  The problem of cosmological production of gravitational waves is discussed in
the framework of an expanding, spatially homogeneous and isotropic FRW type
Universe with time-evolving vacuum energy density. The gravitational wave
equation is established and its modified time-dependent part is analytically
resolved for different epochs in the case of a flat geometry. Unlike the
standard $\Lambda$CDM cosmology (no interacting vacuum), we show that
gravitational waves are produced in the radiation era even in the context of
general relativity. We also show that for all values of the free parameter, the
high frequency modes are damped out even faster than in the standard cosmology
both in the radiation and matter-vacuum dominated epoch. The formation of the
stochastic background of gravitons and the remnant power spectrum generated at
different cosmological eras are also explicitly evaluated. It is argued that
measurements of the CMB polarization (B-modes) and its comparison with the
rigid $\Lambda$CDM model plus the inflationary paradigm may become a crucial
test for dynamical dark energy models in the near future.
