for review only · for review only reconstructing the state nder hierarchy for an interacting...

For Review OnlyReconstructing the statefinder hierarchy for an interacting

cosmology

Journal: Canadian Journal of Physics

Manuscript ID cjp-2019-0021.R2

Manuscript Type: Article

Date Submitted by the Author: 02-Jul-2019

Complete List of Authors: Cueva Solano, Freddy; IFM-UMSNH, Sciciences

Keyword: dark sector, interaction, observational data, geometrical parameters, diagnostic method

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

For Review Only

Reconstructing the statefinder hierarchy for an interacting cosmology

Freddy Cueva SolanoInstituto de Fısica y Matematicas, Universidad Michoacana de San Nicolas de Hidalgo

Edificio C-3, Ciudad Universitaria, CP. 58040, Morelia, Michoacan, Mexico.∗

(Dated: August 26, 2019)

In this letter, we study an interacting cosmology in a spatially flat universe. Here, we havereconstructed the interaction term, Q, between a cold dark matter (DM) fluid and a dark energy(DE) fluid, as well as a time-varying equation of state (EoS) parameter ωDE, and have exploredits evolutionary trajectories in the second members of the statefinder hierarchy, allowing us toestablish differences when they are compared with those obtained in other non-interacting models.Likewise, both Q and ωDE have been modeled in terms of the Chebyshev polynomials. Then,via a Markov-Chain Monte Carlo (MCMC) method, we have constrained the model parameterspace by using a combined analysis of geometric data. Our results show that an interacting modelcan be distinguished well from the standard concordance model at the present time by using anycombination of the second members of the statefinder hierarchy. In this way, different DE scenarioscould be compared and distinguished among them, by using this method.

PACS numbers: 04.20.Cv, 95.36.+x, 98.80.Es,98.80.Jk

I. INTRODUCTION

In the last years different cosmological observations [1–32] have provided strong evidence that the present uni-verse is experiencing an accelerated expansion, which hasbecome one of the most important discovery of the Cos-mology. In order to explain this phenomenon a great va-riety of theoretical models have been proposed. There aremainly two approaches: On the other one, is to assumethe existence of an unknown energy component dubbedDE [33], which is uniformly distributed in the universe,and with negative pressure. This energy has been inter-preted in various forms and widely studied in [34]. Hence,the cosmological constant Λ, is the simplest candidate ofDE. Although, this scenario is physically consistent withthe observational data, it presents anomalies such as thefine-tuning problem and the coincidence problem, respec-tively.On the other hand, another approach is based on a modi-fication of general relativity called modified gravity (MG)[35]. Due to the degeneracy between the space of param-eter and the cosmic expansion, it is difficult to decidewhich above explanation is correct.In this work, we only focus on DE model of the firsttype and will use a new generation of diagnostic parame-ters called statefinder hierarchy, which were constructedto distinguish different DE models and obtain informa-tion about the expansion of the universe. This hierarchyhas been widely used in [36–42]. In the literature, dif-ferent members of statefinder hierarchy have been pro-posed to analyze a possible deviation from a given DEmodel to the standard model. It will be a useful methodto differentiate kinds of dynamical DE models and un-der the assumption that the Friedmann - Robertson -

∗ [email protected], [email protected]

Walker (FRW) metric is still valid. On the other hand,an interacting DE model (IDE) with two different casesis discussed here. Due to the lacking of an underlyingtheory for construct a general term of interaction, Q,between the dark sectors, different ansatzes have beenwidely discussed in [43–48]. So, It has been shown in DEscenarios that Q can affect the background expansionhistory of the universe and could very possibly introducenew features on the evolutionary trajectories of the sec-ond member of the statefinder hierarchy. In the presentwork, we have attempted phenomenological descriptionsforQ and ωDE, by expanding them in terms of the Cheby-shev polynomials Tn, defined in the interval [−1, 1] andwith a divergence-free ωDE at z → −1 [49, 50]. However,that polynomial base was particularly chosen due to itsrapid convergence and better stability than others, bygiving minimal errors [24, 51]. Besides, Q could also beproportional to the DM energy density ρDM and to theHubble parameter H. Here, Q will be restricted from thecriteria exhibit in [52].The focus of this paper is to investigate the effects ofQ and ωDE on the evolutionary trajectories of the IDEin the different statefinder hierarchy planes, and com-pare them with the respective results obtained in non-interacting models.To constrain the parameter spaces of our models, breakthe degeneracy of their parameters and put tighter con-straints on them, we use an analysis combined of JointLight Curve Analy-sis (JLA) type Ia Supernovae (SNeIa) data [1–3], including with Baryon Acoustic Oscilla-tion (BAO) data [5–16], together the Planck distance pri-ors of the Cosmic Microwave Background (CMB) data,[4, 20–22] and the Hubble parameter (H) data obtainedfrom galaxy surveys [23–32].The paper is organized as follows. In Sec. II, wehave described the phenomenological model consideredhere. The reconstruction of cosmological variables usingChebyshev polynomials up to order N = 4 is presented

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in Sec. III. In Sec. IV, we have introduced the differentkinds of statefinder hierarchy. In Sec. V, we show thenecessary condition to avoid the crossing of the phantomdivide line ωDE = −1. In the Sec. VI, we provides adescription of the constraint method and observationaldata. We discuss the results obtained in Sec. VII. Fi-nally, we have summarized the conclusions in Sec. VIII.

II. INTERACTING DARK ENERGY (IDE)MODEL

We assume a spatially flat FRW universe, composedwith four perfect fluids-like, radiation (subscript r),baryonic matter (subscript b), DM and DE, respec-tively. Moreover, we postulate the existence of a non-gravitational coupling in the background between DMand DE (so-called dark sector) and two decoupled sec-tors related to the b and r components, respectively.We also consider that these fluids have EoS parametersPA = ωAρA, A = b, r,DM,DE, where PA and ρA are thecorresponding pressures and the energy densities. Here,we choose ωDM = ωb = 0, ωr = 1/3 and ωDE is a time-varying function. Therefore, the balance equations of ourfluids are respectively,

dρbdz− 3Hρb = 0, (1)

dρrdz− 4Hρr = 0, (2)

dρDMdz

− 3ρDM(1 + z)

= − Q

H(1 + z), (3)

dρDEdz

− 3(1 + ωDE)ρDE(1 + z)

= +Q

H(1 + z), (4)

where the differentiation has been done with respect tothe redshift, z, H denotes the Hubble expansion rateand the quantity Q expresses the interaction between thedark sectors. For simplicity, it is convenient to definethe fractional energy densities ΩA ≡ ρA

