Inflation at the maxima of the potentials

We construct a two-stage inflationary model which can accommodate early inflation at a scale $\Lambda_1$ as well as a second stage of inflation at $\Lambda_2$ with a single scalar field $\phi$. We use a symmetric potential, valid in a frictionless world, in which the two inflationary periods have exactly the same scale, i.e. $\Lambda_1=\Lambda_2$. However, we see today $\Lambda_1 \gg \Lambda_2$ due to the friction terms (expansion of the universe and interaction with matter). These type of models can be motivated from supergravity. Inflation occurs close to the maxima of the potential. As a consequence both inflations are necessarily finite. This opens the interesting possibility that the second inflation has already or is about to end. A first inflation is produced when fluctuations displace the inflaton field from its higher maximum rolling down the potential as in new inflation. Instead of rolling towards a global minimum the inflaton approaches a lower maximum where a second inflation takes place.