On the evolution of primordial gravitational waves: a semi-analytic   detailed approach
  A cosmological gravitational wave background resulting from space-time quantum perturbations at energy scales of $\sim 10^{15}$GeV is expected as a consequence of the general relativity theory in the context of the standard cosmological model. Initial conditions %for the problem are determined during the inflationary (de Sitter) era, at $z \gtrsim 10^{25} $. A semi-analytic method was developed to evolve the system up to the present with no need of simplifying approximations as the thin-horizon (super-adiabatic) or the instantaneous transitions between the successive phases of domain of the different cosmic fluids. The accuracy of such assumptions, broadly employed in the literature, is put in check. Since the physical nature of the fluid (known as dark energy) leading to the accelerated expansion observed in the recent Universe is still uncertain, four categories of models were analyzed: cosmological constant, X-fluid (phantom or not), generalized Chaplygin gas and (a parametric form of) quintessence. The results are conclusive with respect to the insensitivity of gravitational waves to dark energy, due to the recentness of its phase of domain ($z \sim 1 $). The empirical counterparts of the gravitational wave forecasts are still nonexistent for the noise levels and operational frequencies of the experiments already built are inadequate to detect those relics. Perspectives are more promising for space detectors (planned to be sensitive to amplitudes of $\sim 10^{-23}$ at $10^{-3}-1$Hz). The cosmic microwave background is also discussed as an alternative of indirect detection and the energy density scale of inflation is constrained to be smaller than $10^{-10}$ in the analysis here presented. 
M. Soares-Santos, E. M. de Gouveia Dal Pino