Extended ΛCDM model and viscous dark energy: A bayesian analysis

W. J. C. da Silva1, ∗ and R. Silva1, 2, †

1Universidade Federal do Rio Grande do Norte, Departamento de Física, Natal - RN, 59072-970, Brasil2Universidade do Estado do Rio Grande do Norte, Departamento de Física, Mossoró - RN, 59610-210, Brasil

(Dated: May 24, 2019)

We propose an approach considering the nonextensive effects in the context of the Verlinde theory in orderto address an extended cosmological model in the context of viscous dark energy. Specifically, this modelleads to a tiny perturbation in the dynamics of the expansion of the universe through the generalized Friedmannequations so-called the extended ΛCDM model. From the observational test standpoint, we make a Bayesiananalysis of the models of bulk viscosity for dark energy which follows the Eckart theory of bulk viscosity.These models are investigated through the context of both models ΛCDM and extended ΛCDM. The Bayesiananalysis is performed using the data of CMB Distance priors, Baryon Acoustic Oscillations Measurements,Cosmic Chronometers, and SNe Ia distance measurements.

I. INTRODUCTION

The accelerated expansion of the universe has widely beencorroborated by the greater amount of observational data,such as type Ia Supernovae [1, 2], Baryon Acoustic Oscilla-tions (BAO) [3] and Cosmic Microwave Background (CMB)anisotropies [4, 5]. These observations converge to the stan-dard model, the ΛCDM model, where cosmological constantΛ is responsible by acceleration of the universe and CDMrefers to the Cold Dark Matter. Although this model has beenconfirmed as the standard cosmological model, a theoreticalexplanation of the physical mechanism responsible by cosmicacceleration has been a significant challenge in the moderncosmology [6]. From the observational standpoint, there is atension associated with the measurements of Hubble param-eter at z = 0 by CMB anisotropies [4, 5], and Cepheids andSupernovae [7–10]. The other reported tension is related tomeasurements of the growth of matter density fluctuations be-tween late-time observations and CMB anisotropies (see moredetails in Ref. [11]). There are different approaches to solvethese puzzles. Typically, they can be divided into modifiedgeneral relativity [12] and dark energy models [13]. The firstcase assumes modification in the standard general relativitybased on some physical phenomena. The latter case proposesa new description for dark energy, or scalar field within thegeneral relativity framework.

Another idea addressing the dark energy has focused on thefluid description, with the thermodynamics being the core ofthis scenario (see, e.g., [14] and references therein). Manycosmological models, which are extensions of ΛCDM, havetypically addressed dissipative process like the bulk viscosityin order to provide a thermodynamical framework [15]. Morerecently, by considering that the dark energy presents a bulkviscosity mechanism, the dark energy models have been an-alyzed in the context of fluid [16], in the context scalar field[17, 18] and in the modified general relativity framework [19].

On the other hand, the thermodynamics and its microscopicapproach (statistical mechanics and kinetic theory) have been

∗ williamjouse@fisica.ufrn.br† raimundosilva@fisica.ufrn.br

extended in order to face the so-called complex systems [21].By summarizing, the non-additive (or nonextensive) frame-work is based on the parametrization of the entropy formulawhich depends on a free parameter q (also called entropic pa-rameter), and provides the Boltzmann-Gibbs (BG) entropy inthe additive limit q→ 1. Specifically, the nonextensive frame-work is related with the Tsallis entropy, which for a classicalnon-degenerated gas system of point particles reads (unlessexplicitly stated, in our units kB = c = 1)

S q = −

∫f q lnq f d3 p, (1)

where q is quantifying the degree of nonadditivity of S q, f isthe distribution of momentum and lnq( f ) is the nonadditive q-logarithmic function whose inverse is the q-exponential. Bothfunctions are defined by:

lnq( f ) = (1 − q)−1( f 1−q − 1), ( f > 0), (2)

eq( f ) = [1 + (1 − q) f ]1

1−q , eq(lnq f ) = f , (3)

which reduce to the standard expressions in the limit q → 1.The above formulas also imply that for a gas system composedby two subsystems (A,B), the kinetic Tsallis measure verifiesS q(A + B) = S q(A) + S q(B) + (1 − q)S q(A)S q(B). Hence, forq = 1 the logarithm extensive measure associated to the GBapproach is recovered.

In the context of cosmology, many connections have beeninvestigated, e.g. the entropic cosmology for a generalizedblack hole entropy [22], black holes formation [23, 24], themodified Friedmann equations from Verlinde theory [25],the role of the q-statistics on the light dark matter fermions[26], the new perspective for the holographic dark energy[27]. Indeed, there are many connections between cosmologyand nonextensive framework (see, e.g., [28] and referencestherein).

A recent study addressed a connection between dissipativeprocesses and nonextensive framework [29, 30]. The principlebehind this connection is based on the so-called Nonexten-sive/Dissipative Correspondence (NexDC), being associatedwith the microscopic description of the fluid through the Tsal-lis distribution function [31]. Specifically, the NexDC has beenimplemented to describe viscous dark matter [29]. In addi-tion, by using the nonextensive effect in the Verlinde’s theory

arX

iv:1

810.

0375

9v4

[as

tro-

ph.C

O]

22

May

201

9

2

standpoint, a general model was proposed in order to investi-gate viscous dark matter [30].

In this paper, we particularly are interested in the investiga-tion from models of the viscous dark energy, which considerfirst order deviations from equilibrium, i.e., the Eckart the-ory. From the background standpoint, we study these modelsby taking into the account the modified Friedmann equationsbased on the connections between the Tsallis statistics and theVerlinde’s conjecture [25, 30]. Specifically, by using the mod-els of viscous dark energy [16, 17, 29, 30, 32], we follow twodifferent route, namely: i) By considering the standard dy-namic (ΛCDM model), we make a Bayesian analysis in orderto investigate the bulk viscous models for dark energy withdifferent forms of the bulk viscous coefficient and ii) By usinga general dynamic (extended ΛCDM), based on the nonexten-sive effects and Verlinde theory, we repeat the Bayesian anal-ysis in order to investigate those models [16, 17, 29, 30, 32].

The paper follows the sequence: in Sec. II we summarizethe assumptions behind of the generalized Friedmann equa-tions for bulk viscosity process. In Sec. III we introduce theviscous dark energy models [16, 17, 29, 30, 32] consideringthe extended ΛCDM model. In order to constrain parame-ters and compare models, in Sec. IV, we make a BayesianAnalysis based on the data of CMB Distance priors, BaryonAcoustic Oscillations Measurements, Cosmic Chronometers,and SNe Ia distance measurements. The main results and dis-cussion concerning our approach for the viscous dark energymodels are presented in Sec. IV.

