The Cosmic Mach Number as an Environment Measure for the Underlying Dark
  Matter Density Field
  Using cosmological dark matter only simulations of a $(1.6$ Gpc$/h)^3$ volume
from the Legacy simulation project, we calculate Cosmic Mach Numbers (CMN) and
perform a theoretical investigation of their relation with halo properties and
features of the density field to gauge their use as an measure of the
environment.
  CMNs calculated on individual spheres show correlations with both the
overdensity in a region and the density gradient in the direction of the bulk
flow around that region. To reduce the scatter around the median of these
correlations, we introduce a new measure, the rank ordered Cosmic Mach number
($\hat{\mathcal{M}}_g$), which shows a tight correlations with the overdensity
$\delta=\frac{\rho-\bar{\rho}}{\bar{\rho}}$. Measures of the large scale
density gradient as well as other average properties of the halo population in
a region show tight correlations with $\hat{\mathcal{M}}_g$ as well. Our
results in this first empirical study suggest that $\hat{\mathcal{M}}_g$ is an
excellent proxy for the underlying density field and hence environment that can
circumvent reliance on number density counts in a region. For scales between
$10$ and $100 Mpc$/h, Mach numbers calculated using dark matter halos $(>
10^{12}$ M$_{\odot})$ that would typically host massive galaxies are consistent
with theoretical predictions of the linear matter power spectrum at a level of
$10\%$ due to non-linear effects of gravity. At redshifts $z\geq 3$, these
deviations disappear. We also quantify errors due to missing large scale modes
in simulations. Simulations of box size $\leq 1 $ Gpc/$h$ typically predict
CMNs 10-30\% too small on scales of$\sim 100$ Mpc$/h$.
