Tracing galaxy populations through cosmic time: A critical test of
  methods for connecting the same galaxies between different redshifts at $z <
  3$
  Connecting galaxies with their descendants (or progenitors) at different
redshifts can yield strong constraints on galaxy evolution. Observational
studies have historically selected samples of galaxies using a physical
quantity, such as stellar mass, either above a constant limit or at a constant
cumulative number density. Investigation into the efficacy of these selection
methods has not been fully explored. Using a set of four semi-analytical models
based on the output of the Millennium Simulation, we find that selecting
galaxies at a constant number density (in the range $-4.3 < \log\ n\
[\mathrm{Mpc}^{-3}\ h^{3}] < -3.0$) is superior to a constant stellar mass
selected sample, although it still has significant limitations. Recovery of the
average stellar mass, stellar mass density and average star-formation rate is
highly dependent on the choice of number density but can all be recovered to
within $<50\%$ at the commonly employed choice of $\log\ n\ [\mathrm{Mpc}^{-3}\
h^{3}] = -4.0$, corresponding to $\log M_\odot / h \sim 11.2$ at $z=0$, but
this increases at lower mass limits. We show that there is a large scatter
between the location of a given galaxy in a rank ordering based on stellar mass
between different redshifts. We find that the inferred velocity dispersion may
be a better tracer of galaxy properties, although further investigation is
warranted into simulating this property. Finally, we find that over large
redshift ranges selection at a constant number density is more effective in
tracing the progenitors of modern galaxies than vice-versa.
