Eur. Phys. J. C (2013) 73:2424DOI 10.1140/epjc/s10052-013-2424-8

Regular Article - Theoretical Physics

Constraining redshift parametrization parameters of darkenergy: loop quantum gravity as back ground

Ritabrata Biswasa, Ujjal Debnathb

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India

Received: 27 February 2013 / Revised: 12 April 2013 / Published online: 8 May 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract We have assumed that the FRW universe filledwith dark matter (perfect fluid with negligible pressure)along with dark energy in LQC gravity where three parame-terizations have been proposed for the variations of EOS pa-rameter ω(z) and they are Linear, CPL and JBP parameter-izations. From Stern, Stern+BAO and Stern+BAO+CMBjoint data analysis, we have obtained the bounds of the arbi-trary parameters ω0 and ω1 by minimizing the χ2 test. Thebest-fit values and bounds of the parameters are obtained by66 %, 90 % and 99 % confidence levels which are shown in(i) Figs. 1a, 1b, 1c for Stern data analysis, (ii) Figs. 2a, 2b,2c for Stern+BAO joint data analysis and (iii) Figs. 3a, 3b,3c for Stern+BAO+CMB joint data analysis for linear, CPLand JBP models. The distance modulus μ(z) against redshiftz has been drawn in Figs. 4a, 4b, 4c for our predicted the-oretical model of the three models for the best fit values ofthe parameters and the observed SNe Ia Union2 data sam-ple and we have shown that our predicted theoretical threemodels permitted the observational data sets.

1 Introduction

Failure of second quantization of gravity leads to the ideathat our existing theories may missing some fundamentalpoints. No doubt, many theories were been proposed to over-come such a problem. Loop Quantum Gravity (LQG) is in-cluded in such efforts. To be brief, LQG had taken initiativeto replace the general relativistic metric by some parameterswhich are again mostly making after the fields, to be under-stood by gauge theory. The theory and principles of LQG,when applied in the cosmological framework, create a newtheoretical framework of Loop Quantum Cosmology (LQC)[1–3]. LQC with FRW universe was studied in literature.

a e-mail: biswas.ritabrata@gmail.comb e-mail: ujjaldebnath@yahoo.com

The effect of LQG can be described by the modification ofFriedmann equation by adding a term quadratic in density.In LQC, the non-perturbative effects lead to some correc-

tion termρ2

T

ρ1to the standard Friedmann equation. With the

inclusion of this term, the universe bounces quantum me-chanically as the matter energy density reaches the level ofρ1 (order of Planck density). Future singularity appearing inthe standard FRW cosmology can be avoided by loop quan-tum gravity effects. In some cases if the phantom field isinitially rolling down the potential, the loop quantum effecthas no influence on the cosmic late time evolution and theuniverse will accelerate forever with a constant energy ratiobetween the dark energy and dark matter [4].

Ever since the discovery of the accelerated expansionof the universe by the supernova (SN) type Ia observa-tions [5, 6], many efforts have been made to understandthe mechanism of this accelerated expansion. One of theequations governing the dynamics of Friedmann equation,aa

= −4πG(ρ + 3p)/3 says that the accelerated expansion(a > 0) requires the condition (ρ + 3p) < 0. Two possi-bilities give such result. One, a negative density, of coursewhich was not a popular option to think about, while theother was to take the EoS parameter negative. Although dif-ferent observations all pointed to the existence of unknownmatter termed as dark energy (DE hereafter) [7–11], whichviolates the strong energy condition. The property of the DEis that it generates sufficient negative pressure but the natureof it is still a mystery. DE occupies about 73 % of the energyof our universe, while dark matter about 23 % and the usualbaryonic matter 4 %. The simplest candidate of DE is a tinypositive cosmological constant which obeys the equation ofstate (EoS hereafter), ω = −1. But due to the low energyscale rather than the normal scale for constant Λ, the dy-namical Λ was introduced [12]. Again at very early stage ofuniverse the energy scale for varying Λ is not sufficient. Soto avoid this problem, known as cosmic coincidence [13], anew field, called tracker field [14] was prescribed. In simi-lar way there are many models [15, 16] in Einstein gravity

Page 2 of 9 Eur. Phys. J. C (2013) 73:2424

to best fit the data. Yet its require some modifications. Fromthis point of view some alternative models are evolved. Mostof the DE models involve one or more scalar fields with var-ious actions and with or without a scalar field potential [17–19]. Now, as the observational data permits us to have a timevarying EoS, there are a bunch of models characterized bydifferent scalar fields such as a slowly rolling scalar field(quintessence) (−1 < ω < −1/3) [12, 20], k-essence [21],tachyon [22], phantom (ω < −1) [23], ghost condensate [24,25], quintom [26, 27], Chaplygin gas models [28] etc. Somerecent reviews on DE models are described in refs. [11, 29–31].

