  The impact of the existence of gravitons with non-vanishing masses on the B
modes of the Cosmic Microwave Background (CMB) is investigated. We also focus
on putative modifications to the speed of the gravitational waves. We find that
a change of the graviton speed shifts the acoustic peaks of the CMB and then
could be easily constrained. For the case of massive gravity, we show
analytically how the B modes are sourced in a manner differing from the
massless case leading to a plateau at low $l$ in the CMB spectrum. We also
study the case when there are more than one graviton, and when pressure
instabilities are present. The latter would occur in doubly coupled bigravity
in the radiation era. We focus on the case where a massless graviton becomes
tachyonic in the radiation era whilst a massive one remains stable. As the
unstable mode decouples from matter in the radiation era, we find that the
effects of the instability is largely reduced on the spectrum of B-modes as
long as the unstable graviton does not grow into the non-linear regime. In all
cases when both massless and massive gravitons are present, we find that the
B-mode CMB spectrum is characterised by a low $l$ plateau together with a
shifted position for the first few peaks compared to a purely massive graviton
spectrum, a shift which depends on the mixing between the gravitons in their
coupling to matter and could serve as a hint in favour of the existence of
multiple gravitons.
