The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence

Abstract

Hydromagnetic turbulence produced during phase transitions in the early universe can be a powerful source of stochastic gravitational waves (GWs). GWs can be modelled by the linearised spatial part of the Einstein equations sourced by the Reynolds and Maxwell stresses. We have implemented two different GW solvers into the Pencil Code-a code which uses a third order timestep and sixth order finite differences. Using direct numerical integration of the GW equations, we study the appearance of a numerical degradation of the GW amplitude at the highest wavenumbers, which depends on the length of the timestep-even when the Courant-Friedrichs-Lewy condition is ten times below the stability limit. This degradation leads to a numerical error, which is found to scale with the third power of the timestep. A similar degradation is not seen in the magnetic and velocity fields. To mitigate numerical degradation effects, we alternatively use the exact solution of the GW equations under the assumption that the source is constant between subsequent timesteps. This allows us to use a much longer timestep, which cuts the computational cost by a factor of about ten.

The Turbulent Stress Spectrum in the Inertial and Subinertial Ranges

“…The simulations of Roper Pol et al (2020) show that most of the wave generation occurs at the time when the stress has reached maximum amplitude. Subsequent changes of the source hardly contribute to wave production.…”

“…It may be for this reason that the assumption of no time delay is a reasonable one. Under this assumption, we expect that the gravitational wave energy, which is proportional to (∂h ij /∂t) 2 , should be proportional to F (k)/k 2 ; see Roper Pol et al (2020) for details. The extent of the empirically determined departures from this simplistic way of estimating the gravitational wave energy spectrum are not yet fully understood and would need to be determined numerically or analytically, similarly to earlier work using the aeroacoustic approximation of Lighthill (1952a), as already done by Gogoberidze et al (2007) in the context of primordial gravitational waves.…”

Polarization of gravitational waves from helical MHD turbulent sources

“…The details of the numerical setup and application to the electroweak scale are described in refs. [81,82].…”

“…Recently, in refs. [81,82], the authors have described the implementation of a GW solver into the Pencil Code [83] and have presented direct numerical simulations for modeling development and dynamics of primordial hydrodynamic and hydromagnetic turbulence from phase transitions, and subsequent generation of a stochastic GW background, also computed numerically. Their simulations included fully helical sources, but the estimation of the GW polarization degree spectrum, as well as the polarization detection prospects, were not the focus of their studies.…”

“…The driving is then applied during a short time interval (around a 10% of the Hubble time), and then switched off such that turbulence decays for later times. By using suitably scaled variables and conformal time, the governing equations describing the evolution of GWs and turbulent magnetic fields in an expanding universe in the radiation era can be brought into a form that is best suited for numerical simulations [82]. We explore the detectability of the generated GW signal and its polarization with planned space-based GW detectors.…”