A new interacting two fluid model and its consequences German S. Sharov∗ Department of Mathematics, Tver State University, 170002, Sadovyj per. 35, Tver, Russia Subhra Bhattacharya† Department of Mathematics, Presidency University, Kolkata-700073, West Bengal, India Supriya Pan‡ Department of Physical Sciences, Indian Institute of Science Education and Research−Kolkata, Mohanpur−741246, West Bengal, India Rafael C. Nunes§ Departamento de F´ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil 7 Subenoy Chakraborty¶ 1 Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India 0 2 In the background of a homogeneous and isotropic spacetime with zero spatial curvature, we n consider interacting scenarios between two barotropic fluids, one is the pressureless dark matter a (DM) and the other one is dark energy (DE), in which the equation of state (EoS) in DE is either J constant or time dependent. In particular, for constant EoS in DE, we show that the evolution 1 equations for both fluids can be analytically solved. For all these scenarios, the model parameters 3 havebeenconstrainedusingthecurrentastronomicalobservationsfromTypeIaSupernovae,Hubble parameter measurements, and baryon acoustic oscillations distance measurements. Our analysis ] shows that both for constant and variable EoS in DE, a very small but nonzero interaction in the c dark sector is favored while the EoS in DE can predict a slight phantom nature, i.e. the EoS in q - DE can cross the phantom divide line ‘−1’. On the other hand, although the models with variable r EoS describe the observations better, but the Akaike Information Criterion supports models with g minimalnumberofparameters. However,itisfoundthatallthemodelsareveryclosetotheΛCDM [ cosmology. 2 v PACSnumbers: 98.80.-k,95.35.+d,95.36.+x,98.80.Es. 0 Keywords: Cosmologicalparameters;Darkenergy;Darkmatter;Interaction. 8 7 0 1. INTRODUCTION in the last several years [14]. 0 . However, concerning different cosmological theories, 1 Theexplanationofthelate-timeacceleratedexpansion thescenariowheredarkmatter(DM)interactswithdark 0 of the universe [1–9] has become one of the biggest and energy (DE) has gained much attention in the current 7 open problems in modern cosmology today. The most 1 literature. The interaction between DM and DE were reasonable description for this accelerating phase intro- : primarily motivated to address the small value of the v ducessomehypotheticaldarkenergy(DE)fluidcompris- cosmological constant, however later on the models were i X ingabout70%ofthetotalenergydensityoftheuniverse found to provide a reasonable explanation to the cosmic [10]. The cosmological constant, Λ, associated with the r coincidence problem [15, 16]. Recently it has been ar- a zero point energy of the quantum fields, is the main can- gued that the current observational data can favor the didateforDEinagreementwithalargenumberofavail- late-time interaction between DM and DE [17–22]. The able observational data. However, despite of great suc- coupling parameter of the interaction in the dark sec- cess of Λ-cosmology, it presents serious objections in the torhasalsobeenmeasuredbyseveralobservationaldata interface of cosmology and particle physics, such as the [20–29]. In fact, the class of interacting DM − DE mod- cosmological constant problem [11, 12] and the coinci- els could be a promising candidate to resolve the current dence problem [13]. Due to these two serous problems in tensions on σ and the local value of the Hubble con- 8 Λ,severalalternativeshavebeenproposedanddiscussed stant H [21, 30]. Further, it has been contended that 0 the interaction between these dark sectors can influence on the perturbation analysis which results in significant changes in the lowest multipoles of the CMB spectrum ∗Electronicaddress: german.sharov@mail.ru [31, 32]. For a comprehensive analysis on several inter- †Electronicaddress: subhra.maths@presiuniv.ac.in acting models explored in the last couple of years we ‡Electronicaddress: span@iiserkol.ac.in §Electronicaddress: rcnunes@fisica.ufjf.br refer [33–52]. Therefore, based on the analysis on in- ¶Electronicaddress: schakraborty@math.jdvu.ac.in teracting dark energy models with a special indication 2 for a nonzero interaction between the dark sectors from 2. INTERACTING DYNAMICS IN FLAT FLRW the the currently available observational data, it seems promising that the interaction in the dark sectors might Observationsfromcosmicmicrowavebackgroundradi- open some new possibilities in near future. ation and large scale structure predict in good approxi- The