The detectability of cosmological gravitational-wave backgrounds: a rule
  of thumb
  The recent claim by BICEP2 of evidence for primordial gravitational waves
from inflation has focused interest on the potential for early-Universe
cosmology using observations of gravitational waves. In addition to cosmic
microwave background detectors, efforts are underway to carry out
gravitational-wave astronomy over a wide range of frequencies including pulsar
timing arrays (nHz), space-based detectors (mHz), and terrestrial detectors
($\sim$10-2000 Hz). This multiband effort will probe a wide range of times in
the early Universe (each corresponding to a different energy scale), during
which gravitational-wave backgrounds may have been produced through processes
such as phase transitions or preheating. In this letter, we derive a rule of
thumb (not quite so strong as an upper limit) governing the maximum energy
density of cosmological backgrounds. For most cosmological scenarios, we expect
the energy density spectrum to peak at values of
$\Omega_\text{gw}(f)\lesssim10^{-12\pm2}$. We discuss the applicability of this
rule of thumb and the implications for gravitational-wave astronomy.
