Table Of ContentInteracting New Agegraphic Dark Energy in a Cyclic
Universe
K. Saaidi , 1 • H. Sheikhahmadi 2 •
∗† ∗
A. H. Mohammadi 3
∗∗
1
1
0
2
c
e
D
, , 1 Introduction
1 Abstract The main goal of this work is investigation
3 of NADE in the cyclic universe scenario. Since, cyclic Cosmologicalandastronomicalobservationssuchassu-
universe is explained by a phantom phase (ω < 1), it pernovae type Ia observational data (Perlmuter et al.
] −
c is shown when there is no interaction between matter 1997; Reiss et al. 1998; de Bernardis, et al. 2000)
q
and dark energy, ADE and NADE do not produce a andWilkonsonMicrowaveAnisotropicProbe(WMAP)
-
r phantom phase, then can not describe cyclic universe. (Astier et al. 2006;Perlmutter et al. 1999;Peiris et al.
g
Therefore, we study interacting models of ADE and 2003; Riess et al. 2007) imply that the universe is
[
NADE in the modified Friedmann equation. We find undergoing a period of accelerated expansion. Since
1 out that, in the high energy regime, which it is a nec- normal matter can not give rise to accelerated expan-
v
5 essary part of cyclic universe evolution, only NADE sionof the universe,Scientists came up with a solution
candescribethisphantomphaseeraforcyclicuniverse.
7 whichexpressesthatthisexpansionisaresultofanam-
2 Considering deceleration parameter tells us that the biguous fluid named Dark Energy (Bridle et al. 2003;
0 universe has a deceleration phase after an acceleration
Koght et al. 2003).
.
1 phase, and NADE is able to produce a cyclic universe. The theoreticaland experimentalanalysis suggestthat
0 Alsoitisfoundvaluabletostudygeneralizedsecondlaw
the universe consist of %73 dark energy, %23 cold
2 ofthermodynamics. Sincetheloopquantumcorrection
1 dark matter (CDM), and remnant matter is baryons
is taken account in high energy regime, it may not be
: (Fujii and Maeda 2003; Spergel et al. 2003). Unfor-
v suitable to use standard treatment of thermodynam-
tunately the nature and origin of dark energy are am-
i
X ics, so we turn our attention to the result of (Li et al.
biguous up to now, but people have proposed some
2009), which the authors have studied thermodynam-
r candidates to describe dark energy. Amongst the var-
a ics in loop quantum gravity, and we show that which
ious candidates of dark energy to describe accelerated
conditioncansatisfy generalizedsecondlawofthermo-
expansion of the universe, cosmological constant (vac-
dynamics.
uum energy), Λ, with equation of state (EoS) ω = 1
−
Keywords Cyclic universe; Low energy Regime; High is located in central position. However, as it is well
Energy Regime; Agegraphich Dark Energy. known, the cosmological constant proposal has two
famous problems, fine-tuning problem and the cos-
mic coincidence problem (Weinberg 1989; Steinhardt
K.Saaidi∗,†
1997; Carroll 2001; Peebles and Ratra 2003). Some
H.Sheikhahmadi∗
other of dark energy models suggest that dark en-
A.H.Mohammadi∗∗
ergy component can treat as scalar field with dy-
∗Department of Physics, Faculty of Science, University of Kur-
namical EoS. In this scenario the evolution of the
distan,Sanandaj,Iran.
field is very slow, so that kinetic energy density is
†Department of Physics, Kansas State University,116 Cardwell
less than the potential energy density, and this give
Hall,Manhattan, KS66506, USA.
∗∗IslamicAzadUniversity,EvazBranch,FarsProvince,Iran us a negative pressure, responsible to the cosmic ac-
[email protected] celeration (Frieman et al. 1995; Turner and White
[email protected] 1997; Caldwell et al. 1998). Some of scalar field
[email protected] models are as chameleon field (Khoury et al. 2004;
2
Mota and Barrow 2004),quintessence(Q-field)(Liddle, etEaln.qvist and Sloth 2004; Dutta et al. 2010). Another
1999; Peebles and Ratra 1999), and phantom. In the attractive model to describe the nature of dark energy,
phantom field scenario, the parameter of EoS is as within the framework of a fundamental theory origi-
ω < 1, due to existence of a negative kinetic en- natingfromsomeconsiderationsofthefeatureofquan-
−
ergy density of scalar field. It is well known that the tum gravity theory, is called agegraphic dark energy
phantom dark energy model suffers from two kind of (ADE) model (Cai 2007). The ADE assumes that the
problems, ”Big Bang” singularity and ”Big Rip” sin- dark energy comes from the universe components fluc-
gularity, where big bang is related to initial epoch of tuation such as space-time and matter fluctuation (for
universe and big rip is related to a finite future sin- further discussion we refer the reader to (Kim et al.