ρc, Ω?A ≡

ρAρc,0

and

ΩA,0 ≡ ρA,0

ρc,0, where the critical density ρc ≡ 3H2/8πG

and the critical density today ρc,0 ≡ 3H02/8πG being

H0 = 100hKms−1Mpc−1 the current value of H.Likewise, we have taken the relation

∑A ΩA,0 = 1. Here,

the subscript “0” indicates the present value of the quan-tity.In this work, we consider the spatially flat FRW metricwith line element

ds2 = −dt2 + a2(t)δijdxidxj , (5)

where t represents the cosmic time and “a” represents thescale factor of the metric and it is defined in terms of theredshift z as a = (1 + z)−1, from which one can find therelation of H and the cosmic time dt/dz = −1/(1 + z)H.Then, we analyze the ratio between the energy densitiesof DM and DE, defined as R ≡ ρDM/ρDE. From Eqs. (3)

and (4), we obtain [52, 53]

dR

dz=−R

(1 + z)

(3ωDE +

(1 + R)Q

HρDM

). (6)

This Eq. leads to

Q = −(

3ωDE +dR

dz

(1 + z)

R

)HρDM1 + R

. (7)

Due to the fact that the origin and nature of the darkfluids are unknown, it is not possible to derive Q fromfundamental principles. However, we have the freedomof choosing any possible form of Q that satisfies Eqs. (3)and (4) simultaneously. Hence, we propose a phenomeno-logical description for Q as a linear combination of ρDM,H and a time-varying function IQ,

Q ≡ HρDMIQ, IQ ≡N∑n=0

λnTn, (8)

where Tn denotes the Chebyshev polynomials of ordern with n ∈ [0, N ] and N is a positive integer. Here,IQ is defined in terms of Tn and λn are constant andsmall |λn| 1 dimensionless parameters. This poly-nomial base was chosen because it converges rapidly, ismore stable than others and behaves well in any polyno-mial expansion, giving minimal errors [46].On the other hand, the phenomenological ansatz pro-posed for ωDE in [49], is very useful for exploring theproperty of DE with observational data [4]. This ansatzcan explain successfully the past dynamical evolution ofDE but has a problem that ωDE diverges in the far future(z → −1), namely, cannot describe the future evolutionof DE. This is not a physical feature. To avoid such afuture divergence problem [49], in the present article wepropose a novel reconstruction for ωDE in function of theChebyshev polynomials, to explore the whole dynamicalevolution of DE and find certain predictions for the ulti-mate destiny of the universe. This novel ansatz is

ωDE ≡ ω2 + 2N∑m=0

ωmTm2 + z2

. (9)

where Tm denotes the Chebyshev polynomials of order mwith m ∈ [0, N ] and N is a positive integer, ωm are freedimensionless parameters. The polynomial (2+z2)−1 andthe parameter ω2 were included conveniently to simplifythe calculations. Thus, a possible physical descriptionshould be explored.In order to guarantee that Q may be physically accept-able in the dark sectors [52], we equal the right-hand sidesof Eqs. (7) and (8), which becomes

dR

dz=−R

(1 + z)

(IQ(1 + R) + 3ωDE

). (10)

Now, to solve or alleviate of coincidence problem, we re-quire that R tends to a fixed value at late times. This

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leads to the condition dR/dz = 0, which therefore im-plies two stationary solutions R+ = R(z → ∞) =−(1 + 3ωDE/IQ) and R− = R(z → −1) = 0, The firstsolution occurs in the past and the second one happensin the future.

III. RECONSTRUCTION OF COSMOLOGICALVARIABLES USING CHEBYSHEV

POLYNOMIALS UP TO ORDER N = 4

To simplify our analysis and show how the methodworks, we do the reconstruction of IQ, ωDE in terms ofChebyshev polynomials up to order N = 4; namely, thefirst five Chebyshev polynomials, which are

T0(z) = 1, T1(z) = z, T2(z) = (2z2 − 1),

T3(z) = 4z3 − 3z, T4(z) = 8z4 − 8z2 + 1. (11)

Then, by inserting Eqs. (8), (9) and (11) into Eq. (10),we find that R has no analytical solution, and must besolved numerically. Likewise, from Eqs. (1) and (2),there are an analytical solution for just ρb and ρr, respec-tively. Similarly, ρDM will be reconstructed up to orderN = 4 from Eqs. (3), (8) and (11), respectively. More-over, ρDE will be obtained from R, as ρDE = ρDM/R.Therefore, the first Friedmann equation is given by

E2 =H2

H20

= Ωb,0(1 + z)3+Ωr,0(1 + z)

4+Ω?DM(z)(1+R−1)

(12)

in where Ω?DM have been expanded in terms of the firstfive Chebyshev polynomials, as

Ω?DM(z) = (1 + z)3ΩDM,0exp

[−zmax

2

4∑n=0

λnIn(z)

],∫ z

0

Tn(x)

(1 + x)dx ≈ zmax

2

∫ x

−1

Tn(x)

(a1 + a2x)dx ≡ zmax

2In(z),

x ≡ 2z

zmax− 1, a1 ≡ 1 +

zmax2

, a2 ≡zmax

2,

I0(z) =2

zmaxln(1 + z),

I1(z) =2

zmax

(2z

zmax− (2 + zmax)

zmaxln(1 + z)

),

I2(z) =2

zmax

[4z

zmax

(z

zmax− 2

zmax− 2

)+(

1 +6.8284

zmax

)(1 +

1.1716

zmax

)ln(1 + z)

],

I3(z) =2

zmax

[1

z3max

(10.6666[(z − 0.75)2 + 2.4375]−

(2 + zmax)(zmax + 14.9282)(zmax + 1.0718) ln(1 + z)

−6zzmax(4z − 3zmax − 8)

)],

I4(z) =2

zmax

[8

z4max

[(z − 2zmax)(2z + zmax)(2z − zmax)

+5zz2max

]− 16z(2 + zmax)

z4max

[1.3333

((z + 1.5)2 + 0.75

)+3(zmax + 2)(zmax − z)

]− 16z

z2max

(z − 2zmax − 2

)+

(1− 8(1 +

2

zmax)2 + 8(1 +

2

zmax)4

)ln(1 + z)

],

where zmax is the maximum value of z such that x ∈[−1, 1] and |Tn(x)| ≤ 1 and n ∈ [0, 4] [46].From Eqs. (8) and (11) an asymptotic value for IQ canbe found: IQ → ∞ for z 1, IQ = λ0 − λ2 + λ4 forz = 0 and IQ ≈ λ0 − λ1 + λ2 − λ3 + λ4 for z → −1.Similarly, using Eqs. (9) and (11), the following asymp-totic values can be obtained for ωDE: ωDE ≈ 5ω2 forz 1, ωDE ≈ ω0 + ω4 for z = 0 and ωDE ≈ (5/3)ω2 +(2/3)[ω0 − ω1 − ω3 + ω4] for z → −1.For a better analysis, we have compared the IDE modelwith other possible cosmological models. Thus, if Q(z) =0 and ωDE = −1 in Eq. (12) the standard ΛCDM modelis recovered. Similarly, when Q(z) = 0 and ωDE is givenby Eq. (9), the ωDE model is obtained. These non-interacting models have an analytical solution for R.

