Table Of ContentEffect of an external interaction mechanism in solving
agegraphic dark energy problems
A. Aghamohammadia1 • Kh. Saaidib2 •
A. Mohammadic 3 • H. Sheikhahmadibd 4 •
4 T. Golanbarib5 • S. W. Rabieib6
1
0
2
n
a
J
8
1
Abstract Agegraphic dark energy(ADE) and New- Keywords Agegraphicdarkenergy,chameleonicscalar
]
h ADE models have been introduced as two candidates interaction, phantom crossing, coincidence problem,
p
for dark energy to explain the accelerated expansion classical non-stability, data fitting.
-
n phaseoftheUniverse. Inspiteofafewsuitablefeatures
e ofthesemodelssomestudieshaveshownthatthereare
g
severaldrawbacksin them. Thereforein this investiga- 1 Introduction
.
s tion a new version of ADE and New-ADE are studied
c
i whichcanimprovesuchdrawbackswhichappearinthe During two past decades, numerously observational
s
y ordinaryADE and New-ADE scenario. In fact we con- data, such as Supernovae type-Ia (SnIa) (Riess et al.
h sider an interacting model of scalar field with matter 1998; Perlmutter et al. 1999; Riess et al. 2004; Astier
p and after re-deriving some cosmological parameters of 2006), Cosmic Microwave Background (Peiris et al.
[
the model, we find out the best fit for the model. Ac-
2003; Bennett et al. 2003; Spergel et al. 2003) and
1 tually by finding the best fitting for free parameters of so on, faced scientists with this shocking fact that
v the model, we show that our theoretical results are in
the Universe is undergoing an accelerated expansion
8
a good agreement with observationaldata.
0 phase. Some people look for the source of this ac-
6 celeration in the geometrical part of the Hilbert-
2 A.Aghamohammadia Einstein action and have studied the modified grav-
2. Kh. Saaidib ity (Wand 1994;Nojiri et al.2007; Saaidi et al.2012e;
0 A.Mohammadic Saaidi and Aghamohammadi 2011b). On the other
4
H.Sheikhahmadibd hand some researcher believe that the Universe should
1
: T.Golanbarib be dominated by an ambiguous kind of fluid with neg-
v
S.W.Rabieib ativepressure,calleddarkenergy,whichis able to pro-
i
X aSanandaj Branch,IslamicAzadUniversity,Sanandaj, Iran. videsuchanexpansion. Thecosmologicalobservational
r bDepartment of Physics, Faculty of Science, University of Kur- data express that the universe includes 73% dark en-
a
distan,Sanandaj,Iran. ergy,23% dark matter and only 4% baryons,note that
cYoungResearcherClub,LarestanBranch,IslamicAzadUniver- thecontributionofradiationcouldbeignoredagainstto
sity,Lar,Iran. the other components of the Universe. The nature and
dYoung Researcher Club, Sanandaj Branch, Islamic Azad Uni- originofdark energyis unknownfor scientists and this
versity,Sanandaj,Iran.
factmake this kindof fluid to one of the mostpuzzling
[email protected] aspects of the Universe. Up to now, many proposals
[email protected] have been introduced to realized this phenomenon. It
[email protected] seemsthat the best candidate fordarkenergycouldbe
[email protected]@gmail.com cosmological constant which has an equation of state
[email protected] as ω = p/ρ = 1 (Einstein et al. 1917; Weinberg
[email protected] 1989; Sahni et al.−2000; Carroll 2001; Peebles et al.
2003; Padmanabhan 2003). However the cosmolog-
ical constant scenario suffers two well-known prob-
lems, namely the fine-tuning and cosmic coincidence
2
t
problems (Steinhardt 1997). Some other proposals for t = dt. One can realized that in ADE, the age of
0
dark energy are based on dynamical equation of state theuniverseistakenasthelengthmeasure(incontrast
R
which are realized by scalar field mechanism. The idea to the HDE which we take horizon as the length mea-
has provided large classes of scalar field dark energy sure), which this selection in turn solve the casuality
such as quintessence (Peebles et al. 1988; Riess et al. probleminHDE,howevertheoriginADEmodelsuffers
1988;Wetterich1988;Clemson et. al. 2009),k-essence fromaproblemto describethe matter dominantepoch
(Armendariz-Picon et al.2000;Chiba et al.2000;Armendaroifz-tPhiecoUnneitvearls.e. After that Wei and Cai introduced
2001), tachyon (Sen 2002), phantom (Caldwell 2002; a new mechanism as New-ADE which the model uses
Caldwell et al. 2003; Cline et al. 2004; Saidi et al. a conformal time in Friedmann-Lemaˆıtre-Robertson-
2012a), quintom (Elizalde et al. 2004; Nojiri et al. Walker (FLRW) space-time, known as η = dt/a, in-
2005;Anisimov et al.2005),andchameleon(Mota et al. stead of time t. Therefore the energy density is given
R
2004;Khoury et al.2004a,b;Saaidi et al.2011a;Wei et al. as ρ = 3m2n2/η2, where 3n2 is a numerical constant
p
2008b; Saaidi et al. 2012d). (Wei et al. 2008a).
