# Model-independent test for the cosmic distance duality relation with
Pantheon and eBOSS DR16 quasar sample

Bing Xu  School of Electrical and Electronic Engineering, Anhui Science and
Technology University, Bengbu, Anhui 233030, China  Zhenzhen Wang  Department
of Physics, Anhui Normal University, Wuhu, Anhui 241000, China  Kaituo Zhang
Department of Physics, Anhui Normal University, Wuhu, Anhui 241000, China
<EMAIL_ADDRESS>Qihong Huang  School of Physics and Electronic Science,
Zunyi Normal University, Zunyi 563006, Guizhou, China  Jianjian Zhang  School
of Electrical and Electronic Engineering, Anhui Science and Technology
University, Bengbu, Anhui 233030, China

###### Abstract

In this paper, we carry out a new model-independent cosmological test for the
cosmic distance duality relation (CDDR) by combining the latest five baryon
acoustic oscillations (BAO) measurements and the Pantheon type Ia supernova
(SNIa) sample. Particularly, the BAO measurement from extended Baryon
Oscillation Spectroscopic Survey (eBOSS) data release (DR) 16 quasar sample at
effective redshift $z=1.48$ is used, and two methods, i.e. a compressed form
of Pantheon sample and the Artificial Neural Network (ANN) combined with the
binning SNIa method, are applied to overcome the redshift-matching problem.
Our results suggest that the CDDR is compatible with the observations, and the
high-redshift BAO and SNIa data can effectively strengthen the constraints on
the violation parameters of CDDR with the confidence interval decreasing by
more than 20 percent. In addition, we find that the compressed form of
observational data can provide a more rigorous constraint on the CDDR, and
thus can be generalized to the applications of other actual observational data
with limited sample size in the test for CDDR.

##  1 Introduction

Based on three fundamental hypotheses that the spacetime is described by a
metric theory of gravity, photons always travel along null geodesics and their
number is conserved, Etherington (1993, 2007) proved a famous cosmic distance-
duality relation (CDDR), which connects the luminosity distance (LD) $D_{\rm
L}$ and the angular diameter distance (ADD) $D_{\rm A}$ at the same redshift
$z$ through the following identity

$\frac{D_{\rm{L}}(z)}{D_{\rm{A}}(z)}(1+z)^{-2}=1\,.$ (1)

This relation is independent of the Einstein field equations and the nature of
matter, and has been widely used in astronomical observations and modern
cosmology as a fundamental relation. For example, the CDDR has been used to
test the geometrical shape of galaxy clusters (Holanda et al., 2011), the
temperature profile and the gas mass density profile of galaxy clusters (Cao
et al., 2011, 2016). However, a violation of one of the hypotheses leading to
the CDDR might be possible, which may be considered as a signal of exotic
physics (Bassett & Kunz, 2004a, b). Therefore, it is necessary to test the
reliability of the CDDR accurately before applying to various astronomical
theories.

A straightforward approach to test the validity of CDDR is to constrain the
parameterization function
$\eta(z)\equiv{D_{\rm{L}}(z)}/{D_{\rm{A}}(z)}(1+z)^{-2}$ with the LD and ADD
of some objects at the same redshift. Here the function $\eta(z)$ represents
the possible violation of the standard CDDR, and can be parameterized in
distinct forms, such as $\eta(z,\eta_{1})=1+\eta_{1}z$,
$\eta(z,\eta_{2})=1+\eta_{2}z/(1+z)$, and
$\eta(z,\eta_{3})=1+\eta_{3}\mathrm{ln}(1+z)$, where the parameter $\eta_{i}$
can be constrained by observational data. Following this idea, a lot of works
have been devoted to test the validity of CDDR by combining LD data inferred
from the observations of type Ia supernovae (SNIa), HII galaxies, or gamma-ray
burst with ADD data determined from different observations such as the X-ray
plus Sunyaev-Zel’dovich (SZ) effect of galaxy clusters, and the gas mass
fraction measurement in a galaxy cluster (Uzan et al., 2004; De Bernardis et
al., 2006; Holanda et al., 2010, 2011, 2012; Li et al., 2011; Nair et al.,
2011; Meng et al., 2012; Yang et al., 2013; Santos et al., 2015; Hu & Wang,
2018; da Silva et al., 2020; Bora & Desai, 2021), the angular size of ultra-
compact radio sources (Li & Lin, 2018; Liu et al., 2021), and strong
gravitational lensing (Liao et al., 2016; Holanda et al., 2016, 2017; Ruan et
al., 2018; Lima et al, 2021; Qin et al., 2021). These works do not suggest
that the CDDR deviates significantly from the real universe. In addition,
since the baryon acoustic oscillations (BAO) measurement, which is a very
precise experiment and plays an important role in modern cosmology, can
provide more accurate ADD data than the observations described above, it is
also used to test the CDDR and has been proved to be a very powerful tool to
test the relation (Wu et al., 2015). Recently, some works have attempted to
test the CDDR with LD data from different SNIa observations and the ADD data
derived from different BAO measurements. For example, combining the Union2.1
SNIa data (Suzuki et al., 2012) with five BAO ADD data points from the WiggleZ
Dark Energy Survey (Blake et al., 2012), the Sloan Digital Sky Survey (SDSS)
Data Release 7 (DR7) (Xu et al., 2013) and DR11 (Samushia et al., 2014), Wu et
al. (2015) tested the validity of CDDR and found that there was no violation
of CDDR. A similar result was obtained by Lin et al. (2018), in which the
authors tested the validity of CDDR by using the same BAO data points plus the
ADD data from galaxy clusters and the latest SNIa sample, i.e. the Pantheon
sample from the Pan-STARRS1 Medium Deep Survey which is the largest SNIa
sample released to date and consists of 1048 SNIa data covering the redshift
range of $0.01<z<2.3$ (Scolnic et al., 2018). Xu & Huang (2020) updated the
constraints with the Baryon Oscillation Spectroscopic Survey (BOSS) DR12 data
and the Pantheon sample, and they obtained consistent results. However, it
should be noted that although the redshift of the newest Pantheon SNIa sample
is up to $z\sim 2.3$, these tests still suffer from low redshift range, i.e.
$z<1$, due to the lack of observations at higher redshift. Most recently,
eBOSS collaboration provided a precise BAO measurement from the final quasar
sample of BOSS DR16 at effective redshift $z=1.48$ (Neveux et al., 2020; Alam
et al., 2021). In this paper, we therefore plan to check the validity of CDDR
by combining the newest BAO measurements with the Pantheon sample. Using these
data, the tests of CDDR with the ADD data derived from BAO measurement can
reach high redshift range $z\sim 1.5$.

