DBI-essence

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by  Jerome Martin

DBI-essence
J´erˆome Martin∗
Institut d’Astrophysique de Paris, UMR 7095-CNRS,
Universit´e Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, France
Masahide Yamaguchi†
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara 229-8558, Japan
(Dated: February 2, 2008)
Modelswherethedarkenergyisascalarfieldwithanon-standardDirac-Born-Infeld(DBI)kinetic
termareinvestigated. Scalingsolutionsarestudiedandproventobeattractors. Thecorresponding
shape of the brane tension and of the potential is also determined and found to be, as in the
standard case, either exponentials or power-law of the DBI field. In these scenarios, in contrast to
8 the standard situation, the vacuum expectation value of the field at small redshifts can be small
0 in comparison to the Planck mass which could be an advantage from the model building point of
0 view. This situation arises when the present-day value of the Lorentz factor is large, this property
2 being per se interesting. Serious shortcomings are also present such as the fact that, for simple
n potentials, the equation of state appears to be too far from the observational favored value −1.
a Another problem is that, although simple stringy-inspired models precisely lead to the power-law
J shape that has been shown to possess a tracking behavior, the power index turns out to have the
2 wrong sign. Possible solutions to these issues are discussed.
2
PACSnumbers: 98.80.Cq,98.70.Vc

]
h
t I. INTRODUCTION not be done without going beyond the standard model
-
p of particle physics. In particular, it would be very in-
e teresting to achieve this goal in string theory since it is
h Since the discoverythat the expansionofthe Universe
presently our best candidate as a unified theory [18].
[ is presently accelerated [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
1 various suggestions have been made in order to explain Recently, there have been many works aiming at con-
necting stringtheory with inflationwhichis alsoa phase
v this observational fact. Amongst them is the hypothe-
of accelerated expansion (but taking place in the very
5 sis of dark energy, a fluid with a negative pressure rep-
early Universe at a much higher energy scale). For this
7 resenting about 70% of the total energy density in the
3 Universe. The question of the physical nature of the purpose,newideasinstringtheorybasedontheconcept
3 of branes haverevealedthemselves especially fruitful. In
dark energy has, of course, been widely discussed. The
. particular, scenarios where the inflaton is interpreted as
1 most natural candidate, still perfectly compatible with
0 allthedataavailable,isthecosmologicalconstant. How- the distance between two branes moving in the extra di-
8 mensions along a warped throat have given rise to many
ever, the difficulty to reconcile the value of Λ deduced
0 interesting studies [19, 20, 21, 22]. In this article, we
from the observations with the value calculated theo-
: want to investigate whether the same kind of ideas can
v retically [11] (maybe too naively?) has prompted the
Xi study of alternatives. Clearly, a simple scalar field, a lead to sensible dark energy scenarios.
“quintessence” field, is a natural candidate for such an At the technicallevel,scenariosof the type mentioned
r
a alternative [12, 13, 14, 15, 16]. Amongst all the possibil- above lead to scalar field models where the kinetic term
ities, scalarfields with inverse power-lawpotentials have is non-canonical. More precisely, the kinetic term has
attracted lot of interest because, in this case, there is a Dirac-Born-Infeld (DBI) form. Physically, this origi-
a solution of the equations of motion that is an attrac- nates from the fact that the action of the system is pro-
tor[12]. Thismeansthatthepresent-daybehaviorofthe portional to the volume traced out by the brane during
Universe is insensitive to the initial conditions. Usually, its motion. This volume is given by the square-root of
the attractorsolutionis a scalingsolution, i.e. a solution the induced metric which automatically leads to a DBI
forwhichtheenergydensityscalesasapowerofthescale kinetic term. Therefore, as a first step toward a sce-
factor [17]. narioof“DBI-essence”,itisfirstnecessarytounderstand
whether scaling and attractor solutions are still present
If the above mentioned route is correct, then another
when the scalar field has a DBI kinetic term. This ques-
interesting issue is whether a candidate for quintessence
tion constitutes the main target of the present article.
inhighenergyphysicscanbeidentified. Clearly,thiscan-
Thispaperisorganizedasfollows. InSec.II,webriefly
review the scaling properties of a quintessence field with
a standardkinetic term. Then, in Sec. III, we reconsider
∗Electronicaddress: [email protected] this questionbut withaDBI kinetic term. Inparticular,
†Electronicaddress: [email protected] we compare the DBI results with the standard ones. In
2
Sec. IV, we study the behavior of DBI-essence at small where λ is a constant or is of the Ratra-Peebles type
redshifts. Since,inthiscase,thescalarfieldisnolongera (i.e. inverse power-law of the field), namely
test field, this requires numerical computations. Finally,
in Sec. V, we present our conclusions and discuss the V(φ)=M4+αφ−α. (7)
open issues that should be studied in the future.
