Table Of ContentCanonical and phantom scalar fields as an interaction of two perfect fluids
Mauricio Cataldo
∗
Departamento de F´ısica, Facultad de Ciencias, Universidad del B´ıo-B´ıo,
Avenida Collao 1202, Casilla 5-C, Concepci´on, Chile.
Fabiola Ar´evalo and Patricio Mella
† ‡
Departamento de F´ısica, Universidad de Concepci´on,
Casilla 160-C, Concepci´on, Chile.
(Dated: January 16, 2013)
InthisarticleweinvestigateanddevelopspecificaspectsofFriedmann-Robertson-Walker(FRW)
scalarfieldcosmologiesrelatedtotheinterpretationthatcanonicalandphantomscalarfieldsources
may be interpreted as cosmological configurations with a mixture of two interacting barotropic
3 perfect fluids: a matter component ρ1(t) with a stiff equation of state (p1 =ρ1), and an “effective
1 vacuum energy” ρ2(t) with a cosmological constant equation of state (p2 = −ρ2). An important
characteristic ofthisalternativeequivalentformulation in theframework ofinteractingcosmologies
0
2 is that it gives, by choosing a suitable form of the interacting term Q, an approach for obtaining
exact and numerical solutions. The choice of Q merely determines a specific scalar field with its
n
potential, thusallowing to generate closed, open and flat FRW scalar field cosmologies.
a
J
PACSnumbers:
5
1
I. INTRODUCTION minimally coupled to gravity. The evolution of a scalar
] field cosmology is given by
c
q
3k
- It is well known that scalar fields play an important 3H2+ = κρ , (2)
r a2 φ
g role in modeling both the early and late stages in the
[ evolution of the Universe [12, 17–19, 25, 26, 28, 34, 49, ǫφ˙φ¨+3ǫHφ˙2 = φ˙V′, (3)
50,53,54]. Inflationisconsideredanessentialpartofthe
1
where H = a˙/a, κ = 8πG, φ(t) is the scalar field, V(φ)
v description of the early universe [14, 15, 40–42, 48, 56].
itspotential, =d/dt, =d/dφandǫ= 1. TheEq.(3)
1 Itdoessolveclassicalcosmologicalproblemssuchasflat-
· ′ ±
1 ness, horizon and monopole problems, providing at the represents the dynamic equation for the evolution of a
3 same time precise predictions for the primordial density scalar field in a FRW universe. The energy density and
3 pressureofasuchscalarfieldaredefinedbytherelations
inhomogeneities, which are in good agreement with ob-
.
1 servational data [35]. Many different physical fields and
0 mechanismsmaybe responsibleforthis acceleratingcos- ρφ =ǫφ˙2/2+V(φ), pφ =ǫφ˙2/2−V(φ), (4)
3
micexpansion[2,3,5–9,11,20,21,27,31–33,36–38,49],
1 where ǫ = 1 and ǫ = 1 correspond to canonical and
however, it has become a common practice to employ −
: phantom scalar fields respectively.
v scalarfields in order to explainthis earlyacceleratedex-
The correspondingscalar field equation of state (EoS)
i pansion [4]. On the other hand, in recent years, several
X is given by
proposalshavebeenmade,byusingscalarfields,inorder
r
a to explain the late acceleration of the Universe [29] and ǫφ˙2/2 V(φ) φ˙2/2 ǫV(φ)
the coincidence problem. Ingeneral,due to isotropyand ω = − = − . (5)
homogeneityoftheUniverseoncosmologicalscales,most φ ǫφ˙2/2+V(φ) φ˙2/2+ǫV(φ)
of the papers consider self-interacting scalar fields in the
Note that ω variesin the range 1<ω <1 for V >0,
FRW framework. We shall consider here canonical and φ − φ
ǫ=1, and for V <0, ǫ= 1.
phantom scalar fields in the FRW cosmology described −
Notice that, in order to fulfill the isotropy and homo-
by the metric
geneity requirements of FRW models, scalar fields actu-
ally have the form φ = φ(t), V = V(t), so ρ = ρ (t),
φ φ
p = p (t) and ω = ω (t) depend also only on the cos-
φ φ φ φ
dr2 mological time t.
ds2 =dt2 a(t)2 +r2(dθ2+sin2θdϕ2) , (1) In this paper we show that, any self-interacting scalar
− 1 kr2
(cid:18) − (cid:19) field coupled to a FRW cosmology may be equivalently
formulatedasamixture oftwointeractingperfectfluids.
Perfect fluid description for sources plays an impor-
tant role in cosmology, where are considered single fluid
∗[email protected]
†[email protected] configurations, as well as multifluid ones, for describing
‡[email protected] the expansion of the Universe. It is well known that
2
for simplicity, and because it is consistent with much we By putting expressions (7) into Eq. (9), and (8) into
have observed about the universe, usually some epochs Eq. (10), with ω = 1 and ω = 1, we respectively
1 2
−
of the universe evolution are modeled with perfect fluids obtain
and also with mixtures of non interacting perfect fluids.
