The surface densities of molecular gas, $Sigma_{rm H_2}$, and the star formation rate (SFR), $dotSigma_star$, correlate almost linearly on kiloparsec scales in observed star-forming (non-starburst) galaxies. We explore the origin of the linear slope of this correlation using a suite of isolated $L_star$ galaxy simulations. We show that in simulations with efficient feedback, the slope of the $dotSigma_star$-$Sigma_{rm H_2}$ relation on kiloparsec scales is insensitive to the slope of the $dotrho_star$-$rho$ relation assumed at the resolution scale. We also find that the slope on kiloparsec scales depends on the criteria used to identify star-forming gas, with a linear slope arising in simulations that identify star-forming gas using a virial parameter threshold. This behavior can be understood using a simple theoretical model based on conservation of interstellar gas mass as the gas cycles between atomic, molecular, and star-forming states under the influence of feedback and dynamical processes. In particular, we show that the linear slope emerges when feedback efficiently regulates and stirs the evolution of dense, molecular gas. We show that the model also provides insights into the likely origin of the relation between the SFR and molecular gas in real galaxies on different scales.
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