# Study of parametrized dark energy models with a general non-canonical scalar field

## Abstract

In this paper, we consider various dark energy models in the framework of a non-canonical scalar field with a Lagrangian density of the form \(\mathcal{L}(\phi , X)=f(\phi )X{\left( \frac{X}{M^{4}_{Pl}}\right) }^{\alpha -1} - V(\phi )\), which provides the standard canonical scalar field model for \(\alpha =1\) and \(f(\phi )=1\). In this particular non-canonical scalar field model, we carry out the analysis for \(\alpha =2\). We then obtain cosmological solutions for constant as well as variable equation of state parameter (\(\omega _{\phi }(z)\)) for dark energy. We also perform the data analysis for three different functional forms of \(\omega _{\phi }(z)\) by using the combination of SN Ia, BAO, and CMB datasets. We have found that for all the choices of \(\omega _{\phi }(z)\), the SN Ia \(+\) CMB/BAO dataset favors the past decelerated and recent accelerated expansion phase of the universe. Furthermore, using the combined dataset, we have observed that the reconstructed results of \(\omega _{\phi }(z)\) and *q*(*z*) are almost choice independent and the resulting cosmological scenarios are in good agreement with the \(\Lambda \)CDM model (within the \(1\sigma \) confidence contour). We have also derived the form of the potentials for each model and the resulting potentials are found to be a quartic potential for constant \(\omega _{\phi }\) and a polynomial in \(\phi \) for variable \(\omega _{\phi }\).

### Keywords

Dark Energy Scalar Field Lagrangian Density Deceleration Parameter Dark Energy Model## 1 Introduction

One of the biggest challenges in modern cosmology is understanding the nature of the dark energy (DE), which seems to be responsible for the observed accelerated expansion phase of the universe at the present epoch [1, 2]. Among the many candidates for DE, the cosmological constant (\(\Lambda \)) emerges as the most natural and the simplest possibility. However, \(\Lambda \)-cosmology suffers from the so-called “*fine tuning*” and “*cosmic coincidence”* problems [3, 4]. These theoretical problems motivated cosmologists to think beyond the cosmological constant and explore other unknown components which may be responsible for the late-time accelerated expansion phase of the universe. The scalar field models have played an important leading role as a candidate of DE due to its dynamical nature and simplicity. Till now, a variety of scalar field DE models have been proposed, such as quintessence (canonical scalar field), k-essence, phantom, tachyon, dilatonic dark energy, and so on (for details, see Ref. [5] and the references therein). But the origin and nature of DE still remains completely unknown, despite many years of research.