In the first case, the particular solution leading to the
scaling behavior reads
II. SCALING SOLUTIONS WITH A STANDARD 2 t
KINETIC TERM φ(t)= ln , (8)
λ t
(cid:18) 0(cid:19)
We consider a spatially flat Friedmann-Lemaitre- the constantλandthemassscaleM beinglinkedbythe
Robertson-Walker(FLRW)Universecontainingaperfect relation
fluid and a scalar field φ. Assuming that the scalar field
2(1 ω)
and the perfect fluid are separately conserved, the equa- λ2M4t2 = − . (9)
0 1+ω
tions of motion are given by
As is well-known, the quintessence equation of state pa-
κ
H2 = (ρ+ρ ) , (1) rameterisjustgivenbytheequationofstateoftheback-
φ
3
groundperfectfluid,ω =w. Theparticularsolution(8)
φ
ρ˙+3H(ρ+p) = 0, (2)
is important because it is an attractor. This means that
ρ˙φ+3H(ρφ+pφ) = 0, (3) the final(i.e. presentday)evolutionofthe field is in fact
independent of the initial conditions. At the technical
whereH a˙/ais the Hubble parameterandρ andpare level, this can be seen by studying small (linear) per-
≡
respectively the energy density and the pressure of the turbations around the attractor. The eigenvalues of the
perfect fluid. A dot denotes a derivative with respect to perturbations around the critical point can be expressed
cosmictimeandthequantityκisdefinedbyκ 8π/m2. as
≡ Pl
In the following, we assume that the perfect fluid has a
1
constant equation of state parameter ω p/ρ, the two λ = (m 6) (m 6)2+8m(m 6) , (10)
cases of main interest being ω = 1/3 for≡the radiation- ± 2m − ± − −
(cid:20) q (cid:21)
dominated era and ω = 0 for the matter-dominated
where we have defined m 3(1 + ω). Since ω < 1,
era. In this case, the conservation equation (2) can ≡
one has m<6 and the eigenvalues are negative and one
be integrated exactly and leads to the familiar behav-
has a stable spiral point. Moreover, for the particular
ior ρ a−3(1+ω). Moreover, if we further assume that
∝ solution (8), one has
thescalarfieldisatestfieldandthattheevolutionofthe
backgroundgeometryis mainlycontrolledby the perfect d2V 9
fluid, then one has a(t) t2/[3(1+ω)] or, for the Hubble = 1 ω2 H2. (11)
∝ dφ2 2 − φ
parameter, H =2/[3(1+ω)t].
(cid:0) (cid:1)
Let us first briefly remind what are the scaling solu- This is an important formula because it implies that
tions inthe simplecasewherethe scalarfieldhasastan- φ m today. Indeed, V′′ V/φ2 and H2 V/m2
∼ Pl ∼ ∼ Pl
dard kinetic term. In this situation, the energy density when the field starts dominating the energydensity con-
and the pressure are given by the familiar expressions tent of the Universe; equating these two quantities leads
to the above mentioned conclusion. For this reason, a
φ˙2 φ˙2 sensible model building of quintessence is only possible
ρ = +V(φ), p = V(φ). (4)
φ 2 φ 2 − in a supergravity (SUGRA) framework [14, 15]. How-
ever, a well-known difficulty of the exponential case is
If one inserts these expressions into the conservation
that the property ω = ω implies that the scalar field
φ
equation for the scalar field (3), then one obtains the
cannot drive an accelerated expansion. This is why the
Klein-Gordon equation
inverse power-law case seems to be more interesting.
In the case of the Ratra-Peebles potential (7), there
φ¨+3Hφ˙+V′(φ)=0, (5)
also exists an exact particular solution of the Klein-
Gordon equation that is an attractor. It reads
where a prime denotes a derivative with respect to φ.
