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Crossing the phantom divide with a classical Dirac field Mauricio Cataldo1 • Luis P. Chimento2 1 1 0 2 n a J 1 1 Abstract In this paper we consider a spatially flat pressure and is homogeneously distributed and, unlike ] Friedmann-Robertson-Walker(FRW)cosmologicalmodel dark matter, is not concentrated in the galactic halos O with cosmologicalconstant, containing a stiff fluid and nor in the clusters of galaxies. The observational data C a classical Dirac field. The proposed cosmological sce- provide compelling evidence for the existence of dark h. nario describes the evolution of effective dark mat- energy which dominates the present–day Universe and p ter and dark energy components reproducing, with accelerates its expansion. - the help of that effective multifluid configuration, the In principle, any matter content which violates the o r quintessential behavior. We find the value of the strongenergyconditionandpossessesapositiveenergy st scale factor where the effective dark energy component density and negative pressure, may cause the dark en- a crosses the phantom divide. The model we introduce, ergy effect of repulsive gravitation. So the main prob- [ which can be considered as a modified ΛCDM one, is lemofmoderncosmologyistoidentifytheformofdark 2 characterizedbyasetofparameterswhichmaybecon- energy that dominates the Universe today. v strained by the astrophysicalobservations available up Intheliteraturethemostpopularcandidatesarecos- 1 to date. 8 mological constant Λ, quintessence and phantom mat- 2 ter. Their equationof state is givenby w =p/ρ,where Keywords Classical Dirac fields, phantom divide 0 w = 1, w > 1 and w < 1 respectively. Dark en- . − − − 1 ergy composed of just a cosmological term Λ is fully 0 1 Introduction consistent with existing observational data. However, 1 these data do not exclude the possibility of explain- 1 : According to the standard cosmology the total energy ingthe observedaccelerationwiththe helpofphantom v density of the Universe is dominated today by both matter. The cosmological constant can be associated i X dark matter and dark energy densities. The dark mat- with a time independent dark energy density; the en- r ter, which includes all components with nearly vanish- ergy density of quintessence scales down with the cos- a ing pressure, has an attractive gravitational effect like micexpansion,andtheenergydensityofphantommat- usualpressurelessmatterandneitherabsorbsnoremits ter increases with the expansion of the Universe. radiation. The dark energy component in general is Mostly, the attention has been paid to dark en- considered as a kind of vacuum energy with negative ergy as high energy scalar fields, characterized by a time varying equation of state, for which the potential MauricioCataldo of the scalar field plays an important role. Among Departamento de F´ısica, Facultad de Ciencias, Universidad del scalar field models we can enumerate quintessence B´ıo–B´ıo,AvenidaCollao1202,Casilla5-C,Concepcioń, Chile. models (Ratra and Peebles 1988; Chiba et al. 1988; Carroll1998), Chameleonfields(Khoury and Weltman LuisP.Chimento 2004; Brax et al. 2004), K-essence (Myrzakulov 2011; Departamento de F´ısica, Facultad de Ciencias Exactas y Nat- Chiba et al.2000;Armendariz et al.2000;Armendariz1 et al. urales, Universidad de Buenos Aires, Ciudad Universitaria Pa- 2001; Aguirregabiria et al. 2004; Aguirregabiria1et al. bellońI,1428Buenos Aires,Argentina. 