Crossing $w=-1$ by a single scalar on a Dvali-Gabadadze-Porrati brane

0.17 MB·English

by  Hongsheng Zhang

BUN/1022-2006
Crossing w = 1 by a single scalar on a Dvali-Gabadadze-Porrati brane
−
Hongsheng Zhang and Zong-Hong Zhu∗
Department of Astronomy, Beijing Normal University, Beijing 100875, China
Recent typeIa supernovaedataseem tofavor a dark energy model whose equation of state w(z)
crosses−1,whichisamuchmoreamazingproblemthantheaccelerationoftheuniverse. Eitherthe
casethatw(z)evolvesfromabove−1tobelow−1orthecasethatw(z)runsfrombelow−1toabove
−1,is consistent with presentdata. Inthispaperwe show thatit ispossible to realize thecrossing
behavioursofbothofthetwocasesbyonlyasinglescalarfieldinframeofDvali-Gabadadze-Porrati
braneworld. Atthesametimeweprovethattheredoesnotexistscalingsolutioninauniversewith
dust.
I. INTRODUCTION the other is a phantom: they dominate the universe by
7
turns, under this situation the effective EOS can cross
0
1 [10]. As we have mentioned that either the case that
0 −
2 The existence of dark energy is one of the most sig- w(z) evolves from above 1 to below 1 (Case I) or the
− −
nificant cosmological discoveries over the last decades case that w(z) runs from below 1 to above 1 (Case
n − −
[1]. However, the nature of this dark energy remains II)isconsistentwith presentdata. The observationdata
a
J amystery. Variousmodelsofdarkenergyhavebeenpro- mildlyfavordarkenergyofCaseI,butalsoleaveenough
posed, such as a small positive cosmological constant, space for a dark energy of Case II [11]. It is worthy to
0
1 quintessence, k-essence, phantom, holographic dark en- point out that there exists some interacting models, in
ergy, etc., see [2] for recent reviews with fairly complete which the effective EOS of dark energy crosses 1 [12].
−
3 list of references of different dark energy models. A cos- More recently it has been found that crossing 1 within
v mologicalconstantisa simplecandidate fordarkenergy. one scalar field model is possible, the cost is t−he action
4
However, following the more accurate data a more dra- containshigherderivativeterms[13](seealso[14]). Also
3
matic result appears: the recent analysis of the type Ia it is found that such a crossing can be realized without
8
supernovasdataindicatesthatthetime varyingdarken- introducing ordinary scalar or phantom component in a
1
1 ergy gives a better fit than a cosmologicalconstant, and Gauss-Bonnetbraneworldwithinducedgravity,wherea
6 in particular, the equation of state (EOS) parameter w fourdimensionalcurvaturescalaronthebraneandafive
0 (defined as the ratio of pressure to energy density) may dimensional Gauss-Bonnet term in the bulk are present
h/ cross 1 [3]. The dark energy with w < 1 is called [15]. Othertworelatedworksonphantomdividecrossing
− −
p phantomdark energy[4], for whichall energyconditions can be found in [16].
- areviolated. Here,itshouldbenotedthatthepossibility
o
that the dark energy behaves as phantom today is yet In this paper we suggest a possibility with effective
r
t a matter under debate: the observational data, mainly EOS crossing 1 in brane world scenario with only a
s −
a thosecomingfromthetypeIasupernovaeofhighredshift single scalar field. The brane world scenario is now one
: andcosmicmicrowavebackground,mayleadtodifferent of the most important ideas in high energy physics and
v
conclusionsdependingonwhatsamplesareselected,and cosmology. In this scenario, the standard model parti-
i
X whatstatisticalanalysisisapplied[5]. Bycontrast,other cles are confined to the 3-brane, while the gravitation
r researches imply that all classes of dark energy models can propagatein the whole spacetime. As for cosmology
a are comfortably allowed by those very observations [6]. inthe brane worldscenario,many workshave beendone
Presentlyallobservationsseemtonotruleoutthe possi- overthelastseveralyears;forareview,see[17]andrefer-
bility of the existence of matter with w< 1. Even in a ences therein. Brane world models admit a much wider
−
model in whichthe Newton constantis evolvingwith re- range of possibilities for dark energy [18]. In an inter-
spectredshiftz,thebestfitw(z)crossesthephantomdi- esting braneworld model proposed by Dvali, Gabadadze
vide w = 1 [7]. Hence the phenomenological model for andPorrati(DGP)[19]alate-timeself-accelerationsolu-
−
phantom dark energy should be considered seriously. To tion [20] appears naturally. In the DGP model, the bulk
obtain w < 1, scalar field with a negative kinetic term, isaflatMinkowskispacetime,butareducedgravityterm
−
may be a simplest realization. The model with phantom appearsonthebrane,whichistensionless. Inthismodel,