gularity. Since the space-time singularities are invidi- 2008; Nozri and Azizi 2009; Kim 2008; Karami et al
ous for theorists, models that avoid these singularities 2008;Setare 2010). Inthismodel,theageofuniverseis
are attractive. One of these models names cyclic uni- takenas the length measure instead of the horizondis-
versewhichhavereceivedhugeattention(Brown et al. tance, therefore the causality problem which appears
2008; Baum and Frampton 2007; Brown et al. 2008). in the HDE model can be avoided. The ADE model
Presence of ρ2 term with a negative sign in Friedmann suffers from the difficulty to describe the matter dom-
equation (which is used at studying cyclic universe) is inant epoch. The authors of (Wei et al. 2008) have
an effective way to eliminate these singularities. In the introduced a new mechanism to overcome that prob-
cyclic universe scenario, where is based on the phan- lem, which it is called new agegraphic dark energy
tomdarkenergymodel, the universeoscillatesthrough (NADE) model, and its energy density is defined by
a series of expansion and contraction. Universe in this ρΛ = 3n2m2pη−2, where 3n2 is introduced to parame-
scenariohasaveryhighenergydensityatbeginningan terizedsome uncertainties and η is conformaltime and
ending of the expansion, so quantum gravity can not can be written as
be ignored in these stages (Ashtekar. 2006b,a). This
a da
evolution can be result from the modified Friedmann η = , (2)
a2H
equation in the loop quantum cosmology (LQC). In Z0
LQC, the Friedmann equation has been modified to where a is scale factor and H is well-known as Hubble
parameter. Cyclic model of universe, due to avoiding
ρ ρ
2
H = 1 , (1) singularity, and NADE, due to estimating a good ap-
3m2 − ρ
p (cid:18) c(cid:19) proximate of dark energy value and solving causality
where H is the Hubble parameter, m2 is the reduced problem of HDE, have received huge interest. In pre-
p
planckmass(m2 = 1 =2.44 1018GeV), ρistheto- vious works, like (Wei et al. 2008), it was only men-
p 8πG × tioned that interaction NADE can produce phantom,
talofenergydensity, ρ isthe criticalenergydensityas
c
ρ =4√3γ 3m4 =0.82ρ ,whereρ =2.22 1076GeV, however it was not explained that if NADE can pro-
c − p p p × duce cyclic universe and how the quantities like dark
andγ isthedimensionlessBarbero-Immiriziparameter
energydensityparameter,equationofstateparameter,
(Ashtekar et al. 2006c). Wenotice,thiscorrectioncan
and deceleration parameter behave in cyclic universe
solve the singularity problems as follow, when the to-
by taking NADE as component of dark energy. In this
tal energy density reaches the critical density, the uni-
work we motivated to take NADE as dark energy and
verse reaches the maximum at the end of expansion
investigate universe evolution in cyclic model. If it is
that is called ”turnaround point”, and universe arrives
able to stand in phantom area, it may produce cyclic
atsmallestsizeattheendofcontraction,thenwehavea
universe, howeverit is not all of story. Although, pres-
bounce there (Boyle, et al. 2004; Khoury et al. 2001;
enceofthis type ofdarkenergydisplaysanaccelerated
Steinhardt and Turok 2002; Boyle, et al. 2006). We
universe, but the most important thing in cyclic uni-
emphasize the idea of cyclic universe was first intro-
verseis that, after awhile, the universeshouldenter in
duced by Tolman (Tolman 1931, 1934).
adecelerationareaandrecontractinturnaroundpoint.
