Galaxy And Mass Assembly (GAMA): Gas Fuelling of Spiral Galaxies in the Local Universe II. -- Direct Measurement of the Dependencies on Redshift and Host Halo Mass of Stellar Mass Growth in Central Disk Galaxies

We present a detailed analysis of the specific star formation rate -- stellar mass ($\mathrm{sSFR}-M_*$) of $z\le 0.13$ disk central galaxies using a morphologically selected mass-complete sample ($M_* \ge 10^{9.5} M_{\odot}$). Considering samples of grouped and ungrouped galaxies, we find the $\mathrm{sSFR}-M_*$ relations of disk-dominated central galaxies to have no detectable dependence on host dark-matter halo (DMH) mass, even where weak-lensing measurements indicate a difference in halo mass of a factor $\gtrsim5$. We further detect a gradual evolution of the $\mathrm{sSFR}-M_*$ relation of non-grouped (field) central disk galaxies with redshift, even over a $\Delta z \approx 0.04$ ($\approx5\cdot10^{8}\mathrm{yr}$) interval, while the scatter remains constant. This evolution is consistent with extrapolation of the "main-sequence-of-star-forming-galaxies" from previous literature that uses larger redshift baselines and coarser sampling. Taken together, our results present new constraints on the paradigm under which the SFR of galaxies is determined by a self-regulated balance between gas inflows and outflows, and consumption of gas by star-formation in disks, with the inflow being determined by the product of the cosmological accretion rate and a fuelling-efficiency -- $\dot{M}_{\mathrm{b,halo}}\zeta$. In particular, maintaining the paradigm requires $\dot{M}_{\mathrm{b,halo}}\zeta$ to be independent of the mass $M_{\mathrm{halo}}$ of the host DMH. Furthermore, it requires the fuelling-efficiency $\zeta$ to have a strong redshift dependence ($\propto (1+z)^{2.7}$ for $M_*=10^{10.3} M_{\odot}$ over $z=0 - 0.13$), even though no morphological transformation to spheroids can be invoked to explain this in our disk-dominated sample. The physical mechanisms capable of giving rise to such dependencies of $\zeta$ on $M_{\mathrm{halo}}$ and $z$ for disks are unclear.