# Interacting scalar tensor cosmology in light of SNeIa, CMB, BAO and OHD observational data sets

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## Abstract

In this work, an interacting chameleon-like scalar field scenario, by considering SNeIa, CMB, BAO, and OHD data sets, is investigated. In fact, the investigation is realized by introducing an ansatz for the effective dark energy equation of state, which mimics the behavior of chameleon-like models. Based on this assumption, some cosmological parameters, including the Hubble, deceleration, and coincidence parameters, in such a mechanism are analyzed. It is realized that, to estimate the free parameters of a theoretical model, by regarding the systematic errors it is better that the whole of the above observational data sets would be considered. In fact, if one considers SNeIa, CMB, and BAO, but disregards OHD, it maybe leads to different results. Also, to get a better overlap between the contours with the constraint \(\chi _\mathrm{{m}}^2\le 1\), the \(\chi _\mathrm{{T}}^2\) function could be re-weighted. The relative probability functions are plotted for marginalized likelihood \(\mathscr {L} (\Omega _\mathrm{{m0}} ,\omega _1, \beta )\) according to the two dimensional confidence levels 68.3, 90, and \(95.4\,\%\). Meanwhile, the value of the free parameters which maximize the marginalized likelihoods using the above confidence levels are obtained. In addition, based on these calculations the minimum value of \(\chi ^2\) based on the free parameters of the ansatz for the effective dark energy equation of state is achieved.

### Keywords

Dark Energy Scalar Field Cosmic Microwave Background Deceleration Parameter Baryonic Acoustic Oscillation## 1 Introduction

Observational data sets, including the Cosmic Microwave Background (CMB) [1, 2], Supernovae type Ia (SNeIa) [3, 4], Baryonic Acoustic Oscillations (BAO) [5, 6], Observational Hubble Data (OHD) [7, 8], Sloan Digital Sky Survey (SDSS) [9, 10], and Wilkinson Microwave Anisotropy Probe (WMAP) [11, 12], are considered as a criterion for the accuracy of theoretical models. Amongst these constraints, the CMB and SNeIa (because of the abundance of their data sources) attract more attention. It is notable that the SNeIa constraint for high redshift values do not give a good clue to investigate the evolution of the Universe. It is obvious that the results of individual observations give different values for the free parameters of a theoretical model; hence, it is better that, to estimate the best values for the free parameters of the model one considers the whole of the observational data sets, including CMB, SNeIa, BAO, and OHD. This motivated us to study the behavior of the free parameters and their overlaps. Thence, a collective of observations including SNeIa, CMB, BAO, and OHD are considered. Meanwhile the mentioned observational data sets have predicted an ambiguous form of matter which leads to an accelerated phase of present epoch and is well known as dark energy. Based on this ambiguous form of matter, scientists have proposed different proposals up to now. Amongst all of those proposals, the cosmological constant, \(\Lambda \), model attracts the most attention [13, 14]. But this mechanism suffers two well-known drawbacks. The first of them is related to making an estimate of the contribution of quantum fluctuation of the zero point energy, and the second is related to the ratio of \(\Lambda \) and the dark matter energy densities. These problems and also the excellent work by Brans and Dicke [15] motivated scientists to introduce a mechanism in which \(\Lambda \) had a time dependency, namely quintessence [16, 17, 18]. Beside the quintessence mechanism, some proposals which have arisen from quantum gravity or string theory are introduced to estimate the cosmological parameters. For instance, one has the tachyon [19, 20], phantom [21, 22, 23], quintum [24, 25], k-essence [26, 27] proposals. Also some models which have a risen from quantum field fluctuations or space time fluctuations attract much attention to investigate the dark energy concept. For such models, one can 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 a scalar field, in the quintessence model, couples to (non-relativistic) matter it induces a fifth force. When the coupling is of order unity, the results of a strongly coupling scalar field is not in good agreement with local gravity tests (for instance in the solar system). Thus a mechanism should exist suppressing the effect of the fifth force; such a mechanism is capable of reconciling strong coupling models with local experiments, as proposed by Khoury and Weltman [40, 41] and also, separately, by Mota and Barrow [42], namely the chameleon-like model. In this mechanism, one cannot choose an arbitrary Lagrangian for matter, \(L_\mathrm{m}\). To avoid a deviation of the geodesic trajectory, the author of [43] has shown that the best choices are \(L_\mathrm{m}=P\) and \(L_\mathrm{m}=-\rho \), where *P* is the pressure and \(\rho \) is the energy density of matter; for more discussion we refer the reader to [44, 45, 46]. Therefore the main motivation of this work is the investigation of the behavior of an interacting scalar field mechanism; based on these calculations and the, SNeIa, CMB, BAO and OHD data sets, the minimum value of \(\chi ^2\) for the effective dark energy equation of state is achieved. The organization of the paper is as follows: The above brief discussions are a review as regards observational and theoretical motivations; they are considered as an introduction. In Sect. 2, the general theoretical discussions arising from a chameleon-like mechanism related to the cosmological parameters, such as the Hubble, deceleration, and coincidence parameters, will be discussed. In Sect. 3, a brief review of the cosmological data sets is presented. In Sect. 4, the observational data sets including SNeIa, CMB, BAO, and OHD are considered, to estimate the minimum value of \(\chi ^2\) related to the free parameters of an ansatz for the effective dark energy equation of state. Finally, Sect. 5, is dedicated to concluding remarks.

