Probing the cosmic distance duality with strong gravitational lensing and supernovae Ia data

We propose and perform a new test of the cosmic distance-duality relation (CDDR), $D_L(z) / D_A(z) (1 + z)^{2} = 1$, where $D_A$ is the angular diameter distance and $D_L$ is the luminosity distance to a given source at redshift $z$, using strong gravitational lensing (SGL) and type Ia Supernovae (SNe Ia) data. We show that the ratio $D=D_{A_{12}}/D_{A_2}$ and $D^{*}=D_{L_{12}}/D_{L_{2}}$, where the subscripts 1 and 2 correspond, respectively, to redshifts $z_1$ and $z_2$, are linked by $D/D^*=(1+z_1)^2$ if the CDDR is valid. We allow departures from the CDDR by defining two funcions for $\eta(z_1)$, which equals unity when the CDDR is valid. We find that combination of SGL and SNe Ia data favours no violation of the CDDR at 1$\sigma$ confidence level ($\eta(z) \simeq 1$), in complete agreement with other tests and reinforcing the theoretical pillars of the CDDR.