Variations in Integrated Galactic Initial Mass Functions due to Sampling
  Method and Cluster Mass Function
  [abridged] Stars are thought to be formed predominantly in clusters. The
clusters are formed following a cluster initial mass function (CMF) similar to
the stellar initial mass function (IMF). Both the IMF and the CMF favour
low-mass objects. The numerous low-mass clusters will lack high mass stars. If
the integrated galactic initial mass function originates from stars formed in
clusters, the IGIMF could be steeper than the IMF. We investigate how well
constrained this steepening is and how it depends on the choice of sampling
method and CMF. We compare analytic sampling to several implementations of
random sampling of the IMF, and different CMFs. We implement different IGIMFs
into GALEV to obtain colours and metallicities for galaxies. Choosing different
ways of sampling the IMF results in different IGIMFs. Depending on the lower
cluster mass limit and the slope of the cluster mass function, the steepening
varies between very strong and negligible. We find the size of the effect is
continuous as a function of the power-law slope of the CMF, if the CMF extends
to masses smaller than the maximum stellarmass. The number of O-stars detected
by GAIA might help in judging on the importance of the IGIMF effect. The impact
of different IGIMFs on integrated galaxy photometry is small, within the
intrinsic scatter of observed galaxies. Observations of gas fractions and
metallicities could rule out at least the most extreme sampling methods. As we
still do not understand the details of star formation, one sampling method
cannot be favoured over another. Also, the CMF at very low cluster masses is
not well constrained observationally. These uncertainties need to be taken into
account when using an IGIMF, with severe implications for galaxy evolution
models and interpretations of galaxy observations.