In the last few years, the models which include an in- Since New-ADE belongs to a dynamical cosmological
teraction between matter and other components of the constant, therefore one need a dynamical framework
model received more attention. In fact there are two to consider it. One of such a framework is known
different kind of interactions. One of them is an inter- as Brans-Dicke gravity. Another dynamical frame-
nal interactions between the components of matter, in work which attains cosmologists attention is called
whichthefieldconservationrelationissatisfied. Inthis chameleon model that first introduced by (Mota et al.
kind, a term which describes this interaction is manu- 2004) and (Khoury et al. 2004a,b). In this model, it is
ally added to the model. The other one is an external assumedthatthereisascalarfieldwhichnon-minimally
interaction term between scalar field and matter. In is coupled to matter. The coupling causes the scalar
this case the every conservation equation is modified. field gets an effective mass which depends on the local
One type of the second kind is known as chameleon matter density; indeed chameleon scenario is a way to
mechanism. When a scalar field interact with mat- get an effective mass for light scalar field via self in-
ter (visible and invisible matter) through gravity or teraction and interaction with matter. On the other
directly, produces a fifth force on the matter which hand, due to this coupling, the action and field equa-
mayviolatethe weakequivalenceprinciple (WEP)and tion is modified. Other important result is that the
creates a non-geodesic motion (Amendola. 2000,2004; energyconservationequationisgeneralizedaswell. An
Bean et al. 2001). There are some particular ways to interaction term appears in the right hand side of the
trapping the fifth force effect. Some authors believe relation which changes the behavior of matter density
that the scalar field can be coupled differently to visi- and equation of state parameter.
ble and invisible matter of the Universe. So according Both ADE and HDE have the same origin, however it
to this, for suppressing the effects of fifth force, they is argued that ADE has a different IR-cutoff, and dif-
admitthatthescalarfieldcouplesmerelywiththeinvis- ferent IR-cutoff brings different results. In this work
ible matter (Caldera et al. 2009; Valiviita et al. 2010). wewouldliketoconsiderADEandNew-ADE inanin-
Chameleon scenario is another mechanism to circum- teracting scenario. Therefore the effective dark energy
vent this force and solve the WEP problem, refer to in the model is taken equal to the energy density of
(Saaidi 2012b) and references therein for more details. ADE. The generalized conservation relation results in
Another proposal to probe the nature of dark energy a equation of state parameter which has this ability to
is ADE model (Cai2007). Quantummechanics in gen- cross the phantom divide.
eral relativity results in the uncertainty relation called The paper has been planned in the following form: In
Kar`olyhazyrelation. Inthisscenarioitisassumedthat Sec.2wederivethegeneralformoftheevolutionequa-
dark energy comes from both space-time and matter tions and energy conservation equation; In Sec.3 the
field fluctuations. Kar`olyhazy et. al. (Karolyhazy effective dark energy is taken equalto the ADE energy
1966; Karolyhazy et al. 1982, 1986) argued that the density and the behavior of dark energy density pa-
distance t cannot be known better accuracy than rameter, equation of state parameter, sound speed are
δt=βtp23t13 if we take β as a dimension-lessconstant of obtained. In Sec.4 we take New-ADE as the effective
orderunityandt asPlancktime. Followingthiswork, dark energy and we study the behavior of cosmologi-
p
Maziashvili (Maziashvili 2007a,b) deduced that energy cal parameter of the model. In Sec.5 we fit the data
density of metric fluctuation of Minkowski space-time for ADE mechanism and find out the best fit for free
is given by ρ 1/t2t2 m2/t2, where the quantity parameters of the model and Sec.6 is devoted to the
d ∼ p ∼ p
t denotes the age of the Universe and is expressed as conclusion and discussion.
