# What are recent observations telling us in light of improved tests of
distance duality relation?

Tonghua Liu1,2, Shuo Cao2,3 111Corresponding author<EMAIL_ADDRESS>Shuai
Ma4, Yuting Liu2,3, Chenfa Zheng2,3, Jieci Wang5 222Corresponding author:
<EMAIL_ADDRESS>1\. School of Physics and Optoelectronic, Yangtze
University, Jingzhou 434023, China;
2\. Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China;
3\. Department of Astronomy, Beijing Normal University, Beijing 100875, China;
4\. Beijing Academy, Kangyuan Road, Dongba, Beijing 100018, China;
5\. Department of Physics, and Collaborative Innovation Center for Quantum
Effects and Applications, Hunan Normal University, Changsha, Hunan 410081,
China.

###### Abstract

As an exact result required by the Etherington reciprocity theorem, the cosmic
distance duality relation (CDDR), $\eta(z)=D_{L}(z)(1+z)^{-2}/D_{A}(z)=1$
plays an essential part in modern cosmology. In this paper, we present a new
method ($\eta(z_{i})/\eta(z_{j})$) to use the measurements of ultra-compact
structure in radio quasars (QSO) and the latest observations of type Ia
supernova (SN Ia) to test CDDR. By taking the observations directly from SN Ia
and QSOs, one can completely eliminate the uncertainty caused by the
calibration of the absolute magnitudes of standard candles ($M_{B}$) and the
linear sizes of standard rulers ($l_{m}$). Benefit from the absence of
nuisance parameters involved in other currently available methods, our
analysis demonstrates no evidence for the deviation and redshift evolution of
CDDR up to $z=2.3$. The combination of our methodology and the machine
learning Artificial Neural Network (ANN) would produce $10^{-3}$ level
constraints on the violation parameter at high redshifts. Our results indicate
perfect agreement between observations and predictions, supporting the
persisting claims that the Etherington reciprocity theorem could still be the
best description of our universe.

###### pacs:

98.80.-k,98.54.-h,98.54.Aj,06.30.Bp

##  I Introduction

The cosmic distance duality relation (CDDR), as a fundamental relation in
modern cosmology, correlates the luminosity distance $D_{L}(z)$ with the
angular diameter distance $D_{A}(z)$. More specifically, the CDDR indicates
that $D_{L}(z)$ and $D_{A}(z)$ should satisfy the relation of
$D_{L}(z)=D_{A}(z)(1+z)^{2}$ at the same redshift 1933PMag…15..761E ;
2007GReGr..39.1055E . However, the validity of the CDDR depends on three basic
assumptions: i) the space-time is described by metric; ii) the light travels
along the null geodesics between the observer and the source; iii) the photon
number is conserved, i.e., the CDDR will be violated if the number of photons
in the universe is not conserved. Therefore, the validity test of CDDR is, to
some extent, an indirect verification of the accelerated expansion of the
universe (1998AJ….116.1009R, ; 1999ApJ…517..565P, ; 2004ApJ…607..665R, ;
2003ApJ…598..102K, ). In addition, as a fundamental relationship in cosmology,
the CDDR has been widely used in various fields of astronomy, such as the
observations of large-scale distribution of galaxies and the near-uniformity
of the cosmic microwave background radiation (CMBR) temperature
2020A&A…641A…6P , the determination of gas mass density profile and
temperature profile of galaxy clusters 2011SCPMA..54.2260C ;
2016MNRAS.457..281C , as well as the measurements of cosmic curvature with
strong gravitational lensing systems 2020MNRAS.496..708L ; 2017ApJ…834…75X ;
2019MNRAS.483.1104Q .

