Probing the inflationary background of gravitational waves from large to small scales

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.