Non-Relativistic $\tilde{\delta }$ Gravity: A Description of Dark Matter

$\tilde{\delta }$ Gravity is a gravitational field model, where the geometry is governed by two symmetric tensors, $g_{\mu \nu }$ and ${\tilde{g}}_{\mu \nu }$, and new matter fields ($\tilde{\delta }$ Matter fields) are added to the original matter fields. These new components appear motivated by a new symmetry, called $\tilde{\delta }$ symmetry. In previous works, the model is used to explain the expansion of the Universe without Dark Energy. This result and the additional contribution to the mass by $\tilde{\delta }$ Matter are motivations to study the Dark Matter effect with $\tilde{\delta }$ Gravity. In this work, we will derive the Non-Relativistic limit to obtain a correction to the rotation velocity in a galaxy, then we will analyze the most common galaxy density profiles to describe Dark Matter.