In this letter we introduce the non-linear partial differential equation (PDE) $\partial^2_{\tau} \pi \propto (\vec\nabla \pi)^2$ showing a new type of instability. Such equations appear in the effective field theory (EFT) of dark energy for the $k$-essence model as well as in many other theories based on the EFT formalism. We demonstrate the occurrence of instability in the cosmological context using a relativistic $N$-body code, and we study it mathematically in 3+1 dimensions within spherical symmetry. We show that this term dominates for the low speed of sound limit where some important linear terms are suppressed.

翻译：本文提出了一种新的非线性偏微分方程（PDE）$\partial^2_{\tau} \pi \propto (\vec\nabla \pi)^2$，该方程展现了一种新型不稳定性。这类方程出现在暗能量有效场论（EFT）中的$k$-本质模型以及基于EFT形式体系的许多其他理论中。我们利用相对论性$N$-体数值模拟证明了该不稳定性在宇宙学背景下的发生，并在球对称条件下（3+1维）进行了数学分析。研究表明，在声速较低（若干重要线性项被抑制）的极限下，这一项将主导动力学行为。