First-principles gravitational-wave spectrum of a hot plasma at intermediate momenta between the hydrodynamic and particle regimes

A hot plasma in thermal equilibrium emits an irreducible stochastic gravitational-wave (GW) background through thermal fluctuations of its energy-momentum tensor. The emission rate is known from first principles only in two momentum regimes: the hydrodynamic domain, where wavelengths exceed the mean free path and the rate is set by the shear viscosity, and the hard particle domain at graviton momenta $k\sim\pi T$, where the Standard Model rate is known to complete leading order. Kolesova and Laine (2024), forecasting GHz–THz signals from phase transitions soon after inflation, state that in between these domains no computation exists to date and join them by a straight-line interpolation; the same unknown spectrum fixes where phase-transition signals must merge into the thermal background ($f_0\gtrsim100$ GHz today). The problem asks for a controlled first-principles computation of the graviton production rate — equivalently the transverse-traceless anisotropic-stress spectral function — across the intermediate regime, and of the resulting full present-day spectrum.

Background

A first-order cosmological phase transition — one that proceeds by the nucleation and growth of bubbles of a new phase — generates a stochastic gravitational-wave (GW) background, because bubble collisions and the associated bulk fluid motions carry anisotropic stress, the traceless spatial part of the energy-momentum tensor, which is the source of tensor metric perturbations, i.e. GWs. The peak frequency of the signal today is proportional to the transition temperature $T_*$: electroweak-scale transitions fall into the band of LISA (the Laser Interferometer Space Antenna, a planned space-based gravitational-wave detector), while transitions occurring soon after inflation, at temperatures near the maximum allowed by the Hubble rate during reheating, redshift into the GHz–THz range that proposed high-frequency detectors aim to reach (N. Aggarwal et al, Living Rev. Rel. 24 (2021) 4). Kolesova and Laine (Phys. Lett. B 851 (2024) 138553) derived upper limits on such post-inflationary phase-transition signals: the matter-dominated epoch that follows the end of slow-roll inflation — the final, gradual stage of inflationary expansion — suppresses the signal unless all equilibration rates — the inflaton interaction rate $\Upsilon$ and the energy-transfer rate $\Gamma$ — exceed the Hubble rate $H_*$ at the transition; and the characteristic bubble length $\ell_B$ cannot fall below the mean free path $\ell_{\rm free}$, the typical distance a particle travels between scatterings, because at distances shorter than $\ell_{\rm free}$ "bubbles" are nothing but ordinary thermal fluctuations. Consequently the phase-transition signal can at best saturate the $\Delta N_{\rm eff}$ bound (defined below) at present frequencies $f_0<20$ GHz, while at $f_0>100$ GHz it must merge with, and disappear into, the irreducible GW background produced by thermal fluctuations of the plasma itself.

That irreducible background exists because any plasma in thermal equilibrium has fluctuating energy and momentum densities, whose anisotropic stress continuously sources GWs. The production rate is governed by the spectral function $\mathop{\rm Im}G_R^{\,t}(\omega,k)$ — the imaginary part of the retarded, i.e. linear-response, two-point function — of the transverse-traceless (TT) part of the energy-momentum tensor, and it is known from first principles in two momentum regimes. In the hydrodynamic regime, $k\ll T$ (wavelengths much larger than the mean free path), the rate is controlled by the shear viscosity $\eta$, the transport coefficient that damps velocity gradients in viscous hydrodynamics; for the Standard Model (SM) this was computed by Ghiglieri and Laine (JCAP 07 (2015) 022). In the hard particle regime, $k\sim\pi T$, GWs are produced by $2\to2$ particle scatterings with soft gauge-boson exchange; exchanges of soft momenta of order $gT$ (with $g$ a gauge coupling) require Hard Thermal Loop (HTL) resummation, i.e. resummed thermal propagators that restore infrared safety, and Ghiglieri, Jackson, Laine and Zhu (JHEP 07 (2020) 092) completed this computation to leading order for the SM. Integrating that rate over the thermal history bounds the maximal temperature of the radiation epoch through $\Delta N_{\rm eff}$ — the effective number of light relativistic species — reaching $T_{\max}\lesssim2\times10^{17}$ GeV at the $\Delta N_{\rm eff}\sim10^{-3}$ sensitivity corresponding to the current theoretical uncertainty of the Standard Model prediction. Primordial nucleosynthesis and the cosmic microwave background separately constrain $\Delta N_{\rm eff}\lesssim0.2$, and the background has been proposed as a "big-bang thermometer" (Ringwald, Schütte-Engel and Tamarit, JCAP 03 (2021) 054).