3
2 Framework On the other hand, taking variation of the action with
respecttothescalarfieldφleadstothefollowingequa-
We consider the following action tion for scalar field
S = d4x√ g 1R 1∂µφ∂µφ V(φ)+f(φ)Lm , (1) φ¨+3Hφ˙ =−V′(φ)+f′(φ)Lm. (7)
− 2 −2 −
Z (cid:16) (cid:17) Theprimedenotesderivativewithrespecttoscalarfield
where R is Ricci scalar, G is Newtonian gravitational and dot denotes derivative with respect to cosmic time
constant and we take 8πG = 1. φ is the scalar field t. One should specify the matter Lagrangian to more
with a potential V(φ) which has a non-minimal cou- clarifytheaboveequation. Basedon(Bicak et al.1997;
pling with matter sector. The coupling is described in Brown et al. 1993; Brown 1993; Sotiriou et al. 2008;
thelasttermoftheaction,whereL istheLagrangian Harko 2010), the Lagrangian of perfect fluid has two
m
density ofmatter andf(φ) isananalyticfunctionofφ. well-known definition as L1 = p and L2 = ρ, for
m m −
To getthe fieldequations one shouldtakes variation thecasewhenthereisnoanyinteractionbetweenmat-
oftheactionwithrespecttotheindependentvariables. ter and other components of the model. But in this
Taking variation of action with respect to the metric studywherethereisaninteractionbetweenmatterand
leads to the Einstein field equation as scalar field, the Lagrangian density degeneracy is bro-
ken. This means that the Lagrangian density L1 = p
m
and L2 = ρ has different results, so that according
G =f(φ)T + φ φ 1g ( φ)2 g V(φ), to (Samaidi 2−012b) we pick out Lm as p which describe
µν µν µ ν µν µν
∇ ∇ − 2 ∇ − a geodesic motion for perfect fluid.
h i (2) The next important equation is energy conservation
equation. Using Eqs.(5), (6) and (7) one could obtain
where T is matter energy-momentumtensor which is
µν the energy conservation equation for matter as
defined as
2 δ √−gLm d f(φ)ρ +3Hf(φ)(ρ+p)= pf˙(φ). (8)
Tµν = √− g (cid:16) δgµν (cid:17). (3) dt −
(cid:16) (cid:17)
−
Asweexpectedtherelationisnotconservedthatisdue
Herewemerelyassumethatmatterisacombinationof to the interaction of matter and scalar field. Since the
(dark)matter and dark energy as a perfect fluid which energydensityisacombinationofdarkenergyandcold
has a well-known energy-momentum tensor as Tµν = darkmatter,namelyρ=ρm+ρΛ andp=pm+pΛ,the
(ρ +p)uµuν +pgµν, in which ρ = ρm +ρΛ and p = conservation relation could be divided as following
p +p .
m Λ
Moreover to derive evolution equations a metric
d
shouldbesupposedtodescribegeometryofspace-time, f(φ)ρΛ +3Hf(φ)(ρΛ+pΛ)= pΛf˙(φ), (9)
dt −
sowetakeanspatiallyflatcase(k =0)ofFLRWmetric (cid:16) (cid:17)
which is defined as
d
f(φ)ρ +3Hf(φ)(ρ +p )= p f˙(φ). (10)
m m m m
dt −
ds2 = dt2+a2(t) dx2+dy2+z2 , (4) Th(cid:16)e next en(cid:17)ergy density is related to the scalar field
−
(cid:16) (cid:17) densitythatis indicatedasρφ. Accordingtothe Fried-
where a(t) stands for scale factor of the Universe, t is mann equations, the energy density and pressure of
the cosmic time parameter. By substituting the metric scalar field could be defined as
and the definition of matter energy-momentum tensor
in the Eq.(2), the Friedmann equation is given as fol-
1
lows ρφ = φ˙2+V(φ), (11)
2
1
p = φ˙2 V(φ), (12)
φ
1 2 −
3H2 =f(φ)ρ+ φ˙2+V(φ), (5)
2 some manipulation lead one to a conservation relation
for scalar field as
1
2H˙ +3H2 = f(φ)p φ˙2+V(φ). (6)
− − 2 ρ˙ +3H(ρ +p )= p +p f˙(φ). (13)
φ φ φ m Λ
h i
4
In this section we have obtained the required equation tion of state parameter of effective dark energy could
for our analysis. In the following sections we will con- be acquired as
sider the proposed model.
