The σ_s^2- relation in the multi-phase interstellar medium. Exploring the density distribution with the Cloud Factory simulations

The density probability distribution (PDF) of molecular clouds is a crucial component of analytical theories of star formation. In idealised simulations of isothermal turbulence, the width of the density PDF, σ_s^2, is dependent on the sonic Mach number of the medium, The σ_s^2- relation is widely used to connect cloud-scale turbulence to the density PDF, and further to star formation activity, yet its validity within individual phases of the multi-phase interstellar medium (ISM) remains untested. In this study, we evaluate whether the σ_s^2- relation is applicable to individual phases of the ISM. We study the density PDFs of molecular cloud complexes in the Cloud Factory simulations; a suite of detailed zoom-in simulations that self-consistently generate a turbulent, multi-phase ISM. We test whether the σ_s^2- relation holds in the hot ionised medium (HIM), warm ionised medium (WIM), warm neutral medium (WNM), cold neutral medium (CNM), the molecular phase, and the highly shielded molecular phase traced by CO. We find the applicability of the classical σ_s^2- relation to vary between phases and depend strongly on how σ_s^2 and are measured. The relation fails to capture the widths of the WNM and CNM density distributions, with possible contributing factors including non-isothermality and large-scale coherent motions. In contrast, we find the σ_s^2- relation to tentatively hold for the log-normal portion of the H_2 distribution. The width of the CO density PDF is systematically overpredicted by the classical relation, resulting from the selective nature of CO as a molecular gas tracer.