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 Educationand Research−Kolkata, Mohanpur−741246, West Bengal, India

Rafael C. Nunes§

Departamento de Fısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil

Subenoy Chakraborty¶

Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India

In the background of a homogeneous and isotropic spacetime with zero spatial curvature, weconsider interacting scenarios between two barotropic fluids, one is the pressureless dark matter(DM) and the other one is dark energy (DE), in which the equation of state (EoS) in DE is eitherconstant or time dependent. In particular, for constant EoS in DE, we show that the evolutionequations for both fluids can be analytically solved. For all these scenarios, the model parametershave been constrained using the current astronomical observations from Type Ia Supernovae, Hubbleparameter measurements, and baryon acoustic oscillations distance measurements. Our analysisshows that both for constant and variable EoS in DE, a very small but nonzero interaction in thedark sector is favored while the EoS in DE can predict a slight phantom nature, i.e. the EoS inDE can cross the phantom divide line ‘−1’. On the other hand, although the models with variableEoS describe the observations better, but the Akaike Information Criterion supports models withminimal number of parameters. However, it is found that all the models are very close to the ΛCDMcosmology.

PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x, 98.80.Es.Keywords: Cosmological parameters; Dark energy; Dark matter; Interaction.

1. INTRODUCTION

The explanation of the late-time accelerated expansionof the universe [1–9] has become one of the biggest andopen problems in modern cosmology today. The mostreasonable description for this accelerating phase intro-duces some hypothetical dark energy (DE) fluid compris-ing about 70% of the total energy density of the universe[10]. The cosmological constant, Λ, associated with thezero point energy of the quantum fields, is the main can-didate for DE in agreement with a large number of avail-able observational data. However, despite of great suc-cess of Λ-cosmology, it presents serious objections in theinterface of cosmology and particle physics, such as thecosmological constant problem [11, 12] and the coinci-dence problem [13]. Due to these two serous problems inΛ, several alternatives have been proposed and discussed

∗Electronic address: german.sharov@mail.ru†Electronic address: subhra.maths@presiuniv.ac.in‡Electronic address: span@iiserkol.ac.in§Electronic address: rcnunes@fisica.ufjf.br¶Electronic address: schakraborty@math.jdvu.ac.in

in the last several years [14].

However, concerning different cosmological theories,the scenario where dark matter (DM) interacts with darkenergy (DE) has gained much attention in the currentliterature. The interaction between DM and DE wereprimarily motivated to address the small value of thecosmological constant, however later on the models werefound to provide a reasonable explanation to the cosmiccoincidence problem [15, 16]. Recently it has been ar-gued that the current observational data can favor thelate-time interaction between DM and DE [17–22]. Thecoupling parameter of the interaction in the dark sec-tor has also been measured by several observational data[20–29]. In fact, the class of interacting DM − DE mod-els could be a promising candidate to resolve the currenttensions on σ8 and the local value of the Hubble con-stant H0 [21, 30]. Further, it has been contended thatthe interaction between these dark sectors can influenceon the perturbation analysis which results in significantchanges in the lowest multipoles of the CMB spectrum[31, 32]. For a comprehensive analysis on several inter-acting models explored in the last couple of years werefer [33–52]. Therefore, based on the analysis on in-teracting dark energy models with a special indication

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for a nonzero interaction between the dark sectors fromthe the currently available observational data, it seemspromising that the interaction in the dark sectors mightopen some new possibilities in near future.

The interacting dynamics proposes for continuous en-ergy and/or momentum exchange between DM and DEwith the evolution of the universe and modifies the con-servation equations for DM (≡ TDMµν ) and DE (≡ TDEµν )as

∇νTDMµν = Q, and ∇νTDEµν = −Q, (1)

where the function Q characterizes the interaction be-tween the dark sectors in which Q > 0 represents theenergy and/or momentum flow from DE to DM whileQ < 0 means energy and/or momentum flow takes placefrom DM to DE. Thus, essentially, we choose differentfunctional forms for Q (with correct physical dimension)and we solve the corresponding evolution equations. Theprocedure followed is similar to that when one choosesdifferent parametrizations for the EOS in DE. Few devel-opments can be found in the literature where the com-plete dynamics of the interacting model has been ex-plored [48, 53, 54]. Another approach to consider theinteraction between two fluids is for instance the sigmamodels [55, 56].

In the present work, we determine the complete dy-namics of an interacting scenario where DE interactswith pressureless DM through a nongravitational inter-action considered in [57] and we generalize its observa-tional consequences considering that the DE componentis barotropic and its EoS could be either constant or timedependent. We first consider the constant equation ofstate in dark energy and show that the background evolu-tion can be analytically solved which we constrain usingthe combined analysis from Type Ia Supernovae, Hub-ble data, and baryon acoustic oscillation data. Then weapply the same combined analysis of the above observa-tional data to the case when the EoS in DE is evolvingwith the cosmic time.

The work has been organized in the following way:Section 2 presents a brief introduction of the interact-ing dynamics in a spatially flat Friedmann-Lemaıtre-Robertson-Walker (FLRW) universe. In section 3, wepresent the analytic solutions for dark energy and colddark matter for constant equation of state ωd 6= −1 andwhen the dark energy is the cosmological constant whileadditionally, in subsection 3.1, we perform an asymptoticanalysis of the analytic solutions, and then we discussedabout the possible sign of the interaction. In section 4 weintroduce dynamical DE models interacting with CDM.Section 5 contains all the data analysis tools we used inthis study. In the next section 6 we present the obser-vational constraints on the interacting models and themain results of the paper. Finally, we close our work insection 7 with summary and discussions.

2. INTERACTING DYNAMICS IN FLAT FLRW

Observations from cosmic microwave background radi-ation and large scale structure predict in good approxi-mation that our Universe is homogeneous and isotropicin the largest scale, and further it is almost spatiallyflat [10]. This line element describing such a universeis known as the Friedmann-Lemaıtre-Robertson-Walkerline element, which takes the form

ds2 = −dt2 + a2(t)(dx2 + dy2 + dz2), (2)

where a(t) is the scale factor of the universe, t beingthe universal cosmic time. Further, this homogeneousand isotropic principle gives a restriction on the matterdistribution of our Universe. It tells that the matterdistribution should be of perfect fluid type which takesits energy momentum tensor as Tµν = (p+ρ)uµuν+pgµν ,where ρ, p, respectively denote the energy density andthe pressure of the cosmic fluid, and uν is the fourvelocity vector of the fluid components. For our model,we consider the general relativity described by theEinstein’s field equations Gµν = 8πGTµν , and two fluidsas follows:

• Pressureless dark matter (or dust) where ρm is itsenergy density and pm(= 0), is its pressure.

• A dark energy fluid satisfying a barotropic equationof state pd = ωdρd, where ρd, pd, are respectively theenergy density and the pressure of the component, ωd isits equation of state.

