Question #79333

In a theoretical model of quantum gravity, the vacuum acts as a non-linear dispersive medium. The dispersion relation $\omega(k)$ for a wave satisfies the differential equation: $k \frac{d\omega}{dk} - \omega = \frac{h G}{c^2} k^3$ where $h$ is Planck's constant, $G$ is the gravitational constant, and $c$ is the speed of light. In the linear limit ($k \to 0$), the phase velocity approaches $c$. The energy density of these waves is given by $u = \frac{c^4}{G} k^2$. The energy-density flux is defined as $\Phi_E = u v_g$, where $v_g$ is the group velocity. Let $\Phi_0 = \frac{c^8}{h G^2}$ be the fundamental unit of energy-density flux, and $\tilde{k} = k l_p$ be the dimensionless wave number, where $l_p = \sqrt{\frac{h G}{c^3}}$ is the Planck length. Which of the following represents the total energy-density flux $\Phi_E$ in terms of $\Phi_0$ and $\tilde{k}$?