Isotropic universe with almost scale-invariant fourth-order gravity

We study a class of isotropic cosmologies in the fourth-order gravity with Lagrangians of the form L = f(R) + k(G) where R and G are the Ricci and Gauss-Bonnet scalars, respectively. A general discussion is given on the conditions under which this gravitational Lagrangian is scale-invariant or almost scale-invariant. We then apply this general background to the specific case L = αR 2 + β Gln G with constants α, β. We find closed form cosmological solutions for this case. One interesting feature of this choice of f(R) and k(G) is that for very small negative value of the parameter β, the Lagrangian L = R 2/3 + βGln G leads to the replacement of the exact de Sitter solution coming from L = R 2 (which is a local attractor) to an exact, power-law inflation solution a(t) = t p = t −3/β which is also a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of a Gln G-term.