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Interacting scalar tensor cosmology in light of SNeIa, CMB,BAO andOHD observational data sets

Sayed Wrya Rabieia,1, Haidar Sheikhahmadib,1,2, Khaled Saaidic,1,3, AliAghamohammadid,4

1Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran.2Institute for Advance Studies in Basic Sciences (IASBS) Gava Zang, Zanjan 45137-66731, Iran.3Science and Technology Park of Kurdistan, Sanandaj, Iran.4Sanandaj Branch Islamic Azad University, Iran.

Received: date / Accepted: date

Abstract During this work, an interacting chameleon likescalar field scenario, by considering SNeIa, CMB, BAO andOHD data sets is investigated. In fact, the investigation isre-alized by introducing an ansatz for the effective dark energyequation of state, which mimics the behaviour of chameleonlike models. Based on this assumption, some cosmologicalparameters including Hubble, deceleration and coincidenceparameters in such mechanism are analysed. It is realizedthat, to estimate the free parameters of a theoretical model,by regarding the systematic errors it better the whole of theabove observational data sets to be considered. In fact, ifone considers SNeIa, CMB and BAO but disregards OHDit maybe leads to different results. Also to get a better over-lap between the counters with the constraintχ2

m ≤ 1, theχ2T

function could be re-weighted. The relative probability func-tions are plotted for marginalized likelihoodL (Ωm0,ω1,β )according to two dimensional confidence levels 68.3%, 90%and 95.4%. Meanwhile, the value of free parameters whichmaximize the marginalized likelihoods using above confi-dence levels are obtained. In addition, based on these calcu-lations the minimum value ofχ2 based on free parametersof an ansatz for the effective dark energy equation of stateare achieved.

PACS 98.80.-k· 98.80.Es· 95.36.+x

1 Introduction

Observational data sets including Cosmic Microwave Back-ground(CMB) [1,2], Supernovae type Ia (SNeIa) [3,4], Bary-onic Acoustic Oscillations (BAO) [5,6], Observational Hub-ble Data (OHD) [7,8], Sloan Digital Sky Survey (SDSS) [9,

ae-mail: wrabiei@uok.ac.irbe-mail: h.sh.ahmadi@gmail.com/ h.sh.ahmadi@iasbs.ac.irce-mail: ksaaidi@uok.ac.irde-mail: a.aqamohamadi@iausdj.ac.ir

10], and Wilkinson Microwave Anisotropy Probe (WMAP)[11,12], are considered as criterion for accuracy of theo-retical models. Amongst these constraints, the CMB andSNeIa( because of abundance of their data sources) attractmore attention. It is notable, the SNeIa constraint in highredshift values do not give a good clue to investigate theevolution of the Universe. It is obvious the results of in-dividual observations give different values for free param-eters of a theoretical model; hence, it is better that, to es-timate the best quantities for free parameters of the modelone considers the whole of observational data sets includingCMB, SNeIa, BAO and OHD. Therefore, this motivated usto study the behaviour of free parameters and their overlaps.Thence, a collective of observations including SNeIa, CMB,BAO and OHD are considered. Meanwhile the mentionedobservational data sets have predicted an ambiguous formof matter which leads to an accelerated phase of presentepoch and it is well known as dark energy. Based on thisambiguous form of matter, scientists have proposed differ-ent proposals up to now. Amongst all of those proposals,the cosmological constant,Λ , model attracts more attention[13,14]. But this mechanism suffers two well known draw-backs. The first of them is related to estimate the contribu-tion of quantum fluctuation of zero point energy and secondis related to the ratio ofΛ and dark matter energy densi-ties. These problems and also the excellent work by Bransand Dicke [15] motivated scientists to introduce a mecha-nism whichΛ had time dependency, namely quintessence[16,17,18]. Beside the quintessence mechanism, some pro-posals which risen from quantum gravity or string theoryare introduced to estimate cosmological parameters. For in-stance one can refer to tachyon [19,20], phantom [21,22,23], quintum [24,25], k-essense [26,27] and so on. Alsosome models which were risen from quantum field fluctu-ations or space time fluctuations attract more attention toinvestigate dark energy concept. For such models, one can

2

mention Zero Point Quantum Fluctuations(ZPQF) [28,29,30], Holographic Dark Energy(HDE) [31,32,33,34,35,36],Agegraphic Dark Energy (ADE) and new-ADE [37,38,39].If scalar field, in the quintessence model couples to mat-ter (non relativistic) it induces to appear a fifth force. Whenthe coupling is of order unity, the results of strongly cou-pling scalar field is not in good agreement with local grav-ity tests (for instance in solar system). Thus a mechanismshould be exist to suppress the effect of fifth force; suchmechanism is capable to reconcile strong coupling modelswith local experiments was proposed by Khoury and Welt-man [40,41] and also, separately, by Mota and Barrow [42]namely chameleon. In this mechanism, one can not choosean arbitrary Lagrangian for matter,Lm. To avoid deviationof geodesic trajectory, the author of [43] has shown the bestchoices areLm = P andLm = −ρ , whereP is pressure andρ is energy density of matter, for more discussion we re-fer reader to [44,45,46]. Therefore main motivation of thiswork is investigation the behavior of an interacting scalarfield mechanism; based on these calculations and SNeIa,CMB, BAO and OHD data sets the minimum value ofχ2

for the effective dark energy equation of state are achieved.The scheme of the paper is as follows:The above discussions which are brief review about obser-vational and theoretical motivations are considered as intro-duction. In Sec. 2, the general theoretical discussions risenfrom a chameleon like mechanism related to the cosmologi-cal parameters such as Hubble, deceleration and coincidenceparameters will be discussed. In Sec. 3, a brief review aboutcosmological data sets are brought. In Sec. 4, the observa-tional data sets including SNeIa, CMB, BAO and OHD areconsidered, to estimate the minimum value ofχ2 related tofree parameters of an ansatz for the effective dark energyequation of state. And at last, Sec. 5, is dedicated to con-cluding remarks.

2 Conservation and field’s equations in an effectivedark energy scenario

In chameleon like scalar field scenario, the mass of scalarfield is a function of local matter density, so that, it is suf-ficiently large on dense environment. Due to this fact, theequivalence principal (EP) is satisfied in the laboratory [40,41]. In addition, the Brans Dickeω parameter for two ob-servational values ofγ post Newtonian parameter take thevalues of order 104 [47], which satisfies the solar systemconstraint. The chameleon like scenario is defined as

S=

∫

d4x12

√−g(

R− ∂ µϕ ∂µϕ −2V(ϕ)+2 f (ϕ)L)

, (1)

[43,44,48,49,50,51,52]. In this equationg is the determi-nant of the metric,V(ϕ) is a run away potential and the latestterm indicates a non-minimal coupling between scalar field

and matter sector. It should be notedL is the Lagrangian den-sity of matter which consists of both dark matter and darkenergy sectors as perfect fluid [47,48,49,52,53]. It shouldbe noticed that, background is a spatially flat Friedmann-Limatire-Robertson-Walker (FLRW) Universe, with signa-ture (+2). The variation of the action1, with respect to(w.r.t) gµν results the gravitational field equation as

Gµν = f (ϕ)Tµν +T(ϕ)µν , (2)

where the stress-energy density of scalar field expresses

T(ϕ)µν =

(

∇µϕ∇ν ϕ − 12

gµν(∇ϕ)2)

−gµνV(ϕ), (3)

and

Tµν =−2√−g

δ (√−gL)δgµν , (4)

is the definition of stress-energy tensor of matter. By consid-

ering 00 andii components ofT(ϕ)µν , the energy density and

pressure could be achieved. After some algebra the conser-vation equation reads

∇µ(Gµν) = ∇µ

[

f (ϕ)Tµν +Tµν(ϕ)

]

= 0. (5)

In addition, the variation of the action1, w.r.t scalar fieldgives the evolution equation as

ϕ +3Hϕ =−V(ϕ)+∂ f (ϕ)

∂ϕL. (6)

