What does this equation mean?

The detector, not the mode decomposition, is the object this article works with, for a reason that matters later: a detector’s response is what an experiment can actually report, while “the Bogoliubov coefficients of the true vacuum” is not the kind of thing any apparatus reads off a dial. The response functional for a pointlike, linearly coupled two-level (Unruh-DeWitt) detector on worldline x($\tau$) , parametrized by its own proper time $\tau$ , with energy gap $\Omega$ and switching function $\chi(\tau)$ controlling when the interaction is on, is $\mathcal F_x(\Omega) = \int d\tau\, d\tau'\, \chi(\tau)\chi(\tau')\, e^{-i\Omega(\tau-\tau')}\, W\big[x(\tau), x(\tau')\big]$. where W is the field’s two-point Wightman function evaluated along the trajectory [ 5 ] . Three things in this expression are worth fixing in the reader’s mind before anything else happens to it. First, $\tau$ is proper time — the detector’s own clock, the only time variable with unambiguous physical meaning for a single worldline — never the coordinate time of whatever chart the trajectory happens to be written in. Second, $\Omega$ is conjugate to that same proper time; it is an energy gap measured in the detector’s own instantaneous rest frame, not a coordinate-frame frequency. Third, the field state enters only through W , and changing the field state — the ordinary vacuum, a thermal state, the Boulware, Hartle-Hawking, or Unruh state of a black-hole exterior — changes $\mathcal F_x$ even for a fixed trajectory. Keeping these three inputs — proper time, proper-frame gap, and field state — cleanly separated is what prevents an ordinary coordinate-time effect, like gravitational redshift between two static observers at different heights, from being mistaken for a genuine disagreement about particle content. A redshift is a fact about clocks. A response-spectrum disagreement, once clocks are correctly accounted for, is a fact about the field and the worldline together.