Gamma-rays bursts are the brightest individual sources in the Universe (see e.g., Hjorth and Bloom 2012, Berger 2014 for recent reviews), therefore it is more than natural to attempt to use them in cosmology. Those intense flashes of the gamma-ray emission lasting from milliseconds to several hours are not easily affected by the medium between an observer and the source, so they can be easily detected at very high redshifts. The record holder is the gamma-ray burst GRB 090429B at z = 9.4 (Cucchiara et al, 2011), with photometric redshift determination, or GRB 090429B at z = 8.2 (Tanvir et al, 2009), if we limit ourselves to spectroscopically confirmed redshift. Gamma-ray prompted emission is accompanied by afterglow emission at other wavelengths.
However, due to the large burst variety and the critical role of the relativistic boosting of the part of the observed multi-wavelength emission the use of the gamma-ray bursts as distance indicators is rather difficult. The first attempts based on relatively large number of sources was made by Schaefer (2007) who successfully constructed the Hubble diagram in a broad redshift range almost up to z ∼ 7. The method was based on empirical correlations established for sources at z < 2 and then extended to higher redshifts.
Several correlations can be used for that purpose (see e.g., Wang et al, 2015). However, there are two basic problems which are not easy to overcome. First, there is some circularity problem, as there are no low redshift gamma-ray bursts and the scaling of the relations has to be done for intermediate redshift sources where cosmological effects already become important. Second, the number of gamma-ray bursts with well measured parameters is still low, and it increases only slowly with time. Samples with high quality data and low scatter in the measured quantities contain less than 50 objects so far (e.g., Dainotti et al, 2016). In addition, the results can be affected by the weak lensing, and the appropriate statistical corrections should be introduced, as discussed by Wang and Dai (2011).