It is strongly believed that the universe had a rapid exponential expansion phase during a short era in the very early epoch. This is known as *inflation* [6, 7]; it can give a satisfactory explanation to the problems of the Hot Big Bang cosmology (for example, the horizon, flatness, and monopole problems). Generally, cosmologists realized this inflationary scenario by using a single canonical scalar field called the “*inflaton*”, which has a canonical kinetic energy term (\(\frac{{\dot{\phi }}^{2}}{2}\)) in the Lagrangian density. In the literature, there also exist some inflationary models in which the kinetic energy term is different from the standard canonical scalar field case (instead of the standard form \(\frac{{\dot{\phi }}^{2}}{2}\)). Such models are commonly known as the *non-canonical scalar field models of inflation*. Such non-canonical scalar fields have been found to have many attractive features compared to the canonical scalar field case, for example, the slow-roll conditions can be achieved more easily as compared to the canonical case. Many interesting possibilities with these models have been recently studied in the literature (see Refs. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]). It has been first shown in Refs. [8, 9] that the k-essence model (which belongs to an important class of non-canonical scalar field models) is capable of generating inflation in the early epoch. Later, Chiba et al. [10] showed that such models can equally effectively describe a DE scenario. Since the nature of DE is completely unknown, it is quite reasonable to consider a non-canonical scalar field as a candidate for DE component and check for the viability of such models. Within the framework of a non-canonical scalar field, in this work, we shall try to obtain an observationally viable cosmological model to analyze the behavior of the deceleration parameter (*q*) and the equation of state (EoS) parameter (\(\omega _{\phi }\)) for describing the expansion history of the universe. The motivation for this work is discussed in detail in Sect. 2. As already mentioned, as the nature of DE is unknown to us, we eventually have no firm idea regarding whether the EoS parameter of DE is a constant quantity or whether it is dynamical in nature. In this connection, the most effective choice is to assume a specific functional form for the dark energy EoS parameter \(\omega _{\phi }\) as a function of the redshift *z* (for details see Sect. 3.2). To study the non-canonical scalar field DE model in a more general framework, in this paper, we have considered both possibilities. First, we have studied the model for a constant EoS parameter \(\omega _{\phi }\), which is in the range \(-1<\omega _{\phi }<-\frac{1}{3}\) so as to obtain acceleration. Second, we have considered three different choices for \(\omega _{\phi }(z)\) in order to cover a wide range of the DE evolution. We have then solved the field equations and analyzed the respective cosmological scenarios for all the cases. For all the models, the deceleration parameter *q* is found to exhibit an evolution from early deceleration to late-time acceleration phase of the universe. This feature is essential for the structure formation of the universe. For all the models, we have also derived the potential \(V(\phi )\) in terms of the scalar field \(\phi \) by considering a specific parametrization of \(f(\phi )\). In order to compare the theoretical models of DE (for dynamical \(\omega _{\phi }\)) with the observations, we have used the SN Ia, BAO, and CMB dataset to constrain the various model parameters (for details see Appendix A). We have found that the combined dataset favors the \(\Lambda \)CDM model within the \(1\sigma \) confidence contour. We give the detailed results of this work in Sect. 4.

The present paper is organized in the following way. In the next section, we introduce some basic equations of a general non-canonical scalar field model and also discuss the motivation of this work. We then obtain the general solutions of the field equations for a particular choice of the function \(f(\phi )\) and for different forms of the EoS parameter \(\omega _{\phi }\). In Sect. 4, we summarize the results of this work. Finally, the conclusions of this work are presented in Sect. 5. Additionally, for completeness, we perform the combined data analysis in Appendix A and find the observational constraints on \(\omega _{\phi }(z)\) and *q*(*z*) using the SN Ia, BAO, and CMB datasets.

## 2 Basic framework

*R*is the Ricci scalar, and \(\mathcal{L}(\phi ,X)\) is the Lagrangian density, which is an arbitrary function of the scalar field \(\phi \) and its kinetic term

*X*. The kinetic term

*X*is defined as \(X=\frac{1}{2}\partial _\mu \phi \partial ^\mu \phi \), which is a function of time only. The last term, \(S_{m}\), represents the action of the background matter. Throughout this paper we shall work in natural units, such that \(8\pi G=c=1\).

*F*(

*X*) are arbitrary functions of \(\phi \) and

*X*, respectively. \(V(\phi )\) is the potential for the scalar field \(\phi \).

*a*(

*t*) is the scale factor of the universe. With the FRW geometry, the equations of motion take the form

*t*, and \(\rho _{m}\) represents the energy density of the matter component of the universe, \(F_{X} \equiv \frac{\partial F}{\partial X}\) and \(F_{XX}\equiv \frac{\partial ^{2}F}{\partial X^{2}}\).

*general non-canonical scalar field model*[\(\mathcal{L}(\phi , X)=F(X)-V(\phi )\)] when \(f(\phi )=1\). This type of non-canonical scalar field models was proposed by Fang et al. [21]. They studied several aspects of this type of scalar fields for different forms of

