What does this equation mean?

The first term is the field state — the same vacuum, or the same thermal state, or in curved spacetime the same choice among the Boulware, Hartle-Hawking, and Unruh states, since none of those is fixed by the geometry alone. The second is the coupling type: a detector linearly coupled to the field itself and a detector coupled to the field’s proper-time derivative are different instruments on the same worldline, and their response spectra carry different powers of $\Omega$ in the numerator even for identical trajectories — Moustos showed exactly this, that a detector’s early-time response depends on which bilinear of the field it couples to, even though, remarkably, its late-time asymptotic state does not [ 9 ] . The third is the gap $\Omega$ itself, or more precisely the requirement that $\Omega$ be defined against each detector’s own proper time rather than against a shared coordinate time — two detectors at different heights in a static spacetime, or moving at different speeds, experience the same coordinate-time interval as different amounts of proper time, and building $\Omega$ from coordinate time instead of proper time would inject that redshift directly into $D_{\rm det}$ as a spurious “particle disagreement” that is really just two clocks running at different rates. The fourth is the switching profile $\chi(\tau)$ : its total duration and its smoothness both leave fingerprints in $\mathcal F_x(\Omega)$ that have nothing to do with the trajectory, as the next two sections show in detail.