On the self-interaction of dark energy in a ghost-condensate model

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by  Gour Bhattacharya

On the self-interaction of dark energy in a ghost-condensate model
Gour Bhattacharya∗
Department of Physics, Presidency University, 86/1, College Street, Kolkata 700 073, India†
Pradip Mukherjee‡
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†
Inaghost-condensatemodelofdarkenergythecombineddynamicsofthescalarfieldandgravitation
is shown to impose non-trivial restriction on the self-interaction of the scalar field. Using this
3 restriction we show that the choice of a zero self-interaction leads to a situation too restrictive for
1 thegeneralevolutionoftheuniverse. Thisrestriction,obtainedintheform ofaquadraticequation
0 of the scalar potential, is demonstrated to admit real solutions. Also, in the appropriate limit it
2
reproduces thepotential in thephantom cosmology.

n
a PACSnumbers: 98.80.-k,95.36.+x
J

1
2 I. INTRODUCTION ture.

Since very little is known about the nature of dark
]
c Recentcosmologicalobservationsindicatelate-timeac- energy it may appear that the presence or otherwise of
q celeration of the observable universe1,2. Why the evolu- an interaction term in the ghost-condensate model may
-
r tion of the universe is interposed between an early in- not be ascertainedfromany fundamental premise. How-
g flationary phase and the late-time acceleration is a yet- ever, in this letter we show that this issue can be settled
[ unresolved problem. Various theoretical attempts have by demanding a consistent scalar field dynamics. We
1 been undertaken to confront this observational fact. Al- establish here that this consistency requirement imposes
v though the simplest way to explain this behavior is the non-trivialrestrictiononthechoiceoftheself-interaction
6 consideration of a cosmological constant3, the known in the ghost-condensate model. Using this restriction
4 fine-tuning problem4 led to the dark energy paradigm. we show that describing the general evolutionary sce-
7
Here one introduces exotic dark energy component in nario of the universe using a ghost-condensate without
4
the form of scalar fields such as quintessence5–11, k- self-interaction may lead to too restrictive a situaton.
.
1 essence12–14 etc. Quintessence is based on scalar field Specifically,inthebouncinguniversescenario28–34 where
0 models using a canonical field with a slowly varying po- theuniversebouncesfromacontractingtoanexpanding
3
tential. On the other hand the models grouped under k- phase,absenceofself-interactionoftheghost-condensate
1
: essence are characterized by noncanonical kinetic terms. is not admissible at all. Further, that a real solution for
v A key feature of the k-essence models is that the cosmic theself-interactionpotentialiscompatiblewiththeghost
i
X accelerationisrealizedbythekineticenergyofthescalar field dynamics has been demonstraed using the restric-
field. Thepopularmodelsunderthiscategoryincludethe tion obtained here. It may also be noted that in the ap-
r
a phantom model, the ghost condensate model etc4. propriate limit the ghost-condensate model is known to
goovertothe phantommodel. Reassuringly,the restric-
It is well-knownthat the late time cosmic acceleration
tionwehavederived,reproducesthephantompotential35
requires an exotic equation of state ω < 1. Current
DE −3 in the same limit.
observationsallowω < 1 whichcanbe explainedby
DE
−
consideringnegativekineticenergywithafieldpotential. At this point it will be appropriate to describe the
The resulting phantom model15–20 is extensively used to organisation of this letter. In section 2 the ghost con-
confront cosmological observation21–26. This model is densate model is introduced where we include an arbi-
however ridden with various instabilities as its energy trary self-interaction potential. The equations of mo-
density is unbounded. This instability canbe eliminated tion for the scalar field and the scale factor are derived.
in the so-called ghost-condensate models27 by including These equations exhibit the coupling between the scalar
a term quadratic in the kinetic energy. In this context field dynamics and gravity. Expressions for the energy
let us note that to realize the late-time acceleration sce- density and pressure of the dark energy components are
