• Inflation
• Gravitational Waves
• Cosmology
• Back-reaction
• Stochastic background of Gravitational Waves

In this master thesis project we investigate the back-reaction effects of a background of Gravitational Waves (GWs) produced during an early phase in the Universe evolution. Since GWs carry energy and momentum, they contribute to the total energy density at any time, hence affecting the space-time metric itself. Requiring that the early Universe is not dominated by back-reaction effects, i.e. requiring that the energy of GWs is smaller than the "ordinary" one, we obtain some constraints on the tensor spectral parameters, in particular on the spectral tilt and the tensor-to-scalar ratio. The bounds are extracted for each of the main phases of Universe evolution (slow-roll inflation, radiation domination epoch and matter domination epoch, neglecting the present Dark Energy-dominated phase). Moreover, we make a distinction between the energy density contributed by super- and sub-Hubble modes. In fact, in these two regimes GWs show different behaviours: super-Hubble modes lead to a negative contribution to the effective energy density, and to an equation of state of curvature, while the sub-Hubble ones contributes to the effective degrees of freedom of radiation and are therefore constrained by Big Bang nucleosynthesis. This work follows the footprints of a recent paper from Brandenberger and Takahashi, with some improvements - mostly in the consideration of more realistic models of inflation - which however translate in small differences on the final bounds.