Viscous Dark Energy

CHIN.PHYS.LETT.

Vol. 25, No. 10 (2008) 3834

Viscous Dark Energy Models with Variable G and Λ Arbab I. Arbab∗∗ Department of Physics, Faculty of Science, University of Khartoum, PO Box 321, Khartoum 11115, Sudan Department of Physics and Applied Mathematics, Faculty of Applied Sciences and Computer, Omdurman Ahlia University, PO Box 786, Omdurman, Sudan

(Received 27 May 2008) We consider a cosmological model with bulk viscosity η and variable cosmological Λ ∝ ρ−α , alpha = const and gravitational G constants. The model exhibits many interesting cosmological features. Inflation proceeds du to the presence of bulk viscosity and dark energy without requiring the equation of state p = −ρ. During the inflationary era the energy density ρ does not remain constant, as in the de-Sitter type. Moreover, the cosmological and gravitational constants increase exponentially with time, whereas the energy density and viscosity decrease exponentially with time. The rate of mass creation during inflation is found to be very huge suggesting that all matter in the universe is created during inflation.

PACS: 98. 80. −k, 98. 80. Es, 98. 80. Cq, 04. 20. Jb, 04. 50. Kd The present acceleration of the universe as favoured by the supernovae data can be explained by hypothizing some exotic matter dominated the present universe evolution that breaks the strong energy condition, viz., p + 3ρ ≥ 0.[1,2] One variant of this exotic matter is the one violates the null energy condition, viz., p + ρ > 0. Such exotic matter can be modelled by a scalar field as a dark energy having an equation of state p > −ρ or a phantom with an equation of state p < −ρ.[3] The phantom scalar field can be motivated by S-brane arising in string theory.[4,5] Moreover, phantom fields are introduced by bulk viscosity (η) effects that are equivalent to replacing the pressure p by an effective pressure, viz., peff. = p − 3ηH, where H is the Hubble constant. Viscous effects in an expanding universe are connected with dissipations that are attributed to creation of energy (matter) in the universe. Several authors suggested that the bulk viscosity can drive the universe into a period of exponential expansion (inflation).[6−9] This is really the case, as the effect of bulk viscosity in an expanding universe is to decrease the pressure making the total pressure negative. Inflation can also be induced by higher order corrections.[10] In scalar field, inflation stopped by the slow roll-down of the scalar field from the false vacuum to the true vacuum. In this work, we investigate the effect of coupling of gravity with vacuum and viscosity. Provided that a certain conspiracy is maintained, the evolution of the universe can proceed in an attractive way. We have found that such a recipe is possible and leads to interesting features related to the cosmic evolution. Inflation is triggered by the vacuum energy bulk viscosity cooperation. This work generalizes our recent model of phantom and dark energy models.[11] Consider the Einstein-Hilbert action with a cos∗∗ Email:

[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd °

mological constant Λ, ∫ 1 √ S=− d4 x g(R + 2Λ) + Smatter . 16πG

(1)

The variation of the metric with respect to gµν with f (R) = R − 2Λ gives[12] 1 (2) f 0 (R)Rµν − f (R)gµν = −8πGTµν , 2 where Tµν is the energy momentum tensor of the cosmic fluid. For an ideal fluid one has Tµν = (ρ + p)uµ uν + pgµν ,

(3)

where uµ , ρ and p are the velocity, density and pressure of the cosmic fluid. Contracting Eq. (2), using Eq. (3) and taking its 00 components give the equations Rf 0 (R) − 2f (R) + 8πGT = 0, (4) 1 (5) f 0 (R)R00 + f (R) + 8πGT00 = 0, 2 with T00 = ρ, T = ρ−3p and Tij = −p for i, j = 1, 2, 3. For a flat Friedmann–Lematre–Robertson–Walker metric, ( ) ds2 = dt2 − a2 (t) dr2 + r2 (dθ2 + sin2 θdφ2 ) , (a (( a˙ )2 a ¨) ¨) one has R00 = −3 and R = −6 + , where a a a a is the scale factor, so that Eqs. (4) and (5) yield ( a˙ )2 = 8π Gρ + Λ, (6) 3 a ) (a ¨ = − 4π G(ρ + 3p) + Λ, (7) 3 a and the energy conservation equation reads ( a˙ ) (ρ + p) = 0. (8) ρ˙ + 3 a

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Arbab I. Arbab

The pressure p and energy density ρ of an ideal fluid are related by the equation of state, p = ωρ,

ω = const.