2005),G-essence(Yerzhanov et al. 2010;Kulnazarov et al. [email protected] 2010),Chaplygingases(Kamenshik et al. 2001;Bento et al. [email protected] 2002;Chimento 2006),tachyons(Padmanabhan 2002; 2 Padmanabhan and Choudhury 2002;Chimento 2004), of “dust”. In particular, motivated by the fact that phantom dark energy (Caldwell 2002), cosmon mod- the dark matter is generally modelled as a system of els(Bauer et al. 2010;Grande et al. 2006,2007),etc. collisionless particles (Wanget al. 2005; Muller 2005; Ingeneralthecrossingofthephantomdividecannot Rabey 2005; Copeland et al. 2006), we have the pos- beachievedwithauniquescalarfield(Melchiorri et al. sibility of giving to cold dark matter content an origin 2003; Vikman 2005; Sen 2005). This fact has moti- basedonthenatureoftheCDF.Ontheotherhand,the vated a lot of activity oriented towards different ways stiff fluid is an important component because, at early to realize such a crossing (Caldwell and Doran 2006; times, it could describe the shear dominated phase of Cai et al. 2005;Sahni 2005;Sahni and Shtanov 2003; a possible initial anisotropic scenario, dominating the Arefeva et al. 2005; Cai et al. 2010, 2006; McInnes remaining components of the model. 2005; Wei and Zhang 2003; Wei and Tian 2004; Wei The organization of the paper is as follows: In Sec. 2005; Wei et al. 2005; Wei and Cai 2005; Feng et al. II we present the dynamical field equations for a FRW 2005, 2006; Xia et al. 2005; Wu and Yu 2005; Zhang cosmological model with a matter source composed of 2005; Elizalde et al. 2004; Perivolaropoulos 2005; a stiff fluid and a CDF. In Sec. III the behavior of Zhao et al. 2005; Wei and Cai 2006; Li et al. 2004; the dark energy component is studied. In Sec. IV we Huang and Guo 2005; Zhang et al. 2006; Guo et al. conclude with some remarks. 2005;Zhang et al. 2006;Anisimov et al. 2005;Lazkoz and Leon 2006; Deffayet et al. 2010; Lim et al. 2010). For in- stance, in Ref. (Chimento and Lazkoz 2002) was ex- 2 Dynamical field equations ploredthesocalledkinetick-essencemodels(Chimento and Feinstein 2004;Chimento 2004;Scherrer 2004;Chimento et al. We shall adopt a spatially flat, homogeneous and 2005; Chimento and Forte 2006; Aguirregabiria et al. isotropic spacetime described by the FRW metric 2004), i.e. cosmological models with several k-fields in which the Lagrangian does not depend on the fields themselves but only on their derivatives. It was shown ds2 =dt2 a2(t) dx2+dy2+dz2 , (1) − that the dark energy equation of state transits from a (cid:0) (cid:1) where a(t) is the scale factor. The spacetime contains conventionalto a phantomtype matter. Note that for- a cosmic fluid composed by (i) a stiff fluid ρ =ρ /a6 mally,onecangetthephantommatterwiththehelpof s s0 and (ii) a homogeneous classical Dirac field ψ. The a scalar field by switching the sign of kinetic energy of Einstein-Dirac equations are the standard scalar field Lagrangian (Caldwell 2002). Sothattheenergydensityρ = (1/2)Φ2+V(Φ)and ρ ph − 3H2 Λ= s0 +ρ , (2) the pressure pph = (1/2)Φ2 V(Φ) of the phantom − a6 D − − field leads to ρph+pph = −Φ2 <0, violating the weak Γi∇i−α ψ =0, (3) energy condition. (cid:0) (cid:1) In the Universe nearly 70% of the energy is in the where H = a˙/a is the Hubble expansion rate, α is a form of dark energy. Baryonic matter amounts to only constantandthedotdenotedifferentiationwithrespect 3 4%, while the rest of the matter (27 %) is be- tothecosmologicaltime. HereρDrepresentstheenergy − lievedtobeintheformofanon-luminouscomponentof density of the CDF. non-baryonic nature with a dust–like equation of state The dynamical equation for the CDF in curved (w =0)knownascolddarkmatter(CDM).Inthiscase, spacetime