Towards a reliable calculation of relic radiation from primordial gravitational waves

Inflationary gravitational waves, behaving as additional radiation in the Early Universe, can increase the effective number of relativistic species ($N_{\rm eff}$) by a further correction that depends on the integrated energy-density in gravitational waves over all scales. This effect is typically used to constrain (blue-tilted) models of inflation in light of the bounds resulting from the Big Bang Nucleosynthesis. In this paper, we recompute this contribution, discussing some caveats of the state-of-the-art analyses. Through a parametric investigation, we first demonstrate that the calculation is dominated by the ultraviolet frequencies of the integral and therefore by the behavior of the tensor spectrum on scales corresponding to modes that cross the horizon very close to the end of inflation, when the slow-roll dynamics breaks down and the production of gravitational waves becomes strongly model dependent. Motivated by these results, we realize a theoretical Monte Carlo and, working within the framework of the Effective Field Theory of inflation, we investigate the observable predictions of a very broad class of models. For each model, we solve a system of coupled differential equations whose solution completely specifies the evolution of the spectrum up to the end of inflation. We prove the calculation of $\Delta N_{\rm eff}^{\rm GW}$ to be remarkably model-dependent and therefore conclude that accurate analyses are needed to infer reliable information on the inflationary Universe.