Exact models with non-minimal interaction between dark matter and (either phantom or quintessence) dark energy

0.3 MB·English

by  Tame Gonzalez

Exact models with non-minimal interaction between dark matter
and (either phantom or quintessence) dark energy
Tame Gonzalez1,a and Israel Quiros1,b
1Universidad Central de Las Villas, Santa Clara CP 54830, Cuba
(Dated: February 1, 2008)
A method for deriving Friedmann-Robertson-Walker(FRW) solutions developed in Int. J. Mod.
Phys. D5(1996)71-84, is generalized toaccount for models with non-minimalcoupling between the
dark energy and the dark matter. New quintessence and phantom (flat) FRW solutions are found.
Their physical significance is discussed. Additionally, the aforementioned method is modified so
that, ”coincidence free” solutions can be readily derived. Besides, we review some aspects of the
phantom barrier crossing. In this regard we present a model which is free from the coincidence
problem and, at the same time, does the crossing of the phantom barrier ω = −1 at late time.
8 Finally, we give additional comments on the non predictive properties of scalar field cosmological
0 models with or without energy transfer.
0
2 PACSnumbers: 04.20.Jb,04.20.Dw,98.80.-k,98.80.Es,95.30.Sf,95.35.+d

n
a
J

4
I. INTRODUCTION
1

] Although many alternative models of the universe that take account of the present stage of accelerated expansion
c
arebeingstudied,duetotheirsimplicitydarkenergy(DE)modelsareperhapsthepreferredonesinthebibliography.
q
- Notwithstanding these are plagued by many problems, among which is the so called ”cosmic coincidence” (CC)
r problem. Other undesirable features of these models are related with the possibility- already anticipated by the
g
observational data- that the DE equation of state (EOS) parameter might be more negative than minus unity (the
[
limiting value of the vacuum EOS parameter). This led cosmologists to face the so called ”phantom” dark energy,

2 i.e.,aDEcomponentwithnegativekineticenergyterm. Nomatterwhetherornottherearerealchancesforthedark
v
energy(whether vacuum, quintessence or phantom)to exist, the study ofsuchmodels is veryinteresting andit could
9
hint at more compelling possibilities.
8
0 Perhaps, the simplest realization of the dark energy is a minimally coupled (self-interacting) scalar field. Due
2 to their mathematical simplicity, these models- also called as ”quintessence” or ”decaying vacuum” DE- have been
. intensively and detailed studied overdecades. The next step towardsa more complicatedand, may be, more realistic
7
model,istoconsiderthepossibilityofanadditionalnongravitationalinteractionbetweentheDEandthebackground
0
7 cosmic fluid (see reference [1] for a general formalism to describe the coupled quintessence, phantom, non phantom
0 K-essence and Tachyon, etc, and [2] for a coherent review). Otherwise the scalar field DE is coupled non-minimally
: to the background fluid (usually cold dark matter) and, consequently, the background particles do not follow the
v
geodesics of the metric, i.e, these are coupled to a scalar-tensormetric (see below for details).
i
X Althoughexperimentaltestsinthesolarsystemimposesevererestrictionsonthepossibilityofnon-minimalcoupling
r betweenthe DEandordinarymatterfluids (the background)[3], duetothe unknownnatureofthe darkmatter(DM)
a
asthe majorpartofthe background,itispossible tohaveadditional(nongravitational)interactionsbetweenthe DE
component and DM without conflict with the experimental data. Besides, in references [4] and [5] it is shown that
interacting DE models are well consistent with current observational bounds.1 Since there are suggestive arguments
showing that observational bounds on the ”fifth” force do not necessarily imply decoupling of baryons from the
dark energy[7], then baryons might be considered also as part of the background DM that is interacting with the
quintessence field.
Therefore,as itis done inreference[7],we mayconsidera universalcoupling ofthe quintessence fieldto allsortsof
matter (radiation is excluded). Since the arguments given in the appendix of reference [7] to explain this possibility
arealsoapplicablein the casesofinterestin the presentstudy, we referthe interestedreaderto thatreferenceto look
aElectronicaddress: [email protected]
bElectronicaddress: [email protected]
1 It should be pointed out, however, that when the stability of DE potentials in quintessence models is considered, the coupling dark
matter-darkenergyistroublesome[6]. Otherargumentsbasedonoscillationsorexponentialblowupofthematterpowerspectrumthat
isinconsistentwithobservations, ruleouttightcoupling[8].
