Isochrones in primordial magnetic field evolution

During the radiation-dominated era of the Universe, a primordial magnetic field undergoes a turbulent decay while its length scale increases due to an inverse cascade. At later times, the size of the largest processed eddy scales with the Alfv\'en speed and it describes an isochrone that moves toward larger scales with increasing time. Different magnetogenesis mechanisms produce different initial length scales and field strengths, independently of the nominal generation time. However, we show that for any initial field, a proper time can be determined such that the isochrones at early times are parallel to those at late times. We use two-dimensional numerical simulations of decaying MHD turbulence and vary the initial position of the peak of the magnetic energy spectrum. In this case, the evolution is governed by the conservation of anastrophy. A fit to the Alfv\'en time yields an accurate estimate of the factor by which the decay time is longer than the Alfv\'en time, while the offset in the fit provides an additional estimate of the proper time that needs to be added to the nominal time since the beginning of each simulation. We also find that the presence of an initial velocity field of realistic strength helps producing a more straight track. The magnetic field parameters lie on universal isochrones even for early times.