# AIC, BIC, Bayesian evidence against the interacting dark energy model

- 622 Downloads
- 9 Citations

## Abstract

Recent astronomical observations have indicated that the Universe is in a phase of accelerated expansion. While there are many cosmological models which try to explain this phenomenon, we focus on the interacting \(\Lambda \)CDM model where an interaction between the dark energy and dark matter sectors takes place. This model is compared to its simpler alternative—the \(\Lambda \)CDM model. To choose between these models the likelihood ratio test was applied as well as the model comparison methods (employing Occam’s principle): the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Bayesian evidence. Using the current astronomical data: type Ia supernova (Union2.1), \(h(z)\), baryon acoustic oscillation, the Alcock–Paczynski test, and the cosmic microwave background data, we evaluated both models. The analyses based on the AIC indicated that there is less support for the interacting \(\Lambda \)CDM model when compared to the \(\Lambda \)CDM model, while those based on the BIC indicated that there is strong evidence against it in favor of the \(\Lambda \)CDM model. Given the weak or almost non-existing support for the interacting \(\Lambda \)CDM model and bearing in mind Occam’s razor we are inclined to reject this model.

### Keywords

Dark Matter Dark Energy Cosmic Microwave Background Baryon Acoustic Oscillation Dark Energy Component## 1 Introduction

Recent observations of type Ia supernovae (SNIa) provide the main evidence that the current Universe is in an accelerating phase of expansion [1]. Cosmic microwave background (CMB) data indicate that the present Universe has also a negligible space curvature [2]. Therefore if we assume the Friedmann–Robertson–Walker (FRW) model in which the effects of nonhomogeneities are neglected, then the acceleration must be driven by a dark energy component \(X\) (matter fluid violating the strong energy condition \(\rho _{{X}}+3p_{{X}}\ge 0)\). This kind of energy represents roughly 70 % of the matter content of the current Universe. Because the nature as well as mechanism of the cosmological origin of the dark energy component are unknown some alternative theories try to eliminate the dark energy option by modifying the theory of gravity itself. The main prototype of this kind of models is a class of covariant brane models based on the Dvali–Gabadadze–Porrati (DGP) model [3] as generalized to cosmology by Deffayet [4]. The simplest explanation of a dark energy component is the cosmological constant with effective equation of state \(p=-\rho \) but then the problem of its smallness appears and hence its relatively recent dominance. Although the \(\Lambda \)CDM model offers a possibility of explanation of the observational data it is only the effective theory which contains the enigmatic theoretical term—the cosmological constant \(\Lambda \). Numerous other candidates for a dark energy description have also been proposed like the evolving scalar field [5], usually referred as quintessence, the phantom energy [6, 7], the Chaplygin gas [8] model, etc. Some authors believe that the dark energy problem belongs to the quantum gravity domain [9].

Recent Planck observations still favor the standard cosmological model [10], especially for the high multipoles. However, in this model there are some problems with understanding the values of the density parameters for both dark matter and dark energy. The question is why energies of vacuum energy and dark matter are of the same order for the current Universe. The very popular methodology to solve this problem is to treat the coefficient equation of state as a free parameter, i.e. the wCDM model, which should be estimated from the astronomical and astrophysical data. The observations from the CMB and baryon acoustic oscillation (BAO) data sets give \(w_x=-1.13^{+0.24}_{-0.23}\) with 95 % confidence levels [10].

Alternative to this idea of the phantom dark energy mechanism of alleviating the coincidence problem is to consider the interaction between dark matter and dark energy; the interaction model. Many authors investigated observational constraints of the interaction model. Costa et al. [11] concluded that the interaction models become in agreement with the admissible observational data which can provide some argument toward consistency of the measured density parameters. Yang and Xu [12] constrained some interaction models under the choice of an ansatz for the transfer energy mechanism. From this investigation the joined geometrical tests show a stricter constraint on the interaction model if we include information from the large scale structure [\(f\sigma _{8}(z)\) data] of the Universe. These authors have found the interaction rate in the \(3\sigma \) region. This means that the recent cosmic observations favor it but with rather a small interaction between the both dark sectors. However, the measurement of the redshift-space distortion could rule out a large interaction rate in the \(1\sigma \) region. Zhang and Liu [13] using the SNIa observations, \(H(z)\) data (OHD), CMB, and secular Sandage–Loeb obtained the small value of the interacting parameter: \(\delta =-0.019\pm 0.01 (1 \sigma ), \pm 0.02 (2\sigma )\).

