On detecting the cosmic web and its magnetic field

Magnetic fields are known to permeate space over a remarkable range of scales, from meter-large bodies up to ~Mpc wide clusters of galaxies. It is still debated whether their origin is either "primordial", involving their early formation in the homogeneous universe, or "astrophysical", i.e. from feedback into the inter-galactic space by the first dense and highly magnetised structures (e.g. stars, galaxies, AGN). Observing galaxy cluster outskirts and filaments of the cosmic web is keys to test the above scenarios, as the current strength and morphology of their evolved magnetic field should correlate with the initial conditions in the most rarefied environments. In this thesis, we push complementary radio-astronomy techniques (namely direct imaging, Faraday rotation and FRB studies) on dedicated data from new generation instruments, in combination with state-of-the-art cosmological simulations, to constrain the properties of the magnetic fields at ~Mpc scale at low redshift. This work provides relevant advances into observational constraints of magnetic fields in cluster outskirts (e.g. 0.4uG<B_Mpc<10uG in the outskirts of the galaxy cluster A2249) and intra-cluster filaments (B_Mpc<0.25-0.75uG). We explore the feasibility and possible systematics of Faraday rotation studies with new and next generation instrumentation (JVLA, SKA) and provide new perspectives to best exploit their use. We also constrain the population properties of fast radio bursts (FRBs), which can be used as background sources for the study of the large scale structures. Our findings hint to either an evolving luminosity function of FRBs or to the presence of un-diagnosed selection effect in the current avaliable samples. Finally, we plan (and implement) the refurbishment of an old radio telescope, the Northern Cross in Medicina, for dedicated FRB search. We characterise the instrument sensitivity at 408 MHz and estimate detection rates with respect to different complementary set-up, e.g. a follow-up mode of known FRBs and a wide-area survey.