interacting dynamics proposes for continuous en- mation that our Universe is homogeneous and isotropic ergy and/or momentum exchange between DM and DE in the largest scale, and further it is almost spatially with the evolution of the universe and modifies the con- flat [10]. This line element describing such a universe servation equations for DM (≡ TDM) and DE (≡ TDE) is known as the Friedmann-Lemaˆıtre-Robertson-Walker µν µν as line element, which takes the form ds2 =−dt2+a2(t)(dx2+dy2+dz2), (2) ∇νTDM = Q, and ∇νTDE =−Q, (1) µν µν where a(t) is the scale factor of the universe, t being the universal cosmic time. Further, this homogeneous where the function Q characterizes the interaction be- and isotropic principle gives a restriction on the matter tween the dark sectors in which Q > 0 represents the distribution of our Universe. It tells that the matter energy and/or momentum flow from DE to DM while distribution should be of perfect fluid type which takes Q<0 means energy and/or momentum flow takes place itsenergymomentumtensorasT =(p+ρ)u u +pg , from DM to DE. Thus, essentially, we choose different µν µ ν µν where ρ, p, respectively denote the energy density and functional forms for Q (with correct physical dimension) the pressure of the cosmic fluid, and u is the four andwesolvethecorrespondingevolutionequations. The ν velocity vector of the fluid components. For our model, procedure followed is similar to that when one chooses we consider the general relativity described by the differentparametrizationsfortheEOSinDE.Fewdevel- Einstein’s field equations G =8πGT , and two fluids opments can be found in the literature where the com- µν µν as follows: plete dynamics of the interacting model has been explored [48, 53, 54]. Another approach to consider the • Pressureless dark matter (or dust) where ρ is its interaction between two fluids is for instance the sigma m energy density and p (=0), is its pressure. models [55, 56]. m In the present work, we determine the complete dy- • A dark energy fluid satisfying a barotropic equation namics of an interacting scenario where DE interacts of state p = ω ρ , where ρ , p , are respectively the d d d d d with pressureless DM through a nongravitational inter- energy density and the pressure of the component, ω is d action considered in [57] and we generalize its observa- its equation of state. tional consequences considering that the DE component isbarotropicanditsEoScouldbeeitherconstantortime Thus, for a co-moving observer (uµ = δµ), the Ein- dependent. We first consider the constant equation of 0 stein’s field equations are explicitly written as (consider- stateindarkenergyandshowthatthebackgroundevolu- ing 8πG=1) tion can be analytically solved which we constrain using the combined analysis from Type Ia Supernovae, Hub- 1 ble data, and baryon acoustic oscillation data. Then we H2 = 3(ρm+ρd), (3) apply the same combined analysis of the above observa- 1 tional data to the case when the EoS in DE is evolving H˙ =−2(pm+pd+ρm+ρd), (4) with the cosmic time. The work has been organized in the following way: where H = a˙/a, is the Hubble parameter. Further, the Section 2 presents a brief introduction of the interact- conservationequationsforDMandDEfollowsfromeqn. ing dynamics in a spatially flat Friedmann-Lemaˆıtre- (1) Robertson-Walker (FLRW) universe. In section 3, we present the analytic solutions for dark energy and cold ρ˙m+3Hρm = Q, (5) dark matter for constant equation of state ω (cid:54)=−1 and ρ˙ +3H(1+ω )ρ = −Q. (6) d d d d when the dark energy is the cosmological constant while additionally,insubsection3.1,weperformanasymptotic Clearly, equations (5) and (6) offer a new dynamics of analysis of the analytic solutions, and then we discussed the universe with the use of the interaction function Q. aboutthepossiblesignoftheinteraction. Insection4we Therearemanyproposedinteractionsintheliteratureto introduce dynamical DE models interacting with CDM. study the dynamics of the universe, however, the exact Section 5 contains all the data analysis tools we used in functional form of Q is still unknown. From the coupled this study. In the next section 6 we present the obser- continuityequations(5)and(6),weobservethatQcould vational constraints on the interacting models and the be any arbitrary function of the parameters H, ρ , ρ . m d main results of the paper. Finally, we close our work in So, naturally, onecanconstructinfinitelymanyinteract- section 7 with summary and discussions. ing models to understand the dynamics of the universe 3 inthisframework. Inthepresentwork,westartwiththe Equations (11), (12) lead to the following interesting following interaction [57]: possibilities we will discuss now. Q = αH(ρ(cid:48)m+ρ(cid:48)d). (7) • If Q > 0, Eq. (11) implies that, ωmeff < 0 meaning that the effective equation of state of the pressureless whereαisthecouplingparameterwhichisdimensionless, dark matter1 is of exotic type, and it could become a and the ‘ (cid:48) ’ denotes the differentiation with respect to dark energy candidate (EoS < −1/3) if Q > Hρ . On x = ln(a/a ) (a is the present value of the scale factor m 0 0 the other hand, the effective equation of state for dark which has been set to be unity in this work, i.e. a =1). 