In all interaction models the specific ansatz for a model of interaction is postulated. There are infinite many of such models with a different form of interaction and there is some kind of a theoretical bias or degeneracy, coming from the choice of the potential form in scalar field cosmology. Szydlowski [14] proposed the idea of the estimation of the interaction parameter without any ansatz for the model of the interaction.

These theoretical models are consistent with the observations; they are able to explain the phenomenon of the accelerated expansion of the Universe. But should we really prefer such models over the \(\Lambda \)CDM one? All observational constraints show that the \(\Lambda \)CDM model still shows a good fit to the observational data. But from these constraints the small value of the interaction is still admissible. To answer this question we should use some model comparison methods to confront the existing cosmological models having observations at hand. We choose the information and Bayesian criteria of the model selection which are based on Occam’s razor (principle), the well-known and effective instrument in science to obtain a definite answer of whether the interacting \(\Lambda \)CDM model can be rejected.

We could not use this principle directly because the situations when two models explain the observations equally well are rare. But in information theory as well as in the Bayesian theory there are methods for model comparison which include such a rule.

It is worth noting that the complexity of the model is interpreted here as the number of its free parameters that can be adjusted to fit the model to the observations. If models under consideration fit the data equally well according to the Akaike rule the best one is with the smallest number of model parameters (the simplest one in such an approach).

It should be pointed out that the model selection methods presented are widely used in the context of cosmological model comparisons [18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. We should keep in mind that the conclusions based on such quantities depend on the data at hand. Let us mention again the example with the black box. Suppose that we made a few steps toward this box so that we can see the difference between the height of the left and right side of the white box. Our conclusion changes now.

Let us quote the example taken from [30]. Assume that we want to compare the Newtonian and Einsteinian theories in the light of the data coming from a laboratory experiment where general relativistic effects are negligible. In this situation the Bayes factor between Newtonian and Einsteinian theories will be close to unity. But comparing the general relativistic and Newtonian explanations of the deflection of a light ray that just grazes the Sun’s surface gives the Bayes factor \(\sim 10^{10}\) in favor of the first one (and even greater with more accurate data).

We share George Efstathiou’s opinion [41, 42, 43] that there is no sound theoretical basis for considering dynamical dark energy, whereas we are beginning to see an explanation for a small cosmological constant emerging from a more fundamental theory. In our opinion the \(\Lambda \)CDM model has the status of a satisfactory effective theory. Efstathiou argued why the cosmological constant should be given a higher weight as a candidate for a dark energy description than the dynamical dark energy. In this argumentation Occam’s principle is used to point out a more economical model explaining the observational data.

The main aim of this paper is to compare the simplest cosmological model—the \(\Lambda \)CDM model—with its generalization where the interaction between dark energy and matter sectors is allowed using the methods described above.

## 2 Interacting \(\Lambda \)CDM model

The interaction interpretation of the continuity condition (conservation condition) was investigated in the context of the coincidence problem since the paper Zimdahl [44], for recent developments in this area see Olivares et al. [45, 46]; see also Le Delliou et al. [47] for a discussion of recent observational constraints.

We find a very simple interpretation of (11): the evolution of the Universe is equivalent to the motion of a particle of unit mass in the potential well parameterized by the scale factor. In the procedure of the reduction of the problem of the FRW evolution to the problem of the investigation of a dynamical system of a Newtonian type we only assume that the effective energy density satisfies the conservation condition. We do not assume the conservation condition for each energy component (or non-interacting matter sectors).

In the next section we draw a comparison between the above model with the assumption that \(\overline{w_X}(a)=\text {const}=-1\) and the \(\Lambda \)CDM model.

## 3 Data

To estimate the parameters of the two models we used for our purposes the modified CosmoMC code [52, 53] with the implemented nested sampling algorithm Multinest [54, 55].