Now, by substituting equation6 into relation5, two conser-vation equations for scalar field and matter are attained as

∇µ [Tµ0(ϕ)] = f (ϕ)L, (7)

∇µ [ f (ϕ)Tµ0] = − f (ϕ)L, (8)

where overdot denotes derivation w.r.t ordinary cosmic time,t. As it was mentioned in the introduction, the Lagrangian ofmatter is considered asL= L(m)+L(de), [52,54]; where sub-scriptm denotes matter (cold dark matter and baryons) andde refers to dark energy. Then the conservation equationscould be rewritten as

∇µ

[

f (ϕ)Tµ0(m)

]

= − f (ϕ)L(m), (9)

∇µ

[

f (ϕ)Tµ0(de)

]

= − f (ϕ)L(de), (10)

∇µ

[

Tµ0(ϕ)

]

= f (ϕ)(L(m)+L(de)). (11)

By combining the relation2 and above equations, it is easyto receive

∇µ

[

Tµ0(ϕ)+ f (ϕ)Tµ0

(de)

]

= f (ϕ)L(m). (12)

In the next step, by virtue of the definition ofT(ϕ)µν and equa-

tion 2, the Einstein tensor is modified as

Gµν = f (ϕ)[

Tµν(m)

+Tµν(de)+

1f (ϕ)

Tµν(ϕ)

]

. (13)

3

Hereafter, we postulate that both scalar field and dark en-ergy, behave the same as perfect fluid, thence for such per-fect mixture the effective stress-energy tensor is obtained asfollows

Tµν(DE) = Tµν

(de)+1

f (ϕ)Tµν(ϕ) , (14)

where subscriptDE denotes effective dark energy. There-fore using equations7-14, the modified Einstein equationand conservation relations are attained as

Gµν = f (ϕ)[

Tµν(m) +Tµν

(DE)

]

, (15)

∇µ

[

f (ϕ)Tµ0(m)

]

= − f (ϕ)L(m), (16)

∇µ

[

f (ϕ)Tµ0(DE)

]

= f (ϕ)L(m). (17)

It should be noticed that, in the right hand side of aboveequations, onlyL(m) is appeared. In fact it could be con-cluded that the energy, for different components of the Uni-verse is not conserved separately. In refs.[55,56,57], it hasbeen shown that, for perfect fluids, that do not couple di-rectly to the other components of the Universe, there aredifferent Lagrangian densities are equivalent. Namely, onecan find that the two Lagrangian densitiesL(m) = P andL(m) =−ρ give the same stress-energy tensor and the equa-tion of motions for all components of the system are simi-lar as well. But in an interacting case, which matter has aninteraction with scalar field, the Lagrangian degeneracy isbroken. Based on ref.[43], the best choice for such modelsis L(m) = P. Using this definition for Lagrangian of the mat-ter one can obtain

H2 =13

f (ϕ)[

ρm+ρDE

]

, (18)

and alsoddt

[

f (ϕ)ρm

]

+3H f (ϕ)ρm = 0, (19)

ddt

[

f (ϕ)ρDE

]

+3H f (ϕ)[

1+ωDE

]

ρDE = 0, (20)

whereH = a(t)/a(t) is the Hubble parameter,a(t) is scalefactor andωDE, is the EoS parameter of the effective darkenergy and satisfies EoS equation as

PDE = ωDE ×ρDE. (21)

To establish an accurate link between theoretical results andobservations, one can use the red shift parameter,z, insteadof the scale factor; these two cosmological parameters havea relation asa(t0)a(t)

= 1+ z z=−(1+ z)H. (22)

Thus substituting equation22 into 19and20, one finds out

f (ϕ)ρm = f0×ρm0× (1+ z)3, (23)

f (ϕ)ρDE = f0×ρDE0×exp[

∫ z0 31+ωDE(z)

1+z dz]

, (24)

whereρDE0 andρm0 refer to energy densities of dark energyand matter at present time, respectively.

2.1 Hubble parameter

Dimensionless Hubble parameter and density parameters couldbe defined as

E(z) =H(z)H0

, (25)

Ωm0 =ρm0

3H20

, (26)

ΩDE0 =ρDE0

3H20

. (27)

The dimensionless density parameters could be rewritten as

Ωm0 =f0×ρm0

3H20

, (28)

ΩDE0 =f0×ρDE0

3H20

. (29)

Therefore using relations18and25, the dimensionless Hub-ble parameter is obtained as follows

E2(z) = Ωm0(1+ z)3+ΩDE0exp

[

∫ z

03

1+ωDE(z)1+ z

dz

]

.(30)

2.2 Coincidence parameter

The ratio of dark matter and dark energy is defined as coin-cidence parameter and could be obtained as

r =ρm

ρDE(31)

= r0(1+ z)3exp[−3∫ z

0

1+ωDE(z)1+ z

dz].

Also one can obtaindrdz

=−3ωDE(z)

1+ zr(z). (32)

Due to the role of this parameter,r, in the investigation of thecosmic evolution, it attracts more attention in observationalinvestigations. In fact one can observe that, this importanceis arisen from the relation between the EoS parameter andthe evolution ofr.

2.3 Deceleration Parameter

To investigate the acceleration of the Universe, one can usedeceleration parameter which is defined as

q(t) =−1

a(t)H2

d2a(t)dt2

. (33)

The above equation can be rewritten as

q(z) =−1+32

(

(1+ωDE)E2−Ωm0(1+ z)3ωDE

(1+ z)E2

)

. (34)

In present epoch of the Universe evolution, deceleration pa-rameter is determined as

q0 =12+

32

[

1−Ωm0

]

ωDE(0). (35)

4

To solve the above equation we introduce an ansatz for EoSparameter as [54,58]

ωDE(z) =−1+ω0+ω1(1+ z)β , (36)

whereω0, ω1 andβ are free parameters of the model, wherethe minimum value ofχ2 of them will be obtained in fit-ting part. Also it is notable if we chooseβ = 0, the modelreduces to EoS constant models (for instanceΛCDM) [54].By substituting36 in 30, the dimensionless Hubble parame-ter is attained as follows

E2(z;Pi) = Ωm0(1+ z)3+ΩDE0(1+ z)3ω0 × (37)

exp

[

3ω1

β

(

(1+ z)β −1)

]

,

where

z;Pi = Ωm0,ω0,ω1,β, (38)

andPi is a set of free parameters which should be deter-mined using data fitting process. Using equation (36), onecan rewrite the equations23, 24and31, respectively as

f (ϕ)ρm = f0ρm0× (1+ z)3, (39)

f (ϕ)ρDE = f0ρDE0× (1+ z)3ω0 × (40)

exp[

3ω1β(

(1+ z)β −1)

]

,

and

r(z) = r0(1+ z)−3(1−ω0)exp

[

−3ω1

β

(

(1+ z)β −1)

]

. (41)

3 A brief review as to cosmological observational datasets

In this section, we should emphasis that the analysis is re-stricted to the background level, and do not include pertur-bations. In the following, we want to compare our theoreti-cal results with observations. To this end, we consider fourimportant data sets including SNeIa, CMB, BAO and OHD.In some papers, it was claimed OHD, which obtained versusred shift, is comparable with SNeIa data set, for instance werefer reader to reference [7] and references which are there.This subject motivated us to investigate the effects of thisnew data set beside other observations to improve the the-oretical results. As it will be discussed, the results of OHDalthough is not independent of SNeIa and BAO data sets [7]but has not any dependency to CMB. Also there are twoways to study CMB and BAO data point among the full pa-rameter distribution and Gaussian which in follow the latterwill be used.