Nevertheless, there are no observational data confirm- ǫφ˙φ¨+3ǫHφ˙2 = Q(t), (11)
ing that this is the only possible scenario. This means φ˙V = Q(t). (12)
′
that it is theoretically plausible to consider cosmological −
modelscontainingperfectfluidswhichinteractwitheach ItisclearthatEqs.(11)and(12)directlyimplythefulfill-
other[16,30]. Thisalsoappliestothedescriptionofdark ment of the evolutionequation of a scalar field (3). This
energy models for explaining the accelerated expansion showsthatFRWscalarfieldmodelscanbeseenasconfig-
of the universe at the present stage. urationswithamixtureofastiffperfectfluidinteracting
In the case of a two-fluid FRW configuration, the with an “effective vacuum energy”. Note that for a van-
cosmological description of a superposition of two per- ishing interaction term we have the condition φ˙V = 0.
′
fect fluids ρ (t) and ρ (t), with barotropic EoS p (t) = This implies that we are in presence of a cosmological
1 2 1
ω ρ (t) and p (t) = ω ρ (t), allows us to describe an constant (for φ = const with V = V(φ) = const), or in
1 1 2 2 2
effective fluid ρ (t) = ρ (t) + ρ (t) with an effective the presence of a mixture of a stiff matter with a cosmo-
eff 1 2
pressure given by logical constant (for φ = φ(t) with V = const). Thus,
clearly an interaction term Q = 0 allows the cosmolog-
ω1ρ1(t)+ω2ρ2(t) ical constant (p = ρ = const6) to become a dynamical
p (t)=ω (t)ρ (t)= ρ (t). (6) −
eff eff eff ρ (t)+ρ (t) eff quantity, always respecting the EoS p(t)= ρ(t).
1 2
−
Let us remark that to our knowledge this interpreta-
Let us suppose that ρ1(t) > 0, ρ2(t) > 0 and ω1 > ω2, tion for scalar fields, described in terms of two interact-
then the effective state parameter ω (t) varies in the ing fluids, was first mentioned in the Ref. [24], where
eff
range ω2 < ωeff < ω1. Thus, one can mimic the scalar the author finds the group of symmetry transformation
field behavior with a two-fluid description by requiring generated by interacting fluids in spatially flat acceler-
ω1 =1 and ω2 = 1. ated FRW cosmologies, and applies it to a mixture of
−
With this purpose in mind we shall identify the first n non-interacting canonical scalar fields with exponen-
perfect fluid ρ1 with the scalar field φ as tialpotential, inorderto haveassistedinflation. Itmust
be noted also that in the Ref. [23] this interpretation is
ρ (t)=ǫφ˙2/2, p (t)=ǫφ˙2/2, (7)
1 1 applied to a flat FRW cosmology with a self-interacting
conformalscalarfieldΨ. Specifically, itisconsideredthe
while the second fluid ρ2 with the potential V as potential V(Ψ) = λΨ4 +V (V > 0), and it is shown
0 0
that this scalar field may be represented as the inter-
ρ (t)=V(φ), p (t)= V(φ). (8)
2 2 − action of two perfect fluids with EoS p1 = ρ1/3 and
p = ρ = const, representing radiation and vacuum
This identification is based on the fact that both terms, 2 − 2
ǫφ˙2/2 and V, contribute to the right hand side of Ein- energy respectively. However, no one of this papers con-
siders specific interacting terms Q.
stein’s equations, serving as the source of the gravita-
tional field. So, this interpretation means that the first
mattercomponentisaperfectfluidwithastiffEoS,while
II. SOME APPLICATIONS
the second component is an “effective vacuum energy”
with a cosmologicalconstant EoS.
Now, let us consider some exact cosmological scenar-
If we require that each component, ρ (t) and ρ (t),
1 2
ios containing a scalar field φ, described by a potential
satisfies the standard conservation equation separately,
then ρ = C +C a(t) 6 and the scale factor is given V. The question of the choice of the interacting term
eff 1 2 −
by a(t) = [−C1−1C2cosh(√3κC1(t +C))]1/6. This im- rtievmataeindsbcyruccoinals,idsoerwedewinotuhldelliikteertaotumreaksepethciifiscchfooricmesmfoor-
plies that we can not reproduce all the richness of scalar
Q(t)[22,30,39,46,51,52]. Inthefollowing,weshallob-
field cosmologies. In order to do this, one must consider
tainsomeknownexactsolutionsforself-interactingscalar
interacting FRW models.
field FRW cosmologies (with k=-1,0,1).
For two interacting fluids ρ (t) and ρ (t), with
1 2
Weshallbeginbyconsideringaninteractingtermgiven
barotropic EoS ρ (t) = ω p (t) and ρ (t) = ω p (t), we
1 1 1 2 2 2
by
can write for the conservation equations
Q(t)=3αHρ . (13)
ρ˙ +3H(1+ω )ρ = Q(t), (9) 1
1 1 1
ρ˙2+3H(1+ω2)ρ2 = Q(t), (10) where α is a constant parameter. In this case from
−
Eqs.(9)and(10)weobtainthatforω =1andω = 1
1 2
where Q(t) is the interaction term. Note that for Q >0 the energy densities are given by −
wehaveatransferofenergyfromthe fluidρ tofluidρ ,
2 1
and for Q<0 from the fluid ρ to ρ . ρ (a) = C a3(α 2), (14)
1 2 1 1 −
3
ρ2(a) = C2+ C1α a3(α−2), (15) C˜1 > 0 and the energy density ρ1 > 0. For 0 < α < 2
2 α we have a positive potential (ρ > 0) and for α > 2 or
− 2
α<0 we have a negative scalar field potential (ρ <0).
and for ω = 1 and ω = 1 the energy densities are 2
1 − 2 In the framework of this perfect fluid interpretation, we
given by
conclude that for 0<α< 2 (and α >2) the interacting
ρ (a) = C a3α, (16) termQ>0,implyingthattheenergyisbeingtransferred
1 1
from the “effective vacuum energy” ρ to the stiff fluid
C α 2
ρ2(a) = C2a−6− α+1 2a3α. (17) ρ1, while for α < 0 we have that Q < 0 and the energy
goesfromthestifffluidtothe“effectivevacuumenergy”.