An interesting attempt for probing the nature of dark
energy, in the framework of quantum gravity, is the
holographic dark energy (HDE). In the HDE model,
dark energy is a dynamical evolving vacuum energy
2 High and low energy
density that can satisfy the phantom behavior. Au-
thors of (Zhang et al. 2007) have investigated the
Inthis step, we focus onhighenergyregime. Certainly
cyclic universe by HDE (and some interesting work
the quadratic term of energy density in the Friedmann
about HDE have been done (Li 2004; Setare et al.
equation can not be ignored, this term can play a very
2007a,b; Karami and Fehri 2010; Cohen et al. 1999;
3
impressive role in the evolution of universe. The modi- Assumption of interaction between ρm and ρΛ may
fied Friedmann equation is as solveaboveproblem. Byincludinginteraction,thecon-
servation equations are as
ρ
2
3H =ρ 1 , (3)
(cid:18) − ρc(cid:19) ρ˙m+3H(1+ωm)ρm = Q (8)
where ρ is a combination of dark energy density, ρΛ, ρ˙Λ+3H(1+ωΛ)ρΛ = Q, (9)
−
and matter density, ρ . For dark energy dominant in
m
highenergyregime,theaboverelationcanberewritten where Q indicates interaction. Q is taken as Q=ΓρΛ,
with Γ > 0, which means there is transfer of energy
as
from ρΛ to ρm Sheykhi et al. (2010). We take Q as
Q = 3b2H(1+r)ρΛ, where r = ρm. Setting ADE as
3H2 ρΛ 1 ρΛ , (4) the component of dark energy giveρsΛus an ωΛ as
≈ − ρ
(cid:18) c(cid:19)
so,accordingto this relationthe parameterofdark en- 2
ergy may be estimated as ωΛ =−1+ 3n Ω♮Λ−b2(1+r). (10)
q
Ω♮ isalwayslargerthanone,anditbecomesverylarge
Ω♮Λ = 3ρHΛ2 ≈ ρc−ρcρΛ. (5) hataΛvteurbn2a(1ro+unrd)>poi2n√t,Ωs♮Λo.fTorhhisavpirnegdiωctΛs<a l−ar1g,ewveasluheoufoldr
3n
Ω♮ is always larger than one, and it can be very large coupling constant b, while it is in contrast to the ob-
Λ
whenthe universeapproachesto the turnaroundpoint. tained value of another papers such as (Sheykhi et al.
However note that, since we have ρ , although it is 2010).
m
very small, Ω♮Λ can not be infinite. This small value of Now,NADE istakenasρΛ whichthatis the maincase
ρm does not allow ρΛ to reach the exact value of ρc, of this work. Because we use conformal time η instead
then the presence of ρm can prevent the infinite value of T, scale factor appear in relation (10), namely
of Ω♮.
Λ
If we assume there is no interaction between these two
2
components of the universe, conservation equation can ωΛ =−1+ 3na Ω♮Λ−b2(1+r). (11)
be written as q
To have ωΛ < 1, the coupling constant b should obey
−
following relation
ρ˙ +3H(1+ω )ρ = 0
m m m
ρ˙Λ+3H(1+ωΛ)ρΛ = 0. (6)
2
b2 > Ω♮, (12)
BytakingADE,whichisdefinedasρΛ =3n2T−2where 3naq Λ
TωΛdiesnoobtetsaitnimede,aass dark energy component ofuniverse, hineroer,drerhaosf ba2e,enwiegnmoaryedobbteacianusaecoofnρvmen≪ienρtΛv.alIufeΩf♮Λorbeb
in order it could give us phantom dark energy in this
2 regime.
ωΛ =−1+ 3n Ω♮Λ. (7) In Fig.1 ωΛ parameter has been plotted versus r =
q ρm,forthreedifferentvalueofinteractioncouplingcon-
Also,if we replaceNADE insteadofADE, wearrive ρΛ
stant. In expansion phase, by passing time, the quan-
atsameresultbutonlyscalefactor,a,shouldbeadded
in the denominator of second term on the right hand tity r decreases and makes ωΛ larger. Since it still
of relation (7). Since the latest term, in both case, is standsinphantomrange,largervalueofωΛ showsthat
darkenergydensity growsup slower;incontrastto the
alwayspositiveωΛ nevercanbesmallerthan 1. Since
− contraction phase. Also, having stronger interaction
phantomtype ofdark energyis requisite for cyclic uni-
verse(phantomenergydensityisgettinglargerwithin- coupling causes smaller value for ωΛ.