Thus, for a co-moving observer (uµ = δµ0 ), the Ein-stein’s field equations are explicitly written as (consider-ing 8πG = 1)

H2 =1

3(ρm + ρd), (3)

H = −1

2(pm + pd + ρm + ρd), (4)

where H = a/a, is the Hubble parameter. Further, theconservation equations for DM and DE follows from eqn.(1)

ρm + 3Hρm = Q, (5)

ρd + 3H(1 + ωd)ρd = −Q . (6)

Clearly, equations (5) and (6) offer a new dynamics ofthe universe with the use of the interaction function Q.There are many proposed interactions in the literature tostudy the dynamics of the universe, however, the exactfunctional form of Q is still unknown. From the coupledcontinuity equations (5) and (6), we observe that Q couldbe any arbitrary function of the parameters H, ρm, ρd.So, naturally, one can construct infinitely many interact-ing models to understand the dynamics of the universe

3

in this framework. In the present work, we start with thefollowing interaction [57]:

Q = αH(ρ′m + ρ′d). (7)

where α is the coupling parameter which is dimensionless,and the ‘ ′ ’ denotes the differentiation with respect tox = ln (a/a0) (a0 is the present value of the scale factorwhich has been set to be unity in this work, i.e. a0 = 1).In fact, if one uses the conservation equations (5), and(6), it will be readily clear that the interaction (7) hasthe following equivalent form

Q = −3αH(ρm + ρd + ωd ρd) (8)

which directly includes the equation of state parameterof the dark energy component, that means the pressureof the dark energy component. However, inserting (7)into the conservation equations (5) and (6), we decouplethe conservation equations as

(1− α)ρ′m + 3ρm − αρ′d = 0, (9)

(1 + α)ρ′d + +3ρd(1 + ωd) + αρ′m = 0, (10)

where both equations (9) and (10) seem to describe aneffective non-interacting two fluid system, and due to theinteraction, the effective EoS for pressureless DM (ωeffm ),

DE (ωeffd ) take the forms

ωeffm = − Q

3Hρm

= − α

ρm(ρ′m + ρ′d)

=3α

r(1 + r + ωd), (11)

and

ωeffd = ωd +Q

3Hρd,

= ωd +α

ρd(ρ′m + ρ′d),

= ωd − 3α (1 + r + ωd), (12)

where r = ρm/ρd, is the coincidence parameter. Now,from (11), (12), we note that, even if the EoS in DMand DE are constant, but due to the interaction betweenthem, the effective EoS of the two components becomevariable in nature. Now, if we denote the total EoS forthe non-interacting DM-DE system by ωeff = ωm+ωd =ωd (as DM is pressureless, i.e. ωm = 0), then the effective

EoS: ωefft = weffm +ωeffd , of the same system due to thisinteraction becomes

ωefft = ωeff + α

(1

ρd− 1

ρm

)(ρ′m + ρ′d)

= ωd + α

(1

ρd− 1

ρm

)(ρ′m + ρ′d),

= ωd +3α

r(1− r)(1 + r + ωd) (13)

Equations (11), (12) lead to the following interestingpossibilities we will discuss now.

• If Q > 0, Eq. (11) implies that, ωeffm < 0 meaningthat the effective equation of state of the pressurelessdark matter1 is of exotic type, and it could become adark energy candidate (EoS < −1/3) if Q > Hρm. Onthe other hand, the effective equation of state for dark

energy satisfies the following inequality: ωd < ωeffd <∞,which shows that if the EoS in dark energy satisfies therelation ωs ≤ −1, then the effective interactive system is

of course in the quintessence era (ωeffd > −1). However,if the dark energy is of phantom kind, then the aboveinequality could still hold, but in this case, the effectiveEoS in dark energy may lie in the phantom/quintessenceregion depending on the strength of the interaction.

• If Q < 0, then ωeffm > 0 irrespective of anything.

In case of effective dark energy, we find that ωeffd < ωdwhich readily points that, for ωd ≤ −1, the effectiveequation of state is always phantom in nature. On the

other hand, for ωd > −1, ωeffd could cross the phantombarrier line ‘−1’ due to the presence of Q/3Hρd < 0 in(12). But, the possibility of quintessence behavior can’tbe excluded as again we do not specify the strength ofthe interaction.

However, introducing the total energy density of theuniverse, ρt = ρm + ρd, we can express the dark energyand the dark matter density in terms of the total energydensity and its derivative as

ρd = −(ρ′t + 3ρt

3ωd

), (14)

ρm =

(ρ′t + 3(1 + ωd)ρt

3ωd

), (15)

Now, inserting (14) into (10), we obtain the followingsecond order differential equation

ρ′′t + 3

[2 + ωd − αωd −

ω′d3ωd

]ρ′t + 9

[(1 + ωd)−

ω′d3ωd

]ρt = 0,

(16)

which if we could solve might describe the possible dy-namics of the universe. In what follows, we consider theevolution of the universe both for constant and dynamicEoS in DE.

3. CONSTANT EOS IN DE

Let us now consider the case when the EoS in DE isindependent of time, we denote the EoS by ωd. This

1 We define the effective EoS of the pressureless dark matter asthe EoS of the pressureless dark matter plus the energy gaineddue to this interacting process.

4

consideration reduces the differential equation (16) intothe following simplified format

ρ′′t + 3[2 + ωd − αωd

]ρ′t + 9(1 + ωd)ρt = 0, (17)

which solves for ρt as

ρt = ρ1em1x + ρ2e

m2x = ρ1am1 + ρ2a

m2 , (18)

where ρ1, ρ2 are the constants of integration, and,(m1,m2

)are given by

m1 =3

2

[−(2 + ωd − αωd) +

√(1− α)2ω2

d − 4αωd

],

m2 =3

2

[−(2 + ωd − αωd)−

√(1− α)2ω2

d − 4αωd

],

As ωd < −1/3, so,(m1,m2

)are real. The equation (18)

can be written as(H

H0

)2

= Ω1(1 + z)−m1 + Ω2(1 + z)−m2 . (19)

where Ω1 = ρ1/ρc0, Ω2 = ρ2/ρc0 in which ρc0 =3H2

0/8πG. It is interesting to note that, the solution (19)which represents the analytic behavior for two interact-ing fluids, has also been found in the context of two non-interacting fluids where one is Brans-Dicke scalar fieldand the other is the perfect fluid with constant equationof state [58].