3.1 Supernovae type Ia

It is explicit that, supernovae attract more attention in em-pirical cosmology. Whereas they are very luminous, people

interested to consider them, also for instance at closer dis-tances (i.e. lower redshift) they could be used to calculateHubble parameter, and for farther distances (i.e. higher red-shift) they attain an important role to estimate decelerationparameterq. It is obvious there are uncertainties of differentnature: statistical or random errors and systematic errors. Inthis work it is remarkable the systematic errors for SNeIaand OHD are neglected. In reality there is always a limiton statistical accuracy, besides the trivial one that time forrepetitions is limited. The assumption of independence isviolated in a very specific way by so-called systematic er-rors which appear in any realistic experiment. For instanceexperiments in nuclear and particle physics usually extractthe information from a statistical data sample. The precisionof the results then is mainly determined by the number Nof collected reactions. Besides the corresponding well de-fined statistical errors, nearly every measurement is subjectto further uncertainties, the systematic errors, typically as-sociated with auxiliary parameters related to the measuringapparatus, or with model assumptions. The result is typicallypresented in the form

x= 2.34±0.06= 2.34±0.05(stat)±0.03(syst).

The only reason for the separate quotation of the two uncer-tainties is that the size of the systematic uncertainties islesswell known than that of the purely statistical error [60]. Byvirtue of the likelihood functions, one able to estimate theminimum value ofχ2 for the set of parameterspi, as

L (pi,µ0) ∝ exp

[

−12

χ2SNe(pi,µ0)

]

, (42)

where

χ2SNe(pi,µ0) =

557

∑n=1

[µobs(zn)− µth(zn;pi,µ0)]2σ2

n. (43)

In 43, µobs(zn) is the observational distance modulus fornth supernova,σn is the variance of the measurement andµth(zn) is the theoretical distance modulus fornth supernovawhich defined as

µth(zn;pi ,µ0) = 5log10[DL(zn;pi)]+ µ0,

µ0 = 42.38−5log10[h] ,

DL(zn;pi) = (1+ z)∫ z

0

dzE(z;pi)

,

whereDL is the luminous distance andh= 100kms−1Mpc−1.To achieve best fit of free parameters, one can marginalizelikelihood function w.r.tµ0 [59,60]. Thenceχ2

SNe(pi) re-duces to

χ2SNe(pi) = A− B2

C, (44)

whereA, B andC are defined as follows

A=557

∑n=1

[µobs(zn)− µth(zn;pi ,µ0 = 0)]2σ2

n, (45)

5

B=557

∑n=1

µobs(zn)− µth(zn;pi,µ0 = 0)σ2

n, (46)

C=557

∑n=1

1σ2

n. (47)

3.2 Cosmic Microwave Background

According to oscillations appear in matter and radiation fieldsDoppler peaks in radiation (photon) spectrum are produced.Also it should be noted that existence of dark energy, affectsthe place of the Doppler peaks in spectrum diagrams. Todetermine the shift of these peaks, theoretically, CMB shiftparameters is defined as refs. [1,61]

Rth(zrec;pi) =√

Ωm0

f0

∫ zrec

0

dzE(z;pi)

. (48)

In CMB investigations [62], theχ2CMB function versus CMB

shift parameter is

χ2CMB(pi) =

[Robs−Rth(zrec;pi)]2

σ2R

(49)

whereRobs= 1.725,σR = 0.018 andzrec ≈ 1091.3 are ob-servational quantities of CMB shift parameter, uncertaintyof R in σ1 confidence level and recombination redshift, re-spectively refs. [61,1].

3.3 Baryonic Acoustic Oscillations

As in [63] mentioned, because BAO can be considered asa standard length scale at a wide range of redshift it is anuseful candidate for cosmological models testing. The im-portance of BAO mechanism is related to its ability in esti-mation the contents and curvature of the Universe. One canestablish a relation between theoretical BAO parameter,Ath,and dimensionless Hubble parameter, Eq.(30), as

Ath(zb;pi) = (50)√

Ωm0f0

[E(zb;pi)]−1/3[

1zb

∫ zb0

dzE(z;pi)

]2/3,

wherezb = 0.35 [5,6]. Also χ2BAO in BAO mechanism in-

vestigation is as follows

χ2BAO(pi) =

[Aobs−Ath(zrec;pi)]2σ2

A

, (51)

and alsoAobs= 0.469(ns/0.98)−0.35 andns= 0.968, [6,59].It is obvious that, the BAO are detected in the clusteringof the combined 2dFGRS and SDSS main galaxy samples,and measure the distance-redshift relation atz= 0.2. But weconsider BAO in the clustering of the SDSS luminous redgalaxies in which measure the distance-redshift relation atz= 0.35 [64].

3.4 Observational Hubble Data

We suggest that, if people want to investigate the accuracy ofany theoretical model, it is better, maybe, to consider SNeIa,CMB, BAO and OHD together. In [7], it was claimed thatthree different models of dark energy i.e.ΛCDM, ϕCDMand XCDM have been investigated just by consideringH(z)measurement. But they have usedH0 = 68±2.8 andH0 =

73.8±2.4 which risen from SNeIa data [8]. Therefore it isrealized that for the comparison between theoretical resultsand observations only OHD could not be considered. Theχ2

OHD function parameter based on OHD data set is definedas

χ2OHD(pi ,H0) =

28

∑n=1

[Hobs(zn)−H0Eth(zn;pi)]2σ2

n, (52)

after marginalize w.r.tH0, to calculate likelihood,χ2OHD could

be considered as

χ2OHD(pi) = AH − B2

H

CH, (53)

where

AH =28

∑n=1

[Hobs(zn)]2

σ2n

, (54)

BH =28

∑n=1

Hobs(zn)×Eth(zn;pi)σ2

n, (55)

CH =28

∑n=1

[Eth(zn;pi)]2σ2

n. (56)

In above equations subscriptobs is refer to observationalquantities and subscriptth is for theoretical one.

4 Cosmological constraints and data fitting

As it was mentioned, we have introduced an ansatz as equa-tion 36, that consist of three free parameters. Whereω1 indi-cates present time value ofωDE. For more convenience wecan supposeω0 = f0 = 1 and therefore Eq.(36) is reduced to[54,58]

ωDE(z) = ω1(1+ z)β . (57)

Also, the mean square of relative error functionsχ2, nor-mally cause the free parameters plane split in two parts. Peo-ple usually are interested to the regions whichχ2/N ≤ 1,whereN denotes the amount of observational data. Whereaswe useUnion− 2 data set for SNeIa,N for supernovae isNSNe= 557, and also for OHD, CMB and BAO, one hasNOHD = 28, NCMB = 1 andNBAO = 1. Since in this workthree free parameters are appeared, the space of constraintshas three dimensions. Thence for better expression, one canmap figures on two dimensions (in fact it is supposed that,the free parameters are independent) and their values will be

6

analyzed. The common regions for best fitting of all con-straints, play key role in this study. Based on above dis-cussions we plot a couple of free parameters in Figures4- 6. In Figure4 we investigate the constraints onΩm0 inω1 β plate, and also for two constraints SNeIa and OHDminimum points ofχ2 are distinguished. In Figures5 us-ing best value ofω1, the constraints inΩm0 β are obtained,in a similar way for best value ofβ , the behavior of con-straints inω1 Ωm0 surface will be shown. Let us, return ourattention to figure4 again. ForΩm0 = 0.2, the CMB, BAOand OHD have an overlap region, but they are not in agree-ment with SNeIa results. Also for a different quantity, theSNeIa and OHD results could be in agreement with together.This different behavior of constraints indicates that if onewants to compare theoretical results with observations, itisbetter the greatest set of constraints, to be considered. Formore investigation about overlaps and the effects of indi-vidual observations, we plot figures7 and8. In figure7 thebehaviour ofχ2

T = χ2SNe+ χ2

OHD+ χ2CMB + χ2

BAO andχ2T =

χ2SNe+ χ2

CMB + χ2BAO for ∆ χ2

T = 3.53,6.25,8.02 are com-pared. Also in figure8 to investigate degeneracy one canconsiderχ2

