Size of a hydrogen atom in the expanding universe

I take a simple model of the hydrogen atom in a universe without spatial curvature. The Maxwell equations are formulated on the background cosmic spacetime. For a class of cosmic metrics, which includes the de Sitter universe, these equations admit solutions corresponding to an atom whose radius remains strictly constant during the expansion. In the Einstein-de Sitter universe approximate calculations show that the atom expands, but at a rate which is negligible compared with the general cosmic expansion.