Cosmic distance ladder

The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth. The techniques for determining distances to more distant objects are all based on various measured correlations between methods that work at close distances and methods that work at larger distances. Several methods rely on a standard candle, which is an astronomical object that has a known luminosity.
The ladder analogy arises because no single technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distances at the next higher rung.

Direct measurement

At the base of the ladder are fundamental distance measurements, in which distances are determined directly, with no physical assumptions about the nature of the object in question. The precise measurement of stellar positions is part of the discipline of astrometry. Early fundamental distances—such as the radii of the earth, moon and sun, and the distances between them—were well estimated with very low technology by the ancient Greeks.

Astronomical unit

Direct distance measurements are based upon the astronomical unit, which is equal to and historically was conceived as the mean distance between the Earth and the Sun.
Kepler's laws provide precise ratios of the orbit sizes of objects orbiting the Sun, but provide no measurement of the overall scale of the orbit system. Radar is used to measure the distance between the orbits of the Earth and of a second body. From that measurement and the ratio of the two orbit sizes, the size of Earth's orbit is calculated. The Earth's orbit is known with an absolute precision of a few meters and a relative precision of a few parts in 100 billion.
Historically, observations of Venus transits were crucial in determining the AU; in the first half of the 20th century, observations of asteroids were also important. Presently the orbit of Earth is determined with high precision using radar measurements of distances to Venus and other nearby planets and asteroids, and by tracking interplanetary spacecraft in their orbits around the Sun through the Solar System.

Parallax

Standard candles

Almost all astronomical objects used as physical distance indicators belong to a class that has a known brightness. By comparing this known luminosity to an object's observed brightness, the distance to the object can be computed using the inverse-square law. These objects of known brightness are termed standard candles, coined by Henrietta Swan Leavitt.
The brightness of an object can be expressed in terms of its absolute magnitude. This quantity is derived from the logarithm of its luminosity as seen from a distance of 10 parsecs. The apparent magnitude, the magnitude as seen by the observer, can be measured and used with the absolute magnitude to calculate the distance d to the object in parsecs as follows:
or
where m is the apparent magnitude, and M the absolute magnitude. For this to be accurate, both magnitudes must be in the same frequency band and there can be no relative motion in the radial direction.
Some means of correcting for interstellar extinction, which also makes objects appear fainter and more red, is needed, especially if the object lies within a dusty or gaseous region. The difference between an object's absolute and apparent magnitudes is called its distance modulus, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.

Problems

Two problems exist for any class of standard candle. The principal one is calibration, that is the determination of exactly what the absolute magnitude of the candle is. This includes defining the class well enough that members can be recognized, and finding enough members of that class with well-known distances to allow their true absolute magnitude to be determined with enough accuracy. The second problem lies in recognizing members of the class, and not mistakenly using a standard candle calibration on an object which does not belong to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.
A significant issue with standard candles is the recurring question of how standard they are. For example, all observations seem to indicate that Type Ia supernovae that are of known distance have the same brightness, corrected by the shape of the light curve. The basis for this closeness in brightness is discussed below; however, the possibility exists that the distant Type Ia supernovae have different properties than nearby Type Ia supernovae. The use of Type Ia supernovae is crucial in determining the correct cosmological model. If indeed the properties of Type Ia supernovae are different at large distances, i.e. if the extrapolation of their calibration to arbitrary distances is not valid, ignoring this variation can dangerously bias the reconstruction of the cosmological parameters, in particular the reconstruction of the matter density parameter.
That this is not merely a philosophical issue can be seen from the history of distance measurements using Cepheid variables. In the 1950s, Walter Baade discovered that the nearby Cepheid variables used to calibrate the standard candle were of a different type than the ones used to measure distances to nearby galaxies. The nearby Cepheid variables were population I stars with much higher metal content than the distant population II stars. As a result, the population II stars were actually much brighter than believed, and when corrected, this had the effect of doubling the estimates of distances to the globular clusters, the nearby galaxies, and the diameter of the Milky Way.
Most recently kilonovae have been proposed as another type of standard candle. "Since kilonovae explosions are spherical, astronomers could compare the apparent size of a supernova explosion with its actual size as seen by the gas motion, and thus measure the rate of cosmic expansion at different distances."

