  There is ample observational evidence that the star formation rate (SFR)
surface density, Sigma_SFR, is closely correlated with the surface density of
molecular hydrogen, Sigma_H2. This empirical relation holds both for
galaxy-wide averages and for individual >=kpc sized patches of the interstellar
medium (ISM), but appears to degrade substantially at a sub-kpc scale.
Identifying the physical mechanisms that determine the scale-dependent
properties of the observed Sigma_H2-Sigma_SFR relation remains a challenge from
a theoretical perspective. To address this question, we analyze the slope and
scatter of the Sigma_H2-Sigma_SFR relation using a set of cosmological, galaxy
formation simulations with a peak resolution of ~100 pc. These simulations
include a chemical network for molecular hydrogen, a model for the CO emission,
and a simple, stochastic prescription for star formation that operates on ~100
pc scales. Specifically, star formation is modeled as a Poisson process in
which the average SFR is directly proportional to the present mass of H2. The
predictions of our numerical model are in good agreement with the observed
Kennicutt-Schmidt and Sigma_H2-Sigma_SFR relations. We show that observations
based on CO emission are ill suited to reliably measure the slope of the latter
relation at low (<20 M_sun pc^-2) H2 surface densities on sub-kpc scales. Our
models also predict that the inferred Sigma_H2-Sigma_SFR relation steepens at
high H2 surface densities as a result of the surface density dependence of the
CO/H2 conversion factor. Finally, we show that on sub-kpc scales most of the
scatter in the relation is a consequence of discreteness effects in the star
formation process. In contrast, variations of the CO/H2 conversion factor are
responsible for most of the scatter measured on super-kpc scales.
