The role of potential in the ghost-condensate dark energy model

0.19 MB·

by  Gour Bhattacharya

The role of potential in the ghost-condensate dark energy model
Gour Bhattacharya∗
Department of Physics, Presidency University, 86/1, College Street, Kolkata 700 073, India
Pradip Mukherjee† and Amit Singha Roy‡
Department of Physics, Barasat Govt. College, 10 K. N. C. Road, Barasat, Kolkata 700 124, India
Anirban Saha§
Department of Physics, West Bengal State University,
Barasat, North 24 Paraganas, West Bengal, India
We consider the ghost-condensate model of dark energy with a generic potential term. The in-
4 clusion of the potential is shown to give greater freedom in realising the phantom regime. The
1 self-consistency of theanalysis is demonstrated using WMAP7+BAO+Hubbledata.
0
2 PACSnumbers: 98.80.-k,95.36.+x

c
e
D I. INTRODUCTION

5 Recent cosmological observations indicate late-time acceleration of the observable universe1,2. Why the evolution
1
of the universe is interposed between an early inflationary phase and the late-time acceleration is a yet-unresolved

problem. Varioustheoreticalattemptshavebeenundertakentoconfrontthisobservationalfact. Althoughthesimplest
]
c way to explain this behavior is the consideration of a cosmological constant3, the known fine-tuning problem4 led to
q
the dark energy paradigm. Here one introduces exotic dark energy component in the form of scalar fields such as
-
r quintessence6–12, k-essence13–15 etc. Quintessence is basedon scalarfield models using a canonicalfield with a slowly
g
varying potential. On the other hand the models grouped under k-essence are characterized by noncanonical kinetic
[
terms. A key feature of the k-essence models is that the cosmic acceleration is realized by the kinetic energy of the

3 scalar field. The popular models under this category include the phantom model, the ghost-condensate model etc4,5.
v It is well-known that the late time cosmic acceleration requires an exotic equation of state ω < 1. From
5 the seven year Wilkinson Microwave Anisotropy Probe (WMAP7) observations data, distance mDeaEsurem−e3nts from
4
the BAO and the Hubble constant measurements the value of a constant EOS for dark energy has been estimated
7
as ω = 1.10 0.14(68%CL) for flat universe16. Primary results from PAN-STARRS in fact pushes this limit
6 DE − ±
. further17 though the full data is yet to arrive. No scalar field dark energy model with canonical kinetic energy term
1
can achieve ω < 1. For this one has to consider a scalar field theory with negative kinetic energy along with a
0 DE −
field potential. The resulting phantom model18–23 is extensively used to confront cosmologicalobservation24–29.
4
Thephantommodelishoweverriddenwithvariousinstabilitiesasitsenergydensityisunbounded. Thisinstability
1
: canbeeliminatedintheso-calledghost-condensate(GC)models30 byincludingatermquadraticinthekineticenergy.
v
In this context let us note that to realize the late-time acceleration scenario some self-interaction must be present
i
X in the phantom model. In contrast, in the GC models the inclusion of self-interaction potential of the scalar field is
believed to be a matter of choice4. This fact, thoughnot unfamiliar, has not been emphasisedmuch in the literature.
r
a In the present paper we show that by including a potential term in the GC model brings more flexibility in realising
the phantom evolution.
It is well-knownthat the GC model without the potentialresides within the phantomregime for a certain rangeof
valuesofthescalarfieldkineticenergy4. Wewilldemonstrateherethattheserangeiswidenedinpresenceofageneric
potential term. Note that this widening is a consequence of the field theoretic aspects of the present dark energy
model. Also it crucially depends on the positive energy condition. The question arises whether these conditions for
achieving the phantom regime are consistent with the scalar field dynamics or not.
Now the scalar field dynamics is not independent but is coupled with gravity. Usually one assumes a specific
potential and the consequent evolution is studied. But in this paper our objective is to point out the advantage of
including a potential in the GC model for achieving the phantom regime. Thus we start with an arbitrary potential
and exploit a specific feature of the GC action to show that the potential can be expressed in terms of observable
parameters (e.g. H˙) once the evolution of the scale factor is chosen. Naturally we use the phantom power law here.