Here, it should be pointed out that since any deviation from the CDDR can
contribute to the non-conservation of the photon number (Ellis, 2007),
exploring the CDDR is equivalent to testing the cosmic opacity of the
universe. Thus, the parameter of the CDDR can also be constrained by the
parametrization function of optical depth $\tau(z)$ rather than $\eta(z)$ with
their relation being $\eta(z)=e^{\tau(z)/2}$ (Lima et al., 2011). And many
works have been made to perform the constraints on the cosmic opacity by using
various astronomical observations, and the results show that there is no
obvious evidence of an opaque universe (More et al., 2009; Nair et al., 2012;
Chen et al., 2012; Li et al., 2013; Holanda et al., 2013; Liao et al., 2013,
2015; Wang et al., 2017; Ma et al., 2019; Qi et al., 2019b; Wei, 2019; Zhou et
al., 2019; Liu et al., 2020; Fu et al., 2020; Geng et al., 2020; Xu et al.,
2021; He et al., 2022). However, it is worth mentioning that combining the
simulated gravitation waves from DECi-hertz Interferometer Gravitational-wave
Observatory (DECIGO) and the Einstein Telescope (ET) with the observations of
SNIa, HII galaxies and monochromatic X-ray and ultraviolet (UV) luminosity of
quasars, the authors in the references (Liu et al., 2020; Geng et al., 2020)
have tested the cosmic opacity out to high redshifts and found that the
constraint results is slightly sensitive to the parameterization of $\tau(z)$.
In this paper, considering that the expressions of $\eta(z)$ have several
advantages such as a manageable one-dimensional phase space and a good
sensitivity to observational data (Holanda et al., 2011), we only use the
parameterizations of $\eta(z)$ as listed in the previous text to check the
validity of CDDR.

A common issue we will encounter when performing the tests is that
measurements of LD from SNIa data and ADD from BAO data are not available at
the same redshift. In response to this issue, several methods have been
proposed in the literatures. Holanda et al. (2010) and Li et al. (2011)
derived the LD by using the SNIa data whose redshift is closest to the
cluster’s within the range $\Delta z=|z-z_{\rm SNIa}|<0.005$. Meng et al.
(2012) extended the method by binning all SNIa data available in the range
$\Delta z$. Cardone et al. (2012) applied a local regression technique to the
SNIa data at redshift windows of interest with adjustable bandwidth. These
methods fail at $z=1.48$ because the current SNIa data have very few
distributions at high redshifts $z>1$. The values of LD at the redshift of BAO
measurements can also obtain by using the Gaussian Processes (GPs) (Nair et
al., 2015; Rana et al., 2017; Mukherjee & Mukherjee, 2021). However, when the
number of observed events is not enough, GPs are not reliable for the
reconstruction of SNIa (Zhou & Li, 2019) and lead to great uncertainty (Lin et
al., 2018). In addition, Ma & Corasaniti (2018) applied a Bayesian statistical
method to calculate the SNIa luminosity distance moduli at the redshifts of
ADD data from BAO measurements. Although this method can effectively reduce
the uncertainty and consider the correlation between data points, it needs to
assume auxiliary cosmological information to obtain the values of LD and ADD.
In this paper, we use two methods to overcome the redshift-matching problem.
Firstly, we propose a new method to overcome the issue. In this method, a
compressed form of the Pantheon SNIa sample is provided by using a piecewise
linear function of $\mathrm{ln}(z)$ proposed by Betoule et al. (2014), which
is shown that it still remains accurate for cosmological constraints. By this
methods, we can derive the LD values at the redshift of BAO data points self-
consistently from the binned apparent magnitude values based on the linear
function. Most recently, an Artificial Neural Network (ANN) is used to
reconstruct the Hubble diagrams from the observed HII galaxy and radio quasar
samples and then test the CDDR (Liu et al., 2021). The main purpose of using
an ANN is to construct an approximate function that associates input data with
output data. This machine learning reconstruction method is data-driven and
makes no assumptions about the data, which suggests that it is completely
model-independent, and have shown outstanding performance in solving
cosmological problems in both accuracy and efficiency, such as analyzing
gravitational waves (Li et al., 2020; George & Huerta, 2018), estimating
cosmological parameters (Fluri et al., 2019; Wang et al., 2020b, 2021), and
studying the evolution of dark energy models (Escamilla-Rivera et al., 2020).
Specially, it has been shown to be able to reconstruct a safer function than
GPs when there are fewer data points (Wang et al., 2020a). In order to see
whether the conclusions are changed by a different method and improve the
robustness of the test, we will then apply the ANN method to reconstruct the
apparent magnitude-redshift relation $m_{\rm B}(z)$ which is used to test the
CDDR by combining the newest BAO measurements. Furthermore, in our analysis,
the nuisance parameters used in the construction of likelihood function, such
as the absolute B-band magnitude $M_{\rm B}$ and the fiducial value of sound
horizon $r_{\rm d,f}$, are marginalized analytically with flat priors to avoid
the tests that rely on any auxiliary cosmological information.

This paper is organized as follows: we introduce the data and methodology used
in our analysis in Section 2. In section 3, we apply the method to test the
CDDR and yield results. Conclusions and discussions are presented in Section
4.

##  2 Data and Methodology

In this work, we aim to use ADD derived from the newest BAO measurements and
LD derived from the latest Pantheon sample to test the CDDR by constraining
the function $\eta(z)$ parameterized with $\eta_{i}$. So we now introduce
briefly the BAO data used in our analysis. The BAO refers to an overdensity or
clustering of baryonic matter at certain length scales due to the oscillations
of acoustic waves which propagated in the early universe. It happens on some
typical scales and can provide a standard ruler for length scale in cosmology
to explore the expansion history of the universe. The length of this standard
ruler ($\sim$150 Mpc in today’s universe) corresponds to the distance that a
sound wave propagating from a point source at the end of inflation would have
traveled before decoupling. Most recently, the eBOSS Collaboration have
released their final BAO observations and summarized fourteen measurements of
$D_{\rm V}(z)/r_{\rm d}$, $D_{\rm M}(z)/r_{\rm d}$ and $D_{\rm H}(z)/r_{\rm
d}$, covering the redshift range $0.15\leq z\leq 2.33$ (see Tabel III in Ref.
(Alam et al., 2021)). Here, $r_{\rm d}$ is the sound horizon, $D_{\rm V}(z)$,
$D_{\rm M}(z)$ and $D_{\rm H}$ is the spherically-averaged distance, the
comoving ADD and the Hubble distance, respectively. These measurements are
obtained from the final observations of clustering using galaxies, quasars,
and Ly$\alpha$ forests from the completed SDSS lineage of experiments in
large-scale structure, composing of data from SDSS, SDSS-II, BOSS, and eBOSS,
and thus allow us to perform a comprehensive assessment of the cosmological
model and parameter. In this paper, given that the largest redshift of
Pantheon sample $z<2.3$ (Scolnic et al., 2018) and the ADD is associate with
the comoving ADD through the relation $D_{\rm M}=(1+z)D_{\rm A}$, we will only
use the five measurements of $D_{\rm M}(z)/r_{\rm d}$ from the SDSS-III BOSS
DR12 galaxy sample (Alam et al., 2017), the SDSS-IV eBOSS DR16 LRG sample
(Gil-Marín et al., 2020; Bautista et al., 2021), the SDSS-IV eBOSS DR16 ELG
sample (Tamone et al., 2020; de Mattia et al., 2021) and the SDSS-IV eBOSS
DR16 quasar sample (Neveux et al., 2020; Hou et al., 2021), which are
summarized in Table 1. Here, it is necessary to point out that although the
BAO measurements are widely used in cosmological analysis, the so-called
fitting problem still remains a challenge for BAO peak location as a standard
ruler (Ellis & Stoeger, 1987). In particular, the environmental dependence of
the BAO location has recently been detected by Roukema et al. (2015, 2016).
Moreover, Ding et al. (2015) and Zheng et al. (2016) pointed out a noticeable
systematic difference between H(z) measurements based on BAO and those
obtained with differential aging techniques. Since these problems are related
to the calibration of $r_{\rm d}$, we will therefore introduce a fiducial
value of sound horizon $r_{\rm d,f}$ as a nuisance parameter to calculate the
values of ADD from the measurements of BAO and marginalize its influence with
a flat prior in the analysis.