matter has been investigated extensively [8], and a test gravity appears 4-dimensional at short distances but is
of such matters in solar system, see [9]. However, the altered at distance larger than some freely adjustable
EOS of phantom field is always less than 1 and can crossover scale rc through the slow evaporation of the
−
not cross 1. It is easy to understand that if we put 2 gravitonoff our 4-dimensional brane world universe into
−
scalarfieldsintothemodel,oneisanordinaryscalarand an unseen, yet large, fifth dimension. The late-time ac-
celeration is driven by the manifestation of the excruci-
atingly slow leakage of gravity off our four-dimensional
world into an extra dimension. In this scenario the uni-
∗E-mailaddress: [email protected] verse eventually evolves into a de Sitter phase, and the
2
effectiveEOSofdarkenergynevercomesacross 1. The where H = a˙/a is the Hubble parameter, a is the scale
−
behaviourofcrossingw = 1oftheeffectivedarkenergy factor,k isthespatialcurvatureofthethreedimensional
−
on a DGP brane with a cosmological constant and dust maximally symmetric space in the FRW metric on the
has been studied in [21]. brane, and θ = 1 denotes the two branches of DGP
±
Inthepresentpaperwestudysomeinterestingproper- model,ρdenotesthe totalenergydensity,including dust
tiesofthescalar,includingordinaryscalarandphantom, matter and scalar, on the brane,
as dark energy, in the late time universe. We find effec-
tiveEOSofthedarkenergycancross 1onlybyasingle ρ=ρφ+ρdm, (6)
−
scalar: Forordinaryscalar,the EOSrunsfromabove 1
to below 1 ,evolving as dark energy of Case I in t−he and the term ρ0 relates the the strength of the 5-
negative b−ranch, and for phantom, the EOS runs from dimensional gravity with respect to 4-dimensional grav-
below 1 to above 1, evolving as dark energy of Case ity,
− −
II in the positive branch. As a byproduct, we provethat
6µ2
bothforordinaryscalarandphantom,theredoesnotex- ρ = , (7)
istscalingsolutioninauniversewithdustinDGPbrane 0 rc2
world scenario.
In the next section we shall investigate our model in where the cross radius is defined as rc ,κ25µ2.
detail. And in section III, we present the main conclu- Inquintommodel[10], twoscalarfields mustbe intro-
sions and some discussions. duced for the crossing 1 of effective EOS of dark en-
−
ergy. By contrast, we shall show that in our model only
one field is enough for this crossing behaviour by aiding
II. THE MODEL of the 5-dimensional gravity. Therefore, the accelerated
expansion of the universe is due to the combined effect
ofthe scalarand the competition between 4-dimensional
Let us start from the action of the DGP model
gravity and the 5-dimensional gravity.
S =S +S , (1) To explain the the observed evolving of EOS of effec-
bulk brane
tive dark energy, we calculate the equation of state w
where of the effective “dark energy” caused by the scalar field
and term representing brane world effect by comparing
1
S = d5X (5)g (5)R, (2) themodifiedFriedmannequationinthe braneworldsce-
bulk ZM p− 2κ25 nario and the standard Friedmann equation in general
relativity, because all observed features of dark energy
and
are “derived” in general relativity. Note that the Fried-
1 mannequation inthe four dimensionalgeneralrelativity
S = d4x√ g K±+L (g ,ψ) . (3)
brane Z − (cid:20)κ2 brane αβ (cid:21) can be written as
M 5
k 1
Hereκ25 is the 5-dimensionalgravitationalconstant,(5)R H2+ a2 = 3µ2(ρdm+ρde), (8)
is the 5-dimensionalcurvature scalarand the matter La-
grangian in the bulk. xµ (µ = 0,1,2,3) are the induced
wherethe firsttermofRHSofthe aboveequationrepre-
4-dimensional coordinates on the brane, K± is the trace
sentsthe dustmatterandthe secondtermstandsforthe
of extrinsic curvature on either side of the brane and
effective dark energy. Compare (8) with (5), one obtains
L (g ,ψ) is the effective 4-dimensional Lagrangian,
brane αβ the density of effective dark energy,
whichisgivenbyagenericfunctionalofthebranemetric
gαβ and matter fields ψ on the brane. ρ =ρ +ρ +θρ ρ+ρ +θρ (1+ 2ρ)1/2 . (9)
Consider the brane Lagrangian consisting of the fol- de φ 0 0 0 0
(cid:20) ρ (cid:21)
0
lowing terms
Since the dust matter obeys the continuity equation
µ2
and the Bianchi identity keeps valid, dark energy itself
L = R+L +L , (4)
brane m φ
2 satisfies the continuity equation
whereµis4-dimensionalreducedPlanckmass,Rdenotes
dρ
de
the curvature scalaron the brane, Lm stands for the La- dt +3H(ρde+peff)=0, (10)
grangian of other matters on the brane, and L repre-
φ
sents the lagrangian of a scalar confined to the brane. where p denotes the effective pressure of the dark en-
