Conditions on the stability of the external space solutions in a higher‐dimensional quadratic theory of gravity

By using Lyapounov’s direct method we examine the conditions under which stable solutions to the field equations for the scale function of the external space may be derived in the context of a five‐dimensional quadratic theory of gravity. We show that the time evolution of the distance, in a diagram t–R, between our solution to the field equations and a neighbouring one is determined, in the linear approximation, in terms of a second‐order linear differential equation. Asking for bounded solutions of this equation we arrive at a stability criterion for the external scale function solutions, indicating that there exist three types of cosmological evolution of the visible universe which are linearly stable at all times. These are (i) the Milne model, (ii) the spatially flat Friedmann radiation solution, and (iii) the De Sitter inflationary solution.