Instabilities appearing in cosmological effective field theories: when and how?

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 originated from cosmological studies. In particular, we are interested in the equation ∂ t 2 u ( x , t ) = α ( ∂ x u ( x , t ) ) 2 + β ∂ 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 α > 0. We study the nature of this divergence as a function of the parameters α > 0 and β ⩾ 0 . The divergence does not disappear even when β is very large contrary to what one might believe (note that since we consider fixed initial data, α and β cannot be scaled away). But it will take longer to appear as β increases when α 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.