Consequently the kinetic and potential energy are expressed as functions of time. We still require observationaldata
to fix the geometric parameters appearing in these functional relations so that their time-evolutions can be explicitly
obtained. ForthispurposethecombinedWMAP7+BAO+Hubbledatawillbeused. Thepotentialandkineticenergy
are plotted. The plots clearlyshow that the criteriaderivedhere for our model to realizethe phantom evolutionhold
throughout the entire late-time evolution.
2
The organization of this paper is as follows. In section II we briefly review the ghost-condensate model with an
arbitrary potential. The equations of motion for the scalar field and the scale factor are derived. These equations
exhibitthecouplingbetweenthescalarfielddynamicsandgravity. Expressionsfortheenergydensityandpressureof
the darkenergycomponentsarecomputed. TheseexpressionsareusedinsectionIII tofindthecriteriaforthe model
to acquire phantom evolution. In section IV we utilize an obvious algebric consistency which leads to a quadratic
equationinthepotential. Solvingthisthegenericpotentialisexpressesedintermsofmeasurablegeometricquantities.
TofixthesegeometricquantitiesphantompowerlawevolutionisassumedandthecombinedWMAP7+BAO+Hubble
dataisusedinsectionV.Theexplicitetimevariationsofthepotentialandthekineticenergyareobtained. Weprovide
the plots of these quantities throughout the late-time evolution. Remarkably the conditions for the phantom regime
given in section III are observed to hold. FInally we conclude in section VI.
II. THE MODEL
In this section we consider the ghost-condensate model with a self-interaction potential V(φ). The action is given
by
R
S = d4x√ g + + , (1)
− 2k2 Lφ Lm
Z (cid:20) (cid:21)
where
X2
= X+ V (φ) (2)
Lφ − M4 −
1
X = gµν∂ φ∂ φ (3)
µ ν
−2
M is a mass parameter, R the Ricci scalar and G = k2/8π the gravitational constant. The term accounts for
m
L
the total (dark plus baryonic) matter content of the universe, which is assumed to be a barotropic fluid with energy
density ρ and pressure p , and equation-of-state parameter w = p /ρ . We neglect the radiation sector for
m m m m m
simplicity.
Theactiongivenbyequation(1)describesascalarfieldinteractingwithgravity. Invokingthecosmologicalprinciple
one requires the metric to be of the Robertson-Walker (RW) form
dr2
ds2 =dt2 a2(t) +r2dΩ2 , (4)
− 1 Kr2 2
(cid:20) − (cid:21)
where t is the cosmic time, r is the spatial radial coordinate, Ω is the 2-dimensional unit sphere volume, K charac-
2
terizes the curvature of3-dimensionalspaceand a(t) is the scalefactor. The Einstein equations leadto the Freidman
equations
k2 K
H2 = ρ +ρ (5)
3 m φ − a2
k(cid:16)2 (cid:17) K
H˙ = ρ +p +ρ +p + , (6)
− 2 m m φ φ a2
(cid:16) (cid:17)
In the above a dot denotes derivative with respect to t and H a˙/a is the Hubble parameter. In these expressions,
≡
ρ and p are respectively the energy density and pressure of the scalar field. The quantities ρ and p are defined
φ φ φ φ
through the symmetric energy-momentum tensor
2 δ
T(φ) = − √ g (7)
µν √ gδgµν −
−
(cid:0) (cid:1)
A straightforwardcalculation gives
2X
T(φ) =g + 1+ ∂ φ∂ φ (8)
µν µνLφ − M4 µ ν
(cid:18) (cid:19)
Assuming a perfect fluid model we identify
3X2
ρ = X + +V (φ) (9)
φ − M4
X2
p = = X+ V (φ) (10)
φ Lφ − M4 −
3
The equation of motion for the scalar field φ can be derived from the action (1). Due to the isotropy of the FLRW
universe the scalar field is a function of time only. Consequently, its equation of motion reduces to
3φ˙2 φ˙2 dV
1 φ¨+3H 1 φ˙ =0. (11)
− M4! − M4! − dφ
As is well known the same equation of motion follows from the conservation of T . Indeed under isotropy the
µν
equations (9) and (10) reduce to
1 3˙φ4
ρ = φ˙2+ +V (φ) (12)
φ −2 4M4
1 φ˙4
p = φ˙2+ V (φ) (13)
φ −2 4M4 −
From the conservation condition T(φ)µν =0 we get
µ
∇
ρ˙ +3H(ρ +p )=0, (14)
φ φ φ
which, written equivalently in field terms gives equation (11).
