Energy Scale of Lorentz Violation in Rainbow Gravity

We modify the standard relativistic dispersion relation in a way which breaks Lorentz symmetry - the effect is predicted in a high-energy regime of some modern theories of quantum gravity. We show that it is possible to realise this scenario within the framework of Rainbow Gravity which introduces two new energy-dependent functions $f_1(E)$ and $f_2(E)$ into the dispersion relation. Additionally, we assume that the gravitational constant $G$ and the cosmological constant $\Lambda$ also depend on energy $E$ and introduce the scaling function $h(E)$ in order to express this dependence. For cosmological applications we specify the functions $f_1$ and $f_2$ in order to fit massless particles which allows us to derive modified cosmological equations. Finally, by using Hubble+SNIa+BAO(BOSS+Lyman $\alpha$)+CMB data, we constrain the energy scale $E_{LV}$ to be at least of the order of $10^{16}$ GeV at $1\sigma$ which is the GUT scale or even higher $10^{17}$ GeV at $3\sigma$. Our claim is that this energy can be interpreted as the decoupling scale of massless particles from spacetime Lorentz violating effects.