*F*(

*X*). Recently, these types of non-canonical scalar field models have gathered attention due to their simplicity. Unnikrishnan et al. [11] have showed that for non-canonical scalar field models, the slow-roll conditions can be more easily satisfied compared to the canonical inflationary theory. They have shown that such models (with quadratic and quartic potentials) are more consistent with the current observational constraints relative to the canonical inflation. They have also shown that such non-canonical models can drop the tensor-to-scalar ratio rather than their canonical counterparts. In fact, a lot of work has been done in the framework of the non-canonical inflationary scenario in the early epoch [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Furthermore, Franche et al. [25] showed that the non-canonical scalar fields are the most universal case with a general Lagrangian density satisfying certain conditions. These interesting properties of non-canonical scalar field models motivated us to study the cosmological aspects of such fields in a more general framework in the context of dark energy. In the literature, a large number of functional forms of \(\mathcal{L}(\phi , X)\) have been proposed so far; see for example [11, 26, 27, 28]. In our earlier work [22, 23], we have considered a Lagrangian density of the following form:

*z*) as

*H*, \(\rho _{m}\), \(f(\phi )\), \(V(\phi )\), and \(\phi \)). So naturally one has to assume two relationships among the different variables to solve the system of equations.

*f*has the functional form

*q*(

*z*), can also be expressed in terms of

*H*(

*z*) as

We shall now concentrate on the dark energy EoS parameter \(\omega _{\phi }(z)\). If a function of \(\omega _{\phi }(z)\) is given, then we can find the evolution of \(\rho _{\phi }(z)\) from Eq. (21). As a result, we can also find the evolutions of *H*(*z*), *q*(*z*), *V*(*z*), and \(\phi (z)\). Inverting \(\phi (z)\) into \(z(\phi )\) and using Eq. (23), one can then obtain the potential \(V(\phi )\) in terms of \(\phi \). As already mentioned, we have considered a specific parametrization of \(f(\phi )\) and still we need another assumption to match the number of unknown parameters with the number of independent equations. With this freedom, we choose different functional forms for \(\omega _{\phi }(z)\), the equation of state parameter. In the next section, we try to obtain the functional forms of various cosmological parameters for different choices of \(\omega _{\phi }(z)\) and study their cosmological implications.

## 3 Theoretical models

In this section, we shall consider two phenomenological DE models for obtaining the current acceleration of the universe in the framework of a general non-canonical scalar field theory.

### 3.1 Model I: Accelerating universe driven by a constant EoS parameter for dark energy \((-1< \omega _{\phi } < -\frac{1}{3})\)

In this model, we shall investigate the properties of an accelerated expanding universe driven by a non-canonical scalar field dark energy with a constant EoS parameter. Recent observations suggest that the dark energy EoS \(\omega _{\phi }\) is very close to \(-1\) and the approximate bound on \(\omega _{\phi }\) is \(-1.1\le \omega _{\phi }\le -0.9\) [29, 30]. Keeping this limit in mind, we choose a constant \(\omega _{\phi }\) in the limit \(-1< \omega _{\phi } < -\frac{1}{3}\), which ensures that the model does not deviate much from a \(\Lambda \)CDM model.

*H*(

*z*) (as given in the above equation) into Eq. (27), we obtain the deceleration parameter as

*q*(

*z*) against

*z*is shown in Fig. 1 for different values of \(\omega _{\phi }\) (within the range \(-1<\omega _{\phi }<-\frac{1}{3}\)) and \(\Omega _{m 0}=0.3\). Figure 1 shows that

*q*(

*z*) crosses its transition point from its positive value regime to the negative value regime in the recent past, which is consistent with the independent measurements reported by several authors (see Ref. [31] and the references therein).

### 3.2 Model II: Accelerating universe driven by time-dependent EoS parameter for dark energy

*z*. The numerical values of the \(\omega _{n}\)’s can be found by fitting to the observational data. Following the first order expansions in Eq. (34), several authors considered many functional forms for \(x_{n}(z)\) to investigate the evolution of \(\omega _{\phi }(z)\).

- (i)
\(\omega _{\phi }(z)=\omega _{0}~=~\)constant (as we have discussed in model I) for \(x_{0}(z)=1\) and \(x_{n}=0\) (\(n\ge 1\)).

- (ii)
\(\omega _{\phi }(z) = \omega _{0} + \omega _{1} z\) i.e., linear redshift parametrization [32, 33], for \(x_{n}(z)=z^{n}\) with \(n\le 1\).

