Aims: Detecting diffuse synchrotron emission from the cosmic web is still a
challenge for current radio telescopes. We aim to make predictions for the
detectability of cosmic web filaments from simulations. Methods: We present the
first cosmological MHD simulation of a 500 h−1cMpc volume with an
on-the-fly spectral cosmic ray (CR) model. This allows us to follow the
evolution of populations of CR electrons and protons within every resolution
element of the simulation. We model CR injection at shocks, while accounting
for adiabatic changes to the CR population and high energy loss processes of
electrons. The synchrotron emission is then calculated from the aged electron
population, using the simulated magnetic field, as well as different models for
origin and amplification of magnetic fields. We use constrained initial
conditions, which closely resemble the local Universe and compare the results
of the cosmological volume to zoom-in simulation of the Coma cluster, to study
the impact of resolution and turbulent re-acceleration of CRs on the results.
Results: We find consistent injection of CRs at accretion shocks onto cosmic
web filaments and galaxy clusters. This leads to diffuse emission from
filaments of the order Sν≈0.1μJy beam−1 for a potential
LOFAR observation at 144 MHz, when assuming the most optimistic magnetic field
model and the inclusion of an on-the-fly treatment of re-acceleration of
electrons by turbulence. The flux can be increased by up-to two orders of
magnitude for different choices of CR injection parameters. This can bring the
flux within a factor of 10 of the current limits for direct detection. We find
a spectral index of the simulated synchrotron emission from filaments of
{\alpha} {\approx} 1.0 - 1.5.Comment: 23 pages, 12 figures, submitted to A&A. Comments welcome