2
forthedetails. Howeverwewanttomentionthebasicargumentsgiventherein: Apossibleexplanationisthroughthe
”longitudinal coupling” approach to inhomogeneous perturbations of the model. The longitudinal coupling involves
energy transfer between matter and quintessence with no momentum transfer to matter, so that no anomalous
accelerationarises. In consequence,this choiceis notaffected by observationalbounds on”fifth” force exertedonthe
baryons. Othergeneralizationsofthegivenapproachcouldbeconsideredthatdoinvolveananomalousaccelerationof
thebackgroundduetoitscouplingtoquintessence. However,duetothe universalnatureofthecoupling,itcouldnot
be detected by differentialaccelerationexperiments. Another argumentgivenin [7] is that, since the coupling chosen
is of phenomenologicalnature and its validity is restricted to cosmologicalscales (it depends on magnitudes that are
onlywelldefinedinthatsetting),theformofthecouplingatsmallerscalesremainsunspecified. Therequirementsfor
thedifferentcouplingsthatcouldhaveamanifestationatthesescalesarethati)theygivethesameaveragedcoupling
atcosmologicalscales,andii)they meetthe observationalbounds fromthe localexperiments. We complete the afore
mentioned arguments, by noting that these are applicable even if the coupling is not of phenomenological origin like
in the present investigation where the kind of coupling chosen is originated in a scalar-tensor theory of gravity.
Models with non-minimal interaction between the dark energy (whether scalar field modelled or not) and the
background DM are appealing since the cosmic coincidence problem -why the energy densities of dark matter and
of dark energy are of the same order precisely at present?- can be avoided or, at least, smoothed out[9, 10, 11]. In
the present study we will show that these models are compelling, besides, because they can do the crossing of the
phantom barrier (ω = 1) with just a single scalar field, a possibility that is incompatible with models with minimal
−
interaction between the scalar field and the background fluid. In the latter case the above mentioned crossing is
possible only if two or more scalar fields are considered [12, 13, 14].
Many exact Fredmann-Robertson-Walker (FRW) solutions have been found in models in which the background
fluid and the self-interacting scalar field DE have no other interaction than the gravitational one. However, no such
galleryofsolutionsexistsforthecasewherethe couplingbetweenthe backgroundandthescalarfieldisnon-minimal.
The present paper is partially aimed, precisely, at deriving classes of exact FRW solutions in models where, besides
the gravitational interaction between the components of the cosmic fluid, additional non-gravitational interaction is
also considered. To this end we extend a method formerly applied to derive FRW solutions in models with a scalar
field minimally coupled to the background fluid (Ref. [15]) to account for non-minimal coupling also. The classes
of solutions found comprise both the phantom and the quintessence. For the input and coupling functions chosen,
self-interaction potentials of the exponential and sinh like form arise. The importance of these kinds of potential is
noticeable. Exponentialpotentials(seeforinstancereference[16])arefoundinhigher-order[17]orhigher-dimensional
gravity[18]. These arise also in Kaluza-Klein and in string theories and due to non-perturbative effects (gaugino
condensation [19], for instance). Their role in cosmology has been investigated, for instance, in [16, 20]. Sinh-like
potentials have been studied, mainly, in the context of quintessence models of dark energy and as candidates for the
dark matter[21]. To complement the study, in the final part of the paper, we modify the method of reference [15] so
that coincidence free solutions could be derived. Another goal of this paper is to study models which not only are
free of the coincidence problem but, at the same time, do the crossing of the phantom barrier ω = 1 at late times.
−
Wealsogiveadditionalcommentsonthenonpredictivepropertiesofscalarfieldcosmologicalmodelswithorwithout
energy transfer.