- (iii)
\(\omega _{\phi }(z) = \omega _{0} + \omega _{1} \mathrm{log}(1+z)\) i.e., logarithmic parametrization [36], for \(x_{n}(z)=[\mathrm{log}(1+z)]^{n}\) with \(n\le 1\).

- (iv)
\(\omega _{\phi }(z) = \omega _{0} + \omega _{1} \frac{z}{(1+z)}\) i.e., CPL parametrization [34, 35], for \(x_{n}={\left( \frac{z}{1+z}\right) }^{n}\) with \(n=1\).

*q*(

*z*) using Eq. (27) for these different choices in Sect. 4.

\(\bullet \) **Assumption I**

*z*.

*V*as a function of \(\phi \) in Fig. 2 by considering \(\omega _{0}=-0.95\), \(\omega _{1}=0.15\), \(\Omega _{m0}=0.3\), \(f_{0}=1\), and \(\phi _{0}=150\) for this case. Figure 2 shows that the potential \(V(\phi )\) increases initially but becomes almost flat as \(\phi \) increases. The reason behind this seems to be the form of the linear parametrization, which is appropriate only for low redshifts (\(z<<1\)) and diverges for large redshifts. The corresponding expressions for \(V(\phi )\) and \(f(\phi )\) become approximately equal to (for details see Appendix B)

In the context of DE (as it is a late-time phenomenon), the above choice of \(\omega _{\phi }(z)\) has been widely used due to its simplicity and we find for the present parametrization of *f* given in Eq. (25) that the potential turns out to be a polynomial in \(\phi \).

\(\bullet \) **Assumption II**

*H*(

*z*) evolve as

Best-fit values for various model parameters for the analysis of SN Ia \(+\) BAO/CMB dataset. Here, \(\omega _{\phi }(z=0)\) represents the present value of the EoS parameter \(\omega _{\phi }(z)\) in the best-fit models. For this analysis, we have considered \(\Omega _{m0}=0.3\) (for choices I and II) and \(\Omega _{m0}=0.3\), \(A_{0}=3.5\) for choice III

Choice | Best-fit values of | \(\omega _{\phi }(z=0)\) | \({\chi ^{2}_{m}}\) | |
---|---|---|---|---|

model parameters | ||||

I | \(\omega _{0}=-1.01\) | \(\omega _{1}=0.10\) | \(-1.01\) | 599.90 |

II | \(\omega _{2}=-1.19\) | \(\omega _{3}=7\) | \(-1.06\) | 565.43 |

III | \(A_{1}=-0.16\) | \(A_{2}=-0.14\) | \(-1.04\) | 564.86 |

\(\bullet \) **Assumption III**

*V*as a function of \(\phi \) for some specific values of the model parameters (\(A_{0}=3.5\), \(A_{1}=0.2\), \(A_{2}=0.4\), \(\Omega _{m0}=0.3\), \(f_{0}=1\), and \(\phi _{0}=150\)) in Fig. 4. It is evident from Fig. 4 that the potential \(V(\phi )\) always decreases with the scalar field \(\phi \). For the present model, \(V(\phi )\) and \(f(\phi )\) can be explicitly expressed in terms of \(\phi \) as (see Appendix B)

*V*as a polynomial in \(\phi \) in the following manner:

*q*(

*z*) from observations.

## 4 Results

Following the statistical analysis (see Appendix A), in this section, we present the fitting results for different choices of the EoS parameter for DE. Figure 5 shows the \(1\sigma \) and \(2\sigma \) confidence contours for each choice (I, II, and III) using the SN Ia \(+\) BAO/CMB dataset.

The best-fit values of the model parameters and \(\omega _{\phi }(z=0)\) for these different choices are given in Table 1.

Using those best-fit values, we have reconstructed the deceleration parameter *q*(*z*) for each model and the results are plotted in Fig. 6.

