# Reconstructing cosmographic parameters from different cosmological models: case study. Interacting new generalized Chaplygin gas model

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## Abstract

In this paper, we present a method to reconstruct cosmographic parameters, \(\{q,j,s,l\}\) and their evolution from particular cosmological models. In this method inspired by dynamical system approach, we convert the equation of the cosmological model into an equivalent system of first order differential equations which is much easier to solve numerically. Then, we reproduce the cosmographic parameters in terms of these variables. Instead of best fitting the cosmographic parameters with observation directly, we best fit and estimate the parameters and initial conditions of the model by observational, then the cosmographic parameters will be constrained automatically. The advantage of this method is that, it is free of some of shortcomings reconstructing theories with higher-order derivatives. It also enable us to measure not only the current value of cosmographic parameters \(\{q_{0},j_{0},s_{0},l_{0}\}\) the past values, \(\{q_{i},j_{i},s_{i},l_{i}\}\), future values \(\{q_{f},j_{f},s_{f},l_{f}\}\) and their evolution. This can be a useful tool to test, compare and distinguish different cosmological models according to the reconstructed cosmographic parameters at different epoch of the universe.

## 1 Introduction

The publication of Hubble’s 1929 article [1] marked a turning point in understanding the universe. In this interesting report, Hubble exposed the evidence for one of the great discoveries in 20th century science: the expanding universe. As soon as astrophysicists comprehend that Type Ia Supernovae (SNeIa) were standard candles, it appeared evident that their high luminosity should make it possible to extend the Hubble diagram, i.e. a plot of the distance-redshift relation, for interesting distance ranges. Inspired by this appealing consideration, two independent research groups started SNeIa surveys leading to the unexpected discovery that the Universe expansion is picking up speed, rather than slowing down as assumed by the Cosmological Standard Model [2, 3, 4, 5, 6]. This surprising finding has now been confirmed by more recent data coming from SNeIa surveys [7, 8, 9, 10, 11, 12, 13, 14], large scale structure [15, 16, 17, 18, 19] and cosmic microwave background (CMBR) anisotropy spectrum [20, 21, 22, 23, 24, 25, 26]. This remarkable discovery has led cosmologists to hypothesize the presence of unknown form of energy called dark energy (DE), which is an exotic matter with negative pressure [27]. All current observations are consistent with a cosmological constant (CC); while this is in some sense the most economical possibility, the CC has its own theoretical and naturalness problems [28, 29], so it is worthwhile to consider alternatives.

In search for the solution of the riddle of dark energy two different ways are usually chosen. One is to modify Einstein’s theory of gravitation and the second is constructing various dark energy candidates. Thus some dark energy models like quintessence [30, 31] phantom energy [32, 33], k-essence [34], tachyon [35], Chaplygin gas [36] have been proposed. Also unified dark energy (UDE) models as an economical and attractive idea to unify the d ark sector of the universe have been studied by many authors [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Chaplygin gas is one of these UDE models which was introduced by Kamenshchik et al. [53]. Main idea of CG model comes from aerodynamics [54]. Its interesting characteristic is that it has a dual role as it gives density evolution of matter at high redshifts and dark energy at low redshifts [55, 56]. It obeys an equation of state (EOS) as \(p=-\frac{A}{\rho }\) where *p* and \(\rho \) are pressure and density respectively and *A* is a positive constant [57]. However Chaplygin gas model has a problem, in other words it has some inconsistency with observational datas [57, 58, 59], therefore Generalized Chaplygin gas (GCG) was introduced in order to establish an appropriate cosmological model with the equation of state \(p=-\frac{{A}}{\rho ^{\alpha }}\) where \(\alpha \) is a real number and it has a value in range (0, 1) and covers Chaplygin gas for \(\alpha =0\) [55, 56, 60, 61]. By presuming that there is a sufficiently high level of non-linear clustering on small scales, it has recently been shown that the GCG may be consistent with current observational constraints, over a wide region of parameter space [62, 63, 64, 65, 66]. Then, the modified Chaplygin gas (MCG) is proposed as a generalization of the GCG model. Modified Chaplygin gas obeying an equation of state \(p=A\rho -\frac{B}{\rho ^{\alpha }}\). Equation of state of dark energy (\(\omega _{x}\)) still cannot be determined exactly and the observational data show \(\omega _{x}\) is in the range of \((-1.46,-0.78)\) so Zhang et al. have generalized GCG model as possible X-type dark energy with constant \(\omega _{x}\) [67]. Therefore, they propose a new generalized Chaplygin gas (NGCG) scenario. Due to DM and DE are dominant component of universe, it is logical considering interaction between them. Moreover recent observation data from SNIa, CMB and galaxy cluster show this interaction [68, 69], Also The cosmic coincidence problem is solved for interacting models. Both dark energy models and modified gravity models may both fit observation. The question then arises whether the two scenarios can be distinguished. In fact Most of the observational constraints are model-dependent, thus, it not easy to distinguish dark energy models from modified gravity theories even by using the observations. These confusions suggest that a more conservative approach to the problem of the cosmic acceleration. Thus, various model-independent approaches have been proposed in the literature. A well-known one is the CPL parametrization [70, 71]. Another powerful model-independent approach is cosmography

