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We investigate the primordial gravitational waves (PGWs) in the generalscenario where the inflation is preceded by a pre-inflationary stage with theeffective equation of state $w$. Comparing with the results in the usualinflationary models, the power spectrum of PGWs is modified in two aspects: Oneis the mixture of the perturbation modes caused by he presence of thepre-inflationary period, and the other is the thermal initial state formed atthe Planck era of the early Universe. By investigating the observationalimprints of these modifications on the B-mode polarization of cosmic microwavebackground (CMB) radiation, we obtain the constraints on the conformaltemperature of the thermal gravitational-wave background $T<5.01\times10^{-4}$Mpc$^{-1}$ and a tensor-to-scalar ratio $r<0.084$ ($95\%$ confidentlevel), which follows the bounds on total number of e-folds $N>63.5$ for themodel with $w=1/3$, and $N>65.7$ for that with $w=1$. By taking into accountvarious noises and the foreground radiations, we forecast the detectionpossibility of the thermal gravitational-wave background by the future CMBPolmission, and find that if $r>0.01$, the detection is possible as long as$T>1.5\times 10^{-4}$Mpc$^{-1}$. However, the effect of different $w$ is quitesmall, and it seems impossible to determine its value from the potentialobservations of CMBPol mission.

Thermal gravitational-wave background in the general pre-inflationary scenario