Moreover, using (14), (15), the explicit analytic expres-sions for DE and DM density are

ρd = −ρ1(m1 + 3)(1 + z)−m1 + ρ2(m2 + 3)(1 + z)−m2

3ωd(20)

and

ρm =

(1

3ωd

)[ρ1(m1 + 3 + 3ωd)(1 + z)−m1

+ρ2(m2 + 3 + 3ωd)(1 + z)−m2

]. (21)

However, the constants ρ1, ρ2 do not serve only as theconstants of integrations, they have crucial meanings. Inorder to explain this, let us introduce the density param-eters for dark matter and dark energy as Ωm = ρm/ρc(ρc = 3H2/8πG), Ωd = (ρd/ρc) respectively, Thus, theFriedmann equation (3) can be written as Ωm + Ωd = 1.Now, the energy densities at present time can be givenby

ρd0 = −(

1

3ωd0

)[ρ1(m1 + 3) + ρ2(m2 + 3)

], (22)

ρm0 =

(1

3ωd0

)[ρ1(m1 + 3 + 3ωd0) + ρ2(m2 + 3 + 3ωd0)

],

(23)

where ωd0 is the present value of the EoS in DE2. Now,dividing (22) and (23) by ρc0 = 3H2

0/8πG, we get thepresent day density parameters for DE and DM as fol-lows:

Ωd0 = −(

1

3ωd0

)[Ω1(m1 + 3) + Ω2(m2 + 3)

], (24)

Ωm0 =

(1

3ωd0

)[Ω1(m1 + 3 + 3ωd0) + Ω2(m2 + 3 + 3ωd0)

],

(25)

Now, solving for Ω1 and Ω2, we find

Ω1 =(m2 + 3 + 3ωd0)Ωd0 + (m2 + 3)Ωm0

m2 −m1, (26)

Ω2 =(m1 + 3 + 3ωd0)Ωd0 + (m1 + 3)Ωm0

m1 −m2. (27)

•When DE is the cosmological constant

Now, we consider the case when DE is the cosmologicalconstant. Thus, in this case, we have ωd = −1, hence,(m1,m2

)take the forms

(0, −3(1+α)

). The total energy

density takes the form

ρt = ρ1 + ρ2(1 + z)3(1+α), (28)

and the explicit analytic solutions of the energy densityfor the cosmological constant and the energy density ofthe pressureless matter in this interacting picture takethe expressions

ρd = ρ1 − ρ2α(1 + z)3(1+α), (29)

ρm = ρ2(1 + α)(1 + z)3(1+α). (30)

Hence, the Friedmann equation becomes(H

H0

)2

= Ω1 + Ω2(1 + z)3(1+α), (31)

where Ω1 = ρ1/ρc0, Ω2 = ρ2/ρc0, and the solution hasalso been found in the context of two non-interactingperfect fluids where one is the Brans-Dicke scalar fieldand the other is the perfect fluid (in the form of dust)(Paliathanasis, Tsamparlis, Basilakos, & Barrow 2016).Further, we have the following relations:

Ωd0 = Ω1 − αΩ2,

Ωm0 = (1 + α) Ω2.

2 Let us note that since ωd has been considered to be constant,then ωd = ωd0. However, we have used the suffix 0 in ωd to keepthe same structure with the other parameters, for instance, Ωd0,Ωm0 and H0. Therefore, in the scenario of interacting DE withconstant EoS, ωd0 and ωd should be considered to be idential.

5

Further, we can solve for Ω1 and Ω2 as

Ω2 =1

1 + αΩm0,

Ω1 = Ωd0 +α

1 + αΩm0.

It should be noted that, Ωm0 + Ωd0 = Ω1 + Ω2 = 1.

3.1. Asymptotic behavior of the energy densitiesand the sign of the interaction

As the solutions for ρd, ρm include several parameters,so, we shall be very careful, as well as, explicit, whiledescribing the asymptotic behavior of the solutions. Firstof all, we recall that, the roots m1, m2 should be real inorder to realize a feasible state. Hence, for m1, m2 tobe real, we must have (1 − α)2ω2

d − 4αωd > 0. Now,under the low interaction condition, α2 can be neglected,thus, we find that, (1 − 2α)ω2

d − 4αωd > 0. As ωd < 0,we let ωd = − d2 (d ∈ R) which gives a restriction asd2(1 − 2α) + 4α > 0, which is always true (recall thatthe interaction is very low, i.e. the coupling parameteris very very small). Thus, if ωd < 0, then m1, m2 arealways real under the low interaction assumption. Now,if m1 > 0, then√

(1− α)2ω2d − 4αωd > (2 + ωd − αωd)

=⇒ 1 + ωd < 0, i.e. ωd < −1.

Similarly, m1 < 0 =⇒ ωd > −1. On the other hand,if 2 + ωd − αωd < 0, we have ωd < − 2

1−α < −1 (inlow interaction scheme and if we do not restrict on thecoupling parameter, then of course α < 1).

In both cases, m2 < 0, for ωd < −1, or ωd > −1.Hence, we arrive at two different situations: (I) If ωd <−1, then m1 > 0, m2 < 0, (II) If ωd > −1, then m1 < 0,m2 < 0. Now, we calculate the following expressions inorder to analyze the asymptotic behavior of the energydensities.

m1 + 3

3ωd=

1

2

[−(1− α) +

√(1− α)2 +

4α

d2

]> 0

m2 + 3

3ωd=

1

2

[−(1− α) +

√(1− α)2 +

4α

d2

]< 0

Also, we have the following

m1 + 3 + 3ωd3ωd

=1

2

[(1 + α) +

√(1− α)2 +

4α

d2

]> 0

and

m2 + 3 + 3ωd3ωd

=1

2

[(1 + α) +

√(1 + α)2 + 4α

(1

d2− 1

)]> 0 (or < 0) if ωd < −1 (or ωd > −1)

Now, we discuss the three different stages of the universeas follows:

• For z −→ 0, ρm, ρd are finite, and the total energydensity ρt, as well.

• When z is very large, i.e. z −→∞,

ρd −→ −ρ2 (m2 + 3)

3ωd(1 + z)−m2 ;

ρm −→ρ2 (m2 + 3 + 3ωd)

3ωd(1 + z)−m2 ;

ρt −→ ρ2(1 + z)−m2 .

• When z ≈ −1, we consider z = −1 + ε, where ε > 0is any arbitrary infinitesimal quantity, we have

ρd −→ −ρ1(m1 + 3)

3ωdε−3m1 ;

ρm −→ρ1(m1 + 3 + 3ωd)

3ωdε−3m1 ;

ρt −→ ρ1ε−m1 .

For convenience, depending on the signs of the quan-tities, we introduce the following:

m1 + 3

3ωd= β2;

m2 + 3

3ωd= − γ2;

m1 + 3 + 3ωd3ωd

= θ2

m2 + 3 + 3ωd3ωd

= δ2 (if ωd < −1), or = − δ2 (if ωd > −1)

Thus, based on the nature of the dark energy (thatmeans it is of quintessence or of phantom type), it ispossible to express the energy densities of the darksectors in the following manner.

• If ωd < −1, then we have

ρd = −ρ1 β2(1 + z)−m1 + ρ2 γ

2(1 + z)−m2 ,

ρm = ρ1θ2(1 + z)−m1 + ρ2δ

2(1 + z)−m2 .

• If ωd > −1, we have

ρd = −ρ1 β2(1 + z)−m1 + ρ2 γ

2(1 + z)−m2 ,

ρm = ρ1 θ2(1 + z)−m1 − ρ2 δ

2(1 + z)−m2 .