The rest of the paper has been organized in the following way. In the next section II, we explain the details of
the model of interacting components of the cosmic fluid we want to explore, including the field equations, etc. The
model includes the possibility to deal with phantom fields by considering arbitrary sign of the scalar field kinetic
energy. InSectionIII the methodofreference[15](formerlyusedtogenerateexactFRWsolutionsinmodels without
interaction between the components of the cosmic fluid), is generalized to account for models with interaction and
with arbitrarysignofthe kinetic energyof the DE. As anillustrationthe methodis appliedto the simplestcase with
minimal coupling. Section IV is devoted to deriving of exact solutions in models with non-minimal coupling between
the DEandthe backgroundfluid. InsectionVwecommentindetailonthephysicalcontentofsolutionsfoundinthe
former section. In section VI, emphasis is made in explaining how to deal with the coincidence problem in the model
with interaction between the components of the cosmic fluid, subject of the present investigation. For this purpose,
the method of section III is modified so that coincidence free solutions can be derived. In section VII we study a
model that does the crossing of the phantom barrier. Finally, in Section VIII, we summarize the main achievements
and shortcomings of the paper.
II. THE MODEL
Weconsiderthefollowingactionthatisinspiredinascalar-tensortheorywrittenintheEinsteinframe. Thematter
degrees of freedom and the scalar field are coupled through the scalar-tensor metric χ(φ)−1g [22]:
ab
3
R ǫ
S = d4x g ( φ)2 V(φ)
| |{2 − 2 ∇ −
Z
+χp−2(φ) (µ, µ,χ−1g ) . (1)
m ab
L ∇ }
where ǫ = 1 (ǫ = 1 for phantom DE, while ǫ = +1 for quintessence), V(φ) is the scalar field self-interaction
potential, ± is the m−atter Lagrangian (µ is the collective notation for the matter degrees of freedom), and χ(φ)−2
m
L
is the coupling function.
This action could be considered, instead, as an effective theory, implying additional non-gravitational interaction
between the components of the cosmic fluid. When the coupling between the scalar field and the matter is minimal
(no otherinteractionthanthe gravitationalone)χ(φ)=χ =1. The fieldequationsderivablefromthe action(1)are
0
1
R g R=T(φ)+T(m), (2)
µν − 2 µν µν µν
where
T(m) =(ρ +p )u u +p g ,
µν m m µ ν m µν
1
T(φ) =ǫ φ φ g ǫ( φ)2+2V(φ) , (3)
µν ∇µ ∇ν − 2 µν{ ∇ }
are the stress-energy tensors for the ordinary matter degrees of freedom and for the self-interacting scalar field (dark
energy) respectively. These fulfill the following conservation equation:
νT(m)+ νT(φ) =0, (4)
∇ νµ ∇ νµ
so that energy is not separately conserved by each one of the species in the cosmic mixture. Instead, the following
dynamical equations hold:
νT(m) = Q ,
∇ νµ − µ
νT(φ) =Q , (5)
∇ νµ µ
whereQ istheinteractionterm. Thisispreciselythebasicfeatureofinteractingmodels: thereisexchangeofenergy
µ
between the components of the cosmic fluid.
For FRW universe with metric
dr2
ds2 = dt2+a2[ +r2dΩ2], (6)
− 1 kr2
−
wheredΩ2 =dθ2+sin2θdϕ2, andthe spatialcurvaturek = 1ork =0,the FriedmannandRaychaudhuriequations
±
look like
3k
3H2+ =ρ +ρ +Λ, (7)
a2 m φ
and
k
2H˙ 2 = (p +ρ +p +ρ ), (8)
− a2 − m m φ φ
respectively, where ρ = ǫφ˙2/2+V and p = ǫφ˙2/2 V. In the last equation we have assumed p = (γ 1)ρ ,
φ φ φ φ φ
− −
where γ is the scalar field barotropic parameter,which is related with the equation of state (EOS) parameter ω by
φ φ
the relationship γ =ω +1.
