Inflation in maximal gauged supergravities

We discuss the dynamics of multiple scalar fields and the possibility of realistic inflation in the maximal gauged supergravity. In this paper, we address this problem in the framework of recently discovered 1-parameter deformation of ${\rm SO}(4,4)$ and ${\rm SO}(5,3)$ dyonic gaugings, for which the base point of the scalar manifold corresponds to an unstable de Sitter critical point. In the gauge-field frame where the embedding tensor takes the value in the sum of the {\bf 36} and {\bf 36'} representations of ${\rm SL}(8)$, we present a scheme that allows us to derive an analytic expression for the scalar potential. With the help of this formalism, we derive the full potential and gauge coupling functions in analytic forms for the ${\rm SO}(3)\times {\rm SO}(3)$-invariant subsectors of ${\rm SO}(4,4)$ and ${\rm SO}(5,3)$ gaugings, and argue that there exist no new critical points in addition to those discovered so far. For the ${\rm SO}(4,4)$ gauging, we also study the behavior of 6-dimensional scalar fields in this sector near the Dall'Agata-Inverso de Sitter critical point at which the negative eigenvalue of the scalar mass square with the largest modulus goes to zero as the deformation parameter approaches a critical value. We find that when the deformation parameter is taken sufficiently close to the critical value, inflation lasts more than 60 e-folds even if the initial point of the inflaton allows an $O(0.1)$ deviation in Planck units from the Dall'Agata-Inverso critical point. It turns out that the spectral index $n_s$ of the curvature perturbation at the time of the 60 e-folding number is always about 0.96 and within the $1\sigma$ range $n_s=0.9639\pm0.0047$ obtained by Planck, irrespective of the value of the $\eta$ parameter at the critical saddle point. The tensor-scalar ratio predicted by this model is around $10^{-3}$ and is close to the value in the Starobinsky model.