6

Further, due to the analytic solutions, we can simplifythe interaction (7) into

Q

H= 3αH2

0 (m1Ω1am1 +m2Ω2a

m2). (32)

Now, from Eq. (32), we can infer about the sign in Qdepending on the parameters involved.

• If m1 > m2 > 0, then Q > 0 if α > 0; and Q < 0 forα < 0

• Also, if 0 < m1 < m2 <, then Q > 0 if α > 0; andQ < 0 for α < 0

• If m1 < m2 < 0, then Q > 0 when α < 0; and Q < 0for α > 0

• Now when m1m2 < 0, the sign of Q seems dif-ficult to determine. In fact, in this case, we have aninteresting observation which tells that, Q could changeits sign during the evolution of the universe. To illus-trate the idea, let us take the following simplest example:

• An Example: Just for simplicity let ustake m1 = −1, m2 = 1. Therefore, we haveQ = 3HαH2

0

(−Ω1

a + Ω2a). Now, the terms in the

bracket could be once positive and once negative, whichagain depends on the sign of α outside the bracket.

Hence, generalizing this event (i.e. when m1m2 < 0),the equation Q = 0 can have more than one real rootsleading to an oscillatory interacting scenarios in whichonce Q > 0, and once Q < 0.

• Asymptotic behavior when DE is the cosmo-logical constant

Depending on the coupling parameter α, we find thefollowing asymptotic behavior of the energy densitiesboth at present and late-times.

I. When α > 0

In that case, for z −→ 0, ρd −→ ρ1 − αρ2 andρm −→ ρ2(1 + α). On the other hand, for z −→ ∞,ρd −→ −∞, ρm −→ ∞. Further, in the case forz −→ −1, ρd −→ ρ1, ρm −→ 0.

II. When −1 < α < 0

For z −→ 0, ρd −→ ρ1 − αρ2 ≥ 0, andρm −→ ρ2(1 + α) > 0. When z −→ ∞, ρd −→ ∞, andρm −→∞. Further, for z −→ −1, ρd −→ ρ1, ρm −→ 0.

III. When α < −1

For z −→ 0, ρd −→ ρ1 − αρ2 ≥ 0, andρm −→ ρ2(1 + α) < 0. For z −→ ∞, ρd −→ ρ1,and ρm −→ 0. When z −→ −1, we see ρd −→ ∞,ρm −→ −∞.

Now, we can rewrite the interaction in this case as

Q

H= −9α(1 + α)H2

0 Ω2a−3(1+α), (33)

which summarizes the following possibilities:

• If α > 0, then Q < 0, i.e. the energy flows from DEto DM.

• When −1 < α < 0, then Q > 0, indicating the flowof energy from DM to DE.

• Finally, for α < −1, Q < 0 leading to the energyflow from DE to DM.

4. INTERACTING DYNAMICS FOR VARIABLEEQUATION OF STATE IN DE

Now, we consider an interacting scenario where darkenergy fluid has a time varying equation of state. Thisscenario can be considered as a general scenario of in-teracting dynamics. The possibility of time varying darkenergy interacting with pressureless dark matter has beeninvestigated during last couple of years [42, 59, 60]. How-ever, the present interaction is different from the othersin [42, 59, 60] and it becomes clear when the variableequation of state in DE is considered. So, in the cur-rent study along with the constant dark energy equationof state in section 3, we consider the case for dynamicalDE. Let us introduce a generalized equation of state inDE given by [61]

ωd(z) = ωd0 − ωβ(

(1 + z)−β − 1

β

), (34)

where ωd0 is the current value of ωd(z), and ωβ is thefree parameter. We note that the EoS in (34) recov-ers three well known dynamical dark energy equationsof state. For β = 1, we recover the Chevallier-Polarski-Linder (CPL) parametrization [62, 63]

ωd(z) = ωd0 + ω1

(z

1 + z

). (35)

Now, for β = −1, one finds the linear parametrization[64–66],

ωd(z) = ωd0 + ω2 z, (36)

while for β → 0, one realizes the logarithmic parametriza-tion in DE [67]

ωd(z) = ωd0 + ω3 ln(1 + z), (37)

7

where ω3 is the free parameter. Cosequently, for othervalues of β 6= −1, 0, 1, one can generate several equationsof state. Thus, one can study the second order differen-tial equation (16) for the variable EoS in (34) in orderto estimate the cosmological parameters with the use ofcurrent observational data. On the other hand, pluggingthe total energy density ρt = ρm + ρd = 3H2 into the in-teraction (8) for ωd(z) given in eqn. (34), the interactionnow becomes

Q = H3 ψ(z), (38)

where ψ(z) is given by

ψ(z) = − 9α

[1 +

ωd0 − ωβ

((1 + z)−β − 1

β

)Ωd

](39)

and this quantity only determines the sign of Q duringthe evolution of the Universe (since H > 0, for expandinguniverse). Further, current value of the interaction ratecan be found to be Q0 = −9αH3

0 (1 + ωd0 Ωd0) and itssign can be determined once the cosmological parametersare estimated by the current observational data.

Below we shall describe the current observational datato constrain the mentioned interacting models and fromthis point of view we aim to find the most successfuldynamic EoS given in (34) for the current interactingscenario. For this purpose we have to solve numericallythe second order differential equation (16) for ωd(z) in(34).

5. OBSERVATIONAL CONSTRAINTS ON THEMODEL PARAMETERS

In order to constrain the free parameters of themodels presented in the previous sections, we considerthe lastest observational data from Supernovae type Ia(SNIa), Hubble expansion history (OHD), and the bary-onic acoustic oscillation (BAO) distance measurements.The best fit values, with their corresponding uncertain-ties, follow from minimizing the likelihood function L ∝exp(−χ2

tot/2), where χ2tot = χ2

SN +χ2OHD+χ2

BAO. In thefollowing subsections we briefly describe each data set.

5.1. Supernovae Type Ia data

Data from Supernovae was the first indication for ac-celerating universe, and it is very useful to test the cos-mological models. In the present analysis we use 580Supernovae data points from Union 2.1 available in [68]in the redshift interval 0 ≤ z ≤ 1.41. Now, for any Su-pernova, situated at a redshift z, the distance modulus µis given by

µ(z) = 5 log10

(dL(z)

1Mpc

)+ 25, (40)

α

Ωm

0

SNe + BAO

SNe + H(z)

SNe + BAO + H(z)

−0.06 −0.04 −0.02 0 0.02 0.040.2

0.25

0.3

0.35

0.4

ωd

Ωm

0

SNe + BAO

SNe + H(z)

SNe + BAO + H(z)

−1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8

0.2

0.25

0.3

0.35

0.4

SNe + BAO

SNe + H(z)

SNe + BAO + H(z)

ωd

α

−1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8

−0.06

−0.04

−0.02

0

0.02

0.04

FIG. 1: The figure shows the 1σ, 2σ, 3σ confidence-level con-tour plots of the model parameters with their best fit valuesfor the interacting DE model with constant EoS, ωd 6= −1(Eq. (18)), using different combinations of the observationaldata sets as SN+ OHD+BAO (blue filled contours), SN+BAO (red contours), SNe+ OHD (green contours). We notethat the contour lines for SN+ H(z) do not appear properlyin all the plots.

