tokens: Einstein radius, and σ SIS is the velocity dispersion of the star in the periphery of the lens galaxy due to the lens mass distribution in the SIS model. In general, σ SIS does not equal the observed stellar velocity dispersion σ 0 (White Davis 1996 ). To express the difference, researchers use a phenomenological free parameter f e defined by the relation σ SIS = f e σ 0 (Kochanek 1992 ; Ofek et al. 2003 ; Cao et al. 2012 ), where (0.8) 1/2 f e (1.2) 1/2 . In this case, the systematic error is caused by σ 0 as σ SIS , the deviation of the SIS model, the effects of secondary lenses (nearby galaxies), the line-of-sight contamination (Ofek et al. 2003 ), etc. The uncertainty of Equation ( 5 ) can be written by (Liao et al. 2016 )In Equation ( 6 ), and are the fractional uncertainties of σ sis and θ E , respectively. To test the CDDR, the left term of Equation (ref5), R A ( z l , z s ), should be expressed as the ratio of luminosity distance, . We transform R A ( z l , z s ) into the ratio of comoving distance ( D C ) or dimensionless distance H 0 D C / c . In a flat space, the dimensionless distance satisfiesBy using the equation, R A ( z l , z s ) can be written as In a nonflat space, the expression of R A ( z l , z s ) is more complicated; one can refer to Räsänen et al. ( 2015 ). Fortunately, most cosmological tests today support a flat cosmic space (Planck Collaboration et al. 2016 ). In this work, we test the CDD relation in a case where we assume that spacetime is flat. To test the CDDR, Equation ( 1 ) is rewritten by the parameterization of the deviation:Combining Equations ( 9 ) and ( 10 ), R A ( z l , z s ) can be written asBy using Equation ( 4 ), of Equation ( 11 ) can be rewritten aswhere the absolute magnitude of the SNe Ia subsample is offset, z l is the redshift of the lens, and z s is the redshift of the sources. 3. Method and Results In this section, we introduce a distance-deviation consistency method of data selection and show the result. 3.1. Method of Data Selection: A Distance-deviation Consistency Method To test the CDDR, D L and D A at the same redshift z need to be provided simultaneously. However, redshifts of subsamples from the SGLS and the SNe Ia are different. To take full advantage of the data, one needs to pair the subsamples efficiently. In this section, we provide a distance-deviation consistency method to pair subsamples, which outperforms the conventional method. The redshift-difference of subsample pairs in this work is not fixed, and it decreases with the redshifts to ensure the distance deviation of the sources stays the same. The relation between Δ z and coordinate distance with a cosmology model readswhere R h = ct and flat ΛCDM model with Ω m = 0.31 (Planck Collaboration et al. 2016 ) are used. By setting Δ d c / d c equal to 5% and combining Equation ( 13 ), Δ z ( z ) can be calculated:where the distance formulas of the two cosmology model, ΛCDM and R h = ct are used. The R h = ct cosmological model was
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