Conditional probability distributions P(j_ν | ρ) of synchrotron monochromatic emission coefficient j_ν (at rest-frame (emission) frequency ν = 150 MHz) given total matter density ρ for the Hoeft & Brüggen (2007) formalism, as derived from z = 0.025 snapshots of cosmological MHD simulations by Vazza et al. (2019). The dashed lines indicate a j_ν ∝ ρ^23/6 (single shock) scaling relation for various proportionality constants. ρ is shown relative to the current-day mean total matter density , while the weakly constrained electron acceleration efficiency ξ_e is divided out from j_ν. Each conditional is numerically approximated by a probability mass function with 1000 bins, whose edges vary by a constant factor and span 40 orders of magnitude in j_ν.