What are Protoclusters? -- Defining High Redshift Galaxy Clusters and
  Protoclusters
  We explore the structures of protoclusters and their relationship with high
redshift clusters using the Millennium Simulation combined with a semi-analytic
model. We find that protoclusters are very extended, with 90 per cent of their
mass spread across $\sim35\,h^{-1}{\rm Mpc}$ comoving at $z=2$ ($\sim30\,
\rm{arcmin}$). The `main halo', which can manifest as a high redshift cluster
or group, is only a minor feature of the protocluster, containing less than 20
per cent of all protocluster galaxies at $z=2$. Furthermore, many protoclusters
do not contain a main halo that is massive enough to be identified as a high
redshift cluster. Protoclusters exist in a range of evolutionary states at high
redshift, independent of the mass they will evolve to at $z=0$. We show that
the evolutionary state of a protocluster can be approximated by the mass ratio
of the first and second most massive haloes within the protocluster, and the
$z=0$ mass of a protocluster can be estimated to within 0.2 dex accuracy if
both the mass of the main halo and the evolutionary state is known. We also
investigate the biases introduced by only observing star-forming protocluster
members within small fields. The star formation rate required for line-emitting
galaxies to be detected is typically high, which leads to the artificial loss
of low mass galaxies from the protocluster sample. This effect is stronger for
observations of the centre of the protocluster, where the quenched galaxy
fraction is higher. This loss of low mass galaxies, relative to the field,
distorts the size of the galaxy overdensity, which in turn can contribute to
errors in predicting the $z=0$ evolved mass.