eff
Then assuming a mirror symmetry in the bulk, we have ergy. And then we can express the equation of state for
the Friedmann equation on the brane [20], see also [15], the dark energy as
k 1 2ρ p 1 dlnρ
H2+ = ρ+ρ +θρ (1+ )1/2 (5) w = eff = 1+ de , (11)
a2 3µ2 (cid:20) 0 0 ρ (cid:21) de ρ − 3dln(1+z)
0 de
3
where, from (10), be solved exactly in the standard model. Also it has
beenshownthatthe inflationdrivenby ascalarwith ex-
dlnρ 3
de = [ρ +p ponentialpotentialcanexitnaturallyinthewarpedDGP
de de
dln(1+z) ρ
de model [15]. In addition, we know that such exponential
+θ(1+2ρde+ρdm)−1/2(ρ +ρ +p ) ,(12) potentials of scalar fields occur naturally in some funda-
ρ de dm de (cid:21) mentaltheoriessuchasstring/Mtheories. Itis therefore
0
quite interesting to investigate a scalar with such a po-
where p represents the pressure of the dark energy.
de tential in late time universe on a DGP brane. Here we
Here we stress that it is different from p . Clearly, if
eff set
dlnρde is greater than 0, dark energy evolves as phan-
tdolmn(1;+ifz) dlnρde is less than 0, it evolves as quintessence; V =V0e−λφµ. (16)
dln(1+z)
if dlnρde equals 0, it is just cosmological constant. In Here λ is a constant and V0 denotes the initial value of
dln(1+z)
the potential.
a more intuitionistic way, if ρ decreases and then in-
de
Then the derivationofeffective density of darkenergy
creaseswithrespecttoredshift(ortime),orincreasesand
with respective to ln(1+z) reads,
then decreases, which implies that EOS of dark energy
crosses phantom divide. It is known that the equations dρ
of sort (5) and (10) belong to so-called inhomogeneous de = 3[φ˙2
dln(1+z)
EOSofFRWuniverse[22]orFRWuniversewithgeneral
EOS [23], which can cross the phantom divide. +θ(1 + φ˙2+2V +2ρdm)−1/2(φ˙2+ρ )].(17)
dm
In the DGP brane world model, the induced gravity ρ0
correction arises because the localized massive scalars
If θ = 1, both terms of RHS are positive, hence it never
fields on the brane, which couple to bulk gravitons, can
goes to zero at finite time. But if θ = 1, the two terms
generate via quantum loops a localized four-dimensional −
of RHS carry opposite sign, therefore it is possible that
world-volume kinetic term for gravitons [24]. Therefore,
the EOS of dark energy crosses phantom divide. In the
to investigate the behaviour of classical level of such a
following of this subsection we only consider the case of
field is not only interesting for cosmology, but also im-
θ = 1. For a more detailed researchof the evolution of
portant for analysesing the DPG model itself. The be- −
thevariablesinthismodelwewritetheminadynamical
haviour of the scalar in the early universe ,ie, the infla-
system, which can be derived from the Friedmann equa-
tiondrivenbyascalarfieldconfinedtoaDGPbrane,has
tion (5) and continuity equation (10). We first define
been investigated in [25] [26]. A proposal that the same
some new dimensionless variables,
phantom scalar plays the role of early time (phantom)
inflaton and late time dark energy is presented in [27]. φ˙
In view of these progresses, it is valuable to study the x , , (18)
√6µH
behaviour of a scalar field on a DGP brane at late time
universe. y , √V , (19)
In the following two subsections, we shall analyze the √3µH
dlaytneatmimices uonfiavnersoerdoinnaaryDsGcaPlarbraannde,arepshpaencttoivmelyi.n Wthee l , √ρm , (20)
√3µH
show both of them can cross the phantom divide: in the
negative branch for an ordinary scalar; in the positive b , √ρ0 . (21)
branch for a phantom. √3µH
The dynamics of the universe can be described by the
followingdynamicalsystemwiththesenewdimensionless
A. Ordinary scalar field
variables,
For an ordinary scalar, the action in Lφ of (4) reads, x′ = 3αx(2x2+l2)+3x √6λy2, (22)
1 −2 − 2
Lφ = ∂µφ∂µφ V(φ). (13) 3 √6
−2 − y′ = αy(2x2+l2)+ λxy, (23)
−2 2
Varying the action with respect to the metric tensor in
3 3
an FRW universe we reach to l′ = αl(2x2+l2)+ l, (24)
−2 2
1
ρφ = φ˙2+V(φ), (14) b′ = 3αb(2x2+l2), (25)
2
−2
1
pφ = φ˙2 V(φ), (15) where
2 −
whereadotdenotesderivativewithrespecttotime. The x2+y2+l2 −1/2
α,1 1+2 , (26)
exponentialpotentialisanimportantexamplewhichcan −(cid:18) b2 (cid:19)
4
aprimestandsforderivationwithrespecttos, ln(1+
z),andwehavesetk =0,whichisimpliedeither−bythe- 0.8
oretical side (inflation in the early universe) ,or observa-
tion side (CMB fluctuations [28]). One can check this 0.6
system degenerates to a quintessence with dust matter
in standard generalrelativity. Note that the 4 equations
0.4
(22), (23), (24), (25) of this system are not independent.