Note thatinbothcasesforC =0wehavethatρ ρ . Foraphantomscalarfield(i.e. ǫ= 1andthe kinetic
The interpretation is direct:2for ω = 1 and ω 1=∼ 21 term of the scalar field is positive) we−have that C˜ < 0
1 2 1
−
the interacting term is proportional to the stiff matter andthen Eq.(20) implies that ρ <0. For 0<α<2 we
1
energy density (ρ ), while for ω = 1 and ω = 1 the haveanegativepotential(ρ <0)andforα>2orα<0
s 1 2 2
−
interacting term is proportional to effective vacuum en- wehaveapositivescalarfieldpotential(ρ >0). Onthe
2
ergy density (ρ ). other hand, in this case we have that for 0<α<2 (and
v
Inordertofindthescalefactor,wemustusetheFried- α>2)theinteractingtermQ<0sinceρ <0,implying
1
mann equation in the form thattheenergyisbeingtransferredfromthestifffluidρ
1
to the “effective vacuum energy” ρ , while for α < 0 we
2
3k
3H2+ =κρ (a)+κρ (a). (18) have that Q>0 and the energy goes from the “effective
a2 1 2 vacuum energy” to the stiff fluid.
It is remarkable that in this case the canonical and
In general, we can not find the scale factor in analyti-
phantom scalar fields have the same EoS, given by
cal form, so sometimes we shall look for some particular
solutions to Friedmann equation (18).
ω =1 α. (24)
φ
Nowweshalldiscusssolutionsforenergydensities(14) −
and (15). Solutions for energy densities in the form (16) Thus we have that 1 < ωφ < 1 for 0 < α < 2, while
−
and (17) will be discussed in Sec. III. for α > 2 and α < 0 it is verified that ωφ < 1 and
−
ω > 1 respectively. In this case both types of scalar
φ
fields cosmologies have the same scale factor.
A. Exponential potential in flat FRW cosmologies It is interesting to note that we can put for energy
densities (14) and (15) ω = 1 and ω =1 and identify
1 2
−
First, let us consider the case C2 = 0. Thus from ρ1(t) = V(φ),ρ2(t)= ǫφ˙2/2, but in this case we can not
Eqs. (14) and (15) we have that r = ρ2/ρ1 = 2αα. In write the scale factor, for C1 6=0 andC2 6=0, in analyti-
this case, for k =0, the scale factor is given by − cal form. For the case C1 = 0, C2 = 0 the solutions are
6
included in (22) and (23).
a(t)=a0(t+C)2/(6−3α). (19) Notice that for energy densities (16) and (17), with
C =0 we obtain also a scalar field with an exponential
2
Thus the energy densities (14) and (15) take the form
potential.
C˜
1
ρ (t) = , (20)
1 (t+C)2 B. Sinusoidal potential in flat FRW cosmologies
C˜ α
1
ρ (t) = , (21)
2 (2 α)(t+C)2 Letus nowconsiderthe energydensities (14) and(15)
− with C = 0. Thus from the Friedmann equation, with
2
6
whereC˜1 =C1a0−3(2−α). BycomparingEqs.(7)and(20) k =0, we obtain that the scale factor is given by
we findthatǫφ˙2 =C˜ (t+C) 2, implying thatthe scalar
1 − a(t)=a
0
field has the form ×
2
3κ 3(2−α)
φ(t)=± ǫC˜1ln(t+C)+φ0, (22) "cosh r 4 C2(2−α)(t+C)!# ,
q
where φ is an integration constant. Now from Eqs. (8) (25)
0
and (21), and by taking into account Eq. (22), we have
where C is an integration constant, and the constraint
that the potential is given by
C2(α−2) = 2C1a03(α−2) must be fulfilled. Thus we can
V(φ)= C˜1α e±(φ(t)−φ0)/√ǫC˜1. (23) wtimriteetenergy densities as functions of the cosmological
(2 α)
−
C (α 2)
2
Note that for a canonical scalar field (i.e. ǫ = 1 and the ρ1(t) = − , (26)
kinetic term of the scalar field is positive) we have that 2cosh2 34κC2(2−α)(t+C)
(cid:16)q (cid:17)
4
ρ (t) = C ǫ=k = 1andthenρ <0andρ >0fora> 2A2/3,
2 2 1 2
× −
implyingthatthekinetictermisalwaysnegative. Inthis
α 3κ p
1 cosh−2 C2(2 α)(t+C) . (27) casetheinteractingtermisgivenbyQ=4Hρ1,thusfora
" − 2 r 4 − !# canonicalscalarfieldthetransferofenergygoesfromthe
effectivevacuumfluidtothestiffone,andforaphantom
Now we shall connect this interacting solution with a
scalar field the energy is being transferred from the stiff
scalar field. From Eqs. (7) and (26) we have that the
fluid to the effective vacuum.