φ φ
4
The null component of the conservationequationfor the matter degrees of freedom in equation(5), can be written
as
ρ˙ +3H(ρ +p )=Q, (9)
m m m
where the interaction term Q is given by
χ¯˙ d(lnχ¯)
Q=ρ =ρ H[a ], (10)
m m
χ¯ da
and the following “reduced” coupling function χ¯ = χ(3γm−4)/2 (0 γm 2 - the matter barotropic index) has been
≤ ≤
introduced. Equation (9) with Q given by (10) can be readily integrated to yield:
ρ =ρ a−3γmχ¯, (11)
m m0
wherewehaveconsideredthattheordinarymatterdegreesoffreedom(thebackground)areintheformofabarotropic
perfect fluid, so that p =(γ 1)ρ .
m m m
−
For the DE one has, instead (null-component of the second equation in (5)):
ρ˙ +3H(ρ +p )= Q, (12)
φ φ φ
−
or, equivalently;
φ˙[ǫφ¨+3ǫHφ˙+V′]= Q. (13)
−
Equations (7), (10), (11) and (13) represent the basic set of equations of the model of interacting components of
thecosmicfluidweareabouttoinvestigate. Inwhatfollowsweshallapplyamethodforderivingnewexactsolutions
to this set of equations.
III. THE METHOD
Inthis sectionwe willgeneralizethe method developedin reference[15]so thatwe canderiveexactFRW solutions
inmodelswithinteraction,suchasthemodeldetailedintheformersection. Tothisend,letsassumethattherelevant
functions can be given in terms of the scale factor: ρ =ρ (a), ρ =ρ (a), χ¯ = χ¯(a), etc. If we introduce the new
m m φ φ
time variable: dη =a−3dt, then equation (13) can be written as
d ǫ dφ F da dχ¯
[ ( )2+F]=6 ρ a3(2−γm) , (14)
m0
dη 2 dη a dη − dη
wherewehaveintroducedtheinputfunctionF =F(a). Thisfunctionischoseninsuchawaythattheself-interaction
potential V can be rewritten as function of the scale factor in the following form[15]:
F(a)
V(a)= . (15)
a6
Equation (14) can be readily integrated to yield
F dχ¯
a6ρ = da[6 ρ a3(2−γm) ]+C, (16)
φ m0
a − da
Z
where C is an arbitrary integration constant. If we integrate by parts the second term in the right hand side (RHS)
of (16), then we are led with the following equation:
da
a6(ρ +ρ )= 3 [2F +
φ m
a
Z
(2 γ )ρ a3(2−γm)χ¯]+C. (17)
m m0
−
5
We now introduce the functions
G(a) 3H2 =ρ +ρ +Λ 3k/a2, (18)
m φ
≡ −
and
F
L(a) ǫ(ρ V)=ǫ(ρ ), (19)
≡ φ− φ− a6
that will be useful in what follows. Both functions G(a) and L(a) = φ˙2/2 are always non-negative: G(a) 0,
≥
L(a) 0. The cosmological constant Λ can be absorbed into the self-interaction potential V(φ) so we can set Λ=0
≥
without loss of generality.
In what follows,for sake of simplicity and unless the contraryis specified, we choose the spatial curvature k =0, i.
e., we will explore flat FRW cosmologies. Another magnitude of interest, that will be useful in the future, is the DE
barotropic parameter:
2a6L(a)
γ = . (20)
φ a6L(a)+ǫF
After considering equations (16) and (17), the functions G(a) and L(a) can be written in the following form:
3 da
G(a)= [2F +
a6 a
Z
(2 γ )ρ a3(2−γm)χ¯]+C/a6, (21)
m m0
−
and
ǫ
L(a)=ǫG(a) (F +ρ a3(2−γm)χ¯), (22)
− a6 m0
respectively.
Exact solutions can be found in the form of quadratures[15]:
da
∆t= √3 , (23)
± a G(a)
Z
or p
da
∆η = √3 , (24)
± a a6G(a)
Z
and p
da L(a)
∆φ= √6 . (25)
± a sG(a)
Z
Inequations(23),(24)the“ ”signintheRHSmeansbothtimedirectionsand,sinceEinstein’sequations(equations
±
(7), (9), (10) and (12)) are invariant under time inversion then, in what follows, we choose the branch with the “+”
sign in equation (23) (or (24)).
Once the input function F = F(a) and the coupling function χ¯ = χ¯(a) are given, we can find G = G(a) and
L = L(a) through equations (21) and (22) respectively. Then, by use of equations (23) (or (24)) and (25), one is
able to find t = t(a) (or η = η(a)) and φ = φ(a) by direct integration and, by inversion, a = a(t) (or a = a(η)) and
φ=φ(t) (or φ=φ(η)) respectively.
