Hubble multi-scalar inflation
  Multiple field models of inflation exhibit new features than single field models. In this work, we study the hierarchy of parameters based on Hubble expansion rate in curved field space and derive the system of flow equations that describe their evolution. Then we focus on obtaining derivatives of number of $e$-folds with respect to scalar fields during inflation and at hypersurface of the end of inflation. 
Habib Abedi and Amir M. Abbassi
gr-qc