II. BACKGROUND AND ASSUMPTIONS

Let us introduce a phenomenological approach by assum-ing an imperfect fluid. Furthermore, the dissipative process isrelated with an energy source of the FLRW universe. In thisdescription, the momentum-energy tensor reads

T µνT = T µν + ∆T µν, (4)

where T µνT is the total momentum-energy tensor, T µν is

momentum-energy tensor of perfect fluid and ∆T µν repre-sents dissipative process such as heat flux, anisotropic-stressand bulk viscosity. Here, we will consider a homogeneous,isotropic and flat universe, then only dissipative process al-lowed is the bulk viscosity [40]. The simplest approach totreat bulk viscosity process is derived of the Eckart theory,which is a noncausal approach to dissipative phenomena. Inthis concern, the bulk viscosity pressure is given by [59]

∆T µν = Πhµν, (5)

where, hµν = gµν + uµuν is the usual projector onto the localrest space of uµ (four-velocity) and gµν is the FLRW metric.Π is the bulk viscous pressure, which depends on the bulk vis-cosity coefficient and the Hubble parameter, Π = −3ξH. Al-beit this formalism has been widely used at background andperturbative levels [15–17, 19], its fundamental difficulty is

related with its noncausal behavior, i.e, since it admits dissi-pative signals with superluminal velocities [41–45]. A pos-sible solution in order to face this difficult in the cosmolog-ical context, would be use a causal extension of the Eckartframework, which is the so-called Israel-Stewart (IS) theory[42, 46]. Issues on the fundament, which have approached theproblem of causality as well as the Ostrogradsky ghost havebeen addressed by considering the IS theory and Lagrangianformalism [47], however, these issues are yet under debate[48]. Indeed, IS theory presented a better description thanEckart theory and Landau and Lifshitz theories, however likethem, the common behavior is associated to small deviationof equilibrium [42]. Recently, another connection with cos-mology has been proposed through the full causal theory inthe context of the acceleration of the universe [56] and darkmatter and dark energy as a viscous single fluid [49]. Thereare other connections between the full causal theory and cos-mology (see, e.g., [50] and references therein). On the otherhand, even though the Eckart theory has drawbacks at funda-mental level, it is the simplest than the IS theory, being widelyused in order to investigate the accelerating universe with thebulk viscous fluid (see, e.g. Refs.[51–55]). From the earlyinflation standpoint, the Ref. [57] has shown that both thepathological Eckart theory and the truncated IS theory provideinflation1. But the truncated version of IS theory presented aconstant relaxation time, being incorrect for an expanding uni-verse. As we are investigating the viscous dark energy, whichis the component of the dark sector that provides the late-timeacceleration of the expanding universe, we will consider themodels [16, 17, 29, 30, 32] which have used the Eckart for-malism as a first order limit of the IS theory with zero relationtime. Moreover, this framework is a plausible approach to in-vestigate the viscous dark energy since the physical effect oc-curs on the pressure, being a tiny perturbation of the standardΛCDM model.

Now, by choosing a reference frame in which the hydrody-namics four-velocity uµ is unitary, uµuµ = −1, and replacingthe Eq.(5) into Eq.(4), we obtain

T µνT = (ρ + Peff)uµuν + Peffgµν, (6)

where ρ is the energy density, Peff = pk + Π, where pk is thekinetic pressure (equilibrium pressure) and Π = −3ξH.

By considering the conservation equation ∇µT µν = 0 in Eq.

(6) one finds

ρ + 3H(ρ + pk) − 9H2ξ = 0. (7)

This is the energy conservation equation for viscous fluid. Thefunctional form of ξ is fundamental to the dynamics of themodel.

1 From the cosmological perspective, the Ref. [57] has demonstrated that ISapproach converges to the Eckart’s theory, when the collision time-scalein the transport equation of fluid is zero, i.e., the bulk viscous model isnecessarily noncausal and unstable [46].

3

In order to compare some models of viscous dark energythrough the Bayesian Analysis, let us consider the so calledthe extended Friedmann equations [25, 30]2

H2 =8πG

3ρ

(5 − 3q

2

)−

ka2 , (8)

and

aa

= −4π3

(5 − 3q

2

)G(ρ + 3p), (9)

where H = aa is the Hubble parameter, q is the nonextensive

parameter, ρ is the total energy density, p is the pressure ofthe fluid and, k represents the spatial curvature. Here, we willconsider the flat and non-flat Universe k = 0,±1, and the ex-tended expression (9) in order to investigate different modelsof viscous dark energy.

III. VISCOUS DARK ENERGY

In this section, we consider the Friedmann equations ob-tained in the previous one and the dark energy as a fluid withbulk viscosity process. The main contributions to the totalmomentum-energy tensor of the cosmic fluid are the radiation,the baryonic matter, cold dark matter and the viscous dark en-ergy. The radiation, baryons and dark matter are assumed tohave the usual properties of perfect fluids. As each componentof the cosmic fluid is individually conserved, we obtain

ρi + 3H(ρi + pi) = 0, (10)

where i is related to the radiation (r), the baryonic matter (b),cold dark matter (dm). By considering the Eq. (7), the energyconservation of viscous dark energy is given by

ρde + 3H(ρde + pde) = 0, (11)

where ρde the energy density of viscous dark energy, pde =

pk + Π is the effective pressure, pk is the equilibrium pressure,Π = −3ξH, bulk viscosity pressure and ξ is the bulk viscositycoefficient. The choice of bulk viscosity coefficient ξ gener-ates different viscous dark energy models. The general case,

2 In the Verlinde’s conjecture [34], the gravity is explained as an entropicforce caused by changes in the information associated with the positionsof particles. The assumption of the entropic force, together with the Unruhtemperature, provides the derivation of the second law of Newton. More-over, by considering the holographic principle and the equipartition law ofenergy, this approach leads to Newton’s law of gravitation. These ideashave been used in order to propose a thermodynamic derivation of Einsteinequations [35]. In this regards, it was demonstrated in Ref. [25] througharguments of the Refs. [33, 34] that one modification in the field equationscan be obtained simply by assuming the nonextensive equipartition law ofenergy. From the mathematical standpoint, this extended approach leads toan effective gravitational constant, i.e., G → Gq =

5−3q2 G [25, 30].

ξ is not constant, and in the literature there are different ap-proaches to determining how bulk viscosity evolves. We con-sider three different bulk viscosity functions in our analysis:(i) bulk viscosity being proportional to the Hubble parameter,ξ = ξ0H; (ii) bulk viscosity proportional to energy densityand inversely proportional to Hubble parameter; (iii) the usualansatz for the bulk viscosity, a function for thermodynamicalstate, i.e., energy density of the fluid, in the case ξ = ξ(ρde).

A. Model I

The first model analyzed is the bulk viscosity proportionalto the Hubble parameter, i.e, from the Friedmann equation, thebulk viscosity is proportional to the square root of the totalenergy density. This dependency allows us to consider thatbulk viscosity is a function of all the other cosmological fluids.The model was studied in Ref. [16, 20], with the ansatz forbulk viscosity evolution given by

ξ = ξ0H, (12)

and, the effective pressure reads

pde = pde − 3H2ξ0, (13)

where ξ0 is the current value for bulk viscosity and pde =

ωρde. Firstly, we consider parameter of equation of state, ω =

−1, consequently, the effective pressure is pde = −ρde−3ξ0H2.We call this Model Ia. The second case, we consider ω as freeparameter, this model is called Model Ib. Afterwards, combin-ing the Eqs. (8), (10), (11) and (13), the Friedmann equationfor Model Ia is given by

H2

H20

=

(5 − 3q

2

) [Ωb(1 + z)3 + Ωr(1 + z)4 +

Ωdm

1 + ξ(1 + z)3

+

(1 −

Ωdm

1 + ξ

)(1 + z)−3ξ

],

(14)where z is the redshift, Ωdm is the matter density parametertoday and ξ is dimensionless bulk viscosity. For Model Ib, theFriedmann equations reads

H2

H20

=

(5 − 3q

2

) [Ωb(1 + z)3 + Ωr(1 + z)4 +

ωΩdm

ω − ξ(1 + z)3

+

(1 −

ωΩdm

ω − ξ

)(1 + z)3(1+ω−ξ)

].

(15)Also, we consider bulk viscosity effects on the non-flat Uni-verse. To make this, we add curvature density parameter evo-lution in the Model Ia, and we name Model Ic. Friedmannequation for this model is given by

4

H2

H20

=

(5 − 3q

2

) [Ωb(1 + z)3 + Ωr(1 + z)4 +

Ωdm

1 − ξ(1 + z)3

+2Ωk

2 + 3ξ(1 + z)2 +

(1 −

2Ωk

2 + 3ξ−

Ωdm

1 + ξ

)(1 + z)−3ξ

],

(16)where Ωk is the today curvature density parameter. The di-mensionless bulk viscosity parameter is defined by

ξ =8πGξ0

H0, (17)

is valid for all models.

