Numerical convergence of hydrodynamical simulations of galaxy formation:
  the abundance and internal structure of galaxies and their cold dark matter
  haloes
  We address the issue of numerical convergence in cosmological smoothed
particle hydrodynamics simulations using a suite of runs drawn from the EAGLE
project. Our simulations adopt subgrid models that produce realistic galaxy
populations at a fiducial mass and force resolution, but systematically vary
the latter in order to study their impact on galaxy properties. We provide
several analytic criteria that help guide the selection of gravitational
softening for hydrodynamical simulations, and present results from runs that
both adhere to and deviate from them. Unlike dark matter-only simulations,
hydrodynamical simulations exhibit a strong sensitivity to gravitational
softening, and care must be taken when selecting numerical parameters. Our
results--which focus mainly on star formation histories, galaxy stellar mass
functions and sizes--illuminate three main considerations. First, softening
imposes a minimum resolved escape speed, $v_\epsilon$, due to the binding
energy between gas particles. Runs that adopt such small softening lengths that
$v_\epsilon \gt 10\,{\rm km s^{-1}}$ (the sound speed in ionised $\sim
10^4\,{\rm K}$ gas) suffer from reduced effects of photo-heating. Second,
feedback from stars or active galactic nuclei may suffer from numerical
over-cooling if the gravitational softening length is chosen below a critical
value, $\epsilon_{\rm eFB}$. Third, we note that small softening lengths
exacerbate the segregation of stars and dark matter particles in halo centres,
often leading to the counter-intuitive result that galaxy sizes {\em increase}
as softening is reduced. The structure of dark matter haloes in hydrodynamical
runs respond to softening in a way that reflects the sensitivity of their
galaxy populations to numerical parameters.