On the other hand, the accumulation of precise astrophysical observations
allow us to test the validity of CDDR at different redshifts. From the
theoretical point of view, two types of cosmological distances are usually
required in developing CDDR tests, i.e., angular diameter distance $D_{A}$ and
luminosity distance $D_{L}$. For the observations of luminosity distances,
many works turn to luminous sources with known (or standardizable) intrinsic
luminosity in the universe like type-Ia supernova (SN Ia), while the angular
diameter distances are inferred from baryon acoustic oscillations (BAO),
Sunyaev-Zeldovich (SZ) effect of galaxy clusters 2015PhRvD..92b3520W ;
2010ApJ…722L.233H ; 2011RAA….11.1199C , gas mass fraction measurements in
galaxy clusters 2021JCAP…06..052B ; 2012ApJ…745…98M , and strong gravitational
lensing systems 2018ApJ…866…31R ; 2019ApJ…885…70L ; 2019ApJ…887..163M ;
2022ChPhL..39k9801L . However, it is worth noting that angular diameter
distances derived from BAO is puzzled by the so-called fitting problem, which
is a major challenge confronted by the standard BAO peak location with a fixed
comoving ruler of about 105 h-1 Mpc 1987CQGra…4.1697E . Therefore, such
distance estimation is model-dependent to some extent, which inevitably brings
systematic uncertainties and further affects the validity of testing CDDR with
BAO. Meanwhile, galaxy clusters alone are not able to provide a competitive
source of angular diameter distance at different redshifts, suffering from the
large observational uncertainties arising from radio observations of the SZ
effect of galaxy clusters together with X-ray emission. In addition, based on
the observations of SN Ia, it was argued that the nuisance parameters
characterizing SN Ia light-curves also introduce considerable uncertainties to
the final results 2016ApJ…833..240L ; 2017ApJ…838..160W . Therefore, in order
to perform the validity of testing CDDR, one needs to eliminate the effects
and uncertanties caused by the nuisance parameters in both two types of
observational data sets (angular diameter distance $D_{A}$ and luminosity
distance $D_{L}$). In this paper we further analyse the most updated QSO and
SNe Ia data sets. Specially, we present a new approach that harnesses the
ratio $\eta(z_{i})/\eta(z_{j})$ as cosmic observations, based on the
luminosity distance inferred from the latest observations of type Ia supernova
and angular diameter distances obtained from the measurements of ultra-compact
structure in radio quasars. All of the quantities used in the CDDR test come
directly from observations, i.e., the absolute magnitudes of SN Ia and the
linear size of the compact structure in radio quasars need not to be
calibrated. In this way, the effects and uncertainties caused by the nuisance
parameters are eliminated completely. More interestingly, our methodology will
also benefit from the consistent redshift coverage of both samples that can
reach a high redshift range of $z\sim 2.3$.

With the increase both in the depth and quality of cosmological measurements,
new techniques and methods for CDDR tests have also been developed, focusing
on different machine learning algorithms 2021PhRvD.103j3513A ;
2020JCAP…12..019H ; 2021MNRAS.504.3938M ; 2014PhRvD..89d3007M . In this paper,
we will use the Artificial Neural Network (ANN) algorithm to reconstruct the
possible evolution of CDDR with redshifts. Note that such data-driven approach
has no assumptions about the observational data, suggesting its advantage of
being completely model-independent. Summarizing, we will propose an improved
approach to test CDDR directly, which not only effectively avoids the
influence of nuisance parameter on CDDR, but also achieves more stringent
constraints on CDDR in the case of small data samples. The outline of this
paper is given as follow: in Section II we briefly introduce the observations
of ultra-compact structure of radio quasars acting standard rulers and SN Ia
acting as standard candles. The improved methodology of testing CDDR and the
corresponding results are presented in Section III.