0 energysatisfiesthefollowinginequality: ω <ωeff <∞, In fact, if one uses the conservation equations (5), and d d which shows that if the EoS in dark energy satisfies the (6), it will be readily clear that the interaction (7) has relation ω ≤−1, then the effective interactive system is the following equivalent form s of course in the quintessence era (ωeff >−1). However, d Q = −3αH(ρ +ρ +ω ρ ) (8) if the dark energy is of phantom kind, then the above m d d d inequality could still hold, but in this case, the effective which directly includes the equation of state parameter EoSindarkenergymaylieinthephantom/quintessence of the dark energy component, that means the pressure region depending on the strength of the interaction. of the dark energy component. However, inserting (7) into the conservation equations (5) and (6), we decouple • If Q < 0, then ωeff > 0 irrespective of anything. m the conservation equations as In case of effective dark energy, we find that ωeff < ω d d which readily points that, for ω ≤ −1, the effective (1−α)ρ(cid:48) +3ρ −αρ(cid:48) = 0, (9) d m m d equation of state is always phantom in nature. On the (1+α)ρ(cid:48)d++3ρd(1+ωd)+αρ(cid:48)m = 0, (10) other hand, for ω > −1, ωeff could cross the phantom d d barrier line ‘−1’ due to the presence of Q/3Hρ < 0 in where both equations (9) and (10) seem to describe an d (12). But, the possibility of quintessence behavior can’t effectivenon-interactingtwofluidsystem,andduetothe be excluded as again we do not specify the strength of interaction,theeffectiveEoSforpressurelessDM(ωeff), m the interaction. DE (ωeff) take the forms d Q However, introducing the total energy density of the ωeff = − universe, ρ = ρ +ρ , we can express the dark energy m 3Hρ t m d m and the dark matter density in terms of the total energy α = − (ρ(cid:48) +ρ(cid:48)) density and its derivative as ρ m d = 3rαm(1+r+ωd), (11) ρd =−(cid:18)ρ(cid:48)t3+ωd3ρt(cid:19), (14) (cid:18)ρ(cid:48) +3(1+ω )ρ (cid:19) and ρ = t d t , (15) m 3ω d Q ωdeff = ωd+ 3Hρ , Now, inserting (14) into (10), we obtain the following d second order differential equation α = ωd+ ρ (ρ(cid:48)m+ρ(cid:48)d), (cid:20) ω(cid:48) (cid:21) (cid:20) ω(cid:48) (cid:21) d ρ(cid:48)(cid:48)+3 2+ω −αω − d ρ(cid:48) +9 (1+ω )− d ρ =0, = ωd−3α(1+r+ωd), (12) t d d 3ωd t d 3ωd t (16) where r = ρ /ρ , is the coincidence parameter. Now, m d which if we could solve might describe the possible dy- from (11), (12), we note that, even if the EoS in DM namics of the universe. In what follows, we consider the andDEareconstant, butduetotheinteractionbetween evolution of the universe both for constant and dynamic them, the effective EoS of the two components become EoS in DE. variable in nature. Now, if we denote the total EoS for thenon-interactingDM-DEsystembyωeff =ω +ω = m d ω (asDMispressureless,i.e. ω =0),thentheeffective d m 3. CONSTANT EOS IN DE EoS:ωeff =weff+ωeff,ofthesamesystemduetothis t m d interaction becomes Let us now consider the case when the EoS in DE is (cid:18) 1 1 (cid:19) independent of time, we denote the EoS by ω . This ωeff = ωeff +α − (ρ(cid:48) +ρ(cid:48)) d t ρ ρ m d d m (cid:18) (cid:19) 1 1 = ω +α − (ρ(cid:48) +ρ(cid:48)), d ρ ρ m d d m 1 We define the effective EoS of the pressureless dark matter as 3α the EoS of the pressureless dark matter plus the energy gained = ω + (1−r)(1+r+ω ) (13) d r d duetothisinteractingprocess. 