IV. STATEFINDER HIERARCHY

The satefinder hierarchy was introduced initially by[36–42] and represents an alternative method to differen-tiate kinds of cosmological models. Using this methoddivers papers have shown that the standard ΛCDMmodel can be distinguished from other models such asquintessence [54, 55], holographic DE model [42, 56],Ricci DE model [57, 58], Agregating DE models [59],Generalized Chaplyging Gas model [60], DGP branemodel generalized [61], quintom model [62], galileon mod-els [63], interacting DE models [64] etc. For this reason,we perform a Taylor series expansion of the scale factoraround the current epoch, t0, with t− t0 > 0,

a(t) = 1 +∞∑n=1

Ann!

[H0(t− t0)]n, An =an(t)

a(t)Hn, (13)

where an(t) = dna(t)/dtn and A1 = 1. Here, the su-perscript “n” indicate the derivatives with respect to thecosmic time. From here, we have adopted the followingconvention for the coefficients, An [65–67]

A2 = −q, A3 = −j,A4 = s, A5 = l, A6 = m. (14)

These functions are usually denominated as deceleration,jerk, snap, lerk and merk parameters, respectively.In general, for a DE model with EoS parameter ωDE andR, it is convenient to convert the derivatives of the aboveequation from time to redshift, and then, combine those

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functions among themselves, obtaining

A2 = −3ωDE + 1 + R

(1 +

Ωb,0+2Ωr,0

ΩDM,0

)2

[1 + R

(1 +

Ωb,0+Ωr,0

ΩDM,0

)] ,

A3 = q + 2q2 + (1 + z)q′,

A4 = −j(2 + 3q)− (1 + z)j′,

A5 = (1 + z)

[(1 + z)j′′ + (3q′ + (6 + 7q)′

]+ j(2 + 3q)(3 + 4q),

A6 = −(1 + z)l′ − (4 + 5q)l. (15)

where ′ denotes the derivatives with respect to z.Arabsalmani et al [37]; proposed a new generation ofdiagnostic parameters so-called statefinder hierarchy todiscriminate the different DE models. Then, the originalstatefinder hierarchy, Sn, can be defined as

S2 = A2 +3

2ΩM , S3 = A3, S4 = A4 +

9

2ΩM

S5 = A5 − 3ΩM −9

2Ω2M . (16)

For the ΛCDM model, this statefinder hierarchy, Sn, sat-isfies Sn|ΛCDM = 1.Furthermore, from Eqs. (13) and (16), the originalstatefinder hierarchy can be improved sequentially, by

introducing another series of statefinder S(1)n with n ≥ 3,

as follows

S(1)3 = A3, S

(1)4 = A4 + 3(1 + q),

S(1)5 = A5 − 2(4 + 3q)(1 + q). (17)

where the superscript “(1)” is used to distinguish be-

tween S(1)n and Sn. From the above equation, one

can verify that the statefinder hierarchy S(1)n , satisfies

S(1)n |ΛCDM = 1 for n ≥ 3.

Thus, for the ΛCDM model we obtain the following prop-

erty, i. e; Sn, S(1)n |ΛCDM = (1, 1), for n ≥ 3. The val-

ues of this pair will change in other DE models.From Eq. (17), one can construct a second member ofstatefinder hierarchy in the following manner [37]

S(2)n =

S(1)n − 1

3(q − 0.5), ∀n ≥ 3. (18)

From the above equations, for the ΛCDM model we can

find the following properties: Sn, S(2)n |ΛCDM = (1, 0),

and also S(1)n , S

(2)n |ΛCDM = (1, 0) with n ≥ 3. For

other DE models one will get different results.

In this work, we will take the planes S(1)n , S

(1)m ,

S(2)n , S

(2)m , S(1)

n , S(2)m , R, S(1)

n and R, S(2)n with

n,m = 3, 4, 5, to illustrate the effects of the reconstruc-tions of Q and ωDE on the evolutionary trajectories ofthe different members of the statefinder hierarchy.

V. NON CROSSING THE PHANTOM DIVIDELINE ωDE = −1

In this section, we now study the behavior of ωDE forthe present models, indicating the possibility of the noncrossing of the phantom divide line ωDE = −1.Taking the derivative of both sides of Eqs. (12), usingEqs. (1)-(4) and the definition of the fractional energydensities, one can get

ωDE =1

ΩDE

[2

3

(1 + z)

H

dH

dz− 1− Ωr

3

]. (19)

The above equation shows that for the phantom domi-nated universe ωDE < −1, one needs to have dH

dz < 0,for z 6= −1. Similarly, for the quintessence dominateduniverse ωDE > −1, we need to have dH

dz > 0 for z 6= −1.Then, the deceleration parameter is defined by

q = −A2 =

((1 + z)

H

dH

dz− 1

). (20)

To explore the possibility that ωDE does not cross −1, wereplace Eq. (19) into Eq. (20), and using the conditionωDE > −1, for an universe dominated by DE, we find

q > −1. (21)

Substituting −A2 gives by Eq. (15) into Eq. (21) andconsidering that Ωb,0, Ωr,0, ΩDM,0 > 0, we obtain

1 + ωDE > −R

(1 +

4

3

Ωr,0

ΩDM,0+

Ωb,0

ΩDM,0

). (22)

In the ωDE and IDE models, the phantom divide lineωDE = −1 cannot be crossed when 1+ωDE > 0 (necessarycondition). So, to fulfill such requirement, Eq. (22) leadsto the following condition (enough condition)

R < 0, (23)

which implies that if ρDM > 0 then ρDE < 0 is requiredto ensure the positivity of 1+ωDE. Now, we remark thatthe above condition, is enough for ωDE. Thus, ωDE willnever cross −1. By contrast, in the present models, thephantom divide line can be crossed, when 1 + ωDE < 0,i.e, for R > 0 with ρDE > 0.The results of Eqs. (21) and (23) are in agreement withthose found in [68–70]. In where, the authors show thatthe phantom divide line is crossed only when ρDE > 0.

VI. CONSTRAINT METHOD ANDOBSERVATIONAL DATA

A. Constraint method

Here, to simplify our analysis and show how themethod works, we do the reconstruction of IQ, ωDE

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and Ω?DM in terms of Chebyshev polynomials up to or-der N = 4; namely, the first five Chebyshev poly-nomials and constrain the parameter spaces of thepresent models through a modified MCMC method[71]. There are three statistical analyses that we havedone to calculate the best-fit parameters: The firstwas done on a non-interacting model so-called ΛCDMwith six parameters P1 = (ΩDM,0,H0, α, β,M,dM),the second was also made on a non-interacting sce-nario denominated ωDE model with eleven param-eters P2 = (ω0, ω1, ω2, ω3, ω4,ΩDM,0,H0, α, β,M,dM)and an interacting model with sixteen parametersP3 = (λ0, λ1, λ2, λ3, λ4, ω0, ω1, ω2, ω3, ω4,ΩDM,0,H0,α, β,M,dM). Furthermore, the constant priors for themodel parameters were: λ0 = [−1.5×10+2 + 1.5×10+2],λ1 = [−1.5 × 10+2,+1.5 × 10+2], λ2 = [−15,+15],λ3 = [−5,+5], λ4 = [−3,+3], ω0 = [−2.0,−0.3],ω1 = [−1.0,+1.0], ω2 = [−2.0,+0.1], ω3 = [−0.5,+0.1],ω4 = [−0.4,+0.1], ΩDM,0 = [0, 0.7], H0 = [20, 120],α = [−0.2,+0.5], β = [+2.1,+3.8], M = [−20,−17],dM = [−1.0,+1.0]. We have also fixed Ωr,0 = Ωγ,0(1 +0.2271Neff ), where Neff represents the effective num-ber of neutrino species. So, Neff = 3.04 ± 0.18, Ωγ,0 =2.469×10−5h−2 and Ωb,0 = 0.02230h−2 were chosen fromTable 4 in [4].

