  The validity of the anelastic approximation has recently been questioned in
the regime of rapidly-rotating compressible convection in low Prandtl number
fluids (Calkins et al. 2015). Given the broad usage and the high computational
efficiency of sound-proof approaches in this astrophysically relevant regime,
this paper clarifies the conditions for a safe application. The potential of
the alternative pseudo-incompressible approximation is investigated, which in
contrast to the anelastic approximation is shown to never break down for
predicting the point of marginal stability. Its accuracy, however, decreases
close to the parameters corresponding to the failure of the anelastic approach,
which is shown to occur when the sound-crossing time of the domain exceeds a
rotation time scale, i.e. for rotational Mach numbers greater than one.
Concerning the supercritical case, which is naturally characterised by smaller
rotational Mach numbers, we find that the anelastic approximation does not show
unphysical behaviour. Growth rates computed with the linearised anelastic
equations converge toward the corresponding fully compressible values as the
Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations,
produced with our fully compressible and anelastic models and carried out in a
highly supercritical, rotating, compressible, low Prandtl number regime show
good agreement. However, this nonlinear test example is for only a moderately
low convective Rossby number of 0.14.
