Model of inflation independent of the initial conditions, with bounded number of e-folds and nslarger or smaller than one

We study a supergravity model of inflation essentially depending on one parameter which can be identified with the slope of the potential at the origin. In this type of models the inflaton rolls at high energy from negative values and a positive curvature potential. At some point defined by the equation η=1 inflation starts. The potential curvature eventually changes to negative values and inflation ends when η=-1. No spontaneous symmetry breaking type mechanism for inflation of the new type to occur is here required. The model naturally gives a bounded total number of e-folds which is typically close to the required number for observable inflation and it is independent of the initial conditions for the inflaton. The energy scale introduced is fixed by the amplitude of the anisotropies and is of the order of the supersymmetry breaking scale. The model can also accommodate a spectral index bigger or smaller than 1 without extreme fine tuning. We show that it is possible to obtain reasonable numbers for cosmological parameters and, as an example, we reproduce values obtained recently by Tegmark et al., from WMAP and SDSS data alone.