The MeerKAT Fornax Survey IV. A close look at the cluster physics through the densest rotation measure grid

Using the Square Kilometre Array (SKA) mid precursor MeerKAT, we acquired broadband spectro-polarimetric data in the context of the MeerKAT Fornax Survey to study the Fornax cluster's magnetic fields in detail by building the densest rotation measure (RM) grid to date. Here, we present the survey, the analysis, and a discussion of the RM grid properties. We analyzed a circular region centered on the Fornax cluster center with a radius of $\sim1.4^\circ$; that is, $\rm\sim 0.73 R_{vir}$. The mosaics have a resolution of 13arcsec and cover the frequencies between 900\,MHz and 1.4\,GHz, reaching an average noise of 16$\mu$Jy beam$^{-1}$ in total intensity and 3$\mu$Jy beam$^{-1}$ in the Q and U Stokes images. With these data, we detected 508 polarized sources over an area of $\sim$6.35 deg$^2$ corresponding to a density of $\sim$80 polarized sources/deg$^2$. This is the densest RM grid ever built. Of the polarized sources, five are cluster sources. Excluding the cluster sources, we built the Euclidean-normalized differential source counts in polarization and we went a factor of ten deeper than previous surveys. We tentatively detect for the first time an increment in the differential source counts at low polarized flux densities; that is, $\sim$9\,$\mu$Jy at 1.4\,GHz. The average degree of polarization of about 3--4\% suggests that the sub$-\mu$Jansky population is not dominated by star-forming galaxies, typically showing a degree of polarization lower than 1\%. The majority of the polarized sources are Faraday simple; in other words, their polarization plane rotates linearly with the wavelength squared. The RM shows the typical decrement going from the center to the outskirts of the Fornax cluster. However, interesting features are observed both in the RM grid and in the RM radial profiles across different directions. A combination of the ...