  We explore the assumption, widely used in many astrophysical calculations,
that the stellar initial mass function (IMF) is universal across all galaxies.
By considering both a canonical Salpeter-like IMF and a non-universal IMF, we
are able to compare the effect of different IMFs on multiple observables and
derived quantities in astrophysics. Specifically, we consider a non-universal
IMF which varies as a function of the local star formation rate, and explore
the effects on the star formation rate density (SFRD), the extragalactic
background light, the supernova (both core-collapse and thermonuclear) rates,
and the diffuse supernova neutrino background. Our most interesting result is
that our adopted varying IMF leads to much greater uncertainty on the SFRD at
$z \approx 2-4$ than is usually assumed. Indeed, we find a SFRD (inferred using
observed galaxy luminosity distributions) that is a factor of $\gtrsim 3$ lower
than canonical results obtained using a universal Salpeter-like IMF. Secondly,
the non-universal IMF we explore implies a reduction in the supernova
core-collapse rate of a factor of $\sim2$, compared against a universal IMF.
The other potential tracers are only slightly affected by changes to the
properties of the IMF. We find that currently available data do not provide a
clear preference for universal or non-universal IMF. However, improvements to
measurements of the star formation rate and core-collapse supernova rate at
redshifts $z \gtrsim 2$ may offer the best prospects for discernment.
