Formulating E- & T-Model Inflation in Supergravity

We present novel realizations of E- and T-model inflation within Supergravity which are largely associated with the existence of a pole of order one and two respectively in the kinetic term of the inflaton superfield. This pole arises due to the selected logarithmic Kahler potentials K, which parameterize hyperbolic manifolds with scalar curvature related to the coefficient (-N)<0 of a logarithmic term. The associated superpotential W exhibits the same R charge with the inflaton-accompanying superfield and includes all the allowed terms. The role of the inflaton can be played by a gauge singlet or non-singlet superfield. Models with one logarithmic term in K for the inflaton, require N=2, some tuning -- of the order of 10^-5 -- between the terms of W and predict a tensor-to-scalar ratio r at the level of 0.001. The tuning can be totally eluded for more structured K's, with N values increasing with r and spectral index close or even equal to its present central observational value.