Addressing the Hubble tension with cosmic chronometers

Twenty years after the discovery that the expansion of the Universe is accelerating, a new finding is now challenging our understanding of the cosmos. Recent studies have shown that the Hubble constant, the speed of expansion measured today, provides values in significant tension when measured from the Cosmic Microwave Background in the primordial Universe or from Cepheids and Supernovae Type Ia in the local Universe. Whether this tension is hinting towards new physics or some issue in the measurements, is still under debate; but it is clearly calling for new independent cosmological probes to provide additional pieces of evidence to solve this puzzle. This chapter introduces the method of cosmic chronometers, a new emerging cosmological probe that can provide cosmology-independent estimates of the Universe's expansion history. This method is based on the fact that the expansion rate of the Universe can be directly derived from measuring how much the Universe has changed in age between two different redshifts, i.e. by estimating the slope of the age--redshift relation. First, the main ingredients of the method will be discussed, presenting the main equations involved and how to estimate from the observables the needed quantities. After, it will be presented how to reliably select a sample of tracers to map the age evolution of the Universe coherently. Next, different methods to robustly measure the differential age of a population, the fundamental quantity involved in the method, will be reviewed. Finally, the main measurements obtained will be presented, providing forecasts for future surveys and discussing how these data can provide useful feedback to address the Hubble tension....