## 2 Cosmography

*H*as,

*y*as

## 3 Interacting new generalized Chaplygin gas model

*a*,

*A*is a positive constant and the observational data favor \(\omega _{x}\) to have a value in range of \((1.46,-0.78)\) [89, 90, 91, 92]. The energy density of NGCG is expressed as follow

*B*is integration constant. The energy conservation equation is

*c*in

*Q*we have

*c*have a positive value of the order unity [87]. Defining the e-folding

*x*with definition \(x=lna=-ln(1+z)\), where

*z*is redshift parameter. In addition density and pressure of NGCG can be represented by dimensionless parameter \(\chi \) and \(\zeta \) as

## 4 Reconstructing cosmographic parameters from NGCG model

*j*can be rewrite as

*j*can be obtain in terms of new variables as

*j*, the parameter,

*s*, will be

*l*.

## 5 Cosmological constrain

*h*is the Hubble constant \(H_{0}\) in units of 100 km/s/Mpc. Also \(\mu _{i}^{obs}\) is the distance modulus parameters calculated from our model and \(\sigma \) is the estimated error of the \(\mu _{i}^{obs}\). In this paper, we use the Union2 data set, which contains 557 SNIa data. we have obtained the best values as \(\alpha =0.17^{+1}_{-4}\), \(c=0.12^{+0.04}_{-0.03}\)(at \(1\sigma \)), the best initial conditions as \(\chi _{0}=0.651\) and \(\zeta _{0}=-0.702\). and \(h=0.6976\) with the \(\chi ^{2}_{min}=540.8649642\). In Fig. 1, the two dimensional likelihood distribution, \(e^{-\chi ^{2}/2} \), for parameters \(\alpha \) and

*c*and

*h*have been plotted.

*c*and initial conditions, shows that the current effective EoS parameter is \(\omega _{eff}\simeq -0.706\). This value is very close to those obtained for \(\Lambda CDM\) and

*CPL*models in [93, 94]. In Fig. 2 the comparison of the best fitted trajectory for

*NGCG*, \(\Lambda CDM\) and

*CPL*models and their current values have been shown.

In Fig. 3 cosmographical parameters have been plotted for best fitted parameters *c*, \(\alpha \) and different initial conditions. The red dash line shows the plot for best values of \(\alpha \) and *c* and best fitted initial conditions.

*q*(

*z*) focused in transition redshift range in our study shows that the transition redshift \(z_{tr} = 0.825\) which is in good agreement with the recent Busca et al. [111] determination of \(z_{tr} = 0.820.08\), and [95] determination of \(z_{tr} = 0.740.05\) (Fig. 4).

