  The detection of Primordial Gravitational Waves (PGWs) is one of the most
important goals of modern cosmology since PGWs can both provide substantial
evidence for primordial inflation and shed light on its physical nature. Small
scale experiments on gravitational waves such as LIGO/VIRGO and, in future,
LISA and Einstein Telescope (ET), being sensitive to the stochastic background
of gravitational waves, can be used together with the CMB data to constrain the
inflationary parameters. In performing these analyses the primordial tensor
spectrum is usually parametrized with a power law that includes only the
amplitude and a scale independent tilt. In this paper, we investigate the
robustness of assuming the tensor tilt as scale independent. We show that due
to the huge difference in the scales probed by CMB and GWs data, even a small
scale dependence can remarkably affect the shape of the primordial spectrum
possibly breaking the power-law assumption. When the non-linear corrections are
considered the final constraints can be significantly changed. We also study
the scale dependence in two different physical models of inflation providing an
example of negligible scale dependence and an example of non-negligible scale
dependence.