Then, we seekpotentials V(φ) suchthatthe energyden- 2/(α+2)
t
sity of the test scalar field scales as a power-law of the φ=φ , (12)
0
scale factor, namely ρφ a−3(1+ωφ) where ωφ pφ/ρφ (cid:18)t0(cid:19)
∝ ≡
is a constant. It has been established in Ref. [12] that
where the quantity φ is linked to the mass scale M by
0
scalingsolutionsexistifthe potentialhasanexponential
the formula
shape,
2 2 α
M4+αφ−α−2t2 = . (13)
V(φ)=M4e−λφ, (6) 0 0 α(α+2) 1+ω − α+2
(cid:18) (cid:19)
3
Forthis particularsolution,theequationofstateparam- Theexpressions(18)oftheenergydensityandpressureof
eter can be expressed as theDBIfieldshouldbecomparedtotheirstandardcoun-
terpart, see Eqs. (4). As usual, if one inserts Eqs. (18)
αω 2
ω = − . (14) in the conservation equation (3), one obtains the DBI
φ
α+2 Klein-Gordon equation, namely
As expected, in the limit α + , one recovers the ex-
ponential case, ωφ = ω. Ho→weve∞r, the crucial difference φ¨ 3T′(φ)φ˙2+T′(φ)+ 3Hφ˙+ 1 [V′(φ) T′(φ)]=0.
− 2T(φ) γ2 γ3 −
with the exponential potential is that one can now have
(20)
ω <ω,thatis tosay,the scalarfieldenergydensity can
φ
Wenoticethattheequationofmotionforφisquitecom-
now scale more slowly than the background fluid and,
plicated compared to Eq. (5) despite the fact that the
hence, eventually dominates,causing the Universe to ac-
conservation equation has retained its standard form.
celerate. Moreover, the solution (12) is also an attrac-
Let us also compare with other works in the litera-
tor as revealed by a dynamical system analysis. Indeed,
ture. Let us start with K-essence where the action can
theeigenvaluesofsmallperturbationsaroundthecritical
be written as
point read
(2n m 6) (2n m 6)2+8m(n 6) S = d4x√−gp(φ,X) , (21)
λ = − − ± − − − , Z
± q 2m
(15) where X = ( φ)2/2. Clearly the action (17) is a spe-
∇
where we have defined n 3(1+ω ). Again, one can cial case of the above action. However, as first dis-
φ
≡
showthatthereexistsastablespiralpointaslongasboth cussedinRef.[23], K-essenceusuallymeansthatthe po-
eigenvaluesarenegative,whichisequivalentto2n m tential term vanishes and the negative pressure of the
− −
6 < 0. On this attractor, the evolution of the second scalar field is realized only by considering the kinetic
order derivative of the potential is given by term[24, 25, 26, 27]. Onthe otherhand, our modelcan-
not realize the negative pressure without the potential
d2V 9α+1
= 1 ω2 H2, (16) term, as shown below. Therefore, this class of models
dφ2 2 α − φ cannot encompass Eq. (17). Another model related to
(cid:0) (cid:1) the present study is the case where the dark energy field
and one check that this last equation reproduces the
is a tachyon for which the action is given by [28]
corresponding equation in the exponential case when
α + . Again, this prompts a SUGRA treatment
→ ∞ 1
of the model building issue since one still has φ ∼ mPl S =− d4x√−gV(T) 1+ M4gµν∂µT∂νT , (22)
now. Z r
where M is a fundamental scale and V(T) is a poten-
tial which, of course, needs not to be the same function
III. SCALING SOLUTIONS WITH A DBI
as V(φ) in Eq. (17). This class of theory is equivalent
KINETIC TERM
to the case studied here (through a redefinition of the
field) only when the potential in Eq. (17) vanishes. Let
Letusnowconsiderthatthedarkenergyscalarfieldis
us also notice that when V(T) is constant the model is
a Dirac-Born-Infeld (DBI) scalar field. In this case, the
in fact equivalent to the Chaplygin gas with the equa-
action of the field can be written as
tion of state p 1/ρ [29, 30]. Therefore, beside the
∝ −
fact that the searchof scaling solutions has not yet been
φ˙2
S = d4xa3(t) T(φ) 1 +V(φ) T(φ) , investigatedin this type of models, we conclude that the
DBI −  s − T(φ) −  class of scenarios under scrutiny in this paper was not
Z
considered before.
