Measuring the expansion history of the Universe with cosmic chronometers

As revealed by Hubble in 1928, our Universe is expanding. This discovery was fundamental to widening our horizons and our conception of space, and since then determining the rate at which our Universe is expanding has become one of the crucial measurements in cosmology. At the beginning of this century, these measurements revealed the unexpected behavior that this expansion is accelerating and allowed us to have a first glimpse of the dark components that constitute $\sim$95\% of our Universe. Cosmic chronometers represent a novel technique to obtain a cosmology-independent determination of the expansion of the Universe, based on the differential age dating of a population of very massive and passively evolving galaxies. Currently, with this new cosmological probe it is possible to constrain the Hubble parameter with an accuracy of around 5\% at $z\sim0.5$ up to 10-20\% at $z\sim2$. In this Chapter, the cosmic chronometers approach is presented, describing the method and how an optimal sample can be selected; it is then discussed how the most recent measurements of the expansion history of the Universe have been obtained with this approach, as well as the cosmological constraints that can be derived. Particular attention will be given to the systematics involved in this approach and the treatment to properly take them into account. We conclude by presenting forecasts that show how future spectroscopic surveys will significantly boost the accuracy of this method and open the possibility to a percent determination of the Hubble constant, making cosmic chronometers a powerful independent tool to derive information on the expansion history of the Universe.