Modified gravity/entropic gravity correspondence due to graviton mass

Some time ago, it has been suggested that gravitons can acquire mass in the process of spontaneous symmetry breaking of diffeomorphisms through the condensation of scalar fields [Chamseddine and Mukhanov, JHEP, 2010]. Taking this possibility into account, in the present paper, first we show how the graviton mass intricately reshapes the gravitational potential akin to a Yukawa-like potential at large distances. Notably, this long-range force modifies the Newton's law in large distances and might explain the phenomena of dark matter. The most important finding in the present paper is the derivation of a modified Newtons law of gravity by modifying the Verlindes entropic force relation due to the graviton contribution. The graviton contribution to the entropy basically measures the correlation of graviton and matter fields which then reproduces the Bekenstein-Hawking entropy at the horizon. This result shows the dual description of gravity: in the language of quantum information and entropy the gravity can be viewed as an entropic force, however in terms of particles and fields, it can be viewed as a longe range force. Further we have recovered the corrected Einstein field equations as well as the $\Lambda$CDM where dark matter emerges as an apparent effect.