$R^2$--Inflation Derived from 4d Strings, the Role of the Dilaton, and Turning the Swampland into a Mirage

Based on a previously derived superstring model possessing a cosmological sector that mimics Starobinsky inflation, we analyze several questions addressed in the recent literature: the generation of an effective $R^2$-term, the stability of the sgoldstino , the modular symmetry of the inflaton potential and the large distance swampland conjecture. We first show that the presence of the string dilaton stabilizes the sgoldstino direction in the supersymmetric case and no modification of the K\"ahler potential is needed. This is a generic property of a large class of Starobinsky type models within the framework of no-scale supergravity. We then present an explicit example of a string derived inflaton potential where the large values of the inflaton field during inflation imply a decompactification of two extra dimensions, while the scale of inflation is generated by higher order $\alpha'$-corrections via expectation values that cancel the D-term of an anomalous $U(1)$ symmetry and break the modular symmetry of the scalar potential. As a result, the scale of inflation is much lower than the compactification scale which at the end of inflation is fixed at the free-fermionic self-dual point at an (approximate) supersymmetric minimum.