can be obtained using the vierbein formal- if the dark energy is composed just by a cosmological ism. So, Γi are the generalized Dirac matrices, which constant, then this scenario is called ΛCDM model. satisfy the anticommutation relations Below, we analyze a FRW universe having cosmo- Γi,Γk = 2gikI, (4) logical constant and filled with a stiff fluid and a clas- { } − sical Dirac field (CDF). With this matter configura- with the metric tensor gik and I the identity 4 4 ma- tion, we will see that the FRW universe evolves from a × trix. They can be defined in terms of the usual repre- non–accelerated stage at early times to an accelerated sentationoftheflatspace-timeconstantDiracmatrices scenario at late times recovering the standard ΛCDM γi as cosmology. The CDF may be justified by an important property: in a spatially flat homogeneous and γβ Γ0 =γ0, Γβ = , (5) isotropicFRWspacetimeitbehavesasa“perfectfluid” a withanenergydensity,notnecessarilypositivedefinite. This pressureless “perfect fluid” can be seen as a kind 3 where the Dirac matrices γi can be written with the where Pauli matrices σβ as αb2 ρ = , (13) I 0 0 σβ m a3 γ0 =i , γβ =i − (6) 0 I σβ 0 ρ αd2 (cid:18) − (cid:19) (cid:18) (cid:19) ρ = s0 +Λ. (14) x a6 − a3 and Here α, b2 and d2 have dimensions of time−1. 0 1 0 i σ1 = 1 0 ,σ2 = i −0 , Notice that there is some arbitrariness in the dis- (cid:18) (cid:19) (cid:18) (cid:19) tribution of the four components of Eq. (11) between 1 0 dark matter and dark energy, since it is not clear that σ = , (7) 3 0 1 we can represent each dark component with a mix of (cid:18) − (cid:19) positive and negative energy components. To avoid with I the identity 2 2 matrix. The symbol = i this arbitrariness we have included both positive en- × ∇ ∂ +Σ denotesthespinorialcovariantderivatives,being i i ergy components of CDF into dark matter and both the spinorialconnectionΣ definedby Γ =0. Then i i k negative energy components of CDF into dark energy. ∇ it leads to Thisassignmenthasaninvariantmeaningbecause itis 1 preservedunder linear transformations of CDF. Σ =0, Σ = HΓ0Γ . (8) 0 β 2 β This distribution allows us to interpret the positive part of the Dirac energy–momentum tensor as a pres- Coming back to the Dirac equation (3), it takes the sureless matter (giving rise to the total (“true”) ob- form servable matter ρ ); while its negative part, together m 3 with the stiff fluid and the cosmological constant, we Γ0 ∂t+ 2H −α ψ(t)=0. (9) interpretas the effective darkenergy componentρx. It (cid:20) (cid:18) (cid:19) (cid:21) must be noted that this is equivalent to assume that Restricting ourselves to the metric (1), the general so- we have a dust component ρ = ρ a−3 and a CDF M M0 lution of the latter equation consistent with Eq. (2) is ρ =ρ a−3associatedwithanegativeenergydensity D D0 given by (i.e. with d2 >b2) in the dark energy sector. The general solution of the Einstein equation (12) with sources (13) and (14) takes the form b e−iαt 1 ψ(t)= a31/2  bd2∗1ee−iiααtt  (10) a3(t)= α(b22−Λd2) −1+cosh√3Λt  d∗eiαt  h i  2  + ρs0 sinh√3Λt, (15) Λ with arbitrary complex coefficients b , b , d and r 1 2 1 d2. The only nonvanishing component of the energy- where we have set the initial singularity at t = 0. In momentum tensor for the CDF is Fig 1 is shown the behavior of the scale factor and its derivatives. α ρ TD = b 2+ b 2 d 2 d 2 D0, (11) 00 a3 | 1| | 2| −| 1| −| 2| ≡ a3 (cid:0) (cid:1) 3 Dark energy evolution where ρ = α(b2 d2), b2 = b 2 + b 2 and d2 = D0 1 2 − | | | | d 2+ d 2. For positive values of ρ this source for- 1 2 D0 It is