One-loop thermal radiation exchange in gravitational wave power spectrum

The radiation-dominated universe is a key ingredient of the standard Big Bang cosmology. Radiation comprises numerous quantum elementary particles, and the macroscopic behavior of radiation is described by taking the quantum thermal average of its constituents. While the interactions between individual particles and gravitational waves are often neglected in this context, it raises the question of whether these elementary particles interact with gravitational waves in the framework of quantum field theory. To address this question, this paper aims to explore the quantum mechanical aspects of gravitational waves in a universe dominated by a massless scalar field, whose averaged energy-momentum tensor plays the role of background radiation. We establish the equivalence between the classical Einstein equation and the mean-field approximation of the Heisenberg equation in a local thermal state. Beyond the mean-field approximation, we analyze the quantum corrections to gravitational waves, particularly focusing on the thermal radiation loop corrections. Interestingly, we find the 1-loop correction surpasses the tree-level spectrum of primordial gravitational waves, which is $O(\alpha^2)$ where $\alpha=H_{\rm inf}/M_{\rm pl}$ is the ratio of the inflationary Hubble parameter to the Planck mass. Then, to see if this result persists even if we take into account all the higher order loop corrections, the loop expansion is reorganized in the series expansion in $\alpha$. We schematically discuss two-loop diagrams that may give $O(\alpha^2)$ contributions. We leave explicit computations of these diagrams for future studies. Thus, although we cannot claim that the whole loop corrections exceed the tree-level spectrum at the moment, our findings highlight the significance of quantum effects when studying the interaction between radiation and gravitational waves in the cosmological context.