A Distance-deviation Consistency and Model-independent Method to Test the Cosmic Distance–Duality Relation

In this section, we introduce the contents of the Pantheon sample (Scolnic et al. 2018). The Pantheon sample consisting of a total of 1048 SNe Ia in the range of 0.01 < z < 2.3 is constructed by a subset including 279 SNe Ia (0.03 < z < 0.68) from the Pan-STARRS1 (PS1) Medium Deep Survey, the Sloan Digital Sky Survey (SDSS), the SNLS, the various low-z, and Hubble Space Telescope (HST) samples.

On the one hand, the luminosity distances D_L (z) can be determined accurately by multiple light-curve fitters (e.g., Jha et al. 2007; Guy et al. 2010; Burns et al. 2011; Mandel et al. 2011). From the phenomenological point of view, the distance modulus, μ, of an SNe Ia can be extracted from its light curve. In a modified version of the Tripp formula (Tripp 1998), the SALT2 light-curve fit parameters are transformed into distances:

$$\begin{eqnarray}&&\mu ={m}_{B}-M+\alpha {x}_{1}-\beta c+{{\rm{\Delta }}}_{M}+{{\rm{\Delta }}}_{B},\end{eqnarray} \tag{ 2 }$$

where m_B is the apparent magnitude, M is the absolute B-band magnitude of a fiducial SNe Ia with x₁ = 0 and c = 0, Δ_M and Δ_B are distance corrections based on the mass of the host galaxy of the SNe Ia and predicted biases from simulations respectively. α and β are light-curve parameters of relations between the luminosity and the stretch and between the luminosity and the color, respectively. Moreover, Δ_M in the Equation (2) can be written in the form (Scolnic et al. 2018)

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{M}=\gamma {[1+{e}^{(-(m-{m}_{\mathrm{step}})/\tau )}]}^{-1},\end{eqnarray} \tag{ 3 }$$

where γ, m_step, and τ are coefficients to be determined (Scolnic et al. 2018). On the other hand, one assumes that the SNe Ia with identical color, shape, and galactic environment have on average the same intrinsic luminosity for all redshifts (Betoule et al. 2014). According to the definition of the distance modulus, it can be written as

$$\begin{eqnarray}&&\mu =5\mathrm{log}\left(\displaystyle \frac{{D}_{L}}{\mathrm{Mpc}}\right)+25.\end{eqnarray} \tag{ 4 }$$

By using Equations (2) and (4), we can obtain the luminosity distances D_L (z) for the Pantheon SNe Ia Sample.

For the new SGLS sample, we used Amante et al. (2020), which contains 205 SGLS. This sample was generated from several survey projects (e.g., the SLACS, the CASTLES survey, the BELLS, the LSD, etc.). In an SGLS, the light is bent by massive bodies (e.g., galaxy or galaxy cluster) which is predicted by the general theory of relativity. The SGLS is a powerful astrophysical tool, which has been rapidly developed in recent years, to explore the universe and galaxy, especially dark energy (Biesiada 2006; Biesiada et al. 2010; Jullo et al. 2010; Cao et al. 2012, 2015; Magaña et al. 2015, 2018), the CDDR (Liao et al. 2016; Liao 2019), the cosmic acceleration (Tu et al. 2019), calibrating the standard candles (Wen & Liao 2020), and cosmological model comparison (Melia et al. 2015; Leaf & Melia 2018; Tu et al. 2019). In an SGLS, a single galaxy acting as the lens, the Einstein radius depends on three parameters: the angular distance to the source and between the lens and the source, and the mass distribution within the lensing galaxy. A singular isothermal sphere (SIS) model (Ratnatunga et al. 1999) is used to describe the lens galaxy's mass distribution. The ratio of the angular diameter distances between lens and source and between observer and source can be obtained from a special physical model (e.g., SIS model). Because these distances depend on the cosmological metric, the ratio can be used to constrain cosmological parameters.