## 2 Conservation and field’s equations in an effective dark energy scenario

*g*is the determinant of the metric, \(V(\varphi )\) is a run away potential and the last term indicates a non-minimal coupling between scalar field and matter sector. It should be noted that

*L*is the Lagrangian density of matter which consists of both dark matter and dark energy sectors as perfect fluid [47, 48, 49, 52, 53]. It should be noticed that the background is a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe, with signature \((+2)\). The variation of the action (1), with respect to (w.r.t.) \(g_{\mu \nu }\) results in the gravitational field equation:

*ii*components of \(T^{(\varphi )}_{\mu \nu }\), the energy density and pressure could be obtained. After some algebra the conservation equation reads

*t*. As mentioned in the introduction, the Lagrangian of the matter is considered to verify \(L= L_\mathrm{(m)} + L_\mathrm{(de)}\), [52, 54], where the subscript m denotes matter (cold dark matter and baryons) and de refers to dark energy. Then the conservation equations could be rewritten as

*a*(

*t*) is the scale factor, and \(\omega _\mathrm{DE}\) is the EoS parameter of the effective dark energy and satisfies the EoS equation:

*z*, instead of the scale factor; these two cosmological parameters have the relation

### 2.1 Hubble parameter

### 2.2 Coincidence parameter

*r*, in the investigation of the cosmic evolution, it attracts more attention in observational investigations. In fact one can observe that this importance arises from the relation between the EoS parameter and the evolution of

*r*.

### 2.3 Deceleration parameter

## 3 A brief review as to cosmological observational data sets

In this section, we should emphasize that the analysis is restricted to the background level, and we do not include perturbations. In the following, we want to compare our theoretical results with observations. To this end, we consider four important data sets including SNeIa, CMB, BAO, and OHD. In some papers, it was claimed that OHD, as obtained versus red shift, is comparable with the SNeIa data set, for instance we refer reader to Table 1, Ref. [7] and references therein. This subject motivated us to investigate the effects of this new data set, beside other observations, to improve the theoretical results. As will be discussed, the results of OHD, although not independent of the SNeIa and BAO data sets [7], do not have any dependency on CMB. Also there are two ways to study the CMB and BAO data points; we refer to the full parameter distribution and the Gaussian; in the following the latter will be used.

### 3.1 Supernovae type Ia

*q*. It is obvious there are uncertainties of a different nature: statistical or random errors and systematic errors. In this work it is remarkable the systematic errors for SNeIa and OHD are neglected. In reality there is always a limit on the statistical accuracy, besides the trivial one that the time for repetitions is limited. The assumption of independence is violated in a very specific way by so-called systematic errors which appear in any realistic experiment. For instance experiments in nuclear and particle physics usually extract the information from a statistical data sample. The precision of the results then is mainly determined by the number N of collected reactions. Besides the corresponding well-defined statistical errors, nearly every measurement is subject to further uncertainties, the systematic errors, typically associated with auxiliary parameters related to the measuring apparatus, or with model assumptions. The result is typically presented in the form

*n*th supernova, \(\sigma _n\) is the variance of the measurement, and \(\mu _\mathrm{th} (z_n )\) is the theoretical distance modulus for the

*n*th supernova, which is defined as

*A*,

*B*, and

*C*are defined as follows:

### 3.2 Cosmic microwave background

*R*at the \(\sigma _1\) confidence level, and the recombination redshift; see, respectively, Refs. [1, 61].