4 consideration reduces the differential equation (16) into where ω is the present value of the EoS in DE2. Now, d0 the following simplified format dividing (22) and (23) by ρ = 3H2/8πG, we get the c0 0 present day density parameters for DE and DM as fol- ρ(cid:48)t(cid:48)+3(cid:2)2+ωd−αωd(cid:3)ρ(cid:48)t+9(1+ωd)ρt = 0, (17) lows: which solves for ρ as (cid:18) 1 (cid:19)(cid:104) (cid:105) t Ω =− Ω (m +3)+Ω (m +3) , (24) d0 3ω 1 1 2 2 d0 ρt = ρ1em1x+ρ2em2x =ρ1am1 +ρ2am2, (18) (cid:18) 1 (cid:19)(cid:104) (cid:105) Ω = Ω (m +3+3ω )+Ω (m +3+3ω ) , where ρ1, ρ2 are the constants of integration, and, m0 3ωd0 1 1 d0 2 2 d0 (cid:0)m ,m (cid:1) are given by (25) 1 2 3(cid:20) (cid:113) (cid:21) Now, solving for Ω1 and Ω2, we find m = −(2+ω −αω )+ (1−α)2ω2−4αω , 1 2 d d d d (m +3+3ω )Ω +(m +3)Ω m2 = 23(cid:20)−(2+ωd−αωd)−(cid:113)(1−α)2ωd2−4αωd(cid:21), Ω1 = 2 dm02−d0m1 2 m0, (26) (m +3+3ω )Ω +(m +3)Ω Ω = 1 d0 d0 1 m0. (27) As ω <−1/3, so, (cid:0)m ,m (cid:1) are real. The equation (18) 2 m1−m2 d 1 2 can be written as (cid:18) H (cid:19)2 • When DE is the cosmological constant = Ω (1+z)−m1 +Ω (1+z)−m2. (19) H 1 2 0 Now,weconsiderthecasewhenDEisthecosmological where Ω1 = ρ1/ρc0, Ω2 = ρ2/ρc0 in which ρc0 = c(cid:0)onstant(cid:1). Thus, in this(cid:0)case, we hav(cid:1)e ωd = −1, hence, 3H2/8πG. Itisinterestingtonotethat,thesolution(19) m1,m2 taketheforms 0, −3(1+α) . Thetotalenergy 0 which represents the analytic behavior for two interact- density takes the form ingfluids, hasalsobeenfoundinthecontextoftwonon- ρ = ρ˜ +ρ˜ (1+z)3(1+α), (28) interacting fluids where one is Brans-Dicke scalar field t 1 2 and the other is the perfect fluid with constant equation and the explicit analytic solutions of the energy density of state [58]. for the cosmological constant and the energy density of Moreover,using(14),(15),theexplicitanalyticexpres- the pressureless matter in this interacting picture take sions for DE and DM density are the expressions ρ =−ρ1(m1+3)(1+z)−m1 +ρ2(m2+3)(1+z)−m2 ρd = ρ˜1−ρ˜2α(1+z)3(1+α), (29) d 3ωd ρm = ρ˜2(1+α)(1+z)3(1+α). (30) (20) Hence, the Friedmann equation becomes and ρm =(cid:18)3ω1 (cid:19)(cid:104)ρ1(m1+3+3ωd)(1+z)−m1 (cid:18)HH0(cid:19)2 = Ω˜1+Ω˜2(1+z)3(1+α), (31) d (cid:105) +ρ2(m2+3+3ωd)(1+z)−m2 . (21) where Ω˜1 = ρ˜1/ρc0, Ω˜2 = ρ˜2/ρc0, and the solution has also been found in the context of two non-interacting However, the constants ρ , ρ do not serve only as the perfect fluids where one is the Brans-Dicke scalar field 1 2 constantsofintegrations, theyhavecrucialmeanings. In and the other is the perfect fluid (in the form of dust) ordertoexplainthis,letusintroducethedensityparam- (Paliathanasis, Tsamparlis, Basilakos, & Barrow 2016). eters for dark matter and dark energy as Ω = ρ /ρ Further, we have the following relations: m m c (ρ = 3H2/8πG), Ω = (ρ /ρ ) respectively, Thus, the c d d c Friedmann equation (3) can be written as Ωm+Ωd =1. Ωd0 = Ω˜1−αΩ˜2, Now, the energy densities at present time can be given Ω = (1+α)Ω˜ . m0 2 by (cid:18) 1 (cid:19)(cid:104) (cid:105) ρ =− ρ (m +3)+ρ (m +3) , (22) d0 3ωd0 1 1 2 2 2 Let us note that since ωd has been considered to be constant, (cid:18) 1 (cid:19)(cid:104) (cid:105) thenωd=ωd0. However,wehaveusedthesuffix0inωdtokeep ρm0 = 3ω ρ1(m1+3+3ωd0)+ρ2(m2+3+3ωd0) , thesamestructurewiththeotherparameters,forinstance,Ωd0, d0 Ωm0 andH0. Therefore, inthescenarioofinteractingDEwith (23) constantEoS,ω andω shouldbeconsideredtobeidential. d0 d 5 Further, we can solve for Ω˜ and Ω˜ as and 1 2 1 m +3+3ω 1(cid:34) (cid:115) (cid:18) 1 (cid:19)(cid:35) Ω˜ = Ω , 2 d = (1+α)+ (1+α)2+4α −1 2 1+α m0 3ω 2 d2 d α Ω˜1 = Ωd0+ 1+αΩm0. >0 (or <0) if ωd <−1 (or ωd >−1) Now,wediscussthethreedifferentstagesoftheuniverse It should be noted that, Ω +Ω =Ω˜ +Ω˜ =1. m0 d0 1 2 as follows: • For z −→ 0, ρ , ρ are finite, and the total energy m d density ρ , as well. t 3.1. Asymptotic behavior of the energy densities • When z is very large, i.e. z −→∞, and the sign of the interaction Asthesolutionsforρd,ρm includeseveralparameters, ρ −→−ρ2(m2+3)(1+z)−m2; so, we shall be very careful, as well as, explicit, while d 3ω d describingtheasymptoticbehaviorofthesolutions. First ρ (m +3+3ω ) of all, we recall that, the roots m1, m2 should be real in ρm −→ 2 23ω d (1+z)−m2; order to realize a feasible state. Hence, for m , m to d be real, we must have (1 − α)2ω2 − 4αω > 01. N2ow, ρt −→ρ2(1+z)−m2. d d underthelowinteractioncondition,α2 canbeneglected, thus, we find that, (1−2α)ω2−4αω > 0. As ω < 0, • When z ≈−1, we consider z =−1+(cid:15), where (cid:15)>0 d d d we let ω = −d2 (d ∈ R) which gives a restriction as is any arbitrary infinitesimal quantity, we have d d2(1−2α)+4α > 0, which is always true (recall that the interaction is very low, i.e. the coupling parameter ρ (m +3) is very very small). Thus, if ωd < 0, then m1, m2 are ρd −→− 1 3ω1 (cid:15)−3m1; d always real under the low interaction assumption. Now, ρ (m +3+3ω ) if m1 >0, then ρm −→ 1 1 3ω d (cid:15)−3m1; d (cid:113) ρ −→ρ (cid:15)−m1. (1−α)2ω2−4αω >(2+ω −αω ) t 1 d d d d =⇒1+ω <0, i.e. ω <−1. For convenience, depending on the signs of the quan- d d tities, we introduce the following: Similarly, m < 0 =⇒ ω > −1. On the other hand, 1 d if 2 + ω − αω < 0, we have ω < − 2 < −1 (in d d d 1−α low interaction scheme and if we do not restrict on the m +3 m +3 m +3+3ω coupling parameter, then of course α<1). 