Isochrones in primordial magnetic field evolution

Abstract

In the early universe, a primordial magnetic field undergoes a turbulent decay while its length scale increases due to an inverse cascade. The size of the largest processed eddy scales with the Alfv\'en speed and grows with time. In a diagram of Alfv\'en speed vs.\ length scale, all possible solutions must lie on a line through the origin with a slope proportional to the inverse of the present time. In principle, however, such lines can also be defined for earlier times. The lines for earlier times form isochrones that may be observationally accessible, for example through the magnetically driven stochastic gravitational wave background. However, the position and slope of these isochrones is sensitive to the zero point of the time. Here, we show that for any initial magnetic field, a proper time can be determined such that the resulting isochrones at early times are nearly parallel to those at late times, i.e., they have the same slope. 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 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 from the beginning. The magnetic field parameters lie on universal isochrones even for early times. They provide a testable framework for magnetic fields generated at times as early as the end of inflation, starting with the time of reheating.