Symmetry principles of gravitational perturbations in thermal environments

TL;DR

This paper addresses how to specify the thermal state of a cosmological plasma when computing the effective dynamics of gravitational perturbations on an FLRW background. By enforcing diffeomorphism and Weyl Ward identities with retarded boundary conditions, the authors show that a global (grand) canonical ensemble ${D}_G$ is the consistent initial state in a radiation-dominated era, while a metric-perturbed local ensemble ${D}_L$ is ruled out. The resulting effective tensor dynamics includes a memory kernel and yields a plasmon-like graviton mass shift $m^2_{ m eff} = 8 H^2/5$, with no secular growth thanks to large-diffeomorphism matching; this aligns with Weinberg's kinetic theory results. The work clarifies that local equilibrium emerges dynamically from linear response rather than being imposed a priori, and it highlights the need to retain expansion and memory effects in thermal gravitational responses, rather than relying on flat-space HTL intuition.

Abstract

The thermal plasma induces a plasmon-like mass shift for gravitational perturbations, which can modify their dynamics near the horizon scale in the early radiation-dominated universe. However, there are several seemingly reasonable ways to introduce this mass shift, reflecting an ambiguity in how one specifies the initial plasma state on a perturbed FLRW background. Invariance under small diffeomorphisms and Weyl rescalings singles out the (grand) canonical ensemble defined in the decoupling limit of gravitational interactions, while excluding ensembles that violate the Weyl identity, including those perturbed by the metric. Large diffeomorphisms further require the mass shift to vanish in the infrared limit. With this consistent choice, primordial tensor modes exhibit stable damping, in agreement with Weinberg's kinetic theory analysis. This cosmological example indicates a more general picture in which local equilibrium in thermal quantum field theory is not an external input but an emergent, dynamical notion.

Figures (2)

• Figure 1: Schematic illustration of the initial environmental ensembles $\hat{D}_G$ and $\hat{D}_L$. Panel (a) depicts the uniform ensemble $\hat{D}_G$ on an FLRW background, shown as a plane, whereas panel (b) shows the ensemble $\hat{D}_L$ perturbed by the metric. We show that the left panel (i.e., $\hat{D}_G$) is consistent with the symmetries of general relativity.
• Figure 2: Numerical amplitude of the traceless transverse metric perturbation $h_{ij}$ as a function of $x\equiv k\tau$ (radiation era). All curves share the initial conditions $h(x_0)=1$ and $\partial_x h(x_0)=0$ at $x_0=k\tau_0=0.001$. The gray dot dashed curve is the linear solution. The blue dashed curve corresponds to the tachyonic case introduced by the local ensemble $\hat{D}_L$, which is excluded by the diffeomorphism symmetry test. The yellow dotted curve shows the evolution with $\hat{D}_G$ without the memory kernel. This case passes the symmetry test for the diffeomorphism Ward identity, but it does not respect large diffeomorphisms. The purple solid curve shows Weinberg's damping solution obtained with the causal memory kernel and the global ensemble $\hat{D}_G$, which respects all symmetries.