Instabilities Appearing in Cosmological Effective Field theories: When and How?

TL;DR

This paper investigates the finite-time divergence of solutions in a cosmological effective field theory PDE, analyzing how parameters influence the nature and timing of instabilities, with implications for modifying such equations.

Contribution

It provides a detailed analysis of the divergence behavior in a specific nonlinear PDE from cosmology, including parameter effects and extension to higher dimensions.

Findings

Solutions diverge in finite time for positive alpha.

Increasing beta delays divergence but does not prevent it.

Two types of divergence are identified and characterized.

Abstract

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe (note that since we consider fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.