We used the observational data of 580 type Ia supernovae (the Union2.1 compilation [56]), 31 observational data points of the Hubble function from [57, 58, 59, 60, 61, 62, 63, 64, 65, 66] collected in [67], the measurements of BAO from the Sloan Digital Sky Survey (SDSS-III) combined with the 2dF Galaxy Redshift Survey [68, 69, 70, 71], the 6dF Galaxy Survey [72, 73], and the WiggleZ Dark Energy Survey [74, 75, 76]. We also used information coming from determinations of the Hubble function using the Alcock–Paczyński test [77, 78]. This test is very restrictive in the context of modified gravity models.

## 4 Results

### 4.1 The model parameter estimation

The mean of marginalized posterior PDF with 68 % confidence level for the parameters of the models. In the brackets are shown parameter values of the joint posterior probabilities. Estimations were made using the Union2.1, \(h(z)\), BAO, determinations of Hubble function using the Alcock–Paczyński test and the CMB \(R\) data sets

Union2.1 data only | Union2.1, \(h(z)\), BAO, AP data | Union2.1, \(h(z)\), BAO, AP, CMB data | |
---|---|---|---|

Interacting model | |||

\(\Omega _{{m},0} \in \langle 0,1 \rangle \) | \(0.3126^{+0.0064}_{-0.0343} \,(0.2952)\) | \(0.2770^{+0.0119}_{-0.0130} \,(0.2690)\) | \(0.2847^{+0.0107}_{-0.0115} \,(0.2725)\) |

\(\Omega _{\text {int},0} \in \langle -1,1 \rangle \) | \(-0.0232^{+0.1070}_{-0.1018} \,(-0.3492)\) | \(0.0109^{+0.0146}_{-0.0267} \,(0.0734)\) | \(-0.0139^{+0.0244}_{-0.0056} \,(-0.0152)\) |

\(m \in \langle -10,10 \rangle \) | \(-0.2687^{+1.2726}_{-0.3223} \,(-0.0528)\) | \(0.5622^{+0.7790}_{-0.5499} \,(0.9911)\) | \(0.3205^{+0.7826}_{-0.6730} \,(3.7364)\) |

\(h \in \langle 0.6,0.8 \rangle \) | \(0.7004^{+0.0996}_{-0.1004} \,(0.7912)\) | \(0.6949^{+0.0121}_{-0.0148} \,(0.6937)\) | \(0.6957^{+0.0120}_{-0.0147} \,(0.7093)\) |

\(\Lambda \)CDM model | |||

\(\Omega _{{m},0} \in \langle 0,1 \rangle \) | \(0.2956^{+0.0035}_{-0.0034} \,(0.2955)\) | \(0.2777^{+0.0070}_{-0.0073} \,(0.2791)\) | \(0.2912^{+0.0043}_{-0.0045} \,(0.2904)\) |

\(h \in \langle 0.6,0.8 \rangle \) | \(0.7000^{+0.1000}_{-0.1000} \,(0.6053)\) | \(0.6932^{+0.0048}_{-0.0049} \,(0.6922)\) | \(0.6858^{+0.0041}_{-0.0043} \,(0.6849)\) |

The value of the interaction parameter \(\Omega _{\text {int},0}\) is very small for all data sets. Especially the result for the second data set [Union2.1, \(h(z)\), BAO, AP data] indicates that the interaction is probably negligible. There is also no indication of the direction of the interaction if it is a physical effect. While for the Union2.1 data set only the interaction parameter \(\Omega _{\text {int},0}\) is negative and a greater value of \(\Omega _{{m},0}\) in the interacting \(\Lambda \) CDM model implies the flow from the dark energy sector to the matter sector, and for the data set consisting of all data the opposite.

