Interacting new agegraphic dark energy in a cyclic universe

Full text

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Astrophys Space Sci (2012) 338:355–361
DOI 10.1007/s10509-011-0944-y
ORIGINAL ARTICLE
Interacting new agegraphic dark energy in a cyclic universe
K. Saaidi ·H. Sheikhahmadi ·A.H. Mohammadi
Received: 4 November 2011 / Accepted: 21 November 2011 / Published online: 6 December 2011
© Springer Science+Business Media B.V. 2011
Abstract The main goal of this work is investigation of
NADE in the cyclic universe scenario. Since, cyclic uni-
verse is explained by a phantom phase (ω < −1), it is shown
whenthereisnointeractionbetweenmatteranddarkenergy,
ADE and NADE do not produce a phantom phase, then can
not describe cyclic universe. Therefore, we study interacting
models of ADE and NADE in the modifiedFriedmann equa-
tion.We findoutthat,in thehighenergyregime,whichitis a
necessary part of cyclic universe evolution, only NADE can
describe this phantom phase era for cyclic universe. Consid-
ering deceleration parameter tells us that the universe has a
deceleration phase after an acceleration phase, and NADE is
able to produce a cyclic universe. Also it is found valuable
to study generalized second law of thermodynamics. Since
the loop quantum correction is taken account in high energy
regime, it may not be suitable to use standard treatment of
thermodynamics, so we turn our attention to the result of Li
et al. (Adv. High Energy Phys. 2009: 905705, 2009), which
the authors have studied thermodynamics in loop quantum
gravity, and we show that which condition can satisfy gen-
eralized second law of thermodynamics.
K. Saaidi (?) · H. Sheikhahmadi
Department of Physics, Faculty of Science,
University of Kurdistan, Sanandaj, Iran
e-mail: ksaaidi@uok.ac
H. Sheikhahmadi
e-mail: h.sh.ahmadi@uok.ac.ir
K. Saaidi
Department of Physics, Kansas State University,
116 Cardwell Hall, Manhattan, KS 66506, USA
e-mail: ksaaidi@phys.ksu.edu
A.H. Mohammadi
Islamic Azad University, Evaz Branch, Evaz, Fars Province, Iran
e-mail: abolhassanm@gmail.com
Keywords Cyclic universe · Low energy regime · High
energy regime · Agegraphic dark energy
1 Introduction
Cosmological and astronomical observations such as super-
novae type Ia observational data (Perlmutter et al. 1997;
Reiss et al. 1998; de Bernardis et al. 2000) and Wilkonson
Microwave Anisotropic Probe (WMAP) (Astier et al. 2006;
Perlmutter et al. 1999; Peiris et al. 2003; Riess et al. 2007)
imply that the universe is undergoing a period of accelerated
expansion. Since normal matter can not give rise to acceler-
ated expansion of the universe, Scientists came up with a
solution which expresses that this expansion is a result of
an ambiguous fluid named Dark Energy (Bridle et al. 2003;
Koght et al. 2003).
