Phase-Space Constraints on a General Class of Eckart-like Viscous Cosmologies

We study the cosmological dynamics of a general class of viscous dark matter models using a dynamical system analysis. In the Eckart formalism, these viscous cosmologies are characterized by a dimensionless bulk viscous pressure $Ω_Π\equiv Ω_Π(Ω_m)$, where $Ω_m$ is the matter density parameter. Our analysis demonstrates that a viable early radiation era requires the complete absence of viscosity, a proper matter-dominated epoch demands a small viscous pressure, and the only physically admissible late-time attractor is the cosmological constant-dominated fixed point; consequently, the background expansion must closely track standard $Λ$CDM evolution. We validate these general results using the power-law parameterization $Ω_Π= -\hatξ_0 Ω_m^s$, where $\hatξ_0$ and $s$ are real parameters; for $s \geq 1$ and $\hatξ_0 \ll 1$, the model reproduces the $Λ$CDM background, while for $s < 1$ the system encounters a past-time integration boundary where the dark matter density transitions into negative values and its viscous pressure evaluates to nonphysical complex numbers. Since the background expansion essentially duplicates $Λ$CDM, the model is unlikely to resolve the $H_0$ tension. However, large differences arise at the linear perturbative level; indeed, on sub-horizon scales, bulk viscosity causes a scale-dependent dissipative drag on the matter density fluctuations; this increases the Hubble friction, offering a channel to suppress structure growth.