# Beyond the Poles in Attractor Models of Inflation

## TL;DR

This paper provides a geometric interpretation of attractor inflation models with singular kinetic terms, revealing how different domains separated by poles lead to distinct phenomenologies and new late-time behaviors, including boundary effects and multifield dynamics.

## Contribution

It introduces a geometric framework for understanding singular kinetic terms in attractor models, exploring boundary behaviors, multifield generalizations, and implications for initial conditions and late-time cosmology.

## Key findings

- Different domains separated by poles have distinct phenomenologies.
- Scalar fields can reach boundary of field space, affecting late-time behavior.
- Multifield models show poles as 'model walls' influencing inflationary trajectories.

## Abstract

We offer a geometric interpretation of attractor theories with singular kinetic terms as a union of multiple canonical models. We demonstrate that different domains (separated by poles) can drastically differ in their phenomenology. We illustrate this with the help of a "master model" that leads to distinct predictions depending on which side of the pole the field evolves before examining the more realistic example of $\alpha$-attractor models. Such models lead to quintessential inflation within the poles when featuring an exponential potential. However, beyond the poles, we discover a novel behaviour: the scalar field responsible for the early-time acceleration of the Universe may reach the boundary of the field-space manifold, indicating that the theory is incomplete and that a boundary condition must be imposed in order to determine its late-time behaviour. If the evolution of the field is arrested before this happens, however, we discover that quintessence can be achieved without a potential offset. Turning to multifield models with singular kinetic terms, we see that poles generalise straightforwardly to singular curves, which act as "model walls" between distinct pole-free inflationary models. As an example, we study a simple two-field $\alpha$-attractor-inspired model, whose evolution of isocurvature perturbations is sensitive to where the non-canonical field begins its trajectory. We finally discuss initial conditions in attractor theories, where the existence of multiple disconnected canonical models implies that we must make a fundamental choice: in which domain we impose a distribution for the inflaton in order to then determine the likelihood of inflation.