Between these two regimes the rate is not known. The hard computation of Ghiglieri et al (2020) is organized for graviton momenta $k\sim\pi T$ and itself notes that at momenta of order $\alpha_s^2T$ (with $\alpha_s=g_s^2/4\pi$) elementary-particle states must be replaced by hydrodynamic modes; the hydrodynamic expansion in turn is controlled only for wavelengths well above the mean free path. Neither description is parametrically trustworthy in the intermediate window. Kolesova and Laine state the status plainly: "In between these domains, no computation exists to date, and our interpolation is simply a straight line." The spectrum is also known deep in the infrared, $k\ll H$, where the relevant physics is thermal noise during inflation rather than a hot plasma (Klose, Laine and Procacci, JCAP 12 (2022) 020), which does not address this gap. As a result, every current forecast of the thermal background's shape between the viscosity-controlled infrared tail and the scattering-controlled peak — and hence of the frequency and amplitude at which any post-inflationary phase-transition signal merges into it — rests on an uncontrolled interpolation.

Problem Statement

Compute, from first principles, the graviton production rate of an equilibrated ultrarelativistic plasma — equivalently the retarded transverse-traceless stress-tensor spectral function $\mathop{\rm Im}G_R^{\,t}(\omega,k)$ — as a function of $k/T$ across the intermediate regime between the hydrodynamic domain ($k\ll T$, wavelengths much larger than the mean free path) and the hard particle domain ($k\sim\pi T$). The determination must be controlled: leading order in the couplings with the infrared resummations that render it finite, or a nonperturbative method with quantified systematics, and it must demonstrably reduce to the known limits — the shear-viscosity-controlled hydrodynamic rate as $k/T\to0$ and the complete-leading-order scattering result around $k\sim\pi T$. From this rate, assemble the present-day thermal gravitational-wave background spectrum $d\Omega_{\rm gw,0}/d\ln f_0$ over the full frequency range without the straight-line interpolation currently used. Such a spectrum would also place the current estimate that a post-inflationary phase-transition signal merges into the thermal background at $f_0\gtrsim100$ GHz today on a controlled, microphysical footing. A result valid only within one of the already-known regimes, or a new phenomenological interpolation without controlled error estimates, does not answer the question.

References

1. H. Kolesova and M. Laine, Update on gravitational wave signals from post-inflationary phase transitions, Phys. Lett. B 851 (2024) 138553, doi:10.1016/j.physletb.2024.138553, https://arxiv.org/abs/2311.03718
2. J. Ghiglieri and M. Laine, Gravitational wave background from Standard Model physics: qualitative features, JCAP 07 (2015) 022, doi:10.1088/1475-7516/2015/07/022, https://arxiv.org/abs/1504.02569
3. J. Ghiglieri, G. Jackson, M. Laine and Y. Zhu, Gravitational wave background from Standard Model physics: complete leading order, JHEP 07 (2020) 092, doi:10.1007/JHEP07(2020)092, https://arxiv.org/abs/2004.11392
4. P. Klose, M. Laine and S. Procacci, Gravitational wave background from vacuum and thermal fluctuations during axion-like inflation, JCAP 12 (2022) 020, doi:10.1088/1475-7516/2022/12/020, https://arxiv.org/abs/2210.11710
5. J. Ghiglieri, M. Laine, J. Schütte-Engel and E. Speranza, Double-graviton production from Standard Model plasma, JCAP 04 (2024) 062, doi:10.1088/1475-7516/2024/04/062, https://arxiv.org/abs/2401.08766
6. N. Bernal and Y. Xu, Thermal gravitational waves during reheating, JHEP 01 (2025) 137, doi:10.1007/JHEP01(2025)137, https://arxiv.org/abs/2410.21385
7. M. Drewes, Y. Georis, J. Klaric and P. Klose, Upper bound on thermal gravitational wave backgrounds from hidden sectors, JCAP 06 (2024) 073, doi:10.1088/1475-7516/2024/06/073, https://arxiv.org/abs/2312.13855
8. A. Ringwald, J. Schütte-Engel and C. Tamarit, Gravitational waves as a big bang thermometer, JCAP 03 (2021) 054, doi:10.1088/1475-7516/2021/03/054, https://arxiv.org/abs/2011.04731
9. N. Aggarwal, O. D. Aguiar, A. Bauswein, G. Cella, S. Clesse, A. M. Cruise, V. Domcke, D. G. Figueroa, A. Geraci, M. Goryachev, H. Grote, M. Hindmarsh, F. Muia, N. Mukund, D. Ottaway, M. Peloso, F. Quevedo, A. Ricciardone, J. Steinlechner, S. Steinlechner, S. Sun, M. E. Tobar, F. Torrenti, C. Ünal and G. White, Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies, Living Rev. Rel. 24 (2021) 4, doi:10.1007/s41114-021-00032-5, https://arxiv.org/abs/2011.12414
10. M. Laine and S. Procacci, From Inflation to Hot Big Bang, Lecture Notes in Physics vol. 1047 (Springer, 2026), doi:10.1007/978-3-032-09893-1, https://arxiv.org/abs/2607.06983
11. K. Bouzoud and J. Ghiglieri, Thermal axion production at hard and soft momenta, JHEP 01 (2025) 163, doi:10.1007/JHEP01(2025)163, https://arxiv.org/abs/2404.06113