##  II Observational data

###  II.1 Angular diameter distances from radio quasars

We consider extracting angular diameter distance from angular redshift
relation of compact structure of radio quasar. As the most distant and
brightest objects in the Universe, quasars exhibit great potential in studying
early cosmology beyond the limitation of supernovae. Unfortunately quasars
exhibit large dispersion in luminosities at all wavelengths, which makes them
unusable as standard probes for measuring cosmological distances. In the past
decades, great effort have been made to make use of quasars as standard
candles or standard rulers in modern cosmology, such as the Baldwin effect
(1977ApJ…214..679B, ), the Broad Line Region radius-luminosity relation
(2011ApJ…740L..49W, ), the properties of highly accreting quasars
(2013PhRvL.110h1301W, ), and the non-linear relation between the ultraviolet
and X-ray fluxes of the quasar to construct the Hubble diagram
2019NatAs…3..272R ; 2015ApJ…815…33R ; 2017AN….338..329R ; 2020ApJ…899…71L ;
2020MNRAS.496..708L . According to the unified model of active galactic nuclei
and quasars, ultra-compact radio sources are identified as cases in which the
jets are moving relativistically and are close to the line of sight. At any
given frequency, the core is believed to be located in the region of the jet
corresponding to unit optical depth with synchrotron self-absorption being the
dominating process. In the original work of (1993Natur.361..134K, ), an
interesting possibility was discussed that compact radio sources (especially
quasars) constitute another potential class of standard rulers that could be
observed by very long baseline interferometry (VLBI). The VLBI with high
precision can not only accurately locate the radio source, but also measure
the tension angle of the compact radio source at the magnitude of mas. Based
on the subsequent works of 1994ApJ…425..442G ; 2001CQGra..18.1159V , the
linear size of the compact structures in radio sources are related to the
intrinsic luminosity $L$ and the redshift $z$ of the background source

$l_{m}=lL^{\beta}(1+z)^{n},$ (1)

where $l$ represents the linear size scaling factor which describes the
apparent distribution of radio brightness within the core, $\beta$ and $n$
denotes the possible dependence of the intrinsic size on the luminosity and
the redshift, respectively. However, the application of radio sources in
cosmology still suffered from the high dispersion in the observed relations or
the limitation of a poor statistics. With the gradually refined selection
technique and observations, a key step forward was made in the work of
1994ApJ…425..442G , which showed that the linear size dispersion in radio
source with a flat spectral index $(-0.38<\alpha<0.18)$ is greatly reduced.
Based on a sample of 2.29 GHz VLBI survey with 613 milliarcsecond compact
radio sources, 2017JCAP…02..012C ; 2017A&A…606A..15C  selected 120
intermediate-luminosity $(10^{27}$W/Hz$<L<10^{28}$W/Hz) quasars (ILQSOs) with
reliable measurements on the angular size of the compact structure. The final
results demonstrated that ILQSOs are almost independent from redshift and
luminosity $(|n|\simeq 10^{-3},\beta\simeq 10^{-4})$, which means they meet
the requirements expected from standard rulers. However, the crucial question
is what is the intrinsic metric linear size of the quasar source? The previous
analysis roughly estimated that the $l_{m}$ parameter is robustly of the scale
of $\sim 11$ pc 2017JCAP…02..012C . For the sake of the following description,
we take a prior value $l_{m}=11.03\pm 0.25$ pc determined by in a
cosmological-model-independent method 2017A&A…606A..15C . However, in our work
the value of $l_{m}$ does not affect the CDDR test, and we will later propose
an improved CDDR test to eliminate the bias and additional systematic errors
associated with the $l_{m}$ value of calibration.

Figure 1: The scatter plot of the observed angular sizes of 120 radio quasars
(left panel) and the apparent magnitudes of 1048 Pantheon SN Ia (right panel).
The red dotted line denotes the angular sizes calculated from the fiducial
$\Lambda$CDM model ($H_{0}=70.0$ km/s/Mpc, $\Omega_{m}=0.30$).

