A dim light leaves two possibilities
A dim porch lamp might be weak or far away. A photograph alone does not distinguish the two. For a nearby star, observers can compare its apparent direction from different parts of Earth’s orbit. The small annual shift gives a geometric distance. That measured distance then tells us how luminous the star really is.
The next step needs objects that can be recognized in more distant galaxies. Cepheids are stars that repeatedly expand and shrink. Their brightness rises and falls with a measurable period. Longer-period Cepheids tend to be more luminous. Once nearby or geometrically anchored Cepheids set the brightness of this relation, the same relation estimates distances elsewhere.
Reading: Riess et al. · Yuan et al.
A galaxy can contain both kinds of signpost
Some galaxies contain measured Cepheids and have also hosted a Type Ia supernova. The Cepheids estimate the galaxy distance. That distance turns the supernova’s apparent peak brightness into a calibrated luminosity. Astronomers correct supernova brightnesses for measured light-curve shape and color before making this comparison. The useful objects are standardized candles, a name for lights whose remaining brightness differences have been accounted for statistically.
Supernovae are bright enough to be seen much farther away. In that farther sample, a measured redshift gives the recession information and a calibrated brightness gives a distance. Their relation determines the present expansion rate, the Hubble constant. It is expressed as kilometres per second of recession per megaparsec of distance. A megaparsec is about 3.26 million light-years.
Reading: Riess et al.
Change the calibration and watch the direction of the effect
Suppose the adopted geometric anchor is made farther away, although all photographed brightnesses remain fixed. Its Cepheids must then be intrinsically brighter. The Cepheids seen in other galaxies must also be placed farther away to match their unchanged brightness. The calibrated supernovae become brighter too, and the farther supernova distances grow. At the same redshift, a larger distance means a smaller inferred expansion rate.
A tenth of a magnitude is a modest brightness-calibration shift. Applied as a difference between the anchor and the supernova-host Cepheids, it changes the inferred distance scale by about 4.7 percent and the Hubble constant in the opposite direction by about 4.5 percent. The frozen small example used here moves from 66.3 to 63.3. Its broad uncertainty is part of the example, so these numbers should be read as a calibration demonstration.
Reading: Riess et al. · PantheonPlusSH0ES DataRelease
A common error behaves differently from a difference
Now shift every Cepheid brightness by the same amount, both in the anchor and in the supernova hosts. The calibrated Cepheid luminosity shifts along with them. Their differences stay the same, and the inferred host distances and expansion rate stay the same in this model. The location of a possible error matters as much as its size.
The uncertainty of the geometric anchor is shared by every galaxy that uses it. Taking many more Cepheids can sharpen their relative comparison, but those stars all inherit the same uncertain starting length. Averaging many copies of one ruler does not supply a new ruler. A modern fit carries these correlations through all the distances, keeping the common uncertainty attached to the whole chain.
Reading: Riess et al. · Reid
Different stars can test parts of the same chain
The tip of the red giant branch offers another route. Old red giant stars reach a sharp upper edge in brightness before helium ignition changes their evolution. Measuring that edge in a galaxy estimates its distance after the edge luminosity is calibrated. It uses a different stellar population and different crowding and extinction sensitivities from Cepheids.
The routes can still share a geometric anchor and the same distant supernova sample. Their agreement then checks the changed middle part of the chain especially well. The shared beginning and end remain correlated. Reid, Pesce and Riess used the NGC4258 maser distance for just such a comparison in their 2019 paper. The next chapters turn these shared and different pieces into quantities we can calculate.
Reading: Reid · Riess et al.