Relic gravitons and pulsar timing arrays: a theoretical viewpoint

During the last three years the pulsar timing arrays reported a series of repeated evidences of gravitational radiation (with stochastically distributed Fourier amplitudes) at a benchmark frequency of the order of $30$ nHz and characterized by spectral energy densities (in critical units) ranging between $10^{-8}$ and $10^{-9}$. While it is still unclear whether or not these effects are just a consequence of the pristine variation of the space-time curvature, the nature of the underlying physical processes would suggest that the spectral energy density of the relic gravitons in the nHz domain may only depend on the evolution of the comoving horizon at late, intermediate and early times. Along this systematic perspective we first consider the most conventional option, namely a post-inflationary modification of the expansion rate. Given the present constraints on the relic graviton backgrounds, we then show that such a late-time effect is unable to produce the desired hump in the nHz region. We then analyze a modified exit of the relevant wavelengths as it may happen when the gravitons inherit an effective refractive index from the interactions with the geometry. A relatively short inflationary phase leads, in this case, to an excess in the nHz region even if the observational data coming from competing experiments do not pin down exactly the same regions in the parameter space. We finally examine an early stage of increasing curvature and argue that it is not compatible with the observed spectral energy density unless the wavelengths crossing the comoving horizon at early times reenter in a decelerated stage not dominated by radiation.