Beyond the poles in attractor models of inflation

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 α-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 α-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.

Hybrid cosmological attractors

“…where k can be any (integer or not) positive constant. Such models, which were invented in the context of D-brane inflation [42][43][44][45][46][47][48] and pole inflation scenario [6,9,49,50], were recently incorporated in the general α-attractor framework [10].…”

Pole inflation from non-minimal coupling to gravity

“…A more geometric way to see this is by observing that the field space distance dφ 2 E = K E (φ)dφ 2 is infinite if the path of the field traverses a pole: even if the space is one-dimensional, it is not possible to normalise the field when its domain includes poles. As a result, a theory with poles is essentially a collection of multiple canonical theories that cannot communicate with each other [21].…”

CMB targets after the latest Planck data release

“…Meanwhile, for q < 2, the canonical field φ can reach the point φ ¼ 0 within finite time, and therefore the point ρ ¼ 0 is also accessible. This means that the coordinate system covering 0 < ρ < ∞ is incomplete; see [74] for a related discussion. This is not necessarily a problem.…”