Nonlinear stability of Einstein-de Sitter universes

Technical Summary: Nonlinear Stability of Einstein–de Sitter Universes

Problem Statement
The Einstein–de Sitter (EdS) universe is the standard cosmological model describing the cold dark matter-dominated epoch of the universe. It is a spatially homogeneous and isotropic spacetime undergoing decelerated expansion ($a(t) \sim t^{2/3}$) driven by a pressureless fluid (dust). While linear perturbation theory suggests the EdS model is unstable under the Einstein–dust equations (leading to the growth of density perturbations), the question of its nonlinear stability has remained open. Previous results on cosmological stability have largely relied on accelerated expansion (e.g., de Sitter space with $\Lambda > 0$) to provide strong damping that dissipates perturbations. In decelerated regimes, such as the EdS model, the expansion is too slow to naturally dampen nonlinear interactions, and the confinement of gravitational waves on a compact spatial domain ($T^3$) precludes dispersive decay mechanisms. Furthermore, the pressureless nature of dust causes the Einstein and Euler equations to decouple, preventing the fluid from homogenizing. This paper addresses whether the EdS model can be nonlinearly stable under a physically appropriate matter model that retains the EdS asymptotic behavior.

Methodology
The authors employ a bootstrap argument to prove the future global existence and asymptotic stability of solutions to the Einstein–Euler equations on a spatial 3-torus ($T^3$) with a polytropic equation of state, $\bar{p} = C\bar{\rho}^{1+1/n}$, where the polytropic index satisfies $n > 3$. The methodology involves several novel technical steps:

1. Conformal Rescaling and Wave Gauge: The system is reformulated using a conformal transformation with a factor $\tau \sim t^{1/3}$, mapping the EdS background to a perturbation of the Minkowski metric. The authors impose a "sourced wave gauge" (Definition 3.6), which transforms the Einstein equations into a system of damped nonlinear wave equations. Crucially, the gauge source functions are chosen to depend explicitly on the spatial averages of the lapse and fluid density.
2. Decoupling Averages and Oscillations: The analysis separates the evolution variables into spatially averaged components (ODEs) and oscillatory remainders (PDEs). The authors derive a specific combination of the averaged lapse and fluid density, $(\rho_{av} - 12) + 12(h_{00})_{av}$, which decays due to the modified continuity equation. This decay is essential to prevent logarithmic growth in the spatial metric perturbations.
3. Energy Estimates with Correction Terms:
   • Metric: For the metric perturbations, the authors construct corrected wave energies (Definition 6.5) that include lower-order terms to ensure coercivity. A key innovation is commuting the wave equations with the material derivative $\partial_v = v^\mu \partial_\mu$. Since material derivatives of fluid variables exhibit improved decay compared to spatial derivatives (Proposition 4.7), this commutation allows for better control of the source terms in the metric equations.
   • Fluid: For the fluid variables, the authors define a base fluid energy with specific time-dependent weights ($\tau^{6/n}$) to account for the polytropic equation of state. They introduce a correction term involving the divergence of the velocity to generate a coercive bulk term (Proposition 7.9). The condition $n > 3$ is critical here to ensure the correction term is well-behaved and the energy is coercive.
4. Handling Coupling: The strong coupling between the metric and fluid is managed by exploiting the remaining gauge freedom. By choosing the damping parameter $\kappa$ in the wave gauge to be sufficiently large, the authors can absorb critical, slowly decaying source terms that arise from the coupling.

Key Contributions and Results
The main result, stated in Theorem 1.1 and Theorem 4.1, establishes the existence of an open family of nonlinearly stable solutions to the Einstein–Euler equations on $T^3$ with $\Lambda = 0$ and a polytropic equation of state ($n > 3$). Specifically:

• Global Existence: Solutions starting from initial data sufficiently close to the EdS background exist for all future time.
• Asymptotic Stability: As $t \to \infty$, the solutions converge to a spatially homogeneous spacetime with the same leading-order asymptotics as the Einstein–de Sitter universe. The metric components satisfy $|t^{-4/3}\bar{g}_{ij} - g^\infty_{ij}| \to 0$, and the fluid density satisfies $|t^2\bar{\rho} - \rho^\infty| \to 0$.
• Homogenization: The solutions homogenize over time, meaning spatial inhomogeneities decay, and the system converges to a member of the family of spatially homogeneous solutions.
• Sharp Thresholds: The paper identifies $n > 3$ as the necessary condition for this stability mechanism. For $n \le 3$, the authors conjecture instability, noting that fluid homogenization likely fails in that regime.

Significance
The paper resolves the question of whether the Einstein–de Sitter model can be nonlinearly stable. It demonstrates that while the pure dust model (linear equation of state with $K=0$) is unstable, introducing a small pressure via a polytropic equation of state with $n > 3$ provides the necessary mechanism for fluid homogenization and metric stabilization. This result complements the existing list of known nonlinearly stable spacetimes (Minkowski, Kerr, Milne) by providing the first stability result for a physically realistic, matter-dominated cosmological spacetime with compact topology and decelerated expansion. The work suggests that the EdS model is a valid description of the large-scale structure of the matter-dominated epoch, provided the matter possesses a slight stiffness (polytropic pressure) that facilitates homogenization, even if that pressure becomes negligible asymptotically. The methodology offers a blueprint for analyzing other slowly expanding spacetimes where the matter field drives the geometric expansion.