Gauging fine-tuning

We introduce a mathematical framework for quantifying fine-tuning in general physical settings. In particular, we identify two distinct perspectives on fine-tuning-namely, a local and a global perspective- and develop corresponding measures. These measures apply broadly to settings characterized by an arbitrary number of observables whose values are dependent on an arbitrary number of parameters. We illustrate our formalism by quantifying fine-tuning as it arises in two pertinent astrophysical settings: (i) in models where a significant fraction of the dark matter in the Universe is in the form of primordial black holes and (ii) in scenarios that derive the fraction of protons in habitable dark-matter halos from underlying models of cosmic inflation.