What is the amplitude of the Gravitational Waves background expected in the Starobinski model ?

The inflationary model proposed by Starobinski in 1979 predicts an amplitude of the spectrum of primordial gravitational waves, parametrized by the tensor to scalar ratio, of $r=0.0037$ in case of a scalar spectral index of $n_S=0.965$. This amplitude is currently used as a target value in the design of future CMB experiments with the ultimate goal of measuring it at more than five standard deviations. Here we evaluate how stable are the predictions of the Starobinski model on $r$ considering the experimental uncertainties on $n_S$ and the assumption of $Lambda$CDM. We also consider inflationary models where the $R^2$ term in Starobinsky action is generalized to a $R^2p$ term with index $p$ close to unity. We found that current data place a lower limit of $r>0.0013$ at $95 %$ C.L. for the classic Starobinski model, and predict also a running of the scalar index different from zero at more than three standard deviation in the range $dn/dlnk=-0.0006_-0.0001^+0.0002$. A level of gravitational waves of $rsim0.001$ is therefore possible in the Starobinski scenario and it will not be clearly detectable by future CMB missions as LiteBIRD and CMB-S4. When assuming a more general $R^2p$ inflation we found no expected lower limit on $r$, and a running consistent with zero. We found that current data are able to place a tight constraints on the index of $R^2p$ models at $95%$ C.L. i.e. $p= 0.99^+0.02_-0.03$.