To evolution of the universe, various DE models havebeen proposed, all of which must be constrained by astro-nomical observations. In all the models, the EoS param-eter ω plays a key role and can reveal the nature of DEwhich accelerates the Universe. Different EoS lead to dif-ferent dynamical changes and may influence the evolutionof the Universe. The EoS parameter ω and its time deriva-tive with respect to Hubble time are currently constrainedby the distance measurements of the type Ia supernova andthe current observational data constrain the range of EoSas −1.38 < ω < −0.82 [32]. Recently, the combination ofWMAP3 and Supernova Legacy Survey data shows a sig-nificant constraint on the EOS ω = −0.97+0.07

−0.09 for the DE,in a flat universe [33]. Recently some parametrizations forthe variation of EOS parameters ω(z) have been proposeddescribing the DE component.

1. ω(z) = ω0 + ω1z [34]. Here ω0 = −1/3 and ω1 = −0.9with z < 1 when Einstein gravity has been considered.This grows increasingly unsuitable for z > 1. So the fol-lowing model has been proposed. This parametrizationwill be referred to as “Linear parametrization” hereafter.

2. ω(z) = ω0 + ω1z

1+z. This ansatz was first discussed by

Chevallier and Polarski [35] and later studied more elab-orately by Linder [36]. In Einstein gravity the best fit val-ues for this model while fitting with the SN1a gold dataset are ω0 = −1.58 and ω1 = 3.29. This parametrizationwill be named for short “CPL Parametrization” after theproposer Chevallier–Polarski–Lindler.

3. ω(z) = ω0 + ω1z

(1+z)2 [37]. A fairly rapid evolution ofthis EoS allowed so that ω(z) ≥ −1/2 at z > 0.5 isconsistent with the supernovae observation in Einsteingravity. We will call this parametrization “JBP” (Jassal–Bagla–Padmanabhan) parametrization.

Concerning flat universe with effective energy densitiescontributed by dust like dark matter (DM hereafter) and DEonly we need to know the Ωm, normalized dark matter den-sity of the dust-like matter and H(z) to a very high accuracyin order to get a handle on ΩX or ωX , i.e., the DE density ofthe dark energy [38, 39]. This can be a fairly strong degener-acy for determining ωX(z) from observations. TONRY data

set with the 230 data points [40] along with the 23 pointsfrom Barris et al. [41] are valid for z > 0.01. Another dataset consists of all the 156 points in the “gold” sample ofRiess et al. [5], which includes the latest points observedby HST and this covers the redshift range 1 < z < 1.6. InEinstein’s gravity and in the flat model of the FRW uni-verse, one finds ΩΛ + Ωm = 1, which are currently fa-vored strongly by CMBR data (for recent WMAP results,see [42]). In a simple analysis for the most recent RIESSdata set gives a best-fit value of Ωm to be 0.31 ± 0.04.This matches with the value Ωm = 0.29+0.05

−0.03 obtained byRiess et al. [5]. In comparison, the best-fit Ωm for flat mod-els was found to be 0.31 ± 0.08 [38]. The flat concordanceΛCDM model remains an excellent fit to the Union2 datawith the best-fit constant equation of state parameter ω =−0.997+0.050

−0.054(stat)+0.077−0.082(stat+sys together) for a flat uni-

verse, or ω = −1.038+0.056−0.059(stat)+0.093

−0.097(stat+sys together)with curvature [43].

