Testing the cosmic distance duality relation using Type Ia supernovae and radio quasars through model-independent methods

The cosmic distance duality relation (CDDR) is a fundamental relation in modern cosmology [1], which relates the luminosity distance (LD) $ D_{\rm{L}}(z) $ and angular diameter distance (ADD) $ D_{\rm{A}}(z) $ through the identity equation $ {D_{\rm{L}}}={D_{\rm{A}}}{(1+z)}^{2} $ , wherezis the cosmological redshift. This relation relies on three fundamental assumptions: the space-time is described by the metric theory, the light travels along the null geodesics between the source and the observer, and the photon number is conserved. As a fundamental relation, the CDDR has undoubtedly been applied in various research fields of astronomy, such as the large-scale distribution of galaxies and the uniformity of the Cosmic Microwave Background (CMB) temperature [2], as well as the gas mass density distribution and temperature distribution of galaxy clusters [3,4]. In astronomical observations, any violation of the CDDR suggests the presence of new physics or unaccounted errors in the observational data. Therefore, it is necessary to conduct reliable testing on CDDR.

The CDDR test is usually conducted using a parametric method. Three different forms can be used for the parametrization, namely $ \eta(z)=1+\eta_0z $ , $ \eta(z)=1+\eta_0z/ (1+z) $ , and $ \eta(z)=1+\eta_0\ln(1+z) $ , where the $ \eta_{0} $ indicates the possible violation from the CDDR. Considering the advantages of the $ \eta(z) $ expression, such as manageable one-dimensional phase space and good sensitivity to observational data [5], the parameterizations of $ \eta(z) $ listed above were used to test the validity of the CDDR. Some studies have been devoted to testing the validity of CDDR by comparing LD data from observations of Type Ia supernovae (SNIa), HII galaxies, or gamma-ray bursts with the various ADD data from the X-ray plus Sunyaev-Zeldovich (SZ) effect and the gas mass fraction measurements in galaxy clusters [5–17]. The results indicate that within various redshift ranges, the CDDR is consistent with astronomical observations [18–29]. Recent efforts have also explored the use of QSO as standard rulers providing ADD measurements for CDDR constraints. Notably, Qiet al.combined QSO data with simulated future Gravitational Wave observations from the Einstein Telescope, which act as standard sirens for LD [30]. Their work highlighted the potential of future multi-messenger observations to test the CDDR at high redshifts with high precision. In addition, an issue may be that it is difficult to obtain LD and ADD measurements from astronomic observation at the same redshifts. To solve this problem, several methods have been proposed in some literature. Using the galaxy cluster samples [31,32] and SNIa data, Holandaet al.[8] and Liet al.[10] selected the closest one through a selection criterion ( $ \Delta z=|z_{\rm{ADD}}-z_{\rm{SNIa}}|<0.005 $ ) for CDDR test. To minimize statistical errors that could arise from utilizing only a single SNIa data point from all those that meet the selection criteria, Menget al.[12], did not use the closest measurement, but instead used a binning method to bin the available data that meets the selection criterion to derive LD.

SNIa and compact radio quasars (QSO) measurements play important roles in constraining cosmological parameters. It is worth noting that the LD derived from SNIa observations is dependent upon its peak absolute magnitude $ M_{\rm{B}} $ , which is assumed to be a constant value, unaffected by other variables. Recently, efforts have been made to derive the value of $ M_{\rm{B}} $ from a cosmological observations [33–35]. Various values of $ M_{\rm{B}} $ have been determined by combining SNIa data, such as Pantheon, with other observational datasets, including CMB observations, cosmic chronometer data related to the Hubble parameter, and baryon acoustic oscillations (BAO). A discrepancy in the absolute magnitude of SNIa, calibrated by Cepheids, was observed between $ z\leq0.01 $ and $ z>0.01 $ [36,37]. Recent studies [38,39] have also indicated a potential weak evolution of $ M_{\rm{B}} $ . In addition, due to the negligible dependence of the compact structure sizes of intermediate-luminosity quasars on the source luminosity and redshift, these quasars, with very-long-baseline interferometry (VLBI) observations, are potentially promising standard rulers [40]. The ADD obtained from these QSO observations depends on the linear size scaling factorl. Based on different calibration methods and data forl, systematic errors may arise, leading to varying calibration values forl. For example, in Ref. [41], Caoet al.used a cosmological model-independent Gaussian process method to reconstruct Hubble $ H(z) $ measurements from 24 cosmic chronometer data points in the redshift range of 0.7 to 1.2 for the calibration ofl. In Ref. [42], Caoet al.calibrated the value oflusing the flat ΛCDM model with $ Planck $ Collaboration data. In Ref. [43], Caoet al.used Gaussian processes to reconstruct the ADD from the BAO sample to calibratel. These calibration results indicate that the value oflvaries slightly depending on the observational data and cosmological model.