Let us illustrate how the method operates through the study of the simplest example: a model without interaction
χ¯ = χ¯ = 1 (Q = 0). I. e., the DE and the background fluid evolve independently. In this case (see equations (21)
0
and (22)):
6
6 da
G(a)=ρ a−3γm + F(a)+C/6 , (26)
m0 a6{ a }
Z
and
ǫ da
L(a)= 6 F(a) F(a)+C . (27)
a6{ a − }
Z
For simplicity we set C =0. To chose the input function we assume, additionally2
da 1
F(a)= F(a), (28)
a s
Z
where s is some arbitrary constant parameter. The equation (28), in particular, is fulfilled if F(a) = B as, where B
is a constant parameter. Therefore
6B
G(a)=ρ a−3γm + as−6,
m0
s
6 s
L(a)=ǫ( − )Bas−6. (29)
s
Taking into account all of the above, the equation (20) yields to:
6 s
γ = − . (30)
φ
3
Since, for quintessence 0 < γ < 2/3, then 4 < s < 6. For s > 6 (γ < 0) the DE is a phantom field instead. Let us
φ φ
investigatethe mostgeneralsituationwhen γ =(6 s)/3=γ . Inthis casethe solutionsaregivenby the following
m φ
6 −
expressions:
∆t=( 2 ) 3 a3γ2m 2F1[ 3γm ,1,
3γ ρ 2(3γ +s 6) 2
m r m0 m −
1+ 3γm , ( 6B )3γm3γ+ms−6a3γm+s−6], (31)
2(3γ +s 6) − sρ
m m0
−
where F is the hypergeometric function, and
2 1
2ǫ/n 6B
∆φ= arcsinh[ a−k],
pk ssρm0
s 6+3γ 2
m
k = − , n= , (32)
− 2 6 s
−
so, by inversion of (32), one gets a=a(φ) and, considering equation (15), one finds the form of the potential:
V(φ)=V sinh2q[λ∆φ], (33)
0
where V =B(sρ /6B)q, q =1/kn and λ= k/ 2ǫ/n. Note that the constant parameter q can be written also in
0 m0
±
the form q = γ /(γ γ ). Then, if φ were a phantom (γ <0 s>6), q were always positive.
φ m φ φ
− − p ⇒
As explained in [21], this potential is a good quintessencial candidate to be the missing energy in the universe.
Since it behaves like an inverse power-law potential at early times, then this allows to avoid the fine tuning problem.
The parameters V , λ and q, can be determined uniquely by the measuredvalues for the equation of state and of the
0
amount of vacuum energy necessary to obtain a tracker solution [21].
2 Otherassumptions tofixtheinputfunction canalsobemade. Inprinciplethe inputfunctionisarbitrary,the onlyrequirementbeing
simplicityoftheintegralstobetaken.
7
IV. GENERATING SOLUTIONS IN NON-MINIMAL COUPLING CASES
In this section we will apply the method explained in the former section,to generate solutions in models where the
DE(eitherquintessenceorphantom)andthe backgroundfluidshareadditionalnon-gravitationalinteractionthrough
the non-minimal coupling given by the coupling function χ−2 (or, alternatively, the “reduced” coupling function χ¯)
in the action (1). Throughthis section, the constantparametersn, k, etc., are different, in general,from those in the
former section.
As before, for the sake of simplicity, we consider C =0. In this case (see equations (21) and (22)):
3 da
G(a)= [2F +(2 γ )ρ a3(2−γm)χ¯],
a6 a − m m0
Z
F
L(a)=ǫ G(a) ρ a−3γmχ¯ , (34)
{ − a6 − m0 }
and
L(a) F +ρm0a3(2−γm)χ¯
=ǫ 1 . (35)
G(a) { − a6G(a) }
NowweareinpositiontointroducedifferentinputF(a)andcouplingfunctionsχ¯(a)togeneratesolutions. However,
to illustrate the possibilities offered by the method, it will be enough to choose just one given input function and a
couple of coupling functions.