1 =β2; 2 =−γ2; 1 d =θ2 3ω 3ω 3ω d d d In both cases, m < 0, for ω < −1, or ω > −1. 2 d d Hence, we arrive at two different situations: (I) If ω < d m +3+3ω −1, then m >0, m <0, (II) If ω >−1, then m <0, 2 d =δ2 (ifω <−1), or =−δ2 (ifω >−1) 1 2 d 1 3ω d d m < 0. Now, we calculate the following expressions in d 2 order to analyze the asymptotic behavior of the energy Thus, based on the nature of the dark energy (that densities. means it is of quintessence or of phantom type), it is possible to express the energy densities of the dark sectors in the following manner. (cid:34) (cid:114) (cid:35) m +3 1 4α 1 = −(1−α)+ (1−α)2+ >0 3ωd 2 d2 • If ωd <−1, then we have (cid:34) (cid:114) (cid:35) ρd =−ρ1β2(1+z)−m1 +ρ2γ2(1+z)−m2, m +3 1 4α 2 = −(1−α)+ (1−α)2+ <0 ρ =ρ θ2(1+z)−m1 +ρ δ2(1+z)−m2. 3ω 2 d2 m 1 2 d Also, we have the following • If ω >−1, we have d m +3+3ω 1(cid:34) (cid:114) 4α(cid:35) ρ =−ρ β2(1+z)−m1 +ρ γ2(1+z)−m2, 1 d = (1+α)+ (1−α)2+ >0 d 1 2 3ωd 2 d2 ρm =ρ1θ2(1+z)−m1 −ρ2δ2(1+z)−m2. 6 Further, due to the analytic solutions, we can simplify For z −→ 0, ρ −→ ρ˜ − αρ˜ ≥ 0, and d 1 2 the interaction (7) into ρ −→ ρ˜ (1 + α) < 0. For z −→ ∞, ρ −→ ρ˜ , m 2 d 1 and ρ −→ 0. When z −→ −1, we see ρ −→ ∞, m d Q ρ −→−∞. = 3αH2(m Ω am1 +m Ω am2). (32) m H 0 1 1 2 2 Now, we can rewrite the interaction in this case as Now, from Eq. (32), we can infer about the sign in Q Q depending on the parameters involved. = −9α(1+α)H2Ω˜ a−3(1+α), (33) H 0 2 • If m1 >m2 >0, then Q>0 if α>0; and Q<0 for which summarizes the following possibilities: α<0 • If α >0, then Q<0, i.e. the energy flows from DE • Also, if 0 < m < m <, then Q > 0 if α > 0; and to DM. 1 2 Q<0 for α<0 • When −1 < α < 0, then Q > 0, indicating the flow • If m 0 when α<0; and Q<0 of energy from DM to DE. 1 2 for α>0 • Finally, for α < −1, Q < 0 leading to the energy • Now when m m < 0, the sign of Q seems dif- flow from DE to DM. 1 2 ficult to determine. In fact, in this case, we have an interesting observation which tells that, Q could change its sign during the evolution of the universe. To illus- tratetheidea,letustakethefollowingsimplestexample: 4. INTERACTING DYNAMICS FOR VARIABLE EQUATION OF STATE IN DE • An Example: Just for simplicity let us take m = −1, m = 1. Therefore, we have 1 2 Q = 3HαH02(cid:0)−Ωa1 +Ω2a(cid:1). Now, the terms in the Now, we consider an interacting scenario where dark bracket could be once positive and once negative, which energy fluid has a time varying equation of state. This again depends on the sign of α outside the bracket. scenario can be considered as a general scenario of in- teractingdynamics. Thepossibility oftimevaryingdark Hence, generalizing this event (i.e. when m m < 0), energyinteractingwithpressurelessdarkmatterhasbeen 1 2 the equation Q = 0 can have more than one real roots investigatedduringlastcoupleofyears[42,59,60]. How- leading to an oscillatory interacting scenarios in which ever, the present interaction is different from the others once Q>0, and once Q<0. in [42, 59, 60] and it becomes clear when the variable equation of state in DE is considered. So, in the current study along with the constant dark energy equation • Asymptotic behavior when DE is the cosmo- of state in section 3, we consider the case for dynamical logical constant DE. Let us introduce a generalized equation of state in DE given by [61] Depending on the coupling parameter α, we find the (cid:18)(1+z)−β −1(cid:19) following asymptotic behavior of the energy densities ω (z)=ω −ω , (34) d d0 β β both at present and late-times. where ω is the current value of ω (z), and ω is the d0 d β I. When α>0 free parameter. We note that the EoS in (34) recov- ers three well known dynamical dark energy equations In that case, for z −→ 0, ρ −→ ρ˜ − αρ˜ and of state. For β = 1, we recover the Chevallier-Polarski- d 1 2 ρ −→ ρ˜ (1 + α). On the other hand, for z −→ ∞, Linder (CPL) parametrization [62, 63] m 2 ρ −→ −∞, ρ −→ ∞. Further, in the case for d m (cid:18) z (cid:19) z −→−1, ρd −→ρ˜1, ρm −→0. ωd(z)=ωd0+ω1 1+z . (35) II. When −1<α<0 Now, for β = −1, one finds the linear parametrization [64–66], For z −→ 0, ρ −→ ρ˜ − αρ˜ ≥ 0, and d 1 2 ρm −→ ρ˜2(1+α) > 0. When z −→ ∞, ρd −→ ∞, and ωd(z)=ωd0+ω2z, (36) ρ −→∞. Further, for z −→−1, ρ −→ρ˜ , ρ −→0. m d 1 m whileforβ →0,onerealizesthelogarithmicparametriza- tion