How to measure distances in the Universe?

The measurement of distances in our vast observable Universe is fundamental in astrophysics because it then becomes possible to estimate the intrinsic properties of the observed object. Unfortunately it is very difficult to obtain high precision. To measure the great cosmic distances of the universe, astronomers combine several calculation methods on ever more distant objects. The measurement of a nearby object must be done with confidence because the methods are interrelated and each measurement method allows the next one to be calibrated.
The image of the world around us is reconstructed in real time by our brain from two sets of information.
Our two eyes are spaced a few centimeters apart, so the image received by each of these receptors is slightly different. Indeed, for the observed object, each eye gives a position in relation to a more distant and immobile background. An object that does not have the same position relative to the background causes an optical effect called the parallax effect. It is this effect that allows our brain to reconstruct a three-dimensional image in order to apprehend the distance of the object.
In astronomy, the parallax is the angle at which the observed object is seen from two points far apart from each other (the more distant the observation points, the greater the angle of parallax is large, so easy to measure).
For the stars of the solar system, the equatorial radius of the Earth (6,378 km) was chosen. The equatorial parallax is the angle under which an observer located in the center of the observed object sees the terrestrial radius.

This angle directly gives us the distance of the object by a simple trigonometric calculation. For example, for an average distance from the Moon of 384,400 km, the lunar equatorial parallax is ≈57' and the apparent diameter of the Moon is ≈31' or ≈½ degree.
To measure the distance of stars in our Galaxy, the reference is the semi-major axis of Earth's orbit (149,597,870,700 meters or 1 Astronomical Unit). To measure this annual parallax, it suffices to observe the star twice, six months apart. In other words, astronomers measure the annual parallax angle by first measuring the position of a star and then measuring again, 6 months later, when the Earth is opposite in its orbit. However, it is not so easy to measure because the farther the star is from Earth, the weaker the parallax.
For example, for the closest star to the Sun, Proxima Centauri (Alpha Centauri C) located 4,244 light-years away, the parallax is only 0.7 arcseconds, which is tiny, there are 360 degrees in a circle, in each degree there are 60 minutes of arc and in each minute there are 60 seconds of arc. But the precision of the measurement from the Earth only makes it possible to measure a few stars located a few dozen light-years away.
From space, thanks to the Hipparcos satellite (HIgh Precision PARallax COllecting Satellite), the parallaxes of 120,000 stars can be measured with an accuracy of 2 to 4 milliseconds of arc. In 2013, the Gaia satellite took over from Hipparcos.
Gaia (Global Astrometric Interferometer for Astrophysics) has compiled a catalog of one billion stars with an accuracy of up to 10 microarcseconds.

To measure the distance of very distant stars, the parallax method is no longer possible because it is imprecise. Astronomers use another method, that of "standard candles".
In the 1910s at Harvard University, Henrietta Leavitt (1868-1921) classified the Cepheids of the Magellanic Clouds (two neighboring dwarf galaxies of the Milky Way located between 150 and 200 light years). A Cepheid is a star whose brightness varies according to a well-defined period (between 1 and 135 days). It was in 1908 that the first standard candle was unexpectedly discovered, thanks to the intuition of Henrietta Leavitt. She realizes that the periods of the Cepheids are all the greater as they are brilliant. The intrinsic luminosity of classical Cepheids thus increases with their period. In other words, the bigger and brighter a Cepheid is, the slower its pulse rate. She finds a relationship linking the period of variation (time between two maxima or minima) to the apparent luminosity of these stars.

Thus, it suffices to measure the precise distance of one of these Cepheids with the parallax method, to obtain a general relation linking the periods and the absolute luminosities of the Cepheids.
This measurement was carried out for the first time in 1916, at Harvard University by Harlow Shapley (1885-1972) who completed Henrietta Leavitt's discovery. From this date, the Cepheids whose intrinsic luminosity is known, become "standard candles" to measure the distance of distant stars and that of nearby galaxies.
The apparent luminosity of an object therefore depends on its absolute luminosity and its distance. It was by observing variable stars of the Cepheids type that the astronomer Edwin Hubble (1889-1953) measured in 1923 the distance of the spiral galaxy M31 (Andromeda, the closest to the Milky Way ), using the Mount Wilson Observatory Telescope near Pasadena, California.

For distant galaxies, telescopes can no longer distinguish their individual stars. Astronomers must rely on extremely bright objects as bright as a galaxy. The object used is the type 1a supernova.
Supernovae are rare events in our Milky Way, one to three per century, on the other hand on the scale of the universe, we observe them every day.
A type 1a supernova corresponds to the full explosion of a white dwarf having crossed the Chandrasekhar mass limit (when the radius of the star decreases, the mass tends towards a limit of 1.44 solar masses). Due to their physical properties, white dwarfs powered by matter from a close companion cannot exceed this limit.
The type 1a supernova SN 2014J is located in the Cigar Galaxy (Messier 82) at 11.5 ± 0.8 million light-years away.

This standard candle of cosmology will make it possible to determine extragalactic distances.
Because they always explode at the same point (limit mass), type supernovae 1a still have roughly the same intrinsic brightness after reaching their maximum brightness. Type 1a supernovae, very bright and visible at very long distances, therefore serve as standard candles.
To calibrate the method, it is necessary to use type 1a supernovae close enough to be measured by the Cepheid method. Scientists know of a few dozen type 1a supernovae that are close enough.
Type 1a supernovae will make it possible to measure the distance of distant galaxies up to several billion light-years.

For even more distant galaxies, astronomers use the method of redshift of the electromagnetic spectrum absorption lines.
This redshift is used to measure the travel time of light. The "older" the light, the greater the redshift.
Each chemical element or molecule leaves different marks on this spectrum. These traces appear at very specific wavelengths (absorption lines). But if a galaxy moves away from us, its light stretches and the wavelengths of these chemical fingerprints shift to red. This offset is related to the galaxy's distance by Hubble's law. This law states that the further away a galaxy is, the faster it is moving away from us as the universe expands. By measuring the redshift, astronomers were able to spot the first galaxies near the Big Bang.

Thus light reveals to us the distance of near and distant cosmic objects, astronomers now know how to calculate distances to the depths of the observable Universe (about 13 billion light-years).

nota: The redshift (redshift) is a shift towards long wavelengths of the spectral lines and of the entire spectrum of the visible domain. There is a correlation between distance and the redshift of the optical spectra of galaxies. Redshift is the most widely used method for measuring the distance of extragalactic objects. This phenomenon observed in the light of distant astronomical objects is proof of the expansion of the Universe (dilation of space) and of the Big Bang.