If Ω♮ be in order of aξ, where ξ 2, equation (12)
creasingscalefactorsothetotalenergydensityreaches Λ ≤
can be valid in good approximation. Now suppose Ω♮
thecriticalenergydensityinturnaroundpointanduni- Λ
verse begins contraction), selecting ADE and NADE, is in order of a2, for instance Ω♮Λ = αa2. From equa-
without interaction, as a component of dark energy in tion (12) we have b2 > 2√α, and by taking the value
3n
cyclic universe is not suitable (Wei et al. 2008). of (Sheykhi et al. 2010) for b, namely b2 = 0.25, we
4
Ω tion, which governsthe NADE evolutionof universe in
-1.5 high energy regime, can be attained as
-2.0 b2=0.5 Ω♮Λ′ =−2Ω♮Λ(qnΩa♮Λ(Ω♮Λ−1)−3b2(1+r)(21Ω♮Λ−1))
-2.5 b2=1 (17)
b2=2
with attention to (12), it is clearly shown that Ω♮′ =
Λ
r 1Ω˙♮ is positive. So one canrealize that when the uni-
0.02 0.04 0.06 0.08 0.10 H Λ
verse is growing up (decreasing) Ω˙♮ is positive (nega-
Λ
Fig.1 equationofstateparameterofdarkenergyhasbeen tive)whichitdisplaysthatdarkenergydensityparame-
plottedversusthedimensionless quantityr= ρm,forthree terisincreasing(decreasing)bypassingtime. Bypass-
ρΛ
different value of interaction coupling constant. ing the universe from turnaround point, Ω♮ decreases
Λ
and puts the universe in low energy regime.
obtain the value of α as √α< 3n. Another useful cosmological parameter is decelera-
8
tion parameter which can tell us if the expansion of
Now,wewanttoobtaindifferentialequationforΩ♮. In
Λ universeis acceleratingor not. Decelerationparameter
the NADE, ρΛ is given as
is given by
3n2
ρΛ = η2 . (13) q = 1 H˙ , (18)
− − H2
From the definition of dark energy density parameter,
˙
namely Ω♮Λ = 3ρHΛ2, The differential equation for Ω♮Λ is substituting HH2 term in the above relation, we arrive
at
acquired as
3
Ω♮′ = 2Ω♮ H˙ + Ω♮Λ, (14) q = −1+ 2 (1+ωm)Ω♮m+(1+ωΛ)Ω♮Λ
Λ − ΛH2 qan (cid:16) 2ρ (cid:17)
1
× − ρ
(cid:18) c(cid:19)
where prime denote derivative with respect to = 3 2ρ
lna. Taking the time derivative ofmodified FriedmNann = −1+ 2(1+ωΛ)Ω♮Λ 1− ρ . (19)
(cid:18) c(cid:19)
equation, and substituting that in the above equation,
we obtain Deceleration parameter has been plotted versus the
q
H˙ 3 2ρ
H2 = −2 (1+ωm)Ω♮m+(1+ωΛ)Ω♮Λ 1− ρ , b2=0.5
(cid:16) (cid:17)(cid:18) c(cid:19)(15) 10 b2=1
b2=2
since, in the high energy regime, Ω♮ can be ignored
against to Ω♮, therefore one can estimmate 5
Λ
HH˙2 ≈ −23(1+ωΛ)Ω♮Λ 1− 2ρρ , (16) 0.2 0.4 0.6 0.8 1.0 Θ
(cid:18) c(cid:19)
(sfeoerrderfi.v(Sinhgeyakbhoiveeteaqlu.at2io0n10w).e hTahvee udsieffdereHρnc2ti=al Ω3eΩ♮Λq−♮Λu2a1-, bFeiegn. p2lot(teexdpvanerssiouns tehpeocdhim) eDnescioelnelreastsioqnuapnatriatymeθt=er,ρqΛ,,hfaosr
ρc
threedifferent value of interaction coupling constant.
5
q By defining the energy density parameters in the form
of
b2=0.5
10
b2=1 ρ
Ω = i , (20)
b2=2 i 3H2
5 where i refer to matter and dark energy. One can
rearrange the standard Friedmann equation as 1 =
Ωm+ΩΛ, from this relation we find out the maximum
x
0.2 0.4 0.6 0.8 1.0 valueofΩ isone. Now,NADE istakenasdarkenergy
i
component with interaction with matter, so from con-
servationequation relatedto the dark energy and with
Fig. 3 (contractionepoch)Deceleration parameter,q,has the help of above Friedmann equation, one obtain ωΛ
been plotted versus the dimensionless quantity x = 1−θ, as
for threedifferent value of interaction coupling constant.
