LeMMINGs. III. The e-MERLIN Legacy Survey of the Palomar sample. Exploring the origin of nuclear radio emission in active and inactive galaxies through the [O III] -- radio connection

What determines the nuclear radio emission in local galaxies? We combine optical [O III] line emission, robust black hole (BH) mass estimates, and high-resolution e-MERLIN 1.5-GHz data, from the LeMMINGs survey, of a statistically-complete sample of 280 nearby, optically active (LINER and Seyfert) and inactive HII and Absorption line galaxies [ALG]) galaxies. Using [O III] luminosity (L_{rm [O~III]}) as a proxy for the accretion power, local galaxies follow distinct sequences in the optical-radio planes of BH activity, which suggest different origins of the nuclear radio emission for the optical classes. The 1.5-GHz radio luminosity of their parsec-scale cores (L_{rm core}) is found to scale with BH mass (M_{rm BH}) and [O~III] luminosity. Below M_{rm BH} sim10^{6.5} M_{odot}, stellar processes from non-jetted HII galaxies dominate with L_{rm core} propto M_{rm BH}^{0.61pm0.33} and L_{rm core} propto L_{rm [O~III]}^{0.79pm0.30}. Above M_{rm BH} sim10^{6.5} M_{odot}, accretion-driven processes dominate with L_{rm core} propto M_{rm BH}^{1.5-1.65} and L_{rm core} propto L_{rm [O~III]}^{0.99-1.31} for active galaxies: radio-quiet/loud LINERs, Seyferts and jetted HII galaxies always display (although low) signatures of radio-emitting BH activity, with L_{rm 1.5, GHz}gtrsim10^{19.8} W Hz^{-1} and M_{rm BH}gtrsim10^{7} M_{odot}, on a broad range of Eddington-scaled accretion rates (m). Radio-quiet and radio-loud LINERs are powered by low-m discs launching sub-relativistic and relativistic jets, respectively. Low-power slow jets and disc/corona winds from moderately high to high-m discs account for the compact and edge-brightened jets of Seyferts, respectively. Jetted HII galaxies may host weakly active BHs. Fuel-starved BHs and recurrent activity account for ALG properties. [abridged]