ByusingtheFriedmannconstraint,whichcanbederived
0.2
from the Friedmann equation,
x2+y2+l2 1/2 -1.5 -1 -0.5 0.5 1 s
x2+y2+l2+b2 b2 1+2 =1, (27)
− (cid:18) b2 (cid:19) q
thenumberoftheindependentequationscanbe reduced 0.4
to 3. There are two critical points of this system satisfy-
ing x′ =y′ =l′ =b′ =0 appearing at 0.2
s
-1.5 -1 -0.5 0.5 1
x =y =l =0, b=constant; (28)
-0.2
x =y =l =b=0. (29)
-0.4
However, neither of them satisfies the Friedmann con- -0.6
straint (27). Hence we prove that there is no ki-
-0.8
netic energy-potential energy scaling solution or kinetic
energy-potential energy -dust matter scaling solution on -1
aDGPbranewithquintessenceanddust. Sincetheequa-
tion set (22), (23), (24), (25) can not be solved analyti- FIG.1: Forthisfigure,Ωki =0.01,Ωrc =0.01,λ=0.05. (a)
cally, we present some numerical results about the dark β andγ asfunctionsofs,inwhichβ residesonthesolidline,
while γ dwells at the dotted line. The EOS of dark energy
energy density. And one will see that in reasonable re-
crosses −1 at about s = −1.6, or z = 3.9. (b) The cor-
gions of parameters,the EOSof dark energy crosses 1.
− responding deceleration parameter, which crosses 0 at about
The stagnation point of ρde dwells at s=−0.52, or z=0.68.
b l2
2+ =2, (30)
√2+b(cid:18) x2(cid:19)
The most significant parameters from the viewpoint of
which can be derivedfrom (17) and (27). One concludes observations is the deceleration parameter q, which car-
from the above equation that a smaller r , a smaller Ω ries the total effects of cosmic fluids. q is defined as
c m
(which is defined as the present value of the energy den-
sity of dust matter over the critical density), or a larger a¨a
q =
Ωki (which is defined as the present value of the kinetic −a˙2
energy density of the scalar over the critical density) is 3
helpful to shift the stagnation point to lower redshift = 1+ α(2x2+l2), (33)
− 2
region. We show a concrete numerical example of this
crossing behaviours in Fig. 1, Fig. 2, Fig. 3. For conve-
which is also plotted in these figures for corresponding
nienceweintroducethedimensionlessdensityandrateof
density curve of dark energy. In all the figures we set
change with respect to redshift of dark energy as below.
Ω =0.3 but with different Ω , λ, Ω , in which Ω is
m ki rc rc
defined as the present value of the energy density of ρ
ρ Ω 0
β = de = rc x2+y2+b2 over the critical density Ω =ρ /ρ .
ρ b2 (cid:20) rc 0 c
c
FromFig1,2,and3,clearly,theEOSofeffectivedark
x2+y2+l2
b2(1+2 )1/2 , (31) energy crosses 1 as expected. At the same time the
− b2 (cid:21) deceleration pa−rameter is consistent with observations.
As is well known, the EOS of a single scalar in standard
where ρ denotes the present critical density of the uni-
c general relativity never crosses the phantom divide, the
verse, and
induced term, through the “energy density” of r , ρ ,
c 0
plays a critical in this crossing. We see that a small
1 b2 dρ
γ = ρ Ω dsde component of ρ0, i.e. Ωrc =0.01, is successfully to make
c rc the EOSofdarkenergycross 1. The other note is that
= 3 (1+2x2+y2+l2)−1/2(2x2+l2) 2x2 (.32) here the EOS runs from abov−e −1 to below −1 ,which
(cid:20) b2 − (cid:21) describes the dark energy of Case I.
