# Bayesian analysis of bulk viscous matter dominated universe

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## Abstract

In our previous works, we have analyzed the evolution of bulk viscous matter dominated universe with a more general form for bulk viscous coefficient, \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) and also carried out the dynamical system analysis. We found that the model reasonably describes the evolution of the universe if the viscous coefficient is a constant. In the present work we are contrasting this model with the standard \(\varLambda \)CDM model of the universe using the Bayesian method. We have shown that, even though the viscous model gives a reasonable back ground evolution of the universe, the Bayes factor of the model indicates that, it is not so superior over the \(\varLambda \)CDM model, but have a slight advantage over it.

## 1 Introduction

Many observations lead to the conclusion that the present universe is accelerating [1, 2, 3, 4, 5, 6]. The reason for this acceleration was attributed to the dominant presence of a new cosmic component called dark energy. The \(\varLambda \)CDM model came out as the most successful one for explaining this late time acceleration of the universe. In this model the cosmological constant is being considered as the dark energy. But the model is plagued with severe drawbacks. The foremost among them is the cosmological constant problem and is about the discrepancy between the observed and predicted values of the dark energy density, which is of the order of 120. The other is the coincidence problem, the mysterious coincidence between the energy densities of the dark energy and dark matter component during the current epoch of the universe in spite of their completely different evolution history. This motivates a large class of models with varying dark energy density [7, 8, 9, 10, 11, 12, 13, 14]. Perfect fluid models like Chaplygin gas model [15, 16] would be an alternate suggestion, due to their ability to explain both the deceleration and late acceleration by a single cosmic component, which thus effectively leads to a unification of the dark matter and dark energy sectors. There were also attempts to study this phenomenon by modifying the geometry part of the gravity theories, like *f*(*R*) gravity [17, 18, 19], *f*(*T*) gravity [20, 21], Gauss-Bonnet theory [22], Lovelock gravity [23], Horava-Lifshitz gravity [24], scalar–tensor theories [25], braneworld models [26] etc.

As in the case of the Chaplygin gas model, another possibility of the unified description of both dark energy and dark matter arises in the dissipative fluid models. It has been shown that the early inflationary period of the universe can be due to the presence of an imperfect fluid with bulk viscosity [27, 28, 29, 30, 31]. This motivates the study of the dissipative cosmologies in the context of the late acceleration of the universe [32, 33, 34, 35, 36, 37]. In [33], by considering a single cosmic component, which is the dark matter with bulk viscosity, \(\zeta (\rho )=\alpha \rho ^m\) with \(\alpha \) and *m* being constants, the authors have shown that, the universe can make a transition from a decelerating phase to a late accelerating phase and ultimately to a de Sitter epoch. Inspite of this good background evolution, the model have come across with some negative aspects while analysing the structure formation. For instance, in reference [33] with \(\zeta =\alpha \rho ^{-0.4}\) the authors have shown that the density perturbation would rapidly be damped out, which adversely affect mainly the CMBR. It may be due to the power factor of the density \(- \ 0.4\), which was obtained by constraining the model with old supernovae luminosity data by Riess et al. [1]. At around the same time, in reference [35], the authors have considered a constant bulk viscous dark matter dominated universe, with \(0<\zeta <3\) and predicts that the universe began with a Big-Bang, followed by a decelerated expansion epoch and later transition into an accelerated epoch. Later these authors [36] extended their model by taking varying bulk viscosity of the form \(\zeta =\zeta _0+\zeta _1 H\) and found that it shows a background evolution close to that of the standard \(\varLambda \)CDM model. In [38], the authors have shown that the data from Planck CMB observation and different LSS observations prefer small but non-zero amount of viscosity in cold dark matter fluid.

In [39], we have analyzed the evolution of bulk viscous matter dominated universe with a more general form for bulk viscous coefficient, \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\), such that the viscosity depends on both the velocity and acceleration of the expansion, where *a* is the scale factor of expansion of the universe. We have studied the model by contrasting it with Union compilation of supernovae data and found that it predicts the transition to the late accelerating phase. A similar work has been done in reference [40], where they also include non-viscous baryonic matter, which constitute less than \(5\%\) of the total density, however, in constraining the parameters (\(\zeta _{0}\), \(\zeta _{1}\), \(\zeta _{2}\)), the authors fixed either \(\zeta _{1}\) or \(\zeta _{2}\) as zero while evaluating the other. In a later work [41] we found that a general evolution of the universe with all the conventional phases (radiation epoch, matter dominated epoch and later acceleration) can be described without any causality violation only when the bulk viscous coefficient is a constant, \(\zeta =\zeta _{0}\).

