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Physics Letters B

www.elsevier.com/locate/physletb

Cosmography: Supernovae Union2, Baryon Acoustic Oscillation, observational Hubble data and Gamma ray bursts

Lixin Xua b c'*, Yuting Wanga

a Institute of Theoretical Physics, School of Physics & Optoelectronic Technology, Dalian University of Technology, Dalian H6024, PR China b College of Advanced Science & Technology, Dalian University of Technology, Dalian H6024, PR China c Korea Astronomy and Space Science Institute, Yuseong Daedeokdaero 776, Daejeon 305-348, Republic of Korea

ARTICLE INFO

Article history:

Received 12 April 2011

Received in revised form 7 June 2011

Accepted 30 June 2011

Available online 5 July 2011

Editor: S. Hannestad

Keywords: Cosmography

ABSTRACT

In this Letter, a parametrization describing the kinematical state of the universe via cosmographic approach is considered, where the minimum input is the assumption of the cosmological principle, i.e. the Friedmann-Robertson-Walker metric. A distinguished feature is that the result does not depend on any gravity theory and dark energy models. As a result, a series of cosmographic parameters (deceleration parameter q0, jerk parameter j0 and snap parameter s0) are constrained from the cosmic observations which include type Ia supernovae (SN) Union2, the Baryon Acoustic Oscillation (BAO), the observational Hubble data (OHD), the high redshift Gamma ray bursts (GRBs). By using Markov Chain Monte Carlo (MCMC) method, we find the best fit values of cosmographic parameters in 1a regions: H0 = 74.299-4 932, q0 = -0 -386+0 658, j0 = —4.925+f -257 and s0 = -26 .404+2°0964 which are improved remarkably. The values of q0 and j0 are consistent with flat ACDM model in 1a region. But the value of s0 of flat ACDM model will go beyond the 1a region.

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1. Introduction

The kinematical approach to describe the status of universe is interesting for its distinguished feature that it does not rely on any dynamical gravity theory and dark energy models. Then it becomes crucial for its potential ability to distinguish cosmological models when a flood of dark energy models and modified gravity theories are proposed to explain the current accelerated expansion of our universe. This late time accelerated expansion of our universe was firstly revealed by two teams' observation of type Ia supernovae [1,2]. In general, via the Taylor expansion of the scale factor a(t) in terms of cosmic time t, the dimensionless coefficients q0, j0 and s0 named deceleration, jerk and snap parameters are defined respectively, for the detailed forms please see Eqs. (8), (9), (10) in the following. For convenience, they are dubbed as cosmo-graphic parameters. These cosmographic parameters, which current values can be determined by cosmic observations, describe the kinematical status of our universe. For example, the present value of Hubble parameter H0 describes the present expansion rate of our universe, and a negative value of q0 means that our universe

* Corresponding author at: Institute of Theoretical Physics, School of Physics

& Optoelectronic Technology, Dalian University of Technology, Dalian 116024, PR China.

E-mail address: lxxu@dlut.edu.cn (L. Xu).

is undergoing an accelerated expansion. This kind of approach is also called cosmography [3,4], cosmokinetics [5,6], or Friedmann-less cosmology [7,8]. Recently, this approach was considered by using SN in Ref. [9], SN + GRBs in Ref. [10] and SN + OHD + BAO in [11], where the current status of our universe can be read. On the other hand, for a concrete dark energy model or gravity theory, when the Friedmann equation is arrived the corresponding cosmo-graphic parameters can be derived by simple calculation. As a consequence, the corresponding parameter spaces can be fixed from cosmographic parameters space without implementing annoying data fitting procedure. However, the reliability of the cosmographic approach depends crucially on how the cosmographic parameter space is shrunk, in other words, the improvement of the figure of merit (FoM). That is the main motivation of this Letter. In general, when more cosmic observational data sets are added to constrain model parameter space, the more degeneracies between model parameters will be broken. Also the FoM will be improved. So, to investigate the current status of our universe and to improve the FoM, the cosmographic parameters will be determined by more cosmic observations. When the SN and GRBs are used as distance indicators, the Hubble parameter H0 and the absolute magnitudes of SN and GRBs are treated as notorious parameters and marginalized. That is to say, SN and GRBs cannot fix the current value of Hubble parameter H0. That is what the authors have done in Refs. [9,10] where the cosmographic parameters q0, j0 and s0 were investigated. However, the cosmographic parameters permeate in a