B. Observational data

To test the viability of our models and set constraintson the model parameters, we use the following data sets:• The Supernovae (SNe Ia) data: We used the JoinAnalysis Luminous (JLA) [1–3] data composed by 740SNe Ia with hight-quality light curves, which includesamples from z < 0.1 to 0.2 < z < 1.0.The observed distance modulus is modeled by [1–3]

µJLAi = m∗B,i+αx1,i−βCi−M−dM, 1 ≤ i ≤ 740, (24)

where and the parameters m∗B , x1 and C describe the in-trinsic variability in the luminosity of the SNe. Further-more, the nuisance parameters α, β, M and dM charac-terize the global properties of the light-curves of the SNeand are estimated simultaneously with the cosmologicalparameters of interest. Then, the theoretical distancemodulus is

µth(z) ≡ 5log10

[DL(z)

Mpc

]+ 25, (25)

where “th” denotes the theoretical prediction for a SNeat z. The luminosity distance DL(z), is defined as

DL(zhel, zCMB) = (1 + zhel)c

∫ zCMB

0

dz′

H(z′), (26)

where zhel is the heliocentric redshift, zCMB is the CMBrest-frame redshift, c = 2.9999× 105km/s is the speed of

the light. Thus,

µth(zhel, zCMB) = 5 log10

[(1 + zhel

∫ zCMB

0

dz′

E(z′)

]+52.385606− 5 log10(H0). (27)

Then, the χ2 distribution function for the JLA data is

χ2JLA = (∆µi)

t (C−1

Betoule

)ij

(∆µj

), (28)

where ∆µi = µthi −µJLAi is a column vector and C−1Betoule

is the 740× 740 covariance matrix [3].

• Baryon Acoustic Oscillation (BAO) data:The BAO distance measurements can be used toconstrain the distance ratio dz(z) = rs(zd)

DV (z) at differ-

ent redshifts, obtained from different surveys [5–16]listed in Table I. Here, rs(zd) is the comoving soundhorizon size at the baryon drag epoch zd, where thebaryons were released from photons and has beencalculated by [17]. Moreover, the dilation scale is

defined as Dv(z) ≡ 1H0

[(1 + z)2DA

2(z) czE(z)

]1/3, where

DA(z) = c∫ z

0dz′

H(z′) is the angular diameter distance.

Thus, the χ2 is given as

χ2BAO I =

17∑i=1

(dthz (zi)− dobs

z (zi)

σ(zi)

)2

. (29)

z dobsz σz Refs. z dobsz σ Refs.

0.106 0.3360 ±0.0150 [5, 6] 0.350 0.1161 ±0.0146 [13]0.150 0.2232 ±0.0084 [7] 0.440 0.0916 ±0.0071 [9]0.200 0.1905 ±0.0061 [8, 9] 0.570 0.0739 ±0.0043 [14]0.275 0.1390 ±0.0037 [8] 0.570 0.0726 ±0.0014 [11]0.278 0.1394 ±0.0049 [10] 0.600 0.0726 ±0.0034 [9]0.314 0.1239 ±0.0033 [9] 0.730 0.0592 ±0.0032 [9]0.320 0.1181 ±0.0026 [11] 2.340 0.0320 ±0.0021 [15]0.350 0.1097 ±0.0036 [8, 9] 2.360 0.0329 ±0.0017 [16]0.350 0.1126 ±0.0022 [12]

TABLE I. Summary of BAO I data [5–16].

• Cosmic Microwave Backgroung data: We usethe Planck distance priors data extracted from Planck2015 results XIII Cosmological parameters, for the com-bined analysis TT, TF, FF + lowP + lensing [4, 22].From here, we have obtained the values of the shift pa-rameter R(z∗), the angular scale for the sound horizonat photon-decoupling epoch, lA(z∗), and the redshift atphoton-decoupling epoch, z∗. Then, the shift parameterR is defined by [20]

R(z∗) ≡√

ΩM,0

∫ z∗

0

dy

E(y), (30)

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6

z H(z) 1σ Refs. z H(z) 1σ Refs.

0.070 69.0 ±19.6 [23] 0.480 97.0 ±62.0 [30]0.090 69.0 ±12.0 [24] 0.570 87.6 ±7.80 [14]0.120 68.6 ±26.2 [23] 0.570 96.8 ±3.40 [11]0.170 83.0 ±8.0 [24] 0.593 104.0 ±13.0 [25]0.179 75.0 ±4.0 [25] 0.600 87.9 ±6.1 [29]0.199 75.0 ±5.0 [25] 0.680 92.0 ±8.0 [25]0.200 72.9 ±29.6 [23] 0.730 97.3 ±7.0 [29]0.240 79.69 ±2.99 [27] 0.781 105.0 ±12.0 [25]0.270 77.0 ±14.0 [24] 0.875 125.0 ±17.0 [25]0.280 88.8 ±36.6 [23] 0.880 90.0 ±40.0 [30]0.300 81.7 ±6.22 [28] 0.900 117.0 ±23.0 [24]0.340 83.8 ±3.66 [27] 1.037 154.0 ±20.0 [27]0.350 82.7 ±9.1 [13] 1.300 168.0 ±17.0 [24]0.352 83.0 ±14.0 [25] 1.363 160.0 ±33.6 [31]0.3802 83.0 ±13.5 [26] 1.430 177.0 ±18.0 [24]0.400 95.0 ±17.0 [24] 1.530 140.0 ±14.0 [24]0.4247 87.1 ±11.2 [26] 1.750 202.0 ±40.0 [24]0.430 86.45 ±3.97 [27] 1.965 186.5 ±50.4 [31]0.440 82.6 ±7.8 [29] 2.300 224.0 ±8.6 [32]0.4497 92.8 ±12.9 [26] 2.340 222.0 ±8.5 [15]0.4783 80.9 ±9.0 [26] 2.360 226.0 ±9.3 [16]

TABLE II. Shows the H(z) data [11, 13–16, 23–32]

where E(y) is given by Eq. (12) and the redshift z∗ isobtained from [21]

z∗ = 1048

[1+0.00124(Ωb,0h

2)−0.738

][1+g1(ΩM,0h

2)g2],

(31)where

g1 =0.0783(Ωb,0h

2)−0.238

1 + 39.5(Ωb,0h2)0.763, g2 =

0.560

1 + 21.1(Ωb,0h2)1.81.

