Correspondence between modified gravity and 5D Ricci-flat cosmologies
  We study the correspondence between two theoretical frameworks for describing dark energy, $f(R)$ gravity and higher-dimensional Space-Time-Matter (STM) or induced-matter theory. We show that the Hubble expansion parameter in $f(R)$ gravity can be associated with a combination of metric functions in STM theory, and consider a specific example whose properties are consistent with late-time acceleration. 
K. Atazadeh, F. Darabi and H. R. Sepangi