nario some self-interaction must be present in the phan- computed. These expressions are used in section 3 to
tom model. In contrast, in the ghost-condensate models demonstratethattherequirementofconsistencybetween
the inclusion of self-interaction of the scalar field is be- the Friedman equations and the scalar field equations
lieved to be a matter of choice4. This fact, though not imposes nontrivial restriction on the self-interaction po-
unfamiliar, has not been emphasised much in the litera- tential in the form of a quadratic equation. The conse-
2
quences of this is discussed. The concluding remarksare Assuming a perfect fluid model we identify
contained in section 4. We use mostly positive signature
of the metric. 3X2
ρ = X + +V (φ) (9)
φ − M4
X2
II. THE GHOST CONDENSATE MODEL WITH pφ = Lφ =−X + M4 −V (φ) (10)
SELF-INTERACTION OF THE SCALAR FIELD
The equation of motion for the scalar field φ can be de-
Inthissectionweconsidertheghostcondensatemodel rived from the action (1). Due to the isotropy of the
withaself-interactionpotentialV(φ). Theactionisgiven FLRWuniversethescalarfieldisafunctionoftimeonly.
by Consequently, its equation of motion reduces to
S = d4x√−g 2Rk2 +Lφ+Lm , (1) 1 3φ˙2 φ¨+3H 1 φ˙2 φ˙ dV =0. (11)
Z (cid:20) (cid:21) − M4! − M4! − dφ
where
As is well known the same equation of motion follows
X2
Lφ = −X+ M4 −V (φ) (2) ferqoumattiohnesc(o9n)searnvdat(i1o0n)orfedTuµνce. Itnodeed under isotropy the
1
X = gµν∂ φ∂ φ (3)
µ ν
−2 1 3˙φ4
ρ = φ˙2+ +V (φ) (12)
M is a mass parameter, R the Ricci scalar and G = φ −2 4M4
k2/8π thegravitationalconstant. Theterm m accounts 1 φ˙4
for the total (dark plus baryonic) matter coLntent of the pφ = −2φ˙2+ 4M4 −V (φ) (13)
universe, which is assumed to be a barotropic fluid with
energydensityρmandpressurepm,andequation-of-state From the conservation condition T(φ)µν =0 we get
µ
parameterw =p /ρ . Weneglecttheradiationsector ∇
m m m
for simplicity. ρ˙ +3H(ρ +p )=0, (14)
φ φ φ
The action given by equation (1) describes a scalar
field interacting with gravity. Invoking the cosmological which, written equivalently in field terms gives equation
principle one requiresthe metric to be of the Robertson- (11).
Walker (RW) form
To complete the set of differential equations (5), (6),
(14) we include the equation for the evolution of matter
dr2
ds2 =dt2 a2(t) +r2dΩ2 , (4) density
− 1 Kr2 2
(cid:20) − (cid:21)
ρ˙ +3H(1+w )ρ =0, (15)
where t is the cosmic time, r is the spatial radial coor- m m m
dinate, Ω is the 2-dimensional unit sphere volume, K
2
where w = p /ρ is the matter equation of state pa-
characterizes the curvature of 3-dimensional space and m m m
rameter. The solution to equation (15) can immediately
a(t) is the scale factor. The Einstein equations lead to
be written down as
the Friedmann equations
n
H2 = k32 ρm+ρφ − aK2 (5) ρρmm0 =(cid:20)aa((tt0))(cid:21) , (16)
k(cid:16)2 (cid:17) K
H˙ = − 2 ρm+pm+ρφ+pφ + a2, (6) where n=3(1+wm) and ρm0 ≥0 is the value of matter
density at present time t . Now, the set of equations
(cid:16) (cid:17) 0
In the above a dot denotes derivative with respect to t (5), (6), (14) and (15) must give the dynamics of the
and H a˙/a is the Hubble parameter. In these expres- scalar field under gravity in a self-consistent manner. In
≡
sions, ρ and p are respectively the energy density and the next section we demonstrate that this consistency
φ φ
pressureofthe scalarfield. The quantities ρ andp are requirement constrains the self-interaction V (φ) in (1).
φ φ
definedthroughthesymmetricenergy-momentumtensor
2 δ
T(φ) = − √ g (7) III. RESTRICTION ON THE
µν √ gδgµν − SELF-INTERACTION OF THE SCALAR FIELD
−
(cid:0) (cid:1)
A straightforwardcalculation gives
We start by constructing two independent combina-
2X tions of the pressure and energy density of the dark en-
T(φ) =g + 1+ ∂ φ∂ φ (8)
µν µνLφ − M4 µ ν ergy sector in terms of the Hubble parameter H, matter
(cid:18) (cid:19)
3
energy density ρ , matter equation of state parameter Assumingmatterintheformofdust(n=3)inauniverse
m
w and curvature parameter K using (5), (6) and (15) withflatgeometry(K =1),thiscanbefurthersimplified
m
to
2H˙ n 2K
ρφ+pφ =A = − k2 − 3ρm+ k2a2 (17) 3H2+H˙ k2ρ >0 (26)
m
6a¨ ≥ 2
ρ +3p =B = (n 2)ρ (18)
φ φ −k2a − − m Using H˙ +H2 =a¨/a we reexpress this as
Using equations (12) and (13), we rewrite these combi-
1a¨
nations in terms of the ghost condensate field derivative H2 > (27)
φ˙ and potential V (φ): −2a
