  We study the production, spectrum and detectability of gravitational waves in
models of the early Universe where first order phase transitions occur during
inflation. We consider all relevant sources. The self-consistency of the
scenario strongly affects the features of the waves. The spectrum appears to be
mainly sourced by collisions of bubble of the new phases, while plasma dynamics
(turbulence) and the primordial gauge fields connected to the physics of the
transitions are generally subdominant. The amplitude and frequency dependence
of the spectrum for modes that exit the horizon during inflation are different
from those of the waves produced by quantum vacuum oscillations of the metric
or by first order phase transitions not occurring during inflation. A moderate
number of slow (but still successful) phase transitions can leave detectable
marks in the CMBR, but the signal weakens rapidly for faster transitions. When
the number of phase transitions is instead large, the primordial gravitational
waves can be observed both in the CMBR or with LISA (marginally) and especially
DECIGO. We also discuss the nucleosynthesis bound and the constraints it places
on the parameters of the models.