The theoretical and experimental analysis suggest that
the universe consist of 73% dark energy, 23% cold dark
matter (CDM), and remnant matter is baryons (Fujii and
Maeda 2003; Spergel et al. 2003). Unfortunately the na-
ture and origin of dark energy are ambiguous up to now,
but people have proposed some candidates to describe dark
energy. Amongst the various candidates of dark energy to
describe accelerated expansion of the universe, cosmolog-
ical constant (vacuum energy), ?, with equation of state
(EoS) ω = −1 is located in central position. However, as
it is well known, the cosmological constant proposal has
two famous problems, fine-tuning problem and the cos-
mic coincidence problem (Weinberg 1989; Steinhardt 1997;
Carroll 2001; Peebles and Ratra 2003). Some other of dark
energy models suggest that dark energy component can treat
as scalar field with dynamical EoS. In this scenario the
evolution of the field is very slow, so that kinetic energy
density is less than the potential energy density, and this
give us a negative pressure, responsible to the cosmic ac-

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356Astrophys Space Sci (2012) 338:355–361
celeration (Frieman et al. 1995; Turner and White 1997;
Caldwell et al. 1998). Some of scalar field models are as
chameleon field (Khoury and Weltman 2004; Mota and
Barrow 2004), quintessence (Q-field) (Liddle and Scherrer
1999; Peebles and Ratra 1988), and phantom. In the phan-
tom field scenario, the parameter of EoS is as ω < −1, due
to existence of a negative kinetic energy density of scalar
field. It is well known that the phantom dark energy model
suffers from two kind of problems, “Big Bang” singularity
and “Big Rip” singularity, where big bang is related to ini-
tial epoch of universe and big rip is related to a finite fu-
ture singularity. Since the space-time singularities are in-
vidious for theorists, models that avoid these singularities
are attractive. One of these models names cyclic universe
which have received huge attention (Brown et al. 2008;
Baum and Frampton 2007; Sun 2008). Presence of ρ2term
with a negative sign in Friedmann equation (which is used
at studying cyclic universe) is an effective way to eliminate
these singularities. In the cyclic universe scenario, where is
based on the phantom dark energy model, the universe oscil-
latesthrougha series ofexpansionandcontraction.Universe
in this scenario has a very high energy density at beginning
an ending of the expansion, so quantum gravity can not be
ignored in these stages (Ashtekar et al. 2006, 2006a). This
evolution can be result from the modified Friedmann equa-
tion in the loop quantum cosmology (LQC). In LQC, the
Friedmann equation has been modified to
ρ
3m2p
ρc
where H is the Hubble parameter, m2
mass (m2
p=
energy density, ρc is the critical energy density as ρc=
4√3γ−3m4
is the dimensionless Barbero-Immirizi parameter (Ashtekar
2006b). We notice, this correction can solve the singu-
larity problems as follow, when the total energy density
reaches the critical density, the universe reaches the max-
imum at the end of expansion that is called “turnaround
point”, and universe arrives at smallest size at the end of
contraction, then we have a bounce there (Boyle et al. 2004;
Khoury et al. 2001; Steinhardt and Turok 2002; Steinhardt
and Turok 2006). We emphasize the idea of cyclic universe
was first introduced by Tolman (1931, 1934).
An interesting attempt for probing the nature of dark
energy, in the framework of quantum gravity, is the holo-
graphic dark energy (HDE). In the HDE model, dark energy
is a dynamical evolving vacuum energy density that can sat-
isfy the phantom behavior. Authors of Zhang et al. (2007)
have investigated the cyclic universe by HDE, and some in-
teresting work about HDE have been done (Li 2004; Setare
2007a, 2007b; Karami and Fehri 2010; Cohen et al. 1999;
Enqvist and Sloth 2004; Dutta et al. 2010). Another attrac-
tive model to describe the nature of dark energy, within the
H2=
?
1−ρ
?
,
(1)
pis the reduced Planck
1
8πG= 2.44 × 1018GeV), ρ is the total of
p= 0.82ρp, where ρp= 2.22×1076GeV, and γ
framework of a fundamental theory originating from some
considerations of the feature of quantum gravity theory, is
calledagegraphicdarkenergy(ADE)model(Cai2007).The
ADE assumes that the dark energy comes from the universe
components fluctuation such as space-time and matter fluc-
tuation (for further discussion we refer the reader to Kim
et al. (2008a, 2008b), Nozari and Azizi (2009), Karami et al.