In the present paper we concentrate on the Bayesian analysis of the bulk viscous model corresponding to the different forms of the bulk viscous coefficient. We intend to compare these models in general with the standard \(\varLambda \)CDM model and also among themselves. Bayesian model comparison method is commonly used in the context of cosmological model selection [42, 43, 44, 45, 46, 47, 48]. More details regarding Bayesian model comparison are discussed in the Sect. 2.

The paper is organized as follows. In Sect. 2, we discuss the basic details regarding the Bayesian analysis. In Sect. 3, we introduce the bulk viscous model of the universe. In Sect. 4, we extract the bulk viscous parameters corresponding to different cases of our model and discuss the results of the Bayesian analysis of the viscous model and finally, in Sect. 5, we present our conclusion.

## 2 Bayesian model comparison

Various models have been proposed to interpret the cosmological observational data which eventually add to our understanding of the evolution of the universe. So there exist, in fact many models explaining the expected evolution of the universe. Contrasting these models among themselves to select the better ones is essential for understanding the true evolution of the universe. Bayesian statistical approach [48, 49, 50] is an effective tool to compare the new models with the standard \(\varLambda \)CDM model and also among themselves. The basic approach of this method is originated from the theory of random variables. In general, the relative merit of a random variable can be obtained by calculating the basic probability of it among the ensemble of values obtained theoretically or through repeated observations. But in cosmology repeated observations are virtually impossible. Here, what one can often do is to form hypothesis or a theory. For making the decision regarding the viability of such a proposed theory one have to assign certain probability to it in contrast to other theories existing for the same purpose. It is in this stage the Bayesian theory help us, so as to assign probability for a certain hypothesis by considering the observational data already available to us. Due to the acquisition of more data, one can in fact adjust the plausibility of the hypothesis using Bayesian theorem. This method have been adopted by many in the past, for instance, Jaffe [51] and Hobson et al. [52] have analysed the relative merits of certain cosmological models. Also John and Narlikar [48] have compared a simple cosmological model with scale factor \(a(t) \propto t\) with standard and inflationary models of the universe. In many models one does not have a prior knowledge about the model parameters for assigning the corresponding probability and in such cases one often starts with a flat prior for the parameter.

*D*and assuming any other background information

*I*to be true, is given as,

*I*is true and \(p(D|H_i,I)\) is the likelihood for the hypothesis \(H_i\), which is the probability for obtaining the data D provided the hypothesis \(H_i\) and

*I*are true. The factor

*p*(

*D*|

*I*) helps in normalization.

*D*is the likelihood for the model \(M_i\), we re-notate it with \(L(M_i)\), then the Eq. (2) becomes,

*I*does not give any preference to a model over any other, then the prior probabilities becomes equal, so that,

## 3 Bulk viscous FLRW Universe

*a*(

*t*) is the scale factor and an overdot represents the derivative with respect to cosmic time

*t*. We are neglecting the radiation components as it doesn’t have any decisive role in the late evolution of the universe. The conservation equation of the cosmic component is then,

*P*is the normal kinetic pressure and \(\zeta \) is the coefficient of bulk viscosity. The matter component in the late universe is non-relativistic hence it is usually taken as, \(P=0.\) Then the contribution to effective pressure is only due to the negative viscous pressure. The coefficient \(\zeta \) is basically a transport coefficient, hence it would depend on the dynamics of the cosmic fluid. We consider the most general form for the bulk viscous coefficient \(\zeta \), which is a linear combination of the three terms [39, 40, 41, 56, 57],

*H*for the general form of \(\zeta \) as [39],

In reference [41], we have done the phase space analysis of the model with bulk viscous matter as the dominating cosmic component. Three choices for the bulk viscous coefficient, (i) \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\), (ii) \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\) and (iii) \(\zeta =\zeta _{0}\) have been adopted. It was found that for all the three cases, it predicts a prior unstable decelerated epoch, and a later stable accelerating epoch, similar to the de Sitter phase. However, when the radiation component is also taken into account, the model support the conventional evolution of the universe, only for the case \(\zeta =\zeta _{0}\). The other two cases doesn’t predicts a prior radiation dominated phase and conventional decelerated matter dominated phase of the universe respectively.