(9)

The Einstein field equation, with time-dependent G and Λ, then yields two independent equations (6) and (7) having the same form as in the standard model. Hence, we can now allow Λ and G to vary with time, i.e., Λ = Λ(t) and G = G(t). Thought such ansatz breaks Lorentz invariance, energy and momentum are formally conserved. In an expanding isotropic and homogenous universe, such ansatz is admissible, since all cosmic variables are time-dependent only. The Bianchi identity 1 (Rµν − Rg µν ); 2

µ

= −(8πGT µν + Λg µν );

µ

= 0, (10)

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so that

a˙ = Ka−3(1+ω)(n−α) , (20) a N A(n−α) , ω 6= −1. Substituting it in where K = − 3 (1 + ω) Eq. (13) and using Eq. (14), one reaches a = Dt1/3(1+ω)(n−α) ,

where D = A1/3(1+ω) [−N (n−α)]1/3(1+ω)(n−α) , n 6= α. Substituting the above equation into Eq. (16), one finds ρ = [−N (n − α)]−1/(n−α) t−1/(n−α) , n 6= α

(22)

so Eq. (15) becomes Λ = 3β[−N (n − α)]α/(n−α) tα/(n−α) , n 6= α.

(23)

Eq. (18) now reads

with Eqs. (2) and (3) implying that a˙ Λ˙ Gρ˙ + 3(p + ρ)G + ρG˙ + = 0. a 8π

G= (11)

Bulk viscosity can be introduced in a uniform perfect fluid by replacing the pressure term p by an effective pressure, peff defined by peff. = p − 3ηH,

(12)

where η is the coefficient of bulk viscosity. This is normally modelled by the relation η = η0 ρn ,

n, η0 = const.

(13)

Applying Eq. (12) into Eq. (11) and using Eq. (8), one obtains[7] ˙ + Λ˙ = 9η(8πG)H 2 . 8π Gρ (14) We consider here the ansatz Λ = 3β/ρα

β, α = const.

(15)

Integrating Eq. (11), using Eq. (12), we obtain ρ = Aa−3(1+ω) ,

A = const.

(16)

Using Eqs. (6) and (15), Eq. (14) reads ( ρ˙ ) G˙ 3αβ − = 3η0 ρn (8πGρ + 3βρ−α ). (17) G 8πGρ(α+1) ρ Now consider the following functional dependence of the gravitational constant 8πG = Cρ−(α+1) ,

C = const.

(18)

Eqs. (17) and (18) imply that ρ˙ = N ρn−α+1 ,

(21)

N =−

3Cη0 (C + 3β) , C(1 + α) + 3αβ

(19)

C [−N (n − α)](1+α)/(n−α) 8π · t(1+α)/(n−α) , n 6= α.

(24)

Our present model can be compared with the Brans– Dicke (BD) theory in which the gravitational constant 4 + 2ωD 1 is related to the scalar function φ as G = , 3 + 2ωD φ where ωD defines a coupling between the scalar field 2+2ωD

and gravity.[13] Accordingly, one has a ∝ t 4+3ωD and −

2

G ∝ t 4+3ωD . Notice that a comparison with BD model reveals that [ (2n + 2) ] . α=− 1+ 2 + 3ωD This means that the BD model is equivalent to a bulk viscous model with a cosmological constant varying as Λ ∝ ρ−α . Or conversely, our model is equivalent to the BD model where the scalar field is given by 3 + 2ωD 1 φ= , where G is given by Eq. (24). Notice 4 + 2ωD G α − also that one can write Λ ∝ φ 1+α . It is therefore interesting to see that our model is equivalent to the BD model with a cosmological constant of the form shown above. Now we consider the following cases: Case 1. Now let n = α/2 where −1 < α < 0 and 1 + ω > 0. In this case, Eqs. (21) and (22) reduce to a ∝ t−2/3α(1+ω) ,