Table 1: Summary of the dimensionless $D_{\rm M}(z)/r_{\rm d}$ measurements from different BAO samples. $z_{\rm eff}$ | $\mathrm{value}$ | $\mathrm{Survey}$ | $\mathrm{Reference}$
---|---|---|---
$0.38$ | $10.23\pm 0.17$ | $\mathrm{BOSS\,Galaxy}$ | Alam et al. (2017)
$0.51$ | $13.36\pm 0.21$ | $\mathrm{BOSS\,Galaxy}$ | Alam et al. (2017)
$0.70$ | $17.86\pm 0.33$ | $\mathrm{eBOSS\,LRG}$ | Gil-Marín et al. (2020)
$0.85$ | $19.5\pm 1.0$ | $\mathrm{eBOSS\,ELG}$ | Tamone et al. (2020)
$1.48$ | $30.69\pm 0.80$ | $\mathrm{eBOSS\,Quasar}$ | Neveux et al. (2020)

Now we turn our attention on the Pantheon compilation released by the Pan-
STARRS1 Medium Deep Survey (Scolnic et al., 2018), which consists of 1048 SNIa
data covering the redshift range $0.01<z<2.3$. And the observed distance
modulus of each SNIa in this compilation is given by

$\mu_{\rm obs}(z)=m^{*}_{\rm B}-(M_{\rm B}-\alpha X_{1}+\beta\mathcal{C})\,.$
(2)

Here, $m^{*}_{\rm B}$ is the observed peak magnitude in rest frame B-band,
$X_{1}$ is the time stretching of the light-curve, $\mathcal{C}$ is the SNIa
color at maximum brightness, $M_{\rm B}$ is the absolute magnitude, $\alpha$
and $\beta$ are two nuisance parameters, which should be fitted simultaneously
with the cosmological parameters. In order to avoid the dependence of $\alpha$
and $\beta$ on the cosmological model, Kessler & Scolnic (2017) proposed a new
method called BEAMS with Bias Corrections (BBC) to calibrate the SNIa, and the
corrected apparent magnitude $m^{\ast}_{\rm B,corr}=m^{\ast}_{\rm B}+\alpha
X_{1}-\beta C+\Delta_{\rm B}$ for all the SNIa is reported in Ref. (Scolnic et
al., 2018) with $\Delta_{\rm B}$ being the correction term. And the observed
distance modulus is rewritten as

$\mu_{\rm obs}=m^{*}_{\rm B,corr}-M_{\rm B}\,.$ (3)

To test the validity of the CDDR with the BAO measurements and Pantheon SNIa
sample, we must obtain the values of LD from SNIa data at the redshifts of BAO
data. For this purpose, we use two methods to derive the apparent magnitude
values at the redshifts of BAO measurements, in order to achieve the CDDR
testing and obtain convincing results. In the first way, we provide the
cosmological information of the Pantheon SNIa sample in a compressed form with
the method proposed by Betoule et al. (2014). Here, instead of binning the
distance modulus used in (Betoule et al., 2014), the corrected apparent
magnitude is approximated by a piecewise linear function of $\mathrm{ln}(z)$
by defining on each segment $z_{b}\leq z\leq z_{b+1}$ as:

$\overline{m}_{\rm B}(z)=(1-\alpha)m_{{\rm B},b}+\alpha m_{{\rm B},b+1}\,.$
(4)

Here $\alpha=\mathrm{ln}(z/z_{b})/\mathrm{ln}(z_{b+1}/z_{b})$, $m_{{\rm B},b}$
is the apparent magnitude at $z_{b}$. In consideration of the smallest and
largest redshift of the Pantheon sample, we use 36 ln-spaced control points
$z_{b}$ in the redshift range $0.01<z<2.3$. Then using the above interpolation
function to fit the Pantheon data, we can obtain the compressed form by
minimizing the $\chi^{2}$ function,

$\chi^{2}=\left[\mathbf{m}^{*}_{\rm B}-\mathbf{\overline{m}}_{\rm
B}\right]^{\rm T}\cdot\mathbf{Cov}^{-1}\cdot\left[\mathbf{m}^{*}_{\rm
B}-\mathbf{\overline{m}}_{\rm B}\right].$ (5)

Here, $\mathbf{Cov}=\mathbf{D}_{\rm stat}+\mathbf{C}_{\rm sys}$ is the
covariance matrix, where the statistical matrix $\mathbf{D}_{\rm stat}$ has
only diagonal elements and $\mathbf{C}_{\rm sys}$ is the systematic
covariance.

Figure 1:  Left: The exact measurements of apparent magnitude in Pantheon
sample (blue) and its compressed form (red). Right: Comparison of the
cosmological constraints obtained from the exact Pantheon sample likelihood
(filled blue contour) with compressed version (dashed red contour).

In order to constrain the 36 parameter spaces of $m_{\rm B,b}$, we have
modified the publicly available Markov Chain Monte Carlo (MCMC) package
CosmoMC (Lewis & Bridle, 2002) to conduct the calculation. We list the binned
apparent magnitude in Table 2 with their covariance matrix given in Table A1
in the section of Appendix. For a visual comparison, we also plot the exact
measurements of apparent magnitude in Pantheon sample and its compressed form
in the left of Figure 1. One can see that the compressed form of Pantheon
sample can provide a good approximation of the relationship between the
apparent magnitude and redshift revealed by the full exact measurements.
Furthermore, as an illustration, a comparison of the cosmological constraints
obtained from the approximate and full version of the Pantheon likelihood for
the $w$CDM model is shown on the right side of Figure 1. It is clear that the
differences of the constraint results from the approximate and full version
are very small. This means that the compressed form still remains accurate for
cosmological constraints.