B. Model II

Another interesting functional form for bulk viscosity isa ratio between corresponding energy density and expansionrate given by [17]

ξ = 3ξ0

√ρde

H, (18)

where ξ0 is the present-day bulk viscosity and ρde, dark energydensity. The effective pressure for this model is

pde = −ρde − 3ξ0√ρde. (19)

For this ansatz, bulk viscosity of dark energy is insignificant inearly Universe (when dark matter dominates). Any one way,the bulk viscosity increases in late Universe [17].

From Eqs. (8), (10), (11) and and (19) in the flat Universe,the evolution of the Friedmann equation for bulk viscositymodel is given by

H2

H20

=

(5 − 3q

2

) [Ωb(1 + z)3 + Ωr(1 + z)4 + Ωdm(1 + z)3

+ Ωde

(1 −

9ξ2√

Ωdeln(1 + z)

)2],

(20)

where ξ is the dimensionless bulk viscosity coefficient definedby

ξ =

√8πG3H2

0

ξ0. (21)

We use the normalization condition Ωde = 25−3q−Ωb−Ωdm−Ωr.

C. Model III

The last model of bulk viscosity considered in this workwas studied by Refs. [29, 30, 32] in the context of viscous

dark matter. Thus, assuming that the bulk viscosity is givenby

ξ = ξ0

(ρde

ρde0

)α, (22)

where ξ0 is the current value for bulk viscosity, ρde0 is thedensity of the viscous dark energy today and α is constant.We can set α in two values, α = 0 and α = −1/2, for to allevi-ate the integrated Sachs-Wolfe effect [32]. Then, we consideronly α = 0, then, the effective pressure for this model

pde = pde − 3Hξ0, (23)

where pde = ωρde and ξ0 is a constant parameter. The valueof α = 0 has a physical interpretation, means a constant bulkviscosity coefficient.

The Hubble expansion rate H is given in terms of the energydensities Ωi where the subscript i corresponds to each fluid,i.e., dark matter, dark energy, radiation

H2

H20

=

(5 − 3q

2

) [Ωb(1 + z)3 + Ωr(1 + z)4

+ Ωdm(1 + z)3 + Ωde(z)],

(24)

where Ωb, Ωr, Ωdm are baryonic, radiation and dark matterdensity parameter today, respectively. In goal to determinatefunctional form of Ωde, we have to solve conservation equa-tion (11) with the effective pressure, Eq. (23), then

dΩde

dz=

3Ωde

1 + z(1 + ω) −

ξ

1 + z

(5 − 3q2

) [Ωb(1 + z)3

+ Ωr(1 + z)4 + Ωdm(1 + z)3 + Ωde(z)]1/2

,

(25)

where we define a dimensionless bulk viscosity parameter as

ξ =24πGξ0

H0, (26)

in which is valid for Eq. (25). The initial condition for thedifferential equation is Ωde(0) = 2

5−3q −Ωb −Ωdm −Ωr.In the next sections we will present the cosmological data

to do a Bayesian analysis of these viscous dark energy mod-els in two ways: the first analysis will be done by fixing thevalue of q = 1, that is, without nonextensivity and in the sec-ond moment, we will consider the parameter q as free in theanalysis.

IV. DATA CONSTRAINTS AND BAYESIAN ANALYSIS

In this section, we will present the data and technique usedin this work. To constrain parameters and compare models,we perform Bayesian Analysis based on the presented data.

5

Recently, Bayesian Analysis has been extensively used to con-straint and compare cosmological models [30, 60–64]. Inour analysis, we consider CMB Distance priors derived fromPlanck 2015, the eight baryon acoustic oscillations measure-ments [74–78], 24 cosmic chronometers measurements fromRef. [88] and 1048 SNe Ia distance measurements of the Pan-STARRS (Pantheon) dataset [91].

The ΛCDM model is assumed as reference model and isparameterized with following set of cosmological parameters:the dimensionless Hubble constant h, the baryon density pa-rameter, Ωb, the cold dark matter density parameter Ωdm. Theparameters of the other models are listed in the Table I to-gether with their priors. We choose uniform priors on baryonparameter Ωb and cold dark matter parameter Ωdm. For di-mensionless Hubble parameter h we consider a range 10 timeswider than value obtained in Ref. [7]. For curvature parame-ter Ωk and ω, we adopt 1σ values reported by Planck Results[4] and uniform prior, respectively. The prior for bulk viscos-ity is based in recent results [29, 30, 32]. For nonextesivityparameter q we assume the values that agree with Friedmannequation. We fix Ωγh2 = 2.469 × 10−5 [89], Ωrh2 = 1.698Ωγ

[90].The most important quantity for Bayesian model compar-

ison is the Bayesian evidence, or marginal likelihood, andis obtained here by implementing the PyMultiNest [67], aPython interface for MultiNest [68], the Bayesian tool basedon the nested sampling [69] in which calculates the evidence,but still allows constrain parameters with consequence. Weplot the results using GetDist [70].

We following the standard description (see Refs. [30, 60,64, 65]), the posterior distribution P(Θ|D,M) is given by

P(Θ|D,M) =L(D|Θ,M)π(Θ|M)

E(D|M), (27)

where L(D|Θ,M), π(Θ|M) and E(D|M), the likelihood, theprior and Bayesian evidence with Θ denotes the parametersset, D the cosmological data and M the model. The evidencecan be written in the continuous parameter space Ω as

E =

∫Ω

L(D|Θ,M)π(Θ|M)dΘ. (28)

In order to compare two models, Mi and M j, we computethe ratio of the posterior probabilities, given by [65]

P(Mi|D)P(M j|D)

= Bi jP(Mi)P(M j)

, (29)

where Bi j is known as the Bayes factor, defined as

Bi j =Ei

E j. (30)

The Bayes factor of the model i relative to the model j (here,we assumed to be the ΛCDM model). It is emphasized point-ing out that the Bayesian evidence rewards models that bal-ance the quality of fit and complexity [71]. Indeed, the larger

the number of free parameter, not required by the data, thepenalization of the model will be greater than the other. Theusual interpretation of the Bayes factor is related to Jeffreys’scale. We use an alternative version of Jeffreys’ scale sug-gested in Ref. [65].

Table I. The table shows the priors distribution used is this work.

Parameter Model Prior Reference

h All U(0.5584, 0.9064) [7]

Ωb All U(0.0005, 0.1) -

Ωdm All U(0.001, 0.99) -

Ωk Model Ic N(−0.05, 0.05) [4]

ω ωCDM, Model Ib U(−2.0, 0.0) -

ξ All except ΛCDM N(0.0, 0.1) [29, 30, 32]

q All except ΛCDM U(0.8, 1.4) -

A. Baryon Acoustic Oscillations Measurements

The interaction between gravitational force and primordialrelativistic plasma generates acoustic oscillations at the re-combination epoch, which leave their signature in every epochof the Universe. The measurements of BAO provide an inde-pendent standard ruler to constrain cosmological models.