Obviously, if the exact values of $ M_{\rm{B}} $ andlare not determined by astronomical observations, the specific prior values for $ M_{\rm{B}} $ andlmay potentially introduce biases into the constraints on cosmological parameters. Furthermore, the independence of the CDDR test, which relies on the prior values of $ M_{\rm{B}} $ andl, may be questionable, as the values of $ M_{\rm{B}} $ andlare obtained from specific cosmological models. It is worth noting that the measurement error of a single SNIa or QSO measurement is not dependent on the parameter $ M_{\rm{B}} $ orl, thus, theoretically, one can use marginalization methods to eliminate $ M_{\rm{B}} $ orlparameters during statistical analysis. In view of this, it is important to further study the impacts of specific prior values for $ M_{\rm{B}} $ orlon CDDR test, and to develop new testing methods that are not dependent on $ M_{\rm{B}} $ andl, which can improve the reliability of CDDR testing. This is also the main motivation for us to carry out this research.

In this work, we perform the CDDR test by comparing the LD derived from Pantheon+ SNIa data with the ADD from QSO data. The binning method and Artificial Neural Network (ANN) are used to match the SNIa data with the QSO data at the same redshifts. We first investigate the impacts on the CDDR test by considering the specific value of $ M_{\rm{B}} $ andlto derive the LD and ADD. The results indicate that the priors of $ M_{\rm{B}} $ andlmay induce significant biases in the CDDR test. To avoid these biases, we combine $ M_{\rm{B}} $ andlinto a new variableκ, defined as $ \kappa\equiv 10^{M_{\rm{B}} \over 5}\,l $ , and consider it as a nuisance parameter with a flat prior in statistical analysis, thereby marginalizing its impact on CDDR test. Therefore, all of the quantities used in the CDDR test come directly from observations, meaning that the absolute magnitudes from SNIa and the linear size scaling factor from QSO measurements do not need to be calibrated. We demonstrate that CDDR is consistent with the observed data, and the parametric method of testing CDDR is independent of specific cosmological models.

To verify the validity of CDDR, two kinds of cosmic distances are usually required: LD ( $ D_{\rm{L}} $ ) and ADD ( $ D_{\rm{A}} $ ). The LD data in this study are obtained from the Pantheon+ SNIa observational dataset [44]. This dataset combines observations from 18 different survey programs, containing 1701 light curves of 1550 spectroscopically confirmed unique SNIa, with a redshift range of $ 0 < z < 2.3 $ . Pantheon+ employs an improved version of the SALT2 light-curve fitter to calculate the distance modulus, utilizing recalibrated photometric systems and updated training parameters. The analysis applies the Bayesian Estimation Applied to Multiple Species with Bias Corrections method to determine nuisance parameters and correct for distance biases, such as the distance modulus formula $ \mu = m_{\rm{B}}-M_{\rm{B}} $ , where $ m_{\rm{B}} $ is the observed peak apparent magnitude in the rest-frame B-band. Recently, some research has been focused on the possible evolution of $ M_{\rm{B}} $ with redshift. The CMB constraint on the sound horizon forecasts that $ M_{\rm{B}}\sim -19.4\,{\rm{mag}} $ using an inverse distance ladder [35], while the approximation from SH0ES gives that $ M_{\rm{B}}\sim -19.2\,{\rm{mag}} $ [33]. Hence, we firstly investigate the impacts of different prior values of $ M_{\rm{B}} $ on the CDDR test. In this work, we consider two specific priors of $ M_{\rm{B}} $ derived from different observational data sets within various redshift ranges: (a) $ M_{\rm{B}}^{\rm{D20}}={-19.23\pm0.0404\,{\rm{mag}}} $ , obtained from SNIa observation within the relatively low redshift range of $ 0.023 < z < 0.15 $ by Camarena and Marra [33] in ΛCDM, through a de-marginalization of the SH0ES determination [45] (hereafter referred to as $ M_{\rm{B}}^{\rm{D20}} $ ); and (b) $ M_{\rm{B}}^{\rm{B23}}={-19.396\pm0.016\,{\rm{mag}}} $ , obtained by combining SNIa observations with BAO observations [35] (hereafter referred to as $ M_{\rm{B}}^{\rm{B23}} $ ). Taking into consideration the observational uncertainty of $ M_{\rm{B}} $ , the error bar onμcan be represented as $ \sigma_{\mu}=\sqrt{\sigma_{M_{\rm{B}}}^2+\sigma_{m_{\rm{B}}}^2} $ . The relation between the LD $ D_{\rm{L}} $ [46] and the distance modulusμcan be expressed as