A. F =Bas, χ¯=χ¯0 a3γm−n
After this choice of the input functions F and χ¯, straightforwardintegrationin the first equation in (34) yields to:
6B A
G(a)= as−6+ a−n, (36)
s 6 n
−
where n=6 and A=3χ¯ ρ (2 γ ). Meanwhile, Eq. (35) can be written as follows:
0 m0 m
6 −
L (6 s)B as−6+αa−n
=ǫ[ − ],
G 6B as−6+ sA a−n
6−n
s(n 3γ )A
m
α= − , (37)
3(6 n)(2 γ )
m
− −
and Eq(20) for the scalar field barotropic parameter:
(6 s)B as−6+αa−n
γ =2[ − ]. (38)
φ 6B as−6+αa−n
We point out that, since G(a) 0, then the constant parameter n is restricted to be n<6.
≥
1. General case with s6=6
(i) α=0 n=3γ
m
⇒
Inthis case thereis no additional(non-gravitational)interactionbetweenthe scalarfield andthe backgroundfluid:
χ¯=χ¯ , i.e., this is the simplest situation where minimal coupling between the dark energy and the backgroundfluid
0
is considered. According to equation (38) the DE barotropic parameter is related with the constant parameter s
through: 3γ = 6 s. In consequence, for quintessence (0 γ < 2/3) 4 < s 6, meanwhile, for the phantom
φ φ
− ≤ ≤
(γ <0) s>6.
φ
8
Obviously, if one correctly arranges the constants, this case is the same one considered, as a working example, in
the last partof the former section. To simplify writing of the solutions, lets introduce the constantk = n 1. The
6−s −
integral in (25) is now easily taken to yield, for k =0:
6
2 ǫ(6 s) 6(6 n)B
∆φ= − arcsinh[ − Y−k/2], (39)
± k(6 s) sA
p − r
or, after inverting it to get a=a(φ):
V(φ)=V sinh−2/k[λ∆φ],
0
6(6 n)B k(6 s)
V =B[ − ]1/k, λ= − . (40)
0
sA ±2 ǫ(6 s)
−
Equation (23) can now be rewritten as follows: p
1 2s dX
∆t= , (41)
±nrB Z X2p+ 6(6−sAn)B
q
where we have introduced the new variable X = an/2 and the new constant parameter p = k/(k+1). The integral
(41) yields to:
2 3(6 n) 1
∆t= − an/2 F [ ,1,
2 1
±n A 2p
r
1 6p(n 6)B
1+ , − an], (42)
2p sA
where, as before, F is the hypergeometric function.
2 1
(ii) α=0, k =0 (n=6 s)
6 −
The ”reduced”coupling function χ¯ is now of the formχ¯=χ¯0 a3γm+s−6. Since, in this case (see equations (36) and
(37)),
6B+A
G(a)= as−6,
s
L 6 s+α/B
=ǫq, q = − , (43)
G 6+A/B
the evolution of the scale factor evolution in terms of cosmic time is given by
a(t)=a ∆t2/(6−s),
0
6 s 6B+A
a =[( − )2( )]1/(6−s), (44)
0
2 3s
meanwhile the self-interaction potential is given by:
6 s
V(φ)=V e−λ∆φ, λ= − , (45)
0
±√6ǫq
where V = B. Note that there is scaling of the form ρ /ρ = const. In the present case, the barotropic parameter
0 m φ
γ is given by
φ
sB
γ =2[1 ]=const. (46)
φ
− α+6B
9
2. Particular case with s=6
Ifinequations(36-38)oneconsiderss=6,i.e.,V =V =B then, oneisfacedwithasituationwherethe sourcesof
0
gravity are a scalar field without self-interaction potential (only a kinetic energy term present), and the background
perfectfluid(darkmatter,forinstance),both“living”inadeSitterbackgroundspace-time. Itisfixedbytheeffective
cosmologicalconstant Λ=B. The following equations hold:
G(a)=B+Aa−n/(6 n),
−
L α′ a−n 2(n 3γ )A
′ m
=ǫ , α = − . (47)
G 6B+ 66−Ana−n (6−n)(2−γm)
As customary, by taking the integral in equation (23), one obtains the evolution of the scale factor in cosmic time:
n B A
a(t)=a sinh2/n[ ∆t], a =[ ]1/n, (48)
0 0
2 3 (6 n)B
r −
meanwhile, the evolution of the scalar field is given by:
2 ǫα′(6 n)
∆φ= −
±n A ×
r
A
arcsinh[ a−n/2]. (49)
s(6 n)B
−
Thescalarfieldenergydensityiscontributedbytheeffectivecosmologicalconstant3andthescalarfieldkineticenergy
density:
′
α
ρ = a−n+B. (50)
φ
6
′
Noticethat,thequintessencesolution(ǫ=+1)ariseswheneverα >0 0<n<3γ (recallthatn<6),meanwhile
m
the phantom behavior is displayed once 3γ <n<6.4 However,since⇒α′ <0 for the phantom scalar, there is a time
m
t , such that an(t )=an = α′/6B ρ (t )=0. For earliertimes t<t , the energy density of the phantom field is
c c c − ⇒ φ c c
negative definite. This fact alone, rules out the possibility to describe phantom behavior with the help of the present
solution, unless the free parameters are chosen in such a way that t is close enough to the Planck time scale. In this
c
case one might argue that the classical theory of gravity is unable to give an appropriated (perhaps semiclassical or
quantum) description of the cosmic evolution.