5
2 0.8
1.5 0.6
1 0.4
0.5
0.2
s
-0.8 -0.6 -0.4 -0.2 0.2 s
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2
-0.5
-0.2
-1
q q
0.75
s
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2
0.5
0.25 -0.2
s
-0.8 -0.6 -0.4 -0.2 0.2
-0.4
-0.25
-0.5 -0.6
-0.75
FIG. 2: For this figure, Ωki = 0.01, Ωrc = 0.01, λ = 0.5. FIG. 3: For this figure, Ωki = 0.1, Ωrc = 0.01, λ = 0.05.
(a) β and γ as functions of s,in which β resides on thesolid (a) β and γ as functions of s,in which β resides on thesolid
line, while γ dwells at the dotted line. The EOS of dark line, while γ dwells at the dotted line. The EOS of dark
energy crosses −1 at about s = −0.22, or z = 0.25. (b) energy crosses −1 at about s = −0.02, or z = 0.02. (b)
The corresponding deceleration parameter, which crosses 0 The corresponding deceleration parameter, which crosses 0
at about s=−0.40, or z=0.49. at about s=−0.41, or z=0.51.
B. Phantom field
with respective to ln(1+z) becomes,
In this subsection we investigate a phantom field in
DGP model. Most of the discussions are parallel to the
last subsection. For a phantom field, the action in L of
φ
(4) reads, dρde = 3[ φ˙2+θ(1
dln(1+z) −
Lφ = 21∂µφ∂µφ−V(φ). (34) +−φ˙2+2ρV +2ρdm) −1/2 (−φ˙2+ρdm)]. (38)
0
Avariationoftheactionwithrespecttothemetrictensor
in an FRW universe yields,
To study the behaviour of the EOS of dark energy, we
1 first take a look at the signs of the terms of RHS of the
ρφ =−2φ˙2+V(φ), (35) aboveequation. (−φ˙2+ρdm)representsthe totalenergy
1 density of the cosmic fluids, which should be positive.
pφ =−2φ˙2−V(φ). (36) Theterm(1+−φ˙2+2ρV0+2ρdm)−1/2 shouldalsobepositive.
Hence if θ = 1, both terms of RHS are negative: It
−
To compare with the results of the ordinary scalar, here nevergoestozeroatfinite time. Contrarily,ifθ =1,the
we set a same potential as before, two terms of RHS carry opposite sign: The EOS of dark
energy is able to cross phantom divide. In the following
V =V0e−λφµ. (37) ofthepresentsubsectionweconsiderthebranchofθ =1.
Similartothecaseofanordinaryscalar,the dynamics
The ratio of change of effective density of dark energy ofthe universecanbedescribedbythefollowingdynam-
6
ical system,
3 √6 0.8
x′ = α′x( 2x2+l2)+3x λy2, (39)
−2 − − 2
0.6
3 √6
y′ = α′y( 2x2+l2)+ λxy, (40)
−2 − 2 0.4
3 3
l′ = α′l( 2x2+l2)+ l, (41)
−2 − 2 0.2
3
b′ = α′b( 2x2+l2), (42)
−2 − s
-1.25 -1 -0.75 -0.5 -0.25 0.25 0.5
where
q
α′ ,1+ 1+2−x2+y2+l2 −1/2, (43) 0.4
(cid:18) b2 (cid:19)
0.2
,andwehavealsoadoptedthespatialflatnesscondition.
The definitions of x, y, l, b are the same as the last s
-1.25 -1 -0.75 -0.5 -0.25 0.25 0.5
subsection. One can check this system degenerates to a
-0.2
phantomwithdustmatterinstandardgeneralrelativity.
Also the 4 equations (39), (40), (41), (42) of this sys- -0.4
temarenotindependent. NowtheFriedmannconstraint
becomes -0.6
x2+y2+l2 1/2 -0.8
x2+y2+l2+b2+b2 1+2− =1,
− (cid:18) b2 (cid:19)
(44) FIG. 4: For this figure, Ωki = 0.01, Ωrc = 0.01, λ = 0.01.
with which there are 3 independent equations left in (a) β and γ as functions of s,in which β resides on thesolid
this system. Through a similar analysis as the case of line, while γ dwells at the dotted line. The EOS of dark
an ordinary scalar, we can prove that there is no ki- energy crosses −1 at about s = −1.25, or z = 1.49. (b)
The corresponding deceleration parameter, which crosses 0
netic energy-potential energy scaling solution or kinetic
at about s=−0.50, or z=0.65.
energy-potential energy -dust matter scaling solution on
a DGP brane with phantom and dust. The equation set
(39),(40),(41),(42)cannotbesolvedanalytically,there-
foreherewegivesomenumericalresults. Again,onewill
see that in reasonableregions of parameters,the EOSof The deceleration parameter q becomes,
darkenergycrosses 1,butfrombelow 1to above 1.