and the uncertainty in $ D_{\rm{L}} $ can be obtained from the equation

The angular size-distance relationship of QSO is utilized for cosmological inference, originally proposed by Kellermann [47], who attempted to obtain the deceleration parameter using VLBI observations of 79 compact radio sources at 5 GHz. Subsequently, Gurvits [48] extended this method and tried to study the dependence of the observed characteristic sizes of 337 Active Galactic Nuclei (AGNs) at 2.29 GHz on luminosity and redshift [49]. In the following analysis, the angular sizeθof the radio source is refined in Ref. [48] using the visibility modulus $ \Gamma=S_{\rm{c}}/S_{\rm{t}} $ , which can be expressed as $ \theta={2\sqrt{-\ln{\Gamma}\ln2}\over\pi{B}} $ , whereBis the interferometer baseline measured in multiple of wavelengths, and $ S_{\rm{c}} $ and $ S_{\rm{t}} $ are the correlated flux density and total flux density, respectively. The linear size $ l_{\rm{m}} $ of compact structures in radio sources, the intrinsic luminosityL, and the redshiftzof the background source supply the following relationship,

wherelrepresents the linear size scaling factor, describing the apparent distribution of radio brightness within the core,βandnare used to quantify the possible "angular size-luminosity" and "angular size-redshift" relations, respectively. Moreover, for a cosmological rod with intrinsic length, the relation of the angular size-redshift can be expressed as [50]

where $ \theta(z) $ is the observed angular size measured by VLBI techniques. As demonstrated by Ref. [51–53], the VLBI measurement of $ \theta(z) $ is a direct result based on the principles of interferometry (baselines, wavelength) and data analysis. This measurement process is fundamentally independent of cosmological model assumptions regarding cosmic expansion, curvature, or geometry. Therefore, the process of obtaining the observed $ \theta(z) $ itself possesses cosmological model independence. Combining Eq. 3 and Eq. 4, the angular diameter distance $ D_{\rm{A}}(z) $ can be written as

Recently, Cao,et al.found that the linear size scaling factor is almost independent on redshift and intrinsic luminosity ( $ |n|\simeq10^{-3} $ , $ |\beta|\simeq10^{-4} $ ) [41,42]. The sample of 120 intermediate-luminosity radio quasars within redshift range of $ 0.4 < z < 2.8 $ selected in [41] has been widely used in various cosmological studies [40,54–56]. The ADD obtained from the QSO samples has already been used to test the CDDR along with the LD obtained from HII galaxies and supernovae [57,58], and to infer the value of the Hubble constant $ H_0 $ together with the unanchored luminosity from supernovae data [59].

The value of the linear size scaling factorlcan be constrained to $ l=11.19\pm1.64\,{\rm{pc}} $ [42] (hereafter referred to as $ l^{\rm{C17}} $ ) in the flat ΛCDM model with $ Planck $ Collaboration. Then, in the manner of an independent study on cosmological model, Caoet al.obtained $ l={10.86\pm1.58}\,{\rm{pc}} $ [42] by using 36 Hubble data points, some of which were inferred from 30 cosmic chronometers [60–62], while the rest were derived from 6 BAO measurements [63]. Furthermore, Caoet al.obtained the more accurate value ofl[43] by using ADD from the BAO sample [64–66] and 41 Hubble data points, some of which were inferred from 31 passively evolving galaxies [19,61,67–70], while the rest were derived from 10 BAO measurements [64–66,71–74]. The obtained values oflare $ l={11.04\pm0.40}\,{\rm{pc}} $ (hereafter referred to as $ l^{\rm{C19}} $ ) and $ l={11.12\pm0.50}\,{\rm{pc}} $ , respectively. The values oflprovided by different observation data exhibit slight variations. Consequently, the prior values oflmay potentially induce bias in testing the CDDR. In this work, we consider the values of the linear size scaling factors $ l^{\rm{C17}} $ and $ l^{\rm{C19}} $ , which are calibrated using methods that depend on and do not depend on the cosmological model, respectively, to investigate their impacts on the CDDR test.

To test the validity of the CDDR, a straightforward method is to compare the ADD and LD from different observations at the same redshift. Due to the lack of observational ADD and LD data at the same redshift, we bin the SNIa data points satisfying the selection criterion $ \Delta z=|z_{\rm{ADD}}-z_{\rm{SNIa}}|<0.005 $ , as proposed in the literature [8,10,27]. This method, known as the binning method, can be used to avoid statistical errors caused by using only one SNIa data point among those satisfying the selection criterion and has been employed in discussing the CDDR test in the cited Ref. [12,75]. Here, we take the inverse variance weighted average of all selected data. To avoid correlations among the individual CDDR tests, we select the SNIa samples following a procedure. The weighted average LD $ \bar{D_{\rm{L}}} $ and its uncertainty $ \sigma_{D_{\rm{L}}} $ can be obtained using conventional data processing techniques in Chapter 4 of Ref. [76],

Here, $ D_{{\rm{L}}i} $ denotes theith appropriate luminosity distance data points, and $ \sigma_{D_{{\rm{L}}i}} $ corresponds the observational uncertainty.