(2010), Setare (2010). In this model, the age of universe is
taken as the length measure instead of the horizon distance,
therefore the causality problem which appears in the HDE
model can be avoided. The ADE model suffers from the dif-
ficulty to describe the matter dominant epoch. The authors
of Wei and Cai (2008a) have introduced a new mechanism
to overcome that problem, which it is called new agegraphic
dark energy (NADE) model, and its energy density is de-
fined by ρ?= 3n2m2
rameterized some uncertainties and η is conformal time and
can be written as
?a
where a is scale factor and H is well-known as Hubble pa-
rameter. Cyclic model of universe, due to avoiding singu-
larity, and NADE, due to estimating a good approximate of
dark energy value and solving causality problem of HDE,
have received huge interest. In previous works, like Wei and
Cai (2008b), it was only mentioned that interaction NADE
can produce phantom, however it was not explained that if
NADE can produce cyclic universe and how the quantities
like dark energy density parameter, equation of state param-
eter, and deceleration parameter behave in cyclic universe
by taking NADE as component of dark energy. In this work
we motivated to take NADE as dark energy and investigate
universe evolution in cyclic model. If it is able to stand in
phantom area, it may produce cyclic universe, however it
is not all of story. Although, presence of this type of dark
energy displays an accelerated universe, but the most im-
portant thing in cyclic universe is that, after a while, the uni-
verse should enter in a deceleration area and recontract in
turnaround point.
pη−2, where 3n2is introduced to pa-
η =
0
da
a2H,
(2)
2 High and low energy
In this step, we focus on high energy regime. Certainly the
quadratic term of energy density in the Friedmann equation
can not be ignored, this term can play a very impressive role
in the evolution of universe. The modified Friedmann equa-
tion is as
3H2= ρ
ρc
where ρ is a combination of dark energy density, ρ?, and
matter density, ρm. For dark energy dominant in high energy
regime, the above relation can be rewritten as
?
?
1−ρ
?
,
(3)
3H2≈ ρ?
1−ρ?
ρc
?
,
(4)

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Astrophys Space Sci (2012) 338:355–361 357
so, according to this relation the parameter of dark energy
may be estimated as
ρ?
3H2≈
??
?=
ρc
ρc−ρ?.
(5)
??
the universe approaches to the turnaround point. However
note that, since we have ρm, although it is very small, ??
can not be infinite. This small value of ρmdoes not allow ρ?
to reach the exact value of ρc, then the presence of ρmcan
prevent the infinite value of ??
If we assume there is no interaction between these two
components of the universe, conservation equation can be
written as
?is always larger than one, and it can be very large when
?
?.
˙ ρm+3H(1+ωm)ρm= 0
˙ ρ?+3H(1+ω?)ρ?= 0.
By taking ADE, which is defined as ρ?= 3n2T−2where T
denotes time, as dark energy component of universe, ω?is
obtained as
2
3n
Also, if we replace NADE instead of ADE, we arrive
at same result but only scale factor, a, should be added in
the denominator of second term on the right hand of rela-
tion (7). Since the latest term, in both case, is always posi-
tive ω?never can be smaller than −1. Since phantom type
of dark energy is requisite for cyclic universe (phantom en-
ergy density is getting larger with increasing scale factor so
the total energy density reaches the critical energy density
in turnaround point and universe begins contraction), select-
ing ADE and NADE, without interaction, as a component of
dark energy in cyclic universe is not suitable (Wei and Cai
2008b).
Assumption of interaction between ρmand ρ?may solve
above problem. By including interaction, the conservation
equations are as
(6)
ω?= −1+
?
??
?.
(7)
˙ ρm+3H(1+ωm)ρm= Q
(8)
˙ ρ?+3H(1+ω?)ρ?= −Q,
where Q indicates interaction. Q is taken as Q = ?ρ?, with
? > 0, which means there is transfer of energy from ρ?to
ρm (Sheykhi et al. 2010). We take Q as Q = 3b2H(1 +
r)ρ?, where r =ρm
energy gives us an ω?as
2
3n
??
?is always larger than one, and it becomes very large at
turnaround point, so for having ω?< −1, we should have
b2(1 + r) >
(9)
ρ?. Setting ADE as the component of dark
ω?= −1+
?
??
?−b2(1+r).
(10)
2
?
??
3n. This predicts a large value for coupling
?
constant b, while it is in contrast to the obtained value of
another papers such as Sheykhi et al. (2010).
Now, NADE is taken as ρ?which that is the main case
of this work. Because we use conformal time η instead of T ,
scale factor appear in relation (10), namely
ω?= −1+
2
3na
?
??
?−b2(1+r).
(11)
To have ω?< −1, the coupling constant b should obey fol-
lowing relation
b2>
2
3na
?
