String duality transformations in<i>f</i>(<i>R</i>) gravity from Noether symmetry approach

Abstract

We select $f(R)$ gravity models that undergo scale factor duality\ntransformations. As a starting point, we consider the tree-level effective\ngravitational action of bosonic String Theory coupled with the dilaton field.\nThis theory inherits the Busher's duality of its parent String Theory. Using\nconformal transformations of the metric tensor, it is possible to map the\ntree-level dilaton-graviton string effective action into $f(R)$ gravity,\nrelating the dilaton field to the Ricci scalar curvature. Furthermore, the\nduality can be framed under the standard of Noether symmetries and exact\ncosmological solutions are derived. Using suitable changes of variables, the\nstring-based $f(R)$ Lagrangians are shown in cases where the duality\ntransformation becomes a parity inversion.\n