in DE [67] III. When α<−1 ω (z)=ω +ω ln(1+z), (37) d d0 3 7 where ω is the free parameter. Cosequently, for other 3 valuesofβ (cid:54)=−1,0,1,onecangenerateseveralequations 0.4 of state. Thus, one can study the second order differential equation (16) for the variable EoS in (34) in order to estimate the cosmological parameters with the use of 0.35 current observational data. On the other hand, plugging the total energy density ρt =ρm+ρd =3H2 into the in- m0 Ω teraction(8)forωd(z)givenineqn. (34), theinteraction 0.3 now becomes SNe + BAO Q=H3ψ(z), (38) 0.25 SNe + H(z) where ψ(z) is given by SNe + BAO + H(z) (cid:34) (cid:26) (cid:18)(1+z)−β −1(cid:19)(cid:27) (cid:35) 0.2 −0.06 −0.04 −0.02 0 0.02 0.04 ψ(z)=−9α 1+ ωd0−ωβ β Ωd α (39) and this quantity only determines the sign of Q during 0.4 theevolutionoftheUniverse(sinceH >0,forexpanding universe). Further, current value of the interaction rate 0.35 can be found to be Q = −9αH3 (1+ω Ω ) and its 0 0 d0 d0 signcanbedeterminedoncethecosmologicalparameters are estimated by the current observational data. Ωm0 0.3 Belowweshalldescribethecurrentobservationaldata to constrain the mentioned interacting models and from 0.25 SNe + BAO this point of view we aim to find the most successful dynamic EoS given in (34) for the current interacting SNe + H(z) 0.2 scenario. For this purpose we have to solve numerically SNe + BAO + H(z) the second order differential equation (16) for ω (z) in d (34). −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 ω d 5. OBSERVATIONAL CONSTRAINTS ON THE SNe + BAO MODEL PARAMETERS 0.04 SNe + H(z) In order to constrain the free parameters of the 0.02 models presented in the previous sections, we consider the lastest observational data from Supernovae type Ia 0 (SNIa), Hubble expansion history (OHD), and the bary- α onic acoustic oscillation (BAO) distance measurements. −0.02 The best fit values, with their corresponding uncertain- ties, follow from minimizing the likelihood function L∝ −0.04 exp(−χ2 /2),whereχ2 =χ2 +χ2 +χ2 . Inthe tot tot SN OHD BAO following subsections we briefly describe each data set. −0.06 SNe + BAO + H(z) −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 5.1. Supernovae Type Ia data ω d Data from Supernovae was the first indication for accelerating universe, and it is very useful to test the cos- FIG.1: Thefigureshowsthe1σ,2σ,3σ confidence-levelcon- mological models. In the present analysis we use 580 tour plots of the model parameters with their best fit values Supernovae data points from Union 2.1 available in [68] for the interacting DE model with constant EoS, ω (cid:54)= −1 in the redshift interval 0 ≤ z ≤ 1.41. Now, for any Su- d (Eq. (18)), using different combinations of the observational pernova, situatedataredshiftz,thedistancemodulusµ data sets as SN+ OHD+BAO (blue filled contours), SN+ is given by BAO (red contours), SNe+ OHD (green contours). We note that the contour lines for SN+ H(z) do not appear properly (cid:18) (cid:19) d (z) µ(z) = 5log L +25, (40) in all the plots. 10 1Mpc 8 5.3. Baryon Acoustic Oscillation data 0.4 To analyze cosmological models, baryon acoustic oscillations (BAO) data give another useful test. Here, we useBAOdata,i.e. dA(z∗) fromthereferences[75–78], DV(ZBAO) 0.35 wherez istheradiation-matterdecouplingtimegivenby ∗ z ≈1091,d istheco-movingangulardiameterdistance ∗ A Ωm0 given by d = (cid:82)z dz , and, D = (cid:16)d (z)2 z (cid:17)1/3 is 0.3 A 0 H(z) V A H(z) the dilation scale [8]. In table I, we present the observational BAO data. We use the following procedure as in SNe + BAO [79] to calculate the χ2 minimization for BAO data, i.e. 0.25 SNe + H(z) SNe + BAO + H(z) χ2BAO = XBTAOCB−A1OXBAO, (43) 0.2 −0.06 −0.04 −0.02 0 0.02 0.04 where α FIG.2: Thefigureshowsthe1σ,2σ,3σ confidence-levelcon-  dA(z∗) −30.95 DV(0.106) tour plots of the model parameters with their best fit values  dA(z∗) −17.55  for the interacting cosmological constant (Eq. (28)), using  DV(0.2)  different combinations of the observational data sets as SN+  dA(z∗) −10.11 X = DV(0.35)  OSNHeD++OBHADO((gbrleueenficlloendtocuornst)o.urs),SN+BAO(redcontours), BAO  DdVA((0z.∗4)4) −8.44   dA(z∗) −6.69   DV(0.6)  dA(z∗) −5.45 whered (z)=c(1+z)(cid:82)z dz(cid:48) istheluminositydistance. DV(0.73) L 0 H(z(cid:48)) The χ2 function is evaluated as and the inverse covariance matrix C−1 can be found in (Giostri et al. 2012). 580 χ2 (θ ,...)=min (cid:88) ∆µ (cid:0)C−1(cid:1) ∆µ . (41) SN 1 H0 i,j=1 i SN ij j 6. RESULTS OF THE JOINT ANALYSIS where ∆µi = µth(zi,θ1,...)