T = χ2SNe+χ2

OHD+χ2CMB+χ2

BAO andχ2T = χ2

OHD+

χ2CMB + χ2

BAO for ∆ χ2T = 0.1,0.2,0.3. These two figures in-

dicate that although the importance of individual OHD datasurveying (in comparison SNeIa, CMB and BAO) is not soimportant, but it decreases the degeneracy between free pa-rameters of the model. From figures7 and8, it is obviousthat a collective of four constraints has completely differ-ent results in comparison to even three constraints. In thefollowing, by means of observations, we use some customquantities which are considered for better estimation of the-oretical parameters of the model. Since all free parametersof the model are independent, the total likelihood functioncould be introduced as

LT = LSNe×LOHD×LCMB×LBAO, (58)

therefore the totalχ2 function could be achieved as

χ2T = χ2

SNe+ χ2OHD+ χ2

CMB+ χ2BAO. (59)

It is considerable to attain the maximum amount of the prob-ability and the minimum value ofχ2, we should minimizeχ2

T. Also it should be noted, in59all components have sameweight. So the likelihood method is equivalent to this factthat, for instance all measurements which lead to CMB isequal to a supernova explosion!. Afterwards we return tothis problem. Another quantity which could be used for datafitting process is

χ2 =χ2

T

Ndo f(60)

where subscriptdo f is abbreviation of degree of freedom,andNdo f could be defined as the difference between all ob-servational sources and the amount of free parameters. Let’s

explain it in more detail, whereas the amounts of all obser-vations are 557+28+1+1= 587, and the number of freeparameters are 4, by consideringH0, thereforeNdo f, is equalto 583. Also one knows, the acceptable quantity forχ2 is1.05. For more convenient, we now define the average rela-tive error functions as follows

χ2SNe=

χ2SNe

NSNe, (61)

χ2OHD =

χ2OHD

NOHD, (62)

χ2CMB =

χ2CMB

NCMB, (63)

χ2BAO =

χ2BAO

NBAO. (64)

Finally we can introduceχ2m function, which is equal to

maximum ofχ2 functions and it could be considered as

χ2m = maxof

(

χ2SNe, χ

2OHD, χ

2CMB, χ

2BAO

)

. (65)

In fact theχ2m function could be considered as a criterion of

accuracy for the models. Now we want to compare the be-havior of χ2

m and χ2 functions. Without loss the generalityof the model, one can plot the three dimensional shape ofχ2

m and χ2, versus free parameters of the model. These di-agrams help us to find out the best estimation of the freeparameters in comparison to observations; for more clar-ity one can see the figure9. In this figure, the first dia-gram shows the minimum ofχ2

m and χ2 versusΩm0. Alsoin two latest diagrams of figure9, the minimum points aredrown based onω1 andβ respectively. By comparison thebehavior of these relative error functions in Figure9 onecan realize that, there are more points (or neighborhood)in which χ2 < 1, but χ2

m exceeds 1.05. In fact this behav-ior was predictable, because in definition ofχ2, we use thecontribution of all observational data set. So, for examplethe χ2

CMB deviation of best fitting results, could be recom-pense by SNeIa data abundance. We will return to this draw-back, after some discussion about likelihood and relative er-ror functions. For more illustration, we portrait the differ-ent surfaces of three dimensional,(Ωm0,ω1,β ), to (Ωm0,β ),(ω1,Ωm0) and(ω1,β ) surfaces, which are brought in Fig-ures10, 11and12. DiagramsB andC are related to(χ2

T)min,where the subscriptmin, shows the minimum value ofχ2

T .It is notable, in a three dimensional space of free parame-ters, the confidence levels 68.3%, 90% and 95.4% are pro-portional to∆ χ2

T = 3.53,∆ χ2T = 6.25 and∆ χ2

T = 8.02 sur-faces respectively where∆ χ2

T = χ2T − (χ2

T)min. In diagramB of Figures10, 11 and12 the counter lines of confidencelevels are drown and in diagramC, both theχ2

m surfacesand counter lines are brought for more comparison. FromdiagramC it is realized that the confidence level countersexceed theχ2

m regions. From this behaviour it is concludedthat, the theoretical prediction of CMB shift parameter isvery greater than it’s observational quantity. As mentionedheretofore, when the total mean square error function is in-troduced the weight of all constraints was identical and this

7

causes some problems. As a matter of fact, the results oflikelihood’s parameter, equation59, the effect of CMB shiftparameter in comparison to the abundant SNeIa data set isignored. For more information, one can see Table2 and defi-nition of Ndo f. To overcome these problems, we redefineχ2

Tas bellow

χ2T = χ2

SNe+ χ2OHD+3χ2

CMB+3χ2BAO. (66)

It should be noted, in data fitting and maximization of proba-bility quantities these two definitions ofχ2

T, i.e. equations59and66, have not very different. For justifying this claim onecan compare Tables2 and3 which are related to59 and66respectively. But in figures which related to confidence lev-els one can observe that the exceeding of confidence lev-els are reduced, therefore the re-weight of some constraintscan improve the behaviour of the model. For more clarifica-tion one can refer to figures13, 14 and15. Now by meansof 66, we margin the likelihoodL (Ωm0,ω1,β ) w.r.t ω1, βandΩm0 respectively. Also the relative probability functionsL (Ωm0,β ), L (ω1,Ωm0) andL (ω1,β ) in two dimensionalconfidence levels 68.3%, 90% and 95.4% are plotted in Fig-ure 16. For more investigations, we will draw the one di-mensional marginalized likelihood functionsL (Ωm0) ver-susΩm0, L (ω1) based onω1 andL (β ) versusβ in figure17. Meanwhile in Table4 one observes the quantities whichmaximize the marginalized likelihoods using different confi-dence levels by means of confidence levelsσ1 = 68.3% andσ2 = 95.4% .

4.1 Typical example

Now, we define an effective dark energy as combination ofdark energyρde and scalar field density asρDE = ρde+

ρϕ/ f (ϕ). So, the Friedmann equation is rewritten as

3H2 = f (ϕ)(

ρm+ρDE

)

. (67)

An useful parameter in this study is energy density parame-ter Ω . HereΩDE andΩm respectively will be taken equal toΩDE = f (ϕ)ρDE/ρc andΩm = f (ϕ)ρm/ρc, in which ρc isthe critical energy density which is defined asρc = 3H2. Asa result, from the Friedmann equation we haveΩDE +Ωm=

1.To obtain energy conservation equations for effective darkenergy one can achieve the following results

ddt

(

f (ϕ)ρDE

)

+3H f (ϕ)(1+ωDE)ρDE = γρm f (ϕ), (68)

ddt

(

f (ϕ)ρm

)

+3H f (ϕ)(1+ γ)ρm=−γρm f (ϕ), (69)

so that the effective pressure of dark energy is defined aspDE = pΛ + pϕ/ f (ϕ), and one has the effective dark energyequation of state parameter asωDE = pDE/ρDE. Also γ is

the matter equation of state parameter which is defined asγ = pm/ρm. Forγ = constant, integrating of Eq.(69) resultsin the following relation for cold dark matter energy density

ρm =ρ0

em

a3(1+γ) f (1+γ)(ϕ), (70)

whereρ0em= f (1+γ)

0 (ϕ)ρ0m. In this step, we suppose that the

effective dark energy could be defined as ADE, in otherword we assume that

ρDE ≡ ρADE =3n2

T2 , (71)

wheren is a numerical constant andT is cosmic time andthereforeΩDE is obtained asΩDE = f (ϕ)n2/H2T2. Takingthis assumption and using Eq.(68), the equation of state pa-rameter of effective dark energy could be acquired as

ωDE =−1+23

1n

√

ΩDE

f (ϕ)+

f (ϕ)3H f (ϕ)

(

γr −1)

, (72)

wherer is ratio of cold dark matter and effective dark en-ergy, namelyr = ρm/ρDE = Ωm/ΩDE. The interaction termin this model generates an extra term forωDE, which canjustify the phantom divide line crossing. By definition anansatz forωeΛ , it can be considered as

ωeΛ +1= ω0+ω1(1+ z)β . (73)

For fitting the free parameters for ADE in an external scalarfield interaction model, we use the 557 Union-2 sample databaseof SNeIa, andρm = ρradiation+ρbaryon+ρdarkmatter. There-fore in this case the Friemann equation is as

3H2 = f (ϕ)(

ρm+ρDE

)

. (74)

Combining Eqs.(68)-(71), give

3H2 = f (ϕ)( ρ0

em

a3(1+γ) f (1+γ)(ϕ)+

3n2

T2

)

, (75)

whereρ0em is the effective energy density of matter at the

present time. Whereas the 557 Union-2 sample database havecollected from red shift parameter to variousSNeIa, there-fore we rewriteE = H/H0 versuszas

E2 =r0(1+ z)3+(1+ z)3ω0 exp

3ω1β [(1+ z)β −1]

r0+1. (76)

To achieve the best fit for free parameters based on subsec-tion 3.1a minimization method leads to

χ2snmin

(ω0 = 1.1; ω1 =−1.65; β =−2.25), (77)

χ2min = A− B2

C= 542.75, (78)

µ0 = −BC

= 43.1089. (79)

where impliesχ2sn/do f= χ2

snmin/do f = 0.981(do f = 553).

