A new nonlinear instability for scalar fields

Abstract

In this letter we introduce the non-linear partial differential equation (PDE) ${\partial }_{\ddot{I}}^2„\ddot{I}€\propto (\vec{\nabla }\ddot{I}€{)}^2$ 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.

5 pages, 2 figures; data available at https://zenodo.org/record/7950024