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

## TL;DR

This paper argues that modified gravity theories and particle dark matter are fundamentally indistinguishable because they can be mathematically transformed into each other, making experimental differentiation impossible.

## Contribution

It demonstrates that theories based on curvature invariants are equivalent to general relativity with additional fields, blurring the line between modified gravity and dark matter.

## Key findings

- Modified gravity theories can be recast as general relativity with new fields.
- No experimental method can definitively distinguish between dark matter and modified gravity.
- Theories based on curvature invariants are equivalent to general relativity with shifted energy-momentum tensors.

## Abstract

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. This leads to an equivalence between dark matter particles gravitationally coupled to the standard model fields and modified gravity theories designed to account for the dark matter phenomenon. Due to this ambiguity, it is impossible to differentiate experimentally between these theories and any attempt of doing so should be classified as a mere interpretation of the same phenomenon.