Simulating the LOcal Web (SLOW) -- III: Synchrotron emission from the local cosmic web

Aims: Detecting diffuse synchrotron emission from the cosmic web is still a challenge for current radio telescopes. We aim to make predictions about the detectability of cosmic web filaments from simulations. Methods: We present the first cosmological magnetohydrodynamic simulation of a 500 $h^{-1} c$Mpc 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 modeled CR injection at shocks, while accounting for adiabatic changes to the CR population and high-energy-loss processes of electrons. The synchrotron emission was then calculated from the aged electron population, using the simulated magnetic field, as well as different models for the origin and amplification of magnetic fields. We used constrained initial conditions, which closely resemble the local Universe, and compared the results of the cosmological volume to a zoom-in simulation of the Coma cluster, to study the impact of resolution and turbulent reacceleration of CRs on the results. Results: We find a consistent injection of CRs at accretion shocks onto cosmic web filaments and galaxy clusters. This leads to diffuse emission from filaments of the order $S_\nu \approx 0.1 \: \mu$Jy beam$^{-1}$ for a potential LOFAR observation at 144~MHz, when assuming the most optimistic magnetic field model. 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 ten 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 in the LOFAR band.