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The Hyperbolic Geometry of Cosmological Attractors

John Joseph M. Carrasco, Renata Kallosh, Andrei Linde, Diederik Roest

Abstract

Cosmological alpha-attractors give a natural explanation for the spectral index n_s of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future detection of gravity waves. Their embedding into supergravity exploits the hyperbolic geometry of the Poincare disk or half-plane. These geometries are isometric under Mobius transformations, which include the shift symmetry of the inflaton field. We introduce a new Kahler potential frame that explicitly preserves this symmetry, enabling the inflaton to be light. Moreover, we include higher-order curvature deformations, which can stabilize a direction orthogonal to the inflationary trajectory. We illustrate this new framework by stabilizing the single superfield alpha-attractors.

The Hyperbolic Geometry of Cosmological Attractors

Abstract

Cosmological alpha-attractors give a natural explanation for the spectral index n_s of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future detection of gravity waves. Their embedding into supergravity exploits the hyperbolic geometry of the Poincare disk or half-plane. These geometries are isometric under Mobius transformations, which include the shift symmetry of the inflaton field. We introduce a new Kahler potential frame that explicitly preserves this symmetry, enabling the inflaton to be light. Moreover, we include higher-order curvature deformations, which can stabilize a direction orthogonal to the inflationary trajectory. We illustrate this new framework by stabilizing the single superfield alpha-attractors.

Paper Structure

This paper contains 30 equations, 1 figure.

Figures (1)

  • Figure 1: The Planck/BICEP/Keck 2015 constraints on $n_s$ and $r$ with the predictions of a number of models Ade:2015tvaPlanck:2015xua. The yellow lines correspond to the simplest $\alpha$-attractor models for a full range $0<\alpha<\infty$ and $N=50,60$Kallosh:2013yoa. These predictions nicely fit the latest cosmological results for the most natural choice of $\alpha \lesssim O(10)$.