21 Ω f˙(φ)
eΛ
ω = 1+ + γr 1 , (19)
eΛ
3 ADE and scalar field − 3nsf(φ) 3Hf(φ) −
(cid:16) (cid:17)
where r is ratio of cold dark matter and effective dark
In comparison to the ordinary Friedmann equation,
energy, namely r = ρ /ρ = Ω /Ω . The interac-
we define an effective dark energy as combination of m eΛ m eΛ
tionterminthismodelgeneratesanextratermforω ,
dark energy ρ and scalar field density as ρ = ρ + eΛ
Λ eΛ Λ
which can justify the phantom divide line crossing. By
ρ /f(φ), which ρ denotes effective dark energy. So,
φ eΛ
definition an ansatz for ω , it can be considered as
the Friedmann equation is rewritten as eΛ
ω +1=ω +ω (1+z)β. (20)
eΛ 0 1
3H2 =f(φ) ρ +ρ . (14)
m eΛ 3.1 Data fitting
(cid:16) (cid:17)
Anuseful parameterinthis study is energydensity pa-
In this section we want to fit the free parameters for
rameterΩ. HereΩ andΩ respectivelywillbetaken
eΛ m ADE in an external scalar field interaction model. To
equal to Ω = f(φ)ρ /ρ and Ω = f(φ)ρ /ρ , in
eΛ eΛ c m m c achieve this purpose we use the 557 Union II sample
which ρ is the critical energy density which is defined
c database of SnIa, and we assume ρ = ρ +
asρ =3H2. Asaresult,fromtheFriedmannequation m radiation
c ρ +ρ . In this work the effect of coef-
baryon darkmatter
we have Ω +Ω =1.
eΛ m ficient f(φ) which is appeared in Eq.(5), leads us to
To obtain energy conservation equations for effective
introduce some effective density parameters such as
dark energy, using Eqs.(9) and (13), one can achieve
the following results Ω =f(φ)ρ /ρ , Ω =f(φ)ρ /ρ .
eΛ Λ c em m c
d
f(φ)ρ +3Hf(φ)(1+ω )ρ =γρ f˙(φ), (15) Therefore in this case the Friemann equation is as
eΛ eΛ eΛ m
dt
(cid:16) (cid:17) 3H2 =f(φ) ρ +ρ . (21)
m eΛ
d (cid:16) (cid:17)
f(φ)ρm +3Hf(φ)(1+γ)ρm = γρmf˙(φ), (16) Combining Eqs.(15)-(18), give
dt −
(cid:16) (cid:17)
so that the effective pressure of dark energy is defined ρ0 3n2
3H2 =f(φ) em + , (22)
as peΛ = pΛ+pφ/f(φ), and one has the effective dark a3(1+γ)f(1+γ)(φ) T2
energy equation of state parameter as ω = p /ρ . (cid:16) (cid:17)
eΛ eΛ eΛ
Also γ is the matter equationofstate parameterwhich where ρ0 is the effective energy density of matter at
em
isdefinedasγ =p /ρ . Forγ =constant,integrating thepresenttimeandT isthecosmictime. Whereasthe
m m
ofEq.(16)resultsinthefollowingrelationforcolddark 557 Union II sample database have collected from red
matter energy density as shift parameter to various SnIa , therefore we rewrite
E =H/H versus z as
0
ρ0
ρ = em , (17)
m a3(1+γ)f(1+γ)(φ) E2 = r0(1+z)3+(1+z)3ω0exp 3ωβ1[(1+z)β −1] .
where ρ0 = f(1+γ)(φ)ρ0 . In this step, we suppose r0+1n o
em 0 m
thattheeffectivedarkenergycouldbedefinedasADE, (23)
in other word we assume that
To achieve the best-fit for free parameters we should
3n2 compare the theoretical distance modulus, µ , with
th
ρ ρ = , (18)
eΛ ≡ ADE T2 observed, µob, of supernovae. In this model we have
threefreeparametersω , ω andβ. Thedistancemod-
where n is a numerical constant and T is cosmic time 0 1
ulus is defined as
and therefore Ω is obtained as Ω =f(φ)n2/H2T2.
eΛ eΛ
Taking this assumption and using Eq.(15), the equa- µ = 5log [D (z;ω ;ω ;β)]+µ , (24)
th 10 L 0 1 0
µ = µ˜ +µ , (25)
th th 0
(26)
5
where D (z;ω ;ω ;β) is given by with the data since χ2/dof =1.
