Exponential decay for small nonlinear perturbations of expanding flat homogeneous cosmologies

It is shown that during expanding phases of flat homogeneous cosmologies all nonlinear perturbations which are small enough are bounded by an exponentially decaying function, with the exponent being a (negative) fraction of the minimum value the Hubble function takes during the expanding period considered. When the cosmological constant is negative, i.e., in our conventions, when there is sustained inflation, it follows that nonlinear perturbations which are small enough decay exponentially; thus, a cosmic no-hair theorem is established. This result holds for a large class of perfect fluid equations of state, but notably not for very “stiff” fluids such as the pure radiation case.