### 4.2 The likelihood ratio test

The results of the likelihood ratio test for the \(\Lambda \)CDM model (null model) and the \(\Lambda \)CDM interacting model (alternative model). The values of \(\chi ^2_{\text {int}}\), \(\chi ^2_{\Lambda \text {CDM}}\), test statistic \(\lambda \), and the corresponding \(p\) value (\(df=4-2=2\)). Estimations were made using the Union2.1, \(h(z)\), BAO, determinations of the Hubble function using the Alcock–Paczyński test and the CMB \(R\) data sets

Data sets | \(\chi ^2_{\text {int}}/2\) | \(\chi ^2_{\Lambda \text {CDM}}/2\) | \(\lambda \) | \(p\) value |
---|---|---|---|---|

Union2.1 | \(272.5377\) | \(272.5552\) | \(0.0350\) | \(0.9826\) |

Union2.1, \(h(z)\), BAO, AP | \(282.2215\) | \(282.2555\) | \(0.0680\) | \(0.9667\) |

Union2.1, \(h(z)\), BAO, AP, CMB | \(282.3073\) | \(282.4912\) | \(0.3678\) | \(0.8320\) |

### 4.3 The model comparison using the AIC, BIC, and Bayes evidence

Values of the \(\chi ^2\), AIC, \(\Delta \)AIC (with respect to the \(\Lambda \)CDM model), BIC, and Bayes factor. Estimations were made using the Union2.1, \(h(z)\), BAO, determinations of Hubble function using the Alcock–Paczyński test and the CMB \(R\) data sets

Data sets | \(\chi ^2/2\) | AIC | \(\Delta \text {AIC}_{\text {int},\Lambda \text {CDM}}\) | BIC | \(2\ln B_{\Lambda \text {CDM,int}}\) |
---|---|---|---|---|---|

Interacting model | |||||

Union2.1 | \(272.5377\) | \(553.0754\) | \(3.9650\) | \(570.5275\) | \(12.6910\) |

Union2.1, \(h(z)\), BAO, AP | \(282.2215\) | \(572.4430\) | \(3.9320\) | \(590.1683\) | \(12.7947\) |

Union2.1, \(h(z)\), BAO, AP, CMB | \(282.3073\) | \(572.6146\) | \(3.6322\) | \(590.3464\) | \(12.4981\) |

\(\Lambda \)CDM model | |||||

Union2.1 | \(272.5552\) | \(549.1104\) | – | \(557.8365\) | – |

Union2.1, \(h(z)\), BAO, AP | \(282.2555\) | \(568.5110\) | – | \(577.3736\) | – |

Union2.1, \(h(z)\), BAO, AP, CMB | \(282.4912\) | \(568.9824\) | – | \(577.8483\) | – |

Regardless the data set the differences of the AIC quantities are in the interval \((3.4, 4)\) and are a little outside the interval \((4,7)\), which indicates the considerably smaller support for the interacting \(\Lambda \)CDM model. It means that while the \(\Lambda \)CDM model should be preferred over the interacting \(\Lambda \)CDM model, the latter cannot be ruled out.

However, we can arrive at a decisive conclusion employing the Bayes factor. The difference of BIC quantities is greater than 10 and have values in the interval \((12,13)\) for all data sets. Thus, the Bayes factor indicates strong evidence against the interacting \(\Lambda \)CDM model compared to the \(\Lambda \)CDM model. Therefore we are strongly convinced we should reject the interaction between dark energy and dark matter sectors due to Occam’s principle.

## 5 Conclusion

We considered the cosmological model with dark energy represented by the cosmological constant and the model with interaction between dark matter and dark energy (the interacting \(\Lambda \)CDM model). These models were studied statistically using the available astronomical data and then compared using the tools taken from information as well as Bayesian theory. In both cases the model selection is based on Occam’s principle, which states that if two models describe the observations equally well we should choose the simpler one. According to the Akaike and Bayesian information criteria the model complexity is interpreted in terms of the number of free model parameters, while according to the Bayesian evidence a more complex model has a greater volume of the parameter space.

Anyone using the Bayesian methods in astronomy and cosmology should be aware of the ongoing debate not only about pros but also cons of this approach. Efstathiou provided a critique of the evidence ratio approach indicating difficulties in defining models and priors [81]. Jenkins and Peacock [82] called attention to too much noise in the data, which does not allow one to decide to accept or reject a model based solely on whether the evidence ratio reaches some threshold value. That is the reason that we also used the AIC based on information theory.

The observational constraints on the parameter values, which we have obtained, have confirmed previous results that if the interaction between dark energy and matter is a real effect it should be very small. Therefore it seems to be natural to ask whether cosmology with interaction between dark energy and matter is plausible.