L 0 1
Fig.2showcontourplotsforthefreeparametersω and
z 1 1
D (z;ω ;ω ;β)=(1+z) dx. (27) β, it is shown that the best value for these parameters
L 0 1
Z0 E(x;ω0;ω1;β) are 1.86 < ω1 < 1.62 and 2.27 < β < 0.73 in
− − − −
which for stability condition c2 >0, we have taken the
Tocomparingµ withµ weneedtoobtainχ2 which
th ob sn interface between greenand yellow sector, ω = 1.68.
is defined by 1 −
557 [µ (z ) µ (z )]2
χ2 (ω ;ω ;β)= th i − ob i . (28) 3
sn 0 1 σ2
i=1 i
X
2
557 (µ˜ (z ) µ (z ))2 1
th i ob i
A = − , (29)
σ2
Xi=1 i Β 0
557
µ˜ (z ) µ (z )
th i ob i
B = σ−2 , (30) -1
i=1 i
X
557 1 -2
C = , (31)
σ2
i=1 i -3
X
χ2 = A+2µ B+µ2C, (32) -2.5 -2.0 -1.5 -1.0 -0.5 0.0
0 0
Ω
(33) 1
Fig. 2 Contour plots for the free parameters ω1 and β,
showsthatthebestvaluefortheseparametersare−1.86<
In Fig.1, we show a comparison between theoretical ω1<−1.62 and −2.27<β <−0.73.
distancemodulusandobserveddistancemodulusofsu-
pernovaedata. Thered-solidlineindicatesthetheoret-
The evolution of effective dark energy parameter, ω ,
icalvalueofdistancemodulus,µ ,forthebestvalueof eΛ
th
versus z, for ω = 1.1, ω = 1.68 and β = 2.25 in
free parameters ω1 = 1.65, ω0 = 1.1 and β = 2.25. 0 1 − −
− − theFig.3havebeenshown,Itexpressesthatbygrowth
A minimization of this expression leads to
z the parameter get into the phantom phase.
Μ
44 -0.5
42
40 Le -1.0
Ω
38
-1.5
36
34
z -2.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Fig. 1 The observed distance modulus of supernovae Z
(points) and the theoretical predicted distance modulus
(red-solid line) in thecontext of ADEmodel. Fig. 3 The plot shows the evolution of effective dark
energy parameter, ωeΛ, versus z, for ω0 = 1.1, ω1 = −1.68
and β=−2.25.
χ2 (ω = 1.1; ω = 1.65; β = 2.25), (34)
snmin 0 1 − − where ω , ω and β are free parameters of the model
B2 0 1
χ2 = A =542.75, (35) which obtained from data fitting (Campo et al. 2011).
min − C
By using Eqs.(15), (16) and (18) one can obtain
B
µ = =43.1089. (36)
0
−C (z+1)β+2
f(φ)=f t2a−3ω0exp 3ω . (37)
0 1
where implies χ2 /dof= χ2 /dof = 0.981(dof = β+2
sn snmin h i
553). This shows that this model is clearly consistent
6
Here f is the constant of integration. Whereas Itisclearlyseenthatforω =1.1, ω = 1.68, β =
0 0 1
−
2.25,(whichhaveobtainedfromdatafittingproseces)
f˙(φ) 2 −
=3H ω ω (1+z)β+2 . q <0.
0 1
f(φ) 3tH − −
(cid:2) (cid:3)
3.2 Coincidence Problem
Also evolutionof effective dark energy parameter is in-
vestigated by time derivative of Ω = f(φ)n2/H2T2
eΛ
as In this subsection we want to consider one of the cos-
mological problems, namely coincidence problem. The
H˙ 1 f˙(φ)
′ problem addresses that why the ratio of dark matter
Ω = 2Ω + , (38)
eΛ − eΛ H2 HT − 2f(φ)H energy density, ρ , and dark energy density ρ , is of
m eΛ
h i
orderunity inpresenttime. Toindicatethis ratiobyr,
where prime denotes derivative with respect to N =
we have
ln(a).