## 6 Past and future of the cosmographic parameters

*c*, these critical points are simplified as

Comparison of current values of cosmographical for different models

| \(q_{0}\) | \(j_{0}\) | \(s_{0}\) | \(l_{0}\) | References |
---|---|---|---|---|---|

| \( -0.559\) | 0.732 | \( -0.232\) | 3.47 | This study |

\(\Lambda CDM\) | \( -0.588\) | 1 | \( -0.238\) | 2.846 | [93] |

| \( -0.308\) | 0.742 | \( -0.432\) | 2.926 | [93] |

| \( -0.555\) | 0.890 | \( -0.348\) | 3.660 | [93] |

\(CPL\, parametrization\) | \( -0.511\) | 0.342 | \( -2.260\) | 1.383 | [93] |

| \( -0.64\) | 1.02 | \( -0.39\) | 4.05 | [112] |

## 7 Conclusions

In this paper we have presented a method to reconstruct cosmographic parameters from new generalized Chaplygin gas (NGCG) model. It is interesting to note that the most advantage of this method is that it is applicable for various cosmological models, although we have focused on (NGCG) ones. First we have introduced two dimensionless independent variables to simplify the obtained equations \((\chi ,\zeta )\). There are various reasons for doing this, one being that. The numerical solution of Eq. 15 even for simplest cosmological model is afflicted by the large uncertainties on the boundary conditions (i.e., the present day values of the scale factor and its derivatives up to the third order) that have to be set to find out the scale factor, while, introducing the equation of the system in terms of new variables solve this problem. It can help us to convert the equations into an equivalent system of first order differential equations which is much easier to solve numerically ( only the initial condition in first order must be constrained). This also help us to reconstruct cosmographic parameters, (*q*, *j*, *s*, *l*) in terms of new variables \((\chi ,\zeta )\). Thus by fitting the model with observations of type Ia supernovae, parameters of the model, \((\alpha ,c)\) and the initial conditions \((\chi _{0},\zeta _{0} )\), have been best fitted. Thus the cosmographic parameters and equation of state \(\omega _{eff}\) have been best fitted automatically.

The current values of cosmographic have been obtained as \(q_{0}=-0.559,j_{0}=0.732,s_{0}=-0.232,l_{0}=3.47\). Instead of best fitting the cosmographic parameters with observation directly, we best fit and estimate the parameters and initial conditions of the model by observational, then the cosmographic parameters will be constrained automatically. The advantage of this method is that, it is free of some of shortcomings reconstructing theories with higher-order derivatives. It also enable us to measure not only the current value of cosmographic parameters \(\{q_{0},j_{0},s_{0},l_{0}\}\) the past values, \(\{q_{i},j_{i},s_{i},l_{i}\}\), future values \(\{q_{f},j_{f},s_{f},l_{f}\}\) and their evolution. This can be a useful tool to test, compare and distinguish different cosmological models according to the reconstructed cosmographic parameters at different epoch of the universe. For previous studies, one focused on the current values of the cosmographic parameters, to distinguish different cosmological models, while now the past values (here obtained as \(\{q_{i}=0.5,j_{i}=1,s_{i}=-3.5,l_{i}=17.479\}\)) and past values (\(\{q_{f}=-0.916,j_{f}=0.76,s_{f}=0.571,l_{f}=0.379\}\)) can increase our knowledge about cosmological models

Our results for this specific cosmological model, explain the current acceleration of the universe, predict the cosmological deceleration–acceleration transition redshift as \(z_{tr} = 0.825\) which is in good agreement with those obtained in pervious studies based on robust observational supports and are in agreement with results obtained \(\Lambda CDM\) model and *CPL* parametrization. Also the current value of \(\omega _{eff}=-0.706\). This results are comparable with those obtained in [93] for \(\Lambda CDM\), *DGP* and *Cardassian* models and *CPL* parametrization (see Table 1). As a most advantage which distinguishes our study from previous ones, is that, it is possible to estimate cosmographic parameters in past and future.

We have discussed the cosmography of NGCG model so we obtain present value of them. We compare NGCG model with \(\Lambda CDM\), DPG, cardassian, CPL parametrization models.

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