Tocompletethesetofdifferentialequations(5),(6),(14)weincludetheequationfortheevolutionofmatterdensity
ρ˙ +3H(1+w )ρ =0, (15)
m m m
where w = p /ρ is the matter equation of state parameter. The solution to equation (15) can immediately be
m m m
written down as
n
ρ a(t )
m 0
= , (16)
ρ a(t)
m0 (cid:20) (cid:21)
where n = 3(1+w ) and ρ 0 is the value of matter density at present time t . Now, the set of equations (5),
m m0 0
≥
(6), (14) and (15) must give the dynamics of the scalar field under gravity in a self-consistent manner. In the next
section we investigate the criteria for the GC model to realise the phantom evolution.
III. CRITERIA FOR REALISING THE PHANTOM REGIME
The phantom regime is demarcated by ω < 1 where ω is the dark energy equation of state (EoS) parameter
φ φ
−
defined as
P
φ
ω = (17)
φ
ρ
φ
In the present section we investigate the criteria for our model to be in the phantom regime using the definition (17)
onlywithoutrecoursetoactualdynamics. Fromequation(12)and(13),the EoSparameterforthe fieldφisobtained
as
φ˙2 + φ˙4 V(φ)
ω = − 2 4M4 − (18)
φ φ˙2 + 3φ˙4 +V(φ)
− 2 4M4
Defining f φ˙ = φ˙2 3φ˙4 (18) can be cast in the form
2 − 4M4
(cid:16) (cid:17) (cid:16) (cid:17)
φ˙2 1 φ˙2
− M4
ω = 1 (19)
φ − − V ((cid:16)φ) f(φ˙(cid:17))
−
This equation is more suitable to discuss the conditions for achieving the phantom regime.
1. First assume that there is no self-interaction, i.e., V (φ) = 0. The positive energy condition ensures that
ρ = f(φ˙)>0. Thus, for ω < 1 we require 1 φ˙2 >0. These lead to the following bounds4
φ − φ − − M4
(cid:16) (cid:17)
2
M4 <φ˙2 <M4 (20)
3
so that the phantom regime is attained.
4
2. Now suppose, V (φ)=0. From the positive energy condition ρ >0, (see equation (12)) we get
φ
6
V (φ)>f(φ˙) (21)
The only restriction imposed is now φ˙2 <M4. Of course φ˙ is real so we now require
0<φ˙2 <M4 (22)
Comparing the equations (20) and (22) it is clear that inclusion of appropriate self-interaction provides greater
flexibility to realise the phantom domain. In the phantom domain the scale factor evolves according to the phantom
power law4:
β
t t
s
a(t)=a − (23)
0
t t
(cid:18) s− 0(cid:19)
where t and t are the present time and big-rip time18,19 respectively. These parameters are obtained from observa-
0 s
tional data. In this connection it is important to note that the condition (22) is obtained from the definition of the
EoS (17) which in turn follows from the particular energy-momentum tensor obtained from the model (2, 3). Such
quantities have been termed as the ‘physical variables’ in the literature32. In contrast the geometric quantities (e.g.
the Hubble parameterH(t)andits time-derivative)aredeterminedfromobservationsinamodel-independentway32.
Naturally one wonders whether the dynamical evolution of the system according to the phantom power law always
conforms with the condition (22).
At this point, one should note that in general, the dynamical evolution of the fields can not be worked out if the
potential is not specified. However, as emphasised in the introduction, a specific aspect fo the GC model (2, 3)
allows us to express the arbitrary potential in terms of geometric quantities. Consequently, the field variables and
the potential here can be expressed as function of time once the geometric parameters involved in (23) are fixed
from observational data. It will then be possible to answer whether our criteria remains satisfied with the phantom
evolution throughout the late time.