− − −
The stagnation point of ρ inhabits at
de
3
b l2 q = 1+ α′( 2x2+l2), (48)
2+ =2, (45) − 2 −
√2 b(cid:18)− x2(cid:19)
−
which can be derivedfrom (38) and (44). One concludes
which is plotted in the figure for corresponding density
fromthe aboveequationthatasmallerr , asmallerΩ ,
c m
curveofdarkenergy. InthisfigureswealsosetΩ =0.3.
or a larger Ω is helpful to shift the stagnation point to m
ki
lower redshift region, which is the same as the case of
Fig. 4 explicitly illuminates that the EOS of effec-
an ordinary scalar. Then we show a concrete numerical
tive dark energy crosses 1, as expected. At the same
example of the crossing behaviour of this case in Fig. −
time the deceleration parameter is consistent with ob-
4. The dimensionless density and rate of change with
servations. The EOS of a single phantom field in stan-
respect to redshift of dark energy become,
dardgeneralrelativityalwayslessthan 1,therefore,the
−
Ω 5 dimensional gravity plays a critical role in this cross-
β = rc x2+y2+b2 ing, though we see that the “geometric energy density”
b2 (cid:20)−
only takes a small component of the total density, i.e.
x2+y2+l2
+b2(1+2− )1/2 , (46) Ωrc =0.01. Finally,the mostimportantdifference ofthe
b2 (cid:21) dynamicsbetweenanordinaryscalarandthephantomof
the crossingbehaviourlies onthe crossingmanner: Here
and
theEOSrunsfrombelow 1toabove 1,whichcande-
γ =3 (1+2−x2+y2+l2)−1/2( 2x2+l2)+2x2 . scribethedarkenergyofC−aseII,while−theeffectiveEOS
(cid:20)− b2 − (cid:21) ofanordinaryscalarevolvesfromabove 1tobelow 1,
− −
(47) as shown in subsection A.
7
III. CONCLUSIONS AND DISCUSSIONS Fig. 2 shows that the deceleration parameter can ex-
ceed 0.5 in some high redshift region,which implies that
To summarize, this paper displays that in frame of the scalar may enter a kinetic energy dominated phase.
DGPbraneworlditispossibletorealizecrossingw = 1 And hence the energy density of scalar will exceed the
fortheEOSoftheeffectivedarkenergyofasinglescal−ar. density of dust in such region, which can spoil the suc-
For an ordinary scalar in the negative branch, the EOS cessful predictions of structure formation theory. There-
transits fromw> 1 to w < 1 ,whichcandescribe the fore, we should treat the exponential potential as an
dark energy of Cas−e I, while −for a phantom in the posi- approximation of a more appropriate potential, such as
tive branch, the EOS transits from w < 1 to w > 1, the tracker potential [29], in low redshift region. The
whichdescribes dark energyof CaseII. T−he decelerat−ion presentmodelmayfailatveryhighredshiftregion(such
parameter can be consistent with observations. In stan- as z = 1000). How to construct a potential which can
dard general relativity, the EOS of a single scalar never describe both the early universe and the late universe is
crosses 1, therefore, the five dimensional gravity plays our further work.
−
animportantroleinthistransition,althoughthecompe-
tentofthis“geometricenergydensity”ρ overthecritical Acknowledgments: Thisworkwassupportedbythe
0
energydensityisverysmall. Theotherconclusionisthat National Natural Science Foundation of China , under
there does not exist scaling solution in a universe with Grant No. 10533010, the Project-sponsored SRF for
dustonaDGPbrane,neitherinthepositivebranchnor ROCS, SEM of China, and Program for New Century
in the negative branch. Excellent Talents in University (NCET).