It is evident from Fig. 6 that *q*(*z*) shows a smooth transition from a decelerated (\(q>0\)) to an accelerated (\(q<0\)) phase of expansion of the universe at the transition redshift \(z_{t}=0.38\) (for ansatz I), 0.36 (for ansatz II), and 0.43 (for ansatz III) for the best-fit models. These results are in good agreement with those obtained by several authors based on various other considerations [42, 43, 44].

Furthermore, we also show the reconstructed evolution history of the EoS parameter in Fig. 7 for each choice of \(\omega _{\phi }(z)\).

We have also plotted the total EoS parameter, which is defined as \(\omega _\mathrm{tot}(z)=\frac{p_{\phi }}{\rho _{\phi } + \rho _{m}}\), as a function of *z* for these choices (see the inset diagram of Fig. 7). From Table 1, we have found that the current values of \(\omega _{\phi }(z)\) for the best-fit DE models are very close to \(-1\), i.e., the models do not deviate very far from the \(\Lambda \)CDM model (\(\omega _{\Lambda }=-1\)) at the present epoch. However, as indicated in Table 1, the present parametrized model favors a phantom model (\(\omega _{\phi }<-1\)) in the \(2\sigma \) limit and thus requires further attention.

## 5 Conclusions

In this work, we have studied various non-canonical scalar field DE models in a spatially flat, homogeneous, and isotropic FRW space-time. In this framework, we have obtained the general solutions of the field equations for different choices of the EoS parameter. For completeness, we have also investigated how the joint analysis of the SN Ia \(+\) BAO/CMB datasets constrains the redshift evolutions of *q*(*z*) and \(\omega _{\phi }(z)\) for different choices of \(\omega _{\phi }(z)\) (as given in Model II). In Fig. 5, we have also shown the \(1\sigma \) and \(2\sigma \) contour plots of the pairs (\(\omega _{0} , \omega _{1}\)) (upper panel), (\(\omega _{2} , \omega _{3}\)) (middle panel), and (\(A_{1} , A_{2}\)) (lower panel) for the ansatzes I, II, and III, respectively. In this analysis, we have also calculated the best-fit values of the free parameters (as shown by large dots in Fig. 5) and it has been found that the chosen values of these parameters (which were chosen for solving the parametric relations in Appendix B) are well fitted within the \(1\sigma \) confidence contour (as shown by small dots in Fig. 5).

We have shown that the deceleration parameter *q* undergoes a smooth transition from its deceleration phase (\(q>0\), at high *z*) to an acceleration phase (\(q<0\), at low *z*) for all of the considered parametrized models. However, as mentioned in the previous section, the value of \(z_{t}\), where the signature flip of *q* (from the decelerating to an accelerating expansion phase) takes place has been calculated and the results obtained are consistent with the present day cosmological observations. From the SN Ia \(+\) BAO/CMB analysis, we have also found \(q(z=0)=-0.56\), \(-0.64\), and \(-0.60\) for ansatzes I, II, and III, respectively, which also agree very well with the recent observational results.

From Table 1, we have observed that the EoS parameter \(\omega _{\phi }(z=0)\approx -1\), but slightly less than \(-1\) for all three choices (as discussed in Sect. 4). As we have seen \(\omega _{\phi }(z=0)\approx -1\), our models do not deviate very far from the \(\Lambda \)CDM model (see also Fig. 7), which is currently known as the standard model for modern cosmology. In order to gain more physical insight into these time evolutions of the EoS parameter, we have also plotted the reconstructed total EoS parameter \(\omega _\mathrm{tot}(z)\) in Fig. 7 (see the inset diagram of Fig. 7). For each choice, this figure shows that \(\omega _\mathrm{tot}(z)\) attains the required value of \(-\frac{1}{3}\) around \(z=0.62\) (within \(1\sigma \) confidence level) and remains always greater than \(-1\) up to the present epoch. These scenarios also agree very well with the observational data.

