Contribution of long-wavelength gravitational waves to the cosmic microwave background anisotropy

We present an in-depth discussion of the production of gravitational waves from an inflationary phase that could have occurred in the early Universe, giving derivations for the resulting spectrum and energy density. We also consider the large-scale anisotropy in the cosmic microwave background radiation coming from these waves. Assuming that the observed quadrupole anisotropy comes mostly from gravitational waves (consistent with the predictions of a flat spectrum of scalar density perturbations and the measured dipole anisotropy) we describe in detail how to derive a value for the scale of inflation of (1.5-5) × ${10}^{16}$ GeV, which is at a particularly interesting scale for particle physics. This upper limit corresponds to a 95% confidence level upper limit on the scale of inflation assuming only that the quadrupole anisotropy from gravitational waves is not canceled by another source. Direct detection of gravitational waves produced by inflation near this scale will have to wait for the next generation of detectors.