Ultraviolet Structure of Real-time Gravitational Wave Linear Response in a Resonant Scalar Field

Abstract

We study the real-time linear response of gravitational waves in a time-dependent resonant scalar field in a Minkowski background. In the Schwinger-Keldysh formalism, we develop an adiabatic regularization scheme for unequal-time correlation functions and use it to extract the ultraviolet structure of the one-loop response. The leading divergence reproduces the familiar ${\square }^2{h}_{ij}$ structure, whereas the time-dependent background induces additional local divergences proportional to $\square h_{ij}$, ${\partial }_0{h}_{ij}$, and $h_{ij}$. These are renormalized by local counterterms associated with the Weyl-squared term, a time-dependent Ricci-scalar term, and a time-dependent cosmological constant. We also compare the renormalization of the linear response with that of the tadpole stress tensor and find a mismatch beyond leading adiabatic order in the present toy model. By considering a covariant completion of the resonance, we further argue that this mismatch is tied to the off-shell nature of the fixed background, and is expected to disappear once the background is treated on shell.

Comments: 23 pages