Probing the distance-duality relation with high-$z$ data
  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), $D_L(z)/[(1+z)^2D_A(z)]=\eta=1$, 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:
$\eta(z)=1+\eta_0z$, $\eta(z)=1+\eta_0z/(1+z)$, $\eta(z)=(1+z)^{\eta_0}$ 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.