 (17)
As a warm up, let us find the scaling solutions in the
whereT(φ)isthetensionandV(φ)isthepotential. From
simple case where the potential vanishes. As already
this expression, it is easy to obtain the corresponding
mentioned before, this means that we now seek tensions
energy density and pressure of the scalar field. They
T(φ) such that the energy density of the test DBI scalar
read
field scales as a ρφ a−3(1+ωφ). From Eqs. (18), it is
γ 1 ∝
easy to show that γ is, in this case, constant and given
ρ =(γ 1)T(φ)+V(φ), p = − T(φ) V(φ),
φ φ
− γ − by γ = 1/ω . Then, the formula of the energy density,
φ
(18) ρφ = (γ 1)T(φ) a−3(1+ωφ) immediately gives the
where the quantity γ is reminiscent from the usual rela- − ∝
scaling in time of T(φ) which in turn, combined with
tivistic Lorentz factor and is given by φ˙2/T(φ) = (γ2 1)/γ2 and the fact that γ is constant,
−
1 implies that
γ . (19)
≡ 1 φ˙2/T(φ) φ˙ t−(1+ωφ)/(1+ω), (23)
− ∝
q
4
This equation is easily solved. Let us start with ω =ω. initsderivation,onehasneverassumedthatV =0. This
φ
In this case, one has φ lnt and, as a consequence, implies that, asinthe caseofavanishing potential,scal-
∝
ing solutions exist for tensions T(φ) givenby Eq.(24) or
T(φ)=M4e−λφ, (24) Eq. (25). Then, the Klein-Gordon equation (20) can be
usedto determine the potential. Straightforwardmanip-
where λ is a constant. Again, this case is very simi-
ulations lead to
lar to the situation where we have a standard kinetic
term and a exponential potential. As a consequence, V(φ) γ2 1 1 γ
= − , (27)
this model suffers from the standard phenomenological T(φ) γ 1+ω − 1+γ
(cid:18) φ (cid:19)
problems. Since the scalarfieldsexactlytracksthe back-
ground matter, one cannot have a large enough contri- thatis to saythe potentialis proportionalto the tension
butionofdarkenergydensitytodaywithoutspoilingBig and has also the exponential shape or inverse power-law
Bang Nucleosynthesis (BBN). Moreover, the scalar field shape. It is interesting to notice that, when the field is
behavesasmattertodayand,therefore,cannotcausethe ontracksinthestandardkineticcase,thepotentialterm
acceleration of the Universe. isalsoproportionaltothekineticterm,thatistosaythe
On the other hand, if ω = ω, then the scalar field is ratio of the potential term to the kinetic term K φ˙2/2
φ
6 ≡
just a power-law of the cosmic time which implies that is a constant given by V(φ)/K(φ)=2/(1+ω ) 1.
φ
−
T(φ) can be expressed as In the case of an exponential potential, the exact so-
lution (from now on, we put a subscript “e” to de-
T(φ)=M4+αφ−α, (25) note the quantities that are evaluated with the exact
particular solution of the Klein-Gordon equation) reads
where M is a mass scale and ωφ is related to α and the φ (t)=2/λln(t/t )withλ2M4t2 =4γ2/(γ2 1)andthe
background equation of state parameter ω through the e 0 0 e e−
equation of state parameter is ω = ω. In the Ratra-
φ
relation
Peebles case, one as φ (t)=φ (t/t )2/(α+2) with
e 0 0
αω 2 1
ωφ = − = . (26) 4γ2
α+2 γ M4+αφ−α−2t2 = e , (28)
0 0 (α+2)2(γ2 1)
e −
Interestingly enough, as discussed before, this is exactly
the equation obtained when there is a standard kinetic andthe equationofstatehasthestandardformgivenby
term with a Ratra-Peebles potential, see Eq. (14). How- Eq.(14). Itisimportancetonoticethat,becausewedeal
ever,inthepresentcase,ω =1/γ >0. Thismeansthat with a modified Klein-Gordon equation, the expressions
φ
the solution is physically relevant only if ω >2/α which ofλ2M4t2 andM4+αφ−α−2t2 aredifferentfromtheones
0 0 0
excludes the case ω =0, at least for α>0. obtained previously, see Eqs. (9) and (13). Let us also
Sinceitappearsthatthepreviouslydescribedsituation remark that these formulas can either be obtained from
is not satisfactory, we now envisage the case where the therequirementthattheLorentzfactorisconstantorby
potential V(φ) is non vanishing. In order to deal with brute forcecalculationusingthe Klein-Gordonequation.
