The direct detection of a cosmological gravitational wave stochastic background produced by inflation is of great importance for the understanding of early Universe cosmology -- it shall provide a direct test of one of the fundamental, and yet untested, predictions of inflationary theories -- and fundamental physics, by opening an new arena for the phenomenology of fields beyond the standard model of particle physics. The Big Bang Observer, an array of four space-based laser interferometers aimed at observations of gravitational radiation in the frequency window 0.1 Hz - 1 Hz, is currently been proposed as the LISA follow-on mission dedicated to the detection of relic gravitons produced during an inflationary epoch. In this talk I will discuss how such a class of gravitational wave space detectors can provide an effective tool to directly test the slow-roll paradigma and how such experiments can provide complementary information with respect to current and future CMB experiments.