### 3.3 Baryonic acoustic oscillations

### 3.4 Observational Hubble data

*H*(

*z*) measurement, for more details one can refer Table 1. But one has used \(\bar{H}_0= 68 \pm 2.8\) and \(\bar{H}_0 = 73.8\pm 2.4\), which arose from the SNeIa data [8]. Therefore it is realized that for the comparison between theoretical results and observations only OHD could not be considered. The \(\chi _\mathrm{OHD}^2\) function parameter based on the OHD data set is defined as

## 4 Cosmological constraints and data fitting

*N*denotes the amount of observational data. Whereas we use the \(\mathrm{Union}-2\) data set for SNeIa,

*N*for supernovae is \(N_\mathrm{SNe}=557,\) and also for OHD, CMB, and BAO, one has \(N_\mathrm{OHD}=28\), \(N_\mathrm{CMB}=1\), and \(N_\mathrm{BAO}=1\). Since in this work three free parameters appeared, the space of constraints has three dimensions. Thence for clarity, one can map figures on two dimensions (in fact, it is supposed that the free parameters are independent) and their values will be analyzed. The common regions for best fitting of all constraint, play a key role in this study. Based on the above discussions we plot a couple of free parameters in Figs. 1, 2, and 3. In Fig. 1 we investigate the constraints on \(\Omega _{\mathrm{m}0}\) in \(\omega _1 \, \beta \) plate, and also for two constraints SNeIa and OHD minimum points of \(\chi ^2 \) are distinguished. In Fig. 2 using best value of \(\omega _1\), the constraints in \(\Omega _{\mathrm{m}0} \, \beta \) are obtained, in a similar way for best value of \(\beta \), the behavior of constraints in \(\omega _1 \, \Omega _{\mathrm{m}0} \) surface will be shown. Let us, return our attention to Fig. 1 again. For \(\Omega _{\mathrm{m}0}=0.2\), the CMB, BAO and OHD have an overlap region, but they are not in agreement with SNeIa results. Also for a different quantity, the SNeIa and OHD results could be in agreement with together. This different behavior of constraints indicates that if one wants to compare theoretical results with observations, it is better that the greatest set of constraints would be considered. To address overlaps and the effects of individual observations, we plot Figs. 4 and 5. In Fig. 4 the behavior of \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) and \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) for \(\Delta \chi _\mathrm{T} ^2 = 3.53, 6.25, 8.02\) are compared. Also in Fig. 5 to investigate degeneracy one can consider \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) and \(\chi _\mathrm{T} ^2 = \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) for \(\Delta \chi _\mathrm{T} ^2 = 0.1, 0.2, 0.3\). These two figures indicate that, although individual OHD data surveying (in comparison with SNeIa, CMB, and BAO) is not so important, it decreases the degeneracy between the free parameters of the model. From Figs. 4 and 5, it is obvious that a collective of four constraints has completely different results in comparison to even three constraints. In the following, by means of observations, we use some custom values which are considered for better estimation of the theoretical parameters of the model. Since all free parameters of the model are independent, the total likelihood function could be introduced as

*B*and

*C*are related to \((\chi _\mathrm{T} ^2)_\mathrm{min}\), where the subscript min is for the minimum value of \(\chi _{\mathrm{T}}^2\). It is notable, in a three dimensional space of free parameters, that the confidence levels 68.3, 90, and \(95.4\,\%\) are proportional to the \(\Delta \chi _\mathrm{T} ^2 = 3.53\), \(\Delta \chi _\mathrm{T} ^2 = 6.25\), and \(\Delta \chi _\mathrm{T} ^2 = 8.02\) surfaces, respectively, where \(\Delta \chi _\mathrm{T} ^2 =\chi _\mathrm{T} ^2 - (\chi _\mathrm{T} ^2)_\mathrm{min}\). In diagram b of Figs. 7, 8, and 9 the contour lines of the confidence levels are drawn and in diagram c, both the \(\chi _\mathrm{m} ^2\) surfaces and the contour lines are shown for better comparison. From diagram c it is realized that the confidence level contours exceed the \(\chi _\mathrm{m} ^2\) regions. From this behavior it is concluded that the theoretical prediction of the CMB shift parameter is much greater than its observational value. As mentioned, when the total mean square error function is introduced the weights of all constraints were identical and this causes some problems. As a matter of fact, the results of the likelihood’s parameter, Eq. (59), the effect of the CMB shift parameter in comparison with the abundant SNeIa data set is ignored. For more information, see Table 2 and the definition of \(N_\mathrm{dof}\). To overcome these problems, we redefine \(\chi _\mathrm{T}^2\) by

