We already discovered the cosmic background radiation what are the 'leftovers' of electromagnetic radiation of the big bang. Probably gravitational waves of the big bang (if they exists) would be to low to measure at the moment. But according to the unified theorie all forces existed already 'together' in the beginning. So the question is can the really exist today or are gravitational waves just created after a while and only because of matter what is coming together instead of a kind of an 'explosion' like the big bang?
Yes, it is thought that there is a cosmic gravitational wave background, just as there is a cosmic microwave background and presumably a cosmic neutrino background. This background is certainly very weak, and direct detection is likely impossible (or at least unfeasible in the foreseeable future), but evidence of primordial may be visible in other channels.
In fact, in 2014 BICEP2 announced the discovery of indirect evidence of gravity waves from this background, although it was later ruled to instead have been a signal from cosmic dust.