As it will be discussed in the next section, this solution represents an example of a model with transition from
decelerated into accelerated expansion, where the accelerated phase is driven by the combined effect of the kinetic
energy density of the scalar field and of the cosmologicalconstant.
In the next subsection we will investigate cases with the same input function F(a)=B as and a different coupling
function χ¯(a).
B. F =Bas, χ¯=χ¯0 a3γm[a−n+as−6]
We write the relevant functions in this case:
′
6B A
G(a)= as−6+ a−n,
s 6 n
′ −
B =B+A/6, A=3(2 γ )ρ χ¯ , (51)
m m0 0
−
3 Asalreadysaid,thiseffective cosmologicalconstant canbeinterpreted, alternatively,asabackground vacuumfluid.
4 The possibilityn<0is ruled out since, inthis case, according to (50), the negative energy component of the phantom field increases
withtheexpansion.
10
and
L αas−6+β a−n
=ǫ[ ],
G 6B′ as−6+ A a−n
s 6−n
′
(6 s)B γ A
m
α= − ,
s − 6(2 γ )
m
−
(n 3γ )A
m
β = − , (52)
3(6 n)(2 γ )
m
− −
and the barotropic index for the scalar fluid:
αas−6+β a−n
γ =2 . (53)
φ (α+B)as−6+β a−n
We should note that positivity of the energy density (non-negativity of the function G(a)), restricts the parameter
n to be n<6. Classes of exact solutions can be easily found for particular relationships among the free parameters.
1. General case with s6=6
As before, since the input function F(a) = B as V = B as−6. It is also useful to introduce the constant
⇒
k =n/(6 s) 1. Forn>6 s,k >0 meanwhile,for n<6 s,k <0. Straightforwardintegrationin(23)yields to:
− − − −
3(6 n)/A
−k+1
∆t= − Y 2
±2(s 6)(k+1) ×
p−
′
k+1 1 3k+1 6(n 6)B
F [ , , , − Y−k]. (54)
2 1
2k 2 2k sA
Let us try, as before, particular cases where the integral in Eq. (25) is easily taken.
(i) α=0
This choice implies the following relationship involving the constant parameters A, B, s and γ :
m
s 6+3γ
m
B =[ − ]A. (55)
3(2 γ )(6 s)
m
− −
Positivityofthe constantB leadstothe followingrestrictiononthe parameters: 3(2 γ )<s<6. After the choice
m
−
of the relationship between the constant parameters made, if one integrates (25) to find ∆φ and then inverts to get
V =V(φ), one obtains:
V(φ)=V sinh2/k[λ∆φ],
0
6(6 n)B′ (s 6)2A
V =B[ − ]1/k, λ= − . (56)
0
sA ±s6ǫβ(6 n)
−
Since n < 6, and since λ should be real then, for quintessence (ǫ = +1) β > 0 3γ < n < 6, meanwhile for
m
⇒
phantom (ǫ= 1) β <0 n<3γ .
m
− ⇒
(ii) β =0 n=3γ
m
⇒
The coupling function is given now by
χ¯=χ¯ (1+a3γm+s−6). (57)
0