## 4 Bayesian analysis of bulk viscous models

- 1.
\(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\),

where viscosity is depending on both the velocity and acceleration of the expansion of the universe.

- 2.
\(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\),

where viscosity is depending only on velocity of the expansion of the universe apart from a constant additive part \(\zeta _0\). This is equivalent to \(\zeta =\zeta _{0}+\zeta _1\rho ^s\), with \(s=1/2\).

- 3.
\(\zeta =\zeta _{0}\),

where viscosity is pure a constant

- 4.
\(\zeta =\zeta _{1}\frac{\dot{a}}{a}\).

where viscosity only has the velocity dependent term and is equivalent to \(\zeta =\zeta _1\rho ^s\), with \(s=1/2\).

- 5.
\(\zeta =\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\)

where viscosity is depending on acceleration apart from an additive constant.

Best estimates of the bulk viscous parameters, \(H_{0}\) and also \(\chi ^{2}\) minimum value corresponding to the cases 4 and 5 of \(\zeta \). \(\chi ^{2}_{d.o.f}=\frac{\chi ^{2}_{min}}{n-m}\), where \(n=307\), the number of data and *m* is the number of parameters in the model. The subscript d.o.f stands for degrees of freedom. For the best estimation we have use SCP “Union” 307 SNe Ia data sets. The values of parameter corresponding to the first three cases are extracted in [39, 41]

Parameters | Bulk viscous models | |
---|---|---|

\(\zeta =\zeta _{1}\frac{\dot{a}}{a}\) | \(\zeta =\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | |

\(\tilde{\zeta }_0\) | – | 1.275 |

\(\tilde{\zeta }_1\) | 1.683 | – |

\(\tilde{\zeta }_2\) | – | 1.593 |

\(H_0\) | 69.21 | 70.50 |

\(\chi ^2_{min}\) | 319.31 | 310.54 |

\(\chi ^2_{d.o.f}\) | 1.04 | 1.02 |

*n*is the total number of data and \(\mu _{t}\) is the theoretical distance modulus for the k-th supernova with the same redshift \(z_{k}\), which is given as

*m*and

*M*are the apparent and absolute magnitudes of the SNe respectively. \(d_{L}\) is the luminosity distance and is defined as

*c*is the speed of light. After obtaining the \(\chi ^2,\) we evaluate the marginal likelihood, using Eq. (9), and likelihood, using Eq. (7), for all the five cases of the model. We kept \(\varLambda \)CDM model as the reference model in order to compare the bulk viscous models and calculate the Bayes factor using Eq. (4). The marginal likelihood of the parameters \(\tilde{\zeta }\) corresponding to the five cases of bulk viscous models are shown in Figs. 1, 2, 3, 4 and 5, respectively.

Bayes factors with respect to \(\varLambda \)CDM model corresponding to three different priors

Sl. no. | Bulk viscous models | Bayes factor \(B_{i\varLambda }=\frac{L(M_i)}{L(M_\varLambda )}\) | ||
---|---|---|---|---|

\(M_i\) | Prior I | Prior II | Prior III | |

1 | \(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 0.743 | 1.372 | 1.043 |

2 | \(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\) | 1.86 | 2.63 | 3.63 |

3 | \(\zeta _{0}\) | 0.27 | 0.32 | 0.42 |

4 | \(\zeta _{1}\frac{\dot{a}}{a}\) | 0.05 | 0.042 | 0.052 |

5 | \(\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 1.65 | 1.91 | 1.77 |

Best estimates of the Bulk viscous parameters, \(H_{0}\) and also \(\chi ^{2}\) minimum value corresponding to the different cases of \(\zeta \) for high redshift. \(\chi ^{2}_{d.o.f}=\frac{\chi ^{2}_{min}}{n-m}\), where \(n=150\), the number of data and *m* is the number of parameters in the model. The subscript d.o.f stands for degrees of freedom

