Updated constraints and forecasts on primordial tensor modes

We present new, tight, constraints on the cosmological background of gravitational waves (GWs) using the latest measurements of CMB temperature and polarization anisotropies provided by the Planck, BICEP2 and Keck Array experiments. These constraints are further improved when the GW contribution ${N}_{\mathrm{eff}}^{\mathrm{GW}}$ to the effective number of relativistic degrees of freedom ${N}_{\mathrm{eff}}$ is also considered. Parametrizing the tensor spectrum as a power law with tensor-to-scalar ratio $r$, tilt ${n}_{\mathrm{t}}$ and pivot $0.01\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$, and assuming a minimum value of $r=0.001$, we find $r<0.089$, ${n}_{\mathrm{t}}=1.{7}_{\ensuremath{-}2.0}^{+2.1}$ (95% CL, no ${N}_{\mathrm{eff}}^{\mathrm{GW}}$) and $r<0.082$, ${n}_{\mathrm{t}}=\ensuremath{-}0.0{5}_{\ensuremath{-}0.87}^{+0.58}$ (95% CL, with ${N}_{\mathrm{eff}}^{\mathrm{GW}}$). When the recently released 95 GHz data from Keck Array are added to the analysis, the constraints on $r$ are improved to $r<0.067$ (95% CL, no ${N}_{\mathrm{eff}}^{\mathrm{GW}}$), $r<0.061$ (95% CL, with ${N}_{\mathrm{eff}}^{\mathrm{GW}}$). We discuss the limits coming from direct detection experiments such as LIGO-Virgo, pulsar timing (European Pulsar Timing Array) and CMB spectral distortions (FIRAS). Finally, we show future constraints achievable from a COrE-like mission: if the tensor-to-scalar ratio is of order $1{0}^{\ensuremath{-}2}$ and the inflationary consistency relation ${n}_{\mathrm{t}}=\ensuremath{-}r/8$ holds, COrE will be able to constrain ${n}_{\mathrm{t}}$ with an error of 0.16 at 95% CL. In the case that lensing $B$-modes can be subtracted to 10% of their power, a feasible goal for COrE, these limits will be improved to 0.11 at (95% CL).