In this table from left to right *z*, \(H(z) (km\, s^{-1}\, Mpc ^{-1})\), and its uncertainty \(\sigma _{H} (km\, s^{-1}\, Mpc ^{-1})\) in measurement and in related references (by considering the technique which is used) are collected, respectively

| | \(\sigma _{H}\) | References | Technique |
---|---|---|---|---|

0.070 | 69 | 19.6 | [65] | SDSS DR7; \(0<z<0.4\) |

0.100 | 69 | 12 | [66] | ATC; \(0.1< z < 1.8\) |

0.120 | 68.6 | 26.2 | [65] | SDSS DR7; \(0<z<0.4\) |

0.170 | 83 | 8 | [66] | ATC; \(0.1< z < 1.8\) |

0.179 | 75 | 4 | [67] | OHD+CMB; \(0<z<1.75\) |

0.199 | 75 | 5 | [67] | OHD+CMB; \(0<z<1.75\) |

0.200 | 72.9 | 29.6 | [65] | SDSS DR7; \(0<z<0.4\) |

0.270 | 77 | 14 | [66] | ATC; \(0.1< z < 1.8\) |

0.280 | 88.8 | 36.6 | [65] | SDSS DR7; \(0<z<0.4\) |

0.350 | 76.3 | 5.6 | [68] | SDSS DR7 LRGs; \(z=0.35\) |

0.352 | 83 | 14 | [67] | OHD+CMB; \(0<z<1.75\) |

0.400 | 95 | 17 | [66] | ATC; \(0.1< z < 1.8\) |

0.440 | 82.6 | 7.8 | [69] | WiggleZ+H(z); \(z<1.0\) |

0.480 | 97 | 62 | [70] | CMB+OHD; \(0.2< z < 1.0\) |

0.593 | 104 | 13 | [67] | OHD+CMB; \(0<z<1.75\) |

0.600 | 87.9 | 6.1 | [69] | WiggleZ+H(z); \(z<1.0\) |

0.680 | 92 | 8 | [67] | OHD+CMB; \(0<z<1.75\) |

0.730 | 97.3 | 7.0 | [69] | WiggleZ+H(z); \(z<1.0\) |

0.781 | 105 | 12 | [67] | OHD+CMB; \(0<z<1.75\) |

0.875 | 125 | 17 | [67] | OHD+CMB; \(0<z<1.75\) |

0.880 | 90 | 40 | [70] | CMB+OHD; \(0.2<z< 1.0\) |

0.900 | 117 | 23 | [66] | ATC; \(0.1< z < 1.8\) |

1.037 | 154 | 20 | [67] | OHD+CMB; \(0<z<1.75\) |

1.300 | 168 | 17 | [66] | ATC; \(0.1< z < 1.8\) |

1.430 | 177 | 18 | [66] | ATC; \(0.1< z < 1.8\) |

1.530 | 140 | 14 | [66] | ATC; \(0.1< z < 1.8\) |

1.750 | 202 | 40 | [66] | ATC; \(0.1< z < 1.8\) |

2.300 | 224 | 8 | [63] | BAO; \(0.7<z<2.3\) |

In the table, the quantities related to minimum point of \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) are introduced

\(\beta \) | \(\omega _1\) | \(\Omega _{\mathrm{m}0}\) | \(\chi _\mathrm{BAO} ^2\) |
---|---|---|---|

\(-\)0.243 | \(-\)1.053 | 0.272 | \(16 \times 10^{-4}\), |

\(\chi _\mathrm{CMB} ^2\) | \(\chi _\mathrm{OHD} ^2\) | \(\chi _\mathrm{SNe} ^2\) | \((\chi _\mathrm{T} ^2)_\mathrm{min}\) |
---|---|---|---|

\(12 \times 10^{-5}\) | 16.23 | 542.75 | 558.98 |

This table is related to the minimum point of \(\chi _\mathrm{T} ^2 = \chi _\mathrm{{SNe}} ^2 + \chi _\mathrm{{OHD}} ^2 + 3 \chi _\mathrm{{CMB}} ^2 + 3\chi _\mathrm{{BAO}}^2\)