Using the compressed form listed in Table 2 and the interpolation function
given in Equation 4, we can derive the values of $\overline{m}_{\rm B}$ from
the compressed form of Pantheon sample at the redshift of BAO measurements.
Instead of interpolating directly from the two nearest observations, the
values of $\overline{m}_{\rm B}$ obtained in our analysis take into account
the influence of all observations in two adjacent redshift intervals at a
chosen redshift. This can effectively solve the negative impacts caused by the
small sample size, such as great uncertainty. We refer to this method as ‘I’
and summarize the interpolated values of $\overline{m}_{\rm B}$ at the
redshifts of BAO data in Table 3.

Table 2: Binned apparent magnitude fitted to the Pantheon sample. $z_{b}$ | $m_{{\rm B},b}$ | $z_{b}$ | $m_{{\rm B},b}$ | $z_{b}$ | $m_{{\rm B},b}$ | $z_{b}$ | $m_{{\rm B},b}$
---|---|---|---|---|---|---|---
0.010 | $13.909\pm 0.143$ | $0.041$ | $16.862\pm 0.046$ | $0.164$ | $20.078\pm 0.026$ | $0.664$ | $23.633\pm 0.042$
0.012 | $14.131\pm 0.132$ | $0.047$ | $17.234\pm 0.050$ | $0.191$ | $20.530\pm 0.022$ | $0.775$ | $24.082\pm 0.037$
0.014 | $14.604\pm 0.088$ | $0.055$ | $17.536\pm 0.043$ | $0.224$ | $20.882\pm 0.022$ | $0.905$ | $24.503\pm 0.041$
0.016 | $14.751\pm 0.059$ | $0.065$ | $17.965\pm 0.049$ | $0.261$ | $21.231\pm 0.020$ | $1.058$ | $24.893\pm 0.073$
0.019 | $15.209\pm 0.077$ | $0.075$ | $18.294\pm 0.050$ | $0.305$ | $21.663\pm 0.021$ | $1.235$ | $25.433\pm 0.117$
0.022 | $15.482\pm 0.053$ | $0.088$ | $18.695\pm 0.041$ | $0.356$ | $22.042\pm 0.022$ | $1.443$ | $25.572\pm 0.152$
0.025 | $15.839\pm 0.041$ | $0.103$ | $19.094\pm 0.032$ | $0.416$ | $22.469\pm 0.028$ | $1.686$ | $26.248\pm 0.222$
0.030 | $16.204\pm 0.041$ | $0.120$ | $19.386\pm 0.026$ | $0.486$ | $22.836\pm 0.030$ | $1.969$ | $26.125\pm 0.295$
0.035 | $16.550\pm 0.036$ | $0.140$ | $19.825\pm 0.025$ | $0.568$ | $23.272\pm 0.031$ | $2.300$ | $26.967\pm 0.294$
Table 3: the values of $\overline{m}_{\rm B}$ at the redshifts of BAO measurements obtained from two methods. ${\rm method}$ | $0.38$ | $0.51$ | $0.70$ | $0.85$ | $1.48$
---|---|---|---|---|---
${\rm I}$ | $22.218\pm 0.024$ | $22.969\pm 0.030$ | $22.787\pm 0.040$ | $24.331\pm 0.039$ | $25.682\pm 0.164$
${\rm II}$ | $22.236\pm 0.045$ | $22.990\pm 0.044$ | $23.824\pm 0.061$ | $24.317\pm 0.071$ | $25.785\pm 0.180$

In order to see whether the conclusions are changed by a different method and
improve the robustness of the conclusion, we then use Artificial Neural
Network (ANN) to reconstruct the $m_{\rm B}(z)$ function from the observation
data and obtain the value of $m_{\rm B}$ from the reconstructed function to
overcome the redshift-matching problem at high redshifts. In general terms,
ANN is a Deep Learning algorithm consisting of 3 layers - Input, Hidden and
Output. The input layer has $n$ nodes, each of which inputs an independent
variable, followed by the $m$ linked hidden layers and the output layer with
activation function nodes in the basic architecture (Schmidhuber, 2015). By
using adam optimization (Kingma & Ba, 2014), 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. The process can be conveniently described in a vectorized
way (Wang et al., 2020a),

$\bm{z}_{i+1}=\bm{b}_{i+1}+\bm{x}_{i}\bm{W}_{i+1}\,,$ (6)
$\bm{x}_{i+1}=f(\bm{z}_{i+1})\,,$ (7)

where $\bm{x}_{i}$ is the input vector at the $i$th layer, $\bm{b}_{i+1}$, and
$\bm{W}_{i+1}$ are the offset vector and linear weights matrix with learnable
parameter elements, $\bm{z}_{i+1}$ is the output vector after linear
transformation of $\bm{x}_{i}$, and $f$ is the Exponential Linear Unit (ELU)
activation function (Clevert et al., 2015), which has the form

$\displaystyle f(x)=\begin{cases}x&x>0\\\ \alpha(\exp(x)-1)&x\leq
0\end{cases}\,,$ (8)

where $\alpha$ denotes the hyper-parameter that controls the value to which an
ELU saturates for negative net inputs and is set to 1 (Wang et al., 2020a).

The key of ANN is to find the function $f_{W,\mathbf{b}}$ which makes its
output $f_{W,\mathbf{b}}(\mathbf{x})$ as close to the target value
$\mathbf{y}$ as possible. To achieve this goal, the difference between the
predicted value $f_{W,\mathbf{b}}(\mathbf{x})$ of the current network and the
target value $\mathbf{y}$, equal to the defined loss function $\mathcal{L}$,
should be minimum. During the training and evaluation strategy, the weight
matrix of each layer is updated constantly to minimize $\mathcal{L}$.
Depending on the gradient descent method used, ANN reduces the loss value by
continuously moving the loss value in the opposite direction of the current
corresponding gradient. Formally, in a vectorized way (LeCun et al., 2012)

$\displaystyle\frac{\partial\mathcal{L}}{\partial z_{i+1}}$
$\displaystyle=f^{\prime}(z_{i+1})\frac{\partial\mathcal{L}}{\partial
x_{i+1}}\,,$ $\displaystyle\frac{\partial\mathcal{L}}{\partial W_{i+1}}$
$\displaystyle=x_{i}^{T}\frac{\partial\mathcal{L}}{\partial z_{i+1}}\,,$
$\displaystyle\frac{\partial\mathcal{L}}{\partial x_{i}}$
$\displaystyle=W_{i+1}^{T}\frac{\partial\mathcal{L}}{\partial z_{i+1}}\,,$
$\displaystyle\frac{\partial\mathcal{L}}{\partial{b}_{i+1}}$
$\displaystyle=\left(\frac{\partial\mathcal{L}}{\partial z_{i+1}}\right)\,,$
(9)

where the operator $\partial$ stands for partial derivatives, and $f^{\prime}$
is the derivative of the nonlinear function $f$.