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relative larger space. Of course, to describe the kinematical status of our universe well, one has to shrink the parameter space efficiently. Fortunately, when the Hubble parameter H0 is fixed as done in Ref. [11], the parameter space is pinned down effectively. When the snap parameter s0 is included, high redshift observations should be added. So, in this Letter we are going to use SN, BAO, GRBs, OHD to investigate the cosmographic approach. When SN data sets are used, the systematic errors are included. The BAO are detected in the clustering of the combined 2dFGRS and SDSS main galaxy samples, so it is helpful to break the degeneracies between parameters. The OHD data sets are used to fix the Hubble parameter H0. Higher redshift data points are from GRBs where the correlation parameters are calibrated via cosmographic approach synchronously. For the detailed description of these data sets, please see Appendix A.

This Letter is structured as follows. In Section 2, the definition of cosmographic parameters and basic expansions with respect to redshift z are presented, where to consider the convergence issue, the map from z e (0, œ) to y = z/(1 + z) e (0,1) is adopted. To the expansion truncation problem, we compare the expansions with ACDM model in the range of redshift involved in this Letter. The relative departure of Hubble parameter from that of ACDM model is up to 20% at the redshift z ~ 1.75. The difference of distance modulus between the expansion of luminosity distance and that of ACDM model is less than 1.6. Section 3 are the main results of this Letter. To obtain these results, the cosmic observational data sets from SN la, BAO, OHD and GRBs and MCMC method are used. The detailed descriptions are shown in Appendix A. The main points of this Letter are listed as follows: (1) BAO and OHD are used to shrink the model parameter space.1 (2) The calibration of GRBs and constraint to cosmographic parameters are carried out synchronously. In this way the so-called circular problem is removed. We summarize the results in Table 1 and Figs. 2 and 3. Section 4 is a brief conclusion.

2. Cosmographic parameters

The minimum input of the cosmographic approach is the assumption of the cosmological principle, i.e. the FriedmannRobertson-Walker (FRW) metric

dr , + r2(de2 + sin2 9 d02) 1 — kr2 y '

where the parameter k = 1, 0, — 1 denotes spatial curvature for closed, flat and open geometries respectively. In this Letter, we only consider the spatially flat case k = 0.

The Hubble parameter H (z) can be expanded as

ds2 = —dt2 + a (t)

H (z) = H 0 +

1 d2 H

1 d3 H 3 dz3

z3 + •••,

where the subscript '0' denotes the value at the present epoch and z = 1/a(t) — 1. Via the relation

dz = —(1 + z)H(z) , one has

dH dz d2 H dz2

(1 + z)H

= (1 + qo) H 0,

(1 + z)2H2

0 + V (1 + z)2 H (1 + z)2 H3

1 After our work, the papers used BAO and OHD appeared in arXiv: J.Q. Xia et al.,

arXiv:1103.0378 and S. Capozziello et al., arXiv:1104.3096.

= ( jo + 3qo + 2)H 0 — (q0 + 3q0 + 2) H 0 = (jo — o0) H 0,

d3 H dz3

(1 + z)3 H3

H /1 1 dH

— 3 (1 + z)2H2 \1 + z + H~dz

(1 + z)H 2

H2 v dz J

(1 + z)2 (1 + z) H dz

2 fdH\2 1 d2H'

— + 77

= (6 + 12qo + 3q2 + 4 jo — so) H0 — 3(2 + 3q0 + jo)(2 + qo) H0 + (1 + qo)[2 + 2(1 + q0) + 2(1 + qo)2 + q2 — jo] H 0 = [3q3 + 3q0 — jo(3 + 4q0) — s0]H 0, where the cosmographic parameters are defined as follows

da(t) 1

q0 = —

dt a(t) 1 d2a(t) 1

H2 dt2 a(t) 1 d3a(t) 1

H3 dt3 a(t) 1 d4a(t) 1

0 H2 a(t) 1 a(3)(t)

H4 dt4 a(t)

0 H3 a(t) 1 a(4)(t)

H4 a(t)

(8) (9)

Then the Hubble parameter can be rewritten in terms of the cosmographic parameters as

H (z) = H 0{ 1 + (1 + q0)z + (jo — q20) z2 /2

+ {3q3 + 3q2 — j0(3 + 4q0) — S0]z3/6 +•••}.

