Dissipative Future Universe without Big Rip

0.11 MB·English

by  Anil Kumar Yadav

Dissipative Future Universe without Big Rip
Anil Kumar Yadav
Department of Physics, Anand Engineering College, Keetham, Agra-282007, India
E-mail: [email protected],[email protected]
1 Abstract
1
ThepresentstudydealswithdissipativefutureuniversewithoutbigripincontextofEckartformalism.
0
2 The generalized chaplygin gas, characterized by equation of state p = − A1 , has been considered as a
ρα
n model for dark energy due to its dark-energy-like evolution at late time. It is demonstrated that, if the
a cosmic dark energy behaves like a fluid with equation of state p=ωρ; ω <−1, as well as chaplygin gas
J simultaneously thenthebigripproblem doesnotarisesandthescale factorisfoundtoberegularforall

1 time.
1
PACS: 98.80Cq, 98.80 JK

] Keywords : Dissipative effect; Phantom fluid; Big rip; Accelerated universe
c
q
-
r
g
[ Recent observations like CMB anisotropy, supernova and galaxy clustering strongly indicate that our
universe is spatially flat and there exists an exotic cosmic fluid called dark energy with negative pressure,
2
which constitutes about 70 percent of the total energy of universe. The dark energy is usually described by
v
an equation of state (EoS) parameter ω p, the ratio of spatially homogeneous dark-energy pressure p to
5 ≡ ρ
0 its energy density ρ. A value ω < 1 is required for cosmic acceleration. The simplest explanation for dark
0 energyiscosmologicalconstant,for−w3hichω = 1. Theincreasingevidencefromobservationaldataindicates
3 −
that ω lies in a narrow strip around ω = 1 quite likely being less than this value [1] [3]. The regionwhere
. − −
7 EoS parameter ω < 1, is typically referred to as a phantom dark energy universe. The existence of the
0 −
regionwith ω < 1 opens up a number of fundamentalquestions. For instance, the entropyofsuch universe
0 −
is negative. The dominant energy condition (DEC) for phantom fluid is violated, as a rule. The phantom
1
: dominateduniverseendupwithafinite time future singularitycalledbigriporcosmicdoomsday[4,5]. The
v
lastproperty attracted much attention and broughtthe number of speculations upto the explicit calculation
i
X of the rest of the life-time of our universe.
r
a Soon after Caldwell [4] proposed phantom dark energy model with cosmic doomsday of future universe,
cosmologists started making efforts to avoid this problem using ω < 1 [6] [8]. In the braneworld scenario,
− −
SahniandShtanovhasobtainedwell-behavedexpansionforthefutureuniversewithoutbigripproblemwith
ω < 1. Theyhaveshownthataccelerationisatransientphenomenoninthecurrentuniverseandthefuture
−
universe will re-enter matter dominated decelerated phase [9]. It is found that generalrelativity (GR) based
phantom model encounters “sudden future singularity” leading a divergent scale factor, energy density and
pressure at finite time t = t . Thus the classical approach to phantom model exhibits big rip problem. For
s
futuresingularitymodel,curvatureinvariantbecomesverystrongandenergydensityisveryhighneart=t
s
[10]. So, quantum effects should be dominated for t = t < one unit of time (Early universe) [11] [13]
s
| | −
and it is shownthat the an escape from the big rip is possible on making quantum corrections to the energy
density and pressure in Friedmann equations.
IntheframeworkofRobertson-Walkercosmology,Chaplygingas(CG)isalsoconsideredasagoodsource
of dark energy for having negative pressure, given as
A
p= (1)
−ρ
1
where p and ρ are, respectively, pressure and energy density in a co moving reference frame, with ρ > 0; A
is a positive constant.
Moreover, it is only gas having super-symmetry generalization [14] [16]. Bertolami et al [17] have found
−
that generalized Chaplaincy gas (CG) is better fit for latest Supernova data. In case of CG, equation (1) is
modified as
A
p= (2)
−ρα1
where 1 α< .
≤ ∞
For α=1, equation (2) corresponds to equation (1).
OtherapproacheshaveconsidereddissipativeeffectsinCGmodels,usingtheframeworkofElkharttheory
[18]. Thai et al [19], have investigated a viscous GCG, assuming that there is a bulk viscosity in a linear
borotropic fluid and GCG. It is found that the equation of state of GCG can cross the boundary ω = 1.
−
Also in Ref. [20], it is found that a dissipative chaplygin gas can give rise to structurally stable evaluational
scenarios. It is interesting to note that the GCG itself canbehave like a fluid with viscosityin the contextof
Eckart formalism [18]. Fabris et al [21], have investigated an equivalence GCG and dust like fluid. Recently
Cruz et al [22], have studied dissipative generalized chaplygin gas as phantom dark energy and found the
cosmologicalsolutions for GCG with bulk viscosity.
The FRW metric for an homogeneous and isotropic flat universe is given by
ds2 = dt2+a(t)2 dx2+dy2+dz2 (3)
−
(cid:0) (cid:1)
where a(t) is the scale factor and t represents the cosmic time.
The field equations in the presence bulk viscous stresses are
a2 ρ
4 =H2 = (4)
a2 3
a 1
44 = (ρ+3p¯) (5)
a −6
where p¯is the effective pressure given by
p¯=p 3Hξ (6)
−
Here p, ξ are the isotropic pressure and bulk viscous coefficient respectively.
The energy conservation equation is given by
ρ +3H(ρ+p¯)=0 (7)
4
Here, and in what follows the sub indices 4 on a, ρ and elsewhere denote differentiation with respect to t.
Using equations (2), (4) and (6), equation (7) leads to
a A
ρ +3 4 ρ 3Hξ =0 (8)
4 a (cid:18) − ρα1 − (cid:19)
In order to obtain solution of equation (8), we will assume that the viscosity has a power-law dependence
upon the density
ξ =ξ ρn (9)
0
where ξ and n are constant.
0
On using equation (9) in equation (8), we obtain
dρ 3da ρ1+αα A
+ (cid:16) − (cid:17) =3ρn+1ξ (10)
dt a dt ρα1 0
To solve equation (10), we use the transformation ρ =f(ρ)=ρn+1, accordingly equation (10) leads to
4
3(1+α)
ρ1+αα(t)=A+ ρ1+αα A a0 α(1−3ξ0) (11)
(cid:16) 0 − (cid:17)(cid:20)a(t)(cid:21)
2
where ρ =ρ(t ) and a =a(t ); t is the present time.
0 0 0 0 0
In the present model, it is assumed that the dark energy behaves like GCG, obeying equation (2) as well as
fluid with equation of state
p=ωρ (12)
with ω < 1 simultaneously.
−
From equations (2) and (12), we obtain
A
ω(t)= (13)
−ρ(t)1+αα
So, evolution of equation (13) at t=t leads to
0
1+α
A= ω ρ α (14)
− 0 0
with ω =ω(t ).
0 0
Using equation (14), equation (11) leads to
α
3(1+α) 1+α
ρ=ρ0−ω0+(1+ω0)(cid:18)aa(0t)(cid:19)α(1−3ξ0) (15)
 