The BAO measurements are given in terms of angular scaleand the redshift separation, this is obtained from the calcula-tion of the spherical average of the BAO scale measurement,and it is given by [72, 73]

dz =rs(zdrag)DV (z)

, (31)

in which DV (z) is volume-averaged distance given by

DV (z) =

[(1 + z)2DA(z)2 cz

H(z)

]1/3

, (32)

where c is the speed of light and DA is the angular diameterdistance given by

DA(z) =c

1 + z

∫ z

0

dzH(z)

. (33)

rs(zdrag) is the comoving size of the sound horizon calculatedin redshift at the drag epoch defined by

rs(zdrag) =

∫ ∞

zdrag

csdzH(z)

, (34)

in which cs(z) = c√

3(1+R)is the sound speed of the photon-

baryon fluid and R = 34

ΩbΩr

11+z . We consider the redshift at the

drag epoch zdrag given by [73]

6

zdrag =1291(Ωmh2)0.251

1 + 0.659(Ωmh2)0.828

[1 + b1(Ωmh2)b2

], (35)

where b1 = 0.313(Ωmh2)−0.419[1 + 0.607(Ωmh2)−0.674

], b2 =

0.238(Ωmh2)0.223.In this analysis we consider the BAO measurements from

diverse surveys, see the Table II. Furthermore, we also in-clude three measurements from WiggleZ Survey [74]: dz(z =

0.44) = 0.073, dz(z = 0.6) = 0.0726, and dz(z = 0.73) =

0.0592. These measurements are correlated by following in-verse covariance matrix

C−1 =

1040.3 −807.5 336.8−807.5 3720.3 −1551.9336.8 −1551.9 2914.9

. (36)

Table II. BAO distance measurements for each survey considered.

Survey z dz(z) Ref.

6dFGS 0.106 0.3360 ± 0.0150 [75]

MGS 0.15 0.2239 ± 0.0084 [76]

BOSS LOWZ 0.32 0.1181 ± 0.0024 [78]

SDSS(R) 0.35 0.1126 ± 0.0022 [77]

BOSS CMASS 0.57 0.0726 ± 0.0007 [78]

For the WiggleZ data, the chi-squared function is

χ2WiggleZ = DT C−1D, (37)

where D = d obsz −d mod

z and C−1 is the covariance matrix givenby Eq. (36).

The chi-squared function related with each survey is givenby

χ2Survey =

[d obs

z (z) − d modz (z)

σSurvey

]2

, (38)

where d obsz is the observed ratio value, d mod

z is theoretical ratiovalue and σ is the uncertainties in the measurements for eachdata point.

Then, the BAO χ2 function contribution is defined as

χ2BAO = χ2

WiggleZ + χ2Survey. (39)

B. CMB Distance Priors

CMB distance priors can be derived from data, such asPlanck collaboration or WMAP from the full Boltzmann anal-ysis of CMB data. In Refs. [79–81], they discussed the pos-sibility to compress CMB likelihood in few numbers: CMB

shift parameter R [82], the angular scale of the sound horizonat last scattering `A, they are important to deal with the late-time expansion history, and baryon density today Ωbh2, it isimportant to study the late-time Universe but not sensitive tothe cosmological models.

CMB shift parameter is defined as

R =

√ΩmH2

0r(z?)/c, (40)

where r(z?) = cH0

∫ z0

dzE(z) and angular scale of the sound hori-

zon at last scattering

`A = πr(z?)/rs(z?), (41)

where rs(z?) is comoving size of the sound horizon calculatedin the redshift of decoupling epoch given by [83]

z? = 1048[1 + 0.00124(Ωbh2)−0.738][1 + g1(Ωmh2)g2 ], (42)

where

g1 =0.0783Ω−0.238

b

1 + 39.5(Ωbh2)0.763 , (43)

g2 =0.560

1 + 21.1(Ωbh2)1.81 . (44)

Then, the χ2 function of the CMB prior is defined as

χ2CMB = XT

CMB · C−1CMB · XCMB, (45)

where XCMB = (R(z?), `A(z?),Ωbh2) − (Robs, `obsA ,Ωbh2 obs)

with R obs = 1.7488, ` obsA = 301.76, Ωbh2 obs = 0.02228 and

covariance matrix C from Planck Results [4].

C. Cosmic Chronometers

Another analysis considered in this work are the cosmicchronometers obtained through the differential age method.The cosmic chronometer is a method to determine the Hubbleparameter values at different redshifts taking the relative ageof passively evolving galaxies [84–86]. The method calculatesdz/dt and hence the Hubble parameter is given by

H(z) = −1

1 + zdzdt. (46)

Here, the theoretical values of H(z) are given by Eqs. (14),(15), (16), (20), (24). The measurement of dz is obtainedthrough spectroscopic data with high accuracy, then for a pre-cise measurement of the Hubble parameter, it is necessary tomeasure the differential age evolution dt of such galaxies, andhence cosmic chronometers are considered to be model inde-pendent. A detailed description about the cosmic chronome-ter method can be found in Ref. [7, 87]. Here we use the 24

7

measurements of the Hubble parameter in the redshift interval0.1 < z < 1.2, which are listed in Table III [88]. The chooseof this measures is motivate by the following argument,theexpansion history data of the Universe might no be smoothoutside the quoted redshift range [87].

Table III. Estimated values of H(z) obtained using the differential agemethod.

z H(z) z H(z) z H(z)

0.07 69 ± 19.6 0.28 88.8 ± 36.6 0.48 97 ± 62

0.09 69 ± 12 0.352 83 ± 14 0.593 104 ± 13

0.12 68.6 ± 26.2 0.3802 83 ± 13.5 0.68 92 ± 8

0.17 83 ± 8 0.4 95 ± 17 0.781 105 ± 12

0.179 75 ± 4 0.4004 77 ± 10.2 0.875 125 ± 17

0.199 75 ± 5 0.4247 87.1 ± 11.2 0.88 90 ± 40

0.20 72.9 ± 29.6 0.4497 92.8 ± 12.9 0.9 117 ± 23

0.27 77 ± 14 0.4783 80.9 ± 9 1.037 154 ± 20

Then, the χ2 function of the cosmic chronometers is definedas

χ2CC =

24∑i=1

(Hobs(zi) − Hmod(zi)

σiH

)2

, (47)

where the σi uncertainties in the H(z) measurements for eachdata point i.

D. Pantheon Type Ia Supernovae Sample

The Type Ia Supernovae (SNe Ia) data is a relevant tool forunderstanding the actual evolution of the Universe. The Pan-theon sample is the most recent SNe Ia sample, which consistsof 1048 measurements in the redshift range 0.01 < z < 2.3[91]. The observational distance moduli of SNe µobs, can becalculated from

µobs = m∗B + αX1 − βC − MB, (48)

where m∗B, X1 and C are the B-band apparent magnitude, thestretch factor and color parameter, respectively. MB is the ab-solute magnitude. α and β characterize the stretch and color-luminosity relationships, respectively. Commonly, α and βare considered as free parameters and are constrained jointlywith cosmological parameters. Nonetheless, this approach ismodel dependent, thus the distance calibrated by a cosmolog-ical could not be used to constrain parameters. To alleviatethis problem, Ref.[91] proposes a method to calibrated SNe Ianamed BEAMS with Bias Corrections (BBC) [91, 92]. ThePantheon sample is calibrated using the BBC method, reduc-ing the photometric calibration uncertainties (see more details

in Refs.[91, 92]). Then, to calculate the observational distancemoduli we subtract MB from the apparent magnitude m∗B,corrand do not need the color and stretch corrections because nowthey are equal zero.

