Cosmological $\alpha$-attractor models in $\mathcal{N}=1$ supergravity are based on the hyperbolic geometry of a Poincaré disk with the radius square ${\mathcal{R}}^2=3\alpha$. The predictions for the $B$ modes, $r\approx 3\alpha \frac{4}{N^2}$, depend on moduli space geometry and are robust for a rather general class of potentials. Here we notice that starting with M theory compactified on a 7-manifold with $G_2$ holonomy, with a special choice of Betti numbers, one can obtain $d=4$, $\mathcal{N}=1$ supergravity with the rank 7 scalar coset $[\frac{SL(2)}{SO(2)}{]}^7$. In a model where these seven unit size Poincaré disks have identified moduli one finds that $3\alpha =7$. Assuming that the moduli space geometry of the phenomenological models is inherited from this version of M theory, one would predict $r\approx {10}^{-2}$ for $N=53$ $e$-foldings. We also describe the related maximal supergravity and M/string theory models leading to preferred values $3\alpha =1$, 2, 3, 4, 5, 6, 7.

• Received 31 October 2016

DOI:https://doi.org/10.1103/PhysRevD.94.126015

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Published by the American Physical Society