2 √ΩΛ 2
ωΛ = 1+ 3b (1+r), (21)
dimensionless quantity θ = ρΛ for expansion and con- − 3a n −
ρc
tractionepochofuniverseevolutioninFig.2. andFig.3.
maximum value of dark energy density parameter,
respectively (with attentionto this fact that ωΛ stands namelyΩΛ isoneandalsowhenthescalefactorislarge
in phantom range). In the plots three different val-
enough to make the second term on the right hand of
ues for b are taken as b2 = 0.5,1,2; which displays
above relation small in which ωΛ stands in phantom
the strangeofinteractionbetweenmatter anddark en-
range. The sort of dark energy as a component of uni-
ergy. Fig.2. shows that q has negative values that it
versefluidhasbeendetermined,sowewanttobeaware
means the universeis inanaccelerationphase andget-
about the evolution of universe in this stage.
ting larger. Since there is a positive value for Hub-
The deceleration parameter is expressed as
ble parameter, Ω♮ increases too. By passing time and
Λ
increasing dark energy density, q approaches to zero,
3
thereforetheuniversemovesfromaccelerationphaseto q = 1+ (1+ωm)Ωm+(1+ωΛ)ΩΛ , (22)
deceleration phase. However since we are in expansion − 2
(cid:16) (cid:17)
epoch of universe evolution, Ω♮Λ is still getting larger so in the earlier time when ΩΛ converge to zero (and
which causes q grows rapidly (and as we expect larger Ω converge to one) the deceleration parameter is as
m
valuesofinteractioncouplingconstantmakesq togrow q 1 for ω = 1, and q 1 for ω =0 that indicates
m 3 2 m
up faster). Reaching energy density to critical energy a≈decelerationuniverse. W≈hereasinthe latetime when
density, the universe is forced to stop expansion and ΩΛ approachesto one , the decelerationparameter has
change its evolution direction and start to contract in negative sign that introduces an acceleration universe.
turnaroundpoint. Incontractionepoch,theuniverseis
getting smaller more and more with a large value of q
atthebeginningofcontraction,asithasbeenshownin
Fig.3. The Hubble parameter has negative values and 3 Validity of second law of thermodynamics
Ω♮ decreasesthereforedarkenergydensityisdiluted. q
Λ
again takes negative values, however the universe con- So far, we investigated whether NADE can produce a
tinue to contract and decrease more and more, until phantomdarkenergyforcyclicuniverse. Now,itcanbe
it moves to low energy regime and dark energy dilute valuable to study validity of generalized second law of
enough in which matter and radiation become domi- thermodynamic in the model and consider which con-
nant. dition can satisfy generalized second law. Since loop
Up to now, from above discussion in high energy quantum gravity correction enters in Friedmann equa-
regime, we found out that NADE can satisfy a phan- tion which plays an important role in the evolution of
tom fluid for cyclic scenario. However this is not at all universe,especiallyinhighenergyregime,itisnotsuit-
our universe,there is alsoregime relatedto the low en- able one straightforward extends the standard treat-
ergy, where ρ ρc. At this regime we have the usual mentofthermodynamic. Forthis reasonwe use the re-
≪
Friedmann equation for universe, namely sultof(Li et al. 2009)whichthermodynamichasbeen
investigatedinloopquantumcosmologythere. Theau-
2
3H =ρ=ρm+ρΛ. thors of (Li et al. 2009) have performed their work by
6
taking a dynamical apparent horizon as universe hori-
zon. It has been demonstrated that one can have R < 3(1+ωΛ)(1−2α)Ω♮.
A − T Λ
dE =TdS+WdV. (23) I
However,ifα> 1,thetotalentropyvariationisalways
2
However note that, here E = ρ V = ρ 1 ρ V positive.
whichρ iseffectiveenergydenseiftyf andW(cid:16)= −1(ρρc(cid:17) Inlowenergydensity, Ω♮Λ tends to ΩΛ where0<ΩΛ <
P ) (ienffwhich P = P 1 2ρ ρ2 is 2effeecftfiv−e 1, also α approaches to zero, so one arrives at
eff eff − ρc − ρ2c
pressure). The relation can(cid:16)be very(cid:17)useful in this step
3β 4
of the work. Like (Li et al. 2009), we take apparent S˙ =3πβ R + (28)
tot A
horizon (cid:18) TI − TI(cid:19)
1
RA = H. where β = (1+ωm)Ωm +(1+ωΛ)ΩΛ . In radiation
Fromconsideringthevalidityofgeneralizedsecondlaw, andmatter(cid:16)dominant,whereΩΛ converg(cid:17)estozero,β is
positive and we should have
we should sum variationof entropy for both of horizon
and fluid inside the horizon. The horizon entropy
4 3β
T > −
I
R
A
S˙ =2πR R˙ (24)
h A A to generalized second law be valid. In radiation and
matter dominant ω is equal to 1 and 0 respectively.