wellknown that the present observationaldata do | | | | mally behavesasa perfectfluid representinga classical not exclude the existence of the phantom dark energy; dust. However, the CDF will allow us to extend the thusitisinterestingtoconsiderthepossibilityofcross- analysisfornegativevaluesoftheenergydensity. Here, ing the phantom divide. We shall see that the Dirac werestrictourselvestothe physicalsectord2 <b2,this field component can produce a strong variation of the means that ρ 0. D dark energy density ρ , allowing one to have such a ≥ x From Eq. (11) we see that the energy density of the crossing at early times. CDF is given by ρ = ρ /a3. Thus we shall rewrite D D0 Inordertohaveapositiveρ ,wechoosetheparame- x the Friedmann equation (2) in the following form: tersofthemodelsaccordingtothefollowingrestriction: 3H2 =ρ +ρ , (12) m x α2d4 <4Λρ . (16) s0 4 Sincea(t)isanincreasingfunctionoftime,thiseffective darkenergycomponentdecreasesuntilitreachesamin- imumvalueρ =Λ α2d4/4ρ ata =(2ρ /αd2)1/3 xc s0 c s0 − where the dark component crosses the phantom divide and begins to increase with time (see Fig. 2). Fundamentally, the effective dark energy component crosses the phantom divide due to the presence of the d–parameter. In fact, the cosmological evolution of darkenergydependsonthenegativeterm αd2/a3,see − Eq.(14),becauseitproducesaminimumatρ˙ (a )=0, x c showing the importance of considering the CDF as a source of the Einstein equation. Assumingthattotalmatteranddarkenergyarecou- pled only gravitationally, then they are conserved sep- arately, so we have that ρ˙ +3Hρ =0, (17) m m ρ˙ +3H(1+w )ρ =0, (18) x x x wherewehaveassumedtheequationofstatep =w ρ x x x for the dark energy component. Taking into account that the dark energy component has a variable state Fig. 1 We show the behavior of the scale factor (solid parameter w = w (a), we define as a = (ρ /Λ)1/6 x x m s0 line)anditsderivativesa˙ (dottedline)anda¨(dashedline). thevalueofthescalefactorwherethisequationofstate Fromthisitisclearthatthiscosmological scenarioexhibits coincides with the matter one, that is wx =0. So that, anacceleratedexpansionsincethereisastagewherea¨>0. the aboverestrictionα2d4 <4Λρ now becomes a < s0 m a . In terms of the above parameters, a and a , the c c m dark energy state parameter w can be written as x 1 (a/a )6 m w = − . (19) x 1 2(a/a )3+(a/a )6 c m − On the other hand we can consider the state equation for the full source content. For this 2–component system we define the total pressure p = w ρ , where T T T p = p and the total energy density ρ = ρ +ρ . T x T m x This implies that w x w = , (20) T 1+r where r =ρ /ρ . In Fig. 3 we compare the behaviors m x of the state parameters of the dark energy and the full source content. Let us consider in more detail the ratio of energy densitiesofmatteranddarkenergy. Thedefinedabove ratio takes the form αb2a3 r = . (21) Λa6 αd2a3+ρ s0 − It is easy to show that this ratio has a maximum at Fig. 2 We show the behavior of ρx as a function of a = ac = (ρs0/Λ)1/6 (at this point the d2r/da2 < 0). time. The dashed line represents the limit case ac = am, This maximum value is given by i.e. α2d4 = 4Λρs0, while solid and dotted lines represent a typical case satisfying the condition ac > am since αb2 ρ /Λ α2d4 <4Λρs0. s0 r = . (22) max 2ρ αd2 ρ /Λ s0 p s0 − p 5 Itisclearthatforcosmologicalscenarioswherer < max 1thedarkenergycomponentalwaysdominatesoverthe ρ (a=1) ρ αd2+Λ x s0 dark matter during all cosmological evolution. Thus, Ωx,0 = ρ = αb2+ρ− αd2+Λ. (28) crit s0 in order to have stages where dark matter dominates − Now it may be shown that, in general, for a flat FRW over dark energy, we have to require that r > 1. max cosmology the deceleration parameter q is given by So in this case we would have two values for the scale factor where the energy density of dark matter equals aä 1 3p q = = + T . (29) the energy density of the dark energy: −a˙2 2 2ρ T α(b2+d2) α2(b2+d2)2 4Λρs0 1/3 Taking into account that we have pT =px =ωxρx and a± = ± − . (23) ρ =ρ +ρ the deceleration parameter is given by 2Λ T m x " p # 1 3 Thus, for cosmological scenarios where rmax > 1, at q = 2 + 2ωx(1−Ωm), (30) the beginningthe darkenergydominatesoverthe dark matter until the scale factor reaches the value a = a− and evaluating it today (i.e. a = 1) and using (19) we wherethedarkmatterenergydensityequalstheenergy obtain density of dark energy and then it begins to dominate. α(b2 d2)(1 2q )+2ρ (2 q )=2Λ(1+q ). (31) This stage of domination of the dark matter is pro- − − 0 s0 − 0 0 longed until the moment when the scale factor reaches Thus, for accelerated scenarios, q < 0, we require a the value a = a and the dark energy again starts to + positivecosmologicalconstant. Anotherconstraintmay dominate over the dark matter (see Fig. 4). beintroducedbytakingintoaccountthemomentwhen the Universe has started to accelerate again. In other 3.1 Constraints on cosmologicalparameters words,thisisrelatedtothemomentwhentheUniverse starts violating the strong energy condition (ρ+p 0 Inordertoconfrontourmodelandthecosmologicalob- ≥ and ρ +3p 0). So we must require the inequality servations we shall use the constraints on cosmological ≥ ρ+3p < 0. Now from the condition a¨ = 0 and the parametersobtained fromthe analysis ofobservational equivalent Friedmann equation data assuming the Λ-CDM model. This choice is justified by the fact that our model differs from Λ-CDM a¨ 1 = (ρ+3p), (32) only at early times. It must be noted that in general a −6 sucha procedureofusingconstraintsderivedfromafit we conclude that ρ+3p = 0, which implies that ρ + ofonespecificmodelcangiveatbestroughestimations m ρ +3ω ρ =0, obtaining the condition x x x for the parameters of a different cosmologicalmodel. The proposed scenario is characterized by four pa- α(b2 d2)(1+z )3+4ρ (1+z )6 =2Λ, (33) acc s0 acc − rameters which may be constrained by the astrophysical observations available up to date. Since we have where the equation 1/a = (1+z) was used. Here zacc consideredflatFRWcosmologicalscenarios,thedimen- is the value of the redshift when the Universe starts to sionless density parameters are constrained today by accelerate again. In conclusionwe have the four conditions (27), (28), Ωm,0+Ωx,0 =1. (24) (31)and(33)forthefourparametersαb2, ρs0, αd2 and Λ of our model. From Eqs. (12)–(14) we have that Note that from Eqs. (31) and (33) we have that 3H2 = αb2 + ρs0 αd2 +Λ. (25) ρD0 =α(b2 d2)=Kρs0, (34) a3 a6 − a3 − where Evaluating it today (where we set a=1) we have that 4(1+q )z¯2 2(2 q ) ρ =3H2 =αb2+ρ αd2+Λ, (26) K = 0 − − 0 , z¯=(1+zacc)3. (35) crit 0 s0− (1 2q0) (1+q0)z¯ − − so the two dimensionless density parameters are given Since ρ is related to the energy density of the CDF, D0 by we must require that K > 0. So from Eqs. (27), (28), (31) and (33) we have that ρ (a=1) αb2 m Ω = = , (27) m,0 ρcrit αb2+ρs0−αd2+Λ αb2 =3H02Ωm,0, (36) 6 ρ = 6H02 , (37) Inthe Table1weinclude somevaluesobtainedfrom s0 K(2+z¯)+2(1+2z¯2) Eqs. (36)–(39) for the parameters αb2, αd2, ρ , Λ and s0 K corresponding to some given values of the parameters H , q and z (with c = 1 and G = 1). For the 6KH2 0 0 acc d2 =b2 0 , (38) Hubble parameter