The angular size in compact structure and cosmic distance relation for
cosmological inference was first proposed in 1993Natur.361..134K

$D_{A}(z)=\frac{l_{m}}{\theta(z)},$ (2)

where $D_{A}$ is the angular diameter distance, $l_{m}$ is the intrinsic
metric linear size of the source, and $\theta(z)$ is the observed angular
size, which is defined by the modulus of visibility $\Gamma=S_{c}/S_{t}$ in
the literature 1994ApJ…425..442G . The specific definition of angular size is
$\theta(z)=2\sqrt{-\ln\Gamma\ln 2}/\pi B_{\theta}$, where $B_{\theta}$ is
interferometer baseline measured in wavelengths, $S_{c}$ and $S_{t}$ are
correlated flux density and total flux density, respectively 2017A&A…606A..15C
. The sample of raido quasars used in this work is the one described in
2017JCAP…02..012C  with the redshift range between $z=0.462$ and $z=2.73$.
These compact radio sources come from a well-known 2.29 GHz VLBI survey
1985AJ…..90.1599P  (hereafter called P85) with 1398 detected candidates and
917 selected sources. The P85 sample was updated with respect to redshift
2004JCAP…11..007J , which includes 613 compact radio sources that cover the
redshift range up to $0.0035\leq z\leq 3.787$
(http://nrl.northumbria.ac.uk/13109/). These 120 radio quasars have been
carefully selected for cosmological studies and we refer to 2017JCAP…02..012C
for a detailed description of the selection procedure used to turn them into
standard rulers and for an explanation of the calibration method used to
include them in the extensive cosmological analysis (2017arXiv170808867L, ;
2017EPJC…77..891M, ; 2017EPJC…77..502Q, ; 2020ApJ…892..103Z, ). The scatter
diagram of the observed angular sizes for 120 radio quasars is shown in Fig.
1.

###  II.2 Luminosity distances from Type Ia Supernova

In order to carry out the test of CDDR, we need to find another cosmological
probe that can directly provide luminosity distances and satisfy the following
criteria, i.e., the probe should be able to cover roughly the redshift range
of the compact radio quasars. In this work, we seek for SN Ia as a reasonably
empirically well-understood cosmological probe, the usefulness of which to
modern cosmology is well known in revealing the accelerated expansion of the
Universe and placing constraints on cosmological parameters to break parameter
degeneracies. With the rapid growth in the sample size of SN Ia distance
measurements, the analysis and mitigation of systematic uncertainties of Type
Ia Supernova has been considerably improved. However, the application of SN Ia
for cosmology involves so-called ”nuisance” parameters, which need to be
optimized along with the unknown variables in cosmological models and could
potentially affect reliable constraints on cosmological model parameters.

Fortunately, the recent SN Ia sample called Pantheon has been released by the
Pan-STARRS1 (PS1) Medium Deep Survey, which contains 1048 SN Ia measurements
spanning the redshift range $0.01<z<2.3$ (2018ApJ…859..101S, ). Here, we only
summarise the crucial points required by the present work. Benefit from
richness and depth of the sample,the Pantheon catalogue combines the subset of
279 PS1 SN Ia 2014ApJ…795…44R ; 2014ApJ…795…45S  and useful distance
estimations of SN Ia from SDS, SNLS, various low redshift and HST samples
(2018ApJ…859..101S, ). More importantly, compared with the previous SN Ia data
sets (2014A&A…568A..22B, ), the Pantheon sample applies a new approach called
BEAMS with Bias Corrections (BBC) (2017ApJ…836…56K, ), in which the apparent
magnitude is replaced with the corrected apparent magnitude
$m_{B,corr}=m_{B}+\alpha^{\star}\cdot X_{1}-\beta\cdot\mathcal{C}$ for all the
SN Ia 2018ApJ…859..101S . Here, $m_{B}$ is the observed peak magnitude in
rest-frame B band, while $X_{1}$ and $\mathcal{C}$ are the color and light-
curve shape parameters. The two nuisance parameters $\alpha^{\star}$ and
$\beta$ should be fitted simultaneously with the cosmological parameters. It
should be noted that the stretch luminosity parameter $\alpha^{\star}$ and the
color-luminosity parameter $\beta$ are set to zero for the Pantheon sample.
Therefore, the observed distance modulus of SN Ia provides the luminosity
distance as

$D_{L,SN}(z)=10^{(m_{B,corr}(z)-M_{B})/5-5}(Mpc),$ (3)

where $M_{B}$ is the absolute magnitude in B band. For the uncertainty of the
luminosity distance in Pantheon data set, the contribution from photometric
error, distance bias correction, and the peculiar velocity are included in
this analysis 2018ApJ…859..101S . The apparent B-band magnitude for 1048
Pantheon SN Ia of is also illustrated in Fig. 1.