ρ ∝ t2/α ,

(25)

and Eqs. (23), (24) and (13) yield G ∝ t−2(1+α)/α ,

Λ ∝ t−2 ,

η ∝ t,

(26)

where C > 3β > 0, i.e., G > 0. These represent the viscous analogue of the dark energy model.[11] . In particular, a viscous cosmological model with Λ ∝ H 2 is equivalent to a viscous dark energy model with

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Λ ∝ ρ−α if narb. = 1 + α/2, where narb is the index of viscosity in Ref. [7]. Moreover, we have shown that the variation Λ ∝ H 2 is equivalent to Λ ∝ ρ.[14] Case 2. Now let α > 0 and n < α. In this case, Eq.(22) implies that one has a phantom energy solution where the energy density increases with time. However, since the scale factor is an increasing function of time, Eq. (21) with the condition n < α implies that 1 + ω < 0. Since N > 0, one requires here C < 0 so that G < 0, and for β > 0 one has Λ > 0. This solution is found by Ref. [11] and the above solution represents its viscous analogue. It is clear here that though the energy density increases, gravity (decreases) and viscosity (increase) conspire not to allow the phantom energy density to dominate. We notice from Eq. (20) that when n = α > 0, we obtain Cη0 (C + 3β) 1 a˙ =K= . a C(1 + α) + 3αβ (1 + ω)

(27)

This implies that a = Γ exp(Kt),

Γ = const,

(28)

where K > 0, for 1 + ω > 0. Equations (16) yields ρ ∝ exp[−3K(1 + ω)t],

(29)

so that Eqs. (15) and (18) become Λ ∝ exp[3αK(1 + ω)t], G ∝ exp[3K(1 + ω)(1 + α)t],

(30)

and the bulk viscosity, Eq. (13), η ∝ exp[−3αK(1 + ω)t].

(31)

Notice, however, during inflation ω 6= −1 as evident from Eq. (24). This is unlike the standard case where inflation requires ω = −1. It is also remarkable that the cosmological constant during inflation increases exponentially with time, whereas the energy density decreases exponentially with time. Whatever the initial value of the cosmological constant before inflation, its value at the end of inflation will be enormously large. This may explain why Λ was large in the very early universe, as suggested from particle physics considerations. The universe exited from inflation due to the huge growth of the gravitational force that halts the exponential expansion. Thereafter, the universe enters a radiation dominated phase. The large decrease of the bulk viscosity during the inflationary era has allowed the universe to isotropize, and eventually led to the isotropic and homogenous universe we observe today. In the standard de-Sitter model the inflationary expansion is led by the cosmological constant

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(Λ), where the energy density stays constant. We notice from Eqs. (28) and (29) that the mass created (annihilated) during inflation is M ∝ ρa3 ∝ exp(−3Kωt). However, since ω > −1, one has for −1 < ω < 0, a positive mass creation rate. Hence, one would presume that all matter constituting the universe mass was produced during inflation. It is remarkable to notice that inflation is induced by dark energy only. Thus, dark energy played an important role by driving the early universe into an exponential expansion, and the present universe into a accelerated expansion. Hence, the existence of dark energy is very crucial to the evolution of the universe. In summary, we have studied the effect of bulk viscosity on the evolution of dark matter and phantom energies. We have shown that non-viscous dark matter models are equivalent to viscous ones. The increasing bulk viscosity and decreasing gravitational constant do not allow the phantom energy density to condensate. During inflationary era the universe isotropizes and the cosmological constant attained a vary large value. After inflation the cosmological constant decreases with time quadratically (e.g., for n = α/2). This evolution provides a viable mechanism for the smallness of the present cosmological constant, i.e., why today the cosmological constant is vanishingly small compared with its inial value! I am grateful to the Swedish International Development Agency (SIDA) for providing financial support for my visit to Abdus Salam International Center for Theoretical Physics, Trieste, Italy, where this work was carried out. I would like to thank the referees for their critical comments.

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