Self-reproduction in k-inflation

We study cosmological self-reproduction in models of inflation driven by a scalar field ϕ 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_sH^-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 ϕ. 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 Fokker-Planck equations. We also show that there exists a (model-dependent) range ϕ_R<ϕ<ϕ_max⁡ within which large fluctuations are likely to drive the field towards the upper boundary ϕ=ϕ_max⁡, where the semiclassical consideration breaks down. An exit from inflation into reheating without reaching ϕ_max⁡ will occur almost surely (with probability 1) only if the initial value of ϕ is below ϕ_R. In this way, strong self-reproduction effects constrain models of k-inflation.