(32)The angular scale lA for the sound horizon is

lA ≡πDA(z∗)

rs(z∗), (33)

where rs(z∗) is the comoving sound horizon at z∗. From[4, 22], the χ2 is

χ2CMB = (∆xi)

t (C−1

CMB

)ij

(∆xj) , (34)

where ∆xi = xthi − xobs

i is a column vector

xthi (X)− xobs

i =

lA(z∗)− 301.7870R(z∗)− 1.7492z∗ − 1089.990

, (35)

“t” denotes its transpose and (C−1CMB)ij is the inverse

covariance matrix [22] given by

C−1CMB ≡

+162.48 −1529.4 +2.0688−1529.4 +207232 −2866.8+2.0688 −2866.8 +53.572

. (36)

• Hubble observational data: This sample is com-posed by 42 independent measurements of the Hubbleparameter at different redshifts and were derived fromdifferential age dt for passively evolving galaxies with red-shift dz and from the two-points correlation function ofSloan Digital Sky Survey. This sample was taken fromTable III in [26, 72]. Then, the χ2

H function for this dataset is [72]

χ2H ≡

42∑i=1

[Hth(zi)−Hobs(zi)

]2σ2(zi)

, (37)

where Hth denotes the theoretical value of H, Hobs rep-resents its observed value and σ(zi) is the error.In order to put constraints on the model parameters, we

have calculated the overall likelihood Lα e−χ2/2, where

χ2 can be defined by

χ2 = χ2JLA + χ2

BAO + χ2CMB + χ2

H . (38)

VII. RESULTS.

In this work, we have run eight chains for each of thethree models proposed on the computer, and the best-fit parameters with 1σ and 2σ errors, are presented inTables III, IV and V, respectively. From these tables,we can note the fast convergence of the best estimatedwhen the number of Chebyshev polynomials is increasedin the expansions given by Eq. (8) and Eq. (9), respec-tively. Moreover, we also see that the corresponding χ2

min

for the IDE model becomes smaller in comparison withthose obtained in the non-interacting models.The one-dimension probability contours with 1σ and 2σerrors on each parameter of the present models and ob-tained from the combined constraint of geometric data,are plotted in Fig. 1.Due to the two minimums obtained in the IDE model(see Tables III, IV and V), we consider now two differ-ent cases to reconstruct IQ: the case 1 is so-called IDE1with λ2 > 0; by contrast, the case 2 is dubbed IDE2 withλ2 < 0.The evolution of ωDE with respect to redshift and withinthe 1σ error around the best-fit curve for the presentmodels, is presented in the left upper panel of Fig. 2.From here, one can see that in the ωDE and IDE models,the universe evolves from the phantom regime ωDE < −1to the quintessence regime ωDE > −1, and then it be-comes phantom again. Moreover, ωDE crosses the phan-tom divide line −1 [73] twice. In particular, the IDE1(IDE2) case has two crossing points in the 1σ confidenceregion in z = +0.0591+0.1024

−0.0536 (z = +0.0633+0.1038−0.0555) and

z = +1.1794+0.9290−0.3137 (z = +1.1784+0.9288

−0.3163), respectively.Analogously, for the ωDE model these points are respec-tively z = +0.0337+0.1970

−0.0535 and z = +0.5024+0.2827−0.2730. Such

a crossing feature is favored by the data within 1σ error.

Page 6 of 15



For Review Only

7

Parameters ΛCDM ωDE IDE1 IDE2

λ0 × 10+4 N/A N/A +1.120+0.6569+2.1166−0.5709−1.4348 +1.120+0.6569+2.1166

−0.5709−1.4348

λ1 × 10+4 N/A N/A +2.733+0.4108+0.7581−0.5648−1.9464 +2.733+0.4108+0.7581

−0.5648−1.9464

λ2 × 10+5 N/A N/A +2.7112+0.8670+1.6127−3.5277−4.6376 −2.6490+0.5032+1.0396

−1.1769−2.8069

ω0 −1.0 −1.0364+0.0648+0.1140−0.0865−0.1910 −1.0730+0.0644+0.1139

−0.0863−0.1906 −1.0773+0.0645+0.1136−0.0863−0.1918

ω1 N/A +1.150+0.2377+0.5849−0.1230−0.1965 +1.2941+0.2377+0.5848

−0.1230−0.1965 +1.2969+0.2359+0.5849−0.1239−0.1964

ω2 N/A −1.0546+0.1283+0.4803−0.0381−0.0741 −0.6179+0.1281+0.4802

−0.0382−0.0742 −0.6179+0.1281+0.4802−0.0382−0.0742

ΩDM,0 +0.2812+0.0185+0.0478−0.0140−0.0278 +0.2844+0.0118+0.0386

−0.0062−0.0125 +0.2844+0.0118+0.0386−0.0062−0.0125 +0.2844+0.0118+0.0386

−0.0062−0.0125

Ωb,0 +0.0493+0.0018+0.0037−0.0020−0.0040 +0.0494+0.0012+0.0025

−0.0014−0.0030 +0.0494+0.0012+0.0025−0.0014−0.0030 +0.0494+0.0012+0.0025

−0.0014−0.0030

H0 +67.10+1.3038+2.6882−1.3336−2.5704 +67.1480+0.8092+1.8055

−0.9875−1.9252 +67.1487+0.8085+1.8048−0.9882−1.9259 +67.1487+0.8085+1.8048

−0.9882−1.9259

α +0.1360+0.0418+0.0848−0.0403−0.0810 +0.1360+0.1103+0.2289

−0.1205−0.2483 +0.1360+0.1103+0.2289−0.1205−0.2483 +0.1360+0.1103+0.2289

−0.1205−0.2483

β +3.060+0.1031+0.2083−0.1058−0.2069 +3.0780+0.1964+0.3917

−0.1858−0.3708 +3.0780+0.1964+0.3917−0.1858−0.3708 +3.0780+0.1964+0.3917

−0.1858−0.3708

M −19.0324+0.3796+0.7651−0.3888−0.7769 −19.0880+0.5637+1.1179

−0.5527−1.1042 −19.0631+0.5647+1.1099−0.5517−1.0970 −19.0631+0.5647+1.1099

−0.5517−1.0970

dM −0.124+0.2774+0.5409−0.2721−0.5338 −0.1230+0.3711+0.7483

−0.3832−0.7503 −0.1230+0.3711+0.7483−0.3832−0.7503 −0.1230+0.3711+0.7483

−0.3832−0.7503

χ2min 719.8072 707.8162 704.0342 703.2585

TABLE III. Shows the best-fit values of the cosmological parameters for the three models with 1σ and 2σ errors, using Chebyshevpolynomials up to order N = 2 for IQ, ωDE in the ωDE y IDE models, respectively.


