Generalized two-field $\alpha$-attractors from the hyperbolic triply-punctured sphere

Abstract

We study generalized two-field $\alpha$-attractor models whose rescaled scalar manifold is the triply-punctured sphere endowed with its complete hyperbolic metric, whose underlying complex manifold is the modular curve $Y(2)$. Using an explicit embedding into the end compactification, we compute solutions of the cosmological evolution equations for a few globally well-behaved scalar potentials, displaying particular trajectories with inflationary behavior as well as more general cosmological trajectories of surprising complexity. In such models, the orientation-preserving isometry group of the scalar manifold is isomorphic with the permutation group on three elements, acting on $Y(2)$ as the group of anharmonic transformations. When the scalar potential is preserved by this action, $\alpha$-attractor models of this type provide a geometric description of two-field `modular invariant $j$-models' in terms of gravity coupled to a non-linear sigma model with topologically non-trivial target and with a finite (as opposed to discrete but infinite) group of symmetries. The precise relation between the two perspectives is provided by the elliptic modular function $\lambda$, which can be viewed as a field redefinition that eliminates almost all of the countably infinite unphysical ambiguity present in the Poincaré half-plane description of such models.

49 pages

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Cited by in corpus (12)

• On Primordial Black Holes from Rapid Turns in Two-field Models
• Dark Energy from Inspiraling in Field Space
• Generalized two-field $\alpha$-attractor models from geometrically finite hyperbolic surfaces
• Two-field Cosmological $\alpha$-attractors with Noether Symmetry
• Dynamical consistency conditions for rapid turn inflation
• Hidden symmetries of two-field cosmological models
• Modular Inflation at Higher Level $N$
• Large and small field inflation from hyperbolic sigma models
• The infrared behavior of tame two-field cosmological models
• Cosmological flows on hyperbolic surfaces
• Dynamical renormalization and universality in classical multifield cosmological models
• Scaling behavior of observables as a model characteristic in multifield inflation