A huge percentage of studies on DE is carried out in theframework of classical Einstein gravity. But a parallel be-lief, that quantum gravity effects would play a role in theevolution of the universe, exists. Recently, the dynamics ofphantom, quintom and Hessence in loop quantum cosmol-ogy have been studied [44–46] and behaviors different fromthat in the standard FRW cosmology, such the avoidance offuture singularities are found. In this work, we wish to anal-yse the data for linear, CPL and JBP parametrization withthe LQC as a background. The whole treatment of redshiftparametrization of DE EoS was done keeping the Einsteinuniverse in mind. Inclusion of LQC modifies the both sidesof Einstein equation. In such a circumstance it will be inter-esting to find the preferable ranges (suitable with particulardata set) of different parameters of DE EoS. The basic gen-eralized calculations are given in Sect. 2. The main mecha-nisms which will be followed to analyse the data through outis briefly given in 3. Then for Stern data set a brief analysisof data supported values of ω0 and ω1 are given in Sect. 3.1.Joint analysis of Stern + BAO and Stern + BAO + CMBhas been done in Sects. 3.2 and 3.3 respectively. Redshift–Magnitude Observations and the comparison with the the-oretical assumption are in Sect. 4. Lastly, a brief summaryand a conclusion have been included in Sect. 5.

2 Basic calculations

We consider the flat homogeneous and isotropic universe de-scribed by FRW metric, so the modified Einstein field equa-tions in LQC are given by

H 2 = ρ

3

(1 − ρ

ρc

)(1)

Eur. Phys. J. C (2013) 73:2424 Page 3 of 9

and

H = −1

2(ρ + p)

(1 − 2ρ

ρc

)(2)

where H is the Hubble parameter defined as H = aa

with a

is the scale factor. Here ρc = √3π2γ 3G2

� is called the crit-ical loop quantum density, γ is the dimensionless Barbero–Immirzi parameter. In some paper of black hole thermody-namics in LQC [47–49] γ is suggested to be 0.2375. Themotive of the loop quantized cosmological models being thereplacement of the big bang phenomena of classical gen-eral relativity by a quantum bounce in the deep-Planckianregime, it is important to set a maximum matter densitybound. ρc represents exactly this upper bound. Proposingsuch a parameter immediately requires an approximation to-wards an exact value of it. Numerical simulations of quasi-classical states give rise to such values of ρc. The exactlysolvable loop quantization of a flat Friedmann–Lemaitre–Robertson–Walker cosmology sourced by a massless, mini-mally coupled scalar field is only able so far to demonstrateanalytically the existence of the ρc for generic quantumstates [50]. This model predicted the value of ρc to be nearlyequal to 0.41ρp , where ρp is the Planck density. Many moremodels are there with different ρc values.

In Eqs. (1) and (2) ρ = ρm + ρDE and p = pDE. Hereρm is the density of matter (with vanishing pressure) andρDE, pDE are, respectively, the energy density and pressurecontribution of some DE.

Now with DE EoS as pDE = ω(z)ρDE, assuming inde-pendent conservation of DM and DE, we have

ρm + 3Hρm = 0 (3)

and

ρDE + 3H(ρDE + pDE) = 0 (4)

From the first conservation equation (3), using the fact a =1

1+z, z is the linear cosmological redshift parameter, we have

ρm = ρm0(1 + z)3 (5)

and integrating the second conservation equation for ourthree DE models we have three solutions for the density.

2.1 Expressions for linear parametrization

For linear parametrization, integrating Eq. (4), we have theexpression for the DE density

ρDE = ρDE0(1 + z)3(1+ω0) exp

{− 3ω1

1 + z

}(6)

From Eqs. (5) and (6) it is very clear that ρm0 and ρDE0 areboth present time DM and DE densities.

Now taking dimensionless density parameters Ωm0 =ρm0

3H 20

and ΩDE0 = ρDE0

3H 20

we have the expression for Hubble

parameter H in terms of redshift parameter z as follows(8πG = c = 1):

Hlin(z) = H0(1 + z)32

{Ωm0 + ΩDE0(1 + z)3ω0

× exp

{− 3ω1

1 + z

}} 12

×[

1 − 3H 20

ρc

(1 + z)3{Ωm0 + ΩDE0(1 + z)3ω0

× exp

{− 3ω1

1 + z

}}] 12

= H0Elin(z) (7)

where Elin(z) = H/H0 is a function of z.

2.2 Expressions for CPL parametrization

For CPL parametrization, integrating Eq. (4), we have theexpression for dark energy density as

ρDE = ρDE0(1 + z)3(1+ω0+ω1) exp

{3ω1

1 + z

}(8)

From Eq. (8) it is very clear that ρDE0 is present time DMand DE densities.