??
?,
(12)
here, r has been ignored because of ρm? ρ?. If ??
order of a2, we may obtain a convenient value for b in order
it could give us phantom dark energy in this regime.
In Fig. 1 ω?parameter has been plotted versus r =ρm
for three different value of interaction coupling constant. In
expansion phase, by passing time, the quantity r decreases
and makes ω?larger. Since it still stands in phantom range,
larger value of ω?shows that dark energy density grows
up slower; in contrast to the contraction phase. Also, having
stronger interaction coupling causes smaller value for ω?.
If ??
?be in order of aξ, where ξ ≤ 2, (12) can be valid
in good approximation. Now suppose ??
for instance ??
by taking the value of Sheykhi et al. (2010) for b, namely
b2= 0.25, we obtain the value of α as√α <3
Now, we want to obtain differential equation for ??
the NADE, ρ?is given as
?be in
ρ?,
?is in order of a2,
?= αa2. From (12) we have b2>2√α
3n, and
8n.
?. In
ρ?=3n2
η2.
(13)
Fig. 1 Equation of state parameter of dark energy has been plotted
versus the dimensionless quantity r =ρm
interaction coupling constant
ρ?, for three different value of

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358Astrophys Space Sci (2012) 338:355–361
From the definition of dark energy density parameter,
namely ??
?=
quired as
?
an
where prime denote derivative with respect to N = lna.
Taking the time derivative of modified Friedmann equation,
and substituting that in the above equation, we obtain
˙ H
2((1+ωm)??
ρ?
3H2. The differential equation for ??
?is ac-
??
?
?= −2??
?
?˙ H
H2+
??
?
?
,
(14)
H2=−3
m+(1+ω?)??
?)
?
1−2ρ
ρc
?
,
(15)
since, in the high energy regime, ??
to ??
?, therefore one can estimate
˙ H
2(1+ω?)??
mcan be ignored against
H2≈−3
?
?
1−2ρ
ρc
?
,
(16)
(for driving above equation we have usedH2
ρc=??
?−1
3??
?
2, see
Sheykhi et al. 2010). The differential equation, which gov-
erns the NADE evolution of universe in high energy regime,
can be attained as
??
−3b2(1+r)
??
?
?= −2??
?
??
?
na
(??
?1
?−1)
2??
?−1
??
(17)
with attention to (12), it is clearly shown that ??
is positive. So one can realize that when the universe is
growing up (decreasing) ˙??
it displays that dark energy density parameter is increasing
(decreasing) by passing time. By passing the universe from
turnaround point, ??
?decreases and puts the universe in low
energy regime.
Another useful cosmological parameter is deceleration
parameter which can tell us if the expansion of universe is
accelerating or not. Deceleration parameter is given by
˙ H
H2,
˙ H
H2term in the above relation, we arrive at
q = −1+3
?
= −1+3
Deceleration parameter has been plotted versus the di-
mensionless quantity θ =ρ?
tion epoch of universe evolution in Figs. 2 and 3 respec-
tively (with attention to this fact that ω?stands in phantom
?
?=1
H˙??
?
?is positive (negative) which
q = −1−
(18)
substituting
2((1+ωm)??
1−2ρ
ρc
m+(1+ω?)??
?)
×
?
2(1+ω?)??
?
?
1−2ρ
ρc
?
.
(19)
ρcfor expansion and contrac-
Fig. 2 (Expansion epoch) Deceleration parameter, q, has been plotted
versus the dimensionless quantity θ =ρ?
interaction coupling constant
ρc, for three different value of
Fig. 3 (Contraction epoch) Deceleration parameter, q, has been plot-
ted versus the dimensionless quantity x = 1 − θ, for three different
value of interaction coupling constant
range). In the plots three different values for b are taken as
b2= 0.5, 1, 2; which displays the strange of interaction be-
tween matter and dark energy. Figure 2 shows that q has
negative values that it means the universe is in an accelera-
tion phase and getting larger. Since there is a positive value
for Hubble parameter, ??