At the beginning of our model selection analysis we performed the standard likelihood ratio test. This test conclusion was to fail to reject the null hypothesis that there is no interaction between matter and dark energy sectors with the significance level \(\alpha =0.05\). It was the first clue against the interacting \(\Lambda \)CDM model. The \(\Delta \)AIC between both models was less conclusive. While the \(\Lambda \)CDM model was more supported, the interacting \(\Lambda \)CDM cannot be rejected. On the other hand the Bayes factor has given a decisive result; there was a very strong evidence against the interacting \(\Lambda \)CDM model compared to the \(\Lambda \)CDM model. Given the weak or almost non-existing support for the interacting \(\Lambda \)CDM model and bearing in mind Occam’s razor we are inclined to reject this model.

We have also the theoretical argument against the interacting \(\Lambda \)CDM model. If we consider the \(H^2\) formula which is a base for estimation there is a degeneracy because one cannot distinguish the effects of interaction from the effect \(w(z)\)—the case of varying equation of state depending on time or redshift.

## Notes

### Acknowledgments

M. Szydłowski has been supported by the National Science Centre (Narodowe Centrum Nauki) Grant 2013/09/B/ST2/03455. M. Kamionka has been supported by the National Science Centre (Narodowe Centrum Nauki) Grant PRELUDIUM 2012/05/N/ST9/03857. We thank the referee for carefully going through our manuscript.

