Self-reproduction in k-inflation
  We study cosmological self-reproduction in models of inflation driven by a scalar field $\phi$ with a noncanonical kinetic term ($k$-inflation). We develop a general criterion for the existence of attractors and establish conditions selecting a class of $k$-inflation models that admit a unique attractor solution. We then consider quantum fluctuations on the attractor background. We show that the correlation length of the fluctuations is of order $c_{s}H^{-1}$, where $c_{s}$ is the speed of sound. By computing the magnitude of field fluctuations, we determine the coefficients of Fokker-Planck equations describing the probability distribution of the spatially averaged field $\phi$. The field fluctuations are generally large in the inflationary attractor regime; hence, eternal self-reproduction is a generic feature of $k$-inflation. This is established more formally by demonstrating the existence of stationary solutions of the relevant FP equations. We also show that there exists a (model-dependent) range $\phi_{R}<\phi<\phi_{\max}$ within which large fluctuations are likely to drive the field towards the upper boundary $\phi=\phi_{\max}$, where the semiclassical consideration breaks down. An exit from inflation into reheating without reaching $\phi_{\max}$ will occur almost surely (with probability 1) only if the initial value of $\phi$ is below $\phi_{R}$. In this way, strong self-reproduction effects constrain models of $k$-inflation. 
Ferdinand Helmer and Sergei Winitzki