The Astrophysics of Star Formation Across Cosmic Time at $\gtrsim$10 GHz with the Square Kilometre Array

In this chapter, we highlight a number of science investigations that are enabled by the inclusion of Band~5 ($4.6-13.8$ GHz) for SKA1-MID science operations, while focusing on the astrophysics of star formation over cosmic time. For studying the detailed astrophysics of star formation at high-redshift, surveys at frequencies $\gtrsim$10 GHz have the distinct advantage over traditional $\sim$1.4 GHz surveys as they are able to yield higher angular resolution imaging while probing higher rest frame frequencies of galaxies with increasing redshift, where emission of star-forming galaxies becomes dominated by thermal (free-free) radiation. In doing so, surveys carried out at $\gtrsim$10 GHz provide a robust, dust-unbiased measurement of the massive star formation rate by being highly sensitive to the number of ionizing photons that are produced. To access this powerful star formation rate diagnostic requires that Band~5 be available for SKA1-MID. We additionally present a detailed science case for frequency coverage extending up to 30 GHz during full SKA2 operations, as this allows for highly diverse science while additionally providing contiguous frequency coverage between the SKA and ALMA, which will likely be the two most powerful interferometers for the coming decades. To enable this synergy, it is crucial that the dish design of the SKA be flexible enough to include the possibility of being fit with receivers operating up to 30 GHz.