Systematic treatment of the intermediate thermal-correlator regime

Develop a systematic treatment of the intermediate graviton-momentum regime between the transport-relaxation scale and the screening scale, au_{mathrm{tr}}^{-1}lesssim klesssim m_a, where neither the hydrodynamic nor the quasiparticle description is controlled.

Background

The paper describes thermal gravitational-wave production using two controlled descriptions of the thermal stress-tensor correlator. At very low graviton momentum, hydrodynamics applies because the graviton probes timescales longer than the transport relaxation time. At momenta above the screening scale, a quasiparticle and hard-thermal-loop description is used.

Between these scales lies an intermediate momentum interval in which neither approximation is reliable. The authors note that extrapolations of perturbative formulas into this region can become pathological, leaving a systematic calculation of the correlator and the resulting gravitational-wave spectrum unresolved.

References

At weak coupling, \tau_{\rm tr}{-1}\ll m_a, leaving the intermediate momentum region \begin{equation} \tau_{\rm tr}{-1} \lesssim k\lesssim m_a \label{eq:thermal-intermediate-region} \end{equation} which is outside both controlled descriptions. A systematic treatment of this region is not presently available, and perturbative expressions extended into it are known to develop pathological behavior. We therefore leave the intermediate regime unresolved in the present analysis.

— Gravitational Waves from Reheating beyond Instantaneous Thermalization  (2609.01480 - Mukaida et al., 1 Sep 2026) in Section 3.2, immediately following Eq. (thermal-intermediate-region)