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.