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We analyse the response function of an Unruh--DeWitt detector moving withtime-dependent acceleration along a one-dimensional trajectory in Minkowskispacetime. To extract the physics of the process, we propose an adiabaticexpansion of this response function. This expansion is also a useful tool forcomputing the click rate of detectors in general trajectories. The expansion isdone in powers of the time derivatives of the acceleration (jerk, snap, andhigher derivatives). At the lowest order, we recover a Planckian spectrum withtemperature proportional to the acceleration of the detector at each instant ofthe trajectory. Higher orders in the expansion involve powers of thederivatives of the acceleration, with well-behaved spectral coefficients withdifferent shapes. Finally, we illustrate this analysis in the case of aninitially inertial trajectory that acquires a given constant acceleration in afinite time.

Unruh-DeWitt detector event rate for trajectories with time-dependent acceleration