scalar field has the form
1 16ǫ(α 2)
φ(t) φ = ± −
− 0 (α 2) 3κ × D. Quadratic potential in flat FRW cosmologies
− r
arctan e√3κC2/4(α−2)(t+C) , (28) Let us now consider an interacting term given by
(cid:16) (cid:17)
whilefromEqs.(8)and(27),andbytakingontoaccount
Q(t)=3Hα, (34)
Eq. (28), we find that the potential is given by
whereαisaconstantparameter. Forthestateparameter
α 3κ
V(φ)=C2"1− 2 sin2 s4ǫ(α 2)(α−2)(φ−φ0)!#. values ω1 =1 and ω2 =−1 the solution has the form
− α
(29) ρ1(a) = +C1a−6, (35)
2
ρ (a) = C 3αlna. (36)
This typeofpotentialsareconsideredfordescribingnat- 2 2
−
ural/chaotic inflation [1].
Inordertofindanexplicitexpressionforthe scalefactor
let us consider the solution (35) and (36) with C = 0.
1
C. Quadratic potential in non-flat FRW Thus we find
cosmologies
a(t)=a0e−κ4α(t+C)2, (37)
Now we shall consider a solution for energy densities
in the form (14) and (15) with k = 0. This implies that whereC isaconstantofintegration,andtheenergyden-
ρ1+ρ2 = 22Cα1a−6+3α+C2. Letus6considertheparticular sities take the form
solution a(−t) = a0eAt, where a0 and A are constants. α
Thusinordertosatisfythe Friedmannequation(18)the ρ1(t) = , (38)
2
followingconstraintsmustbefulfilled: α=4/3,κC =k
1 3κα2
and κC =3A2, and energy densities take the form ρ (t) = C˜ + (t+C)2, (39)
2 2 2
4
k 2k
κρ1 = a2, κρ2 = a2 +3A2. (30) where C˜2 = C2 3αlna0. By comparing Eqs. (7)
−
and (38) we find that ǫφ˙2 = α, implying that the scalar
By comparing Eqs. (7) and (8) with expressions (30) we field has the form
find that
ǫφ˙2 = 2k , V = 2k + 3A2. (31) φ(t)=±√ǫαt+C3, (40)
κa2 κa2 κ
where C is an integration constant. Now from Eqs. (8)
3
Thus we have that the scalar field has the form and (39) we have that the potential as function of the
cosmologicaltimetisgivenbyV(t)=C˜ +3κα2 (t+C)2.
2ǫk/κ 2 4
φ(t)=φ0± pAa0 e−At, (32) Bφ0ywreewcraitninwgrtihteetshcaelaprofiteenldtia(4l0in)atshφe(fto)rm=±√ǫα(t+C)+
where φ is an integration constant, while the potential
0 3κǫα
is given by V(φ)=C˜ + (φ φ )2. (41)
2 0
4 −
3A2
V(φ)= +ǫA2(φ φ )2. (33) The Eq. (41) is one of the basic potentials considered in
0
κ −
standard cosmology.
Notethatforacanonicalfieldwehavethatǫ=k =1and Note that for ω = 1 and ω = 1 the energy den-
1 2
−
then ρ > 0 and ρ > 0, implying that the kinetic term sities are given by ρ (a) = C + 3αlna and ρ (a) =
1 2 1 2 2
and the potential are always positive magnitudes. On α+C a 6,implyingthatisthesamesolutiondiscussed
−2 1 −
the other hand, for a phantom scalar field we have that above.
5
E. Non linear interacting term samebyusingthescalefactoraasamainvariable. Letus
suppose that the energy densities ρ and ρ are given as
1 2
Now we shall consider a non-linear coupling given by functions of the scale factor a. From Eq. (7) we obtain
2ǫρ (a) = φ(a)a˙, while from the Friedmann equa-
1 ′
ρ ρ ±
Q=3αH 1 2 . (42) tion (2) we can write a˙ = (κ/3)(ρ +ρ )a2 k, thus
ρ +ρ p ± 1 2 −
1 2 obtaining
p
For the state parameter values ω = 1 and ω = 1 the
1 2
− 6ǫρ (a)
energy densities are given by 1
φ(a)= da+φ , (50)
C ±Z sκ[ρ1(a)+ρ2(a)]a2−3k 0
ρ (a) = 1 a3α 6(C a3α 6+C ) α/(α 2), (43)
1 C2 − 1 − 2 − − where φ0 is a constant of integration. In order to write
ρ (a) = (C a3α 6+C ) α/(α 2), (44) the potential as V = V(φ) we can write from (50) the
2 1 − 2 − −
scalefactorasa=a(φ)andintroducingitintoEq.(8)we
can write the potential as a function of the scale factor.
where C and C are constants of integration. Note
1 2
Notethatwecanfindthescalefactorasfunctionofthe
that in this case the energy densities satisfy the relation
ρ1/ρ2 = CC12 a3α−6. cosmologicaltimewiththehelpofthegeneralexpression
Now, in order to find an analytical solution we shall
φ
put α=1. Thus the scale factor is given by t+C = ′ da, (51)
± 2ǫρ (a)
Z 1
C2/3 1/3
a(t)= 2 e±√3κC2(t+C) C1 , (45) where C is an integration cponstant.