this problem, we assume that γ is a constant. Without Let us now study the behavior of small perturbations
this hypothesis, the problem is technically very compli- around the particular solutions. Let us first start with
catedbuttheactualconvincingargumentinfavorofthis the exponential case. For this purpose, we rewrite the
assumption is that the corresponding scaling solutions equationofmotionintermsofu(τ)definedbyu λ(φ
(with γ constant) are attractors, see below. Then, the φ ) and t eτ. If we write p u′, then one obt≡ains th−e
e
≡ ≡
crucial observation is that Eq. (23) is still valid because, system
dp 2 1 3 4 1 1 γ2 1 1 γ
= 5+ p p2 +λ2M4 e−u 1 +λ2M4 e − e 1
dτ − γ21+ω − 2 − 1+ω γ2 − γ2 − γ 1+ω − 1+γ −
(cid:18) (cid:19) (cid:18) e(cid:19) (cid:20) e (cid:18) e(cid:19) (cid:21)
e−u 1 (cid:0) (cid:1)
=0, (29)
× γ3 − γ3
(cid:18) e(cid:19)
du
= p, (30)
dτ
where, now, the quantity γ is no longer a constant and can be written as
γ2 1 p 2 −1/2
γ = 1 e − eu 1+ . (31)
− γ 2
(cid:20) e (cid:16) (cid:17) (cid:21)
5
Asaconsequence,weseethatthecriticalpointis(p,u)= This expression should be compared with Eq. (10). The
(0,0). Notice that, for the criticalpoint, one checks that only difference is the presence of the factor γ in the last
e
γ = γ . We now consider the behavior of small per- term inside the square root. Otherwise, and this is quite
e
turbations (δp,δu) around the critical point (0,0). It is remarkable,theexpressionisthesame. Letusalsonotice
straightforwardto establish that that the condition that the kinetic energy cannot exceed
the total energy is equivalent to n 3(γ +1)/γ . Since
e e
ddτ (cid:18)δδup(cid:19) = 1−11+2ω 2− 1+20ω1+γe2γe2!(cid:18)δδup(cid:19) .(32) tγahueet≥ocmo1an,tdtihictaeiollcnyonnsdait≤tiisofin3e(dnγei≤n+3o1(uγ)r/e2γc+eas1ae≤)n./dγ,We2teihsecsroternfoocnrlueg,deeristthhnaaontt
there is a stable spiral point if n 3(γ2+1)/γ2.
Then, the eigenvalues of this matrix can be expressedas ≤ e e
Let us now turn to the inverse power-law case. This
1
λ± = (m 6) time,thedimensionlessfunctionu(τ)isdefinedbyu(τ)
2m" − φ/φ , the definition of the time τ remaining the sam≡e.
e
Then,astraightforwardcalculationleadstothefollowing
γ2+1
(m 6)2+8m m 3 e . (33) system of equations
±s − (cid:18) − γe2 (cid:19)#
dp 5α+2 2 1 3αp2 4 u 1 1
= + p +αM4+αφ−α−2 u−α−1 u
dτ − α+2 γ21+ω − 2 u − 1+ωα+2 γ2 − γ2 0 −
(cid:18) (cid:19) (cid:18) e(cid:19)
γ2 1 1 γ u−α−1 u (cid:0) (cid:1)
+αM4+αφ−α−2 e − e 1 =0, (34)
0 γ 1+ω − 1+γ − γ3 − γ3
(cid:20) e (cid:18) e(cid:19) (cid:21)(cid:18) e(cid:19)
du
= p, (35)
dτ
where, this time, the Lorentz factor can be written as the square root. Therefore, there is a stable spiral point
if n 3(γ2+1)/γ2 and 2n m 6<0 are satisfied.
uα 4up 4u2 −1/2 Fi≤nally,eon the aettractor,−in th−e exponential case, it is
γ = 1 p2+ + . easy to establish that the following relation holds
( − M4+αφ−0α−2 " α+2 (α+2)2#)
(36) d2V =9γ (1+ω) 1 (1+ω) γe H2. (39)
It is clear form the above system that the critical point dφ2 e − 1+γ
(cid:20) e(cid:21)
is now given by (p,u)=(0,1). The next step is to study
ThisformulaisthegeneralizationofEq.(11). Obviously,
the behavior of small perturbations (δp,1+δu) around
onecanalsoestablishthecorrespondingexpressioninthe
the critical point. One arrives at
inverse power-law case. It reads
d δp α−2 2 2α 2 1+γe2 δp d2V α+1 γ
dτ (cid:18)δu(cid:19) = α+2 −1 1+ω α+2 − 10+ω γe2 !(cid:18)δu(cid:19) . dφ2 =9 α γe(1+ωφ)(cid:20)1−(1+ωφ)1+eγe(cid:21)H2,
(40)
(37)
and this is equivalent to Eq. (16). This has important
consequences for model building. Indeed, if one repeats
As expected,inthe limit α + ,the abovematrixex-
→ ∞ thediscussionafterEq.(11),thenonearrivesatthecon-
actly reproduces the matrix obtained in the exponential
clusion that
case, see Eq. (32). Then, the next step is to determine
the eigenvalues. The result reads φ mPl (41)
∼ √γe
2n m 6
λ± = − − becausethesecondterminthebracketinEq. (40)cannot
2m
exceed unity. Therefore, if γ 1, then the vacuum
e
1 γ2+1 ≫
(2n m 6)2+8m n 3 e (.38) expectation value of the field is not necessarily large in
± 2ms − − (cid:18) − γe2 (cid:19) Planck units. This is certainly an important advantage
of the DBI models over the standard ones as respect to
Thisexpressionshouldbe comparedwithEq.(15). Asit model building issues.
