We set limits on the presence of the synchrotron cosmic web through the cross-correlation of the 2.3 GHz S-PASS survey with a model of the local cosmic web derived from constrained magnetohydrodynamic (MHD) simulations. The MHD simulation assumes cosmologically seeded magnetic fields amplified during large-scale structure formation, and a population of relativistic electrons/positrons from proton-proton collisions within the intergalactic medium. We set a model-dependent 3$sigma$ upper limit on the synchrotron surface brightness of 0.16 mJy arcmin$^{-2}$ at 2.3 GHz in filaments. Extrapolating from magnetic field maps created from the simulation, we infer an upper limit (density-weighted) magnetic field of 0.03 (0.13) $mu$G in filaments at the current epoch, and a limit on the primordial magnetic field (PMF) of B$_{PMF}$~1.0 nG.
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