This condition appears to be too restrictive, in fact in
φ˙4
ρ +p =A = φ˙2+ (19) the decelerating phase (a¨ < 0) this imposes a definite
φ φ − M4 relation between a˙ and a¨. Remembering that the Fried-
3φ˙4 mann equation is of second order in time, there is no a
ρ +3p =B = 2φ˙2+ 2V (φ) (20)
φ φ − 2M4 − priori reason that such constraint holds. Moreover, the
ekpyrotic28–30 and other bouncing theories31–34 of the
Inverting these equations to write φ˙2 and φ˙4 in terms early universe require that spacetime “bounce” from a
of A, B and V (φ) and utilizing the algebraic identity contracting to an expanding phase, perhaps even oscil-
φ˙2 2 =φ˙4 we obtain the following quadratic equation lating cyclically [9, 10]. Clearly, during the switch over
from expanding to contracting phase, a˙ = 0 but a¨ < 0
(cid:16) (cid:17) and thus the condition (27) is violated.
3A M4
V2(φ) + B + V (φ) Theanalysisdetailedabovedemonstratesthatinorder
− 2 4 toapplytheghostcondensatemodelforgeneralevolution
(cid:18) (cid:19)
(3A 2B)2 4M4(A B/2) of the universe a certain self-interaction should always
+ − − − =0 (21) be included. At this point one may wonder whethar the
16
constraining equation (21) on V (φ) at all allows a real
This is the restriction on the choice of potential in the solution. Solving (21) we get
ghost-condensatemodelwhichhasbeenindicatedearlier.
Note that the simple fact that V (φ) has to satisfy a re- 3A 2B M4 M4 M4 12
striction of this form implies that care must be taken in V (φ)= − +A (28)
4 − 8 ± 16 4
assertingtheabsenceoftheself-interactionterm. Wewill (cid:18) (cid:19) (cid:26) (cid:18) (cid:19)(cid:27)
presently discuss this and other issues related to equa-
The reality condition is thus
tion (21) in the following. Meanwhile, observe that in
the limit M4 the equation (21) reduces to M4
→∞ +A 0 (29)
4 ≥
V (φ)=A B/2 (22) (cid:18) (cid:19)
−
That this condition is satisfied in general can be estab-
Substituting for A andB in (22)andsimplifying, we get
lishedexplicitlyifwesubstituteforAfromequation(19)
which gives
1 2K n 6
V (φ)= 3H2+H˙ + + − ρ (23)
k2 (cid:18) a2 (cid:19) 6 m M4 1 M4 2
+A = φ˙2 0 (30)
where equations (17) and (18) have been used. This 4 M4 − 2 ≥
(cid:18) (cid:19) (cid:18) (cid:19)
reproduces the result for the potential in the phantom
model35. This completes our argument in favour of including a
self-interaction potential in ghost-condensate model.
Coming back to equation (21) let us first investigate
the possibility of a vanishing self-interaction. Substitut-
ing V (φ)=0 we get
IV. CONCLUSION
(3A 2B)2
− =(A B/2) (24)
4M4 − We have considered the ghost-condensate model of
dark energy with a self-interaction potential in a gen-
Since the left hand side is positive definite we immedi- eral FLRW universe with curvature K. The combined
ately get the condition dynamics of dark energy and gravity leads to coupled
differential equations involving the universal scale factor
B 1 2K n 6
A = 3H2+H˙ + + − ρ 0 a(t)andthescalarfieldφ. Thestandardbarotropicmat-
− 2 k2 a2 6 m ≥ ter equation of state is assumed. Two independent com-
(cid:18) (cid:19) (cid:18) (cid:19)
(25) bination of the pressure and energy density of the dark
4
energy are expressed in terms of the observable quanti- the bouncing universe scenario disallows such a choice.
ties from the normal matter and gravity sector. These Our analysis thus establishes that in the class of ghost-
combinationsarethen usedto impose a consistencycon- condensate models for general evolution of the universe
dition which leads to a quadratic equation for the self- a self-interaction of the dark energy must be included.
interaction V (φ). This equation is shown to admit real
roots. Also,intheappropriatelimititleadstothephan-
tom model potential35.
Averyinterestingconsequenceariseswhenweexamine
Acknowledgement
the plausibility of the choice of zero self-interaction. Us-
ingthequadraticequationsatisfiedbytheself-interaction
it has been demonstrated that this choice is too restric- The authorswouldliketo thank IUCAA, Pune,where
tive for the general evolution of the scale factor. In fact, part of the work was done.
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† Visiting Associate, InterUniversityCentre for Astronomy (2003).
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‡ Electronic address: [email protected] 043543 (2004).
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