  The Maxwell equations for the electromagnetic potential, supplemented by the
Lorenz gauge condition, are decoupled and solved exactly in de Sitter
space-time studied in static spherical coordinates. There is no source besides
the background. One component of the vector field is expressed, in its radial
part, through the solution of a fourth-order ordinary differential equation
obeying given initial conditions. The other components of the vector field are
then found by acting with lower-order differential operators on the solution of
the fourth-order equation (while the transverse part is decoupled and solved
exactly from the beginning). The whole four-vector potential is eventually
expressed through hypergeometric functions and spherical harmonics. Its radial
part is plotted for given choices of initial conditions. We have thus
completely succeeded in solving the homogeneous vector wave equation for
Maxwell theory in the Lorenz gauge when a de Sitter spacetime is considered,
which is relevant both for inflationary cosmology and gravitational wave
theory. The decoupling technique and analytic formulae and plots are completely
original. This is an important step towards solving exactly the tensor wave
equation in de Sitter space-time, which has important applications to the
theory of gravitational waves about curved backgrounds.