C −
2
(cid:16) (cid:17)
whereC isanintegrationconstant. Thisimpliesthatthe
A. An application for Q=3Hα
energy densities will be given by
C1C2e±√3κC2(t+C) Asasimpleexampleletusconsideragaintheinteract-
ρ1(t) = (e±√3κC2(t+C)−C1)2, (46) ionbgtasicnenarioQ=3Hα. FromEqs.(35),(36)and(50)we
C2e±√3κC2(t+C)
ρ2(t) = e±√3κC2(t+C)−C1. (47) φ(a)−φ0 =± 32κǫα(α+2C2−6αlna), (52)
r
Let us now consider the positive branch, i.e. a(t) =
whereφ isanintegrationconstantandwehaveputC =
CC222/3 e√3κC2(t+C)−C1 1/3. Thus the scalar field and 0p.otTenhtuisa0lbiystgaivkeinngbiynto account Eq. (8) we have that1the
its po(cid:16)tential have the fo(cid:17)rm
α 3κǫα
φ(t) φ = 8ǫarctanh e12√3κC2(t+C) , (48) V(φ)=−2 + 4 (φ−φ0)2. (53)
0
− r3κ √C1 !
Now, from Eqs. (51), (52), and by taking into account
3κ that ρ (a)=α/2, we obtain that
V(φ)=C 1 cosh2 (φ φ ) , (49) 1
2 0
− r8ǫ − !! a(t) e−κ4α(t+C)2. (54)
∼
respectively.
Finally, note that due to the symmetry of the in-
teracting term (42) with respect to energy densities B. An application for Q=3αHρ1
ρ and ρ the solution for ω = 1 and ω = 1 is
1 2 1 2
−
given by ρ (a) = (C a 3α 6 + C ) α/(α+2), ρ (a) = Now, we shall use our procedure to generate a scalar
1 1 − − 2 − 2
sCCa12mae−s3oαl−u6t(ioCn1ao−f3Eαq−s6.+(4C32))−anαd/(α(4+42)).,implyingthatisthe ficoeuldplcionsgmQolo=gy3fαorHinρ1te.raIcntitnhgescfoelnloawrioinsgdewsecrsibheadllbfyoutnhde
the generalsolutionforEqs.(14)and(15)withα=4/3.
From Eq. (50) we find that
III. A PROCEDURE FOR FINDING THE
SCALAR FIELD AS FUNCTION OF THE SCALE 2ǫC1
φ(a) φ =
FACTOR a − 0 ± κC k ×
r 1−
6(κC k)+2 9(κC k)2+3κC (κC k)a2)
1 1 2 1
Up to now we have shown how we can obtain scalar ln − − −
a
field cosmologies by writing a scalar field as a function p !
of the cosmological time t. However one can make the (55)
6
for C1 6=k/κ, and C. An application for Q=3αHρρ11+ρρ22
6ǫk
φ(a)−φ0 =± κ2C a2 (56) Let us now consider the non-linear interacting term
r 2 Q = 3αH ρ1ρ2 . We shall found the general solution for
for C =k/κ. ρ1+ρ2
1 Eqs. (43) and (44) with α=5/3. From Eq. (50) we find
Thus the potential is given by
that
C
V(φ)=C + 1 6ǫ C +C a
2 72(κC k)2 × φ(a) φ = 2 arctanh 1 2 , (64)
1− 2 − 0 − rκ r C1 !
κC1−kφ κC1−kφ
e− 2C1ǫ 12κC2(κC1 k)e 2C1ǫ (57) where φ0 is an integration constant. Consequently, by
q − − q taking into account Eqs. (8) and (44) the potential is
given by
for C =k/κ, and
1
6 κ
κǫC V(φ)= C5sinh10 (φ φ ) . (65)
V(φ)=C2+ 3 2 (φ−φ0)2 (58) − 2 (cid:18)r24ǫ − 0 (cid:19)
for C =k/κ. Now, by using Eqs. (43), (51) and (64) we find for the
1
The scale factor may be found by using Eq. (51) with scale factor
ρ (a)=C /a2. Thus with the help of Eq. (55) we have
1 1 3
t+C =
a(t) = 1 s4κC25 ×
2√κC ×
2 3C2+4C C a+ln (C +C a)2(C1+C2a)2
e κC32 (t+C)−3(κC1−k)e− κC32 (t+C) 1 1 2 (C +h C1 a)2 2 i. (66)
(cid:18) p p (cid:19) 1 2
(59) Lastly, as another example, note that by following the
for C =k/κ, and with the help of Eq. (56) we find same procedure we can directly rederive the solution for
1
6 α=1 discussed at the end of Sec. II.
a(t)=e κC32 (t+C) (60)
for C1 =k/κ. p IV. SOME GENERAL RELATIONS FOR
From this solution we can obtain discussed before in SCALAR FIELDS
the literature scalar field cosmologies. For example the
solution given by Eqs. (58) and (60) is the same solu-
Note that in the framework of FRW scalar field cos-
tion given by Eqs. (30)-(33). They are connected by the
mologies we can obtain some general relations involving
relation κC =3A2.