H is considered both possible val- − α[K(2+z¯)+2(1+2z¯2)] 0 uesH0− forh=0.61andH0+ forh=0.75,andforthe matter dimensionless density parameter we have taken Λ= 3H02z¯(K+4z¯) , (39) Ωm,0 = 0.3. The last two columns represent the age K(2+z¯)+2(1+2z¯2) ofthe Universedetermined fromthe model parameters for h = 0.75 and h = 0.61 respectively. Clearly in the where we have used Eqs. (24) and (34). proposed model there are configurations which are al- Now the four model parameters need to be con- lowed, being their ages larger than the oldest known strained. We do this by using the Eqs. (36)–(39) and stellar ages, since there exist combinations of the pa- byconsideringtheincreasingbulkofobservationaldata rameters αb2, αd2, ρ , Λ which give t > 11-14 Gyr s0 0 that have been accumulated during the past decade. satisfying the stellar population constraints. ThepresentexpansionrateoftheUniverseismeasured by the Hubble constant. From the final results of the 3.2 The effect of the d–parameter Hubble SpaceTelescopeKeyProject(Freedman et al. 2001)tomeasuretheHubbleconstantweknowthatits Nowitisinterestingtogetsomeinsightsconcerningthe present value is constrained to be H = 68 7 kms−1 natureofthed–parameterforstudyingtheeffectonthe 0 Mpc−1 (Chen et al. 2003). Equivalentlyw±ecanwrite cosmologicalevolutionofthenegativeterm αd2/a3 in − H−1 =9.776h−1Gyr,wherehisadimensionlessquan- the Eq. (14) for the dark energy component ρx. It can 0 tity and varies in the range 0.61<h<0.75. beshownthatintheproposedcosmologicalscenariothe Now assuming a flat Universe, i.e. Eq. (24) is valid, dominant energy condition (DEC) is violated thanks Perlmutter et al. (Perlmutter et al. 1999) found that to the presence of this parameter. Effectively, if d = the dimensionless density parameter Ωm,0 may be con- 0 the dark energy state parameter wx and the state strained to be 0.3, implying from Eq. (24) that parameter of the full source content wT are given by ∼ Ω 0.7 (Copeland et al. 2006; Capozzielo 2007; Cax,r0do∼ne et al. 2004), and the present day decelera- w = ρs0−Λa6, w = ρs0−Λa6 , (42) x ρ +Λa6 T ρ +Λa6+αb2a3 tion parameter q may be constrained to be 1 < s0 s0 0 − q0 < 0.64 (Copeland et al. 2006; Capozzielo 2007; respectively. Fromtheseexpressionsweseethatalways − Cardone et al. 2004; Jackson 2004). 1 < w < 1 which implies that now the dark energy x − For consistency we also need to compare the age component satisfies DEC, as well as w . Their general T of the Universe determined from our model with behavior is shown in Fig. 5 (compare with Fig. 3). the age of the oldest stellar populations, requiring However, the fulfilment of the DEC does not imply that the Universe be older than these stellar popu- that the Universe has a decelerated expansion. From lations. Specifically, the age of the Universe t0 is Eq. (32) we may write that constrained to be t > 11-14 Gyr (Copeland et al. 0 6a¨ 2006; Krauss and Chaboyer 2003; Grattonet 2003; = (ρ +3p )= a − T T Imbriani et al. 2004;Hanssen et al. 2004;Jimenez et al. 4ρ α(b2 d2) 1996; Richer et al. 2002; Hansen et al. 2002). s0 + − 2Λ , (43) So let us calculate the age of the Universe from the − a6 a3 − (cid:18) (cid:19) Friedmann equation (25). This equation may be writ- andputtingd=0weseethattheacceleratedexpansion ten as is realized if a > a = ((αb2 + √α2b4+32Λρs0))1/3. acc 4Λ H2 = H02 ρs0+αa3(b2−d2)+a6Λ , (40) Another property of the d–parameter to be considered a6 (cid:18) ρs0+α(b2−d2)+Λ (cid:19) is its effect on the deceleration parameter q0. From Eq.(38),whichisindependentoftheHubbleparameter obtainingEq.