\(\beta \) | \(\omega _1\) | \(\Omega _\mathrm{{m0}}\) | \(\chi _\mathrm{{BAO}} ^2\) |
---|---|---|---|

\(-0.239\) | \(-1.051\) | 0.272 | \(5 \times 10^{-4}\) , |

\(\chi _\mathrm{{CMB}} ^2\) | \(\chi _\mathrm{{OHD}} ^2\) | \(\chi _\mathrm{{SNe}} ^2\) | \((\chi _\mathrm{T} ^2)_\mathrm{{min}}\) |
---|---|---|---|

\(8 \times 10^{-10}\) | 16.23 | 542.75 | 558.98 |

It should be noted that in data fitting and maximization of the probability values the two definitions of \(\chi _\mathrm{T} ^2\), i.e. Eqs. (59) and (66), are not very different. For justifying this claim one can compare Tables 2 and 3, which are related to (59) and (66), respectively. But in figures which are related to confidence levels one can observe that the exceeding of confidence levels is reduced, therefore the re-weight of some constraints can improve the behavior of the model. For more clarification one can refer to Figs. 10, 11, and 12. Now by means of (66), we marginalize the likelihood \(\mathscr {L} (\Omega _{\mathrm{m}0} ,\omega _1 , \beta )\) w.r.t. \(\omega _1\), \(\beta \), and \(\Omega _{\mathrm{m}0}\), respectively. Also the relative probability functions \(\mathscr {L} (\Omega _{\mathrm{m}0}, \beta )\), \(\mathscr {L} (\omega _1, \Omega _{\mathrm{m}0})\), and \(\mathscr {L} (\omega _{1} , \beta )\) in two dimensional confidence levels 68.3, 90, and \(95.4\,\%\) are plotted in Fig. 13. For more investigations, we will draw the one dimensional marginalized likelihood functions \(\mathscr {L}(\Omega _{{\mathrm{m}0}})\) versus \(\Omega _{{\mathrm{m}0}}\), \(\mathscr {L}(\omega _1)\) based on \(\omega _1\) and \(\mathscr {L}(\beta )\) versus \(\beta \) in Fig. 14. Meanwhile in Table 4 one observes the quantities which maximize the marginalized likelihoods using different confidence levels by means of the confidence levels \(\sigma _1 = 68.3\,\%\) and \(\sigma _2 = 95.4\,\%\).

### 4.1 Typical example

*n*is a numerical constant and

*T*is the cosmic time, and therefore \(\Omega _\mathrm{DE}\) is obtained: \(\Omega _\mathrm{DE}={f(\varphi )n^2}/{H^2T^2}\). Taking this assumption and using Eq. (68), the equation of state parameter of the effective dark energy could be obtained:

*r*is the ratio of cold dark matter and effective dark energy, namely \(r={\rho _m}/{\rho _\mathrm{DE}}={\Omega _\mathrm{m}}/{\Omega _\mathrm{DE}}\). The interaction term in this model generates an extra term for \(\omega _\mathrm{DE}\), which can justify the phantom divide line crossing. By the definition of an ansatz for \(\omega _{e \Lambda }\), one can consider

In this table the values which maximize the relative probability functions \(\mathscr {L}(\Omega _\mathrm{{m0}})\), \(\mathscr {L}(\omega _1)\), and \(\mathscr {L}(\beta )\) using the confidence levels \(\sigma _1 = 68.3\,\%\) and \(\sigma _2 = 95.4\,\%\) are calculated. The data sets are includes of SNeIa, CMB, BAO, and OHD in which the weight of \(\chi _\mathrm{CMB}^2\) and \(\chi _\mathrm{BAO}^2\) in the \(\chi _\mathrm{Total}^2\) function is the coefficient 3

\(\sigma _2 ^-\) | \(\sigma _2 ^+ \) | \(\sigma _1 ^-\) | \(\sigma _1 ^+\) | \((\mathscr {L})_\mathrm{{max}}\) | x |
---|---|---|---|---|---|

0.02 | 0.02 | 0.01 | 0.01 | 0.272 | \(\Omega _{\mathrm{m}0}\) |

0.16 | 0.156 | 0.08 | 0.08 | -1.04 | \(\omega _1\) |

0.44 | 0.59 | 0.23 | 0.27 | -0.24 | \(\beta \) |

*z*as

This shows that the model is clearly consistent with the data since \(\chi ^{2}/\mathrm{dof} = 1\).