Viscous models | \(\tilde{\zeta }_0\) | \(\tilde{\zeta }_1\) | \(\tilde{\zeta }_2\) | \(H_0\) | \(\chi ^2_{min}\) | \(\chi ^2_{d.o.f}\) |
---|---|---|---|---|---|---|

\(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 7.36 | \(-\) 4.73 | − 1.25 | 67.41 | 166.88 | 1.135 |

\(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\) | 4.53 | \(-\) 2.53 | – | 67.41 | 166.88 | 1.13 |

\(\zeta _{0}\) | 1.17 | – | – | 63.06 | 167.01 | 1.12 |

\(\zeta _{1}\frac{\dot{a}}{a}\) | – | 0.787 | – | 61.43 | 167.09 | 1.12 |

\(\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 1.28 | – | 1.43 | 67.41 | 166.88 | 1.13 |

Bayes factors with respect to \(\varLambda \)CDM model corresponding to two different priors for high redshift

Sl. no. | Bulk viscous models | Bayes factor \(B_{i\varLambda }=\frac{L(M_i)}{L(M_\varLambda )}\) | |
---|---|---|---|

\(M_i\) | Prior I | Prior II | |

1 | \(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 0.79 | 0.06 |

2 | \(\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\) | 1.044 | 1.4 |

3 | \(\zeta _{0}\) | 1.17 | 1.48 |

4 | \(\zeta _{1}\frac{\dot{a}}{a}\) | 1.13 | 1.42 |

5 | \(\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\) | 0.733 | 0.2685 |

## 5 Conclusion

Bulk viscous models of the universe are important, especially regarding the unification of dark matter and dark energy. In some earlier works we have shown that this class of models reasonably explained the background evolution of the universe. It was also shown from the dynamical behavior that the model describes all the conventional phases of the universe and asymptotically tend towards a stable de Sitter epoch, provided the bulk viscosity of the cosmic fluid is a constant. In the present work we have contrasted the bulk viscous model of the universe with the standard \(\varLambda \)CDM model using the method of Bayesian analysis. We have first extracted the viscous parameters corresponding to the following five cases of bulk viscous model, (1) \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\), (2) \(\zeta =\zeta _{0}+\zeta _{1}\frac{\dot{a}}{a}\), (3) \(\zeta =\zeta _{0}\), (4) \(\zeta =\zeta _{1}\frac{\dot{a}}{a}\), (5) \(\zeta =\zeta _{0}+\zeta _{2}\frac{\ddot{a}}{\dot{a}}\), using the “Union” data of Supernovae type Ia. We have obtained the Bayes factor for all the five cases, see Table 2. For the full supernovae data set, the results indicate that the model corresponding to case 2, i.e., \(\zeta =\zeta _0+\zeta _1 \frac{\dot{a}}{a}\) have a Bayes factor just above 3, and thus have slight advantage over the \(\varLambda \)CDM model compared with other cases. For the model corresponding to cases 1 and 5, the Bayes factor is just above one and can just have a bare mention, in contrast to the standard model. All other cases, especially the case 3, with constant viscosity, seems to fail in standing against the standard model.

For more reliable result, we restrict to supernovae data with relatively high redshifts, \(z>0.5\), which were obtained with less background interference and hence are more reliable in making predictions regarding the evolution near the transition. The results consequent to this have a marked deviation from the previous one, such that the Bayes factor for the constant bulk viscosity \(\zeta =\zeta _0\) (case 3) and models corresponding to cases 2 and 4, where the viscous coefficient depends on the velocity of expansion, are having a slight advantage over other cases when compared with the standard \(\varLambda \)CDM model. Since Bayes factors of the cases 2, 3 and 4, are all in the range \(1<B_{ij}<3,\) it is difficult to discriminate among themselves. However it was shown in reference [41] that only the case 3 will have asymptotically stable end de Sitter phase. Taking account of this, it can be concluded that, among the cases 2, 3 and 4, which are having almost same Bayes factor, the case 3, corresponding to the one with stable end de Sitter phase, can be preferred over the other cases.

## Notes

### Acknowledgements

One of the authors (AS) is thankful to DST for giving financial support through an INSPIRE fellowship and author NDJM is thankful to UGC for financial support through BSR fellowship. Author (TKM) is thankful to IUCAA, Pune for the hospitality, where part of the work has been carried out.

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