Stability of fluids in spacetimes with decelerated expansion (2501.12798v1)

Abstract: We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.

• The paper proves nonlinear stability for homogeneous barotropic fluids in decelerated expanding spacetimes, establishing a condition linking fluid speed of sound and the expansion rate.
• It introduces an innovative universal energy functional, applicable across expansion regimes, used to prove boundedness and decay of perturbed fluid systems.
• The study identifies conditions causing instability for dust and radiation fluids in specific cases, enhancing comprehension of cosmic fluid dynamics in decelerated expansion.

Stability of Fluids in Spacetimes with Decelerated Expansion

This paper presents an investigation into the nonlinear stability of homogeneous barotropic perfect fluid solutions within cosmological spacetimes characterized by decelerated expansion. The research solidifies the understanding of conditions under which fluids remain stable despite the deceleration of the universe's expansion, contributing to the discourse on the behavior of relativistic fluids in cosmological settings.

Mathematical Formulation

The paper is grounded in the relativistic Euler equations, a cornerstone in modeling matter within the framework of general relativity, given by:

$\nabla^{\mu}T_{\mu\nu} = 0$

where $$T_{\mu\nu} = (\rho + p)u_{\mu}u_{\nu} + p g_{\mu\nu}$$ and $p = K \rho$ is the barotropic equation of state. The research focuses on expanding spacetimes of the form:

$ds^2 = -dt^2 + a(t)^2 \delta_{ij} dx^i dx^j$

with a polynomial scale factor $a(t) = t^\alpha$, where $0 < \alpha < 1$, which corresponds to decelerating expansion.

Main Contributions

1. Stability Analysis: The paper provides a rigorous proof of stability for fluids under a specific inequality condition linking the speed of sound $c_s = \sqrt{K}$ and the expansion rate $\alpha$. The condition necessary for stability is:

$K < 1 - \frac{2}{3\alpha}$

This inequality demarcates the boundary between stable and potentially unstable regimes, where the stabilizing friction-type effects due to expansion are countered by fluid characteristics like speed of sound.

1. Energy Functional Approach: An innovative universal energy functional is introduced, applicable across regimes of linear expansion. This energy functional is pivotal in proving boundedness and decay of the perturbed fluid system, thereby demonstrating stability.
2. Special Cases and Instabilities: For dust ($K=0$) and radiation fluids ($K=1/3$), the paper conclusively identifies conditions where small perturbations lead to shock formation, establishing the critical $\alpha = 1/2$ and $\alpha = 1$ as transition points for stability in the dust and radiation regimes, respectively.

Implications

The results enhance comprehension of cosmic fluid dynamics by confirming specific fluid behaviors in the presence of decelerated expansion, thus informing both theoretical models and cosmological simulations. A sharp criterion for fluid stability reinforces existing frameworks and guides future explorations of cosmological fluid dynamics, especially in early universe scenarios or compact spatial topologies.

Future Directions

The insights open potential avenues for further research into broader classes of spacetimes and more complex fluid models. Extending the analysis to fully dynamic spacetimes with backreaction effects, investigating non-polynomial expansion rates, and incorporating additional fluid interactions or dissipation mechanisms remain promising for future pursuits. Additionally, numerical analyses could be employed to further test and refine the delineated stability conditions.

In summary, this paper provides a comprehensive exploration of fluid stability in spacetimes with decelerated expansion, contributing significant theoretical advancements to the field of relativistic fluid dynamics in cosmological contexts.