In the homogeneous model of universe, a scalar field φ(t) with potential V(φ) has energy density
1
ρ = φ2+V(φ) (16)
φ 2 4
and pressure
1
p = φ2 V(φ) (17)
φ 2 4−
Equation (16) and (17) lead to
φ2 =ρ +p (18)
4 φ φ
Using equations (2), (12) and (14), equation (18) reduces to
(1+α) (1+α)
φ2 = ρ α +ρ0 α ω0 (19)
4 ρα1
Equation (15) and (19) lead to
3(1+α)
(1+ω )ρ a0 α(1−3ξ0)
φ2 = 0 0(cid:16)a(t)(cid:17) (20)
4 1
3(1+α) 1+α
(cid:20)−ω0+(1+ω0) aa0 α(1−3ξ0)(cid:21)
(cid:0) (cid:1)
From equation (20), it is clear that for 1+ω >0, φ2 >0, giving positive kinetic energy and for 1+ω <0,
0 4 0
φ2 < 0, giving negative kinetic energy. 1+ω > 0 and 1+ω < 0 are representing the case of quintessence
4 0 0
and phantom fluid dominated universe respectively. Similar result are obtained by Hoyle and Narlikar in C
-field with negative kinetic energy for steady state theory of universe [23].
Now, from equation (4) and (15), we obtain
α
3(1+α) 1+α
aa242 =Ω0H02|ω0|+(1−|ω0|)(cid:18)aa(0t)(cid:19)α(1−3ξ0) (21)
 
where ω = ω, H = 100hkm/s Mpc, present value of the Hubble’s parameter and Ω = ρ0 with
| 0| − 0 0 ρcr,0
ρcr,0 = 83πHG02.
Equation (21) may be written as
α
3(1+α) 2(1+α)
aa4 = Ω0H0|ω0|2(1α+α) 1+ (1−ω|ω0|)(cid:18)aa(0t)(cid:19)α(1−3ξ0) (22)
p | 0|
 
