Episodic Star Formation - II. The Origin of the Fundamental Metallicity Relation

The Fundamental Metallicity Relation (FMR) extends the mass-metallicity relation (MZR) by including star formation rate (SFR) as a third parameter. At fixed stellar mass, galaxies with higher SFRs tend to have lower gas-phase metallicities. We investigate the physical origin of the FMR in the context of episodic star formation (Gui et al. 2026, paper I of this series) by tracing the evolution histories of 8,979 central star-forming galaxies selected at $z=0$ in TNG100 back to $z=1$. We find that the overall MZR scatter at $z=0$ ($\sim0.11$ dex) comprises both long-lived differences among galaxies ($\sim0.09$ dex) and temporal fluctuations within individual galaxies ($\sim 0.07$ dex). The galaxy-to-galaxy component shows an anticorrelation between SFR and gas-phase metallicity, as shown by the observed FMR. As for the consequence from temporal fluctuation, we track individual histories and stack the upward and downward SFR phases of a refined sample. Time-shift analysis and stacked histories show a well-behaved sequence in which the cold non-star-forming gas reservoir increases (decreases) first, followed by a decrease (increase) in gas-phase metallicity and then a rise (decline) in SFR. This ordering is consistent with a physical scenario where metal-poor gas first replenishes the cold reservoir and dilutes the gas before SFR rises to the peak, followed by feedback-driven gas heating/removal, resulting in a shift in metallicity from dilution to enrichment as SFR decreases towards the valley. This naturally explains the anticorrelation between the SFR and gas-phase metallicities as indicated by the observed FMR.