However, the models presented here are restricted because the form of \(f(\phi )\) chosen was ad hoc (as given in Eq. (25)) and did not follow from any principle. In this regard, we have mentioned earlier that we make this choice in order to close the system of equations. With this choice of \(f(\phi )\), we have derived the form of the potential \(V(\phi )\) in terms of \(\phi \) for different models. We have found that Model I leads to a quartic potential, whereas Model II leads to a polynomial potential for each choice of \(\omega _{\phi }(z)\). We have seen that, with a suitable choice of \(V_{i}\)’s for the potential (as given in Eq. (51)), it is possible to reproduce the other well-known potentials in the context of DE. However, many possibilities are opened up to accommodate a physically viable potential for other parametrizations of \(f(\phi )\) or *f*(*H*). Finally, we would like to emphasize that all the considered models provide a deceleration for high redshift and an acceleration for low redshift as required for the structure formation of the universe. However, these results are completely independent of any choice of \(f(\phi )\). With the increase of more good quality observational data at low, intermediate, and high redshifts, the constraints on \(z_{t}\) (or *q*(*z*)) and \(\omega _{\phi }(z)\) are expected to get improved in the near future.

## Notes

### Acknowledgments

One of the authors (AAM) is thankful to Govt. of India for financial support through Maulana Azad National Fellowship. SD wishes to thank IUCAA, Pune, for an associateship program.