(24)whentheexpression(40)isevaluated H and,usingEqs.(35)wecanexpressthedeceleration 0 today. Then the age of the Universe may be written as parameter q as a function of the z obtaining 0 acc ∞ t0 = 0 H(d1z+z) = q (z )= (z¯−1) (2z¯+1)(b2−d2)α−4H02(z¯+1) , 1 ∞ α(Rb2−d2)+ρs0+Λ dz . (41) 0 acc (cid:0) 2H02(2z¯2+1) (cid:1) H0 0 α(b2−d2)(1+z)3+ρs0(1+z)6+Λ (1+z) (44) q R 7 Table 1 In this table we show some values of the model parameters obtained for given H0, q0 and zacc (Ωm,0 =0.3,c= 1,G=1). H0(s−1) q0 Zacc αb2 (s−2) αd2(s−2) ρs0(s−2) Λ(s−2) K H0+,t0 H0−,t0 H0+ -0.68 0.587 5.315×10−36 2.009×10−36 2.368×10−37 2.655×10−35 14 12 (Gyr) 15 (Gyr) H0− -0.68 0.587 3.516×10−36 1.329×10−36 1.566×10−37 1.756×10−35 14 12 (Gyr) 15 (Gyr) H0+ -0.68 0.94 5.315×10−36 1.538×10−36 1.415×10−39 1.210×10−34 2619 11 (Gyr) 13 (Gyr) H0− -0.68 0.94 3.516×10−36 1.017×10−36 9.361×10−40 8.006×10−35 2619 21 (Gyr) 25 (Gyr) H0+ -0.9 0.6 5.315×10−36 5.126×10−36 4.961×10−37 1.751×10−35 0.38 11 (Gyr) 13 (Gyr) H0− -0.9 0.6 3.516×10−36 3.391×10−36 3.281×10−37 1.158×10−35 0.38 11 (Gyr) 13 (Gyr) H0+ -0.9 1 5.315×10−36 4.333×10−36 9.926×10−38 2.217×10−35 9.9 14 (Gyr) 17 (Gyr) H0− -0.9 1 3.516×10−36 2.866×10−36 6.566×10−38 1.466×10−35 9.9 14 (Gyr) 17 (Gyr) Fig. 4 We show the behavior of the ratio r = ρm/ρx as Fig. 3 Weshowthebehaviorof thedarkenergystatepa- a function of the scale factor a. The dashed line repre- rameter wx (dashed line) and the state parameter for the sentstheΛCDMcosmology, wherer∝1/a3. Thesolid line full source content w (solid line). It is clear that the dark represents the case where r > 1 and so at the beginning T energyviolatesthedominantenergyconditionwhilethefull dark energy dominates, then at the range a∈(a−,a+) the source content does not violate it. Note also that the dark dark matter component dominates over dark energy and, energy component remains in the phantom region as it en- for a > a+, the dark energy component dominates again. ters into it after crossing the phantom divide. Thedottedlinerepresentsascenariowherethedarkenergy always dominates overthedark matter, since rmax <1. 8 where, as before, z¯ = (1+z )3. We can see that in acc this case q (z ) rapidly tends to the value 0 acc α 3Ω d2 q0,∞ =−1+ 2H2(b2−d2)=−1+ 2m,0 1− b2 , 0 (cid:18) (cid:19) (45) obtaining, for d = 0 and Ωm,0 = 0.3, the value q0,∞ = 0.55 and, from this value the deceleration parameter − reaches the value q0,∞ = 1 for d b(see Fig. 6). So − ≈ the parameter d directly affects the range of validity of the deceleration parameter which is constrained to be in the range 1 < q < 0. Note that the value 0 − q0,∞ = 1 for d= b is independent of the value of the − dimensionless density parameter Ω . m,0 4 Concluding remarks Sincetodaytheobservationsconstrainthevalueofω to be closeto ω = 1,wehaveconsideredbroadercosmo- − logical scenarios in which the equation of state of dark Fig. 5 We show the behavior of wx and of wT as a func- energy changes with time. The two principal ingredi- tions of the scale factor. The dashed line represents the ents of the model are a stiff fluid which dominates at behaviorofthedarkenergystateparameter,whilethesolid earlytime(Dutta and Scherrer 2010)andaCDF.The line represents the behavior of the state parameter of the positive part of the latter was associated with a dark fullsource content. Both satisfy theDECsincein thiscase matter component while its negative part was consid- d=0. ered as a part of the dark energy component and was responsible for the effective dark energy density cross- ingthephantomdivide(seeandcompareFigs.3and5). At the end, this cosmological model