Self-reproduction in k-inflation

TL;DR

This work develops a general framework for self-reproduction in $k$-inflation by establishing attractor solutions for the inflaton with a noncanonical kinetic term and treating quantum fluctuations via a diffusion (Fokker-Planck) approach. It shows that fluctuations on the sound horizon dominate the stochastic dynamics, yielding a large diffusion coefficient and generic eternal inflation, with a stationary volume distribution concentrated near an upper boundary $\phi_{\max}$. The analysis derives explicit attractor asymptotics, demonstrates the existence of a largest FP eigenvalue $\lambda_{\max}\approx 3H_{\max}$, and identifies a model-dependent threshold $\phi_R$ below which exit to reheating is almost certain, thereby constraining initial conditions. The results illuminate how strong self-reproduction effects impose nontrivial constraints on viable $k$-inflation models and their early-universe dynamics. The framework integrates attractor theory, stochastic inflation, and FP spectral analysis to characterize the crossover between inflationary growth and exit in noncanonical settings.

Abstract

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 $c_{s}H^{-1}$, 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 $φ$. 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 FP 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.

Figures (1)

• Figure 1: A numerically obtained phase plot for the Lagrangian $p(X,\phi)=(-X+X^{2})\arctan\phi$. Trajectories starting at large negative $\phi$ with large positive $v\equiv\dot{\phi}$ approach an attractor solution. The attractor trajectory approximately coincides with the line $v=v_{0}$ for sufficiently large $\left|\phi\right|$.