[1] A. G. Riess et al. , Astron. J. 116, 1009 (1998), los Phys.Rev. D70 (2004) 123529, astro-ph/0410309;
astro-ph/9805201;S.Perlmutteretal.,Astrophys.J.517, Y.H. Wei , gr-qc/0502077; S. Nojiri, S. D. Odintsov
565 (1999), astro-ph/9812133. and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005)
[2] Edmund J. Copeland, M. Sami and Shinji Tsujikawa, [arXiv:hep-th/0501025]; P. Singh, arXiv:gr-qc/0502086;
hep-th/0603057; J. P. Uzan,arXiv:astro-ph/0605313. H.Stefancic,arXiv:astro-ph/0504518;V.K.Onemliand
[3] U. Alam, V. Sahni, T. D. Saini and A. A. Starobin- R. P. Woodard , Phys. Rev. D70:107301,2004; Class.
sky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004) Quant. Grav. 19 (2002) 4607; gr-qc/0612026.
[arXiv:astro-ph/0311364]; U. Alam, V. Sahni [9] J. Martin, C. Schimd and J. P. Uzan, Phys. Rev. Lett.
and A. A. Starobinsky, JCAP 0406, 008 (2004) 96, 061303 (2006) [arXiv:astro-ph/0510208];
[arXiv:astro-ph/0403687]; D. Huterer and A. Cooray, [10] Z. K. Guo, Y. S. Piao, X. Zhang and Y. Z. Zhang,
astro-ph/0404062; Y. Wang and M. Tegmark, arXiv:astro-ph/0608165; B. Feng, X. L. Wang and
astro-ph/0501351; Andrew R Liddle, Pia Mukher- X. M. Zhang, Phys. Lett. B 607, 35 (2005)
jee, David Parkinson and YunWang, astro-ph/0610126. [arXiv:astro-ph/0404224]; M. R. Setare, Phys. Lett.
[4] R.R. Caldwell, Phys.Lett. B545 (2002) 23, B 641, 130 (2006); X. Zhang, Phys. Rev. D 74,
astro-ph/9908168; P. Singh, M. Sami and N. Dadhich, 103505(2006)[arXiv:astro-ph/0609699];Y.f.Cai,H.Li,
Phys.Rev.D68,023522 (2003) [arXiv:hep-th/0305110]. Y.S.PiaoandX.m.Zhang,arXiv:gr-qc/0609039;M.Al-
[5] H. K. Jassal, J. S. Bagla and T. Padmanabhan, imohammadi and H. M. Sadjadi, arXiv:gr-qc/0608016;
arXiv:astro-ph/0601389; H. K. Jassal, J. S. Bagla H. Mohseni Sadjadi and M. Alimohammadi, Phys. Rev.
and T. Padmanabhan, Phys. Rev. D 72 (2005) D 74, 043506 (2006) [arXiv:gr-qc/0605143]; W. Wang,
103503 [arXiv:astro-ph/0506748]; S. Nesseris and Y. X. Gui and Y. Shao, Chin. Phys. Lett. 23 (2006)
L. Perivolaropoulos, arXiv:astro-ph/0610092. 762; X. F. Zhang and T. Qiu, arXiv:astro-ph/0603824;
[6] V. Barger, E. Guarnaccia and D. Marfatia, Phys. Lett. R. Lazkoz and G. Leon, Phys. Lett. B 638, 303 (2006)
B 635 (2006) 61 [arXiv:hep-ph/0512320]. [arXiv:astro-ph/0602590]; X. Zhang, Commun. Theor.
[7] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 73 Phys. 44, 762 (2005); P. x. Wu and H. w. Yu, Int. J.
(2006) 103511 [arXiv:astro-ph/0602053]. Mod. Phys. D 14, 1873 (2005) [arXiv:gr-qc/0509036];
[8] E.Elizalde, S. Nojiri and S.D. Odintsov, Phys. Rev. D G.B.Zhao,J.Q.Xia,M.Li,B.FengandX.Zhang,Phys.
70,2004,043539; S. Nojiri and S. D. Odintsov, Phys. Rev. D 72, 123515 (2005) [arXiv:astro-ph/0507482];
Lett. B 562,2003,147;B. Boisseau, G. Esposito-Farese, H. Wei, R. G. Cai and D. F. Zeng, Class. Quant.
D. Polarski, Alexei A. Starobinsky, Phys. Rev. Lett. 85, Grav. 22, 3189 (2005) [arXiv:hep-th/0501160]; H. Wei
2236(2000);R.Gannouji,D.Polarski,A.Ranquet,A.A. and R. G. Cai, Phys. Rev. D 72 (2005) 123507
Starobinsky JCAP 0609,016 (2006); [arXiv:astro-ph/0509328]; H. Wei and R. G. Cai,
Z.K. Guo, R.G. Cai and Y.Z. Zhang, astro-ph/0412624; Phys. Lett. B 634, 9 (2006) [arXiv:astro-ph/0512018];
R. G. Cai and A. Wang, JCAP 0503, 002 (2005) J. Q. Xia, B. Feng and X. M. Zhang, Mod. Phys. Lett.
[arXiv:hep-th/0411025]; Z.K. Guo and Y.Z. Zhang, A 20, 2409 (2005) [arXiv:astro-ph/0411501]; Z. K. Guo,
Phys.Rev. D71 (2005) 023501, astro-ph/0411524; Z.K. Y. S. Piao, X. M. Zhang and Y. Z. Zhang, Phys.
