The primordial irreducible gravitational-wave background due to quantum vacuum tensor fluctuations produced during inflation spans a large range of frequencies with an almost scale-invariant spectrum but is too low to be detected by the next generation of gravitational-wave interferometers. We show how this signal is enhanced by a short temporary kination era in the cosmological history (less than 10 e-folds), that can arise at any energy scale between a GeV and the inflationary scale $10^{16}$ GeV. We argue that such kination era is naturally generated by a spinning axion before it gets trapped by its potential. It is usually assumed that the axion starts oscillating around its minimum from its initially frozen position. However, the early dynamics of the Peccei-Quinn field can induce a large kinetic energy in the axion field, triggering a kination era, either before or after the axion acquires its mass, leading to a characteristic peak in the primordial gravitational-wave background. This represents a smoking-gun signature of axion physics as no other scalar field dynamics is expected to trigger such a sequence of equations of state in the early universe. We derive the resulting gravitational-wave spectrum, and present the parameter space that leads to such a signal as well as the detectability prospects, in particular at LISA, Einstein Telescope, Cosmic Explorer and Big Bang Observer. We show both model-independent predictions and present as well results for two specific well-motivated UV completions for the QCD axion dark matter where this dynamics is built-in.
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