New exact solutions for power-law inflation Friedmann models

We consider the spatially flat Friedmann model For a ≈ t^p, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = R^m with p = ? (m ? 1) (2m ? 1)/(m ? 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ? in an exponential potential V(?) = exp (μ?) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write The general solutions reads Q(a) = (a^b + C)^f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = ?2 + 1/p, b = (4m ? 5)/(m ? 1). Finally, special solutions for the closed and open Friedmann model are found.