was the case before, the modification introduced by the Finally, it is also worth commenting about the shape
DBI kinetic termis only apparentin the lastterminside of the tension T(φ). From a stringy point of view, the
6
FIG. 1: Evolution of the DBI energy density for different FIG. 2: Evolution of the DBI equation of state for different
initial conditions in the case where the potential is of the initial conditions in the case where the potential is of the
Ratra-Peebles type with α = 4. The value of C is chosen to Ratra-Peebles potential with α = 4. As in Fig. 1, the value
beC ∼3.443whichcorrespondstoγe=20,seeEq.(42). The of C is chosen to beC ∼3.443 which corresponds to γe=20
initial velocity φ˙ini is always chosen such that γini = 5. The and the initial velocity φ˙ini is such that γini = 5. The solid
solid line corresponds to an initial vacuum expectation value line corresponds to an initial vacuum expectation value of
of φini/mPl ∼10−10, the dotted line to φini/mPl ∼10−9 and φini/mPl ∼10−10,thedottedlinetoφini/mPl ∼10−9 andthe
the dashed line to φini/mPl ∼ 10−8. The energy density of dashedlinetoφini/mPl ∼10−8. Thefinal(presentday)value
radiation (dotted-dashedline) and cold dark matter(dotted- of γ is γ0∼3.96 and φ0/mPl ∼1.37. Finally, theequation of
dotted-dashed line) are also represented. state is such that ω0∼−0.42, ω1∼6.68×10−2.
criticalenergydensity, we havetuned the scale M of the
inverse of T(φ) represents the warp factor of the throat
brane tension T(φ), see Eq. (25). The mass scale of the
in which the branes are living. A natural choice [21] is
T(φ) φ4, that is to say α = 4. Therefore, this case potential is determined by Eq. (27) which implies that
∝ − V(φ)=CM4+αφ−α where C is defined by
belongs to the class of tracking models considered here
which is a non trivial result. Unfortunately, the sign of
γ2 1 1 γ
the exponent is not the correctone. Indeed, for α= 4, C e − e
one has ωφ =2ω+1>ω which means that, despite−the ≡ γe (cid:18)1+ωφ − 1+γe(cid:19)
presenceofanattractor,thescalarfieldscalesfasterthan γ2 1 α+2 γ
thebackgroundfluidand,hence,canneverdominatethe = e − e . (42)
γ α(1+ω) − 1+γ
matter content of the Universe. e (cid:20) e(cid:21)
Choosing a value of C is in fact equivalent to choosing
the value of the Lorentz factor on the attractor, γ , dur-
e
ingaphaseofevolutioncharacterizedbythebackground
IV. NUMERICAL CALCULATIONS
equation of state ω. So, for instance, in Fig. 1, we have
chosen γ = 20 and ω = 1/3. This means that the at-
e
In this section, we investigate the behavior of the DBI
tractor solution should be such that γ = 20 during the
scalar field at small redshifts, when it starts dominating e
radiationdominated era. Given Eq.(42) and α=4, this
the matter content of the universe. In this situation,
choice implies that C 3.443. In this case, there is also
the assumption that it is a test field breaks down and ∼
an attractor during the matter dominated era but the
numerical calculations are required.
corresponding value of the Lorentz factor is different. It
We first check that the attractor is observed numer-
is easy to show that it reads
ically. As a representative example, we have chosen to
investigate the case α = 4. In Fig. 1, we have repre-
α(C 1) 8(α+2)
sented the evolution of the DBI energy density for three γcdm = − 1+ 1+ . (43)
different initial conditions (more precisely, the initial ve- e 4 " s α2(C 1)2#
−
locity is always the same and corresponds to an initial
value of the Lorentz factor γ = 5 but different initial In the present case, this gives γcdm 5.438.
ini e ∼
vacuumexpectationvaluesφ areconsidered). Inorder The attractor behavior is clearly seen in Fig. 1. For
ini
to have a DBI energy density today equal to 70% of the initial conditions φ /m 10−9 and φ /m 10−8,
ini Pl ini Pl
∼ ∼
7
the attractor is joined during the radiation-dominated
era while for φ /m 10−10, it is reached during the
ini Pl
∼
matter-dominated era.
