In an expanding Universe, light waves get stretched, increasing their wavelength and shifting them to become more and more red (“redshift”).
This is equivalent to the Doppler shift of an object moving away from us, so we often describe this as the recessional velocity. The redshift can be measured by taking the light of the object and spreading it into a rainbow to measure the amount of light at each separate wavelength. Because we know the chemical fingerprint of elements in galaxies, which includes elements such as hydrogen and oxygen, we know that they emit light at certain wavelengths. In the observed spectra – the range of observed wavelengths of light – we measure the light at redder wavelengths and measure the amount of the redshift.
Importantly, the further the galaxy is from us, the greater the redshift and the larger the inferred recessional velocity. For nearby objects, this is the famous Hubble law: Velocity is proportional to distance. However, as one looks back to galaxies at much earlier periods of time, the relationship between distance and redshift becomes more complicated. This is directly related to the expansion history of the universe. If the universe has more matter in it then the expansion slows down over time, and high-redshift objects appear closer. Conversely, if the dark energy is causing the expansion to accelerate, then the high-redshift objects appear further away. It was the detection of this effect in the brightness of high-redshift supernovae that led to the discovery of dark energy and earned the 2011 Nobel Prize in physics. And it is this detailed relationship of distance and redshift that is the primary science driver for DESI.