For a spatially flat FRW universe, the luminosity distance can also be expanded in terms of redshift z with the cosmographic parameters

dL (z) = cH-1 {z + (1 — q0)z2/2 — (1 — q0 — 3q2 + j0)z3/6 + {2 — 2q0 — 15q2 — 15q3 + 5 J0

+ 10q0 j0 + sa] z4/24 + .

Via the relation dA(z) = dL(z)/(1 + z)2, one has the expansion of

dA (z)

dA (z) = cH—1 {z — (3 + q0 )z2/2 + (11 — J0 + 7q0 + 3q0)z3/6

+ (—50 + 13 j0 — 46q0 + 10 j0q0 — 39q0 — 15q3 + s0 )z4/24 + •••}.

To avoid problems with the convergence of the series for the highest redshift objects, these relations are recast in terms of the new variable y = z/(1 + z) [12,13]

H (y) = H 0{1 + (1 + q0) y + (1 + q0 + jo/2 — q0/2)y2

+ (6 + 3 j0 + 6q0 — 4q0 jo — 3q0 + 3q3 — s0 ) y3/6 + (1 + qo — 2 joqo + 3q0/2 — so/2) y4 + O(y5 )}, (14) dL (y) = cH—1 {y + (3 — qo) y2/2 + (11 — jo — 5qo + 3q2) y3/6

+ (50 — 7 jo — 26qo + 10qo jo + 21 q0 — 15q0 + so) y4/24 + O( y5)},

Fig. 1. The Hubble parameter departure and differences of distance modulus from ACDM model, where Qm0 = 0.27 is fixed.

Table 1

The results of xmin, Ho, q0, jo and s0 in Case I (SN + BAO + GRBs) and Case II (SN + BAO + GRBs + OHD), where d.o.f. denotes the degree of freedom. a and b are parameters from Amati's correlation of GRBs, for their definition please see Eq. (A.13).

xm in/d-o-f-

Case I Case II

656.821/661 670.954/676

74 299+4-932 /4-2 99—4.287

-0-150—0.752 —0.386—0.618

c Q/iQ+10.0999

—5.848—14.412 4.925+6.658

—4.925—7.297

—81268—98.7?8 —26.404+—92.00.99674

_9 522 +0.0909 —9.522—0.104

9.540+0.104 9-54°—0.0999

1 499+0.173 1.499-0.159 1 483+0.187 1483—0.166

dA (y) = cH—1 { y - (1 + q0) y2/2 — (1 + jo — q0 - 3q0) y3/6

+ (-2 + j0 + 2q0 + 10j0Q0 - 3q0

- 15q0 + s0) y4/24 + O(y5)}. (16)

With this new variable, z e (0, m) is mapped into y e (0,1). And the right behavior for series convergence at any distance can be retrieved in principle [12,13]. When the convergence problem is solved, one has to concern the expansion truncation issue. Of course, with higher orders expansion, more accurate approximation would be obtained. However, in this way, one has to introduce more model parameters beyond H0, q0, j0 and s0. How to keep the balance between the free model parameters (or expansion truncation) and comic observational data points is another complicated problem. That is beyond the scope of this Letter. But we'd like to point out that the way out may be the so-call Bayesian evidence method. In fact, we can show the deviations of the expansions from ACDM model. For illustration, with fixed value of Qm0 = 0.27, the relative departure of Hubble parameter from ACDM model (the left panel) and differences of distance modulus to ACDM model (the right panel) are shown in Fig. 1. Actually, in the redshift range (z e [0, 1.75], please see Table 2) of the observational Hubble parameters, the relative departure of ACDM model is up to ~ 20% which is almost the same of order of error bars of OHD. In the right panel of Fig. 1, the difference of distance modulus between the expansion of luminosity distance and that of ACDM model is shown. At high redshift y ~ 1, the departure is larger up to 4. In the redshift range of this Letter, z e [0, 9], the difference of distance modulus is less than 1.6. So, up to the fourth oder of y, these expansions are safe.

