A constraint on turning inflation in supergravity

Abstract

Supergravity inflation is considered in the context of the single-field effective field theory for scalar perturbations arising from multiple-field inflation.\\ \\ It is possible to deduce a single-field effective field theory for the perturbations of multiple-field slow-roll inflation, valid given certain conditions, that gives curvature perturbations a speed of sound, dependent on the turn rate of the inflationary trajectory. On the other hand, $N=1$ supergravity imposes an upper bound on the minimum eigenvalue of the Hessian matrix of the F-term potential, in terms of quantities such as the curvature of the K\"ahler manifold.\\ \\ By the explicit calculation of the minimum eigenvalue for the case of two real scalar fields, it is found that it depends on several quantities such as the slow-roll parameters and the speed of sound, that in the presence of turns it can easily take values of order one, as opposed to the order of the slow-roll parameters, and that it possesses a negative minimum of order one. The case of multiple fields is discussed.\\ \\ If expressed in terms of the speed of sound and under the conditions of validity of the single-field EFT, the inequality indicates that the speed of sound, depending on the curvature of the K\"ahler manifold, cannot be far from one, that is, cannot be much different to canonical single-field inflation, in supergravity. This is a higher bound than the one obtained from current measurements. If the speed of sound is measured to be less than one, classes of supergravity inflation can be discarded.