Using the publicly released code called Reconstruct Functions with ANN
(ReFANN) 111https://github.com/Guo-Jian-Wang/refann (Wang et al., 2020a),
which explicitly describe the ANN method, we reconstruct the apparent
magnitude-redshift relation as a function of logarithmic redshift $\left(\rm
lnz\right)$ and show the function with the estimation of $1\sigma$ confidence
region in Figure 2. From Figure 2, it is easy to see that the uncertainties of
the reconstructed function by ANN are almost equal to that of the
observations, and the $1\sigma$ confidence region reconstructed by ANN can be
considered as the average level of observational error. We refer the reader to
Ref. (Wang et al., 2020a) for further details on this issue. Therefore, it is
difficult to achieve a robust CDDR test by using the $m_{\rm B}$ values
obtained from the reconstructed function directly. Given that the sample size
of Pantheon with redshift $z<1$ is sufficient to apply the binning method
(Meng et al., 2012), we bin the actual observational sample by taking $\lvert
z_{\rm SNIa}-z_{\rm BAO}\rvert<0.005$ to obtain the values of $m_{\rm B}$. Due
to the limited sample size of Pantheon at the redshift around $z=1.48$ and the
failure of binning method, the value with $m_{\rm B}=25.785\pm 0.180$ derived
from the reconstructed function is used. This method is denoted as ‘II’. We
summarize the values obtained by the method of combining the binned SNIa and
ANN at the Table 3.

Figure 2:  The reconstructed function $m_{\rm B}(z)$ with corresponding
$1\sigma$ errors by using ANN (black line), and the exact measurements of
apparent magnitude in Pantheon sample (red).

Using the measurements of $D_{\rm M}/r_{\rm d}$ given in Table 1 and
$\overline{m}_{\rm B}$ in Table 3, we can obtain the best-fitting CDDR
parameters $\eta_{i}$ and the confidence regions by minimizing the following
$\chi^{2}$ function,

$\chi^{2}=\sum^{5}_{i=1}\frac{{\Delta\mu_{i}}^{2}}{s^{2}_{i}}\,,$ (10)

where $\Delta\mu_{i}=\Delta_{i}-\mathcal{M}=5\mathrm{log_{10}}[D_{\rm
M}/r_{\rm d}(1+z)\eta(z)]-\overline{m}_{\rm B}-\mathcal{M}$ with
$\mathcal{M}=M_{\rm B}-5\mathrm{log_{10}}r_{\rm d,f}-25$,
$s^{2}_{i}=\sigma^{2}_{\mu_{i,{\rm BAO}}}+\sigma^{2}_{\mu_{i,{\rm SN}}}$ with
$\sigma_{\mu_{\rm BAO}}=5\sigma_{D_{\rm M}/r_{\rm d}}/\left(D_{\rm M}/r_{\rm
d}\mathrm{ln}10\right)$, and $\sigma_{\mu_{\rm SN}}=\sigma_{\overline{m}_{\rm
B}}$. In our analysis, we marginalize analytically the likelihood function
over $\mathcal{M}$, the combination of $M_{B}$ and $r_{\rm d,f}$, with the
method proposed in (Conley et al., 2011) by assuming a flat prior on
$\mathcal{M}$. And the marginalized $\chi^{2}$ is written as

$\chi^{2}_{\rm marg}=a-\frac{b^{2}}{f}+\mathrm{ln}\frac{f}{2\pi}\,,$ (11)

where $a=\sum\limits_{i=1}^{5}\Delta^{2}_{i}/s^{2}_{i}$,
$b=\sum\limits_{i=1}^{5}\Delta_{i}/s^{2}_{i}$ and
$f=\sum\limits_{i=1}^{5}1/s^{2}_{i}$. Here, it should be pointed out that
although the values of binned apparent magnitude are correlated, it is
difficult to obtain correlations of the values of $\overline{m}_{\rm B}$ used
in the tests. Given that the value of $\sigma_{\mu_{\rm BAO}}$ used in
Equation 10 is greater than the one of $\sigma_{\overline{m}_{\rm B}}$ at the
same redshift and the values of $D_{\rm M}/r_{\rm d}$ are not correlated
except the two data points at low redshift, we have therefore ignored the
correlation of all data points in the analysis.

##  3 Results

Table 4: Summaries of the 68% limits of $\eta_{1}$, $\eta_{2}$ and $\eta_{3}$ from the low-redshift data points (Sub) and the all data points (All) by using two methods. $\mathrm{method\,I}$ | $\eta_{1}$ | $\eta_{2}$ | $\eta_{3}$
---|---|---|---
Sub | $-0.052^{+0.085}_{-0.077}$ | $-0.137^{+0.203}_{-0.177}$ | $-0.085^{+0.132}_{-0.118}$
All | $-0.064^{+0.057}_{-0.052}$ | $-0.181^{+0.160}_{-0.141}$ | $-0.110^{+0.097}_{-0.088}$
$\mathrm{method\,II}$ | $\eta_{1}$ | $\eta_{2}$ | $\eta_{3}$
Sub | $-0.037^{+0.110}_{-0.097}$ | $-0.101^{+0.269}_{-0.225}$ | $-0.061^{+0.173}_{-0.149}$
All | $-0.039^{+0.070}_{-0.062}$ | $-0.119^{+0.207}_{-0.176}$ | $-0.070^{+0.122}_{-0.107}$

Since that the low redshift $(z<1)$ SNIa plus BAO data have been used to test
the validity of CDDR in the literatures (Wu et al., 2015; Lin et al., 2018; Xu
& Huang, 2020), for a comparison, we perform the constraints using the first
four data points (Sub) and all five data points (All) listed in Table 3,
respectively, in order to test the constraint ability of high redshift BAO
measurement on the CDDR.

Figure 3: The likelihood distributions of $\eta_{1}$, $\eta_{2}$ and
$\eta_{3}$ from the low-redshift data points (Sub) and the all data
points(All) by using method I.

Figure 4: The likelihood distributions of $\eta_{1}$, $\eta_{2}$ and
$\eta_{3}$ from the low-redshift data points (Sub) and the all data
points(All) by using method II.