IV. THE POTENTIAL FROM GEOMETRIC QUANTITIES
In this section we will exploit the structure of the model (2, 3) to establish an algebraic identity which will enable
us to express the generic potential in terms of geometric quantities. We start by constructing two independent
combinations of the pressure and energy density of the dark energy sector in terms of the Hubble parameter H,
matterenergydensityρ ,matterequationofstateparameterw andcurvatureparameterK using(5), (6)and(15)
m m
2H˙ n 2K
ρ +p =A = ρ + (24)
φ φ − k2 − 3 m k2a2
6a¨
ρ +3p =B = (n 2)ρ (25)
φ φ −k2a − − m
Using equations (12) and (13), we rewrite these combinations in terms of the ghost-condensate field derivative φ˙ and
potential V (φ):
φ˙4
ρ +p =A = φ˙2+ (26)
φ φ − M4
3φ˙4
ρ +3p =B = 2φ˙2+ 2V (φ) (27)
φ φ − 2M4 −
Inverting the equations (26, 27) we can write φ˙2 and φ˙4 in terms of A, B and V (φ) as
φ˙2 =3A 2B 4V (φ) (28)
− −
φ˙4 =2M4[(2A B) 2V (φ)] (29)
− −
5
Note that there is an obvious suggestion lurking behind the equations (28) and (29), namely, the algebraic identity
2
φ˙2 =φ˙4 (30)
(cid:16) (cid:17)
If one substitutes both sides of the identity fromequations (28) and(29)an equationis obtainedwhich contains only
geometric quantities, except for the potential. Thus it allows us to express the arbitrary potential in terms of these
geometric quantities. Note further that the statement holds because we have already agreed to assume the phantom
power law with the geometric parameters appearing in it fixed by observational data. This is clearly the unique
feature of the ghost-condensate model (2, 3) which has been referred to in the above.
Utilizing the identity we obtain the following equation quadratic in V (φ)
3A M4
V2(φ) + B + V (φ)
− 2 4
(cid:18) (cid:19)
(3A 2B)2 4M4(A B/2)
+ − − − =0 (31)
16
At this point one may ask whethar the constraining equation (31) on V (φ) at all allows a real solution. Solving
(31) we get
1
3A 2B M4 M4 M4 2
V (φ)= − +A (32)
4 − 8 ± 16 4
(cid:18) (cid:19) (cid:26) (cid:18) (cid:19)(cid:27)
The reality condition is thus
M4
+A 0 (33)
4 ≥
(cid:18) (cid:19)
Thatthis conditionis satisfiedingeneralcanbeestablishedexplicitlyifwesubstitute forAfromequation(26)which
gives
M4 1 M4 2
+A = φ˙2 0 (34)
4 M4 − 2 ≥
(cid:18) (cid:19) (cid:18) (cid:19)
Since from physical consideration the interaction potential is required to be real the above observation indicates the
consistency of our formalism.
In the next section we will utilize the solution (32) to express the geneic potential as a function of time employing
the phantom power law. This is the point of departure of our work from the existing works with the GC model
available in the literature. This, as has been explained in the above, suits our purpose of showing that inclusion of a
potential widens the allowedrange of kinetic energy of the GC model to realise the phantom regime. Needless to say
it is imparetive to varify that the criteria identified above are consistent with the dynamical evolution.
V. OUR MODEL AND THE PHANTOM EVOLUTION
In this section we will verify the validity of the criteria (21, 22) in the phantom evolution scenario. Assuming a
phantom power law the time evolutions of both the potential and kinetic energies will be studied. To get explicit
time variations of these quantities we require the values of various parameters appearing therein. These parameters
include the phantompower law exponent, the big rip time as well as the presentvalues of energydensity etc. We use
the combined WMAP7+BAO+Hubble as well as WMAP7 data16 as standard data set31. Also in our model there is
a free parameter M, the value of which will be estimated self-consistently using the same ovservationaldata.
6
A. Consequence of the phantom power law
We will now find explicit expressions of the potential and kinetic energies as functions of time. The potential is
already given by equation (32). Substituting A and B from (24) and (25) respectively we get33
3H2 3H˙ ρ M4
m
V(φ)= +
"8πG 16πG − 4 − 8 #
1
1 H˙ M4 M4 2
ρ + (35)
m
±2" −4πG − 4 ! 4 #
Now solving equation (26) kinetic energy term is obtained as
1
M4 H˙ M4 2
φ˙2 = ρ + M4 (36)
m
2 ∓" −4πG − 4 ! #
The choice of signs in the equations (35) and (36) should be noted. This choice is done so as to satisfy (27).
