Self-reproduction in $k$-inflation

We study cosmological self-reproduction in models of inflation driven by a scalar field $\varphi$ 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 $\varphi$. 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 ${\varphi }_R<\varphi <{\varphi }_{\max }$ within which large fluctuations are likely to drive the field towards the upper boundary $\varphi ={\varphi }_{\max }$, where the semiclassical consideration breaks down. An exit from inflation into reheating without reaching ${\varphi }_{\max }$ will occur almost surely (with probability 1) only if the initial value of $\varphi$ is below ${\varphi }_R$. In this way, strong self-reproduction effects constrain models of $k$-inflation.

Figure 1

A numerically obtained phase plot for the Lagrangian $p(X,\varphi )=(-X+X^2)\arctan \varphi$. Trajectories starting at large negative $\varphi$ with large positive $v\equiv \dot{\varphi }$ approach an attractor solution. The attractor trajectory approximately coincides with the line $v=v_0$ for sufficiently large $|\varphi |$.