2 φ and V for specific interacting terms Q. In general, in
OntheotherhandforκC =(k+1/3)thescalefactor
1 thisalternativeequivalentformulationofscalarfieldsthe
takes the form a(t)=(1/κC2)sinh( κC2/3(t+C)) barotropic perfect fluids ρ1 and ρ2 with EoS ω1,2 = 1
Lastly,letusconsiderasolutionforenergydensitiesin or ω = 1 must fulfill Eqs. (9) and (10). If the int±er-
the form (16) and (17). For α = 7p/3 from eq. (50) we actio1n,2term∓ has the form Q = 3αHρ then the solution
− 1
havethatφ(a)−φ0 =± κ(−6C61ǫaC2+1C2a3)da,obtaining may be written through ρ1(t) in the following form:
for the scalar field R q ω2+1 αρ1(t)
ρ2(t) = C1ρ1(t)ω1+1−α + , (67)
ǫ 6C +C a ω ω α
φ(a) φ = 2 arctan − 1 2 . (61) 1− 2−
− 0 ± rκ r 6C1 a(t) = C2ρ1(t)−ω1+11−α. (68)
Thus by taking into account Eq. (16) we conclude that
This formhasthe advantageto provideus witha wayto
the potential has the form relate the kinetic term φ˙2 and scalar field potential V.
C C7 Let us first consider that the barotropic perfect fluids
V(φ)= 1 2 . (62)
[6C1(1+tan2( 4κǫφ))]7 ρ1 and ρ2 are a stiff fluid ρs and an effective vacuum ρv,
respectively. Therefore, by putting ω =1 and ω = 1
1 2
By using Eq. (51) we find for thepscale factor into Eqs. (67) we obtain −
a(C a 6C ) α
t+C =± √122κ−C3 1 (2C22a2+15C1C2a+135C12) ρ2(t)=C1+ 2 αρ1(t). (69)
p 2 −
Thus,bytakingintoaccounttheEqs.(7)and(8)wemay
135C3 3 [C a 3C + C a(C a 6C )]2
1 ln 2 − 1 2 2 − 1 . rewrite Eq. (69) in the form
± 4 sκC27 pC2 αǫ
(63) V(t)=C1+ φ˙2. (70)
2(2 α)
−
7
On the other hand, if now the barotropic perfect fluids inducing an accelerated expansion for ω < 1/3. In
φ −
ρ and ρ are respectively an effective vacuum fluid and thisworkwehaveshownthatisotropicandhomogeneous
1 2
a stiff one, then by putting ω = 1 and ω = 1 into scalar field cosmologies may be interpreted as cosmolog-
1 2
−
Eqs. (67) we obtain ical configurations with a mixture of a stiff perfect fluid
interacting with an “effective vacuum energy”. In the
ρ2(t)=C1ρ1(t)−2/α αρ1(t). (71) absence of an interaction, we still have an accelerated
− 2+α expansion, due to the presence of a cosmological con-
stant (φ = const and V = V(φ) = const), or to the
Now by taking into accountthe Eqs.(7) and(8) we may
presence of a mixture of a stiff matter with a cosmo-
rewrite Eq. (71) in the form
logical constant (φ = φ(t) and V = const). Therefore
the interaction term Q allows the cosmological constant
ǫφ˙2 αV(t)
=C V(t) 2/α . (72) (p = ρ = const) to become a dynamical quantity, al-
2 1 − − 2+α ways −respecting the EoS p(t) = ρ(t). An important
−
feature of this interpretationis that it gives anapproach
Thus any scalar field φ, which satisfies conditions (70)
to obtain analytical solutions. By selecting a suitable Q
or (72) has an alternative equivalent formulation in the
one easily integrates the equations for FRW scalar field
frameworkofinteractingcosmologieswithaninteraction
cosmologies in closed form. We explicitly obtain exact
term of the form Q=3αHρ or Q=3αHρ .
s v solutions in cosmological time t, and the potentials can
For interacting scenarios described by the interaction
be written as functions of the scalar field φ, provided
Q = 3Hα the solution may be written through ρ (t) in
1 that the scalar field and its potential are invertible func-
the form
tions. As we have illustrated in previous sections many
α C solutionssparselyfoundintheliteratureresultfromspe-
1
ρ2(t)=C2 ln , (73) cificconsideredchoicesoftheinteractingtermQ. Onthe
− 2 ρ (t) α/2
(cid:18) 1 − (cid:19) other hand, it is useful to note that Eqs. (11) and (12)
1/6
C1 imply that we can determine the form of the interacting
a(t)= , (74)
ρ (t) α/2 term Q for a given scalar field φ with its potential V.
(cid:18) 1 − (cid:19)
Fromthediscussedinterpretationwehavethatcosmo-
for ω = 1, ω = 1 and C = 0. Thus, by taking into
1 2 − 1 6 logicalmodelsusedfordescribingthepresentaccelerated
account the Eqs. (7) and (8) we may rewrite Eq. (73) in
expansionofthe Universe and filled with a quintessence,
the form
in the form of a scalar field, and a dark matter compo-
α 2C nent may be interpreted as cosmologies filled with three
1
V(t)=C2− 2 ln ǫφ˙2 α . (75) fluids: a dustcomponent, a stiff perfect fluidand an“ef-
(cid:18) − (cid:19) fective vacuumenergy”. Inthis casewecanhavemodels
wherethedarkmattercomponentdoesnotinteractwith
Note that for the case C = 0 the relation (41) must be
1
any other component, satisfying the standard conserva-
considered for V and φ.
