We use cosmological hydrodynamical galaxy formation simulations from the NIHAO project to investigate the impact of the threshold for star formation on the response of the dark matter (DM) halo to baryonic processes. The fiducial NIHAO threshold, $n=10, {rm cm}^{-3}$, results in strong expansion of the DM halo in galaxies with stellar masses in the range $10^{7.5} < M_{star} < 10^{9.5} M_{odot}$. We find that lower thresholds such as $n=0.1$ (as employed by the EAGLE/APOSTLE and Illustris/AURIGA projects) do not result in significant halo expansion at any mass scale. Halo expansion driven by supernova feedback requires significant fluctuations in the local gas fraction on sub-dynamical times (i.e., < 50 Myr at galaxy half-light radii), which are themselves caused by variability in the star formation rate. At one per cent of the virial radius, simulations with $n=10$ have gas fractions of $simeq 0.2$ and variations of $simeq 0.1$, while $n=0.1$ simulations have order of magnitude lower gas fractions and hence do not expand the halo. The observed DM circular velocities of nearby dwarf galaxies are inconsistent with CDM simulations with $n=0.1$ and $n=1$, but in reasonable agreement with $n=10$. Star formation rates are more variable for higher $n$, lower galaxy masses, and when star formation is measured on shorter time scales. For example, simulations with $n=10$ have up to 0.4 dex higher scatter in specific star formation rates than simulations with $n=0.1$. Thus observationally constraining the sub-grid model for star formation, and hence the nature of DM, should be possible in the near future.
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