A critical threshold for the cosmological Euler-Poisson system

TL;DR

This work analyzes the classical Euler-Poisson system on an expanding Newtonian cosmology with scale factor $a(t)=t^{α}$. It establishes a universal critical threshold $α_{crit}=2/3$ separating a global-stability regime from a shock-forming regime for small near-homogeneous data, using expansion-normalized variables and a corrected energy method. A rigorous small-data global existence result is proved for $α>2/3$ alongside a detailed higher-order energy framework, while numerical simulations in a 1D reduced model provide complementary evidence of shock formation for $α\leq 2/3$ and help locate the critical line. The findings highlight a fundamental difference between non-relativistic and relativistic fluid dynamics in cosmological settings and relate the threshold to the matter-dominated epoch, with implications for understanding structure formation in the Universe.

Abstract

We consider the gravitational Euler-Poisson system with a linear equation of state on an expanding cosmological model of the Universe. The expansion of the spatial sections introduces an additional dissipating effect in the Euler equation. We prescribe the expansion rate of space by a scale factor $a(t)=t^α$ with $α\in(0,1)$, which describes the growth of length scales over time. This model is regularly applied in cosmology to study classical fluids in an expanding Universe. We study the behaviour of solutions to this system arising from small, near-homogeneous initial data and discover a \emph{critical} change of behaviour near the expansion rate $α=2/3$, which corresponds to the matter-dominated regime in cosmology. In particular, we prove that for $α>2/3$ the fluid variables are global in time and remain small provided they are sufficiently small in a suitable norm initially. In the complementary regime $α\leq2/3$, we present numerical evidence for shock formation of solutions to the Euler equation for arbitrarily small initial data. In combination, this establishes the existence of a critical stability threshold for barotropic fluids in expanding domains. In contrast to our previous work on the corresponding relativistic system, the threshold in the classical system considered here is independent of the speed of sound of the fluid. This establishes that fluids in cosmology behave fundamentally different in the non-relativistic regime than in the relativistic one.

Figures (5)

• Figure 1: Time-evolution of the Sobolev norms of solutions with initial data of type \ref{['init']} with $\varepsilon$ of size as indicated in the respective legend. The data is taken for the specific case of $\alpha=0.8$ and $K=1/6$.
• Figure 2: Time-evolution of the Sobolev norms of solutions with initial data of type \ref{['init']} with $\varepsilon$ of size as indicated in the respective legend. The data is taken for the specific case of $\alpha=0.3$ and $K=1/6$.
• Figure 3: Left: Breaking time depending on size of the initial data for different expansion rates. Center: slope of the breaking time function depending on the expansion rate. Right: Critical expansion rate for different values of $K$.
• Figure 4: These pictures depict the physical characteristics for $\alpha=0.8$ and $K=1/6$. The amplitude of the left and right plot is given by $\varepsilon=1$ and $\varepsilon=1/4$, respectively. Note that, for the sake of presentation, the flow of the characteristics is plotted in the universal covering space of $S_{1}$, i.e., $\mathbb{R}$.
• Figure 5: These pictures depict the physical characteristics for $\alpha=0.3$ and $K=1/6$. The amplitude from left to right and top to bottom is given by $\varepsilon=1$, $\varepsilon=1/2$, $\varepsilon=1/8$ and $\varepsilon=1/32$, respectively.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Definition 1
• Lemma 3.1: Coercivity
• proof
• Lemma 3.2: First-order energy estimate
• proof
• Remark 3
• Lemma 3.3: Moser-type estimates
• Lemma 3.4