What is modified gravity and how to differentiate it from particle dark matter?

Understand

An obvious criterion to classify theories of modified gravity is to identify their gravitational degrees of freedom and their coupling to the metric and the matter sector.

• Using this simple idea, we show that any theory which depends on the curvature invariants is equivalent to general relativity in the presence of new fields that are gravitationally coupled to the energy-momentum tensor.
• We show that they can be shifted into a new energy-momentum tensor.
• There is no a priori reason to identify these new fields as gravitational degrees of freedom or matter fields.