  We use the HiZELS narrow-band H-alpha survey in combination with CANDELS,
UKIDSS and WIRDS near-infrared imaging, to investigate the morphologies, merger
rates and sizes of a sample of H-alpha emitting galaxies in the redshift range
z=0.40 - 2.23, an epoch encompassing the rise to the peak of the star formation
rate density. Merger rates are estimated from space- and ground-based imaging
using the M20 coefficient. To account for the increase in the specific
star-formation rate (sSFR) of the star forming `main-sequence' with redshift,
we normalise the star-formation rate of galaxies at each epoch to the typical
value derived from the H-alpha luminosity function. Once this trend in sSFR is
removed we see no evidence for an increase in the number density of
star-forming galaxies or the merger rate with redshift. We thus conclude that
neither is the main driver of the enhanced star-formation rate density at
z=1-2, with secular processes such as instabilities within efficiently fuelled,
gas-rich discs or multiple minor mergers the most likely alternatives. However,
we find that 40-50% of starburst galaxies, those with enhanced specific star
formation at their epoch, are major mergers and this fraction is redshift
independent. Finally, we find the surprising result that the typical size of a
star-forming galaxy of a given mass does not evolve across the redshift range
considered, suggesting a universal size-mass relation. Taken in combination,
these results indicate a star-forming galaxy population that is statistically
similar in physical size, merger rate and mass over the 6 Gyr covered in this
study, despite the increase in typical sSFR.
