New exact solutions for power–law inflation Friedmann models

We consider the spatially flat Friedmann model ds = dt − a(t)(dx + dy + dz) . For a ≈ tp, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = Rm 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 Φ in an exponential potential V (Φ) = exp(μΦ) 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 ds = a ( Q(a)da − dx − dy − dz ) . The general solutions reads Q(a) = (ab + 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.