### References

- 1.
- 2.D. Spergel et al., Astrophys. J. Suppl.
**170**, 377 (2007). doi: 10.1086/513700 CrossRefADSGoogle Scholar - 3.G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B
**485**, 208 (2000). doi: 10.1016/S0370-2693(00)00669-9 CrossRefADSMATHGoogle Scholar - 4.C. Deffayet, Phys. Lett. B
**502**, 199 (2001). doi: 10.1016/S0370-2693(01)00160-5 CrossRefADSMATHGoogle Scholar - 5.
- 6.R. Caldwell, Phys. Lett. B
**545**, 23 (2002). doi: 10.1016/S0370-2693(02)02589-3 CrossRefADSGoogle Scholar - 7.M.P. Dabrowski, T. Stachowiak, M. Szydlowski, Phys. Rev. D
**68**, 103519 (2003). doi: 10.1103/PhysRevD.68.103519 CrossRefADSGoogle Scholar - 8.A.Y. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B
**511**, 265 (2001). doi: 10.1016/S0370-2693(01)00571-8 CrossRefADSMATHGoogle Scholar - 9.E. Witten, in
*Sources and Detection of Dark Matter and Dark Energy in the Universe*, ed. by D.B. Cline (Springer, New York, 2004), pp. 27–36Google Scholar - 10.P. Ade et al., Astron. Astrophys. (2014). doi: 10.1051/0004-6361/201321591 MATHGoogle Scholar
- 11.A.A. Costa, X.D. Xu, B. Wang, E.G.M. Ferreira, E. Abdalla, Phys. Rev. D
**89**, 103531 (2014). doi: 10.1103/PhysRevD.89.103531 CrossRefADSGoogle Scholar - 12.W. Yang, L. Xu, Phys. Rev. D
**89**, 083517 (2014). doi: 10.1103/PhysRevD.89.083517 CrossRefADSGoogle Scholar - 13.M.J. Zhang, W.B. Liu, Eur. Phys. J. C
**74**, 2863 (2014). doi: 10.1140/epjc/s10052-014-2863-x CrossRefADSGoogle Scholar - 14.M. Szydlowski, Phys. Lett. B
**632**, 1 (2006). doi: 10.1016/j.physletb.2005.10.039 CrossRefADSGoogle Scholar - 15.D.J.C. MacKay,
*Information Theory, Inference, and Learning Algorithms*(Cambridge University Press, Cambridge, 2003)MATHGoogle Scholar - 16.H. Akaike, IEEE Trans. Autom. Control
**19**, 716 (1974). doi: 10.1109/TAC.1974.1100705 CrossRefADSMATHGoogle Scholar - 17.H. Jeffreys,
*Theory of Probability*, 3rd edn. (Oxford University Press, Oxford, 1961)MATHGoogle Scholar - 18.R. Trotta, Mon. Not. R. Astron. Soc.
**378**, 72 (2007). doi: 10.1111/j.1365-2966.2007.11738.x CrossRefADSGoogle Scholar - 19.A.R. Liddle, P. Mukherjee, D. Parkinson, Y. Wang, Phys. Rev. D
**74**, 123506 (2006). doi: 10.1103/PhysRevD.74.123506 CrossRefADSGoogle Scholar - 20.
- 21.M. Hobson, C. McLachlan, Mon. Not. R. Astron. Soc.
**338**, 765 (2003). doi: 10.1046/j.1365-8711.2003.06094.x CrossRefADSGoogle Scholar - 22.M. Beltran, J. Garcia-Bellido, J. Lesgourgues, A.R. Liddle, A. Slosar, Phys. Rev. D
**71**, 063532 (2005). doi: 10.1103/PhysRevD.71.063532 CrossRefADSGoogle Scholar - 23.P. Mukherjee, D. Parkinson, A.R. Liddle, Astrophys. J.
**638**, L51 (2006). doi: 10.1086/501068 CrossRefADSGoogle Scholar - 24.P. Mukherjee, D. Parkinson, P.S. Corasaniti, A.R. Liddle, M. Kunz, Mon. Not. R. Astron. Soc.
**369**, 1725 (2006). doi: 10.1111/j.1365-2966.2006.10427.x CrossRefADSGoogle Scholar - 25.A. Niarchou, A.H. Jaffe, L. Pogosian, Phys. Rev. D
**69**, 063515 (2004). doi: 10.1103/PhysRevD.69.063515 CrossRefADSGoogle Scholar - 26.A.R. Liddle, Mon. Not. R. Astron. Soc.
**351**, L49 (2004). doi: 10.1111/j.1365-2966.2004.08033.x CrossRefADSGoogle Scholar - 27.T.D. Saini, J. Weller, S. Bridle, Mon. Not. R. Astron. Soc.
**348**, 603 (2004). doi: 10.1111/j.1365-2966.2004.07391.x CrossRefADSGoogle Scholar - 28.M.V. John, J. Narlikar, Phys. Rev. D
**65**, 043506 (2002). doi: 10.1103/PhysRevD.65.043506 CrossRefADSGoogle Scholar - 29.D. Parkinson, S. Tsujikawa, B.A. Bassett, L. Amendola, Phys. Rev. D
**71**, 063524 (2005). doi: 10.1103/PhysRevD.71.063524 CrossRefADSGoogle Scholar - 30.
- 31.P. Serra, A. Heavens, A. Melchiorri, Mon. Not. R. Astron. Soc.
