Gravitational waves propagating into Friedmann--Robertson--Walker   universes
  We consider space-times with two isometries which represent gravitational waves with distinct wavefronts which propagate into exact Friedmann-Robertson-Walker (FRW) universes. The geometry of possible wavefronts is analysed in detail in all three types of FRW models. In the spatially flat and open universes, the wavefronts can be planar or cylindrical; in the closed case they are toroidal. Exact solutions are given which describe gravitational waves propagating into the FRW universes with a fluid with a stiff equation of state. It is shown that the plane-fronted waves may include impulsive or step (shock) components, while the cylindrical waves in the spatially flat and open universes and the toroidal waves in closed universes may contain steps. In general, wavefronts may exist which have an arbitrary finite degree of smoothness. In all cases, the waves are backscattered. The head-on collision of such waves is also briefly mentioned. 
Jiri Bicak and Jerry B. Griffiths