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New constraints on primordial gravitational waves from Planck 2015

We show that the new precise measurements of Cosmic Microwave Background (CMB) temperature and polarization anisotropies made by the Planck satellite significantly improves previous constraints on the cosmic gravitational waves background (CGWB) at frequencies $f>10^{-15}$ Hz. On scales smaller than the horizon at the time of decoupling, primordial gravitational waves contribute to the total radiation content of the Universe. Considering adiabatic perturbations, CGWB affects temperature and polarization CMB power spectra and matter power spectrum in a manner identical to relativistic particles. Considering the latest Planck results we constrain the CGWB energy density to $Ω_{\rm gw} h^2 <1.7\times 10^{-6} $ at 95\% CL. Combining CMB power spectra with lensing, BAO and primordial Deuterium abundance observations, we obtain $Ω_{\rm gw} h^2 <1.2\times 10^{-6} $ at 95\% CL, improving previous Planck bounds by a factor 3 and the recent direct upper limit from the LIGO and VIRGO experiments a factor 2. A combined analysis of future satellite missions as COrE and EUCLID could improve current bound by more than an order of magnitude.