Comparison of two-atom cross spectra in de Sitter spacetime, uniformly accelerated Minkowski vacuum, and a thermal bath

Abstract

We compare two identical two-level atoms coupled to a four-dimensional massless conformally coupled scalar field in three settings: comoving geodesics in the de Sitter Bunch–Davies vacuum, transversely separated uniformly accelerated trajectories in the Minkowski vacuum, and static trajectories in a Minkowski thermal bath. When

and

$$\beta =2\pi /H,$$ β = 2 π / H ,

their local Wightman kernels and spectra agree in both the excitation and de-excitation sectors. At finite separation, the thermal cross spectrum contains a

$$\operatorname {sinc}(\Omega L)$$ sinc ( Ω L )

factor, whereas the two vacuum settings share a hyperbolic geometric factor. At fixed nonzero gap, their large-distance spectral envelopes scale as

$$L^{-1}$$ L - 1

and

$$L^{-2},$$ L - 2 ,

respectively. For each fixed midpoint time T , the comoving de Sitter cross kernel equals the accelerated kernel with

$$a=H$$ a = H

and

$$L=r(T).$$ L = r ( T ) .

This equality concerns the full relative-time dependence at that midpoint; its Fourier transform is a Wigner-type cross spectrum and is not an instantaneous detector transition rate. We derive the corresponding negative-frequency spectra and express finite-time responses using smooth switching functions. Finite switching samples a range of midpoint times, so matching the two vacuum kernels at one midpoint does not generally match their cross responses over a measurement interval. These relations depend on the specified conformal scalar field and trajectory geometries.