Now taking dimensionless density parameter ΩDE0 =ρDE0

3H 20

we have the expression for Hubble parameter H in

terms of redshift parameter z as follows:

HCPL(z) = H0(1 + z)32

{Ωm0 + ΩDE0(1 + z)3(ω0+ω1)

× exp

{3ω1

1 + z

}} 12[

1 − 3H 20

ρc

(1 + z)3

×{Ωm0 + ΩDE0(1 + z)3(ω0+ω1)

× exp

{3ω1

1 + z

}}] 12

= H0ECPL(z) (9)

where ECPL(z) = H/H0 is a function of z.

2.3 Expressions for JBP parametrization

For JBP parametrization, integrating Eq. (4), we have theexpression for DE density as

ρDE = ρDE0(1 + z)3(1+ω0) exp

{− 3ω1

1 + z+ 6ω1

(1 + z)2

}(10)

From Eq. (10) it is very clear that ρDE0 is present timeDM and DE density.

Now taking dimensionless density parameter ΩDE0 =ρDE0

3H 20

we have the expression for Hubble parameter H in

terms of redshift parameter z as follows:

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HJBP(z) = H0(1 + z)32

{Ωm0 + ΩDE0(1 + z)3ω0

× exp

{− 3ω1

1 + z+ 6ω1

(1 + z)2

}} 12

×[

1 − 3H 20

ρc

(1 + z)3{Ωm0 + ΩDE0(1 + z)3ω0

× exp

{− 3ω1

1 + z+ 6ω1

(1 + z)2

}}] 12

= H0EJBP(z) (11)

where EJBP(z) = H/H0 is a function of z.

3 Observational data analysis

In the following subsections, we present the observa-tional data analysis mechanism for Stern, Stern+BAO andStern+BAO+CMB observations. We use the χ2 minimumtest from theoretical Hubble parameter with the observeddata set and find the best fit values of unknown parametersfor different confidence levels.

3.1 Analysis for Stern Data

Using observed value of Hubble parameter at different red-shifts (12 data points) listed in observed Hubble data by[51] we analyze the model. The Hubble parameter H(z) andthe standard error σ(z) for different values of redshift z aregiven in Table 1. For this purpose we first form the χ2 statis-tics (rather we must say the least square statistics as we havesufficient data points to perform actual χ2 test) as a sum ofstandard normal distribution as follows: For any data set wewill calculate the minimum χ2, with the formula

χ2Stern =

∑ (H(z) − Hobs(z))2

σ 2(z)(12)

Table 1 The Hubble parameter H(z) and the standard error σ(z) fordifferent values of redshift z

z H(z) σ (z)

0 73 ±8

0.1 69 ±12

0.17 83 ±8

0.27 77 ±14

0.4 95 ±17.4

0.48 90 ±60

0.88 97 ±40.4

0.9 117 ±23

1.3 168 ±17.4

1.43 177 ±18.2

1.53 140 ±14

1.75 202 ±40.4

Here, for different redshifts the theoretical and observa-tional values of the Hubble parameter is given as H(z) andHobs(z). The corresponding error term is given as σ(z). Thisis, however, given in Table 1. In this statistics, the nuisanceparameter is given by Hobs which can be safely marginal-ized. The present value of the Hubble parameter, H0 is beenfixed at 72 ± 8 km s−1 Mpc−1. Considering H0 to have afixed prior distribution we will proceed. This mechanismhas recently been also discussed by several authors [52–57]in very simple way. Here we shall determine the parametersω0 and ω1 from minimizing the above distribution χ2. Theprobability distribution function in terms of the parametersω0 and ω1 can be written as

L =∫

e− 12 χ2

SternP(H0) dH0 (13)

where P(H0) is the prior distribution function for H0. Wenow plot the graph for different confidence levels (like 66 %,90 %, and 99 %).

For our models with this data set we can calculate thebest fit values of ω0 and ω1. We have plotted difference con-fidence contours in the Figs. 1a–1c. The values are writtendown in the Table 2. It is to be noted as we make our EoScomplicated if we fix ω1, the σ -contour in ω0’s directionbecomes narrowed in case of CPL. Even the converse state-ment is seemed to be true too. Anyway, as we decrease thenegativity of the ω0 to satisfy the observational data we needto increase the negativity of ω1 (i.e., we have to decreaseω1) in case of linear parametrization. For linear parametriza-tion lower ω1 is allowing a wider ω0 range (a more negativerange rather, we should say) to study inside the 1 − σ con-tour. The trend is just opposite in the case of CPL. JBP’s1 − σ contour is almost uniform in both sides for ω0 but forω1 it allows more less values to stay inside it.