3
10
9
8
7
a(t)
6
dH(t)
5
4
3
2
1
0
0 20 40 60 80 100
t
Figure 1: The plot of scale factor (a) and horizon distance (d ) versus time (t).
H
3(1+α)
Neglecting the higher powers of (1−|ω0|) a0 α(1−3ξ0), equation (22) leads to
|ω0| (cid:16)a(t)(cid:17)
3(1+α)
aa4 = Ω0H0|ω0|2(1α+α) 1+ 2α(1(1+−α|)ω0ω|) (cid:18)aa(0t)(cid:19)α(1−3ξ0) (23)
p | 0|
 
Integrating equation (23), we obtain
α(1−3ξ0)
a(t)= a0 3(1+α)
(cid:18)(2(1+α)ω )(cid:19) ×
0
| |
α(1−3ξ0)
α 3(1+α)
(α+2(1+α)ω0 )e6H0|ω0|2(1+α)√Ω0(t−t0) α(1 ω0 ) (24)
(cid:20) | | − −| | (cid:21)
Fromequation(24),itisclearthatast ,a(t) whichissupportedbyrecentobservationofSupernova
→∞ →∞
Ia [24, 25] and WMAP [26, 27]. Therefore the present model is free from finite time future singularity.
Now the horizon distance is obtained as
α(1−3ξ0)
d (t)= 3(1+α)a(t) 2(1+α)|ω0| 3(1+α)
H
α(1 3ξ )a (cid:18)α+(α+2)ω (cid:19) ×
0 0 0
− | |
e(cid:20)6H0|ω0|2(1α+α)√Ω0α(31(−1+3ξα0))t(cid:21) (25)
Fig. 1 depicts the variation of scale factor (a) and horizon distance (d ) versus cosmic time, as a represen-
H
tativecasewith appropriatechoiceofconstantsandother physicalparameters. Fromequation(24)and(25)
it is clearthat d (t)>a(t) i. e. horizongrowsmore rapidly than scale factor which is clearlyshownin Fig.
H
1.
In this case, the Hubble distance is given by
3(1+α)
H−1 = √Ω0H0|1ω0|2(1α+α) 1− 2α(1(1+−α|)ω|0ω|0)|(cid:18)aa(0t)(cid:19)α(1−3ξ0) (26)
 
4
0.2238
0.2238
H−1
0.2238
0.2238
0 0.05 0.1 0.15 0.2
t
Figure 2: The plot of Hubble distance (H 1) versus time (t).
−
Equation(26)isshowingthegrowthofHubbledistance(H 1)withtimesuchthatH 1 1 =
− − α
0 as t . This behaviour of H 1 is clearly depicted in Fig. 2. Thus in present ca→se,Ht0h√eΩ0g|aωl0a|x2(i1e+sα)wi6ll
−
→ ∞
not disappear when t , avoiding big rip singularity. Therefore, one can conclude that if phantom fluid
→ ∞
behaves like GCG and fluid with p = ωρ simultaneously then the future accelerated expansion of universe
will free from catastropic situation like big rip. Equation (15) may be written as
α
3(1+α) 1+α
ρ=ρ0|ω0|+(1−|ω0|)(cid:18)aa(0t)(cid:19)α(1−3ξ0) (27)
 
α
From equation (27), it is clear that as t , ρ ρ0 ω0 1+α >ρ0 (since t , a(t) ). Thus one can
→∞ → | | →∞ →∞
concludethatenergydensityincreaseswithtime,contrarytootherphantommodelshavingfuturesingularity
at t = t [4, 11]. Based on Ia Supernova data, singh et al [28] have estimated ω for model in the range
s 0
2.4< ω < 1.74 up to 95 percent confidence level. Taking this estimate as an example, with α = 1, ρ
0
i−s found in the−range 1.31ρ <ρ <1.54ρ and with α=2, ρ is found in the range 1.44ρ <ρ <1.78ρ∞
0 0 0 0
∞ ∞ ∞
and so on. This does not yields much increase in energy density as t but if the future experiments
→ ∞
supports large value of ω then ρ will be high.
0
| | ∞
It is interesting to see here that present model is derived by Bulk viscosity in the context of Eckart
formalism [18]. We see that big rip problem does not arises in the present model. In Refs [10] [13], for
−
models with future singularity escape from big rip, is demonstrated using quantum corrections in the field
equations near t = t . However the present study deals phantom cosmology with accelerated expansion
s
without catastrophic situations using classical approach. It is also seen that the model of future universe
presented by Srivastava [29] is a particular case of the model presented in this letter.
Acknowledgements
Author would like to thanks The Institute of Mathematical Science (IMSc), Chennai, India for providing
facility and support where part of this work was carried out.
5
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