We explore the dynamics of multi-field models of inflation in which the field-space metric is a hyperbolic manifold of constant curvature. Such models are known as α-attractors and their single-field regimes have been extensively studied in the context of inflation and supergravity. We find a variety of multi-field inflationary trajectories in different regions of parameter space, which is spanned by the mass parameters and the hyperbolic curvature. Amongst these is a novel dynamical attractor along the boundary of the Poincare disc which we dub "angular inflation". We calculate the evolution of adiabatic and isocurvature fluctuations during this regime and show that, while isocurvature modes decay during this phase, the duration of the angular inflation period can shift the single-field predictions of α-attractors. arXiv:1803.09841v1 [hep-th]

Multi-field inflation with a curved scalar geometry has been found to support background trajectories that violate the slow-roll, slow-turn conditions and thus have the potential to evade the swampland constraints. In order to understand how generic this novel behaviour is and what conditions lead to it, we perform a classification of dynamical attractors of two-field inflation that are of the scaling type. Scaling solutions form a oneparameter generalization of De Sitter solutions with a constant value of the first Hubble flow parameter and, as we argue and demonstrate, form a natural starting point for the study of non-slow-roll slow-turn behaviour.All scaling solutions can be classified as critical points of a specific dynamical system. We recover known multi-field inflationary attractors as approximate scaling solutions and classify their stability using dynamical system techniques. In particular, we discover that dynamical bifurcations play an integral role in the transition between geodesic and nongeodesic motion and discuss the ability of scaling solutions to describe realistic multi-field models. We revisit the criteria for background stability and show cases where the usual criteria found in the literature do not capture the background evolution of the system. arXiv:1903.06116v2 [hep-th] 29 Jul 2019 Contents 7 Summary and Discussion 32 A Coordinate transformations and isometries 34 B Hurwitz-Routh stability criterion 34 C Bifurcations in dynamical systems 35 1 There has been a growing interesting for inflationary models involving gauge fields (see e.g. Refs. [13-17]); however, in such cases, a scalar degree of freedom that takes the role of the inflaton can be identified. 2 There are a few examples in the literature which admit general analytical solutions [18-28]. These are completely integrable systems where the mini-super Lagrangian has Noether symmetries, and have been classified in Refs. [29-31].

Recent years have seen the introduction of various multi-field inflationary scenarios in which the curvature and geodesics of the scalar manifold play a crucial role. We outline a simple description that unifies these different proposals and discuss their stability criteria. We demonstrate how the underlying dynamics is governed by an effective potential, whose critical points and bifurcations determine the late-time behaviour of the system, thus unifying hyperinflation, angular, orbital and side-tracked inflation. Interestingly, we show that hyperinflation is a special case of side-tracked inflation, relying on the enhanced isometries of the hyperbolic manifold. We provide the explicit coordinate transformation that maps the two models into each other. Finally, we relax the assumption of a field-space isometry along the inflationary direction that has been considered a prerequisite in the literature so far. We explicitly construct inflationary solutions that do not proceed along a field-space isometry or geodesic and use them to discuss stability criteria.

We investigate the observational signatures of many-field inflation and present analytic expressions for the spectral index as a function of the prior. For a given prior we employ the central limit theorem and the horizon crossing approximation to derive universal predictions, as found previously. However, we also find a specific dependence on the prior choice for initial conditions that has not been seen in previous studies. Our main focus is on quadratic inflation, for which the initial conditions statistics decouple from those of the mass distribution, while other monomials are also briefly discussed. We verify the validity of our calculations by comparing to full numerical simulations with 10 2 fields using the transport method.

We study the dynamics of a cosmological model with a perfect fluid and 𝒩 fields on a hyperbolic field space interacting via a symmetric potential. We list all late-time solutions, investigate their stability and briefly discuss predictions of the theory. Moreover, for the case of two scalar fields and an exponential potential we prove that the field equations are Liouville integrable and we provide for the first time the general solution for a region of the parameter space.