Using the phantom power law we find
β
H = (37)
−t t
s
−
β
H˙ = (38)
−(t t)2
s
−
Substituting these and restoring to S.I. units eqation (35) and (36) become,
V(t)= 3β2c2 3βc2 1ρm0c2 ts−t0 3β MS.I.4
8πG(ts−t)2 − 16πG(ts−t)2 − 4 a03 ts−t − 8
(cid:20) (cid:16) (cid:17) (cid:21)
1
1 βc2 ρm0c2 ts−t0 3β + MS.I.4 MS.I.4 2 (39)
±2 4πG(ts−t)2 − a03 ts−t 4 4
(cid:20)(cid:18) (cid:16) (cid:17) (cid:19) (cid:21)
and
1
M 4 βc2 ρ c2 t t 3β M 4 2
φ˙2 = S.I. m0 s− 0 + S.I. M 4 (40)
2 ∓" 4πG(ts−t)2 − a03 (cid:18) ts−t (cid:19) 4 ! S.I. #
where M = M(in ev) 1.62 10−2. Equations (39) and (40) are the desired time variations of the potential and
S.I.
× ×
kinetic energies if the phantom power law is imposed.
Toproceedfurther inputfromthe observationaldatais required. Thiswillenableus todetermine the valuesofthe
different geometric parameters appearing in the expressions of above (39, 40). It will then be possible to check the
validity of the conditions (21, 22). However, before invoking the observational data a consistency check is necessary.
This involves the verification whether the reconstructed potential and kinetic energy (39, 40) satisfy equation (11),
the eqution of motion of the scalar field. The necessary calculations for the consistency check will be given in the
next subsection.
B. A consistency check
Since V is given as function of t we use the chain rule of differentiation to write
dV 1dV
= (41)
dt φ˙ dφ
Substituting this in (11) and after a few steps of calculation we get the equivalent of (11) as
1dφ˙2 3 dφ˙4 φ˙2 dV
+3H 1 φ˙2 = (42)
2 dt − 4M4 dt − M4! dt
7
The above form of (11) can be readily used to verify whether the reconstructed potential (39) is consistent with the
equation of motion for φ. Using (40) in the left hand side (L.H.S) of (42) we get
βc2 3ρm0βc2 ts−t0 3β M 4
1 2πG(ts−t)3 − a03(ts−t) ts−t S.I.
L.H.S. = ∓4 ((cid:20)(cid:18)βc2 ρm0c2 ts−t0(cid:16)3β +(cid:17)MS.I(cid:19).4)M (cid:21)4 21
4πG(ts−t)2 − a03 ts−t 4 S.I.
(cid:20) (cid:16) (cid:17) (cid:21)
3 βc2 3ρ βc2 t t 3β
m0 s 0
−
− 4" 2πG(ts−t)3 − a03(ts−t)(cid:18) ts−t (cid:19) !#
βc2 3ρm0βc2 ts−t0 3β M 4
3 2πG(ts−t)3 − a03(ts−t) ts−t S.I.
±8 ((cid:20)(cid:18)βc2 ρm0c2 ts−t0(cid:16)3β +(cid:17)MS.I(cid:19).4)M (cid:21)4 21
4πG(ts−t)2 − a03 ts−t 4 S.I.
(cid:20) (cid:21)
(cid:16) (cid:17)
3β2c2 3ρ βc2 t t 3β
m0 s 0
+ − (43)
4πG(t t)3 − a 3(t t) t t
s− 0 s− (cid:18) s− (cid:19)
Arranging terms, this may be re-expressed as
3β2c2 3βc2 3 ρ βc2 t t 3β
m0 s 0
L.H.S. = −
"4πG(ts−t)3 − 8πG(ts−t)3 − 4a03(ts−t)(cid:18) ts−t (cid:19) #
βc2 3ρm0βc2 ts−t0 3β M 4
1 2πG(ts−t)3 − a03(ts−t) ts−t S.I.
±8 ((cid:20)(cid:18)βc2 ρm0c2 ts−t0(cid:16)3β +(cid:17)MS.I(cid:19).4)M (cid:21)4 21 (44)
4πG(ts−t)2 − a03 ts−t 4 S.I.
