Spatial power spectra of dust across the Local Group: No constraint on disc scale height

We analyse 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 similar to 3 times the FWHM, (2) bright compact regions (30 Doradus), and (3) the global morphology of the tracer (an exponential CO disc).