The theoretical distance modulus µth for a given supernovain redshift z is expressed as

µth = 5 log10dL

Mpc+ 25, (49)

where dL = (c/H0)DL is the luminosity distance, with c isthe speed of light, H0 is the Hubble constant. Hubble-freeluminosity distance is given by

DL = (1 + zhel)∫ zCMB

0

dzE(z)

, (50)

where E(z) = H(z)/H0, zCMB and zhel is the dimensionlessHubble parameter, is the CMB frame and heliocentric redshift,respectively. From Eq. (48) with α and β equal zero, theobserved distance moduli reads [91]

µobs = m∗B −M, (51)

with m∗B the B-band apparent magnitude and M is nuisanceparameter that encompasses absolute magnitude MB and theHubble constant H0. The χ2 function from Pantheon data isgiven by

χ2Pan = XT

Pan · C−1Pan · XPan, (52)

where for the i-th SNe Ia, XPan = µobs,i − µth,i and C is thecovariance matrix. We can rewrite Eq. (52) as

χ2Pan = ∆mT · C−1 · ∆m, (53)

with ∆m = mB − mmod, and

mmod = 5 log10 DL +M, (54)

in which H0 in dL can be absorbed into M, while the totalcovariance matrix C is given by

C = Dstat + Csys, (55)

where Dstat is the diagonal covariance matrix of the staticaluncertainties and Csys, is the covariance matrix of systematicserrors [91]. The nuisance parameterM could be marginalizedfollowing steps in Ref. [93]. Then, after the marginalizationoverM, we define the following quantities

a = ∆mT · C−1 · ∆m, (56)b = ∆mT · C−1 · 1, (57)c = 1T · C−1 · 1, (58)

8

Table IV. Confidence limits for the cosmological parameters using the BAO + CMB + CC + SNe Ia. The columns show the constraints oneach model whereas the rows show each parameter considering in this analysis. In the last rows we have the Bayesian evidence, Bayes’ factorand the interpretation.

Parameter ΛCDM ωCDM Model Ia Model Ib Model Ic Model II Model III

h 0.675 ± 0.005 0.679 ± 0.008 0.675 ± 0.006 0.681 ± 0.009 0.686 ± 0.009 0.679 ± 0.008 0.675 ± 0.005

Ωb 0.049 ± 0.001 0.048 ± 0.001 0.049 ± 0.001 0.048 ± 0.001 0.047 ± 0.001 0.048 ± 0.001 0.049 ± 0.001

Ωdm 0.267 ± 0.007 0.266 ± 0.007 0.267 ± 0.006 0.264 ± 0.007 0.267 ± 0.006 0.266 ± 0.007 0.266 ± 0.006

Ωk − − − − −0.053 ± 0.036 − −

ω − −1.027 ± 0.040 − −1.059 ± 0.059 − − −1.012 ± 0.035

ξ − − −0.0004 ± 0.008 −0.0097 ± 0.013 −0.026 ± 0.020 0.012 ± 0.019 −0.002 ± 0.008

lnE −534.675 ± 0.025 −537.491 ± 0.008 −537.168 ± 0.008 −539.232 ± 0.006 −537.060 ± 0.029 −536.199 ± 0.007 535.902 ± 0.599

ln B − −2.816 ± 0.008 −2.493 ± 0.008 −4.557 ± 0.006 −2.385 ± 0.029 −1.524 ± 0.007 −1.227 ± 0.599

Interpretation − Moderate Weak Moderate Weak Weak Weak

Table V. Confidence limits for the cosmological parameters using the BAO + CMB + CC + SNe Ia. The columns show the constraints on eachextended model, with q as free parameter, whereas the rows show each parameter considering in this analysis. In the last rows we have theBayesian evidence, Bayes’ factor and the interpretation.

Parameter ΛCDM ωCDM Model Ia Model Ib Model Ic Model II Model III

h 0.675+0.011−0.011 0.681+0.017

−0.017 0.662+0.014−0.015 0.664+0.017

−0.017 0.686+0.019−0.018 0.681+0.017

−0.017 0.676+0.016−0.017

Ωb 0.049 ± 0.002 0.048+0.003−0.002 0.051+0.002

−0.002 0.050+0.003−0.002 0.047+0.003

−0.002 0.048+0.003−0.002 0.049+0.003

−0.002

Ωdm 0.268+0.015−0.014 0.268+0.015

−0.014 0.291+0.026−0.024 0.285+0.035

−0.032 0.267+0.013−0.012 0.268+0.015

−0.014 0.274+0.026−0.025

Ωk − − − − −0.052+0.067−0.062 − −

ω − −1.051+0.095−0.097 − −1.04+0.14

−0.15 − − −1.07+0.14−0.15

ξ − − 0.032+0.068−0.066 0.010 ± 0.10 −0.026+0.037

−0.036 0.022+0.046−0.046 0.010+0.082

−0.089

q 0.998+0.013−0.013 0.993+0.016

−0.015 0.977+0.045−0.046 0.986+0.055

−0.058 1.10+0.28−0.29 0.994+0.016

−0.016 0.975+0.061−0.061

lnE −543.921 ± 0.007 −546.230 ± 0.009 −543.831 ± 0.055 −545.839 ± 0.234 −541.955 ± 0.069 −544.855 ± 0.062 −545.852 ± 0.058

ln B − −2.301 ± 0.009 0.090 ± 0.055 −1.918 ± 0.234 1.966 ± 0.069 −0.933 ± 0.062 −1.231 ± 0.058

Interpretation − Moderate Inconclusive Weak Weak (favored) Inconclusive Weak

where ∆m = mB −mmod and 1 is a vector of unitary elements,finally, the χ2 function is reads

χ2Pan = a −

b2

c+ ln

c2π. (59)

For joint analysis, we consider the likelihood of each probe,

namely Ljoint = LBAO × LCC × LCMB × LPan.

V. RESULTS AND CONCLUSIONS

In this work, we investigated some viscous dark energymodels in the context of ΛCDM and the extend ΛCDM

9

−0.05 0.00ξ

0.045

0.051

Ωb

0.24

0.27

Ωdm

−1.2

−1.1

−1.0

−0.9

ω

0.66 0.69h

−0.05

0.00

ξ

0.045 0.051Ωb

0.24 0.27Ωdm

−1.2 −1.1 −1.0 −0.9ω

Model Ib

ωCDM

Model III

Figure 1. Confidence regions and PDFs for the parameters h, Ωb, Ωdm, Ωk, ω and ξ, for all the models studied considering combining dataBAO + CMB + CC + SNe Ia.

model. To analyze these models, we performed a Bayesiananalysis in terms of the Jeffreys’ Scale that evaluating thestrength of evidence when comparing models [65]. To achievethis analysis, we adopted the prior described in Table I andconsidered distinct background data such as CMB priors dis-tance, BAO measurements, cosmic chronometers, PantheonType Ia Supernova.

The main results of joint analysis (CMB + CC + BAO +

SNe Ia) for q fixed were summarized in Table IV, including

the mean and corresponding 1σ error of parameters for eachmodel. In the Figs. 1 and 2 show the posterior distributionsand 1σ, 2σ and 3σ contours regions for models studied. Inthe Table IV, the dimensionless Hubble parameter convergedfor value obtained in the last Planck results [4, 5]. It is easyto see that Ωb and Ωdm were little affected by the test con-sidered. For the Model Ic, we found that the spatial curva-ture Ωk = −0.053 ± 0.036 was not compatible, 1σ, with spa-tially flat Universe. For the Model Ib and Model III, we got

10

−0.08 0.00 0.08ξ

0.0425

0.0475

0.0525

Ωb

0.24

0.27

Ωdm

−0.15

−0.05

0.05

Ωk

0.66 0.68 0.70 0.72h

−0.08

0.00

0.08

ξ

0.0425 0.0475 0.0525Ωb

0.24 0.27Ωdm

−0.15 −0.05 0.05Ωk

Model Ic

ΛCDM

Model Ia

Model II

Figure 2. Confidence regions and PDFs for the parameters h, Ωb, Ωdm, Ωk, ω and ξ, for all the models studied considering combining dataBAO + CMB + CC + SNe Ia.