In figure1, we show a comparison between theoretical dis-tance modulus and observed distance modulus of supernovae

8

0.2 0.4 0.6 0.8 1.0 1.2 1.4z

34

36

38

40

42

44

Μ

Fig. 1 The observed distance modulus of supernovae (points) and thetheoretical predicted distance modulus (red-solid line) in the context ofADE model.

-2.5 -2.0 -1.5 -1.0 -0.5 0.0-3

-2

-1

0

1

2

3

Ω1

Β

Fig. 2 Contour plots for the free parametersω1 andβ , shows that thebest value for these parameters are−1.86< ω1 <−1.62 and−2.27<β <−0.73.

data. The red-solid line indicates the theoretical value ofdis-tance modulus,µth, for the best value of free parametersω1 = −1.65,ω0 = 1.1 andβ = −2.25. This shows that themodel is clearly consistent with the data sinceχ2/do f = 1.Fig.2 show contour plots for the free parametersω1 andβ , it is shown that the best value for these parameters are−1.86< ω1 <−1.62 and−2.27< β <−0.73 in which forstability conditionc2 > 0, we have taken the interface be-tween green and yellow sector,ω1 =−1.68.

The evolution of effective dark energy parameter,ωDE, ver-susz, for ω0 = 1.1, ω1 = −1.68 andβ = −2.25 have beenshown in Fig.3. This show that by growingz the parameterget into the phantom phase.Hereω0, ω1 andβ are free parameters of the model whichobtained from data fitting. It is clear that, if the form of darkenergy density is given the coupling function,f (ϕ), couldbe easily determined. For instance by using Eqs.(68), (69)and (71) one can obtain

f (ϕ) = f0t2a−3ω0 exp[

3ω1(z+1)β+2

β +2

]

, (80)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-2.0

-1.5

-1.0

-0.5

Z

ΩeL

Fig. 3 The plot shows the evolution of effective dark energy parame-ter,ωDE, versusz, for ω0 = 1.1, ω1 =−1.68 andβ =−2.25.

here f0 is the constant of integration. Whereas

f (ϕ)f (ϕ)

= 3H[ 23tH

−ω0−ω1(1+ z)β+2].

A significant result of observational data is accelerated ex-pansion of the Universe. A good cosmological model shouldbe able to describe this acceleration. An useful quantity toinvestigate this property of the Universe is deceleration pa-rameter which defined asq= −1− H/H2. Using Eqs.(67),(68) and (69), one achieves the deceleration parameter gives

q =−1+ 32

[

1−ω0−ω1(1+ z)β]

×(

D0

1+(1+z)3(1−ω0)(t)exp[−3ω1(1+z)β

β ]

)

, (81)

whereD0 is the constant of integration. It is clearly seenthat forω0 = 1.1, ω1 =−1.68, β =−2.25, (which haveobtained from data fitting processes)q< 0.

5 Conclusion and Discussion

Interacting models which contain an external interaction be-tween matter and scalar fields attract more attentions. Suchmechanisms are capable to suppress the fifth force and alsoare in good agrement with observations. Using such pow-erful mechanism we have attained some cosmological pa-rameters consist of coincidence and deceleration parame-ters. For instance based on Table2, and equations32 and34 it is clear thatr(z) is a decreasing function andq hastaken negative values for different amounts ofz. Consid-ering a suitable ansatz for EoS parameter of effective darkenergy dimensionless Hubble parameter is obtained. So bymeans of SNeIa, CMB, BAO and OHD data sets the min-imum value ofχ2 for the free parameters of the model areachieved. To estimate the free parameters of an ansatz for theeffective dark energy equation of state, the whole of the ob-servational data sets have been considered. For more detailsone can compare the results of figures7 and8, and the re-sults of typical example, subsection4.1. Also for getting bet-ter overlap between the counters with the constraintχ2

m ≤ 1,

9

theχ2T function have been re-weighted. Meanwhile the rela-

tive probability functions have plotted for marginalized like-lihood L (Ωm0,ω1,β ) according to two dimensional confi-dence levels 68.3%, 90% and 95.4%. In addition the valueof free parameters which maximize the marginalized likeli-hoods using above confidence levels have obtained. Basedon above discussions a couple of free parameters in figures4 - 6, have plotted. In figure4, the constraints onΩm0 inω1 β plate have investigated; and also, for two constraintsSNeIa and OHD minimum points ofχ2 have distinguished.In figures5, using best value ofω1, the constraints inΩm0 βsurface are obtained. In a similar way, for best value ofβ ,the behavior of constraints inω1 Ωm0 plane have shown.Also based on figure4, for Ωm0 = 0.2, the CMB, BAO andOHD have an overlap region, but they are not in agreementwith SNeIa results; where one possible explanation wouldbe an incompatibility among the data sets. Also, for a dif-ferent values one can find a region which SNeIa and OHDare in more agreement against CMB and BAO. This differ-ent behavior of constraints indicates that, if one wants tocompare theoretical and observational results, it maybe bet-ter the greatest set of constraints to be considered. For moreinvestigation about overlaps and the effects on individualob-servations, the figures7 and8, have been plotted. In figure7, the behaviour ofχ2

T = χ2SNe+ χ2

OHD+ χ2CMB + χ2

BAO andχ2

T = χ2SNe+ χ2

CMB+ χ2BAO for ∆ χ2

T = 3.53,6.25,8.02, havecompared. Also in figure8, we have consideredχ2

T = χ2SNe+

χ2OHD + χ2

CMB + χ2BAO andχ2

T = χ2OHD+ χ2

CMB + χ2BAO, for

∆ χ2T = 0.1,0.2,0.3, to investigate the degeneracy. These two

figures indicate that although the importance of individualOHD data surveying in cosmological investigations (in com-parison SNeIa, CMB and BAO) is not so important but itdecreases degeneracy between free parameters.

Acknowledgements H. Sheikhahmadi would like to thank Iran’s Na-tional Elites Foundation for financially support during this work.

References

1. G. Efstathiou and J. R. Bond,Cosmic Confusion: Degeneraciesamong Cosmological Parameters Derived from Measurements of Mi-crowave Background Anisotropies, Mon. Not. Roy. Astron. Soc 304(1999) 75.

2. J. Dunkley, et al.Five-Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Bayesian Estimation of CMB PolarizationMaps, Astrophys. J701(2009) 1804.

3. A. G. Riess et al.,Observational Evidence from Supernovae for anAccelerating Universe and a Cosmological Constant, Astron. J 116(1998) 1009.

4. S. Perlmutter, et al.,Measurements ofΩ and Λ from 42 High-Redshift Supernovae, Astrophys. J517(1999) 565.

5. D. J. Eisenstein and W. Hu,Baryonic Features in the Matter Trans-fer Function, Astrophys. J.496(1998) 605.

6. M. Shoji, D. Jeong and E. Komatsu,Extracting Angular DiameterDistance and Expansion Rate of the Universe from Two-dimensionalGalaxy Power Spectrum at High Redshifts: Baryon Acoustic Oscil-lation Fitting versus Full Modeling, Astrophys. J.693(2009) 1404.

