A test of cosmic distance duality relation using SPT-SZ galaxy clusters,
  Type Ia supernovae, and cosmic chronometers
  We carry out a test of the cosmic distance duality relation using a sample of
52 SPT-SZ clusters, along with X-ray measurements from XMM-Newton. To carry out
this test, we need an estimate of the luminosity distance ($D_L$) at the
redshift of the cluster. For this purpose, we use three independent methods:
directly using $D_L$ from the closest Type Ia Supernovae from the Union 2.1
sample, non-parametric reconstruction of $D_L$ using the same Union 2.1 sample,
and finally using $H(z)$ measurements from cosmic chronometers and
reconstructing $D_L$ using Gaussian Process regression. We use four different
functions to characterize the deviations from CDDR. All our results for these
($4 \times 3$) analyses are consistent with CDDR to within 1$\sigma$.