Only 31 QSO data points satisfy the selection criterion. The distributions of the QSO data and SNIa data derived from different priors ofland $ M_{\rm{B}} $ are shown inFig. 1.

Figure 1.(color online) In binning method, the sample catalogs of the observed $ D_{\rm{A}}(1+z)^2 $ distribution from the QSO data points and the corresponding LD $ D_{\rm{L}} $ from Pantheon+ data obtained with the priors of $ M_{\rm{B}}^{\rm{D20}} $ (upper panel), $ M_{\rm{B}}^{\rm{B23}} $ (bottom panel), $ l^{\rm{C17}} $ (left panel) and $ l^{\rm{C19}} $ (right panel), respectively.

It is important to note that when using selection criteria, one must be aware of the errors caused by the mismatch between SNIa and QSO data points. Additionally, most of the available QSO data points are excluded due to not meeting the selection criterion, as the density distribution of SNIa data differs from that of QSO data in certain redshift regions. To improve the robustness of QSO data when testing CDDR, we employ the ANN to reconstruct the smoothing $ m_{\rm{B}}(z) $ function from the Pantheon+ SNIa observations. Therefore, each ADD obtained from the QSO sample located within the redshift range of Pantheon+ SNIa has a corresponding LD of SNIa at the same redshift.

An ANN is usually a Deep Learning algorithm consisting of three layers: an input layer, a hidden layer, and an output layer. The input layer comprisesnnodes, each of which corresponds to an independent variable, followed byminterconnected hidden layers and the output layer with activation function nodes in the basic architecture [77]. ANN estimates the error gradient from observations in the training dataset, and then updates the model weights and bias estimates during back propagation process to iterate toward an optimal solution through the Adam optimization [78]. The ANN process can be described by vectorization representation, and more details can be found in Refs. [79–81].

We use the publicly available code, named Reconstructing Functions Using Artificial Neural Networks (ReFANN) ¹ [79], to reconstruct the function of apparent magnitude $ m_{\rm{B}} $ versus redshiftz, as shown inFig. 2. It is easy to find that the uncertainty obtained from the ANN-reconstructed function are close to the observational uncertainty, and the reconstructed $ 1\sigma $ CL of the $ m_{\rm{B}} $ can be considered as the average level of observational error. The LD $ D_{\rm{L}} $ corresponding to ADD $ D_{\rm{A}} $ from QSO data points can be obtained through the smoothing function $ m_{\rm{B}}(z) $ reconstructed by ANN. For the QSO samples, 116 QSO data points within the redshift range of SNIa observation $ 0 < z < 2.26 $ can be matched with those from SNIa observation at the same redshift, and the remaining 4 QSO samples that are not within this redshift range are discarded. The distributions of the QSO data and reconstructed SNIa data derived from different priors of $ M_{\rm{B}} $ andlare shown inFig. 3.

Figure 2.(color online) The distributions of the reconstructed $ m_{\rm{B}}(z) $ function with the corresponding $ 1\sigma $ errors with the ANN (black line), and the measurements of apparent magnitude from the Pantheon+ samples (red).

Figure 3.(color online) The sample catalogs of the observed $ D_{\rm{A}}(1+z)^2 $ distribution from the QSO data points derived with the priors of $ l^{\rm{C17}} $ (left panel) and $ l^{\rm{C19}} $ (right panel), and the LD $ D_{\rm{L}} $ curves from Pantheon+ data derived with the priors of $ M_{\rm{B}}^{\rm{D20}} $ (upper panel) and $ M_{\rm{B}}^{\rm{B23}} $ (bottom panel).

We adopt the $ \eta(z) $ function to verify any possible deviations from the CDDR at any redshift by comparing the $ D_{\rm{L}} $ from SNIa and the $ D_{\rm{A}} $ from QSO measurements. The $ \eta(z) $ can be obtained through the following expression:

At any redshift, $ \eta(z)\neq 1 $ indicates a deviation between the CDDR and astronomical observations.

We adopt three types of parameterizations for $ \eta(z) $ : the linear form P1: $ \eta(z)=1+\eta_0z $ , and two non-linear forms P2: $ \eta(z)=1+\eta_0z/(1+z) $ , and P3: $ \eta(z)=1+\eta_0\ln(1+z) $ . The observed $ \eta_{\rm{obs}} (z) $ is obtained from Eq. 8, and the corresponding error can be written as

Thus, we obtain

Here,Nrepresents the number of available QSO data points obtained the binning method or ANN. The constraint results on $ \eta_0 $ are shown inFig. 4,Fig. 5, andTable 1. It is evident that the results obtained from the parametric method depend on the prior values of $ M_{\rm{B}} $ orl. Thus, specific prior values of $ M_{\rm{B}} $ orlcause biases in the CDDR test, if their true values are not determined by astronomic observations.

