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We study how the shape of the spectrum of primordial gravitational waves canbe constrained by future experiments looking at the B-mode of the CosmicMicrowave Background (CMB) polarization. We implement a Principal ComponentAnalysis (PCA) including the effects of diffuse foreground residuals, followingcomponent separation, in the uncertainty of CMB angular power spectra, andtaking into account the gravitational lensing by Large Scale Structure. Weperform our study by considering the capabilities of future B-mode CMBexperiments such as LiteBIRD, the Simons Observatory (SO) and Stage-IV(CMB-S4), in particular in detecting deviations of the primordial tensorspectrum from the scale-invariant behavior. We find that diffuse foregroundresiduals impact substantially both the derivation of the PCA basis and thecorresponding constraining power, in all cases. In particular, depending onwhich experimental specifications and which value $r$ of tensor-to-scalar ratiofor cosmological perturbations are considered, adding foregrounds residuals candetermine an increase as large as a factor $\sim 4$ both on the uncertainty on$r$ and on the recovery of the PCA modes. We study the limitations of themethodology, including the effect of physicality priors on the PCA, which wequantify via a Monte Carlo Markov chain (MCMC) analysis of the combinedcosmological and PCA power spectrum parameter space.

Principal component analysis of the primordial tensor power spectrum