**379**, 169 (2007). doi: 10.1111/j.1365-2966.2007.11924.x CrossRefADSGoogle Scholar - 32.
- 33.W. Godlowski, M. Szydlowski, Phys. Lett. B
**623**, 10 (2005). doi: 10.1016/j.physletb.2005.07.044 CrossRefADSGoogle Scholar - 34.M. Szydlowski, A. Kurek, A. Krawiec, Phys. Lett. B
**642**, 171 (2006). doi: 10.1016/j.physletb.2006.09.052 CrossRefADSGoogle Scholar - 35.M. Szydlowski, W. Godlowski, Phys. Lett. B
**633**, 427 (2006). doi: 10.1016/j.physletb.2005.12.049 CrossRefADSMATHGoogle Scholar - 36.M. Kunz, R. Trotta, D. Parkinson, Phys. Rev. D
**74**, 023503 (2006). doi: 10.1103/PhysRevD.74.023503 CrossRefADSGoogle Scholar - 37.D. Parkinson, P. Mukherjee, A.R. Liddle, Phys. Rev. D
**73**, 123523 (2006). doi: 10.1103/PhysRevD.73.123523 CrossRefADSGoogle Scholar - 38.R. Trotta, Mon. Not. R. Astron. Soc.
**378**, 819 (2007). doi: 10.1111/j.1365-2966.2007.11861.x CrossRefADSGoogle Scholar - 39.A.R. Liddle, Mon. Not. R. Astron. Soc.
**377**, L74 (2007). doi: 10.1111/j.1745-3933.2007.00306.x CrossRefADSGoogle Scholar - 40.
- 41.
- 42.S. Chongchitnan, G. Efstathiou, Phys. Rev. D
**76**, 043508 (2007). doi: 10.1103/PhysRevD.76.043508 CrossRefADSGoogle Scholar - 43.M. Szydlowski, A. Krawiec, W. Czaja, Phys. Rev. E
**72**, 036221 (2005). doi: 10.1103/PhysRevE.72.036221 CrossRefADSGoogle Scholar - 44.W. Zimdahl, Int. J. Mod. Phys. D
**14**, 2319 (2005). doi: 10.1142/S0218271805007784 CrossRefADSMATHGoogle Scholar - 45.G. Olivares, F. Atrio-Barandela, D. Pavon, Phys. Rev. D
**74**, 043521 (2006). doi: 10.1103/PhysRevD.74.043521 CrossRefADSGoogle Scholar - 46.G. Olivares, F. Atrio-Barandela, D. Pavon, Phys. Rev. D
**77**, 063513 (2008). doi: 10.1103/PhysRevD.77.063513 CrossRefADSGoogle Scholar - 47.M. Le Delliou, O. Bertolami, F. Gil Pedro, AIP Conf. Proc.
**957**, 421 (2007). doi: 10.1063/1.2823818 - 48.M. Szydlowski, T. Stachowiak, R. Wojtak, Phys. Rev. D
**73**, 063516 (2006). doi: 10.1103/PhysRevD.73.063516 CrossRefADSGoogle Scholar - 49.K. Freese, M. Lewis, Phys. Lett. B
**540**, 1 (2002). doi: 10.1016/S0370-2693(02)02122-6 CrossRefADSMATHGoogle Scholar - 50.W. Godlowski, M. Szydlowski, A. Krawiec, Astrophys. J.
**605**, 599 (2004). doi: 10.1086/382669 CrossRefADSGoogle Scholar - 51.F. Costa, J. Alcaniz, J. Maia, Phys. Rev. D
**77**, 083516 (2008). doi: 10.1103/PhysRevD.77.083516 CrossRefADSGoogle Scholar - 52.A. Lewis, CosmoMC. http://cosmologist.info/cosmomc/
- 53.A. Lewis, S. Bridle, Phys. Rev. D
**66**, 103511 (2002). doi: 10.1103/PhysRevD.66.103511 CrossRefADSGoogle Scholar - 54.F. Feroz, M. Hobson, Mon. Not. R. Astron. Soc.
**384**, 449 (2008). doi: 10.1111/j.1365-2966.2007.12353.x CrossRefADSGoogle Scholar - 55.F. Feroz, M. Hobson, M. Bridges, Mon. Not. R. Astron. Soc.
**398**, 1601 (2009). doi: 10.1111/j.1365-2966.2009.14548.x CrossRefADSGoogle Scholar - 56.N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah et al., Astrophys. J.
**746**, 85 (2012). doi: 10.1088/0004-637X/746/1/85 CrossRefADSGoogle Scholar - 57.
- 58.J. Simon, L. Verde, R. Jimenez, Phys. Rev. D