7. O. Farooq and B. Ratra, ,Hubble parameter measurement con-straints on the cosmological deceleration-acceleration transitionredshift, arXiv:1301.5243 (2013).

8. G. Chen, J. R. Gott and B. Ratra,Non-Gaussian Error Distri-bution of Hubble Constant Measurements, Publ. Astron. Soc.Pac.115(2003) 1269.

9. M. Tegmark, et al.,Cosmological parameters from SDSS andWMAP, Phys. Rev.D 69 (2004) 103501.

10. J. Sollerman, et al.,First-Year Sloan Digital Sky Survey-II (SDSS-II) Supernova Results: Constraints on Non-Standard CosmologicalModels, Astrophysical Journal703(2009) 1374.

11. D. N. Spergel, et al.,First-Year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Determination of CosmologicalPa-rameters, Astrophys. J. suppl148(2003) 175.

12. G. Hinshaw, et al.,Five-Year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Data Processing, Sky Maps, and Ba-sic Results, Astrophys. J. Suppl180(2009) 225.

13. S. Weinberg,The cosmological constant problem, Rev. Mod. Phys.61 (1989) 1.

14. S. M. Carrol,The Cosmological Constant, Living Rev. Relativity4(2001) 1.

15. C. Brans and R. H. Dicke,Mach’s Principle and a RelativisticTheory of Gravitation, Phys. Rev124(1961) 925.

16. Y. Fujii, Origin of the gravitational constant and particle massesin scale invariant scalar-tensor theory, Phys. Rev. D26 (1982) 2580.

17. S. M. Carroll,Quintessence and the rest of the world, Phys. Rev.Lett.81 (1998) 3067.

18. E. N. Saridakis, S. V. Sushkov,Quintessence and phantom cosmol-ogy with non-minimal derivative coupling, Phys. Rev. D.81 (2010)083510.

19. M. R. Garousi,Tachyon couplings on non-BPS D-branes andDirac-Born-Infeld action, Nucl. Phys. B584(2000) 284.

20. D. Kutasov and V. Niarchos,Tachyon effective actions in openstring theory, Nucl.Phys. B666(2003) 56.

21. J. M. Cline, S. Jeon, and G. D. Moore,The phantom menaced:Constraints on low-energy effective ghosts, Phys. Rev. D70 (2004)043543.

22. P. Singh, M. Sami, and N. Dadhich,Cosmological dynamics ofphantom field, Phys. Rev. D68 (2003) 023522.

23. X. Chen, Y. Gong, E. N. Saridakis,Phase-space analysis of inter-acting phantom cosmology, JCAP0904, 2009 001.

24. H. Wei, R. G. Cai and D. F. Zeng,Hessence: A New View of Quin-tom Dark Energy, Class.Quant.Grav22 (2005) 3189.

25. D. Kutasov and V. Niarchos,Cosmological Evolution of a Quin-tom Model of Dark Energy, Phys. Lett. B608(2005) 177.

26. M. Malquarti, E. J. Copeland, and A. R. Liddle,k-essence and thecoincidence problem, Phys. Rev. D.68 (2003) 023512.

27. C. Bonvin, C. Caprini, and R. Durrer,A no-go theorem for k-essence dark energy, Phys. Rev. Lett97 (2006) 081303.

28. N. Birrell and P. Davies,Quantum fields in curved space.(Cam-bridge University Press, 1982);

29. V. Mukhanov and S. Winitzki,Introduction to quantum effects ingravity. (Cambridge University Press, Cambridge, 2007).

30. H. Sheikhahmadi, A. Aghamohammadi and Kh. SaaidiVac-uum quantum fluctuations in quasi de Sitter background arXiv: gr-qc/1407.0125.

31. L. Susskind,Strings, Black Holes and Lorentz Contraction, arXiv:hep-th/9308139 ;

32. L. Susskind,The World as a Hologram, Arxiv:hep-th/9409089.33. D. Kutasov and V. Niarchos,Some Speculations About Black Hole

Entropy In String Theory, Arxiv:hep-th/9309145.34. B. Guberina, R. Horvat, and H. Nikolic,Nonsaturated Holo-

graphic Dark Energy, Axiv:astro-ph/0611299.35. A. Aghamohammadi, K. Saaidi, and M. R. Setare,Holographic

dark energy with time depend gravitational constant in the non-flatHorava-Lifshiitz cosmology, Astrophys. Space. Sci.332, 2011 503.

10

36. A. Aghamohammadi, K. Saaidi,Holographic dark energy andf(R) gravity, Physica Scripta83, 2011 025902.

37. R. G. Cai,A Dark Energy Model Characterized by the Age of theUniverse, Phys. Lett. B.657(2007) 228.

38. H. Wei and R. G. Cai,A New Model of Agegraphic Dark Energy,Phys. Lett. B.660(2008) 113.

39. H. Wei and R. G. Cai,Cosmological Constraints on New Age-graphic Dark Energy, Phys. Lett. B.663(2008) 1.

40. J. Khoury and A. Weltman,Chameleon cosmology, Phys. Rev. D69 (2004) 044026.

41. J. Khoury and A. Weltman,Chameleon Fields: Awaiting Surprisesfor Tests of Gravity in Space, Phys. Rev. Lett.93 (2004) 171104.

42. D. F. Mota and J. D. Barrow,Tachyon effective actions in openstring theory, Phys. Lett. B581(2004) 141.

43. Kh. Saaidi,(Non-)geodesic motion in chameleon Brans Dickemodel, Astrophys. Space. Sci.345(2013) 431.

44. S.W. Hawking and G. F. R. Ellis,The Large Scale Structure ofSpacetime,(Cambridge University Press, Cambridge (1973)).

45. G. W. Gibbons and S.W. Hawking,Action integrals and partitionfunctions in quantum gravity, Phys. Rev. D.15 (1997) 2752.

46. J. D. Brown, and J. W. York,Microcanonical functional integralfor the gravitational field, Phys. Rev. D.47 (1993) 1420.

47. Kh. Saaidi, A. Mohammadi, H. Sheikhahmadi,γ parameter andSolar System constraint in chameleon Brans Dicke theory, Phys. Rev.D. 83, 2011 104019.

48. Kh. Saaidi, H. Sheikhahmadi, T. Golanbari, and S.W. Rabiei, Onthe holographic dark energy in chameleon scalar-tensor cosmology,Astrophys. Space. Sci.348, 2013 233.

49. N. Banerjee, S. Das and K. GangulyChameleon field and the latetime acceleration of the universe, Pramana74L481 2010.

50. S. Das and N. Banerjee,Brans-Dicke Scalar Field as a Chameleon,, Phys. Rev. D.78043512 (2008).

51. H. Farajollahi, A. Salehi,Attractors, Statefinders and Obser-vational Measurement for Chameleonic Brans–Dicke Cosmology,JCAP10112010 006.

52. H. Farajollahi, A. Salehi, F. Tayebi,Entropy and statefinder diag-nosis in chameleon cosmology, Astrophys. S pace. Sci.335, 2011629.

53. Kh. Saaid, H. Sheikahmadi and J. Afzali,Chameleon mechanismwith a new potential, Astrophys. Space. Sci.3332011 501.

54. A. Aghamohammadi, Kh. Saaidi, A. Mohammadi, H. Sheikhah-madi, T. Golanbari, and S.W. Rabiei,Effect of an external interactionmechanism in solving agegraphic dark energy problems, Astrophys.Space. Sci.345, 2013, 17.

55. S. Carroll,Quintessence and the Rest of the World: SuppressingLong-Range Interactions Phys. Rev. Lett.81, ( 1998) 3067.