In Fig. 2, we have represented the evolution of the
equation of state for the same situation. Again, the at-
tractor behavior is clearly noticed. We can even check
numerically that, on the attractor, Eq. (14) is valid.
Since we consider a model with α = 4, the DBI equa-
tion of state during the radiation dominated era should
be ω 0.11. Clearly, this is what is obtained in
φ
≃ −
Fig. 2. The present day value of the equation of state
is ω 0.42 (the derivative of the equation of state
0
at van≃ish−ing redshift being ω 6.68 10−2). The
1
≃ ×
corresponding value for a scalar field with a standard
kinetic term and the same Ratra-Peebles potential is
ω 0.487. Firstly, and contrary to a naive expec-
0
≃ −
tation, the equation of state is not pushed towards 1.
−
Therefore,itseemsthatwedonotgainanythingincom-
parison with the model with a standard kinetic term. FIG. 3: Evolution of the Lorentz factor for different initial
Secondly, the value obtained seems to be too large given conditions for the Ratra-Peebles potential with α = 4. The
the constraintsavailableonω . Evenif oneshouldput a value of C is chosen to be C ∼ 3.443 which corresponds to
damperontheseconstraintssi0ncetheyhavenotbeenob- γe=20andtheinitialvelocityφ˙ini ischosensuchthatγini=
5. Thesolidlinecorrespondstoaninitialvacuumexpectation
tained for the model under considerations here (usually,
aclesaimrlpylenolatwvaolfidthfoerfothrme mωo=deωl0u+ndωe1rzcoanresiudseerdat,iwonhihcehreis, vathanelduLetohorefeφndtianzsi/hfmaecdPtollir∼nies10tγo−01φ∼0i,n3ti/.h9me6dPlot∼ted10l−in8e.toThφeinifi/nmaPllv∼alu1e0−o9f
and this can cause a “bias problem”, see Ref. [31]), the
valueissofarfrom 1thatthemodelisprobablyintrou-
−
ble from the observational point of view. This is clearly
tor is represented for different values of γ . We notice
a very serious problem for the class of models studied e
that the larger γ , the larger the present day value γ
in the present article. One possibility is to decrease the e 0
and the smaller φ . For instance, for γ =1000, one ob-
value of α. For instance, α = 0.3 implies ω 0.9. 0 e
Ofcourse,thecorrespondingmodelseemscontφriv∼ed−and, tains γ0 ∼176and φ0/mPl ∼0.2<1. This is certainly a
desirable feature since, usually, vacuum expectation val-
in addition, in this case, a small value of the equation
ues of the order of the Planck mass are at the origin of
of state would also be obtained with a standard kinetic
many serious problems as, for instance, a coupling with
term. Another possibility would be to consider other
the observable sector which violates the constraint on
shapes for the tension andthe potential. The new shape
the presence of a fifth force and/or on the weak equiv-
of T(φ) and V(φ) should approximativelyreduces to the
alence principle [32, 33, 34, 35]. On the other hand, if
inverse power-law shape at large redshifts such that the
the vacuum expectation value remains small in compar-
attractor behavior is preserved and should differ from
ison with the Planck mass, then it could be difficult to
it at small redshifts in order to obtain an equation of
use the SUGRA potential model to push the equationof
state closer to 1. A typical example is provided by the
− state towards 1. Therefore, we face again the “no-go
SUGRApotential[14,15]. However,weshowbelowthat −
theorem” discussed recently in Refs. [33, 34, 35]: what
even this solution could not work in the DBI case.
is interesting from the cosmological point (a large vac-
Let us now study the evolution of the Lorentz factor uum expectation value in order to have ω close to 1)
0
γ. It is represented in Fig. 3 for different initial con- seems to be incompatible with local tests of gravity−(a
ditions, similar to the ones considered in the previous large vacuum expectation value usually means a strong
figures. In particular, the attractor value, valid during coupling with ordinary matter).
the radiation dominated era, γ = 20, is clearly seen
e
on this plot. The value γcdm 5.4, valid during the
e ∼
matter dominated era, can also be noticed. An interest-
V. CONCLUSIONS
ing point is that the present value of the Lorentz factor
is far from 1. For the case under consideration, it is
γ 3.96. This means that the non-standard kinetic Inthis section,werecapourmainfindingsanddiscuss
0
∼
termstillplaysaroleeventoday. Asnoticedearlier,this furtherissuesthatshouldbeinvestigated. Wehavestud-
can have important implications for model building is- iedscenarioswherethedarkenergyisascalarfieldwitha
sues since a large γ implies a small vacuum expectation DBI kinetic term. We haveshownthat, if the braneten-
e
value of the field. We have studied this point in more sion and the potential possess either an exponential or a
details in Fig. 4 where the evolution of the Lorentz fac- power law shape, then there exists scaling solutions that
8
also an interesting feature of the models under scrutiny.