−µobs(zi), θk, CSN are re- In this section we shall present the main observational spectively the discrepancy between theory and observa- results extracted from the scenarios of interacting dark tions,modelparameterstobefitted,andthe580×580co- energy when dark energy equation of state is either con- variancematrix[68]. Weusethefunction(41),marginal- stant or evolving with the cosmic time. In case of con- ized over the Hubble constant H0. stant EoS in DE we have considered two separate cosmological models one is the interacting DE model with ω (cid:54)= −1, and interacting cosmological constant where d for both the models the background evolution equations 5.2. Hubble data are analytically solved. On the other hand, for the variable EoS in DE we have considered a general EoS given Hubbleparametermeasurementsarealsousefultocon- in (34) and solved the second order differential equation strain the parameters in dark energy models. Two inde- (16) numerically. In the following we shall describe the pendent methods have been widely used to measure the observational constraints on the model parameters for Hubbleparametervaluesatdifferentredshifts. Oneisthe the interacting dark energy models. differential age of some galaxy, and the other one is the radialBAOsizemethods. Hereweuse30OHDpointsfor ouranalysisfromRefs. [69–74], obtainedwiththediffer- 6.1. Constant equation of state in DE ential age method. To constrain the model parameters, we apply the χ2 function by the following method We fit both the interacting models with constant EoS using the latest observational data from Supernovae 30 (cid:20) (cid:21) χ2 = (cid:88) Hobs(zi)−Hth(zi,θj) , (42) Type Ia, observed Hubble parameter data, and the OHD σ2 i=1 i baryon acoustic oscillations distance measurements. We note that for all interacting models, we have marginal- whereθ standsfordifferentmodelparameterstobecon- ized χ2 for all values of H . Let us describe the j tot 0 strained. observational constraints and the main cosmological 9 TABLE I: Values of dA(z∗) for different values of z . DV(zBAO) BAO z 0.106 0.2 0.35 0.44 0.6 0.73 BAO dA(z∗) 30.95 ± 1.46 17.55 ± 0.60 10.11 ± 0.37 8.44 ± 0.67 6.69 ± 0.33 5.45 ± 0.31 DV(zBAO) features of the models separately as follows. search the most suitable value of β in EoS (34), in other words,thebestchoiceamongthemostpopularscenarios • For interacting dark energy, where ω (cid:54)= −1, (35), (36), (37). These scenarios are particular cases of d χ2 (Ω ,α,ω )=562.39, ω =−1.095+0.352, Ω = the general EoS (34), if β =1, −1, 0 correspondingly; in min m0 d0 d0 −0.421 m0 0.320+0.093, α=−0.010+0.01 . Here acceptable values of all these cases we diminish P to P = 4. To investigate −0.147 −0.046 α, i.e its priors are limited to α < 0, because for posi- thegeneralizedEoS(34),wecalculatehowtheminimum tive α we have solutions with negative dark energy den- min χ2tot (over all other parameters) depends on sity ρ given in eqn. (20) at early stages of evolution. Ωm0,α,ωd0,ωβ d β. This dependence is shown in the top-left panel of fig- They may be considered as singular solutions, though ure3. Weseethatthebestvaluesofminχ2 correspond tot the total density ρ = ρ +ρ remains positive and the m d to large positive β (moreover, negative values of β lead scale factor a(t) behaves regularly. Also, the reduced toadditionalsingularities),soweshouldchoosetheCPL χ2 = χ2 /d.o.f = 0.917 (d.o.f = degrees of freedom = min parametrization(35)(Chevallier&Polarski2001;Linder N −P, where N is total number of data points; P is to- 2003), corresponding to β = 1, as the most successful tal number of independent model parameters). In figure scenario of interacting DE with variable EoS. For this Fig. 1 we have shown the contour plots at 1σ, 2σ and 3σ scenario we draw the contour plots for various quantities confidence levels with different combinations of the ob- fortheobservationaldataSN+OHD+BAOinotherpan- servational data sets, namely, SN+BAO, SN+OHD and els of Fig. 3. The correspondent estimations of minχ2 tot SN+OHD+BAO. and model parameters are presented in table II. As we Therefore, from the analysis of this model, it is seen observe, the interacting CPL model shows that the cur- that this interaction can describe the phantom nature of rent dark energy EoS has a slight phantom nature (as the universe. Further, concerning the interaction Q in ω =−1.018<−1), which is although very close to the d0 eqn. (32), using the best fit values of the cosmological Λ-cosmology. Ontheotherhand,usingtheobservational parameters, one can see that m1 >0, but m2 <0, which data into eqns. (38) and (39), it is easy to see that at indicates