From atoms to stars: Modelling $\mathrm{H}_2$ formation and its impact on galactic evolution

We present a sub-grid model for star formation in galaxy simulations, incorporating molecular hydrogen ($\mathrm{H}_2$) production via dust grain condensation and its destruction through star formation and photodissociation. Implemented within the magnetohydrodynamical code AREPO, our model tracks the non-equilibrium mass fractions of molecular, ionised, and atomic hydrogen, as well as a stellar component, by solving a system of differential equations governing mass exchange between these phases. Star formation is treated with a variable rate dependent on the local $\mathrm{H}_2$ abundance, which itself varies in a complex way with key quantities such as gas density and metallicity. Testing the model in a cosmological simulation of a Milky Way-mass galaxy, we obtain a well-defined spiral structure at $z = 0$, including a gas disc twice the size of the stellar one, alongside a realistic star formation history. Our results show a broad range of star formation efficiencies per free-fall time, from as low as $0.001\%$ at high redshift to values between $0.1\%$ and $10\%$ for ages $\gtrsim 3-4 \, \mathrm{Gyr}$. These findings align well with observational estimates and simulations of a turbulent interstellar medium. Notably, our model reproduces a star formation rate versus molecular hydrogen surface densities relation akin to the molecular Kennicutt-Schmidt law. Furthermore, we find that the star formation efficiency varies with density and metallicity, providing an alternative to fixed-efficiency assumptions and enabling comparisons with more detailed star formation models. Comparing different star formation prescriptions, we find that in models that link star formation to $\mathrm{H}_2$, star formation onset is $\sim \! 500 \, \mathrm{Myr}$ later than those relying solely on total or cold gas density.