λ0 × 10+4 N/A N/A +1.120+0.6569+2.1166−0.5709−1.4348 +1.120+0.6569+2.1166

−0.5709−1.4348

λ1 × 10+4 N/A N/A +2.733+0.4108+0.7581−0.5648−1.9464 +2.733+0.4108+0.7581

−0.5648−1.9464

λ2 × 10+5 N/A N/A +2.7112+0.8670+1.6127−3.5277−4.6376 −2.6490+0.5032+1.0396

−1.1769−2.8069

λ3 × 10+7 N/A N/A −1.64.0+1.0272+2.5487−1.1537−3.7332 −2.0+1.0272+2.5487

−1.1537−3.7332

λ4 N/A N/A N/A N/A

ω0 −1.0 −1.0364+0.0648+0.1140−0.0865−0.1910 −1.0730+0.0644+0.1139

−0.0863−0.1906 −1.0773+0.0645+0.1136−0.0863−0.1918

ω1 N/A +1.150+0.2377+0.5849−0.1230−0.1965 +1.2941+0.2377+0.5848

−0.1230−0.1965 +1.2969+0.2359+0.5849−0.1239−0.1964

ω2 N/A −1.0546+0.1283+0.4803−0.0381−0.0741 −0.6179+0.1281+0.4802

−0.0382−0.0742 −0.6179+0.1281+0.4802−0.0382−0.0742

ω3 N/A −0.0025+0.1220+0.8554−0.0381−0.0689 −0.0054+0.1255+0.4522

−0.0383−0.0710 −0.0030+0.1299+0.4778−0.03833−0.0728

ω4 N/A N/A N/A N/A

ΩDM,0 +0.2812+0.0185+0.0478−0.0140−0.0278 +0.2844+0.0118+0.0386

−0.0062−0.0125 +0.2844+0.0118+0.0386−0.0062−0.0125 +0.2844+0.0118+0.0386

−0.0062−0.0125

Ωb,0 +0.0493+0.0018+0.0037−0.0020−0.0040 +0.0494+0.0012+0.0025

−0.0014−0.0030 +0.0494+0.0012+0.0025−0.0014−0.0030 +0.0494+0.0012+0.0025

−0.0014−0.0030

H0 +67.10+1.3038+2.6882−1.3336−2.5704 +67.1480+0.8092+1.8055

−0.9875−1.9252 +67.1487+0.8085+1.8048−0.9882−1.9259 +67.1487+0.8085+1.8048

−0.9882−1.9259

α +0.1360+0.0418+0.0848−0.0403−0.0810 +0.1360+0.1103+0.2289

−0.1205−0.2483 +0.1360+0.1103+0.2289−0.1205−0.2483 +0.1360+0.1103+0.2289

−0.1205−0.2483

β +3.060+0.1031+0.2083−0.1058−0.2069 +3.0780+0.1964+0.3917

−0.1858−0.3708 +3.0780+0.1964+0.3917−0.1858−0.3708 +3.0780+0.1964+0.3917

−0.1858−0.3708

M −19.0324+0.3796+0.7651−0.3888−0.7769 −19.0880+0.5637+1.1179

−0.5527−1.1042 −19.0631+0.5647+1.1099−0.5517−1.0970 −19.0631+0.5647+1.1099

−0.5517−1.0970

dM −0.124+0.2774+0.5409−0.2721−0.5338 −0.1230+0.3711+0.7483

−0.3832−0.7503 −0.1230+0.3711+0.7483−0.3832−0.7503 −0.1230+0.3711+0.7483

−0.3832−0.7503

χ2min 719.8072 707.8161 704.0341 703.2582

TABLE IV. Depicts the best-fit values of the cosmological parameters for the three models with 1σ and 2σ errors, usingChebyshev polynomials up to order N = 3 for IQ, ωDE in the ωDE y IDE models, respectively.

Likewise, our fitting results show that the evolution ofωDE in the ωDE and IDE models are very close to eachother, in particular, they are close to −1 today. Theseresults imply that ωDE shows a phantom nature todayand are in excellent agreement with the constraints at1σ confidence region obtained by [4].The evolution of IQ along z and within the 1σ erroraround the best-fit curve for the IDE model is shown inthe right upper panel of Fig. 2. From where, we see thatIQ can change its sign throughout its evolution. Now,from Eqs. (3) and (4), we conveniently establish the fol-lowing convention: I+ denotes an energy transfer fromDE to DM while I− denotes an energy transfer from DM

to DE. From here, we have found a change from I+ toI− and vice versa. This change of sign is linked to thecrossing of the line, IQ = 0, which is also favored by thedata at 1σ error. The IDE model shows three crossingpoints in z = −0.3370+0.0553

−0.1734 (IDE1), z = −0.4673+0.0874−0.1181

(IDE2) and z = +5.6077+2.2901−2.3950 (IDE2), respectively.

The fitting results indicate that IQ is stronger at earlytimes and weaker at later times, namely, IQ remains small

today, being IQ,0 = +8.5875× 10−5+5.6690×10−5

−2.2592×10−5 for the

case IDE1 and IQ,0 = +13.8237× 10−5+6.0544×10−5

−4.5652×10−5 forthe case IDE2, respectively. These results are consistentat 1σ error with those reported in [46, 74, 75]. However,

