Inflaton field potential producing an exactly flat spectrum of adiabatic perturbations

Presented in this letter is the exact solution of the problem of finding the potential of an inflaton scalar field for which adiabatic perturbations generated during a de Sitter (inflationary) stage in the early Universe have an exactly flat (or, the Harrison-Zeldovich) initial spectrum. This solution lies outside the scope of the slow-roll approximation and higher-order corrections to it. The potential found depends on two arbitrary physical constants, one of which determines the amplitude of the perturbations. For small (zero) values of the other constant, a long (infinite) inflationary stage with slow rolling of the inflaton field exists.