?increases too. By passing time
and increasing dark energy density, q approaches to zero,
therefore the universe moves from acceleration phase to de-
celeration phase. However since we are in expansion epoch
of universe evolution, ??
?is still getting larger which causes
q grows rapidly (and as we expect larger values of interac-
tion coupling constant makes q to grow up faster). Reach-
ing energy density to critical energy density, the universe
is forced to stop expansion and change its evolution direc-
tion and start to contract in turnaround point. In contraction
epoch, the universe is getting smaller more and more with
a large value of q at the beginning of contraction, as it has
been shown in Fig. 3. The Hubble parameter has negative
values and ??
?decreases therefore dark energy density is
diluted. q again takes negative values, however the universe
continue to contract and decrease more and more, until it

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Astrophys Space Sci (2012) 338:355–361 359
moves to low energy regime and dark energy dilute enough
in which matter and radiation become dominant.
Up to now, from above discussion in high energy regime,
we found out that NADE can satisfy a phantom fluid for
cyclic scenario. However this is not at all our universe, there
is also regime related to the low energy, where ρ ? ρc. At
this regime we have the usual Friedmann equation for uni-
verse, namely
3H2= ρ = ρm+ρ?.
By defining the energy density parameters in the form of
ρi
3H2,
where i refer to matter and dark energy. One can rearrange
the standard Friedmann equation as 1 = ?m+ ??, from
this relation we find out the maximum value of ?iis one.
Now, NADE is taken as dark energy component with inter-
action with matter, so from conservation equation related to
the dark energy and with the help of above Friedmann equa-
tion, one obtain ω?as
2
3an
maximum value of dark energy density parameter, namely
??is one and also when the scale factor is large enough
to make the second term on the right hand of above rela-
tion small in which ω?stands in phantom range. The sort
of dark energy as a component of universe fluid has been
determined, so we want to be aware about the evolution of
universe in this stage.
The deceleration parameter is expressed as
q = −1+3
so in the earlier time when ??converge to zero (and ?m
converge to one) the deceleration parameter is as q ≈ 1 for
ωm=1
universe. Whereas in the late time when ??approaches to
one , the deceleration parameter has negative sign that intro-
duces an acceleration universe.
?i=
(20)
ω?= −1+
√??
−3b2(1+r),
(21)
2((1+ωm)?m+(1+ω?)??),
(22)
3, and q ≈1
2for ωm= 0 that indicates a deceleration
3 Validity of second law of thermodynamics
So far, we investigated whether NADE can produce a phan-
tom dark energy for cyclic universe. Now, it can be valuable
to study validity of generalized second law of thermody-
namic in the model and consider which condition can satisfy
generalized second law. Since loop quantum gravity correc-
tion enters in Friedmann equation which plays an important
role in the evolution of universe, especially in high energy
regime, it is not suitable one straightforward extends the
standard treatment of thermodynamic. For this reason we
use the result of Li and Zhu (2009) which thermodynamic
has been investigated in loop quantum cosmology there. The
authors of Li and Zhu (2009) have performed their work by
taking a dynamical apparent horizon as universe horizon. It
has been demonstrated that one can have
dE = TdS +WdV.
However note that, here E = ρeffV = ρ(1 −
ρeff is effective energy density and W =1
(in which Peff= P(1−2ρ
relation can be very useful in this step of the work. Like (Li
and Zhu 2009), we take apparent horizon
(23)
ρ
ρc)V which
2(ρeff− Peff)
ρc)−ρ2
ρ2
cis effective pressure). The
RA=1
From considering the validity of generalized second law, we
should sum variation of entropy for both of horizon and fluid
inside the horizon. The horizon entropy
H.
˙Sh= 2πRA˙ RA.
For the relation (23), the entropy variation of fluid inside the
horizon can be acquired as
(24)
TI˙SI= V ˙ ρeff+(ρeff+Peff)
Notethat,thereisanequilibriumbetweenthetemperatureof
horizonandmatterinsidethehorizon.UsingmodifiedFried-
mann equation and conservation relations and also with the
help of the definition of horizon and energy density param-
eters, the total entropy variation is resulted as
2
˙V.
