The infrared behavior of tame two-field cosmological models

We study the first order infared behavior of tame hyperbolizable two-field cosmological models, defined as those classical two-field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable Riemann surface $(\Sigma,\mathcal{G})$ and whose scalar potential $\Phi$ admits a positive and Morse extension to the end compactification of $\Sigma$. We achieve this by determining the universal forms of the asymptotic gradient flow of the classical effective potential $V$ with respect to the uniformizing metric $G$ near all interior critical points and ends of $\Sigma$, finding that some of the latter act like fictitious but exotic stationary points of the gradient flow. We also compare these results with numerical studies of cosmological orbits. For critical cusp ends, we find that cosmological curves have transient quasiperiodic behavior but are eventually attracted or repelled by the cusp along principal geodesic orbits determined by the extended effective potential. This behavior is approximated in the infrared by that of gradient flow curves near the cusp.