In an SIS model of the SGLS, the distance ratio R^A (z_l , z_s ) (${D}_{l}^{A}s/{D}_{s}^{A}$) is related to observables in the following way (Biesiada et al. 2010),

$$\begin{eqnarray}&&{R}^{A}({z}_{l},{z}_{s})=\displaystyle \frac{{c}^{2}{\theta }_{E}}{4\pi {\sigma }_{\mathrm{SIS}}^{2}},\end{eqnarray} \tag{ 5 }$$

where c is the speed of light, θ_E is the 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 σ₀ (White & Davis 1996). To express the difference, researchers use a phenomenological free parameter f_e defined by the relation σ_SIS = f_e σ₀ (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 σ₀ 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)

$$\begin{eqnarray}&&{\sigma }_{{R}^{A}}({z}_{l},{z}_{s})={R}^{A}({z}_{l},{z}_{s})\sqrt{{\left(4{\delta }_{{\sigma }_{{sis}}}\right)}^{2}+{\left({\delta }_{{\theta }_{E}}\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

In Equation (6), ${\delta }_{{\sigma }_{{sis}}}$ and ${\delta }_{{\theta }_{E}}$ 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, ${D}_{{ls}}^{A}/{D}_{s}^{A}$. We transform R^A (z_l , z_s ) into the ratio of comoving distance (D_C ) or dimensionless distance H₀ D_C /c. In a flat space, the dimensionless distance satisfies

$$\begin{eqnarray}&&d({z}_{l},{z}_{s})=d({z}_{s})-d({z}_{l}).\end{eqnarray} \tag{ 7 }$$

By using the equation,

$$\begin{eqnarray}&&{D}_{A}(z)=\displaystyle \frac{{D}_{C}(z)}{1+z}=\displaystyle \frac{{H}_{0}d(z)}{c(1+z)},\end{eqnarray} \tag{ 8 }$$

R^A (z_l , z_s ) can be written as

$$\begin{eqnarray}&&{R}^{A}({z}_{l},{z}_{s})=1-\displaystyle \frac{(1+{z}_{l}){D}_{A}({z}_{l})}{(1+{z}_{s}){D}_{A}({z}_{s})}.\end{eqnarray} \tag{ 9 }$$

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:

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{L}}{{D}_{A}}{\left(1+z\right)}^{-2}=1+\eta z.\end{eqnarray} \tag{ 10 }$$

Combining Equations (9) and (10), R^A (z_l , z_s ) can be written as

$$\begin{eqnarray}&&{R}_{0}^{A}({z}_{l},{z}_{s})=1-\displaystyle \frac{(1+\eta {z}_{s})(1+{z}_{s}){d}_{L}({z}_{l})}{(1+\eta {z}_{l})(1+{z}_{l}){d}_{L}({z}_{s})}.\end{eqnarray} \tag{ 11 }$$

By using Equation (4), $\tfrac{{d}_{L}({z}_{l})}{{d}_{L}({z}_{s})}$ of Equation (11) can be rewritten as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{lg}\left[\displaystyle \frac{{d}_{L}({z}_{l})}{{d}_{L}({z}_{s})}\right] & = & 0.2\{{m}_{B}({z}_{l})-{m}_{B}({z}_{s})+\alpha [x({z}_{l})-x({z}_{s})]\\ & & -\,\beta [c({z}_{l})-c({z}_{s})]+{{\rm{\Delta }}}_{M}({z}_{l})-{{\rm{\Delta }}}_{M}({z}_{s})\},\end{array}\end{eqnarray} \tag{ 12 }$$

where 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.

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 Δz and coordinate distance with a cosmology model reads

$$\begin{eqnarray}&&{\rm{\Delta }}{d}_{c}^{\mathrm{model}}(z)={d}_{c}^{\mathrm{model}}(z+{\rm{\Delta }}{z}^{\mathrm{model}})-{d}_{c}^{\mathrm{model}}(z),\end{eqnarray} \tag{ 13 }$$

where R_h = ct and flat ΛCDM model with Ω_m = 0.31 (Planck Collaboration et al. 2016) are used. By setting Δd_c /d_c equal to 5% and combining Equation (13), Δz(z) can be calculated:

$$\begin{eqnarray}&&{\rm{\Delta }}z(z)={\min }\{{\rm{\Delta }}{z}^{{\rm{\Lambda }}{CDM}}(z),{\rm{\Delta }}{z}^{{R}_{h}={ct}}(z)\},\end{eqnarray} \tag{ 14 }$$

where the distance formulas of the two cosmology model, ΛCDM and R_h = ct are used. The R_h = ct cosmological model was proposed in Melia (2007). In the R_h = ct universe, the luminosity distance D_L can be written by

$$\begin{eqnarray}&&{D}_{L}^{{R}_{h}={ct}}(z)=\displaystyle \frac{c}{{H}_{0}}(1+z)\mathrm{ln}(1+z).\end{eqnarray} \tag{ 15 }$$