that during the evolution of the universe, Q present time, i.e. at z =0, Q>0. should change its sign, but however, the observational data tells that for this scenario, Q > 0, for any z ≥ 0, that means the energy flows from DE to DM. 6.3. Model selection • For interacting Λ with ω = −1, we found d In order to test the quality of fit of the present models χ2 (Ω ,α) = 562.55, hence, reduced χ2 = min m0 byinformationcriteria,weapplytheAkaike Information χ2 /d.o.f =0.919. ItisachievedforΩ =0.294+0.060, min m0 −0.054 Criterion (AIC) [80]. The AIC is defined as α = 0.0035+0.002. In Fig. 2 we have shown the contour plot for−(0Ω.0m280,α) at 1σ, 2σ and 3σ confidence levels AIC =−2lnL+2k =χ2min+2k, (44) forthreedifferentcombinationsoftheobservationaldata where L = exp(cid:0)−χ2 /2(cid:1) is the maximum likelihood sets,namelySN+BAO,SN+OHDandSN+OHD+BAO. min function, k is the number of model parameters and N The positive optimal value of α includes singular behav- denotes the total number of data points used in the sta- ior of DE density ρ . In this case from eqn. (33) one d tistical analysis. However, to quantify any cosmologi- finds that Q < 0, that means, there is an energy flow cal model, a reference model is needed and ΛCDM is of from DM to DE. If we forbid solutions with α > 0, the course the best choice for that. Now, for any concerned ΛCDM case (α=0) will be more preferable. model M, other than the reference model (denoted by R), from the difference ∆AIC =AIC −AIC , we M,R M R arrive at the following conclusions as described in [81]: 6.2. Variable equation of state in DE (i) If ∆AIC ≤2, then the concerned model has sub- M,R stantial support with respect to the reference model (i.e. We consider the scenario of interacting dark energy it has evidence to be a good cosmological model), (ii) with variable EoS in the form (34) (Barboza, Alcaniz, 4 ≤ ∆AIC ≤ 7 indicates less support with respect M,R Zhu&Silva2009). Thismodelhas3freeparametersω , to the reference model, and finally, (iii) ∆AIC ≥ 10 d0 M,R ω , β in addition to Ω , and α (since for all models we means that the model has no support, in fact it has no β m0 marginalize over H ). This number of model parameters use in principle. For simplicity we denote the interact- 0 P = 5 is too large, if we take into account information ingdarkenergymodelwhereω (cid:54)=−1byM ;interacting d 1 criteria, described in the next section. So we have to cosmologicalconstantbyM andinteractingCPLasM . 2 3 10 TABLE II: The table summarizes the best fit values of the model parameters with their errors bars at 1σ confidence level using the observational data SN+OHD+ BAO. M is the interacting DE with constant EoS (ω (cid:54)=−1); M is the interacting 1 d 2 cosmological constant; M is the interacting DE with CPL parametrization (35). 3 Models χ2 (θ ) Model parameters χ2 /d.o.f AIC |∆AIC | min i min 1,i ΛCDM 562.58 Ω =0.299+0.035 0.915 564.58 0 m0 −0.033 M 562.39 ω =−1.095+0.352, Ω =0.320+0.093, α=−0.010+0.01 0.917 568.39 3.81 1 d −0.421 m0 −0.147 −0.046 M 562.55 Ω =0.294+0.060, α=0.0035+0.002 0.916 566.55 1.97 2 m0 −0.054 −0.028 M 562.21 ω =−1.018+0.465, Ω =0.378+0.074, α=−0.026+0.026, 0.919 570.21 5.63 3 d0 −0.46 m0 −0.066 −0.031 ω =−1.74+2.76 1 −4.22 0.45 0.45 562.4 0.4 0.4 2χtot 562.3 0 0 m m n Ω Ω mi 0.35 0.35 562.2 0.3 0.3 562.1 −1 0 1 2 −0.06 −0.04 −0.02 0 −1.4 −1.2 −1 −0.8 −0.6 β α ω d0 0 0 0 −0.02 −2 −2 α ω1 ω1 −0.04 −4 −4 −0.06 −1.4 −1.2 −1 −0.8 −0.6 −0.06 −0.04 −0.02 0 −1.4 −1.2 −1 −0.8 −0.6 ω α ω d0 d0 FIG. 3: In the top-left panel we show the dependence of minχ2 on the free parameter ‘β’ of the general EoS given in (34) tot usingtheobservationaldataSN+OHD+BAO.Itshowsthatβ =1,i.e. theinteractingmodelwithCPLparametrization(eqn. (35))istheviableinteractingmodelwithvariableEoSinDEincomparedtotheothers. Theotherpanelsrepresentthecontour plots at 1σ, 2σ confidence levels for various quatities of the interacting DE model with CPL parametrization (eqn. (35), i.e. β =1) using the same observational data SN+OHD+BAO. Therefore, from our analysis (see table II), we find that erence model. ∆AIC = 1.97 < 2 ∆AIC = 3.81 < 4, and M1,R M2,R ∆AIC = 5.63 > 4. Therefore, from AIC it is ev- M3,R ident that M is the most favored interacting model in 2 7. SUMMARY AND DISCUSSIONS compared to the reference model ΛCDM. However, the model M is also supported by AIC analysis, while the 1 interacting CPL has less support in compared to the ref- In this work, considering a spatially flat FLRW universe, we have described the dynamics of two barotropic

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