Figure 2: The CDDR parameter $\eta(z)$ from the observations of radio quasars
and SN Ia.

##  III Methodology and results

From the theoretical point of view, in order to directly test the DDR from
observations, the following parameterized form is commonly used

$\eta(z)=\frac{D_{L}(z)}{D_{A}(z)(1+z)^{2}},$ (4)

the likelihood of which is expected to peak at one in order to satisfy the
CDDR. By combining Eqs. (2) and (3) to Eq. (4), one can rewrite the above
expression as

$\eta(z)=\frac{\theta(z)10^{(m_{B,corr}(z)-M_{B})/5-5}}{l_{m}(1+z)^{2}}.$ (5)

The difficulty of testing CDDR lies in the fact that the angular diameter
distance from an radio quasar should be observed at the same redshift as SN
Ia. In the previous work for example in 2011RAA….11.1199C , it was pointed out
that the CDDR test could be significantly affected by the particular choice of
the selection criteria for a given pair of data sets. Following the redshift
selection criterion widely used in the literature (within the redshift range
of $0.01<z<2.3$) 2011ApJ…729L..14L ; 2016ApJ…822…74L ; 2021EPJC…81..903L , the
redshifts of SN Ia sample are carefully chosen to coincide with the associated
quasar sample demanding that the difference in redshift is smaller than 0.005.
By performing such selection criterion that could hopefully ease the
systematic errors brought by redshift inconsistency, only 37 pairs of data
sets are obtained from the Pantheon and ILQSO sample. Combining these quasar
data together with the Pantheon SN Ia sample, we obtain the CDDR parameter
$\eta(z)$ shown in Fig. 2. More specifically, the total uncertainties of
$\eta(z)$ are calculated from the standard uncertainty propagation formula,
based on the uncorrelated uncertainties of observables including the observed
angular size errors $\sigma_{\theta}$, corrected apparent magnitude errors
$\sigma_{m_{B,corr}}$, as well as additional systematic errors introduced from
the calibrations of absolute magnitude ($M_{B}$) of SN Ia and linear size
($l_{m}$) of radio quasars. To better illustrate the statistical significance
of our results, we first use the weighted mean statistics (1993ComPh…7..415B,
) to evaluate

$\eta=\frac{\Sigma_{i}\big{(}\eta_{i}/\sigma_{\eta_{i}}^{2}\big{)}}{\Sigma_{i}\big{(}1/\sigma_{\eta_{i}}^{2}\big{)}},\,\,\,\,\,\,\,\,\,\sigma^{2}_{\eta}=\frac{1}{\Sigma_{i}\big{(}1/\sigma_{\eta_{i}}^{2}\big{)}},$
(6)