We first focus on the results obtained using method I and show the
marginalized likelihood distributions of CDDR parameters $\eta_{i}$ are shown
in Figure 3 with the corresponding 68% limits summarizing in Table 4. From
Figure 3 and Table 4, one can see that when the low redshift data is used, the
CDDR is valid at $1\sigma$ CL for all three parameterizations with
$\eta_{1}=-0.052^{+0.085}_{-0.077}$, $\eta_{2}=-0.137^{+0.203}_{-0.177}$,
$\eta_{3}=-0.085^{+0.132}_{-0.118}$. However, once the data point at redshift
$z=1.48$ is combined, the different results are obtained. In this case,
$\eta_{1}=-0.064^{+0.057}_{-0.052}$, $\eta_{2}=-0.181^{+0.160}_{-0.141}$,
$\eta_{3}=-0.110^{+0.097}_{-0.088}$, respectively, which means that the CDDR
is valid at $2\sigma$ CL for all three parameterizations. This suggests that
the tests of CDDR are not sensitive to the parametrization of $\eta(z)$, and
there is no obvious evidence for violation of the CDDR. In addition, it is
easy to see that by comparing the results obtained from the low redshift data
with $\Delta\eta_{1}=0.162$, $\Delta\eta_{2}=0.380$, $\Delta\eta_{3}=0.250$,
there is a significant improvement on the constraints on $\eta_{i}$ with
$\Delta\eta_{1}=0.109$, $\Delta\eta_{2}=0.301$, $\Delta\eta_{3}=0.185$, and
the confidence intervals are reduced by 33%, 21%, 26% for the three
parameterizations respectively when the high redshift data is added.

We then turn to the results in Figure 4 and Table 4 obtained by using method
II. Similar to the results obtained by using method I, the CDDR is consistent
well with the SNIa and BAO observations. From the low redshift data,
$\eta_{1}=-0.037^{+0.110}_{-0.097}$, $\eta_{2}=-0.101^{+0.269}_{-0.225}$,
$\eta_{3}=-0.061^{+0.173}_{-0.149}$, and $\eta_{1}=-0.039^{+0.070}_{-0.062}$,
$\eta_{2}=-0.119^{+0.207}_{-0.176}$, $\eta_{3}=-0.070^{+0.122}_{-0.107}$ from
all data. In other words, the high redshift data improves the constraints on
the violation parameters from $\Delta\eta_{1}=0.207$, $\Delta\eta_{2}=0.494$,
$\Delta\eta_{3}=0.322$ to $\Delta\eta_{1}=0.132$, $\Delta\eta_{2}=0.383$,
$\Delta\eta_{3}=0.229$ with the precisions increasing by 36%, 22% and 29%
corresponding to the first, second and third parameterized forms. In addition,
we can see that each $1\sigma$ error is obviously larger than that from the
method I. This is the advantage of method I that the observations can make a
more rigorous constraint on the CDDR as more SNIa data are used in this method
to derive more precise values of $m_{\rm B}$ at the BAO redshifts.

In order to highlight the constraint ability of the newest BAO measurements,
it is necessary to compare our results with the previous constraints on
$\eta_{i}$ from different data sets of SNIa and BAO. Combining five BAO data
covering the redshift $0.44\leq z\leq 0.57$ with the binned SNIa data in the
range $\Delta z=|z_{\rm BAO}-z_{\rm SNIa}|<0.005$ from Union2.1 sample, Wu et
al. (2015) tested the CDDR with the first two parameterizations used in this
paper. They found that when the dimensionless Hubble constant $h$ is
marginalized with a flat prior, the constraints on $\eta_{i}$ is very weak
with $\eta_{1}=-0.174^{+0.253}_{-0.199}$ and
$\eta_{2}=-0.409^{+0.529}_{-0.381}$. Using the latest Pantheon sample and BOSS
DR12 BAO measurements covering the redshift $0.31\leq z\leq 0.72$, Xu & Huang
(2020) updated the constraints, and found that $\eta_{1}=-0.07\pm 0.12$,
$\eta_{2}=-0.20\pm 0.27$ and $\eta_{3}=-0.12\pm 0.18$, which revealed more
stringent constraints. Furthermore, Lin et al. (2018) tested the CDDR by
combining the ADD data from galaxy clusters and BAO measurements with the
Pantheon sample. In their work, besides the five BAO data used in (Wu et al.,
2015), the authors added one BAO data point at redshift $z=2.34$. However,
they found that the results are almost unaffected when the high redshift BAO
data is added, because that the reconstructed $\mu-z$ function with the
Gaussian processes has large uncertainty at high redshift, and the best-
fitting values are $\eta_{1}=-0.04\pm 0.12$ and $\eta_{2}=-0.05\pm 0.22$.
Through these comparisons, one can conclude that newest BAO data, especially
the eBOSS DR16 data at effective redshift $z=1.48$, could effectively
strengthen the constraints on the violation parameters of CDDR.

Finally, it is worth comparing our results with some tests that have been
already studied by using other high-redshift astrophysical probes to obtain
ADD previously. In particular, many efforts have been made to perform robust
tests of CDDR by combining the ADDs derived from ultra-compact structure in
radio quasars (Cao et al., 2017) with the LDs obtained from Pantheon SNIa
sample (He et al., 2022), the relation between the UV and X-ray luminosities
of quasars (Zheng et al., 2020), observations of HII galaxies (Liu et al.,
2021), and simulated gravitational wave data (Qi et al., 2019a), respectively.
In these works, the authors tested CDDR in the redshift range $z>2$, and found
that the validity of CDDR was in good agreement with the current observational
data. More notably, their results showed that these combinations of
observational data could derive robust constraints on the violation parameter
at the precision of $10^{-2}$ or higher, whereas in our work, the constraint
results obtained from current SNIa and BAO data do not achieve such accuracy.
However, given that those works all use simulated data to solve the redshift
matching problem, while we directly use five data points derived from the
actual observational data, we can expect a better validity check of the CDDR
as more high-precision, high-redshift observations of available BAO and SNIa.

##  4 Conclusion and Discussions

The CDDR plays a fundamental role in astronomical observations and modern
cosmology, while it may be violated if one of the assumptions underlying this
relation is not true. In this paper, we have proposed a new model-independent
test for the CDDR with the Pantheon SNIa sample and the newest BAO
measurements including the eBOSS DR16 quasar sample at effective redshift
$z=1.48$. In our analysis, three parameterized forms
$\eta\left(z,\eta_{1}\right)=1+\eta_{1}z$,
$\eta\left(z,\eta_{2}\right)=1+\eta_{2}z/(1+z)$, and
$\eta\left(z,\eta_{3}\right)=1+\eta_{3}{\rm ln}(1+z)$ are used to describe the
possible violation of CDDR. In particular, two methods are used to derive the
values of $m_{\rm B}$ at the redshifts of BAO measurements to overcome the
redshift-matching problem. We first provide a compressed form of the Pantheon
SNIa sample by using a piecewise linear function of $\mathrm{ln}(z)$, which
shows that it still remains accurate for cosmological constraints, to derive
the values of $m_{\rm B}$ at the redshifts of BAO measurements. We then obtain
the values by using the binning SNIa method at redshift $z<1$ and the ANN
method at $z=1.48$.