Standard siren

s originating from the inspiral phase of compact binary systems, such as neutron stars or black holes, have the useful property that energy emitted as gravitational radiation comes exclusively from the orbital energy of the pair, and the resultant shrinking of their orbits is directly observable as an increase in the frequency of the emitted gravitational waves. To leading order, the rate of change of frequency is given by
where is the gravitational constant, is the speed of light, and is a single number called the chirp mass of the system, a combination of the masses of the two objects
By observing the waveform, the chirp mass can be computed and thence the power of the gravitational waves. Thus, such a gravitational wave source is a standard siren of known loudness.
Just as with standard candles, given the emitted and received amplitudes, the inverse-square law determines the distance to the source. There are some differences with standard candles, however. Gravitational waves are not emitted isotropically, but measuring the polarisation of the wave provides enough information to determine the angle of emission. Gravitational wave detectors also have anisotropic antenna patterns, so the position of the source on the sky relative to the detectors is needed to determine the angle of reception.
Generally, if a wave is detected by a network of three detectors at different locations, the network will measure enough information to make these corrections and obtain the distance. Also unlike standard candles, gravitational waves need no calibration against other distance measures. The measurement of distance does of course require the calibration of the gravitational wave detectors, but then the distance is fundamentally given as a multiple of the wavelength of the laser light being used in the gravitational wave interferometer.
There are other considerations that limit the accuracy of this distance, besides detector calibration. Fortunately, gravitational waves are not subject to extinction due to an intervening absorbing medium. But they are subject to gravitational lensing, in the same way as light. If a signal is strongly lensed, then it might be received as multiple events, separated in time, the analogue of multiple images of a quasar, for example. Less easy to discern and control for is the effect of weak lensing, where the signal's path through space is affected by many small magnification and demagnification events. This will be important for signals originating at cosmological redshifts greater than 1. It is difficult for detector networks to measure the polarization of a signal accurately if the binary system is observed nearly face-on. Such signals suffer significantly larger errors in the distance measurement. Unfortunately, binaries radiate most strongly perpendicular to the orbital plane, so face-on signals are intrinsically stronger and the most commonly observed.
If the binary consists of a pair of neutron stars, their merger will be accompanied by a kilonova/hypernova explosion that may allow the position to be accurately identified by electromagnetic telescopes. In such cases, the redshift of the host galaxy allows a determination of the Hubble constant. This was the case for GW170817, which was used to make the first such measurement. Even if no electromagnetic counterpart can be identified for an ensemble of signals, it is possible to use a statistical method to infer the value of.

Standard ruler

Another class of physical distance indicator is the standard ruler. In 2008, galaxy diameters have been proposed as a possible standard ruler for cosmological parameter determination. More recently the physical scale imprinted by baryon acoustic oscillations in the early universe has been used.
In the early universe the baryons and photons scatter off each other, and form a tightly coupled fluid that can support sound waves. The waves are sourced by primordial density perturbations, and travel at speed that can be predicted from the baryon density and other cosmological parameters.
The total distance that these sound waves can travel before recombination determines a fixed scale, which simply expands with the universe after recombination. BAO therefore provide a standard ruler that can be measured in galaxy surveys from the effect of baryons on the clustering of galaxies. The method requires an extensive galaxy survey in order to make this scale visible, but has been measured with percent-level precision. The scale does depend on cosmological parameters like the baryon and matter densities, and the number of neutrinos, so distances based on BAO are more dependent on cosmological model than those based on local measurements.
Light echos can be also used as standard rulers, although it is challenging to correctly measure the source geometry.