(cid:20) (cid:16) (cid:17) (cid:21)
Now from a straightforwarddifferentiation of V(t) using (39)we derive
dV 3β2c2 3βc2 3 ρ βc2 t t 3β
m0 s 0
= −
dt "4πG(ts−t)3 − 8πG(ts−t)3 − 4a03(ts−t)(cid:18) ts−t (cid:19) #
βc2 3ρm0βc2 ts−t0 3β M 4
1 2πG(ts−t)3 − a03(ts−t) ts−t S.I.
±8 (cid:20)(cid:18) (cid:16)3β (cid:17) (cid:19) (cid:21) 21 (45)
( βc2 ρm0c2 ts−t0 + MS.I.4)M 4
4πG(ts−t)2 − a03 ts−t 4 S.I.
(cid:20) (cid:16) (cid:17) (cid:21)
which is nothing but the R.H.S. of (42). A comparision of (44) with (45) shows that (42) is satisfied. Hence the
reconstructed potential is consistent with the equation of motion of the scalar field.
C. Input from the Observational data
We take into accountthe combined cosmic microwavebackground(CMB), baryonacoustic oscillations(BAO) and
observational Hubble data (H ) as well as the CMB-WMAP7 dataset seperately. The relevant results are tabulated
0
in TABLE I. The usual density parameter is Ω = 8πGρ /(3H2) and it is assumed to contain the baryonic matter
m m
Ω and cold dark matter Ω parts: Ω =Ω +Ω . Using the expressionof the critical density ρ = 3H2, the
b CDM m b CDM c 8πG
matter density at the present time can be found as ρ =Ω ρ . We also set a to 1.
m0 m0 c0 0
From the phantom power law we get
β = H (t t ) (46)
0 s 0
− −
Assuming a flat geometry and that at late times the phantom dark energy will dominate the universe t can be
s
expressed as19
ts t0+ 2 1+wDE −1H0−1(1 Ωm0)−21 (47)
≃ 3| | −
Inderivingtheaboveformulaithasbeenassumedthatatlatetimesthe darkenergyEoSparameterw approaches
DE
a constant value. The values of the derived parameters are given in TABLE II.
8
Parameter WMAP7+BAO+H0 WMAP7
t0 13.78±0.11 Gyr [(4.33±0.04)×1017 sec] 13.71±0.13 Gyr[(4.32±0.04)×1017 sec]
H0 70.2+−11..34 km/s/Mpc 71.4±2.5 km/s/Mpc
Ωb0 0.0455±0.0016 0.0445±0.0028
ΩCDM0 0.227±0.014 0.217±0.026
TABLE I: Maximum likelihood values for the observed cosmological parameters in 1σ confidencelevel16.
Parameter WMAP7+BAO+H0 WMAP7
β −6.51+0.24 6.5±0.4
−0.25
ρm0 2.52+−00..2254×10−27kg/m3 2.50+−00..4482×10−27kg/m3
ρc0 9.3+−00..34×10−27kg/m3 9.58+−00..6686×10−27kg/m3
ts 104.5+−12..90Gyr[(3.30±0.06)×1018 sec] 102.3±3.5Gyr[(3.23±0.11)×1018 sec]
TABLEII: Corresponding maximum likelihood values of the derived parameters .
Wearenowalmostinapositiontocalculatenumericalvaluesofvariousquantitiesasfunctionoftime. Butonelast
point is still missing. We requireto fix the parameterM in the ghost-condensatemodel. The value of this parameter
should be chosen so that the quantity within square root in (39) and (40) is positive ensuring real values for the
potentialandkinetic energies. We findthatM =1ev is agoodchoice. Alsowe choosethe upper signin(39)in order
to ensure positive potential energy. As a consequence the upper sign in equation (40) is selected (see the discussion
under equation(36)).
