  A number of recent studies have proposed that the stellar initial mass
function (IMF) of early type galaxies varies systematically as a function of
galaxy mass, with higher mass galaxies having bottom heavy IMFs. These bottom
heavy IMFs have more low-mass stars relative to the number of high mass stars,
and therefore naturally result in proportionally fewer neutron stars and black
holes. In this paper, we specifically predict the variation in the number of
black holes and neutron stars based on the power-law IMF variation required to
reproduce the observed mass-to-light ratio trends with galaxy mass. We then
test whether such variations are observed by studying the field low-mass X-ray
binary populations (LMXBs) of nearby early-type galaxies. In these binaries, a
neutron star or black hole accretes matter from a low-mass donor star. Their
number is therefore expected to scale with the number of black holes and
neutron stars present in a galaxy. We find that the number of LMXBs per K-band
light is similar among the galaxies in our sample. These data therefore
demonstrate the uniformity of the slope of the IMF from massive stars down to
those now dominating the K-band light, and are consistent with an invariant
IMF. Our results are inconsistent with an IMF which varies from a
Kroupa/Chabrier like IMF for low mass galaxies to a steep power-law IMF (with
slope $x$=2.8) for high mass galaxies. We discuss how these observations
constrain the possible forms of the IMF variations and how future Chandra
observations can enable sharper tests of the IMF.