8

α

Ωm

0

SNe + BAO

SNe + H(z)

SNe + BAO + H(z)

−0.06 −0.04 −0.02 0 0.02 0.040.2

0.25

0.3

0.35

0.4

FIG. 2: The figure shows the 1σ, 2σ, 3σ confidence-level con-tour plots of the model parameters with their best fit valuesfor the interacting cosmological constant (Eq. (28)), usingdifferent combinations of the observational data sets as SN+OHD+BAO (blue filled contours), SN+ BAO (red contours),SNe+ OHD (green contours).

where dL(z) = c(1+z)∫ z

0dz′

H(z′) is the luminosity distance.

The χ2 function is evaluated as

χ2SN (θ1, . . . ) = min

H0

580∑i,j=1

∆µi(C−1SN

)ij

∆µj . (41)

where ∆µi = µth(zi, θ1, . . . ) − µobs(zi), θk, CSN are re-spectively the discrepancy between theory and observa-tions, model parameters to be fitted, and the 580×580 co-variance matrix [68]. We use the function (41), marginal-ized over the Hubble constant H0.

5.2. Hubble data

Hubble parameter measurements are also useful to con-strain the parameters in dark energy models. Two inde-pendent methods have been widely used to measure theHubble parameter values at different redshifts. One is thedifferential age of some galaxy, and the other one is theradial BAO size methods. Here we use 30 OHD points forour analysis from Refs. [69–74], obtained with the differ-ential age method. To constrain the model parameters,we apply the χ2 function by the following method

χ2OHD =

30∑i=1

[Hobs(zi)−Hth(zi, θj)

σ2i

], (42)

where θj stands for different model parameters to be con-strained.

5.3. Baryon Acoustic Oscillation data

To analyze cosmological models, baryon acoustic os-cillations (BAO) data give another useful test. Here, we

use BAO data, i.e. dA(z∗)DV (ZBAO) from the references [75–78],

where z∗ is the radiation-matter decoupling time given byz∗ ≈ 1091, dA is the co-moving angular diameter distance

given by dA =∫ z

0dzH(z) , and, DV =

(dA(z)2 z

H(z)

)1/3

is

the dilation scale [8]. In table I, we present the observa-tional BAO data. We use the following procedure as in[79] to calculate the χ2 minimization for BAO data, i.e.

χ2BAO = XT

BAOC−1BAOXBAO, (43)

where

XBAO =

dA(z∗)DV (0.106) − 30.95dA(z∗)DV (0.2) − 17.55dA(z∗)DV (0.35) − 10.11dA(z∗)DV (0.44) − 8.44dA(z∗)DV (0.6) − 6.69dA(z∗)DV (0.73) − 5.45

and the inverse covariance matrix C−1 can be found in(Giostri et al. 2012).

6. RESULTS OF THE JOINT ANALYSIS

In this section we shall present the main observationalresults extracted from the scenarios of interacting darkenergy when dark energy equation of state is either con-stant or evolving with the cosmic time. In case of con-stant EoS in DE we have considered two separate cos-mological models one is the interacting DE model withωd 6= −1, and interacting cosmological constant wherefor both the models the background evolution equationsare analytically solved. On the other hand, for the vari-able EoS in DE we have considered a general EoS givenin (34) and solved the second order differential equation(16) numerically. In the following we shall describe theobservational constraints on the model parameters forthe interacting dark energy models.

6.1. Constant equation of state in DE

We fit both the interacting models with constant EoSusing the latest observational data from SupernovaeType Ia, observed Hubble parameter data, and thebaryon acoustic oscillations distance measurements. Wenote that for all interacting models, we have marginal-ized χ2

tot for all values of H0. Let us describe theobservational constraints and the main cosmological

9

TABLE I: Values of dA(z∗)DV (zBAO)

for different values of zBAO.

zBAO 0.106 0.2 0.35 0.44 0.6 0.73dA(z∗)

DV (zBAO)30.95 ± 1.46 17.55 ± 0.60 10.11 ± 0.37 8.44 ± 0.67 6.69 ± 0.33 5.45 ± 0.31

features of the models separately as follows.

• For interacting dark energy, where ωd 6= −1,χ2min(Ωm0, α, ωd0) = 562.39, ωd0 = −1.095+0.352

−0.421, Ωm0 =

0.320+0.093−0.147, α = −0.010+0.01

−0.046. Here acceptable values ofα, i.e its priors are limited to α < 0, because for posi-tive α we have solutions with negative dark energy den-sity ρd given in eqn. (20) at early stages of evolution.They may be considered as singular solutions, thoughthe total density ρ = ρm + ρd remains positive and thescale factor a(t) behaves regularly. Also, the reducedχ2 = χ2

min/d.o.f = 0.917 (d.o.f = degrees of freedom =N − P , where N is total number of data points; P is to-tal number of independent model parameters). In figureFig. 1 we have shown the contour plots at 1σ, 2σ and 3σconfidence levels with different combinations of the ob-servational data sets, namely, SN+BAO, SN+OHD andSN+OHD+BAO.

Therefore, from the analysis of this model, it is seenthat this interaction can describe the phantom nature ofthe universe. Further, concerning the interaction Q ineqn. (32), using the best fit values of the cosmologicalparameters, one can see that m1 > 0, but m2 < 0, whichindicates that during the evolution of the universe, Qshould change its sign, but however, the observationaldata tells that for this scenario, Q > 0, for any z ≥ 0,that means the energy flows from DE to DM.

• For interacting Λ with ωd = −1, we foundχ2min(Ωm0, α) = 562.55, hence, reduced χ2 =χ2min/d.o.f = 0.919. It is achieved for Ωm0 = 0.294+0.060

−0.054,

α = 0.0035+0.002−0.028. In Fig. 2 we have shown the con-

tour plot for (Ωm0, α) at 1σ, 2σ and 3σ confidence levelsfor three different combinations of the observational datasets, namely SN+BAO, SN+OHD and SN+OHD+BAO.The positive optimal value of α includes singular behav-ior of DE density ρd. In this case from eqn. (33) onefinds that Q < 0, that means, there is an energy flowfrom DM to DE. If we forbid solutions with α > 0, theΛCDM case (α = 0) will be more preferable.