UsingvaluesfromTABLEIandTABLEIIinequations(39)wegetexpressionsofV(t)fortheWMAP7+BAO+H
0
dataset as:
6.823 10−9 5.24 10−10 0.661 10−19
V(t)= × + × × 8.61 10−9
(3.3 t)2 (3.3 t)2 − (3.3 t)−19.53 − ×
(cid:20) − − − (cid:21)
1
1 6.987 10−10 2.642 10−19 2
+ × × +1.72 10−8 1.72 10−8 (48)
2 − (3.3 t)2 − (3.3 t)−19.53 × ×
(cid:20)(cid:18) − − (cid:19) (cid:21)
and for the WMAP7 dataset as:
6.802 10−9 5.232 10−10 1.088 10−19
V(t)= × + × × 8.61 10−9
(3.23 t)2 (3.23 t)2 − (3.23 t)−19.5 − ×
(cid:20) − − − (cid:21)
1
1 6.976 10−10 4.352 10−19 2
+ ( × × +1.72 10−8)1.72 10−8 (49)
2 − (3.23 t)2 − (3.23 t)−19.5 × ×
(cid:20) − − (cid:21)
Similarly for the kinetic energy term we get from equation (40)
1
6.987 10−10 2.642 10−19 2
φ˙2 =3.444 10−8 ( × × +1.72 10−8)6.89 10−8 (50)
× − − (3.3 t)2 − (3.3 t)−19.53 × ×
(cid:20) − − (cid:21)
and
1
6.976 10−10 4.352 10−19 2
φ˙2 =3.444 10−8 ( × × +1.72 10−8)6.89 10−8 (51)
× − − (3.23 t)2 − (3.23 t)−19.5 × ×
(cid:20) − − (cid:21)
for the WMAP7+BAO+H and WMAP7 dataset respectively.
0
InFigs1and2theevolutionofthepotentialenergyandthekineticenergytermareshowngraphicallyagainsttime.
As expected, these quantities shoot up as the big rip time is approached. Superposed on the plot of potential energy
inFig. 1isthefuntionf(φ˙). Itcanbeclearlyseenthatthecondition(21)issatisfiedthroughoutthefutureevolution.
AgainfromFig. 2weobservethatthecondition(22)isalsosatisfiedbecauseφ˙2 alwaysliesbelowM4 =6.89 10−08.
×
Thus we see that the criteria of realising the phantom regime (21, 22) are satisfied by the present model.
9
VI. CONCLUSION
Recent observations16,17 indicate that there is a fair possibility of the late-time universe to follow the phantom
evolution. The ghost condensate (GC) model is a dark energy model which realises the samke phantom evolution
while eradicating some of the critical problems of the original phantom model. The inclution of a self-interaction in
this model appears to be a matter of choice in the literature4. In this paper we have considered a ghost condensate
(GC) model with an arbitrary potential term in a flat FLRW universe. The standard barotropic matter equation
of state is assumed. Keeping the potential arbitrary we have derived new conditions for this model to realise the
phantom regime. These include a condition on the potential energy (coming from the positive energy condition)
and another condition on the allowed range of the kinetic energy so that the EoS parameter satisfies the phantom
limit. This computation shows that the inclusion of a generic self-interaction widens the range of kinetic energy
for achieving the phantom evolution. Naturally the question comes whether these new conditions derived here are
maintained throughout the late time evolution of the universe. Now one has to start with a definite potential to
trace the dynamics of any system. Since the purpose of the present paper is to stress the inclusion of a potential
in the GC model we do not assume any specific functional form of the potential apriori. We observed that the
structure of the ghost-condensate model gives a non-trivial significance to the obvious identity (30). This allowed
us to express the arbitrary potential in terms of the observable geometric quantities. These geometric quantities are
model independent32 and are determined by observations.
As recent observations16,17 indicate that there is a fair possibility of the late-time universe to follow the phantom
evolutionwehaveassumedthephantompowerlawforthescalefactor. Todeterminetheparametersappearinginthe
powerlawwehaveusedthecombinedWMAP7+BAO+HubbleaswellasWMAP7dataset. Consequently,weobtained
the potential and kinetic energies as functions of time. We have plotted the function f φ˙ (see equation (21)) along
with V (φ) in Fig. 1 and the line φ˙2 = M4 along with φ˙2 in Fig. 2. These plots revea(cid:16)led(cid:17)that the conditions which
were derived earlier for our model to realise the phantom regime holds throughout the future evolution. Naturally
the equation of state parameter ω should remain phantom-like. The variation of ω against time, shown in Fig. 3,
φ φ
clearly exibits the same.
Acknowledgement
The authors thank the referee for his useful comments. AS acknowledges the support by DST SERB under Grant
No. SR/FTP/PS-208/2012.
∗ Electronic address: [email protected]
† Electronic address: [email protected]
‡ Electronic address: [email protected]
§ Electronic address: [email protected]
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