Figure 16 shows contour plots for the free parameters \(\omega _{1}\) and \(\beta \); it is shown that the best value for these parameters are \(-1.86<\omega _{1}<-1.62\) and \(-2.27<\beta <-0.73\) in which for the stability condition \(c^2 >0\) we have taken the interface between the green and yellow sector, \(\omega _1=-1.68\).

The evolution of the effective dark energy parameter, \(\omega _\mathrm{DE}\), versus *z*, for \(\omega _0= 1.1\), \(\omega _1=-1.68\), and \(\beta =-2.25 \) has been shown in Fig. 17. This shows that by increasing *z* the parameter gets into the phantom phase.

## 5 Conclusion and discussion

Interacting models which contain an external interaction between matter and scalar fields attract more attention. Such mechanisms are capable to suppress the fifth force and also are in good agreement with observations. Using such a powerful mechanism we have found some cosmological parameters referring to the coincidence and deceleration parameters. For instance based on Table 2, and Eqs. (32) and (34) it is clear that *r*(*z*) is a decreasing function and *q* has taken negative values for different values of *z*. Considering a suitable ansatz for the EoS parameter of effective dark energy, the dimensionless Hubble parameter is obtained. So by means of SNeIa, CMB, BAO, and OHD data sets the minimum value of \(\chi ^2\) for the free parameters of the model are obtained. To estimate the free parameters of an ansatz for the effective dark energy equation of state, the whole of the observational data sets have been considered. For more details one can compare the results of Figs. 4 and 5 with the results of a typical example, see Sect. 4.1. Also for getting a better overlap between the contours with the constraint \(\chi _\mathrm{m}^2\le 1\), the \(\chi _\mathrm{T}^2\) function has been re-weighted. Meanwhile the relative probability functions have been plotted for the marginalized likelihood \(\mathscr {L} (\Omega _{\mathrm{m}0} ,\omega _1 , \beta )\) according to the two dimensional confidence levels 68.3, 90, and \(95.4\,\%\). In addition the values of the free parameters which maximize the marginalized likelihoods using the above confidence levels have been obtained. Based on the above discussions a couple of free parameters have been plotted in Figs. 1, 2, and 3. In Fig. 1, the constraints on \(\Omega _{\mathrm{m}0}\) in the \(\omega _1 \, \beta \) plane have been investigated; and also, for the two constraints the SNeIa and OHD minimum points of \(\chi ^2 \) have been distinguished. In Fig. 2, using the best value of \(\omega _1\), the constraints in the \(\Omega _{\mathrm{m}0} \, \beta \) plane are obtained. In a similar way, for the best value of \(\beta \), the behavior of the constraints in the \(\omega _1 \, \Omega _{\mathrm{m}0}\) plane is shown. Also based on Fig. 1, for \(\Omega _{\mathrm{m}0}=0.2\), the CMB, BAO, and OHD have an overlap region, but they are not in agreement with the SNeIa results; one possible explanation would be incompatibility among the data sets. Also, for different values one can find a region in which SNeIa and OHD are in better agreement against CMB and BAO. This different behavior of the constraints indicates that if one wants to compare theoretical and observational results, it may be better that the greatest set of constraints would be considered. For more investigation of the overlaps and the effects on individual observations, Figs. 4 and 5 have been plotted. In Fig. 4, the behavior of \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) and \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) for \(\Delta \chi _\mathrm{T} ^2 = 3.53, 6.25, 8.02\), have been compared. Also in Fig. 5, we have considered \(\chi _\mathrm{T} ^2 = \chi _\mathrm{SNe} ^2 + \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\) and \(\chi _\mathrm{T} ^2 = \chi _\mathrm{OHD} ^2 + \chi _\mathrm{CMB} ^2 + \chi _\mathrm{BAO} ^2\), for \(\Delta \chi _\mathrm{T} ^2 = 0.1, 0.2, 0.3\), to investigate the degeneracy. These two figures indicate that although individual OHD data surveying in cosmological investigations (in comparison with SNeIa, CMB, and BAO) is not so important it decreases the degeneracy between the free parameters.

## Notes

### Acknowledgments

H. Sheikhahmadi would like to thank Iran’s National Elites Foundation for financially support during this work.

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