Guo, Y.S. Piao, X. Zhang and Y.Z. Zhang, Phys.Lett. Lett. B 608, 177 (2005) [arXiv:astro-ph/0410654];
B608(2005)177-182,astro-ph/0410654;S.M.Carroll,A. B. Feng, M. Li, Y. S. Piao and X. Zhang, Phys.
DeFeliceandM.TroddenPhys.Rev.D71(2005)023525, Lett. B 634, 101 (2006) [arXiv:astro-ph/0407432]; Y.
astro-ph/0408081; S. Nesseris and L. Perivolaropou- Cai, H. Li, Y. Piao, and X. Zhang, gr-qc/0609039;
8
R. Lazkoz and G. Leon, Phys. Lett. B 638, 303 (2006) astro-ph/0510068.
[arXiv:astro-ph/0602590]; W. Zhao, Phys. Rev. D 73 [20] C. Deffayet, Phys. Lett. B 502, 199 (2001); C. Deffayet,
(2006) 123509 [arXiv:astro-ph/0604460]. G.Dvali, and G. Gabadadze, Phys. Rev. D 65, 044023
[11] J.Q.Xia,G.B.Zhao,B.Feng,H.LiandX.Zhang,Phys. (2002); C.Deffayet, S. J. Landau, J. Raux, M. Zaldar-
Rev. D 73, 063521 (2006) [arXiv:astro-ph/0511625]; riaga, and P. Astier, Phys. Rev.D 66, 024019 (2002);
G. B. Zhao, J. Q. Xia, B. Feng and X. Zhang, [21] L. P. Chimento, R. Lazkoz, R. Maartens and I. Quiros,
arXiv:astro-ph/0603621. JCAP 0609, 004 (2006) [arXiv:astro-ph/0605450];
[12] H. S. Zhang and Z. H. Zhu, Phys. Rev. D 73, 043518 R. Lazkoz, R. Maartens and E. Majerotto, Phys. Rev.
(2006); H. Wei and R. G. Cai, Phys. Rev. D 73, 083002 D 74, 083510 (2006) [arXiv:astro-ph/0605701].
(2006) [22] S. Nojiri and S. D. Odintsov, Phys. Rev. D 72, 023003
[13] M. z. Li, B. Feng and X. m. Zhang, (2005) [arXiv:hep-th/0505215].
arXiv:hep-ph/0503268. [23] S. Nojiri and S. D. Odintsov, Phys. Rev. D 70, 103522
[14] A. Anisimov, E. Babichev and A. Vikman, (2004) [arXiv:hep-th/0408170].
arXiv:astro-ph/0504560. [24] D.M. Capper, NuovoCim. A 25 1975 29.
[15] R. G. Cai, H. s. Zhang and A. Wang, Commun. Theor. [25] R. G. Cai and H. s. Zhang, JCAP 0408, 017 (2004)
Phys.44, 948 (2005) [arXiv:hep-th/0505186]. [arXiv:hep-th/0403234];
[16] A.A. Andrianov, F. Cannata and A. Y. Kamenshchik, [26] E. Papantonopoulos and V. Zamarias, JCAP 0410, 001
Phys.Rev.D72, 043531,2005; F.Cannata and A. Y. Ka- (2004) [arXiv:gr-qc/0403090].
menshchik, , gr-qc/0603129; Pantelis S. Apostolopoulos [27] S.Capozziello, S.Nojiri and S.D.Odintsov,Phys.Lett.
and Nikolaos Tetradis , Phys. Rev.D 74 (2006) 064021. B 632 (2006) 597 [arXiv:hep-th/0507182]; S. Nojiri and
[17] R.Maartens, gr-qc/0312059. S.D. Odintsov,Gen.Rel.Grav. 38 (2006) 1285.
[18] V. Sahni and Y. Shtanov, JCAP 0311 (2003) 014, [28] D. N. Spergel et al.,arXiv:astro-ph/0603449.
astro-ph/0202346. [29] I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev.
[19] G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. Lett. 82, 896 (1999).
B485 (2000) 208 ,hep-th/0005016; G. Dvali and G.
Gabadadze, Phys. Rev. D 63, 065007 (2001); A. Lue,