Figure 4.(color online) In binning method, the likelihood distribution functions obtained with the priors of $ M_{\rm{B}}^{\rm{D20}} $ (upper panel), $ M_{\rm{B}}^{\rm{B23}} $ (bottom panel), $ l^{\rm{C17}} $ (left panel) and $ l^{\rm{C19}} $ (right panel).

Figure 5.(color online) In ANN, the likelihood distribution functions obtained with the priors of $ M_{\rm{B}}^{\rm{D20}} $ (upper panel), $ M_{\rm{B}}^{\rm{B23}} $ (bottom panel), $ l^{\rm{C17}} $ (left panel) and $ l^{\rm{C19}} $ (right panel).

parmetrization	P1: $ 1+\eta_0 {z} $	P2: $ 1+\eta_0 {z\over(1+z)} $	P3: $ 1+\eta_0 {\ln(1+z)} $
$ \eta_0^{\rm {A\,C}\dagger} $	$ {-0.183\pm0.034\pm0.067\pm0.101} $	$ {-0.352\pm0.067\pm0.134\pm0.201} $	$ {-0.261\pm0.049\pm0.097\pm0.146} $
$ \eta_0^{\rm {A\,C}\ddagger} $	$ {-0.118\pm0.013\pm0.026\pm0.039} $	$ {-0.290\pm0.032\pm0.064\pm0.096} $	$ {-0.194\pm0.021\pm0.043\pm0.064} $
$ \eta_0^{\rm {B\,C}\dagger} $	$ {-0.115\pm0.036\pm0.072\pm0.109} $	$ {-0.207\pm0.072\pm0.144\pm0.216} $	$ {-0.158\pm0.052\pm0.105\pm0.158} $
$ \eta_0^{\rm {B\,C}\ddagger} $	$ {-0.072\pm0.014\pm0.028\pm0.042} $	$ {-0.173\pm0.034\pm0.069\pm0.103} $	$ {-0.116\pm0.023\pm0.046\pm0.069} $
$ \eta_0^{\rm {A\,D}\dagger} $	$ {-0.097\pm0.020\pm0.039\pm0.059} $	$ {-0.186\pm0.036\pm0.073\pm0.109} $	$ {-0.138\pm0.027\pm0.055\pm0.082} $
$ \eta_0^{\rm {A\,D}\ddagger} $	$ {-0.079\pm0.010\pm0.019\pm0.029} $	$ {-0.190\pm0.023\pm0.046\pm0.069} $	$ {-0.128\pm0.015\pm0.031\pm0.046} $
$ \eta_0^{\rm {B\,D}\dagger} $	$ {-0.008\pm0.021\pm0.042\pm0.063} $	$ {-0.013\pm0.039\pm0.078\pm0.117} $	$ {-0.010\pm0.029\pm0.059\pm0.088} $
$ \eta_0^{\rm {B\,D}\ddagger} $	$ {-0.027\pm0.010\pm0.021\pm0.031} $	$ {-0.060\pm0.025\pm0.049\pm0.074} $	$ {-0.042\pm0.017\pm0.033\pm0.050} $
$ {\eta_0}^{\star\dagger} $	$ {-0.100\pm^{0.047}_{0.044}\pm^{0.098}_{0.085}\pm^{0.152}_{0.124}} $	$ {-0.380\pm^{0.138}_{0.122}\pm^{0.294}_{0.231}\pm^{0.475}_{0.330}} $	$ {-0.197\pm^{0.083}_{0.076}\pm^{0.174}_{0.145}\pm^{0.275}_{0.210}} $
$ {\eta_0}^{\star\ddagger} $	$ {-0.042\pm^{0.026}_{0.025}\pm^{0.054}_{0.048}\pm^{0.084}_{0.069}} $	$ {-0.171\pm^{0.112}_{0.099}\pm^{0.238}_{0.188}\pm^{0.384}_{0.268}} $	$ {-0.088\pm^{0.055}_{0.051}\pm^{0.115}_{0.097}\pm^{0.182}_{0.141}} $

Table 1.The maximum likelihood estimation results for the parameterizations with the binning method and ANN. The $ \eta_0 $ is represented by the best fit value $ \eta_{0,{\rm{best}}} \pm 1\sigma \pm 2\sigma \pm 3\sigma $ for each dataset. The superscripts A, B, C, and D represent the cases obtained from $ M_{\rm{B}}^{\rm{D20}} $ , $ M_{\rm{B}}^{\rm{B23}} $ , $ l^{\rm{C17}} $ , and $ l^{\rm{C19}} $ , respectively. The superscript $ \star $ denotes the results obtained from the flat marginalization for $ M_{\rm{B}} $ andl, and $ \dagger $ and $ \ddagger $ denote the results obtained from the binning method and ANN, respectively.

