Test of the cosmic distance duality relation for arbitrary spatial
  curvature
  The cosmic distance duality relation (CDDR), eta(z)=(1+z)^2 d_A(z)/d_L(z)=1,
is one of the most fundamental and crucial formulae in cosmology. This relation
couples the luminosity and angular diameter distances, two of the most often
used measures of structure in the Universe. We here propose a new
model-independent method to test this relation, using strong gravitational
lensing (SGL) and the high-redshift quasar Hubble diagram reconstructed with a
Bezier parametric fit. We carry out this test without pre-assuming a zero
spatial curvature, adopting instead the value Omega_K=0.001 +/- 0.002 optimized
by Planck in order to improve the reliability of our result. We parametrize the
CDDR using eta(z)=1 + eta_0 z, 1 + eta_1 z + eta_2 z^2 and 1 + eta_3 z/(1+z),
and consider both the SIS and non-SIS lens models for the strong lensing. Our
best fit results are: eta_0=-0.021^{+0.068}_{-0.048},
eta_1=-0.404^{+0.123}_{-0.090}, eta_2=0.106^{+0.028}_{-0.034}, and
eta_3=-0.507^{+0.193}_{-0.133} for the SIS model, and
eta_0=-0.109^{+0.044}_{-0.031} for the non-SIS model. The measured eta(z),
based on the Planck parameter Omega_K, is essentially consistent with the value
(=1) expected if the CDDR were fully respected. For the sake of comparison, we
also carry out the test for other values of Omega_K, but find that deviations
of spatial flatness beyond the Planck optimization are in even greater tension
with the CDDR. Future measurements of SGL may improve the statistics and alter
this result but, as of now, we conclude that the CDDR favours a flat Universe.