where $\eta$ stands for the weighted mean and $\sigma_{\eta}$ is its
corresponding uncertainty of CDDR parameter. Such statistical method has been
widely applied in meta-analysis to integrate the results of independent
measurements (2019NatSR…911608C, ). Our assessments for weighted mean and
corresponding uncertainty are $Mean(\eta(z))=0.991(\pm 0.147)$, which is in
perfect agreement with the results of previous works (2016ApJ…833..240L, ;
2017ApJ…838..160W, ; 2021PhRvD.103j3513A, ; 2020JCAP…12..019H, ;
2021MNRAS.504.3938M, ), indicates that there is no evidence for the CDDR
violation. Given the possible invalidity of Gaussian distribution of the
errors, we also use a robust median statistics (2012msma.book…..F, ) to
evaluate the measurements of $\eta(z)$. Moreover, if there are extreme values
and outliers in the sequence, it is better to use the median as the
representative value. When making a total number of $N$ measurements, one
might naturally expect that there is a 50% chance that each measurement is
higher/lower than the true median. Therefore, the probability that $n$-th
observation is higher than the median follows the binomial distribution:
$P=2^{-N}N!/[n!(N-n)!]$ (2001ApJ…549….1G, ). Similarly, we can define the
68.3$\%$ confidence interval with median statistics. In the framework of such
non-parametric approach, the resulting constraint on the CDDR parameter
becomes $Med(\eta(z))=1.117(\pm 0.328)$ with the median value and the absolute
deviation. Therefore, the conclusion of CDDR validity ($\eta(z)=1$) seems
robust within $1\sigma$ confidence interval.

Figure 3: The $\eta(z_{i})/\eta(z_{j})$ two-point diagnostics calculated on
the observations of radio quasars and SN Ia (left panel). The reconstructed
$\eta(z_{i})/\eta(z_{j})$ two-point diagnostics with ANN machine learning
algorithm (right panel).

Figure 4: The scatter plot of the CDDR parameter $\eta_{1}/\eta_{0}$ (left
panel) and constraints on the CDDR parameters $(\eta_{0},\eta_{1})$ (right
panel), in the framework of $\eta(z_{i})/\eta(z_{j})$ two-point diagnostics.

Due to the ambiguous interpretation of the compact structure size in radio
quasars and the absolute B-band magnitude of SN Ia whose value is determined
by the host stellar mass, the linear size parameter $l_{m}$ and the absolute
magnitude $M_{B}$ are hard to determine precisely. In fact, the uncertainty of
CDDR measurements shown in Fig. 2 is dominated by the calibration of two
nuisance parameters. In order to eliminate the influence of these two nuisance
parameters, we propose an improved approach by introducing the ratio of CDDR
parameter

$\eta(z_{i})/\eta(z_{j})=\frac{\theta(z_{i})(1+z_{j})^{2}}{\theta(z_{j})(1+z_{i})^{2}}10^{\Delta
m_{B,corr}/5},$ (7)

where $\Delta m_{B,corr}=m_{B,corr}(z_{i})-m_{B,corr}(z_{j})$ is the
difference of corrected apparent magnitude between arbitrary two SN Ia data
points. If one defines the ratio $\eta(z_{i})/\eta(z_{j})$, where $i,j$ denote
the order numbers of the radio quasars and SN Ia, then such quantity does not
depend on the nuisance parameters and it does not introduce any uncertainty to
the results. Note that if we have observational data at $n$ different
redshifts, then we can get $n(n-1)/2$ data pairs. The uncertainty of
$\eta_{ij}=\eta(z_{i})/\eta(z_{j})$ is calculated using the standard error
propagation formula, which is related to the uncorrelated uncertainties of the
observed angular size $\sigma_{\theta}$ and corrected apparent magnitude
$\sigma_{m_{B,corr}}$. More importantly, our approach successfully eliminate
the nuisance parameters $M_{B}$ and $l_{m}$, which brings benefits in
alleviating the systematics caused by precise determination of these
parameters. These are the apparent merits of our methodology. Our approach was
inspired by the two-point diagnostic approach, which has been extensively
applied to quantify the difference between the cosmological constant
($\Lambda$CDM) and other dark energy models (including evolving dark energy)
2012PhRvD..86j3527S ; 2016ApJ…825…17Z ; 2018arXiv180309106Z .

