Low energy Lorentz violation from high energy modified dispersion in inertial and circular motion

We consider an Unruh-DeWitt detector in inertial and circular motion in Minkowski spacetime of arbitrary dimension, coupled to a quantised scalar field with the Lorentz-violating dispersion relation $\omega = |{\bf k}|\, f (|{\bf k}|/M_{\star})$, where $M_{\star}$ is the Lorentz-breaking scale. Assuming that $f$ dips below unity somewhere, we show that an inertial detector experiences large low energy Lorentz violations in all spacetime dimensions greater than two, generalising previous results in four dimensions. For a detector in circular motion, we show that a similar low energy Lorentz violation occurs in three spacetime dimensions, and we lay the analytic groundwork for examining circular motion in all dimensions greater than three, generalising previous work by Stargen, Kajuri and Sriramkumar in four dimensions. The circular motion results may be relevant for the prospects of observing the circular motion Unruh effect in analogue laboratory systems.