  Primordial magnetic fields could explain the large-scale magnetic fields
present in the Universe. Inflation and phase transitions in the early Universe
could give rise to such fields with unique characteristics. We investigate the
magneto-hydrodynamic evolution of these magnetogenesis scenarios with
cosmological simulations. We evolve inflation-generated magnetic fields either
as (i) uniform (homogeneous) or as (ii) scale-invariant stochastic fields, and
phase transition-generated ones either as (iii) helical or as (iv) non-helical
fields from the radiation-dominated epoch. We find that the final distribution
of magnetic fields in the simulated cosmic web shows a dependence on the
initial strength and the topology of the seed field. Thus, the observed field
configuration retains information on the initial conditions at the moment of
the field generation. If detected, primordial magnetic field observations would
open a new window for indirect probes of the early universe. The differences
between the competing models are revealed on the scale of galaxy clusters,
bridges, as well as filaments and voids. The distinctive spectral evolution of
different seed fields produces imprints on the correlation length today. We
discuss how the differences between rotation measures from highly ionized
regions can potentially be probed with forthcoming surveys.