The results show that the tests of CDDR are not sensitive to the
parametrization of $\eta(z)$. And there is no obvious violation of the CDDR.
Moreover, it is observed that the high redshift BAO and SNIa data can
effectively strengthen the constraints on the parameters of CDDR. The
confidence intervals of the violation parameter in the three parameterizations
are decreased by 33%, 21%, 26% in the framework of first method, while they
are reduced by 36%, 22%, 29% in the second method, respectively. Furthermore,
by comparing the results obtained from the two methods, we find that applying
the method of compressed form of Pantheon sample, we can use more actual SNIa
data to derive a more precise value of $m_{\rm B}$ and thus get more rigorous
constraints on the violation parameter.

As the final remarks, due to the lack of high redshift SNIa sample, the test
of CDDR with BAO and SNIa data is hard to reach a higher redshift, though
eBOSS collaboration has release one other high redshift data, i.e. the
measurement with Ly$\rm\alpha$ absorption and quasars from BOSS and eBOSS at
an effective redshift $z=2.33$ (du Mas des Bourboux et al., 2020). We can
expect that the validity of the CDDR will be better checked with the more
accurate and wider redshift range measurements of BAO and SNIa from the
further observations, such as the Euclid Satellite (Amendola et al., 2018) and
Dark Energy Spectroscopic Instrument (DESI, Aghamousa et al. (2016)).
Furthermore, the method of compressed form of observations proposed in this
paper presents a new idea of overcoming the redshift-matching problem in high
redshift range. We therefore expect that the method can be extended to other
high redshift standard candles with limited sample size, such as gamma ray
burst and HII galaxies, to test the CDDR, which is an interesting topic for
future investigation.

We thank the anonymous reviewer for his/her very enlightening comments. We are
very grateful to Guojian Wang for his introduction of the ReFANN code. This
work was supported by the National Natural Science Foundation of China under
grants Nos. 11505004, 11865018, and 12192221, and the Anhui Provincial Natural
Science Foundation of China (1508085QA17).

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##  Appendix A The covariance matrix of binned apparent magnitude parameters

The covariance matrix of 36 binned apparent magnitude parameters $m_{\rm B,b}$
obtained by running CosmoMC is given in Table A1.

