In this seminar I will present the first direct numerical simulation of gravitational wave turbulence (Galtier & Nazarenko, PRL 127, 131101, 2021). General relativity equations are solved numerically in a periodic box with a diagonal metric tensor depending on two space coordinates only (Hadad-Zakharov metric) and with an additional small-scale
dissipative term. We limit ourselves to weak gravitational waves and to a freely decaying turbulence. We find that an initial metric excitation at intermediate wave number leads to a dual cascade of energy and wave action. When the direct energy cascade reaches the dissipative scales, a transition is observed in the temporal evolution of energy from a plateau to a power-law decay, while the inverse cascade front continues to propagate toward low wave numbers. The wavenumber and frequency-wavenumber spectra are found to be compatible with the analytical theory of weak wave turbulence (Galtier & Nazarenko, PRL 119, 221101, 2017) and the characteristic timescale of the dual cascade is
that expected for four-wave resonant interactions. The simulation reveals that an initially weak gravitational wave turbulence tends to become strong as the inverse cascade of wave action progresses with a selective amplification of the fluctuations g11 and g22. To conclude, I will briefly outline the relevance of strong gravitational wave turbulence to the early universe as a plausible natural mechanism for inflation.