We present high-precision experimental and numerical studies of the Nusselt number $Nu$ as functions of the Rayleigh number $Ra$ in geostrophic rotating convection with domain aspect ratio ${Gamma}$ varying from 0.4 to 3.8 and the Ekman number Ek from $2.0{times}10^{-7}$ to $2.7{times}10^{-5}$. The heat-transport data $Nu(Ra)$ reveal a gradual transition from buoyancy-dominated to geostrophic convection at large $Ek$, whereas the transition becomes sharp with decreasing $Ek$. We determine the power-law scaling of $Nu{sim}Ra^{gamma}$, and show that the boundary flows give rise to pronounced enhancement of $Nu$ in a broad range of the geostrophic regime, leading to reduction of the scaling exponent ${gamma}$ in small ${Gamma}$ cells. The present work provides new insight into the heat-transport scaling in geostrophic convection and may explain the discrepancies observed in previous studies.
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