$\setcounter{MaxMatrixCols}{36}\begin{pmatrix}$203884$&$-52108$&$20037$&$-370$&$
4708 $&$2351 $&$2861 $&$ 2619 $&$2786 $&$ 3303 $&$ 2692$&$ 2675 $&$ 561 $&$
2694 $&$ -909 $&$-405 $&$ -282 $&$-1082 $&$-777 $&$ -943 $&$ -414 $&$ -144 $&$
-32 $&$ -841 $&$ -1417 $&$ 291 $&$ -1732 $&$-911 $&$ -1627 $&$ -1142 $&$ 680
$&$ 572 $&$3094 $&$ 3949 $&$ 2467 $&$ 4738$\\\ $ $&$173352 $&$ -43188 $&$17031
$&$ -5196 $&$ 5404$&$2332$&$ 2652$&$ 2488 $&$3113 $&$1905 $&$2768$&$ 1324
$&$797$&$ -962 $&$-482 $&$-186 $&$-650 $&$-1513 $&$-964$&$ -525$&$ 452
$&$-394$&$ -382 $&$-1286 $&$56 $&$-2119$&$ -1080$&$ -1784$&$ -487$&$ 879$&$
1421$&$ 1311$&$ 2460 $&$4089$&$ -276$\\\ $ $&$ $&$77426$&$ -17286$&$ 14544$&$
-167$&$ 3000$&$ 2258$&$ 2767 $&$ 3767 $&$1849$&$ 2504 $&$1318 $&$ 2469$&$ -894
$&$-270 $&$-139$&$ -803$&$ -1051 $&$ -871 $&$-427$&$ -275 $&$52 $&$-752 $&$
-1044 $&$ 54 $&$ -1961$&$ -531 $&$-2295$&$ -758 $&$ -316 $&$395$&$ 675 $&$4616
$&$-452 $&$ 4355 $\\\ $ $&$ $&$ $&$ 34537$&$ -14612 $&$ 8038 $&$1414$&$ 2761
$&$2200 $&$ 2337 $&$1921 $&$2215 $&$1179 $&$ 1554 $&$-632 $&$-572 $&$-464 $&$
-760 $&$-1322 $&$-976 $&$ -649 $&$ 13 $&$-435 $&$-393 $&$-1001 $&$ 33 $&$
-1623 $&$ -1351 $&$ -1600 $&$ -354 $&$872 $&$594 $&$ 2828 $&$2235 $&$ 4016$&$
2628$\\\ $ $&$ $&$ $&$ $&$58692 $&$ -6054 $&$4238 $&$ 1148 $&$2185 $&$1128 $&$
772 $&$939 $&$115 $&$ 862 $&$-1425$&$ -1031 $&$-423 $&$ -702 $&$-1129 $&$
-1026 $&$ -731 $&$-144 $&$-421 $&$ -196 $&$-857 $&$176 $&$-712 $&$ -1201$&$
-846 $&$-401 $&$ 1169 $&$ 1583 $&$1373 $&$3621 $&$1674 $&$ 2137$\\\ $ $&$ $&$
$&$ $&$ $&$28229 $&$ -3112$&$ 3304 $&$1613 $&$ 1225 $&$ 1142 $&$1169 $&$443
$&$979 $&$ -1034$&$ -728 $&$-414 $&$ -688 $&$ -1138 $&$ -930 $&$-650$&$ -129
$&$ -448 $&$-275$&$ -691 $&$87 $&$-1228 $&$ -950 $&$-875$&$ -226 $&$1209$&$
1011 $&$2292 $&$2750 $&$ 3186 $&$ 3863$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$16952$&$
-2567 $&$3003 $&$ 1863 $&$ 1288 $&$1681 $&$ 846 $&$1168$&$ -614 $&$-214$&$
-240 $&$ -551 $&$-773 $&$ -449$&$ -458 $&$-52$&$ -128 $&$ -601$&$ -760 $&$31
$&$-1226 $&$-411 $&$ -1399 $&$ -502 $&$889 $&$ -20 $&$ 2123 $&$2599 $&$2211
$&$3547$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ 16613$&$-2115 $&$2756 $&$ 632 $&$
1008 $&$511 $&$ 870 $&$-450 $&$ -253$&$ -365 $&$-507 $&$-916 $&$-615$&$ -383
$&$ 52 $&$-31 $&$-356 $&$ -810$&$ 54 $&$-1231 $&$ -346 $&$ -1326 $&$-206 $&$
698 $&$1210 $&$ 2498 $&$ 2907 $&$ 3060 $&$ 3295$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ 13173 $&$ -1131 $&$ 1980 $&$1201 $&$ 522 $&$1154 $&$ -696 $&$-199 $&$
-327 $&$-666 $&$-910 $&$-690 $&$ -497 $&$-36 $&$ -123 $&$-469 $&$ -729 $&$168
$&$ -889 $&$-424 $&$ -1127 $&$ -239 $&$931 $&$ 663 $&$ 1443 $&$2335 $&$2143
$&$1982$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$21518$&$ -1913$&$ 4007$&$
1419$&$ 2100 $&$-481 $&$-316 $&$ -261 $&$-545$&$ -950 $&$-658 $&$ -458
$&$-106$&$ -352 $&$-414 $&$-852$&$ -83 $&$-988 $&$ -547 $&$-1415$&$ -583$&$
214$&$ 485$&$ 679 $&$2047$&$ 1649 $&$1614 $\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$24705 $&$-492 $&$1642 $&$1465 $&$ -834 $&$-713$&$ 112 $&$-472$&$
-595 $&$-418 $&$ -424 $&$ -88 $&$-420 $&$ 153 $&$-901 $&$ 222$&$ -483 $&$-1601
$&$-425 $&$ -487 $&$-288$&$ -1973 $&$-307$&$ -2116 $&$ -2838 $&$ -2060 $\\\ $
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$18883$&$ -1433 $&$ 3199 $&$-432
$&$-314 $&$186 $&$ -341$&$ -368 $&$-755 $&$-237 $&$-130$&$ -137 $&$-433$&$
-712 $&$-163$&$ -896 $&$-730 $&$-1297 $&$ -1246 $&$ -485 $&$-265 $&$293
$&$-159 $&$ -410 $&$460 $\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$23573$&$ -3649 $&$ 196 $&$-242 $&$ 60 $&$-131$&$ -332 $&$-95 $&$ -290 $&$98
$&$90 $&$-475 $&$ -276 $&$-273 $&$-523 $&$ -302 $&$-1185 $&$-1050 $&$-257 $&$
190 $&$1703 $&$ -246$&$ 386 $&$ -234$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ 25154 $&$-3435$&$577 $&$ 217 $&$-608 $&$ -368$&$ -387 $&$-102
$&$277 $&$117 $&$ -455$&$ -1144 $&$-279 $&$ -654 $&$ 126 $&$-1863$&$ -1126 $&$
-631 $&$ -1347 $&$ -501 $&$1101$&$ 1353 $&$ 1584 $\\\ $ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$16493$&$ -1266 $&$856$&$ 343 $&$903 $&$ 397
$&$495 $&$76 $&$209 $&$ 164 $&$ -109 $&$-416 $&$206 $&$579 $&$-235 $&$ -855
$&$-1491 $&$-869 $&$ -2292 $&$-2046 $&$-1905 $&$ -3326 $\\\ $ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ 10143 $&$-970 $&$ 800 $&$357
$&$422$&$ 455 $&$ 180 $&$322 $&$-123 $&$ -61 $&$-440 $&$ -182 $&$2 $&$-717
$&$-856 $&$ -1140 $&$ -1315 $&$ -862 $&$53 $&$36 $&$-527$\\\ $ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$6687 $&$ -1148 $&$ 1021 $&$111
$&$301 $&$188$&$177$&$ 114 $&$-51 $&$ -356 $&$-49 $&$ -123 $&$-134 $&$ -798
$&$-1233 $&$-807 $&$ -1283 $&$ -1327 $&$ -1756 $&$ -360 $\\\ $ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$6295 $&$ -888 $&$1040$&$
414 $&$203 $&$343 $&$ 185$&$ 373 $&$-512 $&$ 36 $&$-224 $&$ -202$&$ -912 $&$
-1885$&$ -1833 $&$ -2394 $&$-2306$&$ -2521 $&$ -3224$\\\ $ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$6624 $&$ -999 $&$ 1038
$&$56 $&$243 $&$265 $&$ 510 $&$ -554 $&$174 $&$ -246 $&$ -9 $&$ -1012 $&$
-1852 $&$ -987 $&$-2536 $&$ -1783 $&$ -2572 $&$ -2392 $\\\ $ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$4840 $&$-698 $&$
504 $&$205 $&$287 $&$ 187 $&$-372 $&$56 $&$-123 $&$ -123 $&$-626 $&$ -1388 $&$
-2277 $&$ -1713 $&$ -2848 $&$-1476 $&$-2627 $\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$5006 $&$ -786 $&$649 $&$ 98
$&$250 $&$-539 $&$-248 $&$ -484 $&$-513 $&$-822 $&$ -1930 $&$ -2462 $&$ -3491
$&$ -1662 $&$-3183 $&$-2022 $\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$3932 $&$ -607 $&$ 265 $&$-25 $&$-255
$&$ -328 $&$-195 $&$-547 $&$-799 $&$ -1078 $&$ -2075 $&$ -2016 $&$ -1970
$&$-2032 $&$-2180$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$4419$&$-1124$&$509$&$-454$&$-295$&$-213$&$-622$&$-621$&$-1236$&$-1467$&$-1576$&$-1795$&$-1104$&$-1291$\\\
$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$
$&$4684$&$-941$&$511$&$325$&$-238$&$322$&$-366$&$-897$&$-1832$&$-2795$&$-2824$&$-3094$&$-3059$\\\
$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$
$&$7959$&$-1992$&$1090$&$-366$&$614$&$-467$&$-1329$&$-1507$&$-3123$&$-3675$&$-2914$&$-2453$\\\
$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$
$&$8938$&$-1264$&$1473$&$300$&$916$&$1325$&$1124$&$1483$&$-422$&$668$&$581$\\\
$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$9366$&$-2470$&$2386 $&$337 $&$327$&$123 $&$-818
$&$-1642$&$-1103$&$-1996$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$17279$&$-3461$&$2239$&$489$&$2921$&$2683$&$2022$&$3483$&$2904$\\\ $ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$13398$&$-1070$&$4028$&$1684$&$920$&$-251
$&$926$&$852$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$16903$&$-5699$&$7044$&$4796$&$6705$&$5288$&$5750$\\\ $ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$53196$&$4874$&$18735$&$10287$&$17295$&$14601$\\\ $ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$136322$&$-55347$&$66087$&$-3182$&$29635$\\\ $ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$232004$&$-71141$&$94864$&$23007$\\\ $ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$490753$&$-256627$&$82484$\\\ $ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$872142$&$-77339$\\\ $ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$863104$\\\
\end{pmatrix}\times 10^{-7}\qquad$

Table A1: Covariance matrix of the binned apparent magnitudes.