This model is a Friedmann–Robertson–Walker (FRW) cosmological model, which is obeying the cosmological principle and Weyl's postulate (Melia 2007; Melia & Shevchuk 2012). In an R_h = ct universe, space expands at a constant rate, rather than an accelerating rate. Theoretically, there are some controversies with this model that focus on the zero active mass condition ρ+3p = 0 (for more details, see Kim et al. 2016; Melia 2016, 2017). In terms of the fitting of observational data, the model performs relatively well. The creator of this model himself and his collaborators have done extensive work comparing it with the standard model using many different types of observations, and they have found that the R_h = ct model is better than the Standard Model (e.g., Melia 2013; Wei et al. 2015; Fatuzzo & Melia 2017; Melia et al. 2018; Melia 2019). Additional work also illustrates this point (Yu & Wang 2014; Yuan & Wang 2015). Though there is also a lot of work that argues against this model (e.g., Tutusaus et al. 2016; Shafer 2015). Despite the theoretical controversy, this model has gained some support for the data, and we can use this model jointly with the standard model to select the data.

For a given SGLS subsample, the redshift–dimensionless distance relation, Equation (14), is used to acquire a suitable redshift deviation interval. Then, the subsamples of the SNe Ia with redshifts within the interval are selected as the candidates. Finally, the subsample with the smallest redshift deviation is selected.

Our method outperforms the conventional method in two ways: (1) the information of subsamples at high redshifts is conserved. Pairing subsamples with a slight difference of redshift is a simple and commonly used method, for example, Δz = 0.005 (Holanda et al. 2010, 2012, 2016; Li et al. 2011; Nair et al. 2011), Δz = 0.006 (Goncalves et al. 2012), and Δz = 0.003 (Liao 2019). Gaussian process (GP) reconstruction is also a usable method (Ruan et al. 2018; Zhang 2019), and the linear interpolation method has been chosen by some researchers as well (Liang et al. 2013; Hu & Wang 2018). These methods reduce the systematic error to an extent but do not consider the distance-deviation consistency. The relation between the distance and the redshift is nonlinear: the same redshift deviation at higher redshift has a smaller distance deviation. For example, some researchers set Δz = 0.005, with the uncertainty of dimensionless distance being 5% at z ∼ 0.1, which is the minimum redshift of the subsample we select, but smaller than 1% at z ∼ 1 for selecting data with a general cosmology model; thus, the selecting uncertainty is not the consistency of distance deviation. The selecting uncertainty of their methods are all ignored, and if they did take them into account in their fitting, they had to introduce a cosmological model so that their methods were no longer model-independent. In our approach, although two cosmological models are introduced, they are used to jointly pick the data, breaking the dependence on a single cosmological model when picking the data, and are not introduced into the χ² function. In other words, the final parameter fit results are independent of the cosmological model. Cao et al. (2017) used a similar method for selecting the data with a ΛCDM model to fit the parameters of the ultra-compact radio quasars. At high redshift, in general, subsamples are sparse and matching pairs of subsamples is difficult if a fixed Δz is used. The distance-deviation consistency method selects more subsamples at high redshifts.

(2) More subsample pairs are selected. For example, in the work of Liao et al. (2016), they select only 60 pairs of samples to test the CDDR. According to Figure 1, at z = 0.11, the previously used Δz = 0.005 and the curve of Δd/d in this method intersect. So Δd/d = 5% is the maximum allowable uncertainty of dimensionless distance. In other words, if the error of selecting exceeds 5%, the statistical error of our results must be higher than that of previous work. If the error is significantly lower than 5%, the number of data pairs we choose will not be significantly improved. With the method of fixed Δd/d = 5%, the number of pairs can reach 68. The data utilization increases by 13.3%. In this work, we have obtained 120 pairs of samples with redshifts from 0.11 to 2.16.