Recently, Liuet al.[82] used the fraction division $ \eta(z_{i})/\eta(z_{j}) $ to eliminate the impacts of $ M_{\rm{B}} $ andlon CDDR test, and the results indicated agreement between the CDDR and observations. More recently, using the latest five BAO measurements and the Pantheon SNIa sample, Xuet al.[28] obtained CDDR test independently of the peak absolute magnitude $ M_{\rm{B}} $ and the sound horizon scale $ r_{\rm{s}} $ from transverse BAO measurements by marginalizing analytically the likelihood function over the combination of $ M_{\rm{B}} $ and $ r_{\rm{s}} $ . Since the uncertainty in an individual SNIa or QSO measurement is independent of $ M_{\rm{B}} $ orl, these parameters can be removed from the fits by analytically marginalizing over them in the analysis. Following the process in Ref. [28], we treat the fiducial values of $ M_{\rm{B}} $ andlas nuisance parameters to determine the LD $ D_{\rm{L}} $ and ADD $ D_{\rm{A}} $ , and then marginalize their effect by using a flat prior in the statistic analysis. The likelihood distribution $ \chi^{\prime\,2} $ can be rewritten as

Here, $ \alpha_i=\eta(z_i) $ , $ \beta_i=10^{({m_{{\rm{B}},i}\over 5}-5)}\theta_{{\rm{QSO}},i}(1+z_i)^{-2} $ , $ \kappa=(10^{M_{\rm{B}} \over 5}l) $ , and

Thus, following the approach described in Refs. [28,83,84], the marginalized $ \chi^{\prime\,2} $ in Eq. 11 can be rewritten as:

where $ A=\sum \alpha_i^2/(\beta_i^2{\sigma^{\prime\,2}_{{\eta_{{\rm{obs}},i}}}}) $ , $ B=\sum \alpha_i/(\beta_i{\sigma^{\prime\,2}_{{\eta_{{\rm{obs}},i}}}}) $ , and $ C=\sum 1/{\sigma^{\prime\,2}_{{\eta_{{\rm{obs}},i}}}} $ .

It is evident that all of the quantities used in the CDDR test come directly from observations, and $ \chi_{\rm{M}}^{\prime\,2} $ in Eq. 13 is independent of parameters such as $ M_{\rm{B}} $ andl. In this way, we can remove $ M_{\rm{B}} $ andlfrom the fit by analytically marginalizing them in Eq. 11. Since this test is based on the observed data and does not require any assumptions about cosmological models, the parametric method used to test CDDR is independent of the cosmological model. The results are shown inFig. 6andTable 1. To compare the capability of QSO data with that of other astronomic observational data in testing CDDR, we list the results of the constraints on $ \eta_0 $ obtained from different observational data sets inTable 2.

Figure 6.(color online) The likelihood distribution obtained with flat priors onκusing the binning method (left) and ANN (right).

Dataset used	P1: $ 1+\eta_0 {z} $	P2: $ 1+\eta_0 {z\over(1+z)} $	P3: $ 1+\eta_0 {\ln(1+z)} $
$ {\rm {Y_{SZ}}}-{\rm {Y_{X}}}\, {\rm {ratio}}+H(z)(\rm Prior) $ [17]	$ {0.008{\pm{0.05}}} $	$ {0.019{\pm{0.11}}} $	$ {0.013{\pm{0.07}}} $
$ {\rm {SNIa }}+{\rm {BAO}(\rm Prior)} $ [85]	$ {-0.038{\pm{0.037}}} $	$ {-0.059{\pm{0.055}}} $	$ {-0.048{\pm{0.046}}} $
$ {\rm {GMF}}+{\rm {SNIa}}+{T_{\rm{CMB}}}(\rm Prior) $ [86]	$ {-0.020{\pm{0.027}}} $	$ {-0.041{\pm{0.042}}} $
$ {\rm {SNIa }}+{\rm {BAO}(Marg)} $ [87]	$ {-0.07{\pm{0.12}}} $	$ {-0.20{\pm{0.27}}} $	$ {-0.12{\pm{0.18}}} $
$ {\rm {SNIa }}+{\rm {BAO}(Marg)} $ [85]	$ {0.041{\pm^{0.123}_{0.109}}} $	$ {0.082{\pm^{0.246}_{0.214}}} $	$ {0.059{\pm^{0.174}_{0.159}}} $
$ {\rm {SNIa }}+{\rm {BAO}(Marg)} $ [28]	$ {-0.037{\pm^{0.110}_{0.097}}} $	$ {-0.101{\pm^{0.269}_{0.225}}} $	$ {-0.061{\pm^{0.173}_{0.149}}} $

Table 2.Summary of the constraints on parameter $\eta_0$ with different data sets. "Prior" represents the results obtained using certain parameters with specific priors, and "Marg" represents the results obtained by marginalizing certain parameters with a flat prior.