3.2 Joint analysis with Stern+BAO data sets

We will follow the pathway shown by Eisenstein et al. [58]for joint analysis, the Baryon Acoustic Oscillation (BAO)peak parameter value. Here we will follow their approach.Sloan Digital Sky Survey (SDSS) survey is one of the firstredshift surveys by which the BAO signal has been directlydetected at a scale ∼100 MPc. For low redshift (0 < z <

0.35) we will check for the BAO peak to determine the DE

Table 2 (Stern): The best fit values of ω0 and ω1 and the minimumvalues of χ2 for different redshift parametrization models

DE Models Best Fit values

ω0 ω1 χ2

Linear −1.0893 −0.0185958 7.53881

CPL −1.76995 0.73844 7.13254

JBP −1.20841 1.39154 8.36388

Eur. Phys. J. C (2013) 73:2424 Page 5 of 9

Fig. 1 (a), (b) and (c) show that the variation of ω0 with ω1 for differ-ent confidence levels. The 66 % (solid, blue, the innermost contour),90 % (dashed, red, next to the innermost contour), and 99 % (dashed,

black, the outermost contour) contours are plotted in these figures forthe H(z)–z (Stern) analysis (for Linear, CPL and JBP parameteriza-tions, respectively)

parameters. The BAO peak parameters might be defined as

A =√

Ωm

E(z1)13

(1

z1

∫ z1

0

dz

E(z)

) 23

(14)

Here E(z) = H(z)/H0 is the normalized Hubble parameter,the redshift z1 = 0.35 is the typical redshift of the SDSSsample and the integration term is the dimensionless co-moving distance to the redshift z1. The value of the pa-rameter A for the flat model of the universe is given byA = 0.469 ± 0.017 using SDSS data [58] from the lumi-nous red galaxies survey. Now the χ2 function for the BAOmeasurement can be written as

χ2BAO = (A − 0.469)2

(0.017)2(15)

Now the total joint data analysis (Stern+BAO) for the χ2

function may be defined by

χ2total = χ2

Stern + χ2BAO (16)

According to our analysis the joint scheme gives the bestfit values of ω0 and ω1 in Table 3. Finally we draw the con-tours ω0 vs. ω1 for the 66 % (solid, blue), 90 % (dashed,red), and 99 % (dashed, black) confidence limits depicted inFigs. 2a to 2c.

The basic characteristics of the σ contours resembleswith the case of Stern data analysis. But the prominent

Table 3 (Stern+BAO): The best fit values of ω0 and ω1 and the min-imum values of χ2 for different redshift parametrization models

DE Models Best Fit values

ω0 ω1 χ2

Linear −0.5710 −0.40198 155.292

CPL −1.1835 0.265463 7.79739

JBP −1.39099 1.52028 7.78142

change we can find that if we fix any one of ω0 or ω1 thenthe other gets very less chance to move. Varying both theseparameters slowly we may obtain a χ2 value which will bevery close to the minimum χ2.

3.3 Joint analysis with Stern + BAO + CMB data sets

One interesting geometrical probe of DE can be determinedby the angular scale of the first acoustic peak through angu-lar scale of the sound horizon at the surface of last scatteringwhich is encoded in the CMB power spectrum Cosmic Mi-crowave Background (CMB) shift parameter is defined by[59–61]. It is not sensitive with respect to perturbations butit is suitable to constrain model parameters. The CMB powerspectrum first peak is the shift parameter which is given by

R = √Ωm

∫ z2

0

dz

E(z)(17)

where z2 is the value of redshift at the last scattering surface.From WMAP7 data of the work of Komatsu et al. [62] thevalue of the parameter has obtained as R = 1.726±0.018 atthe redshift z = 1091.3. Now the χ2 function for the CMBmeasurement can be written as

χ2CMB = (R − 1.726)2

(0.018)2(18)

Now when we consider three cosmological tests together,the total joint data analysis (Stern+BAO+CMB) for the χ2

function may be defined by

χ2TOTAL = χ2

Stern + χ2BAO + χ2

CMB (19)

