PROOF OF THE COSMIC NO-HAIR CONJECTURE FOR SOME QUADRATIC HOMOGENEOUSCOSMOLOGIES

We prove the cosmic no-hair conjecture for orthogonal, initially expan ding Bianchi cosmologies in the R + beta R-2 theory with matter satisf ying the strong and dominant energy conditions using the conformally e quivalent Einstein field equations, with the scalar field having the f ull self-interacting potential. We assume that the universe is initial ly on the Bat plateau of the potential and its initial kinetic energy is negligible with respect to its potential energy. We show, in partic ular, that the Blanch IX universe asymptotically approaches de Sitter space provided that initially the scalar 3-curvature does not exceed t he potential of the scalar field associated with the conformal transfo rmation. Our proof relies on rigorous estimates of the possible bounds of the so-called Moss-Sahni function which obeys certain differential inequalities, and a nontrivial argument which connects the behaviour of this function to the evolution of the spatial part of the scalar cu rvature.