Qiet al.also used QSO to measure ADD, and combined them with simulated gravitational wave data from the future Einstein Telescope as a source of LD, to test the CDDR [30]. Compared to their work, our main research focus is investigating potential biases in CDDR tests caused by specific priors on $ M_{\rm{B}} $ from SNIa andlfrom QSO measurements. We quantitatively demonstrate how different choices of these priors impact CDDR constraints. While Qiet al.used calibrated values oflto test CDDR, our principal contribution lies in developing and implementing a method that decouples the constraints from specific priors on both $ M_{\rm{B}} $ of SNIa andlof QSO. Technically, we construct a composite parameter $ \kappa\equiv 10^{M_{\rm{B}} \over 5}\,l $ and establish a prior-independent CDDR testing framework through Bayesian marginalization by adopting flat priors onκ, thus eliminating the calibration dependence on both $ M_{\rm{B}} $ andl.

FromFig. 4andTable 1, it can be seen that through the binning method, CDDR is consistent with the observed data at various confidence levels (CLs) depending on the combination of $ M_{\rm{B}} $ andl. Specifically, it is consistent at $ 1\sigma $ CL with the combination of $ M_{\rm{B}}^{\rm{B23}} $ and $ l^{\rm{C19}} $ . However, when considering the other combinations of $ M_{\rm{B}} $ andlvalues, specifically $ M_{\rm{B}}^{\rm{B23}} $ and $ l^{\rm{C17}} $ , $ M_{\rm{B}}^{\rm{D20}} $ and $ l^{\rm{C17}} $ , as well as $ M_{\rm{B}}^{\rm{D20}} $ and $ l^{\rm{C19}} $ , the CDDR is not consistent with the observed data at $ 3\sigma $ CL. Similarly, fromFig. 5andTable 1, through the ANN method, CDDR is only consistent with the observed data at $ 3\sigma $ CL with the combination of $ M_{\rm{B}}^{\rm{B23}} $ and $ l^{\rm{C19}} $ . Nevertheless, CDDR is not consistent with the observed data at $ 3\sigma $ CL with the other three combinations. For the Binning method with P1 parameterization, results detailed inTable 1and illustrated inFig. 4show that the choice of $ M_{\rm{B}} $ orlpriors significantly shifts the best-fit $ \eta_0 $ . Systematic comparisons, fixing one prior while varying the other, reveal shifts in $ \eta_0 $ ranging from 0.068 to 0.107. These shifts correspond to statistical significance between $ 1.4\sigma $ and $ 3.1\sigma $ , demonstrating a significant dependence of the $ \eta_0 $ constraints on the chosen priors with this method. However, marginalizing over the combined parameterκeffectively removes this dependence on the $ M_{\rm{B}} $ andlpriors. The marginalized result is independent of the $ M_{\rm{B}} $ andlprior choice, thus isolating the calibration uncertainty. Even though this comes at the expense of an increase in statistical uncertainties.

FromFig. 4,Fig. 5, andTable 1, the parametrization P1 imposes the most rigorous constraints on testing CDDR, although the result of the CDDR test is nearly independent of the parametrization of $ \eta(z) $ . Now, we compare the capability of QSO measurements to constrain parameter $ \eta_0 $ with that of other astronomical observations obtained under specific prior conditions of cosmological variables. With the binning method, the QSO measurements improve the accuracy of $ \eta_0 $ by approximately 60% at $ 1\sigma $ CL when compared to results obtained from the South Pole Telescope-SZ clusters and X-ray measurements from Multi-mirror Mission-Newton [17], where the prior of $ M_{\rm{B}} $ and $ H_0 $ are utilized; and about 45% at $ 1\sigma $ CL when compared to results obtained from Pantheon samples with transverse BAO measurements [85], where the CDDR tests were conducted with specific priors of $ M_{\rm{B}} $ or $ r_{\rm{s}} $ . Our results are also roughly 25% more stringent than the result from the x-ray GMF of galaxy clusters jointly with SNIa and CMB temperature [86], where $ M_{\rm{B}} $ is fixed to derive the LD. Using the ANN, while testing the CDDR with more available QSO measurements, the uncertainties of $ \eta_0 $ at the $ 1\sigma $ CL are improved by approximately 50% when compared to results obtained from the binning method.