56. T. Damour and A. M. Polyakov,The String Dilaton and a LeastCoupling Principle, Nucl. Phys. B423, ( 1994) 532.

57. C. F. Will,Theory and Experiment in Gravitational Physics(Cam-bridge: CUP, 1981).

58. S. del Campo, J. C. Fabris, R. Herrera, W. ZimdahlOn holo-graphic dark-energy models, Phys.Rev.D83, 2011 123006.

59. R. Amanullah, et al.,Spectra and Light Curves of Six Type Ia Su-pernovae at0.511< z< 1.12 and the Union2 Compilation, Astro-phys. J716(2010) 712.

60. G. Bohm and G. Zech,Introduction to Statistics and Data Analysisfor Physicists(Verlag Deutsches Elektronen-Synchrotron, 2010).

61. Y. Wang and P. Mukherjee,Robust Dark Energy Constraintsfrom Supernovae, Galaxy Clustering, and Three-Year Wilkinson Mi-crowave Anisotropy Probe Observations, Astrophys. J.650 (2006)1.

62. E. Komatsu et al.,Seven-Year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Cosmological Interpretation,Astro-phys. J. Suppl. Ser.192(2011) 18.

63. N. G. Busca, et al.,Baryon Acoustic Oscillations in the Ly− forestof BOSS quasars, arXiv:1211.2616 (2012).

64. F. C. Solano, U. NucamendiReconstruction of the interaction termbetween dark matter and dark energy using SNeIa, BAO, CMB, H(z)and X-ray gas mass fraction, arXiv:1207.0250 [astro-ph.CO].

65. C. Zhang et al.,Four New Observational H(z) Data From Lumi-nous Red Galaxies of Sloan Digital Sky Survey Data Release Seven,arXiv: 1207.4541 [astro-ph.CO] (2012).

66. J. Simon, , L. Verde and R. Jimenez,Constraints on the redshiftdependence of the dark energy potential, Phys. Rev. D71 (2005)123001.

67. M. Moresco, et al.,New constraints on cosmological parametersand neutrino properties using the expansion rate of the Universe toz1.75, J. Cosmology Astropart. Phys.07 (2012) 053.

68. C. H. Chuang and Y. Wang,Modeling the Anisotropic Two-PointGalaxy Correlation Function on Small Scales and Improved Mea-surements of H(z), DA(z), and f(z) sigma8(z) from the Sloan Dig-ital Sky Survey DR7 Luminous Red Galaxies, arXiv:1209.0210[astro.ph-CO] (2012).

69. C. Blake et al.,The WiggleZ Dark Energy Survey: joint measure-ments of the expansion and growth history at z< 1, MNRAS,425(2012) 405.

70. D. Stern et al.,Cosmic chronometers: constraining the equation ofstate of dark energy. I: H(z) measurements J. Cosmology Astropart.Phys.,02 (2010) 008.

11

Table 1 In this table from left to rightz, H(z)(kms−1 Mpc−1), it’s un-certainty σH(kms−1 Mpc−1) in measurement and related references(by considering the technique which is used) are collected,respec-tively.

z H(z) σH References Techneque

0.070 69 19.6 [65] SDSS DR7; 0< z< 0.40.100 69 12 [66] ATC; 0.1< z< 1.80.120 68.6 26.2 [65] SDSS DR7; 0< z< 0.40.170 83 8 [66] ATC; 0.1< z< 1.80.179 75 4 [67] OHD+CMB; 0< z< 1.750.199 75 5 [67] OHD+CMB; 0< z< 1.750.200 72.9 29.6 [65] SDSS DR7; 0< z< 0.40.270 77 14 [66] ATC; 0.1< z< 1.80.280 88.8 36.6 [65] SDSS DR7; 0< z< 0.40.350 76.3 5.6 [68] SDSS DR7 LRGs; z=0.350.352 83 14 [67] OHD+CMB; 0< z< 1.750.400 95 17 [66] ATC; 0.1< z< 1.80.440 82.6 7.8 [69] WiggleZ+H(z);z< 1.00.480 97 62 [70] CMB+OHD; 0.2< z< 1.00.593 104 13 [67] OHD+CMB; 0< z< 1.750.600 87.9 6.1 [69] WiggleZ+H(z);z< 1.00.680 92 8 [67] OHD+CMB; 0< z< 1.750.730 97.3 7.0 [69] WiggleZ+H(z);z< 1.00.781 105 12 [67] OHD+CMB; 0< z< 1.750.875 125 17 [67] OHD+CMB; 0< z< 1.750.880 90 40 [70] CMB+OHD; 0.2< z< 1.00.900 117 23 [66] ATC; 0.1< z< 1.81.037 154 20 [67] OHD+CMB; 0< z< 1.751.300 168 17 [66] ATC; 0.1< z< 1.81.430 177 18 [66] ATC; 0.1< z< 1.81.530 140 14 [66] ATC; 0.1< z< 1.81.750 202 40 [66] ATC; 0.1< z< 1.82.300 224 8 [63] BAO; 0.7< z< 2.3

Fig. 4 The counter lines ofχ2SNe= 557 (brown),χ2

OHD = 28.0 (green),χ2

CMB= 1.0 (red), andχ2BAO= 1.0(blue) for of Ωm0 = 0.26 are plotted.

Also for two constraints SNeIa and OHD minimum points ofχ2 aredistinguished. The dashed lines refer to the counter lines which aregreater of the minimum points only unity.

Fig. 5 The counter lines ofχ2SNe= 557 (brown),χ2

OHD = 28.0 (green),χ2

CMB = 1.0 (red), andχ2BAO = 1.0(blue) for ω1 = −1.1 are plotted.

Also for two constraints SNeIa and OHD minimum point ofχ2 aredistinguished. The dashed lines refer to the counter lines which aregreater of the minimum points only unity.

Fig. 6 The counter lines ofχ2SNe= 557 (brown),χ2

OHD = 28.0 (green),χ2

CMB = 1.0 (red), andχ2BAO = 1.0(blue) for β = −0.25 are plotted.

Also for two constraints SNeIa and OHD minimum point ofχ2 aredistinguished. The dashed lines refer to the counter lines which aregreater of the minimum points only unity.

Table 2 In the table, the quantities related to minimum point ofχ2T =

χ2SNe+ χ2

OHD+ χ2CMB + χ2

BAO are introduced.

β ω1 Ωm0 χ2BAO

−0.243 −1.053 0.272 16×10−4,

χ2CMB χ2

OHD χ2SNe (χ2

T)min

12×10−5 16.23 542.75 558.98

12

0.23 0.25 0.27 0.29 0.31

-1.

-0.5

0.

0.5

1.

1.5

Wm0

Β

HAL ΧHSNeIa+CMB + BAO L2 — æ

ΧHSNeIa+CMB + BAO +OHDL2 -- æ

................................

:DΧT

2 = 3.53 — ,--

DΧT2 = 6.25 — ,--

DΧT2 = 8.02 — ,--

-1.4 -1.2 -1. -0.80.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HBL

-1.4 -1.2 -1. -0.8

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HCL

Fig. 7 In this figure the behaviour ofχ2T = χ2

SNe+ χ2OHD + χ2

CMB +χ2

BAO (dashed counters)andχ2T = χ2

SNe+χ2CMB+χ2

BAO (solid counters)for ∆ χ2

T = 3.53(inner loops),6.25(middle loops),8.02(outer loop)are compared. The minimum points of these twoχ2

T functions are dis-tinguished by Solid points.

0.22 0.24 0.26 0.28

-1.

-0.8

-0.6

-0.4

-0.2

0.

Wm0

Β

HAL ΧHCMB + BAO +OHDL2 — æ

ΧHSNeIa+CMB + BAO +OHDL2 -- æ

................................

:DΧT

2 = 0.1 — ,--

DΧT2 = 0.2 — ,--

DΧT2 = 0.3 — ,--

-1.8 -1.6 -1.4 -1.2 -1.

0.22

0.24

0.26

0.28

Ω1

Wm

0

HBL

-1.8 -1.6 -1.4 -1.2 -1.

-1.

-0.8

-0.6

-0.4

-0.2

0.