### References

- 1.A.G. Riess et al., Astron. J.
**116**, 1009 (1998)CrossRefADSGoogle Scholar - 2.S. Perlmutter et al., Astrophys. J.
**517**, 565 (1999)CrossRefADSGoogle Scholar - 3.S. Weinberg, Rev. Mod. Phys.
**61**, 1 (1989)MathSciNetCrossRefADSGoogle Scholar - 4.S. Carroll, Living Rev. Relat.
**4**, 1 (2001)ADSGoogle Scholar - 5.E.J. Copeland, M. Sami, S. Tsujikawa, IJMP D
**15**, 1753 (2006)MathSciNetCrossRefADSGoogle Scholar - 6.A.H. Guth, Phys. Rev. D
**23**, 347 (1981)CrossRefADSGoogle Scholar - 7.A.D. Linde, Phys. Lett. B
**129**, 177 (1983)MathSciNetCrossRefADSGoogle Scholar - 8.C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B
**458**, 209 (1999)MathSciNetCrossRefADSGoogle Scholar - 9.J. Garriga, V.F. Mukhanov, Phys. Lett. B
**458**, 219 (1999)MathSciNetCrossRefADSGoogle Scholar - 10.T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D
**62**, 023511 (2000)Google Scholar - 11.
- 12.M. Fairbairn, M.H.G. Tytgat, Phys. Lett. B
**546**, 1 (2002)CrossRefADSGoogle Scholar - 13.D.A. Steer, F. Vernizzi, Phys. Rev. D
**70**, 043527 (2004)MathSciNetCrossRefADSGoogle Scholar - 14.L.P. Chimento, Phys. Rev. D
**69**, 123517 (2004)MathSciNetCrossRefADSGoogle Scholar - 15.R.J. Scherrer, Phys. Rev. Lett.
**93**, 011301 (2004)CrossRefADSGoogle Scholar - 16.D. Bertacca, S. Matarrese, M. Pietroni, Mod. Phys. Lett. A
**22**, 2893 (2007)Google Scholar - 17.G. Panotopoulos, Phys. Rev. D
**76**, 127302 (2007)Google Scholar - 18.N. Bose, A.S. Majumdar, Phys. Rev. D
**80**, 103508 (2009)CrossRefADSGoogle Scholar - 19.J. De-Santiago, J.L. Cervantes-Cota, Phys. Rev. D
**83**, 063502 (2011)CrossRefADSGoogle Scholar - 20.T. Golanbari et al., Phys. Rev. D
**89**, 103529 (2014)Google Scholar - 21.W. Fang et al., Class. Quant. Grav.
**24**, 3799 (2007)CrossRefADSGoogle Scholar - 22.S. Das, A.A. Mamon, Astrophys. Space Sci.
**355**, 371 (2015). arXiv:1407.1666 [gr-qc]CrossRefADSGoogle Scholar - 23.A.A. Mamon, S. Das, Eur. Phys. J. C
**75**, 244 (2015). arXiv:1503.06280 [gr-qc]CrossRefADSGoogle Scholar - 24.A. Melchiorri et al., Phys. Rev. D
**68**, 043509 (2003)CrossRefADSGoogle Scholar - 25.P. Franche et al., Phys. Rev. D
**81**, 123526 (2010)CrossRefADSGoogle Scholar - 26.C. Armendariz-Picon, E.A. Lim, J. Cosmol. Astropart. Phys.
**0508**, 007 (2005)CrossRefADSGoogle Scholar - 27.V. Mukhanov, A. Vikman, J. Cosmol. Astropart. Phys.
**0602**, 004 (2006)CrossRefADSGoogle Scholar - 28.S. Unnikrishnan, Phys. Rev. D
**78**, 063007 (2008)CrossRefADSGoogle Scholar - 29.W.M. Wood-Vasey et al., Astrophys. J.
**666**, 694 (2007)CrossRefADSGoogle Scholar - 30.T.M. Davis et al., Astrophys. J.
**666**, 716 (2007)CrossRefADSGoogle Scholar - 31.A. A. Mamon, S. Das, Int. J. Mod. Phys. D
**25**, 1650032 (2016). arXiv:1507.00531 - 32.D. Huterer, M.S. Turner, Phys. Rev. D
**60**, 081301 (1999)CrossRefADSGoogle Scholar - 33.J. Weller, A. Albrecht, Phys. Rev. Lett.
**86**, 1939 (2001)CrossRefADSGoogle Scholar - 34.M. Chevallier, D. Polarski, Int. J. Mod. Phys. D
**10**, 213 (2001)CrossRefADSGoogle Scholar - 35.E.V. Linder, Phys. Rev. Lett.
**90**, 091301 (2003)CrossRefADSGoogle Scholar - 36.G. Efstathiou, MNRAS
**342**, 810 (2000)Google Scholar - 37.E.M. Barboza, J.S. Alcaniz, Phys. Lett. B
**666**, 415 (2008)CrossRefADSGoogle Scholar - 38.C.-J. Feng et al., arXiv:1206.0063 [astro-ph.CO]
- 39.U. Alam, V. Sahni, T.D. Saini, A.A. Starobinski, MNRAS
**354**, 275 (2004)CrossRefADSGoogle Scholar - 40.U. Alam, V. Sahni, A.A. Starobinski, JCAP
**0406**, 008 (2004)CrossRefADSGoogle Scholar - 41.J. Weller, A. Albrecht, Phys. Rev. D
**65**, 103512 (2002)CrossRefADSGoogle Scholar - 42.A.G. Riess et al., Astrophys. J.
**607**, 665 (2004)CrossRefADSGoogle Scholar - 43.A.G. Riess et al., ApJ
**659**, 98 (2007)CrossRefADSGoogle Scholar - 44.J.V. Cunha, J.A.S. Lima, arXiv:0805.1261 [astro-ph]
- 45.N. Suzuki et al., Astrophys. J.
**746**, 85 (2012). arXiv:1105.3470 [astro-ph.CO]CrossRefADSGoogle Scholar - 46.S. Nesseris, L. Perivolaropoulos, Phys. Rev. D
**72**, 123519 (2005)CrossRefADSGoogle Scholar - 47.F. Beutler et al., Mon. Not. R. Astron. Soc.
**416**, 3017 (2011)CrossRefADSGoogle Scholar - 48.W.J. Percival et al., Mon. Not. R. Astron. Soc.
**401**, 2148 (2010)CrossRefADSGoogle Scholar - 49.C. Blake et al., Mon. Not. R. Astron. Soc.
**418**, 1707 (2011)CrossRefADSGoogle Scholar - 50.N. Jarosik et al., Astrophys. J. Suppl.
**192**, 14 (2011)CrossRefADSGoogle Scholar - 51.R. Goistri et al., JCAP
**03**, 027 (2012)CrossRefADSGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}