Page 7 of 15



For Review Only

8


λ0 × 10+4 N/A N/A +1.120+0.6569+2.1166−0.5709−1.4348 +1.120+0.6569+2.1166

−0.5709−1.4348

λ1 × 10+4 N/A N/A +2.733+0.4108+0.7581−0.5648−1.9464 +2.733+0.4108+0.7581

−0.5648−1.9464

λ2 × 10+5 N/A N/A +2.7112+0.8670+1.6127−3.5277−4.6376 −2.6490+0.5032+1.0396

−1.1769−2.8069

λ3 × 10+7 N/A N/A −2.0+1.0272+2.5487−1.1537−3.7332 −2.0+1.0272+2.5487

−1.1537−3.7332

λ4 × 10+8 N/A N/A −1.0+0.5111+1.2823−0.5710−1.8883 −1.0+0.5111+1.2823

−0.5710−1.8883

ω0 −1.0 −1.0364+0.0648+0.1140−0.0865−0.1910 −1.0730+0.0644+0.1139

−0.0863−0.1906 −1.0773+0.0645+0.1136−0.0863−0.1918

ω1 N/A +1.150+0.2377+0.5849−0.1230−0.1965 +1.2941+0.2377+0.5848

−0.1230−0.1965 +1.2969+0.2359+0.5849−0.1239−0.1964

ω2 N/A −1.0546+0.1283+0.4803−0.0381−0.0741 −0.6179+0.1281+0.4802

−0.0382−0.0742 −0.6179+0.1281+0.4802−0.0382−0.0742

ω3 N/A −0.0015+0.1220+0.8554−0.0381−0.0689 −0.0020+0.1255+0.4522

−0.0383−0.0710 −0.0020+0.1299+0.4778−0.03833−0.0728

ω4 × 10+5 N/A −7.0+0.9447+1.3880−2.5405−6.4488 +3.0+1.0579+2.7301

−0.4127−0.5940 +4.0+1.4279+3.6641−0.5665−0.7881

ΩDM,0 +0.2812+0.0185+0.0478−0.0140−0.0278 +0.2844+0.0118+0.0386

−0.0062−0.0125 +0.2844+0.0118+0.0386−0.0062−0.0125 +0.2844+0.0118+0.0386

−0.0062−0.0125

Ωb,0 +0.0493+0.0018+0.0037−0.0020−0.0040 +0.0494+0.0012+0.0025

−0.0014−0.0030 +0.0494+0.0012+0.0025−0.0014−0.0030 +0.0494+0.0012+0.0025

−0.0014−0.0030

H0 +67.10+1.3038+2.6882−1.3336−2.5704 +67.1480+0.8092+1.8055

−0.9875−1.9252 +67.1487+0.8085+1.8048−0.9882−1.9259 +67.1487+0.8085+1.8048

−0.9882−1.9259

α +0.1360+0.0418+0.0848−0.0403−0.0810 +0.1360+0.1103+0.2289

−0.1205−0.2483 +0.1360+0.1103+0.2289−0.1205−0.2483 +0.1360+0.1103+0.2289

−0.1205−0.2483

β +3.060+0.1031+0.2083−0.1058−0.2069 +3.0780+0.1964+0.3917

−0.1858−0.3708 +3.0780+0.1964+0.3917−0.1858−0.3708 +3.0780+0.1964+0.3917

−0.1858−0.3708

M −19.0324+0.3796+0.7651−0.3888−0.7769 −19.0880+0.5637+1.1179

−0.5527−1.1042 −19.0631+0.5647+1.1099−0.5517−1.0970 −19.0631+0.5647+1.1099

−0.5517−1.0970

dM −0.124+0.2774+0.5409−0.2721−0.5338 −0.1230+0.3711+0.7483

−0.3832−0.7503 −0.1230+0.3711+0.7483−0.3832−0.7503 −0.1230+0.3711+0.7483

−0.3832−0.7503

χ2min 719.8072 707.8136 704.0338 703.2508

TABLE V. Displays the best-fit values of the cosmological parameters for the three models with 1σ and 2σ errors, usingChebyshev polynomials up to order N = 4 for IQ, ωDE in the ωDE y IDE models, respectively.

0

1

2

3

4

5

6

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

∆Χ

2(α

)

α

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

2.6 2.8 3 3.2 3.4 3.6 3.8

∆Χ

2(β

)

β

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

-20.5 -20 -19.5 -19 -18.5 -18 -17.5

∆Χ

2(Μ

)

Μ

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

-1 -0.5 0 0.5 1

∆Χ

2(d

M)

dM

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

64 65 66 67 68 69 70 71

∆Χ

2(Η

0)

Η0

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

0.24 0.26 0.28 0.3 0.32 0.34

∆Χ

2(Ω

DM

,0)

ΩDM,0

ΛCDMωDEIDE1σ2σ

0

1

2

3

4

5

6

-0.0001 0 0.0001 0.0002 0.0003 0.0004 0.0005

∆Χ

2(λ

0)

λ0

IDE1σ2σ

0

1

2

3

4

5

6

-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003 0.0004

∆Χ

2(λ

1)

λ1

IDE1σ2σ

0

1

2

3

4

5

6

-4e-05 -2e-05 0 2e-05 4e-05

∆Χ

2(λ

2)

λ2

IDE1σ2σ

0

1

2

3

4

5

6

7

-1.3 -1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9

∆Χ

2(ω

0)

ω0

ωDEIDE 1IDE 2

1σ2σ

0

1

2

3

4

5

6

7

1 1.2 1.4 1.6 1.8 2

∆Χ

2(ω

1)

ω1

ωDEIDE 1IDE 2

1σ2σ

0

1

2

3

4

5

6

7

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

∆Χ

2(ω

2)

ω2

ωDEIDE 1IDE 2

1σ2σ

0

1

2

3

4

5

6

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

∆Χ

2(ω

3)

ω3

ωDEIDE 1IDE 2

1σ2σ

0

1

2

3

4

5

6

-0.00015 -0.0001 -5e-05 0 5e-05 0.0001

∆Χ

2(ω

4)

ω4

ωDEIDE 1IDE 2

1σ2σ

0

1

2

3

4

5

6

-8e-07 -7e-07 -6e-07 -5e-07 -4e-07 -3e-07 -2e-07 -1e-07 0 1e-07

∆Χ

2(λ

3)

λ3

IDE1σ2σ

0

1

2

3

4

5

6

-4e-08 -3e-08 -2e-08 -1e-08 0 1e-08

∆Χ

2(λ

4)

λ4

IDE1σ2σ

FIG. 1. Displays the one-dimension probability contours of the parameter space at 1σ and 2σ errors. Besides ∆χ2 = χ2 − χ2min.

our outcomes are smaller with tighter constraints. Thisdiscrepancy may be due to the ansatz chosen for IQ andthe used data.The left middle panel of Fig. 2 depicts the evolution ofthe energy densities of DM and DE for the present mod-els. From here, we note that ρDE is definitely positiveat all the range of redshifts considered in the data sam-

ple. In spite of the energy exchange between the twodark components of the universe, unwanted future sin-gularity for ρDE in the IDE model have been found atz → −1, which is commonly called big-rip [76]. More-over, in these models, we also note that the graphs forρDM are overlapped during their evolution. Evolution ofthe fractional energy density of DM and DE are depicted

Page 8 of 15



For Review Only

9

FIG. 2. Shows the background evolution of ωDE (left above panel), IQ (right above panel), ρDM and ρDE (left middle panel),fractional energy density ΩDM and ΩDE (right middle panel), R (left below panel) and q (right below panel) along z for thepresent models. Here, we have fixed the best-fit values of Table V and have considered the respective constraints at 1σ errorfor the ωDE and IDE models.

in the right middle panel of Fig. 2. Here, we also notethat in determined redshift intervals, the values of ΩDM(ΩDE) in the IDE model have the possibility of beingsmaller (larger) that the corresponding values predictedby the non-interacting models at 1σ error. These fea-tures represent the impact of the reconstruction of IQ.The left below panel of Fig. 2 shows the backgroundevolution of R which exhibits a scaling behavior at latetimes (keeping constant) of the universe. These resultssignificantly alleviate the coincidence problem, but theydo not solve it in full. Hence, we see that R in the twocases of the IDE model are essentially overlapped during

their evolution. By contrast, the situation is oppositefor the other models. The cosmic evolution of q alongz for the three scenarios within 1σ confidence level andin the range −1 ≤ z ≤ 5 are plotted in the right be-low panel of Fig. 2. Hence, it is evident that q showsa transition from decelerated phase to accelerated phaseat the transition redshift, zt, defined from q(zt) = 0.For ΛCDM q = +0.595+0.030

−0.040, q = +0.580+0.270−0.255 for

the ωDE model, for the IDE1 model q = +0.540+1.165−0.220

and q = 0.545+1.160−0.215 for the IDE2 model, respectively.