6.2. Variable equation of state in DE

We consider the scenario of interacting dark energywith variable EoS in the form (34) (Barboza, Alcaniz,Zhu & Silva 2009). This model has 3 free parameters ωd0,ωβ , β in addition to Ωm0, and α (since for all models wemarginalize over H0). This number of model parametersP = 5 is too large, if we take into account informationcriteria, described in the next section. So we have to

search the most suitable value of β in EoS (34), in otherwords, the best choice among the most popular scenarios(35), (36), (37). These scenarios are particular cases ofthe general EoS (34), if β = 1, −1, 0 correspondingly; inall these cases we diminish P to P = 4. To investigatethe generalized EoS (34), we calculate how the minimum

minΩm0,α,ωd0,ωβ

χ2tot (over all other parameters) depends on

β. This dependence is shown in the top-left panel of fig-ure 3. We see that the best values of minχ2

tot correspondto large positive β (moreover, negative values of β leadto additional singularities), so we should choose the CPLparametrization (35) (Chevallier & Polarski 2001; Linder2003), corresponding to β = 1, as the most successfulscenario of interacting DE with variable EoS. For thisscenario we draw the contour plots for various quantitiesfor the observational data SN+OHD+BAO in other pan-els of Fig. 3. The correspondent estimations of minχ2

tot

and model parameters are presented in table II. As weobserve, the interacting CPL model shows that the cur-rent dark energy EoS has a slight phantom nature (asωd0 = −1.018 < −1), which is although very close to theΛ-cosmology. On the other hand, using the observationaldata into eqns. (38) and (39), it is easy to see that atpresent time, i.e. at z = 0, Q > 0.

6.3. Model selection

In order to test the quality of fit of the present modelsby information criteria, we apply the Akaike InformationCriterion (AIC) [80]. The AIC is defined as

AIC = −2 lnL+ 2k = χ2min + 2k, (44)

where L = exp(−χ2

min/2)

is the maximum likelihoodfunction, k is the number of model parameters and Ndenotes the total number of data points used in the sta-tistical analysis. However, to quantify any cosmologi-cal model, a reference model is needed and ΛCDM is ofcourse the best choice for that. Now, for any concernedmodel M , other than the reference model (denoted byR), from the difference ∆AICM,R = AICM −AICR, wearrive at the following conclusions as described in [81]:(i) If ∆AICM,R ≤ 2, then the concerned model has sub-stantial support with respect to the reference model (i.e.it has evidence to be a good cosmological model), (ii)4 ≤ ∆AICM,R ≤ 7 indicates less support with respectto the reference model, and finally, (iii) ∆AICM,R ≥ 10means that the model has no support, in fact it has nouse in principle. For simplicity we denote the interact-ing dark energy model where ωd 6= −1 by M1; interactingcosmological constant by M2 and interacting CPL as M3.

10

TABLE II: The table summarizes the best fit values of the model parameters with their errors bars at 1σ confidence levelusing the observational data SN+OHD+ BAO. M1 is the interacting DE with constant EoS (ωd 6= −1); M2 is the interactingcosmological constant; M3 is the interacting DE with CPL parametrization (35).

Models χ2min(θi) Model parameters χ2

min/d.o.f AIC |∆AIC1,i|ΛCDM 562.58 Ωm0 = 0.299+0.035

−0.033 0.915 564.58 0

M1 562.39 ωd = −1.095+0.352−0.421, Ωm0 = 0.320+0.093

−0.147, α = −0.010+0.01−0.046 0.917 568.39 3.81

M2 562.55 Ωm0 = 0.294+0.060−0.054, α = 0.0035+0.002

−0.028 0.916 566.55 1.97

M3 562.21 ωd0 = −1.018+0.465−0.46 , Ωm0 = 0.378+0.074

−0.066, α = −0.026+0.026−0.031, 0.919 570.21 5.63

ω1 = −1.74+2.76−4.22

−1 0 1 2

562.1

562.2

562.3

562.4

β

min

χ2 to

t

α

Ωm

0

−0.06 −0.04 −0.02 0

0.3

0.35

0.4

0.45

ωd0

α

−1.4 −1.2 −1 −0.8 −0.6

−0.06

−0.04

−0.02

0

ωd0

Ωm

0

−1.4 −1.2 −1 −0.8 −0.6

0.3

0.35

0.4

0.45

α

ω1

−0.06 −0.04 −0.02 0

−4

−2

0

ωd0

ω1

−1.4 −1.2 −1 −0.8 −0.6

−4

−2

0

FIG. 3: In the top-left panel we show the dependence of minχ2tot on the free parameter ‘β’ of the general EoS given in (34)

using the observational data SN+OHD+BAO. It shows that β = 1, i.e. the interacting model with CPL parametrization (eqn.(35)) is the viable interacting model with variable EoS in DE in compared to the others. The other panels represent the contourplots 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∆AICM1,R = 1.97 < 2 ∆AICM2,R = 3.81 < 4, and∆AICM3,R = 5.63 > 4. Therefore, from AIC it is ev-ident that M2 is the most favored interacting model incompared to the reference model ΛCDM. However, themodel M1 is also supported by AIC analysis, while theinteracting CPL has less support in compared to the ref-

erence model.

7. SUMMARY AND DISCUSSIONS

In this work, considering a spatially flat FLRW uni-verse, we have described the dynamics of two barotropic

11

cosmic fluids, namely, pressureless DM and DE whichare interacting with each other. The interaction is non-gravitational and this essentially indicates an exchange ofenergy and/or momentum between the interacting com-ponents, and hence determines the dominant characteramong the components in the cosmic sector during theevolution of the universe. We specify the nongravita-tional interaction given in equations (7) or (8) character-ized by a single coupling parameter ‘α’, and the EoS inDE has been considered to be either constant or variablewith the cosmic evolution.

We show that if the dark energy equation of state issupposed to be constant with the evolution of the uni-verse, then the evolution equations for CDM and DE, andhence the Hubble parameter can be completely solved asa function of the scale factor/redshift, where the stan-dard evolution equations ρm ∝ a−3, ρd ∝ a−3(1+ωd), arerecovered under the no-interaction limit, that means forα → 0. We present two separate sets of solutions whenωd 6= −1, and ωd = −1 (i.e. cosmological constant). Ad-ditionally, we present an asymptotic analysis of the solu-tions in order to evaluate their behavior in the extremelimit, and analyzed the nature of the interaction Q whichpredicts a possible sign change during its evolution.

All the models (for constant and variable EoS in DE)have been constrained with the latest observational datafrom (i) Supernovae Type Ia, (ii) observed Hubble pa-rameters, and (iii) Baryon acoustic oscillations distancemeasurements. The results have been summarized in ta-ble II which shows that for the interacting model withωd 6= −1, the observational data slightly favor ωd < −1,that means the dark energy EoS crosses the phantom di-vide line ‘−1’. The crossing of phantom divide line is notnew in the literature, see for instance [10, 82–86]. Conse-quently, the observational constraints further show thatQ might have transient nature, that means during theuniverse evolution it may change its sign, but for z ≥ 0,Q > 0. On the other hand, for interacting cosmologicalconstant (i.e. ωd = −1) the observational data prefera nonzero coupling parameter which is positive and veryclose to zero and consequently from (33), this model givesQ < 0.