ω = −1.059 ± 0.059 and ω = −1.012 ± 0.035 respectively,these results were still compatible with the standard cosmol-ogy (ω = −1) but with very slightly preference for a phantomcosmology (ω < −1) [94]. In order to relieve the H0 tension,we calculated the discrepancy between our results and Hub-ble constant local value [9]. The values were 3.48σ for theModel Ia, 2.92σ for the Model Ib, 2.65σ for the Model Ic,3.11σ and 3.55σ for the Models II and III, respectively. Weconcluded that some models (excluding the Model Ia) stud-

ied in this work alleviate the H0 tension with emphasis on theModel Ic, which has the lowest value of discrepancy.

The Figs. 3 and 4 showed the posterior distributions and1σ, 2σ and 3σ contours regions for extended models studiedwith q as a free parameter. In the Table V, we showed that thevalues of h obtained for Models Ia and Ib are lightly smallerthan those obtained in the last Planck results. For Models Iaand Ib, with addition of the parameter q in the analysis, theproportion of Ωdm was slightly bigger than when q was fixed.

11

For the Model Ic, we found that the results are compatiblewith first analysis (q is fixed). We noted that by adding theparameter q the value of bulk viscosity for the Models Ia , IIand III was slightly increased. Again, in order to relieve theH0 tension, we calculated the discrepancy between our resultsand Hubble constant local value [9]. The values were 5.29σfor the Model Ia, 4.08σ for the Model Ib, 2.47σ for the ModelIc, 3.11σ and 3.64σ for the Models II and III, respectively.We concluded that models Ia and Ib not alleviate the H0 ten-sion, on the other hand, the Model Ic has the lowest value ofdiscrepancy.

For comparison of models, we calculated Bayes’ factorconsidering ΛCDM as the reference model. The values ob-tained for the logarithm of evidence, the logarithm of Bayes’factor and interpretation of Bayes’ factor from the Jeffreys’scale were shown in Table IV. By considering these data used,ωCDM was disfavored with a moderate evidence in relation tothe ΛCDM model. We also noticed that Model Ib had an un-favorable moderately evidence, with ln B = −4.557 ± 0.006.Regarding the other models, for the Model Ia, we obtainedln B = −2.493 ± 0.008, for Model Ic, ln B = −2.385 ± 0.029,Model II, ln B = −1.524 ± 0.007 and for Model III, ln B =

−1.227±0.559, we found a disfavored weakly evidence. Fromthe Bayesian comparison model analysis point of view and thedata considered, we concluded that the viscous models studiedin this work are ruled out.

Now, by considering the analysis with q as a free param-eter, the values obtained for the logarithm of evidence, thelogarithm of Bayes’ factor and interpretation of Bayes’ factorfrom the Jeffreys’ scale were shown in Table V. Then, fromjoint analysis, ωCDM and Model Ic were disfavored with amoderate and weak evidence, respectively, in relation to the

ΛCDM model. The Models Ia and II we can not made anyconclusions about the evidence of this model in comparisonto extended ΛCDM model. We found the positive logarithmof Bayes’ factor (ln B = 1.966 ± 0.069) that indicated a weakevidence in favor of Model Ic.

In summary, we showed that the viscous dark energy iscompatible with the cosmological observations. However,the statistical constraints on the model parameters imply thatthe standard ΛCDM is recovered, i.e., bulk viscosity is zero.Moreover, we concluded from Bayesian analysis standpointthat our model has disfavored moderate and weak evidencecompared with ΛCDM. We concluded that ΛCDM still hasthe best efficiency to explain the data used in this work; thisconclusion is dependent on either by analyzing the parame-ters, or the Bayesian evidence.

Finally, it is worth emphasizing that in order to obtain arobust formulation (theoretical and observational), we needto investigate both background expansion and perturbationseffects. This issue will be investigated in the future work.

ACKNOWLEDGMENTS

The authors thank Brazilian scientific and financial sup-port federal agencies, CAPES and CNPq. RS thanks CNPq(Grant No. 303613/2015-7) for financial support. Thiswork was supported by High-Performance Computing Cen-ter (NPAD)/UFRN. W. J. C. da Silva thanks Antonella Cid forher assistance in the Bayesian analysis and Jailson Alcaniz fordiscussions and support in Rio de Janeiro. W. J. C. da Silvathanks Observatório Nacional (ON) for hospitality during thedevelopment of this work. We thanks G.M. Viswanathan forfeedback.

[1] A. G. Riess et al. (Supernova Search Team), Astron. J. 116,1009 (1998).

[2] S. Perlmutter et al. (Supernova Cosmology Project), Astrophys.J.517, 565 (1999).

[3] D. H. Weinberg et al., Phys. Rep. 530, 87 (2013).[4] P. A. R. Ade et al. (Planck Collaboration), Astron. Astrophys.

594, A13 (2016); P. A. R. Ade et al. (Planck Collaboration)Astron. Astrophys. 594, A14 (2016).

[5] N. Aghanim et al. (Planck Collaboration), “Planck 2018 re-sults.” arXiv:1807.06209 [astroph.CO].

[6] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989); V. Sahni and A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), astro-ph/9904398; T. Padmanabhan, Phys. Rept. 380, 235 (2003),arXiv:hep-th/0212290.

[7] A. G. Riess et al., The Astrophysical Journal, 826, 56 (2016).[8] A. G. Riess et al., The Astrophysical Journal, 855, 136 (2018).[9] A. G. Riess et al., The Astrophysical Journal, 861, 2 (2018).

[10] J. L. Bernal, L. Verde and A. G. Riess, Journ. Cosmol. As-tropart. Phys. 1610, 019 (2016).

[11] E. Macaulay, I. K. Wehus, and H. K. Eriksen, Phys. Rev.Lett. 111, 161301 (2013), 1303.6583; W. L. Freedman, Nat.Astron. 1, 0169 (2017), 1706.02739; A. Amon et al. (2017),1711.10999.

[12] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, Physics Re-ports 513, 1 (2012).

[13] L. Amendola et al., Living Rev. Rel., 21, 2 (2018).[14] J. A. S. Lima, J. S. Alcaniz, Phys. Lett. B 600, 191 (2004); I.

Brevik, S. Nojiri, S. D. Odintsov, L. Vanzo, Phys. Rev. D 70,043520 (2004); R. Bousso, Phys. Rev. D 71, 064024 (2005); G.Izquierdo, D. Pavon, Phys. Lett. B 633, 420 (2006); R. Silva, J.S. Alcaniz, J. A. S. Lima, Int. J. Mod. Phys. D 16, 469 (2007);S. H. Pereira, J. A. S. Lima, Phys. Lett. B 669, 266 (2008); D.Pavon, B. Wang, Gen. Rel. Grav. 41, 1 (2009); R. Silva, R. S.Goncalves, J. S. Alcaniz, H. H. B. Silva, Astr. and Astrophys.537, A11 (2012); H. H. B. Silva, R. Silva, R. S. Gonalves, Z.H. Zhu, J. S. Alcaniz, Phys. Rev. D 88, 127302 (2013); E. M.Barboza, R. C. Nunes, E .M. C. Abreu, J. A. Neto, Phys. Rev.D 92, 083526 (2015); J. E. Gonzalez, H. H. B. Silva, R. Silva,and J. S. Alcaniz, Eur. Phys. J. C 78, 730 (2018); M. Cruz,S. Lepe, and S. D. Odintsov, Phys. Rev. D 98, 083515 (2018)arXiv:1808.03825[gr-qc]; F. Contreras, N. Cruz, E. Elizalde, E.González, S. Odintsov arXiv:1808.06546[gr-qc].

[15] Ya. B. Zeldovich, Sov. Phys. JETP Lett. 12, 307 (1970); G. L.Murphy, Phys. Rev. D 8, 4231 (1973); B. L. Hu, Phys. Lett.A 90, 375 (1982); J. A. S. Lima and A. S. M. Germano, Phys.Lett. A 170, 373 (1992); W. Zimdahl, D. J. Schwarz, A. B.