becomes accelerated recovering the standard ΛCDM cosmology. In general, the model may be seen as a continua- tion of the inflation era. In the inflationary paradigm the scalar field Φ, driven by the potential V(Φ), gen- erates the inflationary stage. In the slow roll limit Φ˙2 << V(Φ), with ω 1, we have a superlumi- Φ ≈ − nal expansion while in the kinetic–energy dominated limit Φ˙2 >>V(Φ), with ω 1, we havea stiff matter Φ ≈ scenariocharacterizedbyasubluminalexpansion. Tak- ingintoaccountthatourmodelhasavariableequation of state, we can think of it as a transient model which interpolatessmoothlybetweendifferentbarotropiceras as,forinstance,radiationdominatedera,matterdomi- natederaansoon. Inotherwords,fromEqs.(19)–(21) we see that w 1 (and w 1) for a 0 imply- x → T → → ing that the energy density ρ behaves like 1/a6 and x matching,afterinflation,withthekinetic–energymode of the scalar field ρ 1/a6. Now from Eq. (19) we Φ ∝ seethatρx passesthrougharadiationdominatedstage Fig. 6 We show the behavior of the deceleration param- (i.e. ωx =1/3) for eter q0 as a function of zacc for three values of the parameter d (0,1/2,1) with Ωm,0 = 0.3 and b = 1. We can see 3 6 thatthedeceleration parameter rapidlytendstothevalues a a a a = m m + 8+ m , (46) (−0.55,−0.6625,−1) respectively. rad 22/3 (cid:18)ac (cid:19) s (cid:18)ac (cid:19)    9 behaving like ρ 1/a4 and dominating over ρ . x ≈ rad m After that, at a = a , we have w = 0 and the effec- m x tive dark component behaves as a pressureless source thus obtaining a matter–dominated stage. Finally the modelevolvesfromthisstatetoavacuum-energydom- inated scenario. It is interesting to note that in the a matter–dominated stage, if the condition (16) is ful- filled, then the total energy density is given by ρ = T α(b2 d2)/a3 +2Λand,ifa =a thenρ =αb2/a3 , − m m c T m being ρ =0, implying that we have at this stage only x the dark matter component. The above results indicate that a cosmological scenario based on a CDF component and the effective multifluid configurationρ can, in certain cases,repro- x duce the quintessentialbehavior(see Figs.3and5). In fact, the state parameter of the total matter content 1 < w < 1 is constrained the same as is the state − T parameterofthescalarfieldinquintessencemodels. In thismannerweavoidtheuseofscalarfieldsandpartic- ularclassesofpotentials fordescribingthe darkenergy component. Finally, all the parameters of the model have been expressed in terms of the observable quantities which may be constrained by the astrophysical observational data. In effect, in Table 1 some values of the model parameters αb2, αd2, ρ and Λ were included, which s0 correspondto some given values of the parameters H , 0 q ,z andΩ constrainedbyastrophysicalobserva- 0 acc m,0 tions. 5 acknowledgements LPC acknowledges the hospitality of the Physics De- partment of Universidad del B´ıo-B´ıo where a part of this work was done. The authors thank Paul Min- ning for carefully reading this manuscript. This work was partially supported by CONICYT through grants FONDECYT N0 1080530(MC), MECESUP USA0108 (MC and LPC) and Project X224 by the University of Buenos Aires along with Project 5169 by the CON- ICET (LPC). It was also supported by the Dirección de Investigación de la Universidad del B´ıo–B´ıo (MC) and Consejo Nacional de Investigaciones Cient´ıficas y Técnicas (LPC). 10 References Feng,B., Wang, X.L. & Zhang, X.M., Phys.Lett. B 607, 35 (2005) [arXiv:astro-ph/0404224]. Aguirregabiria, J.M., Chimento, L.P. & Lazkoz, R., Phys. Freedman, W. 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