Hence, we note that, for the three models, zt is locatedat 0.5 < zt < 1.0, which is consistent at 1σ error with

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FIG. 3. Shows the evolutional trajectory of the present models in the S(1)n , z and S(2)

n , z planes with n = 3, 4, 5, respectively.

the results presented in [26, 77–83].In addition, for the three models, we find that the 1σconfidence regions of q are different in the future. So, inthe ωDE model, q is slightly smaller at z < 0 in compar-ison with the results found in the other models.In this work, we substitute the best fits values of themodel parameters given by Table V, and then we plotthe evolutionary trajectory of the present models in the

S(1)n , z and S(2)

n , z planes with n = 3, 4, 5, as areshown in Fig. 3. Likewise, any combination possible ofthe evolutionary trajectories of these scenarios, in the

different S(1)n , S

(1)m , S(2)

n , S(2)m and S(1)

n , S(2)m planes

with n,m = 3, 4, 5 are presented in Figs. 4 and 5, re-spectively. Furthermore, their current values are marked

as cyan, orange and brown round dots for the two casesIDE1 and IDE2 of the IDE model and for the ωDE model,

respectively. The fixed point S(1)n , S

(2)n = 1, 0 corre-

sponds to ΛCDM and is denoted as a black point; more-over, the difference between two models is determinedby the distance between their fixed points at the presenttime. When compare the different planes of Figs. 4 and5, it is evident that all the three models can be well dis-criminated from each other at the present time. Then,one can also observe that the two cases of the IDE modelare overlapped sometimes in the different S(1)

n , S(1)m ,

S(2)n , S

(2)m and S(1)

n , S(2)m planes with n,m = 3, 4, 5,

which means that them may not be well distinguishedduring their cosmic evolution, but they can be distin-

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FIG. 4. Displays the trajectories of the present models in the different S(1)n , S

(1)m and S(2)

n , S(2)m planes with n,m = 3, 4, 5.

The present time in the different models is shown as a dot and the arrows indicate the evolution direction with respect to time.

guished from the remaining models at the present epoch.In particular, the trajectories of the ωDE model in the

different S(1)n , S

(1)m , S(2)

n , S(2)m and S(1)

n , S(2)m planes

with n,m = 3, 4, 5 are different in comparison to theother models at the present time. Thus, from Figs. 4and 5, one can find that the three models can be well dis-tinguished from each other and from the ΛCDM modelat the present time. Likewise, we also note from the

S(1)3 , S

(2)3 , S

(1)4 , S

(2)4 and S(1)

5 , S(2)4 panels of Fig. 5

that the IDE model can go through the fixed point 1, 0,which corresponds to the ΛCDM model more than twice.We have also plotted in Fig. 6 the evolution trajectories

of these models in the S(1)n ,R and S(2)

n ,R planes to

analyze the possible deviations from the ΛCDM modelat the present time. These results are in agreement at1σ error with those presented in [36–42].As we can see from Figs. 4, 5 and 6, the present values ofthe second members of the statefinder hierarchy can beused to establish differences among the three DE models.

VIII. CONCLUSIONS

In this present work, we are interested in reconstruct-ing the evolutionary trajectories of three different DEmodels in distinct statefinder hierarchy planes. For this

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FIG. 5. Shows the trajectories of the present models in the different S(1)n , S

(2)m planes with n,m = 3, 4, 5. The present time

in the different models is shown as a dot and the arrows indicate the evolution direction with respect to time.

reason, we examined an interacting DE model (IDE) fillswith two interacting components such as DM and DE,together with two non-interacting components decoupledfrom the dark sectors such as baryons and radiation.Here, we propose an interaction Q proportional to theDM energy density, to the Hubble parameter H, andto a time-varying function, IQ, expanded in terms ofthe Chebyshev polynomials Tn, defined in the interval[−1, 1]. Besides, we also reconstruct a non-constant ωDE,in function of that polynomial base. These ansatzeshave been proposed so that their cosmic evolution arefree of divergences at the present and future times,respectively. Making use of a joint analysis of geometric

probes including JLA + BAO + CMB + H data andusing the MCMC method, we constrain the parameterspace of the IDE model and compared it with the resultsobtained from two different non-interacting models,presented in Tables III, IV and V, respectively.Likewise, from the upper panels of Fig. 2, our fittingresults show that ωDE crosses −1 twice. Similarly, IQ

can cross twice the line Q = 0 as well. These crossingfeatures are favored by the data at 1σ error.On the other hand, the combined impact of both Qand ωDE on the evolution of ρDM , ρDE , ΩDM , ΩDE ,R and q are shown in the middle and below panels ofthis Figure. Hence, in the middle panels, we note that

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FIG. 6. Depicts the trajectories of the present models in the different S(1)n , R and S(2)

n , R planes with n = 3, 4, 5. Thepresent time in the different models is shown as a dot and the arrows indicate the evolution direction with respect to time.

ρDE is definitely positive at all the range of redshifts,and in spite of the energy exchange between the twodark components of the universe, an unwanted futuresingularity for ρDE in the IDE model have been found atz → −1, which is commonly called big-rip. Moreover, wealso see that in determined redshift intervals, the valuesof ΩDM (ΩDE) in the IDE model have the possibilityof being smaller (larger) that the corresponding valuespredicted by the non-interacting models at 1σ error.These features represent the impact of the reconstructionof IQ. Then, from the left below panel of this Figure,it is evident that R, exhibits a scaling behavior in lateand future times of the universe. Furthermore, the right

below panel of this Figure, shows that q in the presentmodels, experiences a transition from decelerated phaseto accelerated phase, and that, the universe becomes lessaccelerated in a far future for the ωDE and IDE models,respect to the predicted behavior by the ΛCDM model.In Figs. 4 and 5, we plotted the evolutionary trajec-tories of the present models in the different statefinderhierarchy planes, respectively. So, the difference amongthese scenarios is measured by the respective distancesamong their current fixed points. According to theresults presented previously, the current values of the

ωDE and IDE models in the S(1)n , S

(2)m , S(1)

n , R and

S(2)n , R planes with n,m = 3, 4, 5, deviate far from

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14

that found in the ΛCDM model, namely, the distanceamong them can be measured. Thus, the reconstructionof the second members of the statefinder hierarchy allowus to discriminate the ωDE and IDE models from theΛCDM model.In addition, we also see that our evolutionary trajectoriesof the statefinder hierarchy agree within 1σ error withthose obtained in [36–42]. We have confirmed that theansatzes for Q and ωDE in terms of z are successful andvalid to reconstruct this statefinder hierarchy. In thissense, the IDE model can be compared and distinguished

from the ΛCDM and ωDE models, by using the presentvalues of the statefinder hierarchy, given by Figs. 4, 5and 6, respectively. We believe that the two ansatzesproposed for Q and ωDE in terms of the Chebyshevpolynomials are very successful to explore the dynamicalevolution of DE and have shown that they can beemployed to reconstruct the second members of thehierarchy statefinder. We suggest that those ansatzesshould be further investigated.

Acknowledgments The author is indebted to the Institute of Physics

and Mathematics (IFM-UMSNH) for its hospitality and support.

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