A resolved warm/dense gas Schmidt-Kennicutt relationship in a binary HyLIRG at $z=2.41$

Hyperluminous infrared galaxies (HyLIRGs) provide crucial "long lever arm" constraints on galaxy evolution. H-ATLAS $J084933.4+021443$, a $z=2.41$ binary HyLIRG with at least two additional luminous companion galaxies, is thus an optimal test-ground for studies of star formation and galaxy evolution during "cosmic noon". We have used ALMA to obtain resolved imaging and kinematics of atomic and molecular emission lines, and rest-frame $340$ to $1160$GHz continuum emission, for the known luminous component galaxies in H-ATLAS $J084933.4+021443$: W, T, M, C. All four component galaxies are spatially ($\sim 0 .\!\!^{''} 3$ or $2.5$kpc) resolved in CO J:7-6, [C$_\mathrm{I}$] 2-1, H$_2$O and the millimetre (mm) to sub-mm continuum. Rotation-dominated gas kinematics is confirmed in W and T. The significant extension to component T, in gas and continuum, along its kinematic minor axis, is attributable to its lensing magnification. Spatially resolved sub-mm spectral energy distributions reveal that component W is well fit with greybody emission from dust at a single temperature over the full extent of the galaxy, despite it containing a powerful AGN, while component T requires an additional component of hotter nuclear dust and additional sources of emission in the mm. We confirm that [C$_\mathrm{I}$] 2-1 can be used as a rough tracer of warm/dense molecular gas in extreme systems, though the [C$_\mathrm{I}$] 2-1/CO luminosity ratio increases sub-linearly. We obtain an exquisite and unprecedented resolved ($2.5$-kpc-scale) "warm/dense molecular gas" Schmidt-Kennicutt (SK) relationship for components W and T. Gas exhaustion times for all apertures in W (T) are $1-4$Gyr ($0.5-2$Gyr). Both W and T follow a resolved "warm/dense gas" SK relationship with power law $n\sim1.7$, significantly steeper than the $n\sim1$ found previously via "cold" molecular gas in nearby "normal" star-forming galaxies.