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

We analyse the response function of an Unruh--DeWitt detector moving with time-dependent acceleration along a one-dimensional trajectory in Minkowski spacetime. To extract the physics of the process, we propose an adiabatic expansion of this response function. This expansion is also a useful tool for computing the click rate of detectors in general trajectories. The expansion is done in powers of the time derivatives of the acceleration (jerk, snap, and higher derivatives). At the lowest order, we recover a Planckian spectrum with temperature proportional to the acceleration of the detector at each instant of the trajectory. Higher orders in the expansion involve powers of the derivatives of the acceleration, with well-behaved spectral coefficients with different shapes. Finally, we illustrate this analysis in the case of an initially inertial trajectory that acquires a given constant acceleration in a finite time.