As the reader has noticed the Taylor expansion is up to snap parameter s0, with these cosmographic parameters the Hubble parameter is of the order z3. However, dL (z) and dA (z) are of the order z4. This is really from the fact that the Hubble parameter has contained one order derivative of time t. When it is up to the same order of dL (z) and dA (z), an extra new parameter has to be introduced. So we will classify the data sets on hand into two cases with (Case I: SN + BAO + GRBs) or without (Case II:

SN + BAO + GRBs + OHD) the observational Hubble data. Another reason is that the cosmic observational data sets of SN and GRBs do not have constraint to Hubble parameter H0. That can be seen clearly from the left panel of Fig. 2 in this Letter. So, to fix the current value of Hubble parameter, the OHD data sets should be added. The reader can also see that the BAO data set is helpful to shrink the parameter space.

3. Results and discussion

In our calculations, we have taken the total likelihood function L a e-x /2 to be the products of the separate likelihoods of SN (with systematic errors), BAO, GRBs and OHD. Then we get x2

x 2 = xSN + xBAO + xGRBs + xOHD, (17)

where the separate likelihoods of SN, BAO, GRBs, OHD and the current observational data sets used in this Letter are shown in Appendix A.

In our analysis, we perform a global fitting to determine the cosmographic parameters using the MCMC method. Our code is based on the publicly available CosmoMC package [14]. The results are shown in Table 1 and Fig. 2. And the evolution curves of the Hubble parameter and distance modulus with respect to redshift z are shown in Fig. 3 where the best fitted values of model parameters are adopted from the third row of Table 1.

One can clearly see that when the observational Hubble data are used the 1a error parameters space is shrunk remarkably. Put in other words, the figure of merit is improved tremendously. It is really from the fact that the Hubble parameter H is expressed in terms of z or y with combined cosmographic parameters coefficients. Also, from the second row of Table 1, one has noticed that the BAO data set is helpful to break the degeneracy and shrink the parameter space. We can test the reliability by comparing the result with spatially flat ACDM model. For the spatially flat ACDM model, we can easily find the corresponding deceleration, jerk and snap parameters respectively

-1 -0.5 0 -10 -5 0 -30-20-10 0 70 75 Чп in sn Hn

.65-9.55-9.45 1.4 1.6 a b

Fig. 2. The 1-D marginalized distribution and 2-D contours of model parameter spaces with 1a, 2a regions. Left panel: Case I: SN + BAO + GRBs. Right panel: Case 1 SN + BAO + GRBs + OHD.

Fig. 3. The Hubble parameter and distance modulus with respect to redshift z, where the best fitted model parameter values in the third row of Table 1 are adopted.

qo = 2 ßmö — 1,

jo = 1,

so = 1 — 2 ^mo-

When ¿2m0 varies in the range ¿2m0 e [0, 1], q0 and s0 will be in the ranges q0 e [-1, 0.5] and s0 e [-3.5, 1] respectively. For comparing the best fit values of cosmographic parameters in Case II with the spatially flat ACDM model, where the same data sets combination is used to constrain the flat ACDM model, one finds

the corresponding result: Qm0 = 0.2 70-0.0305, a = -9.398+0.0723

and b = 1.602+°'135. One can clearly see that for the best fit value of Qm0 = 0.270. in flat ACDM model the derived q0 = -0.595 and j0 = 1 are consistent with the results obtained from cosmographic approach in 1a region. However, the value of s0 = -0.215 of flat ACDM model is out the range of the 1a region of cosmographic approach. As discussed in Ref. [11], once the parameterized deceleration parameter q(z) = q0 + q1 z/(1 + z) [15] is known, one can find the relation q1 = -q0 - 2q2 - j0. Also one can find other interesting relations, for example the relations between the modified gravity theory, DGP brane world model, w = constant, CPL parameterized equation of state of dark energy [13] and cos-

mographic parameters were investigated in Ref. [16], see also in Ref. [11].