Now the best fit values of ω0 and ω1 for joint analysis ofBAO and CMB with Stern observational data support thetheoretical range of the parameters given in Table 4. The66 % (solid, blue), 90 % (dashed, red) and 99 % (dashed,black) contours are plotted in Figs. 3a–3c. In this analy-sis the 1σ , 2σ , and 3 σ contours are, however, different

Page 6 of 9 Eur. Phys. J. C (2013) 73:2424

Fig. 2 (a), (b) and (c) show that the variation of ω0 with ω1 for differ-ent confidence levels. The 66 % (solid, blue, the innermost contour),90 % (dashed, red, next to the innermost contour), and 99 % (dashed,

black, the outermost contour) contours are plotted in these figures forthe H(z)–z (Stern+BAO) analysis (for Linear, CPL and JBP parame-terizations, respectively)

Fig. 3 (a), (b) and (c) show that the variation of ω0 with ω1 for differ-ent confidence levels. The 66 % (solid, blue, the innermost contour),90 % (dashed, red, next to the innermost contour), and 99 % (dashed,

black, the outermost contour) contours are plotted in these figures forthe H(z)–z (Stern+BAO+CMB) analysis (for Linear, CPL and JBPparameterizations, respectively)

Table 4 (Stern+BAO+CMB): The best fit values of ω0 and ω1 andthe minimum values of χ2 for different redshift parametrization models

DE Models Best Fit values

ω0 ω1 χ2

Linear −0.5753 −0.4033 5597.64

CPL −1.74706 0.413568 9166.93

JBP −1.4728 −1.58013 0165.12

from the previous two studies. Here we can see for linearparametrization the left side of the best fit, i.e., the low ω0,high ω1 region is highly constrained as compared to the highω0 low ω1 part. The same feature holds for CPL. But as com-pared to the linear parametrization the left side is weaklyconstrained here. Continuing the same pattern JBP has amore weakly constructed left hand side than CPL.

4 Redshift–magnitude observations from supernovaetype Ia

The Supernova Type Ia experiments provided the main ev-idence for the existence of DE. Since 1995, two teams ofHigh-z Supernova Search and the Supernova CosmologyProject have discovered several type Ia supernovas at thehigh redshifts [5, 6]. The observations directly measure thedistance modulus of a supernova and its redshift z [7, 63].Now, we take recent observational data, including SNe Ia,which consists of 557 data points and belongs to the Union2sample [43].

From the observations, the luminosity distance dL(z) de-termines the dark energy density and is defined by

dL(z) = (1 + z)H0

∫ z

0

dz′

H(z′)(20)

Eur. Phys. J. C (2013) 73:2424 Page 7 of 9

the apparent magnitude m of a supernova and its redshift z

are directly measured from the observations. The apparentmagnitude μ (the distance modulus, the distance betweenabsolute and apparent luminosity of a distance object forsupernovae) is related to the luminosity distance dL of thesupernova by the relation:

μ(z) = 5 log10

[dL(z)/H0

1 MPc

]+ 25 (21)

The best fit of distance modulus as a function μ(z) of red-shift z for our theoretical model and the Supernova Type IaUnion2 sample are drawn in Figs. 4a, 4b and 4c. It is veryclear that for low redshifts z < 0.6 all the parameterizationsare efficient enough to explain the Observational data (in thebackground of the LQC). Even 0.6 < z < 0.8 our μ(z) canbe treated as an average. But beyond that it is overestimatedas compared to the observational values.

5 Brief summary and concluding remarks

We proposed here the FRW universe filled with DM (per-fect fluid with negligible pressure) along with dark energy inLQC gravity. For describing DE components, three param-eterizations have been proposed for the variations of EOSparameter ω(z) and they are Linear, CPL and JBP parame-terizations. We present the Hubble parameter H in terms ofthe observable parameters Ωm0, ΩDE0, H0 with the redshiftz and the other parameters like ρc,ω0, and ω1. We have cho-sen the observed values of Ωm0 = 0.28, ΩDE0 = 0.72, andH0 = 72 Km s−1 Mpc−1. From Stern data set (12 points),we have obtained the bounds of the arbitrary parameters ω0

and ω1 (Table 2) by minimizing the χ2 test and by fixingthe other parameter ρc. The likelihood curves for ω0 and ω1