In order to gain insight concerning the two-point diagnostics calculated for
every combination of pairs taken from the full QSO+SN Ia data. We display
these diagnostics together with their uncertainties as a function of redshift
difference $\Delta z=|z_{i}-z_{j}|$ in the left panel of Fig. 3. Benefit from
the improved methodology, the QSO/SN Ia pairs satisfying irrespective of the
redshift selection criteria have a massive growth. One can see that there are
some interesting features regarding the uncertainties of the two-point
diagnostics, i.e., they are apparently non-Gaussian. In order to test further
the validity and efficiency of our method, we use two approaches to produce a
summary statistics of two-point diagnostics calculated on the data sets. The
first is to use the weighted mean statistical method. In order to ensure that
each data point is uncorrelated, the weighted mean formula for the
$\eta(z_{i})/\eta(z_{j})$ diagnostic should be rewritten as 2016ApJ…825…17Z

$\eta_{ij}=\frac{\Sigma_{i=1}^{n-1}\Sigma_{j=i+1}^{n}\big{(}\eta_{ij}/\sigma_{\eta_{ij}}^{2}\big{)}}{\Sigma_{i=1}^{n-1}\Sigma_{j=i+1}^{n}\big{(}1/\sigma_{\eta_{ij}}^{2}\big{)}},\,\,\,\,\,\,\,\,\,\sigma^{2}_{\eta_{ij}}=\frac{1}{\Sigma_{i=1}^{n-1}\Sigma_{j=i+1}^{n}\big{(}1/\sigma_{\eta_{ij}}^{2}\big{)}}.$
(8)

The weighted mean value and corresponding uncertainty is
$Mean(\eta(z_{i})/\eta(z_{j}))=0.968\pm 0.031$, which suggests that the
weighted mean of this diagnostic is compatible with CDDR within the
observational uncertainty. Actually, benefit from the absence of nuisance
parameters involved in other currently available methods, our methodology
produces more stringent constraints on CDDR (with the precision of $10^{-2}$)
at the current observational data level. The second approach is the median
statistics method, which is an appropriate measure in light of the non-
Gaussian error distribution. The validity of CDDR at $z\sim 2.3$, with the 68%
confidence intervals of the median $Med(\eta(z_{i})/\eta(z_{j}))=0.998(\pm
0.436)$, seems much more justified than the previous one drawn from the
weighted mean. Therefore, the results of $\eta(z_{i})/\eta(z_{j})$ showed in
this paper demonstrate no evidence for the deviation from CDDR irrespective of
the statistical method used. This is one of the unambiguous conclusions in our
work. However, one should also be aware of the disadvantage of the above
method, i.e., the ratio of CDDR parameter $\eta(z_{i})/\eta(z_{j})$ should be
constant and exactly equal to one if the CDDR is the true one. However, the
CDDR can be violated even if the ratio is exactly equal to one. In order to
fully explore the consequences of our proposed $\eta(z_{i})/\eta(z_{j})$
diagnostics, we adopt an explicit parameterization
$\eta(z)=\eta_{0}+\eta_{1}z$ to better illustrate what our results imply for
the redshift-evolution of CDDR parameter. Thus, the ratio of CDDR parameter
can be rewritten as

$\frac{\eta_{1}}{\eta_{0}}=(\frac{\Delta
z}{1-\frac{\theta(z_{i})(1+z_{j})^{2}}{\theta(z_{j})(1+z_{i})^{2}}10^{\Delta
m_{B,corr}/5}}-z_{j})^{-1},$ (9)

which should be equal to zero if there is no redshift evolution of CDDR. The
measurements of these diagnostics as a function of redshift difference $\Delta
z$ are shown in Fig. 4. Furthermore, we also use a Python Markov Chain Monte
Carlo (MCMC) module (2013PASP..125..306F, ) to obtain fits on the two CDDR
parameters, by minimizing the $\chi^{2}$ objective function

$\chi^{2}=\frac{2}{n(n-1)}\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{(\eta_{ij}^{th}-\eta_{ij}^{obs})}{\sigma^{2}_{\eta_{ij}}}.$
(10)

