Non-linear causal bulk viscosity in unified dark matter cosmologies

Abstract We propose a bulk viscous unified dark matter scenario based on a non-linear extension of the full causal Israel–Stewart theory. This framework allows the viscous fluid to remain far from equilibrium – an essential feature for a physically consistent description of viscosity-driven accelerated expansion. We adopt the standard parametrization for the bulk viscosity, $$\xi =\xi _{0}\ \rho _{m}^{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mspace/> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , treating s as a free parameter (in contrast to most previous works), and study the model in a spatially flat Friedmann–Robertson–Walker background. By reformulating the cosmological equations as an autonomous dynamical system, we obtain both asymptotic analytical solutions and a numerical characterization of the phase space. At early times, the viscous component can mimic a stiff fluid, while at intermediate epochs it behaves like dark matter. With a suitable choice of dynamical variables, the system admits three distinct classes of late-time attractors. Two of them are separated by a basin-boundary saddle point: (i) a generic quintessence solution for $$s = 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , which encompasses a de Sitter-like behavior when $$\xi _{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> satisfies a specific relation involving the nonlinear parameters; (ii) a global exact de Sitter attractor for $$s &lt; 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ; and (iii) a phantom-like solution that emerges for $$s \ge 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . In contrast to the generic $$s\ne 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> case, the $$s=1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> scenario exhibits a qualitatively different stability structure, allowing de Sitter and phantom attractors to coexist. All solutions respect entropy production, and cosmic acceleration emerges independently of $$\xi _{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , relaxing the strong bounds, $$\xi _{0}\sim \mathcal {O}(1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∼</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , required in Eckart-based viscous models.