  The Faraday rotation effect, quantified by the Rotation Measure (RM), is a
powerful probe of the large-scale magnetization of the Universe - tracing
magnetic fields not only on galaxy and galaxy cluster scales but also in the
intergalactic Medium (IGM; referred to as $\mathrm{RM}_{\text{IGM}}$). The
redshift dependence of the latter has extensively been explored with
observations. It has also been shown that this relation can help to distinguish
between different large-scale magnetization scenarios. We study the evolution
of this $\mathrm{RM}_{\text{IGM}}$ for different primordial magnetogenesis
scenarios to search for the imprints of primordial magnetic fields (PMFs;
magnetic fields originating in the early Universe) on the redshift-dependence
of $\mathrm{RM}_{\text{IGM}}$. We use cosmological magnetohydrodynamic (MHD)
simulations for evolving PMFs during large-scale structure formation, coupled
to the light cone analysis to produce a realistic statistical sample of mock
$\mathrm{RM}_{\text{IGM}}$ images. We study the predicted behavior for the
cosmic evolution of $\mathrm{RM}_{\text{IGM}}$ for different correlation
lengths of PMFs, and provide fitting functions for their dependence on
redshifts. We compare these mock RM trends with the recent analysis of the the
LOw-Frequency ARray (LOFAR) RM Grid and find that large-scale-correlated PMFs
should have (comoving) strengths $\lesssim 0.75$ nanoGauss, if originated
during inflation with the scale invariant spectrum and (comoving) correlation
length $\sim 19$ cMpc/h or $ \lesssim 30$ nanoGauss if they originated during
phase-transition epochs with the comoving correlation length $\sim 1$ cMpc/h.
Our findings agree with previous observations and confirm the results of
semi-analytical studies, showing that upper limits on the PMF strength decrease
as their coherence scales increase.
