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We study the correspondence between two theoretical frameworks for describingdark energy, $f(R)$ gravity and higher-dimensional Space-Time-Matter (STM) orinduced-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-timeacceleration.

Correspondence between modified gravity and 5D Ricci-flat cosmologies