Using Friedmann Eq.(14), energy conservation rela- ρ Ω
m m
r= = . (42)
tions, Eqs.(15) and (16), and effective dark energy pa-
ρ Ω
′ eΛ eΛ
rameter, Ω , is rewritten as
eΛ
To investigate the behavior of r, we use Eqs.(15), (16)
′ β
Ω =3Ω (1 Ω ) 1 ω ω (1+z) . (39) and (19) and therefore attain
eΛ eΛ − eΛ − 0− 1
h i
By integrating of this equation one can find out r=r (1+z)3(1−ω0)exp −3ω1 (1+z)β 1 . (43)
0
β −
Ω = D0 . (40) h (cid:0) (cid:1)i
eΛ 1+(1+z)3(1−ω0)exp[−3ω1(1+z)β] Where r0 is the ratio at present time. The behavior
β of coincidence parameter, r, has been plotted in Fig.5
Where D is the constant of integration. The behav- which illustrates the variation of r versus red-shift z.
0
ior of effective dark energy density parameter has been As one expect the plot shows that the parameter de-
plotted in Fig.4. It can be seen that by passing time creases by passing time. It means that as the universe
the effective energy density parameter is saturated to is getting larger, the effective energy density overcome
0.83. matterenergydensity. Basedonthe previousresultwe
knowthatΩ approachto 1means thatΩ approach
eΛ m
0.80 to zero, therefore the result of this section is consistent
0.75 with the previous works.
0.70
L
We 0.65
1.5
0.60
0.55
1.0
0.50 r
0.0 0.5 1.0 1.5
0.5
a
Fig. 4 The plot shows the behavior of effective energy 0.0
density parameter, ΩeΛ, versus a, for ω0 = 1.1, ω1 = -0.5 0.0 0.5 1.0
−1.68, β =−2.25. Z
An significant result of observational data is acceler- Fig. 5 This figure shows the behavior of r versus z, for
ated expansion of the Universe. A good cosmologi- ω0=1.1, ω1 =−1.68, β =−2.25.
cal model should be able to describe this acceleration.
An useful quantity to investigate this property of the
Universe is deceleration parameter which defined as 3.3 Square adiabatic sound speed
q = 1 H˙/H2. Using Eqs.(14), (15) and (16), one
− −
achieves the deceleration parameter gives Another interesting subject which could be addressed
here is sound speed. This parameter is useful to in-
q =−1+ 23 1−ω0−ω1(1+z)β × vestigate the classical stability of the models, there-
h i fore if sound speed be equal to a positive quantity the
D0 . (41)
1+(1+z)3(1−ω0)(t)exp[−3ω1(β1+z)β] proposal can be considered as a viable model. In our
(cid:16) (cid:17)
7
model the squared adiabatic sound speed is defined as 4 New-ADE and scalar field
c2 = dp /dρ , where ρ and p are the effective
s eΛ eΛ eΛ eΛ
dark energy density and pressure respectively. Given; In New-ADE a conformal time, η, is substituted in the
the equation of state and effective dark energy param- ADE relation instead of cosmological time T. So en-
eters, c2 could be expressed as follow ergy density is rewritten as ρ = 3n2/η2 where
s NADE
the conformal time is defined as η = dt/a. In this
ρ
c2s =ωeΛ+ω˙eΛρ˙eeΛΛ, (44) section we take the effective dark energRy equivalent to
the New-ADE, so we have
therefore using Eqs.(15)-(37), (44) one can see
3n2
ρ ρ = . (47)
c2 =ω + HT(1+Z)dωeΛ. (45) eΛ ≡ ADE η2
s eΛ 2 dz
According to the previous section, the effective dark
Given, the equation HT that is appeared in c2 as fol- energy parameter is specified as Ω♮ = n2f(φ)/H2η2.
eΛ
lowing Thisdefinitionofρ andenergyconservationrelation,
eΛ
z dz´ Eq.(15) lead us to derive the equation of state param-
HT =E(z) H0To , (46) eter as
− E(z) 1+z
h Z0 i
the behavior of HT is plotted(cid:0)in Fig(cid:1).6 ω♮ = 1+ 2 1 Ω♮eΛ f˙(φ) . (48)
eΛ − 3ansf(φ) − 3Hf(φ)
1.0
Using Eq.(19) for this case leads to
0.8
˙ β
f(t) 1 3ω 3ω (1+z)
0.6 = 0 1 (49)
T 2Hf(t) ηH − 2 − 2
H
0.4
In comparisonwith the formalNew-ADE and previous
0.2
sections there is an extra term and scale factor in the
0.0 denominator of the second term, hence the model able
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 to justify the easier phantom divide line crossing. It
Z is interesting to consider evolutionof effective dark en-
ergy parameter as well. Doing the same process as the
Fig. 6 BehaviorofHT hasbeendepictedversusredshift
previous section lead one to the following equations
Z.
