Bulk Viscous Cosmological Model with Interacting Dark Fluids

Abstract

We study a cosmological model for a spatially flat Universe whose constituents are a dark energy field and a matter field comprising baryons and dark matter. The constituents are assumed to interact with each other, and a non-equilibrium pressure is introduced to account for irreversible processes. We take the non-equilibrium pressure to be proportional to the Hubble parameter within the framework of a first-order thermodynamic theory. The dark energy and matter fields are coupled by their barotropic indexes, which depend on the ratio between their energy densities. We adjust the free parameters of the model to optimize the fits to the Hubble parameter data. We compare the viscous model with the non-viscous one, and show that the irreversible processes cause the dark-energy and matter-density parameters to become equal and the decelerated–accelerated transition to occur at earlier times. Furthermore, the density and deceleration parameters and the distance modulus have the correct behavior, consistent with a viable scenario of the present status of the Universe.

Appendix: Bayesian Inference

In a statistical sense, a physical model may be thought of as described by a set of parameters. These parameters may be determined in many ways; most commonly, one resorts to Bayesian inference, a well-established statistical inference procedure, which estimates model parameters on the basis of evidence. The main purpose of this section is to present a brief introduction to the subject.

For a given model and data set, Bayesian inference employs a probability distribution called posterior probability to summarize all uncertainty. This probability distribution is proportional to a prior probability distribution (or simply the prior) and a likelihood function. The later, denoted by $\mathcal{P}(\mathbf{D}|\mathbf{\theta})$, is usually defined as the unnormalized probability density of measuring the data D = {D ₁,D ₂,...,D _n } for a given model $\mathcal{M}$ in terms of its parameters θ = { θ ₁ ,θ ₂ ,...,θ _n }. For our purposes, it suffices to assume that the measured values are normally distributed around their true value, so that

$$ \mathcal{P}(\mathbf{D}|\mathbf{\theta}) \propto \exp \left[-\chi^{2}(\mathbf{\theta})/2 \right].\label{eq:4} $$

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The posterior $\mathcal{P}(\theta|\mathbf{D})$ is determined by Bayes’ theorem:

$$ \mathcal{P}(\theta|\mathbf{D})=\frac{\mathcal{P}(\mathbf{D}|\theta) \mathcal{P}(\theta)}{\int {\rm d}\theta \mathcal{P}(\mathbf{D}|\theta)\mathcal{P}(\theta)},\label{eq:5} $$

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where $\mathcal{P}(\theta)$ denotes the prior probability distribution. The prior carries all previous knowledge of the parameters before the measurements were made.

Bayesian inference estimates parameters by maximizing the posterior $\mathcal{P}(\theta|\mathbf{D})$. This is in contrast with the frequentist approach, which maximizes the likelihood $\mathcal{P}(\mathbf{D}|\mathbf{\theta})$. Nevertheless, whenever the so-called uninformative priors are considered, both frameworks lead to the same conclusions. If the measured data are independent from each other as well as Gaussian distributed around their true value, D(θ), then maximizing the likelihood $\mathcal{P}(\mathbf{D}|\mathbf{\theta})$ is equivalent to minimizing the chi-square function

$$ \chi^{2}(\theta) \equiv \left(\mathbf{D}^{\rm obs}-\mathbf{D}(\theta)\right) C^{-1}\left(\mathbf{D}^{\rm obs}-\mathbf{D}(\theta)\right)^{T},\label{eq:6} $$

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where C is the covariance matrix given by the experimental errors. For uncorrelated data, $C_{ij}=\delta_{ij}\sigma^{2}_{i}$ and

$$ \chi^{2}(\theta) \equiv \sum\limits_{i=1}^{n}\left(\frac{D^{\rm obs}-D(\theta)}{\sigma^{2}_{i}}\right)^{2}, \label{eq:7} $$

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where σ _i denotes the experimental errors.

In Bayesian inference, confidence intervals are drawn around the maximal likelihood point, which yields the best fit parameters. It is conventionally used 1σ and 2σ confidence regions with 68.3% and 95.4% probability, respectively, for the true value of parameters. These regions are mathematically defined by the inequalities

$$ \chi^{2}(\theta)-\chi^{2}(\theta_{\rm bf}) \leq 2.3,\label{eq:8} $$

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for the 1σ range and

$$ \chi^{2}(\theta)-\chi^{2}(\theta_{\rm bf}) \leq 6.17,\label{eq:9} $$

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for the 2σ range, where θ _bf denotes the best-fit value of parameters.