Numerical Simulations of Gravitational Waves from Early-Universe Turbulence

TL;DR

The paper uses direct 3D MHD simulations to compute the gravitational-wave background from early-Universe turbulence without relying on simplifying analytic assumptions. By solving the normalized GW equation alongside MHD dynamics with diverse driving mechanisms, it reveals universal low-frequency behavior and specific spectral scalings in both the inertial and subinertial ranges, and it connects these results to present-day observables relevant for LISA. The findings show that the GW signal strength scales quadratically with the turbulent energy and that detectability by LISA depends sensitively on the driving process and energy fraction converted into turbulence or magnetic fields, with acoustic forcing often yielding higher GW outputs. These insights refine prior analytic expectations, quantify the spectral shapes, and sharpen the prospects for using LISA to probe electroweak-scale phase transitions and primordial magnetic fields.

Abstract

We perform direct numerical simulations of magnetohydrodynamic turbulence in the early universe and numerically compute the resulting stochastic background of gravitational waves and relic magnetic fields. These simulations do not make the simplifying assumptions of earlier analytic work. If the turbulence is assumed to have an energy-carrying scale that is about a hundredth of the Hubble radius at the time of generation, as expected in a first-order phase transition, the peak of gravitational wave power will be in the mHz frequency range for a signal produced at the electroweak scale. The efficiency of gravitational wave (GW) production varies significantly with how the turbulence is driven. Detectability of turbulence at the electroweak scale by the planned Laser Interferometer Space Antenna (LISA) requires anywhere from 0.1% to 10% of the thermal plasma energy density to be in plasma motions or magnetic fields, depending on the model of the driving process. Our results predict a new universal form below the spectral peak frequency that is shallower than previously thought. This implies larger values of the GW energy spectra in the low-frequency range. This extends the range where turbulence is detectable with LISA to lower frequencies, corresponding to higher energy scales than the assumed energy-carrying scale.

Figures (8)

• Figure 1: Magnetic and GW energy spectra for run ini2 averaged over late times ($t > 1.1$), after the GW spectrum has started to fluctuate around a steady state.
• Figure 2: Time evolution of the magnetic and GW energy spectra amplified by a factor of $k_\ast$ for run ini2. Also shown are the results for a domain larger by a factor of 10 (red) and 50 (blue). All runs have $1152^3$ mesh points.
• Figure 3: GW spectral energy versus time for four values of $k$, demonstrating the $k^2$ scaling at early times for run ini2.
• Figure 4: Spectra of $h_0^2{\Omega}_{\rm GW}(f)$ and $h_{\rm c}(f)$ evaluated at the present time, along with the LISA power law sensitivity curve (green dot-dashed line) to a stochastic GW background assuming four years of mission, and a threshold signal-to-noise ratio of 10 RCC18SC19Caprini:2019pxz. See Table \ref{['Tsummary']} for details of runs ini1--3.
• Figure 5: Evolution of $\Omega_{\rm M, K}$ (top) and ${\Omega}_{\rm GW}$ (bottom) for runs with initial energy (ini1--3) and runs where energy is driven through monochromatic forcing (hel1--2 and ac1). Note that the energy densities are normalized with the radiation energy density at the time of generation.