Nonlinear lagrangians and the isotropy of the universe

We analyse the question of whether, in higher derivative gravity theories, the present isotropic state of the universe could have been obtained by starting from “arbitrary” initial conditions. We show that the Collins-Hawking conclusion holds in this extended case, that is, the set of spatially homogeneous cosmologies which can approach isotropy at late times is of measure zero in the space of all spatially homogeneous universe models. This involves an examination of a generalized form of the Raychaudhuri equation valid in a general higher-order theory of gravity in four spacetime dimensions. We show that this equation is conformally equivalent to what we call a Raychaudhuri system, that is, to the usual Raychaudhuri equation in general relativity coupled to a scalar-wave equation satisfied by the scalar field generated by the conformal transformation with the usual effective potential. This result is also used to prove directly in higher-order gravity the known forms of cosmic no-hair theorems in the Sitter and power-law inflation in general relativity. Generalizations to arbitrary dimensionality and also the addition of other curvature invariants are discussed.