Critical temperature of metallic hydrogen at a pressure of $500$ GPa

The Eliashberg theory generalized for electron-phonon systems with a variable electron density of states is used to study $T_c$ in the $\mathrm{I41/AMD}$ phase of metallic hydrogen under compression taking into account the frequency behavior of the renormalization of the mass of the electron and the chemical potential. The phonon contribution to the anomalous electron Green's function is considered. Pairing is taken into account within the entire electron band rather than in a narrow region near the Fermi surface. The frequency and temperature dependences of the complex renormalization of the mass $\mathrm{Re}\, Z(\omega)$, as well as the density of states $N(\omega)$ renormalized by the electron-phonon coupling and the spectral function of the electron-phonon coupling, which are obtained in calculations, are used to calculate the anomalous electron Green's function. The frequency dependence of the real and imaginary parts of the order parameter in the $\mathrm{I41/AMD}$ phase is obtained. The solution of the system of Eliashberg equations gives the value $T_c=217$ K in the $\mathrm{I41/AMD}$ phase of hydrogen at a pressure of $500$ GPa.