  In the primordial plasma, at temperatures above the scale of electroweak
symmetry breaking, the presence of chiral asymmetries is expected to induce the
development of helical hypermagnetic fields through the phenomenon of chiral
plasma instability. It results in magnetohydrodynamic turbulence due to the
high conductivity and low viscosity and sources gravitational waves that
survive in the universe today as a stochastic polarized gravitational wave
background. In this article, we show that this scenario only relies on Standard
Model physics, and therefore the observable signatures, namely the relic
magnetic field and gravitational background, are linked to a single parameter
controlling the initial chiral asymmetry. We estimate the magnetic field and
gravitational wave spectra, and validate these estimates with 3D numerical
simulations.