Now, in case of interacting dynamical DE, we haveconsidered a general equation of state classified by a soleparameter ‘β’ (see eqn. (34)) which recovers three knownand most used DE parametrizations, namely for β = 1one has CPL parametrization [62, 63]; β = −1 gives lin-ear parametrization [64–66]; β → 0 provides the loga-rithmic parametrization [67] and consequently, one can

find several new DE parametrizations with other val-ues of β 6= −1, 0, 1. Now, from the χ2 analysis, us-ing the observational data SN+OHD+ BAO, we foundthat the present interaction prefers CPL parametriza-tion from the other models (see the top left panel in Fig.3) and hence we fixed our aim to interacting DE withCPL parametrization and analyzed the cosmological sce-nario. The constraints on the free model parameters havebeen summarized in table II. The contour plots of variousquantities in interacting CPL model have been shown inother panels of figure 3 for the joint analysis SN+OHD+BAO. The analysis shows that the observational data al-lows a nonzero coupling parameter which is negative butclose to zero, and it also favors a slight phantom natureof DE similar to the case of interacting DE with constantωd 6= −1. Additionally, at present epoch, i.e. at z = 0,the interaction is positive (Q > 0).

Moreover, based on the Akaike Information Criterion(AIC) [80], we see that the interacting models with con-stant EoS are favored than the interacting dynamicaldark energy with CPL parametrization. In fact, amongthe three interacting models, the AIC information cri-terion selects interacting model with ωd = −1 to bethe best fitted cosmological model with the observationaldata.

Summarizing, for the current interaction model be-tween DE and DM the observational data depict anonzero but very small interaction between the dark sec-tors and on the other hand, although all the interactionmodels are qualitatively very close to the ΛCDM cosmol-ogy, but the models with constant EoS in DE are favoredby the AIC criterion than the model with varibale EoSin DE.

Acknowledgments

The authors thank the referee for some effective andilluminating comments to improve the work. SP acknowl-edges Science and Engineering Research Board (SERB),Govt. of India, for awarding National Post-Doctoral Fel-lowship (File No: PDF/2015/000640) and the Depart-ment of Mathematics, Jadavpur University where a partof the work was completed. SC thanks IUCAA, Pune,India, for their warm hospitality while working on thisproject. Also, SP thanks Dr. A. Paliathanasis and Dr.N. Tamanini for helpful discussions.

[1] Perlmutter S. et al., 1999, ApJ, 517, 565, arXiv:astro-ph/9812133

[2] Riess A. G. et al., 1998, A&A, 116, 1009, arXiv:astro-ph/9805201

[3] de Bernardis P. et al., 2000, Nature, 404, 955,arXiv:astro-ph/0004404

[4] Percival W. J. et al., 2001, MNRAS, 327, 1291,arXiv:astro-ph/0105252

[5] Spergel D. N. et al., 2003, ApJS, 148, 175, arXiv:astro-ph/0302209

[6] Jain B., Taylor A., 2003, Phys. Rev. Lett, 91, 141302,arXiv:astro-ph/0306046

12

[7] Tegmark M. et al., 2004, Phys. Rev. D, 69, 103501,arXiv:astro-ph/0310723

[8] Eisenstein D. J. et al., 2005, ApJ, 633, 560, arXiv:astro-ph/0501171

[9] Komatsu E. et al., 2011, ApJS, 192, 18, arXiv:1001.4538[astro-ph.CO]

[10] Planck collaboration XIII, 2016, A&A 594, A13,arXiv:1502.01589 [astro-ph.CO]

[11] Weinberg S., 1989, Rev. Mod. Phys, 61, 1[12] Padmanabhan T., 2003, Phys. Rept, 380 235, arXiv:hep-

th/0212290[13] Steinhardt P. J. et al., 2003, Phil. Trans. Roy. Soc. Lond.

A, 361, 2497[14] Amendola L., Tsujikawa S., 2010, Dark Energy: The-

ory and Observations, Cambridge University Press, Cam-bridge, UK

[15] Bolotin Y. L., Kostenko A., Lemets O. A., & YerokhinD. A., 2015, Int. J. Mod. Phys. D, 24, 1530007,arXiv:1310.0085 [astro-ph.CO]

[16] Wang B., Abdalla E., Barandela F. A., Pavon D., 2016,Rept. Prog. Phys, 79, 096901, arXiv:1603.08299 [astro-ph.CO]

[17] Salvatelli V., Said N., Bruni M., Melchiorri A., WandsD., Phys. Rev. Lett, 2014, 113 181301, arXiv:1406.7297[astro-ph.CO]

[18] Sola J., Valent A. G., Perez J. C., 2015, ApJ, 811, L14,arXiv:1506.05793 [gr-qc]

[19] Sola J., Perez J. C., Valent A. G., Nunes R. C.,arXiv:1606.00450 [gr-qc]

[20] Nunes R. C., Pan S., Saridakis. E. N., 2016, Phys. Rev.D, 94, 023508, arXiv:1605.01712 [astro-ph.CO]

[21] Kumar S., Nunes R. C., 2016, Phys. Rev. D 94, 123511(2016), arXiv:1608.02454 [astro-ph.CO]

[22] Bruck C., Mifsud J., Morrice J., arXiv:1609.09855 [astro-ph.CO]

[23] Yang T., Guo Z. K., Cai R. G., 2015, Phys. Rev. D, 91,123533, arXiv:1505.04443 [astro-ph.CO]

[24] Yang W., Xu L., 2014, Phys. Rev. D, 89, 083517,arXiv:1401.1286 [astro-ph.CO]

[25] Yang W., Xu L., 2014, Phys. Rev. D, 90, 083532,arXiv:1409.5533 [astro-ph.CO]

[26] Yang W., Li H., Wu Y., Lu J., 2016, JCAP, 10, 007,arXiv:1608.07039 [astro-ph.CO]

[27] Xia D. -M., Wang S., 2016, MNRAS, 463, 952,arXiv:1608.04545 [astro-ph.CO]

[28] Caprini C., Tamanini N., 2016, JCAP, 10, 006,arXiv:1607.08755 [astro-ph.CO]

[29] Murgia R., Gariazzo S., Fornengo N., 2016, JCAP, 04,014, arXiv:1602.01765 [astro-ph.CO]

[30] Pourtsidou A., Tram T., 2016, Phys. Rev. D 94, 043518,arXiv:1604.04222 [astro-ph.CO]

[31] Zimdahl W., 2005, Int. J. Mod. Phys. D, 14, 2319,arXiv:gr-qc/0505056

[32] Wang B. , Zang J., Lin C.-Y., Abdalla E., Micheletti S.,2007, Nucl. Phys. B, 778, 69, arXiv:astro-ph/0607126

[33] Amendola L., 2000, Phys. Rev. D, 62, 043511,arXiv:astro-ph/9908023

[34] Billyard A. P., Coley A. A., 2000, Phys. Rev. D, 61,083503, arXiv:astro-ph/9908224

[35] Zimdahl W., Pavon D., Chimento L.P., 2001, Phys. Lett.B, 521, 133, arXiv:astro-ph/0105479

[36] Dalal N. et al., 2001, Phys. Rev. Lett, 87, 141302,arXiv:astro-ph/0105317

[37] Herrera R., Pavon D., Zimdahl W., 2004, Gen. Relt.