Hence, Fig.7, shows behavior of sound speed, c2 versus Ω♮′eΛ =3Ω♮eΛ(1−Ω♮eΛ) 1+(ω1−ω0)+ω1(1+z)β , (50)
red shift z. It is indicated that always c2 >0. h i
therefore
0.30
Ω♮
0.25 Ω♮eΛ = 1+(1+z)3(ω0−10)exp[−3βω1(1+z)β], (51)
0.20
2C 0.15 where Ω♮ is the constant of integration. It easily can
0
0.10 be realized that we have the same situation as the pre-
vious section. The differences come up due to appear
0.05
of scale factor in the denominator of first term on the
0.00
righthandsideoftherelationandalsoinaterminthe
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
relation of Ω♮ .
eΛ
Z
On the other side, to specify the phase of the Uni-
Fig. 7 Behavior of sound speed c2 has been depicted
verse,weconsiderdecelerationparameter. UsingFried-
versus red shift Z.
mann equationand energy conservationequations lead
ustothefollowingconsequencefordecelerationparam-
As it is regardedby Fig.7, this model solve the clas- eter
sical non stability problem of the originalADE mecha-
1 3
nism. q = + ω♮ Ω♮
2 2 eΛ eΛ
8
ThedifferencebetweenEqs.(51)and(41),isduetoΩ this purpose we use the 557 Union II sample data-set
eΛ
and Ω♮ . Therefore based on the our results the decel- ofSnIa, to find the best fit for our models we combine
eΛ
erationparametergetsanegativevalueanddescribean radiation, baryonic and dark matter as matter compo-
accelerated expansion in agreement with observational nent. Thebest-fitforourmodelhasshownthatχ2 will
data. be minimize for ω = 1.65, ω = 1.1 and β = 2.25
1 0
− −
andthismodelhasagoodagreementwithobservational
data as well.
5 Conclusion
6 Acknowledgement
In this work, an external interacting type of ADE and
NADE has been picked out to consider the evolution Theauthorswouldliketothanksforanonymousreferee
of some cosmological parameters such as equation of for their helpful advices, which cause to improve the
state parameter, deceleration parameter, evolution of results of the model.
energy density parameter and sound speed. The gen-
eralformofthe evolutionequationsandenergyconser-
vation equation have been obtained. It is realized that
in comparison to some other scalar field models, this
model provides a generalized form of the energy con-
servationequationthatexpressestherighthandsideof
the relationis not equalzero. After that, the evolution
equations and energy conservation equation have been
rearranged to define an effective dark energy. Then
the effective dark energy has been taken equal to the
ADE and New-ADE to find out the behavior of men-
tioned parameters respectively. The acquired relations
forequationofstateparameterexpressthatincontrast
totheoriginmodel,ω couldremaininphantomrange
eΛ
for both cases of ADE and New-ADE, which in New-
ADE this result happen easer due to the presence of
the Universe scale factor in the denominator of some
terms. Moreover we turned our attention to the de-
celeration parameter to obtain the expansion phase of
the Universe in the model. The results represent that
in agreement with observational data, the model could
displayanacceleratedexpansionphasefortheUniverse.
Alsotheevolutionequationfordarkenergydensitypa-
rameter was acquired in both cases of ADE and New-
ADE.Theresultsshownthatbypassingtime,the den-
sity parameter increases and approaches to 1 at late
time,whichexpressesthatdarkenergyovercomesmat-
ter density. From the relation it could be realized that
when the parameter equal one, it ceases growing.
Inaddition,Itwasfoundoutinterestingtoconsiderthe
behaviorofc2 (soundspeed)andr (the ratioofmatter
s
density and dark energy density) in the model. There-
fore a shortly comments about these two subjects have
been presented in this work. In Fig.3 we have shown
that the rate time of r parameter smoothly decreases,
and the square of sound speed could get positive value
which have been explained in subsection.3.2 and is one
of the advantages of the model. As well we have fitted
the free parameters for ADE in our model. To getting
9
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