on $r_p$ also implies that the red-tilted spectral index can only be very small $-n_t \leq 0.008 \ll 1$. The tensor spectrum is therefore very close to be exactly scale-invariant, at least at around the CMB scales. Actually, in the absence of running of the spectral index, the amplitude of the tensor spectrum would fall only by a factor $\sim (10^{26})^{-0.008} \sim 0.6$ during the $\log(e^{60}) \sim $ 26 orders of magnitude separating the CMB scales and the Hubble radius at the end of inflation. Therefore, for simplicity, we will consider from now on an exact scale-invariant inflationary spectrum, as this gives an excellent approximation. We will comment on deviations from this assumption in Sect. \[s:spectraltilt\].

From a theoretical perspective, it is convenient to work with the power spectrum $\Delta_h^2(k)$, as it is precisely this quantity that is predicted by inflation to be approximately scale invariant. During the evolution of the Universe after inflation, when the tensor modes cross inside the Hubble radius, they become a stochastic background of gravitational waves (GWs). In order to quantify the ability of GW direct detection experiments to measure the inflationary GW background, it is customary to express the amount of GWs in terms 