\left<|h_k(\tau)|^2\right>\,,
\label{deltah}\end{aligned}$$ where $\left<...\right>$ denotes a statistical ensemble average.

Inflationary spectrum
---------------------

In this paper we focus on the tensor modes generated during inflation as they were spatially stretched past the inflationary Hubble radius. At the end of inflation, these modes represent superhorizon tensor perturbations with an almost scale invariant power spectrum [@Caprini:2018mtu] $$\begin{aligned}
\Delta_{h,\text{inf}}^2(k)\simeq \frac{2}{\pi^2}\left(\frac{H_{\text{inf}}}{ m_{\text{p}}}\right)^2\left({k\over k_p}\right)^{n_t} \label{inf}\,,\end{aligned}$$ with $n_t$ a spectral tilt, $k_{p}$ a pivot scale, and $H_{\rm inf}$ the Hubble rate when the mode $k_p$ exited the horizon during inflation. The presence of the tilt stems from the fact that the inflationary phase cannot be perfectly [*de Sitter*]{}. Nevertheless, the spectrum is expected to be only slightly red-tilted in slow-roll inflation, with the spectral index ’slow-roll suppressed’ as $$\begin{aligned}
\label{eq:consistencyCondition}
n_t \simeq -2\epsilon \simeq -{r_p\over8} \,,\end{aligned}$$ where $r_{p}$ is the tensor-to-scalar ratio evaluated at the scale $k_p$, constrained by the most recent analysis of the B-mode polarization anisotropies of the CMB at a scale $k_p/a_0 = 0.002~{\rm Mpc}^{-1}$, as $r_p \leq 0.064$ [@Akrami:2018odb; @Ade:2018gkx]. This bound actually implies an upper bound on the energy scale of inflation, which must be constrained as $$\begin{aligned}
\label{eq:Hmax}
H_{\rm inf} \lesssim H_{\rm max} \simeq 6.6\cdot 10^{13}\,{\rm GeV}\,.\end{aligned}$$ Furthermore, the upper bound 