This paper studies how a spatially fixed but time-changing random environment affects a massive "spectator" field during inflation. Cosmologists use oscillations in primordial correlations as a kind of spectroscope for heavy particles present in the early universe. The authors ask what happens to those oscillatory "cosmological collider" signals when the heavy field is driven by a random source whose pattern in space is frozen (quenched) while its overall strength follows a simple power-law in time.

The setup is simple and solvable. The heavy field is coupled linearly to a spatial random amplitude h(x) multiplied by a deterministic time weight (-Hη)^p, where H is the Hubble scale, η is conformal time, and p labels how the source turns off toward late times. For each fixed spatial realization of the source the linear equation can be solved exactly. The forced, or retarded, response is expressed in terms of Lommel functions (special functions that solve inhomogeneous Bessel-type equations). A Mellin–Barnes representation is used to separate that forced response from the usual homogeneous, massive-field behavior.

After averaging over Gaussian disorder in space, the authors find a clean result: the random source adds a purely statistical piece to the field propagator but does not shift the spectral function or move the poles that encode the heavy-field mass. In other words, the disorder changes amplitudes and phases of the signals seen in correlators, but it does not change the masses that set the oscillation frequency. The exchange contribution that builds the collider signal factorizes exactly into one sector that comes from the forced (disorder-driven) response and two standard massive-field branches. A nonanalytic folded contribution cancels out in this construction.

The time profile of the source matters. If the source persists unchanged to late times (the persistent case, p = 0), the forced response contains a static piece that makes the trispectrum (a four-point correlator) finite but produces a local late-time logarithm in the bispectrum (a three-point correlator). That logarithm is an endpoint effect tied to the source being switched on indefinitely. If the source decays (p > 0), the endpoint becomes integrable and the problematic logarithm is removed. For sufficiently fast decay the nonanalytic "massive clock" oscillations from the heavy field can dominate the squeezed limit of correlators. For spatial white noise (a momentum-independent disorder with f(k) = f_0 H^3) a special value p = 3/2 eliminates an explicit exchanged-scale dependence and can make the disorder contribution interfere destructively with the usual collapsed trispectrum branch.