When testing the CDDR using a flat prior of $ \kappa\equiv 10^{M_{\rm{B}}\over 5}l $ , CDDR is consistent with the observed data at $ 3\sigma $ CL with the binning method; and at $ 2\sigma $ CL with the ANN method. The constraints on $ \eta_0 $ obtained from the flat prior ofκare much weaker than those obtained from the specific priors of $ M_{\rm{B}} $ andl, due to marginalizingκwith a flat prior in our analysis. Specifically, the methods with specific priors in the binning method and ANN method provide 55% and 60% tighter constraints on $ \eta_0 $ respectively compared to the marginalization method. To assess the ability of testing CDDR from QSO measurements, it is valuable to compare our results with previous constraints on $ \eta_0 $ from other observational data by marginalizing certain parameters with a flat prior. With the binning method, the QSO measurements improve the accuracy of $ \eta_0 $ by approximately 60% at $ 1\sigma $ CL when compared to the results obtained from the Pantheon samples and BOSS DR12 BAO measurements within the redshift range $ 0.31\leq{z}\leq0.72 $ [87] and from Pantheon samples with transverse BAO measurements [85], where $ M_{\rm{B}} $ and $ r_{\rm{s}} $ were marginalized. The constraints on $ \eta_0 $ are roughly 55% more stringent than the result from 5 BAO measurements utilizing the extended Baryon Oscillation Spectroscopic Survey data release 16 quasar samples in conjunction with the Pantheon SNIa samples [28], where $ M_{\rm{B}} $ and $ r_{\rm{s}} $ were marginalized. As for the results obtained from the ANN method, the uncertainties of $ \eta_0 $ at $ 1\sigma $ CL are reduced by approximately 40% when compared to the results obtained from the binning method. Therefore, the QSO measurements demonstrate a superior capability in testing CDDR compared to BAO observations, which have been recognized as powerful tools for testing CDDR [75,87].

The CDDR plays an important role in astronomical observations and modern cosmology, and any deviation from the CDDR may indicate new physical signals. SNIa and QSO measurements can be regarded as effective observational data for testing the CDDR. However, due to the uncertainty in the absolute magnitude $ M_{\rm{B}} $ and the linear size scaling factorl, which are constrained by different astronomical observations and cosmological models, it is necessary to investigate the impact of the prior values of $ M_{\rm{B}} $ andlon the CDDR test, and to verify the validity of the CDDR using new methods.

In this work, we test the CDDR by comparing the LD derived from the Pantheon+ SNIa compilation with ADD from QSO measurements, using parametric methods. We employ the binning method and ANN to match the SNIa data with the QSO measurements at the same redshift, and adopt the function $ \eta(z)=D_{\rm{L}}(z)/D_{\rm{A}}(z)(1 + z)^{-2} $ to probe the possible deviations from the CDDR at any redshift. Two specific prior values of $ M_{\rm{B}} $ andlare used to obtain the LDs from the SNIa observations and the ADDs from QSO measurements, respectively. The results show that the specific prior values of $ M_{\rm{B}} $ andlcause significant biases in the CDDR test, if the astronomical observations do not provide accurate values for $ M_{\rm{B}} $ andl.

To avoid the bias in the CDDR test caused by the prior values of $ M_{\rm{B}} $ andl, we treat the fiducial values of $ M_{\rm{B}} $ andlas nuisance parameters to determine the LD $ {D_{\rm{L}}} $ and ADD $ {D_{\rm{A}}} $ . We then marginalize their impacts on the CDDR test by applying a flat prior on the new variableκ $ \equiv $ $ 10^{M_{\rm{B}}\over 5}l $ in the statistical analysis. This marginalization approach eliminates the necessity for the calibration of $ M_{\rm{B}} $ andlvalues. Thus, the methodology for testing the CDDR remains independent of the underlying cosmological model assumptions. Due to the ANN method allowing for the inclusion of more QSO data points, it provides 50% and 40% tighter constraints on $ \eta_0 $ compared to the Binning method when using specific priors on $ M_{\rm{B}} $ andland the marginalization method, respectively. Our results indicate no violation of the CDDR. However, the capability of the QSO measurements to test CDDR is reduced compared to the results obtained from specific values of $ M_{\rm{B}} $ andl, due to marginalizingκwith a flat prior in our analysis. In comparison to the previous results, the capability of QSO measurements to test the CDDR is much stronger than that of other previous astronomic observations, regardless of whether the method used is dependent on $ M_{\rm{B}} $ andlor not. It is noteworthy that the method for testing the CDDR in this work is not only independent of the cosmological model but also independent of the prior values of the absolute magnitude $ M_{\rm{B}} $ and the linear size scaling factorl. Therefore, QSO measurement can serve as a powerful tool for testing CDDR independently of cosmological model.