PDFchem: A new fast method to determine ISM properties and infer environmental parameters using probability distributions

Determining the atomic and molecular content of the interstellar medium (ISM) as a function of environmental parameters is of fundamental importance to understand the star-formation process across the epochs. Although there exist various three-dimensional hydro-chemical codes modelling the ISM at different scales and redshifts, they are computationally expensive and inefficient for studies over a large parameter space. Building on our earlier approach, we present PDFchem, a novel algorithm that models the cold ISM at moderate and large scales using functions connecting the quantities of the local ($A_{\rm V,eff}$) and the observed ($A_{\rm V,obs}$) visual extinctions, and the local number density, $n_{\rm H}$, with probability density functions (PDF) of $A_{\rm V,obs}$ on cloud scales typically tens-to-hundreds of pc as an input. For any given $A_{\rm V,obs}$-PDF, sampled with thousands of clouds, the algorithm instantly computes the average abundances of the most important species (HI, H$_2$, CII, CI, CO, OH, OH$^+$, H$_2$O$^+$, CH, HCO$^+$) and performs radiative transfer calculations to estimate the average emission of the most commonly observed lines ([CII]~$158\mu$m, both [CI] fine-structure lines and the first five rotational transitions of $^{12}$CO). We examine two $A_{\rm V,obs}$-PDFs corresponding to a non star-forming and a star-forming ISM region, under a variety of environmental parameters combinations. These cover FUV intensities in the range of $\chi/\chi_0=10^{-1}-10^3$, cosmic-ray ionization rates in the range of $\zeta_{\rm CR}=10^{-17}-10^{-13}\,{\rm s}^{-1}$ and metallicities in the range of $Z=0.1-2\,{\rm Z}_{\odot}$. PDFchem is fast, easy to use, reproduces the PDR quantities of the time-consuming hydrodynamical models and can be used directly with observed data to understand the evolution of the cold ISM chemistry.