are smooth for the JBP case only. Otherwise they are notuniform. ω0 has a tendency to be more negative than the

least square value whereas ω1 possesses exactly the oppo-site nature for the case of linear parametrization. Next dueto joint analysis of BAO and CMB observations, we havealso obtained the best fit values and the bounds of the pa-rameters (ω0,ω1) (Table 3 and 4) by fixing some other pa-rameter ρc. The best-fit values and bounds of the parametersare obtained by 66 %, 90 %, and 99 % confidence levels,which are shown in (i) Figs. 1a, 1b, 1c for Stern data anal-ysis, (ii) Figs. 2a, 2b, 2c for Stern+BAO joint data analysisand (iii) Figs. 3a, 3b, 3c for Stern+BAO+CMB joint dataanalysis for linear, CPL and JBP models. The σ contours ofCPL and JBP are quite same, stating the fact that more neg-ative ω0 is likely with more positive ω1. The general trendsof 1σ , 2σ and 3σ contours for ω0 and ω1 are a bit the samefor Stern+BAO showing the fact that for linear parametriza-tion the more negative ω0 require more negative ω1 whereasfor CPL and JBP more positive ω1 is required. The distancemodulus μ(z) against redshift z has been drawn in Figs. 4a,4b, 4c for our theoretical model of the three models for thebest fit values of the parameters and the observed SNe IaUnion2 data sample. Here we show that our predicted the-oretical three models permitted the observational data sets.Only Stern data leaves a wide range for both ω0 and ω1 forall the parameterizations. JBP allows more wide range thanCPL or linear parametrization do. Adding BAO Peak anal-ysis, the range of both ω0 and ω1 reduces, e.g., for JBP the1σ confidence range of ω0 is greater than 34.3 and less than34.45. Addition of CMB term reduces the lower and upperfreedom of ω0 and ω1, respectively. However, very fine tun-ing is not achieved.

We will look upon the values of the ρc determined fromthis data for different dark energy models. ρc being the max-imum limit of the density should have a lower limit till time.However, as we started our work with not very large num-ber of data points our distribution is not smooth or a pre-viously known curve to fit this data is absent. So we are

Fig. 4 (a), (b) and (c) show the variation of μ(z) with z for Linear, CPL and JBP parameterizations, respectively (solid lines). The dots denotethe Union Sample (Color figure online)

Page 8 of 9 Eur. Phys. J. C (2013) 73:2424

Fig. 5 (a), (b) and (c) show the variation of ρc with z for Stern,Stern+BAO and Stern+BAO+CMB parameterizations, respectively.The plus denotes the values for Linear parametrization. Cross and star

denote the CPL and JBP parametrizations, respectively. The units ofmass and length are taken to be gms and cms, respectively

only taking the discrete points. These diagrams entail thefact that Stern gives a higher ρc infimum than others. Fig-ures 5a, 5b and 5c show the variation of ρc with z for Stern,Stern+BAO and Stern+BAO+CMB parameterizations, re-spectively. The units of mass and length are taken to be gmsand cms, respectively. Stern+BAO+CMB gives very lowdensity at present epoch with the JBP model (almost a fewkilograms per cubic centimeters only!). Value of ρc derived,with linear parametrization taken into account, does not varywith the different data analyzing methods. In any case a gen-eral trend for all the model is that as we go to the past, ρc

increases, Showing early epoch ρc is greater than ρc0 (atthe present time, say). Though an exact ratio cannot be pro-vided due to the fact that we have been used data so far upto z ≤ 1.75.

Some literature [64] can be found where the Bayesianfactor calculation for CPL and JBP says that JBP is morepreferred than CPL. The study in [64] also supports the factthat there is a bump at z ∼ 0.6 and as z ∼ 1 the accuracy ofthe current data is not enough to place effective constraints

on different parameterizations. Their analysis also says thatthere is a chance to get a tightest constraint situation at z =0.3. Even in our analysis we can see the Union 2 samplecluster does match most with μ(z) at z ∼ 3.

Acknowledgements RB thanks ISRO grant “ISRO/RES/2/367/10–11” for providing Research Associate Fellowship and Department ofPhysics, IISc, Bangalore as a part of this work was done there dur-ing a Post Doctoral period. RB also thanks CSIR project “Dark En-ergy Models and Accelerating Universe” (No. 03(1206)/12/EMR–II)for awarding Research Associate fellowship.

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