  Primordial gravitational waves constitute a promising probe of the very early
Universe and the laws of gravity. We study in this work changes to tensor-mode
perturbations (TMPs) that can arise in various proposed modified gravity (MG)
theories. These include additional friction effects, nonstandard dispersion
relations involving a massive graviton, a modified speed, and a small-scale
modification. We introduce a physically motivated parameterization of these
effects and use current available data to obtain exclusion regions in the
parameter spaces. Taking into account the foreground subtraction, we then
perform a forecast analysis focusing on tensor-mode MG parameters as
constrained by future experiments COrE, Stage-IV and PIXIE. For a fiducial
value of the tensor-to-scalar ratio $r=0.01$, we find that an additional
friction of $3.5\sim4.5\%$ compared to GR will be detected at $3$-$\sigma$ by
these experiments, while a decrease in friction will be more difficult to
detect. The speed of gravitational waves needs to be by $5\sim15\%$ different
from the speed of light for detection. We find that the minimum detectable
graviton mass is about $7.8\sim9.7\times 10^{-33}$ eV, which is of the same
order of magnitude as the graviton mass that allows massive gravity theories to
produce cosmic acceleration. Finally, we study TMPs in MG during inflation
using our parameterization. We find that the tensor spectral index is related
to $r$ and the friction parameter $\nu_0$ by $n_T=-3\nu_0-r/8$. Assuming that
the friction parameter is unchanged throughout the history of the Universe and
is much larger than $r$, future experiments will be able to distinguish this MG
consistency relation from the standard inflation one, and thus can be used as a
further test of MG. In summary, TMPs and cosmic-microwave-background B-mode
polarization provide a complementary avenue to test gravity theories.
(Abridged)