4. Conclusion

In this Letter, the cosmographic approach is reconsidered by using cosmic observational data which include SN Union2, BAO, GRBs and OHD via MCMC method. We find the best fit values of cosmographic parameters in 1a ranges: H0 = 74.29 9+4237, q0 =

—0.386—0.658, j0 = —4.925+7.657 and s0 = — 26.4O4+2°0974 which

are improved remarkably. Comparing with the spatially flat ACDM model, one can find out that the derived values of q0 and j0 in flat ACDM are consistent with the results obtained from cosmo-graphic approach in 1 a region. But the value of s0 of flat ACDM model is out of the 1a region of cosmographic best fit value. As investigated, the BAO data set is helpful to shrink the parameter space. When the OHD data sets are added, the parameters space is improved remarkably. The reason is from the fact that the Hubble parameter H is expressed in terms of z or y with combined cosmographic parameters coefficients. In summary, the main points of this Letter are that (1) BAO and OHD are helpful to shrink the parameter space. (2) The calibration of GRBs and constraint to cos-mographic parameters are carried out synchronously. It is away from the so-called circular problem.

Acknowledgements

This work is supported by NSF (10703001) and the Fundamental Research Funds for the Central Universities (DUT10LK31). We thank Dr. V. Vitagliano for his correspondence and anonymous referee for the constructive and helpful comments.

Appendix A. Cosmic observational data sets

A.1. Type Ia supernovae

Recently, SCP (Supernova Cosmology Project) Collaboration released their Union2 dataset which consists of 557 SN Ia [17]. The distance modulus fi(z) is defined as

ßth(z) = 5log10[ d l (z)] + m0,

where dL (z) is the Hubble-free luminosity distance H0dL (z)/c = H0dA(z)(1 + z)2/c, with H0 the Hubble constant, and /x0 = 42.38 — 5log10 h through the re-normalized quantity h as H0 = 100h kms-1 Mpc-1. Where dL(z) is defined as

dL (Z) = (1 + z)r(z), c

r(z) =

where E2(z) = H2 (z)/H2 Additionally, the observed distance moduli i^obs(zi) of SN Ia at zi are

ßobs(zi) = mobs(zi) - M,

where M is their absolute magnitudes.

For the SN Ia dataset, the best fit values of the parameters ps can be determined by a likelihood analysis, based on the calculation of

x 2 (ps M/) _Y^{^obs(zi) - Vth( ps, zi )}2

{5log10[dL(Ps, zi)] - mobs(zi) + M'}

where M' = fz0 + M is a nuisance parameter which includes the absolute magnitude and the parameter h. The nuisance parameter M' can be marginalized over analytically [18] as

X2 (Ps) = -2ln I exp

- 2 X 2(Ps, M')

resulting to

X 2 = A---+ ln

B2 , f C

A = ^{5l°S10[dl(Ps, zi)] - mobs(zi)} • Cov-

{5logw[dl(Ps, zj)] - mobs(zj

B = Cov-1 •{ 5log10 d (Ps, zj)] - mobs(zj i

C = £ Cov--1,

where Cov-1 is the inverse of covariance matrix with or without systematic errors. One can find the details in Ref. [17] and the web site2 where the covariance matrix with or without systematic errors are included. Relation (A.4) has a minimum at the nuisance parameter value M' = B/C, which contains information of the values of h and M. Therefore, one can extract the values of h and M provided the knowledge of one of them. Finally, the expression

x2n(Ps, B/C) = A - (B2/C),

which coincides to Eq. (A.5) up to a constant, is often used in the likelihood analysis [18,19]. Thus in this case the results will not be affected by a flat M' distribution. It worths noting that the results will be different with or without the systematic errors. In this work, all results are obtained with systematic errors.

A.2. BAO

The BAO are detected in the clustering of the combined 2dFGRS and SDSS main galaxy samples, and measure the distance-redshift relation at z = 0.2. BAO in the clustering of the SDSS luminous red galaxies measure the distance-redshift relation at z = 0.35. The observed scale of the BAO calculated from these samples and from the combined sample are jointly analyzed using estimates of the correlated errors, to constrain the form of the distance measure Dv (z) [20-22]

Dv (z) =

(1 + z)2 DA (z)

where DA(z) is the proper (not comoving) angular diameter distance which has the following relation with dL(z)

Da (Z) =

dL (z) (1 + z)2 •

Matching the BAO to have the same measured scale at all redshifts then gives [23]

DV (035)/DV (02 = 1736 ± 0^065^ Then, the xiAO (Ps) is given as

[Dv(035)/Dv(02) - 1-736]2

XBAO(Ps)=

A.3. Gamma ray bursts

0^0652

(A.10)

(A.11)

Following [24], we consider the well-known Amati's Epi-Eiso correlation [25-28] in GRBs, where EVj = Epobs(1 + z) is the cos-mological rest-frame spectral peak energy, and Eiso is the isotropic energy

EiSo = 4n d2Sbolo/0 + z)

(A.12)

in which dL and Sbolo are the luminosity distance and the bolo-metric fluence of the GRBs respectively. Following [24], we rewrite the Amati's relation as

log — = a + b log ~P,i •

erg 300 keV

(A.13)

In [29], the correlation parameters were calibrated via cos-mographic approach. Following this method, we take correlation parameters a and b as free parameters when GRBs is used as a

http://supernova.lbl.gov/Union/.

