Large and small field inflation from hyperbolic sigma models

Long standing themes in inflation include the issue of large field vs. small as well question what fraction phase space leads to sufficient inflation, and furthermore is compatible with experimental data. In present paper these issues are discussed context modular a specialization framework automorphic nonlinear $\sigma$-models associated homogeneous spaces $G/K$ which continuous shift symmetry group $G$ weakly broken discrete subgroups $\Gamma$. The target theories inherit curved structure from $G$, case invariant hyperbolic geometry. It shown that this class models both inflationary trajectories within single model. analyzes concrete model $j$-inflation, nontrivial inflaton interactions. describes some detail initial conditions, including systematic analysis several phenomenological functions on space, leading constraints curvature scalar by upcoming experiments, discussion scaling behavior spectral index, finite volume attractor comparison geodesics. tensor-ratio shows $j$-inflation an interesting for ground satellite experiments.