12

0.90 0.95 1.00 1.05q

0.045

0.050

0.055

Ωb

0.24

0.27

0.30

0.33

Ωdm

−1.2

−0.9

ω

−0.16

0.00

0.16

ξ

0.64 0.66 0.68 0.70h

0.90

0.95

1.00

1.05

q

0.045 0.050 0.055Ωb

0.24 0.27 0.30 0.33Ωdm

−1.2 −0.9ω

−0.16 0.00 0.16ξ

Model Ib

ωCDM

Model III

Figure 3. Confidence regions and PDFs for the parameters h, Ωb, Ωdm, Ωk, ω, ξ and q, for all the models studied considering combining dataBAO + CMB + CC + SNe Ia.

Balakin, D. Pavon Phys. Rev. D, 64, 063501 (2001); S. No-jiri and S. D. Odintsov Phys. Rev. D 72, 023003 (2005); S.Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri, and S. D.Odintsov Phys. Rev. D 73, 043512 (2006); R. Colistete Jr., J. C.Fabris, J. Tossa, W. Zimdahl Phys. Rev. D, 76, 103516 (2007);I. Brevik, E. Elizalde, S. Nojiri, and S. D. Odintsov Phys. Rev.D 84, 103508 (2011); J. A. S. Lima, J. F. Jesus, F. A. Oliveira,J. Cosmol. Astropart. Phys. 1011, 027 (2010); S. Basilakos, J.A. S. Lima, Phys. Rev. D 82, 023504 (2010); W. S. Hipolito-Ricaldi, H. E. S. Velten, W. Zimdahl Phys. Rev. D, 82, 063507

(2010); J. F. Jesus, F. A. Oliveira, S. Basilakos, J. A. S. Lima,Phys. Rev. D 84, 063511 (2011); H. E. S. Velten, J. Wang, X.Meng Phys. Rev. D, 88 123504 (2013); T. Harko, Phys. Rev.D 90, 044067 (2014); S. Floerchinger, N.Tetradis and U. A.Wiedemann, Phys. Rev. Lett. 114, 091301 (2015); I. Brevik, O.Gron, J. de Haro, S. D. Odintsov and E. N. Saridakis, IJMPD26, 1730024 (2017); S. Anand, P. Chaubal, A. Mazumdar andS. Mohanty Journ. Cosmol. Astropart. Phys., 11, 005 (2017).

[16] D. Wang,Y.-J. Yan, X.-H. Meng, Eur. Phys. J. C 77, 660 (2017).[17] B. Mostaghel, H. Moshafi and S. M. S. Movahed, Eur. Phys.

13

0.8 1.0 1.2 1.4q

0.045

0.051

Ωb

0.25

0.30

Ωdm

−0.15

−0.05

0.05

Ωk

0.00

0.12

ξ

0.64 0.66 0.68 0.70 0.72h

0.8

1.0

1.2

1.4

q

0.045 0.051Ωb

0.25 0.30Ωdm

−0.15 −0.05 0.05Ωk

0.00 0.12ξ

Model Ic

ΛCDM

Model Ia

Model II

Figure 4. Confidence regions and PDFs for the parameters h, Ωb, Ωdm, Ωk, ω, ξ and q, for all the extended models studied consideringcombining data BAO + CMB + CC + SNe Ia.

J. C, 77, 541 (2017); B. Mostaghel, H. Moshafi and S. M. S.Movahed, Mon. Not. Roy. Astron. Soc.481,1799–1808 (2018).

[18] J.-S. Gagnon, J. Lesgourgues, Journ. Cosmol. Astropart. Phys.,09, 26 (2011).

[19] H. Amirhashchi, Phys. Rev. D 96, 123507 (2017).[20] A. Avelino, Y. Leyva, L. Arturo Ureña-López, Phys. Rev. D 88,

123004 (2013).[21] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy: Inter-

disciplinary Applications, Oxford University Press, New York,

(2004).[22] N. Komatsu, S. Kimura, Phys. Rev. D 88, 083534 (2013); N.

Komatsu and S. Kimura, Phys. Rev. D 87, 043531 (2013); N.Komatsu, S. Kimura, Phys. Rev. D 89, 123501 (2014); N. Ko-matsu and S. Kimura, Phys. Rev. D 90, 123516 (2014); N. Ko-matsu and S. Kimura, Phys. Rev. D 93, 043530 (2016)

[23] H. P. Oliveira, I. D. Soares, Phys. Rev. D 71, 124034 (2005).[24] T. S. Biro, V. G. Czinner, Phys. Lett. B 726, 861 (2013).[25] R. C. Nunes, et al. Journ. Cosm. Astrophs. Part. 08, 051 (2016);

14

E. M. Barboza Jr., R. C. Nunes, E. M. C. Abreu, J. AnaniasNeto, Physica A 436, 301 (2015); E. M. C. Abreu, J. Ana-nias Neto, A. C .R. Mendes, W. Oliveira, Physica A 392, 5154(2013); J.A. Neto, Physica A 391, 4320 (2012).

[26] A. Guha, J. Selvaganapathy and P. K. Das, Phys. Rev. D 95,015001 (2017); A. Guha and P. K. Das, J. High Energy Physics6, 1-18 (2018).

[27] N. Komatsu, Eur. Phys. J. C 77, 229 (2017); M.A. Zadeh, A.Sheykhi, H. Moradpour and K. Bamba, Phys. Lett. B 781, 195-200 (2018); A.S. Jahromi, S. A. Moosavi, H. Moradpour, J.P. M. Graca, I. P. Lobo, I. G. Salako and A. Jawad, Phys.Lett. B 780, 21-24 (2018); S. Ghaffari, H. Moradpour, I. P.Lobo, J. P. M. Graca and V. B. Bezerra Eur. Phys. J. C 78, 706(2018), arXiv:1807.04637 [gr-qc]; E. N. Saridakis, K. Bambaand R. Myrzakulov, Journ. Cosmol. Astropart. Phys. 1812, no.12, 012 (2018), arXiv:1806.01301 [gr-qc]; H. Moradpour, S. A.Moosavi, I. P. Lobo, J. P. M. Graca, A. Jawad and I. G. SalakoEur. Phys. J. C 78, 829 (2018), arXiv:1803.02195 [physics.gen-ph].

[28] V. G. Czinner and F. C. Mena, Phys. Lett. B 758, 9 (2016); V. G.Czinner and H. Iguchi, Eur. Phys. J. C 77, 892 (2017); A. Ma-jhi, Phys. Lett. B 775, 32 (2017); A. Lymperis and E. N. Sari-dakis, Eur. Phys. J. C 78, no. 12, 993 (2018), arXiv:1806.04614[gr-qc] (2018); R. D’Agostino, arXiv:1903.03836 [gr-qc]; S.Nojiri, S. D. Odintsov and E. N. Saridakis, Eur. Phys. J. C 79,no. 3, 242 (2019), arXiv:1903.03098 [gr-qc]; G. Varshney, U.K. Sharma and A. Pradhan, New Astron. 70, 36 (2019); M.Abdollahi Zadeh, A. Sheykhi, H. Moradpour and K. Bamba,arXiv:1901.05298 [physics.gen-ph].

[29] H. S. Gimenes, G. M. Viswanathan, R. Silva, Physica A 494,331 (2018).

[30] W. J. C. da Silva, H. S. Gimenes, and R. Silva, AstroparticlePhysics 105, 37-43 (2019), arXiv:1809.07797 [astro-ph.CO].

[31] T. Osada and G. Wilk Phys. Rev. C 77, 044903 (2008); ErratumPhys. Rev. C 78, 069903 (2008); T. Osada, G. Wilk, Indian J.Phys 85, 941 (2011).

