Hydrogen atom and its energy level shifts in de Sitter universe

We investigate the influence of a cosmological constant on the energy levels of a one-electron atom in Fermi normal coordinates. The non-relativistic nS, nP energy levels and the relativistic 1S _1/2, 2S _1/2, 2P _1/2, 2P _3/2 energy levels are calculated in terms of the Riemann tensor. These energy level shifts are non-zero which indicate that the first order gravitational perturbations can partly remove the degeneracy of the studied states in de Sitter space. We show that it is not possible to use the hydrogen atom as a probe of background curvature of the universe. In the second order of perturbation, for 1S state, an upper limit for the energy shifts is obtained.