Measurement of the distance to extragalactic objects is an important task. For objects at large distances the measurement of the redshift was frequently considered as the measurement of the distance but precision cosmology requires the knowledge of both the distance and the redshift.
The measurement of the distances to extragalactic objects was for years based on a carefully set subsequent steps of the cosmic ladder: direct distance measurement methods allowed to formulate scaling relations which then could be applied to more distant objects, and with a few such steps cosmological distances were achieved. The fundamental direct method of measuring the distance in astronomy is to measure the trigonometric parallax of a given source, reflecting the motion of the Earth around the Sun. Hipparcos mission measured parallaxes up to a few hundred parsecs, Hubble Space Telescope reached up to 5 kpc in dedicated observations. Now Gaia mission is collecting the data. It will cover with precisions of 10% a volume of radius 10 kpc, which encompasses < 1% of the stars in our Galaxy by number, but by volume is very large and will cover a large fraction of the Milky Way with reasonable accuracy. Parallaxes of the Cepheid stars, combined with measurements in nearby galaxies, calibrate the Leavitt Law for Cepheids, and Cepheids are accessible to 30 Mpc with the Hubble Space Telescope. Further out, SNe Ia calibrated with the help of the Cepheids take over, covering cosmological distances up to the redshift ∼ 2. This method replaced older methods of covering most distant part of the ladder, like Tully-Fisher relation.
Modern astronomy still strongly relies on indirect measurements which in turn rely in some intermediate steps but in the aim to meet expectations of the precision cosmology the intermediate steps are refined, new shortcut developed and some of the basically indirect methods might be potentially used as direct methods under some physically well motivated assumptions. In the Local Group the use of the eclipsing binary stars allows to skip the step of the parallax measurement. A very important direct distance measurement is possible for the spatially resolved water maser in the galaxy NGC 4258. The impressive jump directly to very large distances opened with observations of gravitational lensing of quasars and the measurement of the time delays between the quasar images which allows their use as a ruler. Another direct method can be applied to galaxy clusters. This method uses the Sunyeav-Zeldovich effect. Extremely important direct methods are the Barion Acoustic Oscillation method and the observations of the fluctuations of the Cosmic Microwave Background. In both cases the (statistical) analysis of the sizes of the fluctuations allows to get directly the cosmological constraints. Another statistical method to obtain the cosmological constraints is gravitational lensing. Now, with firm detection of gravitational waves, the possibility of the direct determination of the distance to the gravitational wave source from the shape of its signal also gains importance. All these methods are described below.
We do not follow the strict definition of indirect method but in this section we concentrate on predominantly indirect methods which can be applied to single specific extragalactic objects like supernovae, galaxies, galaxy clusters, active galaxies or gamma-ray bursts. Global statistical methods, including the results based on the Cosmic Microwave Background, will be discussed in a separate Chapter. We also do not include in this Chapter strong gravitational lensing since it will be more natural to address this issue in the Chapter where weak lensing is discussed.
We outline the methods shortly, with references to the basic books or reviews, and we concentrate on most recent developments in the field. Whenever possible, we discuss the issue of the method cross-calibrations.
The order of the methods' presentation is roughly adjusted to their past and current popularity and role in astronomy, but not necessarily reflects their potential for development in the near future.