Inflaton condensate fragmentation: Analytical conditions and application to $\alpha$-attractor models

We study the stability of an inflaton condensate in the presence of attractive inflaton self-interactions, in order to determine analytical conditions on the self-interaction couplings under which the condensate undergoes fragmentation. As an application of our results, we consider the stability of the inflaton condensate in $\alpha$-attractor inflation. We show that the stability of the condensate depends upon the value of $\alpha$. For the T-model with $n = 1$, we find that it is unstable for $\alpha \lesssim 10^{-4}$. In this case it is expected that inflation will be followed by an oscillon-dominated era.