For interacting scenarios described by Q=3αH ρ1ρ2 tion equation, while other two components interact with
ρ1+ρ2 eachother,orscenarioswhereallthreefluidmattercom-
with ω = 1 and ω = 1 the solution may be written
1 2 − ponents interact with each other.
through ρ (t) in the form
2
Since the behavior of any scalar field cosmology can
ρ (t)(2 α)/α C be emulated by a mixture of two interacting perfect flu-
2 − 2
ρ1(t) = (cid:18) C2 − (cid:19)ρ2(t), (76) itdhsatwtihthe beeqhuaavtiioornooffpsetrafetectωfl1u,2id=int±e1ra,cittinbgeccoomsmeosloclgeiaers
ρ2(t)(2−α)/α C2 1/(3α−6) is richer than the one of scalar field cosmologies,since it
a(t) = − . (77)
(cid:18) C1 (cid:19) canbe consideredalsostateparametervaluesω1,2 6=±1.
Thisismoreclearforcosmologieswhichcrossesthephan-
Thus,bytakingintoaccounttheEqs.(7)and(8)wemay tom divide. In general, the crossing of the phantom di-
rewrite Eq. (76) in the form vide cannot be achieved with a unique scalar field. The
simplestmodelisconstructedwiththehelpoftwoscalar
ǫ φ˙2 = V(t)(2−α)/α−C2 V(t). (78) fields, namely one canonical and one phantom, and the
2 C DE is attributed to their combination [10]. However,
(cid:18) 2 (cid:19)
such a crossing can be done with the help of two inter-
acting[55]ornon-interacting[43]perfectfluidswithstate
V. CONCLUSIONS parameters ω < 1 and ω > 1.
1 2
− −
Lastly, let us note that there are in the literature var-
WithintheframeworkofFRWcosmologiesscalarfields ious studies dealing with relations between fluids and
play a crucial role in the description of accelerated ex- scalar fields. For example in Ref. [4] such relations are
pansion of the Universe. This mainly because the scalar studied in the framework of dark energy cosmological
field state parameter varies in the range 1 < ω < 1, models. However, the proposed procedure gives a way
− φ
8
to express a fluid model as an explicit single or multiple uum one, can be applied in particular to the scalar field
scalarfieldtheories. Insomesense,thisistheoppositeto solutionsdiscussedinRefs.[13,44,45,47]. Thisispossi-
theinterpretationofascalarfielddiscussedinthispaper. blesincetheauthorsdonotseparateascalarfieldsource
In Refs. [13, 44, 45, 47] also are studied relations be- into two interacting perfect fluids, as we proposed. This
tween fluids and scalar fields. Nevertheless, in these pa- work is currently in progress and should be available in
pers it is developed a reconstruction program for scalar the near future.
fieldcosmologiesinEinsteintheoryandintheframework
of modified gravities such as scalar-tensor theory, f(R),
F(G) and string-inspired, scalar-Gauss-Bonnet gravity.
VI. ACKNOWLEDGEMENTS
Inthisscheme,theauthorsuseageneratingfunctionf in
ordertofindcosmologicalmodelswithoneormorescalar
fields. It is interesting to note that in the mentioned ref- This work was supported by CONICYT through
erencesthegravity-scalarsystemcontainsapotentialand Grants FONDECYT N0 1080530 (MC), N0 21070949
a scalar coupling function in front of kinetic term. Our (FA)andN0 21070462(PM)andbyDirecci´ondeInvesti-
equivalent presentation for scalar fields, stating that a gaci´on de la Universidad del B´ıo–B´ıo through grants N0
scalar field in FRW cosmology may be interpreted as a DIUBB 121007 2/R and N0 GI121407/VBC (MC, FA,
mixture of aninteractingstiff fluid with aneffective vac- PM).
[1] Alabidi, L. and Lyth, D. H.: JCAP 0605, 016 (2006) Mod. Phys.D 15, 1753 (2006)
(2010) [27] Cruz, N., Lepe, S. and Pena, F.: Phys. Lett. B 646,
[2] Avelino,A.andNucamendi,U.: JCAP1008,009(2010) 177 (2007)
[3] Avelino,A.and Nucamendi,U.: JCAP0904,006(2009) [28] D’Ambroise, J.: arXiv:1005.1410 [gr-qc]
[4] Bamba, K. Capozziello, S., Nojiri S. ’i. and Odintsov, [29] Elizalde, E., Nojiri, S. ’i. and Odintsov, S. D.: Phys.
S.D.: Astrophys.SpaceSci. 342, 155 (2012) Rev. D 70, 043539 (2004)
[5] Bamba,K.,Geng,C.Q.,Nojiri,S.’i.and Odintsov,S.D.: [30] Farajollahi, H., Mohamadi, N. and Amiri, H.: Mod.