For the relation (23), the entropy variation of fluid in- m 3
side the horizon can be acquired as This expresses that the right term is positive. Passing
time, the darkenergyeventually dominates andΩΛ in-
creases to one, so β < 0. The generalized second law
(ρ +P )
T S˙ =Vρ˙ + eff eff V˙ (25) leads us to the below condition for temperature
I I eff
2
4 3β
Note that, there is anequilibrium betweenthe temper- T −
I
≤ R
ature of horizon and matter inside the horizon. Us- A
ing modified Friedmann equation and conservation re- andsince β is negativewe canhavea positivevalue for
lations and also with the help of the definition of hori- temperature. So the generalizedsecondlaw of thermo-
zon and energy density parameters, the total entropy dynamicscanbevalid,anditcanbe aconfirmationfor
variation is resulted as the model.
2ρ
S˙tot = 3π (1+ωm)Ωm+(1+ωΛ)Ω♮Λ 1− ρ
(cid:16) (cid:17)(cid:18) c(cid:19) 4 Conclusion
3
RA+ (1+ωm)Ωm+(1+ωΛ)ΩΛ
T
( I During this work, in brief, it was shown that both of
(cid:16) (cid:17)
ADE and NADE, without interaction with matter in
2ρ 4
1 (26) bothhighandlowenergyregime,cannotsatisfyphan-
×(cid:18) − ρc(cid:19)− TI) tom dark energy which it is a necessary condition for
cyclic universe scenario. So, as it was the main goal of
Now, we turn our attention to the high energy regime
the work, an interaction between the fluid components
where Ω can be ignored against to the larg value of
m
Ω♮. Define α = ρ in which 0 < α < 1, then the total of the universe was supposed. At first step, we turned
Λ ρc ourattentiontothehighenergyregime,andwerealized
entropy variation is reorganized as
that , in ADE case for providing a phantom fluid there
mustbealargevalueforcouplingconstantbwhichitis
S˙tot = 3π(1+ωΛ)(1 2α)Ω♮Λ not compatible with the results of previous works, but
−
NADE canproduce a phantom fluid for cyclic universe
3 4
×(RA+ TI(1+ωΛ)(1−2α)Ω♮Λ− TI),(27) hwahvielesdaasrakξewnehregryedξensit2y. paIrtamweatsere,xpnlaamineeldy Ωth♮Λa,t,bien-
≤
where (1+ωΛ) < 0. If α < 12, for making valid the eexnperagnysioinncrpehaassees,beyqudaetciorenasoifngstarte=paρrma.meHteorwoefvedrarink
second law, there should be ρΛ
7
contraction phase, the quantity r increases, so ωΛ de-
creases. Incontractionphasewithattentiontothisfact
thatdarkenergystandsinphantomrange,smallerval-
ues of ωΛ express that dark energy density decreases
fasterandletuniversetoentermatterdominantphase.
Moreover larger value of interaction coupling constant
produce smaller valus for ωΛ.
Considering differential equation of Ω♮ exhibited that
Λ
after passing the universe from turnaround point (in
contracting phase), Ω♮ decreases and after a while the
Λ
universe is putted in low energy regime, then matter
andradiationarebeallowedtobe dominate. Bycom-
puting deceleration parameter, it was found out that
there is negative values forq whichshowsthat the uni-
verse is in an accelerated expansion phase. Increasing
dark energy density enters q in positive area and it is
made larger. Then, the universe moves from an ac-
celerated expansion phase to a deceleration phase and
it is slowing down until it stops expansion and starts
contraction in turnaround point. So it is realized that
the model is able to create a deceleration phase and
start contractionafter an accelerated expansion phase.
In next step, we investigated the low energy regime of
universe where it can be possible to ignore loop quan-
tum correction in Friedmann relation and go back to
usual form. Investigation shew us that, in low energy
regime, ωΛ stands inphantomrange. Also,considering
deceleration parameter displayed that universe expan-
sion decelerates in both of matter and radiation domi-
nant, where ΩΛ is small enough, and accelerates by in-
creasingΩΛtowardone. Inthelastpartofthework,we
studied the validity of generalized second law of ther-
modynamic in both case of high and low energy. In
high energy it was determined that when α = ρ > 1
ρc 2
the total entropy variation is always positive. At low
energy, we shew that to make valid the law, a bound
for temperature is imposed.
5 Acknowledgments
The work of Kh. Saaidi have been supported finan-
cially by University of Kurdistan, Sanandaj, Iran, and
he would like thank to the University of Kurdistan for
supporting him in sabbatical period.
8
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