Table 2

The observational H (z) data [44,45].

z 0 0.1 0.17 0.27 0.4 0.48 0.88 0.9 1.30 1.43 1.53 1.75

H(z) (kms-1 Mpc-1 ) 74.2 69 83 77 95 97 90 117 168 177 140 202

1a uncertainty ±3.6 ±12 ±8 ±14 ±17 ±60 ±40 ±23 ±17 ±18 ±14 ±40

cosmic constraint. We fit the Amati's relation through the minimization x2 given by [24]

X0RBs(Ps) — yi

yi - a - bxi

i °y,i + b2ali + '

Xi — log

300 keV

yi — log- — log

4n Sbolo,\

+ 2logd L

where dL is defined as [30] d l = H o(1 + z)r(z)/c,

(A.14)

(A.15) (A.16)

(A.17)

and the errors are calculated by using the error propagation law [31]:

y,i ■

ln 10Epi

CTsbolo,i

ln10Sbolo,i '

(A.18) (A.19)

Here N = 109 GRBs data points are taken from [32]. The x2 is large and dominated by the systematic errors, and the statistical errors on a and b are small. In general the systematic error asys can be derived by required x2 = v (the degrees of freedom) [24]. Here, we take the value of o^ = 0.324 from Table 1. of the case of Qm0 = 0.27 in Ref. [33]. In fact, the concrete value does affect the results concluded in this Letter. At last, the total error is a2ot = ajtat + oS2ys. It would be noticed that in our case, the best fit value of a will be less than 2 log(c/H0) in the definition of luminosity distance dL = (1 + z)r(z) [30].

A.4. Observational Hubble data

The observational Hubble data are based on differential ages of the galaxies [34]. In [35], Jimenez et al. obtained an independent estimate for the Hubble parameter using the method developed in [34], and used it to constrain the EOS of dark energy. The Hubble parameter depending on the differential ages as a function of redshift z can be written in the form of

H(z) — -

1 dz 1 + z dt '

(A.20)

So, once dz/dt is known, H(z) is obtained directly [36]. By using the differential ages of passively-evolving galaxies from the Gemini Deep Deep Survey (GDDS) [37] and archival data [38-43], Simon et al. obtained H(z) in the range of 0.1 < z < 1.8 [36]. In [44], Stern et al. used the new data of the differential ages of passively-evolving galaxies at 0.35 < z < 1 from Keck observations, SPICES survey and VVDS survey. The twelve observational Hubble data from [36,44,45] are list in Table 2. Here, we use the value of Hubble constant H0 = 74.2 ± 3.6 kms—1 Mpc—1, which is obtained by observing 240 long-period Cepheids in [45]. As pointed out in [45], the systematic uncertainties have been greatly reduced by the unprecedented homogeneity in the periods and metallicity of these Cepheids. For all Cepheids, the same instrument and filters are used to reduce the systematic uncertainty related to flux

calibration. In addition, in [46], the authors took the BAO scale as a standard ruler in the radial direction, called "Peak Method", obtaining three more additional data: H(z = 0.24) = 79.69 ± 2.32, H(z = 0.34) = 83.8 ± 2.96, and H(z = 0.43) = 86.45 ± 3.27, which are model and scale independent. Here, we just consider the statistical errors.

The best fit values of the model parameters are determined by minimizing

xshd(Ps ) — J2

[Hth(Ps; zi) - Hobs(zi)]2

CT 2(zi )

(A.21)

where ps denotes the parameters contained in the model, Hth is the predicted value for the Hubble parameter, Hobs is the observed value, a(zj) is the standard deviation measurement uncertainty, and the summation is over the 15 observational Hubble data points at redshifts zj. The OHD was firstly used to constrain cosmological model in [47].

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