*4.1. Distance Indicators*

At the beginning of our review, we emphasized how distances in cosmology are relevant to compute GRB luminosity/energy. A further step consists of noticing the distance measurements are classifiable by


Standard cosmology shows how to relate the redshift to metric distances in both of the above cases. The machinery of dynamical distance indicators involves tightly packing all the ingredients of cosmological physics. We thus require the cosmological principle to hold in an expanding universe in the context of general relativity. Despite it being obvious, there is no direct analogy to classical dynamical distance indicators, as the laboratory in which measurements are obtained is moving as well. Precision cosmology would enrich data during the incoming years, as future surveys will provide resources of data to constrain and refine our understanding about distances and cosmological parameters.

Using current data catalogs, it appears evident that GRBs can be significantly investigated once the calibration of the correlation functions are deduced from absolute confidence. Recently, techniques of non-calibration have been more often used, overcoming the problem of standardizing GRBs that are, as known, not perfect standard candles for cosmological distance tests. Later on, we confront the calibration and non-calibration procedures, emphasizing how to single out the most promising treatment to handle GRBs in cosmology.

#### *4.2. Standard Candles*

Above, we stated astrophysical distances are crucial for picturing the current universe. Though essential, estimating cosmic distances mainly remains a complicated prerogative. In view of the above classification, the distance estimation passes through the use of *standard candles*. These objects hold the fundamental property of relating the intrinsic luminosity, namely *L*, to some known property, enabling one to get constraints over it. Once the luminosity is known, the distance can be computed accordingly.

A standard procedure is to get measures of the energy emitted from astrophysical objects. The energy bounds are obtained in a precise time interval, say ∆*t* and by virtue of *E* = *L* · ∆*t*, i.e., the relation between luminosity and energy, it is possible to get distances from the energy itself, through a well-consolidated strategy, reported below.

Detectors are able to catch fractions *E<sup>d</sup>* of the emitted energy *E*, which is proportional to the ratio between the detector area *A* and the spherical shell 4*πd* 2 *L* in which one defines the cosmic distance *dL*, i.e.,

$$E\_{\mathbf{d}} = \frac{EA}{4\pi d\_L^2}.\tag{19}$$

A general relation for *dL*(*z*) is written as

$$d\_L(z, \theta) = c \left(1 + z\right) \int\_0^z \frac{\mathrm{d}z'}{H(z', \theta)}\tag{20}$$

where the set of free parameters to constrain is indicated by *θ*. Exploring a given cosmological model is equivalent to obtaining *θ*.

Thereby, combining the aforementioned quantities, we obtain the energy per unit detector area *A* and per unitary time ∆*t*, which defines the flux expressed by

$$F = \frac{E\_\mathrm{d}}{A\,\Delta t} = \frac{L}{4\pi d\_\mathrm{L}^2}.\tag{21}$$

As we highlighted, the luminosity *L* is known for standard candles, thus one can measure *F* in order to get a given astrophysical object distance.
