Conformal transformation route to gravity's rainbow

Conformal transformation as a mathematical tool has been used in many areas of gravitational physics. In this paper, we would consider the gravity's rainbow, in which the metric could be treated as a conformal rescaling of the original metric. By using the conformal transformation technique, we get a specific form of modified Newton's constant and cosmological constant in gravity's rainbow, which implies that the total vacuum energy is dependent on probe energy. Moreover, the result shows that the Einstein gravity's rainbow could be described by an energy-dependent $f(E,\tilde R)$ gravity. At last, we study the $f(R)$ gravity, when the gravity's rainbow is considered, it can also be described as another energy-dependent $\tilde f(E,\tilde R)$ gravity.
Comment: 16 pages; accepted for publication in General Relativity and Gravitation