Grav, 36, 2161, arXiv:astro-ph/0404086[38] Amendola L., Quercellini C., 2003, Phys. Rev. D, 68,

023514, arXiv:astro-ph/0303228[39] Mangano G., Miele G., Pettorino V., 2003, Mod. Phys.

Lett. A, 18, 831, arXiv:astro-ph/0212518[40] Pavon D., Zimdahl W., 2005, Phys. Lett. B, 628, 206,

arXiv:gr-qc/0505020[41] Chimento L. P., Jakubi A. S., Pavon D., Zimdahl W.,

2003, Phys. Rev. D, 67, 083513, arXiv:astro-ph/0303145[42] He J. -H., Wang B., 2008, JCAP, 0806, 010,

arXiv:0801.4233 [astro-ph][43] Quartin M. et al., 2008, JCAP, 0805, 007,

arXiv:0802.0546 [astro-ph][44] Bohmer C. G., Caldera-Cabral G., Lazkoz R., Maartens

R., 2008, Phys. Rev. D, 78, 023505, arXiv:0801.1565 [gr-qc]

[45] Caldera-Cabral G., Maartens R., Urena-Lopez L. A.,2009, Phys. Rev. D, 79, 063518 (2009), arXiv:0812.1827[gr-qc]

[46] Nojiri S., Odintsov S. D., 2010, Phys. Lett. B, 636, 44[47] Hashim M., Bertacca D., Maartens R., 2014, Phys. Rev.

D, 90, 103518, arXiv:1409.4933 [astro-ph.CO][48] Pan S., Bhattachrya S., Chakraborty S., 2015, MNRAS,

452, 3038, arXiv:1210.0396 [gr-qc][49] Tamanini N., 2015, Phys. Rev. D, 92, 043524,

arXiv:1504.07397 [gr-qc][50] Skordis C., Pourtsidou A., Copeland E. J., 2015, Phys.

Rev. D, 91, 083537, arXiv:1502.07297 [astro-ph.CO][51] Marra V., 2016, Phys. Dark. Univ. 13, 25,

arXiv:1506.05523 [astro-ph.CO][52] Mukherjee A., Banerjee N., 2017, Class. Quant. Grav.

34, 035016, arXiv:1610.04419 [astro-ph.CO][53] Chimento L. P., 2010, Phys. Rev. D, 81, 043525,

arXiv:0911.5687 [astro-ph.CO][54] Chimento L. P., 2012, AIP. Conf. Proc, 1471, 30,

arXiv:1204.5797 [gr-qc][55] Chervon S. V., 2013, Quantum Matter, 2, 71,

arXiv:1403.7452 [gr-qc][56] Paliathanasis A., Tsamparlis M., 2014, Phys. Rev. D, 90,

043529, arXiv:1408.1798 [gr-qc][57] Sanchez I. E., 2014, Gen. Relt. Grav, 46, 1769,

arXiv:1405.1291 [gr-qc][58] Paliathanasis A., Tsamparlis M., Basilakos S., Barrow

J. D., 2016, Phys. Rev. D, 93, 043528, arXiv:1511.00439[gr-qc]

[59] Wang J. S., Wang F. Y., 2014, A&A, 564, A137,arXiv:1403.4318 [astro-ph.CO]

[60] Nunes R. C., Barboza E. M., 2014, Gen. Relt. Grav, 46,1820, arXiv:1404.1620 [astro-ph.CO]

[61] Barboza E. M., Alcaniz J. S., Zhu Z.-H., Silva R., 2009,Phys. Rev. D, 80, 043521, arXiv:0905.4052 [astro-ph.CO]

[62] Chevallier M., Polarski D., 2001, Int. J. Mod. Phys. D,10, 213, arXiv:gr-qc/0009008

[63] Linder E. V., 2003, Phys. Rev. Lett, 90, 091301,arXiv:astro-ph/0208512

[64] Cooray A. R., Huterer D., 1999, ApJ, 513, L95,arXiv:astro-ph/9901097

[65] Astier P., 2001, Phys. Lett. B, 500, 8, arXiv:astro-ph/0008306

[66] Weller J., Albrecht A., 2002, Phys. Rev D, 65, 103512,arXiv:astro-ph/0106079

[67] Efstathiou G., 1999, MNRAS, 310, 842, arXiv:astro-ph/9904356

[68] Suzuki N. et al., 2012, ApJ, 746, 85, arXiv:1105.3470

13

[astro-ph.CO][69] Simon J., Verde L., Jimenez R., 2005, Phys. Rev. D, 71,

123001, arXiv:astro-ph/0412269[70] Stern D. et al., 2010, JCAP, 1002, 008, arXiv:0907.3149

[astro-ph.CO][71] Zhang C. et al., 2014, Research in Astronomy and Astro-

physics, 14, 1221, arXiv:1207.4541 [astro-ph.CO][72] Moresco M. et al., 2012, JCAP, 1208, 006,

arXiv:1201.3609 [astro-ph.CO][73] Moresco M., 2015, MNRAS, 450 L16, arXiv:1503.01116

[astro-ph.CO][74] Moresco M. et al., 2016, JCAP, 1605, 014,

arXiv:1601.01701 [astro-ph.CO][75] Blake C. et al., 2011, MNRAS, 418, 1707,

arXiv:1108.2635 [astro-ph.CO][76] Percival W. J. et al., 2010, MNRAS, 401, 2148[77] Beutler F. et al., 2011, MNRAS, 416, 3017,

arXiv:1106.3366 [astro-ph.CO][78] Jarosik N. et al., 2011, ApJS, 192, 14, arXiv:1001.4744

[astro-ph.CO][79] Giostri R. et al., 2012, JCAP, 1203, 027, arXiv:1203.3213

[astro-ph.CO][80] Akaike H., 1974, IEEE Transactions on Automatic Con-

trol, 19, 716[81] de la Cruz-Dombriz A., Dunsby P. K. S., Luongo O.,

Reverberi L., 2016, to appear in JCAP, arXiv:1608.03746[gr-qc]

[82] Planck collaboration XVI, 2014, A&A, 571, A16,arXiv:1303.5076 [astro-ph.CO]

[83] Rest A. et al., 2014, ApJ, 795, 44, arXiv:1310.3828[astro-ph.CO]

[84] Xia J.-Q., Li H., Zhang X., 2013, Phys. Rev. D, 88,063501, arXiv:1308.0188 [astro-ph.CO]

[85] Cheng C., Huang Q.-G., 2014, Phys. Rev. D, 89, 043003,arXiv:1306.4091 [astro-ph.CO]

[86] Shafer D. L., Huterer D., 2014, Phys. Rev. D, 89, 063510,arXiv:1312.1688 [astro-ph.CO]