept only linear terms of them.

The age of our universe at redshift z is given by t(z) = T(z)/H0, where

oo — ln(1+z)

f dzZ f dx

T(z) = /,-,. = I TZK, (32)

(1 + z') E (z)

is the so-called dimensionless age parameter. For a flat CDM universe dominated by matter (Qm0 = 1), t0 = 2/(3H0), and according to the observations of the Hubble Space Telescope Key project, the present Hubble parameter is constrained to be H0 = 9.776h—1, 0.64 < h < 0.80, which is consistent with the conclusions arising from observations of the CMB and large scale structure. This gives t0 & 8-10 Gyr, which does not satisfy the stellar age bound: t0 > 11-12 Gyr, namely, the age of the universe should be longer than any objects living in the universe. For the ACDM model, in which Qm0 & 0.27, the age parameter is

T (z) =

(1 + z')[&m0 (1 + Z)3 + (1 — ^m0)l1/2'

hence, it easily satisfied the constraint t0 > 11-12 Gyr, because the value of Qm0 is smaller than that in the CDM model, consequently, the integrand in (33) is bigger when z is large, and thus, the total age of the universe is going to be longer in the ACDM model. However, the age of the universe at z = 3.91 in the ACDM model is about t(3.91) ^ 0.12H—1 ^ 1.44-1.79 Gyr which is still younger than the old object APM 08279 + 5255 observed recently [17] at the same redshift with age 2.1 Gyr. In the RDE model without viscosity, t(3.91) ^ 0.10H-1 ^ 1.26-1.57 Gyr with a = 0.46, so the universe is also younger than APM 08279 . 5255.

We plot the total age and its evolution with different values of a in Figs. 1 and 2, where three black points denote the age of some old objects in the universe: LBDS 53W091 [18] at z = 1.43 with age t = 3.5 Gyr, LBDS 53W069 [19] at z = 1.55 with age t =

Fig. 3. The evolution of age of the universe in the viscous RDE model with a = 0.46 and Tm = 0 (solid), 0.03 (dashed), 0.06 (dot-dashed). Here h = 0.72 and Qm0 = 0.27 are used.

4.0 Gyr and APM 08279 + 5255 at z = 3.91 with age t = 2.1 Gyr. It seems that matter was diluted so fast that makes the universe younger than these old objects in the RDE model.

However, this age problem can be alleviated in the viscous RDE model. We plot the evolution of age of the universe with different values of a in this model, see Fig. 3 and it indicates the viscosity could really alleviate the age problem. And actually, only the viscosity of matter affects the evolution of age of the universe, so it could alleviate the age problem in other cosmological models.

In conclusion, we have investigated the Ricci dark energy model when the bulk viscosity ZR = TRyfp is taken into account in this Letter. The energy conservation equations will have additional terms proportional to the bulk viscosity in this case. However, in this model, the evolution of the universe only depends on the bulk viscosity Zy = xy\fP of ordinary fluids with equation of state p = (y — 1)p, and it does not depend on ZR. The RDE model suffers the age problem since the age of the universe should be longer than any objects living in the universe. It seems that the problem is caused by the fact that matter is diluted too fast. When one considers the viscosity of matter, it changes the energy conservation equation for the matter, consequently, it makes matter diluted a little bit slower, and then the age problem is alleviated.

Considering the viscosity of fluid is a next step from the idea one to treat fluid more realized, since the real fluid should have

the viscous properties when it flows, so it is very interesting and worth further studying.

Acknowledgements

This work is supported by National Science Foundation of China grant Nos. 10847153 and 10671128. C.J.F. would like to thank Chang-Jun Gao, Qing-Guo Huang, Miao Li, Xin-He Meng and Shuang Wang for useful discussions.

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