[32] H. E. S. Velten and D. J. Schwarz, Journ. Cosm. Astrophs. Part.9, 016 (2011); H. E. S. Velten and D. J. Schwarz Phys. Rev. D86, 083501 (2012).

[33] R.-G. Cai, L-M. Cao, and N. Ohta, Phys. Rev. D 81, 061501(R)(2010).

[34] E. Verlinde, J. High Energy Physics 1104, 029 (2011).[35] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).[36] A. R. Plastino and J. A. S. Lima, Phys. Lett. A 260, 46 (1999).[37] R. Silva, A. R. Plastino and J. A. S. Lima, Phys. Lett. A 249,

401 (1998).[38] J. A. S. Lima, R. Silva, and A. R. Plastino, Phys. Rev. Lett. 86,

2938 (2001).[39] Z. B .B. de Oliveira and R. Silva, Ann. of Physics 375, 227

(2016); A. P. Santos, R. Silva, J. S. Alcaniz, J. A. S. Lima, Ann.of Physics 386, 158 (2017).

[40] S. Weinberg, The Astrophysical Journal, 168, 175–194 (1971).[41] R. Maartens, Class. Quantum Grav. 12, 1455 (1995).[42] W. Israel, Ann. Phys. (N.Y.) 100, 310 (1976); W. Israel and J.

M. Stewart, Ann. Phys. (N.Y:) 118, 341 (1979).[43] M. M. Disconzi, T. W. Kephart, R. J. Scherrer, Phys. Rev. D,

91, 043532 (2015).[44] B. Li, J. D. Barrow, Phys. Rev. D, 79, 103521 (2009).[45] D. Pavon, J. Bafaluy, D. Jou, Class. Quantum Grav., 8, 347

(1991).[46] R. Maartens, Causal thermodynamics in relativity, arXiv:astro-

ph/9609119 (third chapter).[47] D. Montenegro and G. Torrieri, Phys. Rev. D, 94, 065042

(2016).

[48] L. Rezzolla, O. Zanotti, Relativistic Hydrodynamics. OxfordUniv. Press, Oxford (2013).

[49] O. F. Piattella, J. C. Fabris, W. Zimdahl, JCAP, 1105, 029(2011).

[50] M. Cruz, N. Cruz and S. Lepe, Phys. Lett. B, 769, 159 (2017).[51] J. C. Fabris, S. V. B. Goncalves, R. de Sá Ribeiro, Gen. Relativ.

Gravit. 38, 495 (2006).[52] G. M. Kremer, F. P. Devecchi, Phys. Rev. D 67, 047301 (2003).[53] M. Cataldo, N. Cruz, S. Lepe, Phys. Lett. B 619, 5 (2005).[54] M. G. Hu, X. H. Meng, Phys. Lett. B 635, 186 (2006).[55] A. Sasidharan and T. K. Mathew, Eur. Phys. J. C 75, 348 (2015).[56] N. D. J. Mohana, A. Sasidharanb, T. K. Mathew, Eur. Phys. J.

C 77, 849 (2017).[57] W. A. Hiscock, J. Salmonson, Phys. Rev., D 43, 3249 (1991).[58] A. Sasidharan, N. D. Jerin Mohan, M. V. John, T. K. Mathew,

Eur. Phys. J. C, 78, 628 (2018).[59] C. Eckart, Phys. Rev. D 58, 919 (1940).[60] B. Santos, N. C. Devi, and J. S. Alcaniz, Phys. Rev. D 95,

123514 (2017).[61] M. A. Santos, M. Benetti, J. Alcaniz, F. A. Brito, and R. Silva,

Journ. Cosmol. Astropart. Phys. 1803, 023 (2018).[62] S. Santos da Costa, M. Benetti, and J. S. Alcaniz, Journ. Cos-

mol. Astropart. Phys. 1803, 004 (2018).[63] U. Andrade, C. A. P. Bengaly, J. S. Alcaniz, B. Santos, Phys.

Rev. D 97, 083518 (2018).[64] A. Cid, B. Santos, C. Pigozzo, T. Ferreira, J. S. Alcaniz,

arXiv:1805.02107 [astro-ph.CO].[65] R. Trotta, Contemp. Phys. 49, 71 (2008), arXiv:0803.4089

[astro-ph].[66] H. Jeffreys, Theory of Probability, 3rd ed. (Oxford Univ. Press,

Oxford, England, 1961).[67] J. Buchner et. al., Astron. and Astrophy., 564, A125 (2014).

https://github.com/JohannesBuchner/PyMultiNest.[68] F. Feroz and M. P. Hobson, Mon. Not. Roy. Astron. Soc. 384,

449 (2008); F. Feroz, M. P. Hobson, and M. Bridges, Mon.Not. Roy. Astron. Soc. 398, 1601 (2009); F. Feroz, M. P.Hobson, E. Cameron, and A. N. Pettitt, arXiv e-prints (2013),arXiv:1306.2144 [astro-ph.IM].

[69] J. Skilling, AIP Conf. Proc. 735, 395 (2004).[70] A. Lewis and S. Bridle, “GetDist”. https://github.

com/cmbant/getdist and http://cosmologist.info/cosmomc/readme_gui.html. Accessed July 2018.

[71] E. V. Linder and R. Miquel, International Journal of ModernPhysics D, 17, 12 (2008).

[72] D. J. Eisenstein et al. (SDSS Collaboration), The AstrophysicalJournal, 633, 2 (2005).

[73] D. J. Eisenstein and W. Hu, The Astrophysical Journal, 496,605 (1998).

[74] C. Blake et al., Mon. Not. R. Astron. Soc. 425, 405 (2012).[75] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-

Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson,Mon. Not. R. Astron. Soc. 416, 3017 (2011).

[76] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden,and M. Manera, Mon. Not. R. Astron. Soc. 449, 835 (2015).

[77] L. Anderson et al.(BOSS), Mon. Not. R. Astron. Soc. 441, 24(2014).

[78] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J.Cuesta, K. T. Mehta, and E. Kazin, Mon. Not. R. Astron. Soc.427, 2132 (2012).

[79] A. Kosowsky, M. Milosavljevic, R. Jimenez, Phys. Rev. D, 66,063007 (2002).

[80] Y. Wang, P. Mukherjee, Phys. Rev. D, 76, 103533 (2007).[81] G. Efstathiou, J. Bond, Mon. Not. Roy. Astron. Soc., 304, 75

(1999).

15

[82] P. Mukherjee, M. Kunz, D. Parkinson, Y. Wang, Phys. Rev. D,78, 083529 (2008).

[83] W. Hu, N. Sugiyama, The Astrophysical Journal, 471, 542(1996).

[84] R. Jimenez and A. Loeb, Astrophys. J. 573, 37 (2002).[85] G. Bruzual and S. Charlot, Mon. Not. Roy. Astron. Soc. 344,

1000 (2003).[86] J. Simon, L. Verde, and R. Jimenez, Phys. Rev. D 71, 123001

(2005).[87] L. Verde, P. Protopapas, and R. Jimenez, Phys. Dark Univ. 5-6,

307 (2014).[88] M. Moresco et al., Journ. Cosmol. Astropart. Phys. 1208, 006

(2012).[89] E. Komatsu et al. (WMAP), Astrophys. J. Suppl. 192, 18 (2011)[90] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D.

Serpico, Nucl. Phys. B 729, 221 (2005).[91] D. M. Scolnic et al, The Astrophysical Journal, 859, 101 (2018).[92] J. Marriner, J. P. Bernstein, R. Kessler et al., Astrophys. J., 740,

72 (2011).[93] A. Conley et al. The Astrophysical Journal Supplement Series

192, 1 (2010).[94] R. R. Caldwell, M. Kamionkowski, and N. N. Weinberg, Phys.

Rev. Lett., 91, 071301 (2003).