Tracers of the ionization fraction in dense and translucent gas: I.
  Automated exploitation of massive astrochemical model grids
  The ionization fraction plays a key role in the physics and chemistry of the
neutral interstellar medium, from controlling the coupling of the gas to the
magnetic field to allowing fast ion-neutral reactions that drive interstellar
chemistry. Most estimations of the ionization fraction have relied on
deuterated species such as DCO+, whose detection is limited to dense cores
representing an extremely small fraction of the volume of the giant molecular
clouds they are part of. As large field-of-view hyperspectral maps become
available, new tracers may be found. We search for the best observable tracers
of the ionization fraction based on a grid of astrochemical models. We build
grids of models that sample randomly a large space of physical conditions
(unobservable quantities such as gas density, temperature, etc.) and compute
the corresponding observables (line intensities, column densities) and the
ionization fraction. We estimate the predictive power of each potential tracer
by training a Random Forest model to predict the ionization fraction from that
tracer, based on these model grids. In both translucent medium and cold dense
medium conditions, several observable tracers with very good predictive power
for the ionization fraction are found. Several tracers in cold dense medium
conditions are found to be better and more widely applicable than the
traditional DCO+/HCO+ ratio. We also provide simpler analytical fits for
estimating the ionization fraction from the best tracers, and for estimating
the associated uncertainties. We discuss the limitations of the present study
and select a few recommended tracers in both types of conditions. The method
presented here is very general and can be applied to the measurement of any
other quantity of interest (cosmic ray flux, elemental abundances, etc.) from
any type of model (PDR models, time-dependent chemical models, etc.).
(abridged)