Ω1

Β

HCL

Fig. 8 In this plot we considerχ2T = χ2

SNe+ χ2OHD + χ2

CMB + χ2BAO

(dashed counters) andχ2T = χ2

OHD+ χ2CMB + χ2

BAO (solid counters) for∆ χ2

T = 0.1(inner loops),0.2(middle loops),0.3(outer loop) to investi-gate degeneracy in this work. This Figure and Figure7 indicate thatalthough the importance of individual OHD data surveying incosmo-logical investigations (in comparison SNe Ia, CMB and BAO) is notso important but it causes decreasing degeneracy between free param-eters of the model. The minimum points of these twoχ2

T functions aredistinguished by Solid points.

0.23 0.25 0.27 0.29 0.31 0.33

0.960.981.001.021.041.061.08

Wm0

Χ2

Χm2

Χ 2

-1.6 -1.4 -1.2 -1. -0.8 -0.6

0.960.981.001.021.041.061.08

Ω1Χ

2

Χm2

Χ 2

-1. -0.5 0. 0.5 1. 1.50.960.981.001.021.041.061.081.10

Β

Χ2

Χm2

Χ 2

Fig. 9 In above diagrams the minimum quantity ofχ2 (blue line) andχ2

m (red-upper-line) versusΩm0, ω1 andβ parameters have been drownrespectively.

Table 3 This table is related to minimum point ofχ2T = χ2

SNe+χ2OHD+

3χ2CMB +3χ2

BAO.

β ω1 Ωm0 χ2BAO

−0.239 −1.051 0.272 5×10−4 ,

χ2CMB χ2

OHD χ2SNe (χ2

T)min

8×10−10 16.23 542.75 558.98

13

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNeIa+CMB + BAO +OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HBL

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HCL

Fig. 10 In diagram (A), the image of χ2m ≤ 1 on the

(Ωm0,β ) surface are portrait. In (B) the minimum pointof χ2

T = χ2SNe + χ2

OHD + χ2CMB + χ2

BAO and the shadow of∆ χ2

T = 3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaceson the(Ωm0,β ) plate, are plotted. In part(C) both diagrams(A) and(B) are brought to compare the results.

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNeIa+CMB +BAO +OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HBL

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HCL

Fig. 11 In diagram (A), the image of χ2m ≤ 1 on the

(ω1,Ωm0) surface are portrait. In (B) the minimum pointof χ2

T = χ2SNe + χ2

OHD + χ2CMB + χ2

BAO and the shadow of∆ χ2

T = 3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaceson the(ω1,Ωm0) plate, are drawn. In part(C) both diagrams(A) and(B) are considered for more comparison.

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNeIa+CMB +BAO +OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HBL

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HCL

Fig. 12 In diagram (A), the image of χ2m ≤ 1 on the

(ω1,β ) surface are portrait. In (B) the minimum pointof χ2

T = χ2SNe + χ2

OHD + χ2CMB + χ2

BAO and the shadow of∆ χ2

T = 3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaceson the (ω1,β ) plate , as counter lines, are drawn. In part(C) bothdiagrams(A) and(B) are compared.

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNeIa+3 CMB +3 BAO+OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HBL

0.23 0.25 0.27 0.29 0.31

-1.0

-0.5

0.0

0.5

1.0

1.5

Wm0

Β

HCL

Fig. 13 In diagram (A), the image ofχ2m ≤ 1 on the (Ωm0,β )

surface are portrait. In(B) the minimum point of χ2T =

χ2SNe + χ2

OHD + 3χ2CMB + 3χ2

BAO and the shadow of∆ χ2T =

3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaces on the(Ωm0,β ) surface, as counter lines, are drawn. In part(C) both diagrams(A) and(B) have been brought for more comparison.

14

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNe+3 CMB + 3 BAO+OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HBL

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HCL

Fig. 14 In diagram (A), the image ofχ2m ≤ 1 on the (ω1,Ωm0)

surface are portrait. In(B) the minimum point of χ2T =

χ2SNe + χ2

OHD + 3χ2CMB + 3χ2

BAO and the shadow of∆ χ2T =

3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaces on the(ω1,Ωm0) plate, as counter lines, are drawn. In part(C) both diagrams(A) and(B) are brought for comparison.

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HAL Region of Χm2 £ 1.0 à

............................

ΧT2 = ΧHSNe+3 CMB + 3 BAO+OHDL

2

:DΧT

2 = 3.53 —

DΧT2 = 6.25 —

DΧT2 = 8.02 —

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HBL

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HCL

Fig. 15 In diagram (A), the image of χ2m ≤ 1 on the (ω1,β )

surface are portrait. In(B) the minimum point of χ2T =

χ2SNe + χ2

OHD + 3χ2CMB + 3χ2

BAO and the shadow of∆ χ2T =

3.53(inner loop),6.25(middle loop),8.02(outer loop) surfaces on the(ω1,β ) plate, as counter lines, are drawn. In part(C) both diagrams(A) and(B) are collected for comparison.

0.23 0.25 0.27 0.29 0.31

-1.

-0.5

0.

0.5

1.

1.5

Wm0

Β

HAL SNe+ 3CMB + 3BAO + OHD

A: LHWm0,ΒL

B: LHΩ1,Wm0L

C: LHΩ1,ΒL

C. Levels of : 68.3%

90.0%

95.4%

-1.4 -1.2 -1. -0.8 -0.60.23

0.25

0.27

0.29

0.31

Ω1

Wm

0

HBL

-1.4 -1.2 -1. -0.8 -0.6

-1.

-0.5

0.

0.5

1.

1.5

Ω1

Β

HCL

Fig. 16 In the above two dimensional likelihood diagrams, the 68.3%confidence level (dotted line), 90.0% confidence level (green-dashed-line) and 95.45% confidence level (red-solid-line) after marginalizationon theω1, β andΩm0 free parameters are plotted. Note that in this fig-ure we useχ2

T = χ2SNe+ χ2

OHD+3χ2CMB +3χ2

BAO and also the shadowof ∆ χ2

T = 3.53(inner loop),6.25(middle loop),8.02(outer loop) sur-faces.

Table 4 In this table the quantities which maximize the relative prob-ability functionsL (Ωm0), L (ω1) andL (β ) using confidence levelsσ1 = 68.3% andσ2 = 95.4% are calculated. The data sets are includesof SNeIa, CMB, BAOandOHD in which the weight ofχ2

CMB andχ2BAO

in χ2Total function is the coefficient 3.

σ−2 σ+

2 σ−1 σ+

1 (L )max x

0.02 0.02 0.01 0.01 0.272 Ωm0

0.16 0.156 0.08 0.08 -1.04 ω1

0.44 0.59 0.23 0.27 -0.24 β

15

0.24 0.26 0.28 0.30 0.32

0

0.2

0.4

0.6

0.8

1.

Wm0

LHW

m0L

68.3%

95.4%

MarginalizedRelativeLikelihoodforthedatasetof:SNeIa+3´CMB+3´BAO+OHD

Β=-0.24-0.23,-0.44

+0.27,+0.59

Ω1=-1.04-0.08,-0.16

+0.08,+0.156

Wm0=0.272-0.01,-0.02

+0.01,+0.02

-1.4 -1.2 -1. -0.8 -0.6

0

0.2

0.4

0.6

0.8

1.

Ω1

LHΩ1L 68.3%

95.4%

-1. -0.5 0. 0.5 1. 1.5

0

0.2

0.4

0.6

0.8

1.

Β

LHΒL 68.3%

95.4%

Fig. 17 In above diagrams the relative likelihoods are plotted. The68.3% and 95.4% confidence levels are distinguished as brown dashedline and red dot-dash line respectively. It should be noted that inthese figure we useχ2

T = χ2SNe+ χ2

OHD + 3χ2CMB + 3χ2

BAO. Also the

best fits of the free parameters are asβ = −0.24+0.27,+0.59−0.23,−0.44, ω1 =

−1.04+0.08,+0.156−0.08,−0.16 andΩm0 = 0.272+0.01,+0.02

−0.01,−0.02.