Forexample,thisimpliesthatthesoundvelocitysquared
c2cansignificantlydeviatefrom1incontrasttothestan-
s
dard case. Indeed, in the DBI case, the sound velocity
squared c2 is given by the following expression
s
∂p ∂ρ −1 1
c2 = = , (44)
s ∂X ∂X γ2
where p = p(X,φ), ρ (cid:18)= ρ((cid:19)X,φ) (aend X = φ˙2/2).
A dark energy component with c2 1 implies less
s ≪
power on large scales and, hence, could account for
the low multipoles of the cosmic microwave background
anisotropies. Moreover,this wouldalso produce peculiar
features in the matter power spectrum as discussed in
Refs. [36, 37, 38]. All these properties could be used to
distinguishtheDBImodelsfromthestandardones. One
more general grounds, it is clear that a complete calcu-
lation of the dark energy perturbations could bring new
FIG. 4: Evolution of the Lorentz factor for different values insights on the model.
of γe in the case where the potential is of the Ratra-Peebles Finally,anotherinterestingissueisthatofthecoupling
type with α = 4. The values considered are γe = 20 (solid of dark energy with the rest of the world. As already
line), γe = 100 (dotted line), γe = 500 (dashed line) and
mentioned, this is usually a problem for quintessence
γe =1000 (dotted-dashed line). The corresponding values of
theLorentzfactortodayareγ0 ∼3.96,γ0∼17.87,γ0∼88.56 because a small mass means a force with a very long
and γ0 ∼ 176.17 respectively. The final vacuum expectation range, see Refs. [33, 34, 35]. In some scenarios, this also
values of the DBI field are φ0/mPl ∼ 1.37, φ0/mPl ∼ 0.66, implies variation of the constants, as, for instance, the
φ0/mPl ∼0.29 and φ0/mPl ∼0.2 respectively. fine structure constant. However, in the present con-
text, the couplings are totally different. For example,
thecouplingwiththe electromagneticfieldis ofthe form
areattractors. Moreover,iftheLorentzfactorislargeto- det(g +F ), where, here, g is the induced met-
µν µν µν
day, then the vacuum expectation value of the field can ric on the brane. Therefore, one can maybe expect this
p
besmallincomparisonwiththePlanckmass. Letusalso issue to be less problematic than in the standard case.
notice that the fact that the scaling solutions obtained More work is clearly needed in order to draw definitive
inthis articlecorrespondtobranetensions ofthe power- conclusions on these matters.
law form is fairly remarkable since this is precisely what
happensinsimplestring-inspiredmodels. Unfortunately,
oneneeds α<0 forwhichthe darkenergydensityscales
Acknowledgments
faster than that of the background. Maybe, the most
problematic aspects of the scenario is the fact that the
equation of state today is too far from 1. We would like to thank P. Brax, T. Chiba, K. Ichiki,
−
Inorderto improvethe abovedescribedsituation, one S. Mukohyama and J. Yokoyama for the useful dis-
probablyneedsmorecomplicatedstringinspiredmodels. cussions. M. Y. would like to thank M. Lemoine,
In particular, one needs shapes of T(φ) and V(φ) that, J. Martin, and P. Peter for kind hospitality at Institut
forφ mPl/√γe areofthepowerlawform(withα>0) d’Astrophysique de Paris where this work was initiated.
≪
inordertopreservetheattractor,and,forφ mPl/√γe, J. M. would like to thank the RESCEU (Tokyo Uni-
≫
deviate from this form in order to push the equation of versity) for warm hospitality. This work was partially
state towards 1. Let us recall at this stage that this is supported by CNRS-JSPS Bilateral Joint Project “The
−
exactly what the SUGRA model does, the characteristic Early Universe: a precision laboratory for high energy
scale being the Planck mass instead of mPl/√γe. physics.” M. Y. is supported in part by JSPS Grant-in-
Thefactthatthepresent-dayvalueofγ canbelargeis Aid for Scientific ResearchNos. 18740157and 19340054.
e
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