[Paper Review] Interpretation of bulk viscosity as the generalized Chaplygin gas

Marek Szydłowski, Adam Krawiec|arXiv (Cornell University)|Jun 26, 2020

Phase Equilibria and Thermodynamics1 references4 citations

TL;DR

This paper proposes that bulk viscosity in dark matter can naturally produce the equation of state of the generalized Chaplygin gas, unifying dark matter and dark energy without a cosmological constant. The authors show that viscous cosmological models with $ m < 1/2 $ are structurally stable and topologically equivalent to the $\Lambda$CDM model, with dynamics reducible to a conservative Newtonian-type system via potential-based phase space analysis.

ABSTRACT

The cosmological observations suggest that the presently accelerating universe should be filled by an exotic form of matter, violating the strong energy condition, of unknown nature and origin. We propose the viscous dark matter of a source of acceleration in the form of Chaplygin gas which is characterized by equation of state in the phenomenological form $p=-\frac{A}{ρ^α}$, where $p$ and $ρ$ are pressure and energy density respectively ($A$ and $α$ are constants). Chaplygin gas is interpreted in terms of viscous matter and without the cosmological constant. The acceleration effect is caused only by viscosity in this class of cosmological models. We show that bulk viscosity effects introduced to the standard FRW cosmology give rise to the natural unification of both dark matter and dark energy. We show that dust viscous cosmological models are structurally stable if $m &lt; 1/2$ ($1+α=1/2-m$).

Motivation & Objective

• To explore the cosmological implications of bulk viscosity in dark matter as an alternative to the cosmological constant.
• To establish a theoretical link between bulk viscosity and the generalized Chaplygin gas equation of state $ p = -A/\rho^\alpha $.
• To demonstrate that viscous dark matter alone can drive cosmic acceleration without invoking dark energy or $\Lambda$.
• To show that the resulting cosmological model is structurally stable and topologically equivalent to the $\Lambda$CDM model under specific parameter constraints.
• To reduce the dissipative FRW dynamics with bulk viscosity to a conservative Newtonian-type dynamical system for phase space analysis.

Proposed method

• Formulate the equation of state for viscous matter as $ p = B\rho - A/\rho^\alpha $, with $ B=0 $ to recover the generalized Chaplygin gas form.
• Use the Eckart approach to model bulk viscosity with $ \xi(\rho) = \beta\rho^m $, linking $ m $ to the Chaplygin gas parameter $ \alpha $ via $ 1 + \alpha = 1/2 - m $.
• Reduce the full FRW cosmological dynamics with bulk viscosity to a one-dimensional conservative system using Belinskii-Khalatnikov parameterization.
• Construct a potential function of the scale factor to analyze the dynamics as a particle motion in a potential well, enabling phase portrait analysis.
• Classify evolutionary paths via energy-level contours and study topological equivalence to $\Lambda$CDM using homeomorphism of trajectories.
• Apply structural stability analysis to determine the range of $ m $ for which the viscous model remains dynamically equivalent to $\Lambda$CDM.

Experimental results

Research questions

• RQ1 Can bulk viscosity in dark matter naturally produce the equation of state of the generalized Chaplygin gas?
• RQ2 What is the range of viscosity parameters $ m $ for which the viscous cosmological model remains structurally stable?
• RQ3 How does the inclusion of bulk viscosity lead to the unification of dark matter and dark energy without a cosmological constant?
• RQ4 To what extent is the phase space structure of the viscous Chaplygin model topologically equivalent to that of the $\Lambda$CDM model?
• RQ5 Can the dynamics of a dissipative FRW model with bulk viscosity be reduced to a conservative Newtonian-type system?

Key findings

• The generalized Chaplygin gas equation of state $ p = -A/\rho^\alpha $ arises naturally from bulk viscosity with $ \xi(\rho) = \beta\rho^m $, where $ m \in [-3/2, -1/2) $ corresponds to $ \alpha \in (0,1] $.
• Cosmological models with viscous matter and no cosmological constant are structurally stable if $ m < 1/2 $, which corresponds to $ \alpha \in (0,1] $.
• The dynamics of the viscous FRW model can be reduced to a conservative Newtonian-type system via a potential function of the scale factor.
• Phase portraits of the viscous model are topologically equivalent to those of the $\Lambda$CDM model when $ m < 1/2 $, with trajectories related by a homeomorphism preserving time direction.
• The viscous Chaplygin fluid dominates in the long-term evolution, making the two-fluid model (viscous Chaplygin + baryonic matter) asymptotically equivalent to a single-fluid Chaplygin model.
• The model is consistent with current astronomical data, as the energy density evolution $ \rho_{\text{Chapl}} = \rho_{\text{Chapl,0}}[A_s + (1 - A_s)a^{-3(1+\alpha)}]^{1/(1+\alpha)} $ satisfies conservation and observational constraints.