(25)
˙Stot= 3π((1+ωm)?m+(1+ω?)??
?
×
ρc
?)
?
1−2ρ
ρc
?
×
RA+3
?
TI((1+ωm)?m+(1+ω?)??)
1−2ρ
TI
?
−4
?
.
(26)
Now, we turn our attention to the high energy regime where
?mcan be ignored against to the large value of ??
α =ρ
is reorganized as
?. Define
ρcin which 0 < α < 1, then the total entropy variation
˙Stot= 3π(1+ω?)(1−2α)??
×
?
?
RA+3
TI(1+ω?)(1−2α)??
?−4
TI
?
,
(27)
where (1 + ω?) < 0. If α <1
law, there should be
2, for making valid the second
RA< −3(1+ω?)(1−2α)
TI
??
?.
However, if α >1
itive.
2, the total entropy variation is always pos-

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360Astrophys Space Sci (2012) 338:355–361
In low energy density, ??
1, also α approaches to zero, so one arrives at
?
where β = ((1 + ωm)?m+ (1 + ω?)??). In radiation and
matter dominant, where ??converges to zero, β is positive
and we should have
TI>4−3β
RA
to generalized second law be valid. In radiation and matter
dominant ωmis equalto1
that the right term is positive. Passing time, the dark energy
eventually dominates and ??increases to one, so β < 0.
The generalized second law leads us to the below condition
for temperature
?tends to ??where 0 < ??<
˙Stot= 3πβRA+3β
TI
−4
TI
?
(28)
3and 0 respectively.This expresses
TI≤4−3β
and since β is negative we can have a positive value for tem-
perature. So the generalized second law of thermodynamics
can be valid, and it can be a confirmation for the model.
RA
4 Conclusion
During this work, in brief, it was shown that both of ADE
and NADE, without interaction with matter in both high
and low energy regime, can not satisfy phantom dark en-
ergy which it is a necessary condition for cyclic universe
scenario. So, as it was the main goal of the work, an in-
teraction between the fluid components of the universe was
supposed. At first step, we turned our attention to the high
energy regime, and we realized that, in ADE case for pro-
viding a phantom fluid there must be a large value for cou-
plingconstant b whichitisnotcompatiblewiththeresultsof
previous works, but NADE can produce a phantom fluid for
cyclic universe while dark energy density parameter, namely
??
?, behaves as aξwhere ξ ≤ 2. It was explained that, in
expansion phase, equation of state parameter of dark en-
ergy increases by decreasing r =ρm
tion phase, the quantity r increases, so ω? decreases. In
contraction phase with attention to this fact that dark energy
stands in phantom range, smaller values of ω?express that
dark energy density decreases faster and let universe to enter
matter dominant phase. Moreover larger value of interaction
coupling constant produce smaller values for ω?.
Consideringdifferentialequationof ??
ter passing the universe from turnaround point (in contract-
ing phase), ??
?decreases and after a while the universe is
putted in low energy regime, then matter and radiation are
be allowed to be dominate. By computing deceleration pa-
rameter, it was found out that there is negative values for q
ρ?. However in contrac-
?exhibitedthataf-
which shows that the universe 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
accelerated 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 contraction after an ac-
celerated expansion phase. In next step, we investigated the
low energy regime of universe where it can be possible to ig-
nore loop quantum correction in Friedmann relation and go
back to usual form. Investigation shew us that, in low energy
regime, ω?stands in phantom range. Also, considering de-
celeration parameter displayed that universe expansion de-
celerates in both of matter and radiation dominant, where
??is small enough, and accelerates by increasing ??to-
ward one. In the last part of the work, we studied the validity
of generalized second law of thermodynamic in both case of
high and low energy. In high energy it was determined that
when α =ρ
itive. At low energy, we shew that to make valid the law, a
bound for temperature is imposed.
ρc>1
2the total entropy variation is always pos-
Acknowledgements
nancially by University of Kurdistan, Sanandaj, Iran, and he would like
thank to the University of Kurdistan for supporting him in sabbatical
period.
The work of Kh. Saaidi have been supported fi-
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