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We consider dynamical axion phenomena in topological superconductors andsuperfluids in three spatial dimensions in terms of the gravitoelectromagnetictopological action, in which the axion field couples with mechanical rotationunder finite temperature gradient. The dynamical axion is induced by relativephase fluctuations between topological and s-wave superconducting orders. Weshow that an antisymmetric spin-orbit interaction which induces parity-mixingof Cooper pairs enlarges the parameter region in which the dynamical axionfluctuation appears as a low-energy excitation. We propose that the dynamicalaxion increases the moment of inertia, and in the case of ac mechanicalrotation, i.e. a shaking motion with a finite frequency $\omega$, as $\omega$approaches the dynamical axion fluctuation mass, the observation of this effectbecomes feasible.

Dynamical axion in topological superconductors and superfluids