**71**, 123001 (2005). doi: 10.1103/PhysRevD.71.123001 CrossRefADSGoogle Scholar - 59.E. Gaztanaga, A. Cabre, L. Hui, Mon. Not. R. Astron. Soc.
**399**, 1663 (2009). doi: 10.1111/j.1365-2966.2009.15405.x CrossRefADSGoogle Scholar - 60.D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, S.A. Stanford, JCAP
**1002**, 008 (2010). doi: 10.1088/1475-7516/2010/02/008 CrossRefADSGoogle Scholar - 61.M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti, G. Zamorani et al., JCAP
**1208**, 006 (2012). doi: 10.1088/1475-7516/2012/08/006 CrossRefADSGoogle Scholar - 62.N.G. Busca, T. Delubac, J. Rich, S. Bailey, A. Font-Ribera et al., Astron. Astrophys.
**552**, A96 (2013). doi: 10.1051/0004-6361/201220724 CrossRefADSGoogle Scholar - 63.C. Zhang, H. Zhang, S. Yuan, T.J. Zhang, Y.C. Sun, Res. Astron. Astrophys.
**14**, 1221 (2014). doi: 10.1088/1674-4527/14/10/002 CrossRefADSGoogle Scholar - 64.C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch et al., Mon. Not. R. Astron. Soc.
**425**, 405 (2012). doi: 10.1111/j.1365-2966.2012.21473.x CrossRefADSGoogle Scholar - 65.C.H. Chuang, Y. Wang, Mon. Not. R. Astron. Soc.
**435**, 255 (2013). doi: 10.1093/mnras/stt1290 CrossRefADSGoogle Scholar - 66.L. Anderson, E. Aubourg, S. Bailey, F. Beutler, A.S. Bolton et al., Mon. Not. R. Astron. Soc.
**439**, 83 (2013). doi: 10.1093/mnras/stt2206 CrossRefADSGoogle Scholar - 67.Y. Chen, C.Q. Geng, S. Cao, Y.M. Huang, Z.H. Zhu (2013). arXiv:1312.1443 [astro-ph.CO]
- 68.D.J. Eisenstein et al., Astrophys. J.
**633**, 560 (2005). doi: 10.1086/466512 CrossRefADSGoogle Scholar - 69.W.J. Percival et al., Mon. Not. R. Astron. Soc.
**401**, 2148 (2010). doi: 10.1111/j.1365-2966.2009.15812.x CrossRefADSGoogle Scholar - 70.D.J. Eisenstein et al., Astron. J.
**142**, 72 (2011). doi: 10.1088/0004-6256/142/3/72 - 71.C.P. Ahn et al., Astrophys. J. Suppl.
**211**, 17 (2014). doi: 10.1088/0067-0049/211/2/17 - 72.D.H. Jones, M.A. Read, W. Saunders, M. Colless, T. Jarrett et al., Mon. Not. R. Astron. Soc.
**399**, 683 (2009). doi: 10.1111/j.1365-2966.2009.15338.x - 73.F. Beutler, C. Blake, M. Colless, D.H. Jones, L. Staveley-Smith et al., Mon. Not. R. Astron. Soc.
**416**, 3017 (2011). doi: 10.1111/j.1365-2966.2011.19250.x CrossRefADSGoogle Scholar - 74.M.J. Drinkwater, R.J. Jurek, C. Blake, D. Woods, K.A. Pimbblet et al., Mon. Not. R. Astron. Soc.
**401**, 1429 (2010). doi: 10.1111/j.1365-2966.2009.15754.x CrossRefADSGoogle Scholar - 75.C. Blake, E. Kazin, F. Beutler, T. Davis, D. Parkinson et al., Mon. Not. R. Astron. Soc.
**418**, 1707 (2011). doi: 10.1111/j.1365-2966.2011.19592.x CrossRefADSGoogle Scholar - 76.C. Blake, T. Davis, G. Poole, D. Parkinson, S. Brough et al., Mon. Not. R. Astron. Soc.
**415**, 2892 (2011). doi: 10.1111/j.1365-2966.2011.19077.x CrossRefADSGoogle Scholar - 77.
- 78.C. Blake, K. Glazebrook, T. Davis, S. Brough, M. Colless et al., Mon. Not. R. Astron. Soc.
**418**, 1725 (2011). doi: 10.1111/j.1365-2966.2011.19606.x - 79.J. Bond, G. Efstathiou, M. Tegmark, Mon. Not. R. Astron. Soc.
**291**, L33 (1997)ADSGoogle Scholar - 80.H. Li, J.Q. Xia, Phys. Lett. B
**726**, 549 (2013). doi: 10.1016/j.physletb.2013.09.005 CrossRefADSGoogle Scholar - 81.G. Efstathiou, Mon. Not. R. Astron. Soc.
**388**, 1314 (2008). doi: 10.1111/j.1365-2966.2008.13498.x ADSGoogle Scholar - 82.C. Jenkins, J. Peacock, Mon. Not. R. Astron. Soc.
**413**, 2895 (2011). doi: 10.1111/j.1365-2966.2011.18361.x CrossRefADSGoogle Scholar - 83.

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP^{3} / License Version CC BY 4.0.