Estimates of maximum energy density of cosmological gravitational-wave backgrounds

The recent claim by BICEP2 of evidence for primordial gravitational waves has focused interest on the potential for early-Universe cosmology using gravitational waves. In addition to cosmic microwave background detectors, efforts are underway to carry out gravitational-wave astronomy with pulsar timing arrays, space-based detectors, and terrestrial detectors. These efforts will probe a wide range of times in the early Universe, during which backgrounds may have been produced through processes such as phase transitions or preheating. We derive a rule of thumb (not so strong as an upper limit) governing the maximum energy density of cosmological backgrounds. For most scenarios, we expect the energy density spectrum to peak at values of Omega(gw)(f) less than or similar to 10(-12 +/- 2). We discuss the applicability of this rule and the implications for gravitational-wave astronomy.