Spatial Power Spectra of Dust across the Local Group: No Constraint on
  Disc Scale Height
  We analyze the 1D spatial power spectra of dust surface density and mid to
far-infrared emission at $24-500\,\mu$m in the LMC, SMC, M31, and M33. By
forward-modelling the point-spread-function (PSF) on the power spectrum, we
find that nearly all power spectra have a single power-law and point source
component. A broken power-law model is only favoured for the LMC 24 $\mu$m MIPS
power spectrum and is due to intense dust heating in 30 Doradus. We also test
for local power spectrum variations by splitting the LMC and SMC maps into
$820$ pc boxes. We find significant variations in the power-law index with no
strong evidence for breaks. The lack of a ubiquitous break suggests that the
spatial power spectrum does not constrain the disc scale height. This
contradicts claims of a break where the turbulent motion changes from 3D to 2D.
The power spectrum indices in the LMC, SMC, and M31 are similar ($2.0-2.5$).
M33 has a flatter power spectrum ($1.3$), similar to more distant spiral
galaxies with a centrally-concentrated H$_{2}$ distribution. We compare the
power spectra of HI, CO, and dust in M31 and M33 and find that HI power spectra
are consistently flatter than CO power spectra. These results cast doubt on the
idea that the spatial power spectrum traces large scale turbulent motion in
nearby galaxies. Instead, we find that the spatial power spectrum is influenced
by (1) the PSF on scales below $\sim3$ times the FWHM, (2) bright compact
regions (30 Doradus), and (3) the global morphology of the tracer (an
exponential CO disc).
