Measurements of strong gravitational lensing jointly with type Ia supernovae ( SNe Ia ) observations have been used to test the validity of the cosmic distance duality relation ( CDDR ) , D _ { L } ( z ) / [ ( 1 + z ) ^ { 2 } D _ { A } ( z ) ] = \eta = 1 , where D _ { L } ( z ) and D _ { A } ( z ) are the luminosity and the angular diameter distances to a given redshift z , respectively .
However , several lensing systems lie in the interval 1.4 \leq z \leq 3.6 i.e. , beyond the redshift range of current SNe Ia compilations ( z \approx 1.50 ) , which prevents this kind of test to be fully explored .
In this paper , we circumvent this problem by testing the CDDR considering observations of strong gravitational lensing along with SNe Ia and a subsample from the latest gamma-ray burst distance modulus data , whose redshift range is 0.033 \leq z \leq 9.3 .
We parameterize their luminosity distances with a second degree polynomial function and search for possible deviations from the CDDR validity by using four different \eta ( z ) functions : \eta ( z ) = 1 + \eta _ { 0 } z , \eta ( z ) = 1 + \eta _ { 0 } z / ( 1 + z ) , \eta ( z ) = ( 1 + z ) ^ { \eta _ { 0 } } and \eta ( z ) = 1 + \eta _ { 0 } \ln ( 1 + z ) .
Unlike previous tests done at redshifts lower than 1.50 , the likelihood for \eta _ { 0 } depends strongly on the \eta ( z ) function considered , but we find no significant deviation from the CDDR validity ( \eta _ { 0 } = 0 ) .
However , our analyses also point to the fact that caution is needed when one fits data in higher redshifts to test the CDDR as well as a better understanding of the mass distribution of lenses also is required for more accurate results .