  We present three-dimensional direct numerical simulations of the production
of magnetic fields and gravitational waves (GWs) in the early Universe during a
low energy scale matter-dominated post-inflationary reheating era, and during
the early subsequent radiative era, which is strongly turbulent. The parameters
of the model are determined such that it avoids a number of known physical
problems and produces magnetic energy densities between 0.2% and 2% of the
critical energy density at the end of reheating. During the subsequent
development of a turbulent magnetohydrodynamic cascade, magnetic fields and GWs
develop a spectrum that extends to higher frequencies in the millihertz
(nanohertz) range for models with reheating temperatures of around 100 GeV (150
MeV) at the beginning of the radiation-dominated era. However, even though the
turbulent cascade is fully developed, the GW spectrum shows a sharp drop for
frequencies above the peak value. This suggests that the turbulence is less
efficient in driving GWs than previously thought. The peaks of the resulting GW
spectra may well be in the range accessible to space interferometers, pulsar
timing arrays, and other facilities.