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A comparison between our method with a fixed Δd/d and the conventional method with a fixed Δz are shown in Figures 1 and 2. In Figure 1, the deviation, Δd/d, falls rapidly with the increase of redshift with Δz fixed. The main advantage of a fixed Δd/d is shown in Figure 2. To conclude, on the one hand, our method maintains more information about subsamples at high redshifts. On the other hand, our method selects more subsample pairs and thus reduces the systematic error.

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To determine the parameters, we minimize the χ² function. By using Equations (5) and (11), the χ² function can be written as

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{1}^{120}\left(\displaystyle \frac{{\left({R}^{A}\left({z}_{l},{z}_{s}\right)-{R}_{0}^{A}\left({z}_{l},{z}_{s}\right)\right)}^{2}}{{\sigma }_{{R}^{A}({z}_{l},{z}_{s})}^{2}+{\sigma }_{{R}_{0}^{A}({z}_{l},{z}_{s})}^{2}+{\sigma }_{{sel}}^{2}}\right),\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{{R}^{A}({z}_{l},{z}_{s})}$ is the uncertainty of the SGLS with the SIS model, ${\sigma }_{{R}_{0}^{A}({z}_{l},{z}_{s})}$ is the error caused by the uncertainty of the distance modulus of SNe Ia, which is related to the uncertainty of observed data (e.g., m_B ), and σ_sel is the uncertainty of data selection, which is related to artificial selection. By using Equation (11), the uncertainty of data selection can be written as

$$\begin{eqnarray}&&{\sigma }_{{sel}}=\displaystyle \frac{(1+\eta {z}_{s})(1+{z}_{s}){d}_{L}({z}_{l})}{(1+\eta {z}_{l})(1+{z}_{l}){d}_{L}({z}_{s})}* \sqrt{{\left(\displaystyle \frac{{\rm{\Delta }}{d}_{l}}{{d}_{l}}\right)}^{2}+{\left(\displaystyle \frac{{\rm{\Delta }}{d}_{s}}{{d}_{s}}\right)}^{2}},\end{eqnarray} \tag{ 17 }$$

where $\tfrac{{\rm{\Delta }}{d}_{l}}{{d}_{l}}=\tfrac{{\rm{\Delta }}{d}_{s}}{{d}_{s}}=\tfrac{{\rm{\Delta }}d}{d}=5 \%$.

We use the Markov Chain Monte Carlo (MCMC) method to constrain the parameters in Equation (17). The EMCEE (Foreman-Mackey et al. 2013) python package is used. In order to execute the MCMC process, we need to provide the prior values first. In Scolnic et al. (2018), the best value and the 1σ confidence level of the parameters are shown. But in our work, we take an SIS model of SGLS to calibrating these parameters, which will differ. We take the prior interval that completely covers the range of the values from Scolnic et al. (2018). The prior probability for parameters P(α, β, f_e , η, γ, m_step, τ) is the product of the prior probability of each parameter. The prior probability is assumed to be uniform distributions: P(α) = U[−0.2, 0.2], P(β) = U[2, 6], P(f_e ) = U[0.5, 1.5], P(η) = U[−0.5, 1.5], P(γ) = U[0, 0.3], P(M_step) = U[5, 15], P(τ) = U[0.001, 1]. In Pantheon samples, the errors include both statistic and systematic deviation. The systematic error is relative to all data points and appears as a huge covariance matrix. In this work, some of the SNe Ia are selected, but only the statistic error is considered.

The result is shown in Figure 3 and Table 1. Triangle contours are plotted by using the open-source python package Getdist. One can see from Figure 3 that the best-fitted center value is $\eta ={0.047}_{-0.151}^{+0.190}$, which is at 1σ confidence level. The result indicates that the CDDR is in agreement with the observations and there are no signs of violation in light of SN Ia and SL data.

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Table 1. Constraints on the Coefficients of Light-curve Parameters and η at the 1σ Confidence Levels

Parameter	Value
α	${0.001}_{-0.061}^{+0.061}$
β	${5.283}_{-0.464}^{+0.417}$
fe	${1.046}_{-0.019}^{+0.020}$
η	${0.047}_{-0.151}^{+0.190}$
γ	${0.141}_{-0.070}^{+0.080}$
mstep	${10.055}_{-0.148}^{+0.177}$
τ	${0.134}_{-0.073}^{+0.048}$
χ2	117.430
χ2/d. o. f	117.430/113

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