Phys.Rev.D 79, 083014 (2009) Phys. Lett. A 25, 2579 (2010)
[6] Banijamali,A.andFazlpour,B.: JCAP1201,039(2012) [31] Felippe, L., Rodrigues, S. and Opher, R.: Phys.Rev.D
[7] Banijamali, A.andFazlpour,B.: Phys.Lett.B703,366 82, 023501 (2010)
(2011) [32] Felippe,L., Sun,C.B., Wang,J.-L.and Li,X.-Z.: Int.
[8] Brevik, I., Elizalde, E., Nojiri, S. ’i. and Odintsov, S.D.: J. Mod. Phys. D 18, 1303 (2009)
Phys.Rev.D 84, 103508 (2011) [33] Ghosh,R., Chattopadhyay,S.and Debnath,U.: Int.J.
[9] Brevik,I.,Gorbunova,O.andSaez-Gomez,D.: Gen.Rel. Theor. Phys. 51, 589 (2012)
Grav. 42, 1513 (2010) [34] Giovannini, M.: Int.J. Mod. Phys.A 22, 2697 (2007)
[10] Cai, Y.F., Saridakis, E.N., Setare, .R. and Xia, J.Q.: [35] Hinshaw, G. et al. [WMAP Collaboration]: Astrophys.
Phys.Rept.493, 1 (2010) J. Suppl.180, 225 (2009)
[11] Cai,Y.F.andWang,J.: Class. Quant.Grav.25,165014 [36] Ito,Y.and Nojiri,S.’i.: Phys.Rev.D79,103008(2009)
(2008) [37] Jamil, M. and Farooq, M. U.: Int. J. Theor. Phys. 49,
[12] Capozziello, S., Piedipalumbo, E., Rubano, C. and 42 (2010)
P.Scudellaro: Phys.Rev.D 80, 104030 (2009) [38] Jamil,M., Farooq,M.U.and Rashid,M.A.: Eur.Phys.
[13] Capozziello, S., Nojiri, S.’i. and Odintsov,S. D:Phys. J. C 61, 471 (2009)
Lett.B 632, 597 (2006) [39] Kofinas, G., Panotopoulos, G. and Tomaras.T. N.:
[14] Cardenas., V. H.: Phys.Rev. D 75, 083512 (2007) JHEP 0601, 107 (2006)
[15] Cardenas., V. H.: Phys.Rev. D 73, 103512 (2006) [40] Kofman.L., Linde,A.D.and Starobinsky,A.A.: Phys.
[16] Cataldo, M, Mella, P., Minning, P. and Saavedra, J.: Rev. Lett.73, 3195 (1994)
Phys.Lett. B 662, 314 (2008) [41] Linde, A. D.: Phys. Rev.D 49, 748 (1994)
[17] Cataldo,M.and delCampo,S.: Phys.Rev.D62,023501 [42] Linde, A. D.: Phys. Lett. B129, 177 (1983)
(2000) [43] Nesseris, S. and Perivolaropoulos, L.: JCAP 0701, 018
[18] Charters,T.and Mimoso,J.P.: JCAP1008,022(2010) (2007)
[19] Charters,T.and Mimoso,J.P.: JCAP1008,022(2010) [44] Nojiri, V. and Odintsov, S. D.: Phys. Rept. 505, 59
[20] Chattopadhyay,S.: TheEuropeanPhysicalJournalPlus (2011)
126, 130 (2011) [45] Nojiri,S.’i.and Odintsov,S.D.: J.Phys.Conf.Ser.66,
[21] Chattopadhyay,S. and Debnath, U.: Can. J. Phys. 88, 012005 (2007)
933 (2010) [46] Nojiri, S. ’i. and Odintsov, S. D.: Phys. Lett. B 639,
[22] Chattopadhyay, S. and Debnath, U.: Astrophys. Space 144 (2006)
Sci. 326, 155 (2010) [47] Nojiri, S. ’i. and Odintsov S. D.: Gen. Rel. Grav. 38,
[23] Chimento, L.: Phys.Lett. B 633, 9 (2006) 1285 (2006)
[24] Chimento, L.: Phys.Rev.D 65, 063517 (2002) [48] Nojiri, S. ’i. and Odintsov, S. D.: Phys. Rev. D 68,
[25] Faraoni, V.: Phys.Lett. B 703, 228 (2011) 123512 (2003)
[26] Copeland, E. J., Sami, M. and Tsujikawa, S.: Int. J. [49] Ren,J.and Meng,X.H.: Int.J.Mod.Phys.D161341,
9
(2007) [54] Williams, F.L., Kevrekidis,P.G., Christodoulakis, T.,
[50] Ren,J. and Meng, X.H.: Phys.Lett. B 636, 5 (2006) Helias, C., Papadopoulos, G. O. and Grammenos, T.:
[51] Setare,M.R.and Sheykhi,A.: Int.J.Mod.Phys.D19, gr-qc/0408056
1205 (2010) [55] Zhang, H.: arXiv:0909.3013 [astro-ph.CO]
[52] Tomaras, T. N.: arXiv:hep-ph/0610412 [56] Zimdahl, W., Pavon, D. and Maartens, R.: Phys. Rev.
[53] Wang, F. and Yang, J. M.: Eur. Phys. J. C 45, 815 D 55, 4681 (1997)
(2006)