In Fig. 4 we also plot the one-dimensional marginalized distributions and two-
dimensional constraint contours for the CDDR parameters, with the best-fit
values of $\eta_{0}=0.952^{+0.019}_{-0.019}$ and
$\eta_{1}=0.023^{+0.053}_{-0.054}$, respectively. It is worth to comment that
on the one hand, our methodology produces a possible deviation from the
expected value of CDDR parameter ($\eta_{0}=1$) up to $z\sim 2.3$. However,
our results are still marginally consistent with the CDDR validity within
$2\sigma$ C.L., which is in full agreement with other recent tests involving
cosmological data. A summary of the current constraints on the $\eta_{0}$ from
different cosmological observables can be found in Ref. 2022EPJC…82..115H . On
the other hand, the CDDR remains redshift independent ($\eta_{1}=0$) within
$1\sigma$ C.L., supporting the persisting claims that the Etherington
reciprocity theorem could still be the best description of our universe.

Figure 5: The CDDR parameters $\eta(z)$ and $\eta(z_{i})/\eta(z_{j})$
calculated from the two statistical methods as weighted mean and the median
statistics. Bands display the 68.3% confidence regions.

There are many ways the above findings could be improved. For instance, it is
still interesting to see whether those conclusions may be changed with machine
learning algorithms, which have shown their excellent potential in addressing
cosmological issues and constraining cosmological parameters
2018PhRvD..98l3518F ; 2019PhRvD.100f3514F ; 2019MNRAS.490.1843R ;
2020ApJS..249…25W . More importantly, as a completely data driven approach,
the Artificial Neural Network (ANN) method does not assume random variables
that satisfy the Gaussian distribution. The main purpose of an ANN (which
consists of an input layer, one or more hidden layers and an output layer) is
to construct an approximate function $f_{\mathbf{W},\mathbf{b}}(\mathbf{x})$
(in which $\mathbf{W}$ and $\mathbf{b}$ are linear weights matrix and the
offset vector) that correlates the input vector $\mathbf{x}$ with the output
vector $\mathbf{y}$ 2015arXiv151107289C . According to the difference between
the predicted value $f_{\mathbf{W},\mathbf{b}}(\mathbf{x})$ of the current
network and the target value $\mathbf{y}$, the weight matrix of each layer
needs to be constantly updated for minimize the difference, which is defined
by a loss function $\mathcal{L}$ 2014arXiv1412.6980K . An issue that needs
clarification is the achievable $1\sigma$ confidence region for the
reconstructed function, which depends on both the actual errors and the cost
function. Following the detailed discussion in 2021MNRAS.501.5714W , a
complete artificial neural network has the following parts: firstly, the
weight is randomly initialized in the neural network; Secondly, the output
value is compared with the expected output value, and the cost function is
used to calculate the error; Thirdly, the error is propagated back to the
neural network and the weight is set according to this information; Fourthly,
repeat steps two to four for each input value in the training set; Finally,
when the entire training set is sent to the neural network, the entire
training is complete. The recent analysis has demonstrated the effectiveness
of ANN acting as “universal approximator” to produce representative
uncertainties of the observations, especially in high-precision test of CDDR
in both electromagnetic and gravitational wave domain 2021EPJC…81..903L . In
particular, Euclid collaboration improved the precision of CDDR test by
approximately a factor of six, based on machine learning reconstruction using
genetic algorithms 2020A&A…644A..80M .

Using the publicly released code called Reconstruct Functions with ANN
333ReFANN; https://github.com/Guo-Jian-Wang/refann., we perform the
reconstruction of the parameter $\eta(z_{i})/\eta(z_{j})$ based on the current
$\eta(z_{i})/\eta(z_{j})$ two-point diagnostics. The reconstructed functions
with corresponding 1$\sigma$ uncertainties, which can be considered as the
average level of observational error are given in right panel of Fig. 3.
Working on the reconstructed 1000 $\eta(z_{i})/\eta(z_{j})$ points with ANN,
we obtain $Mean(\eta(z_{i})/\eta(z_{j}))=0.998(\pm 0.003)$ and
$Med(\eta(z_{i})/\eta(z_{j}))=0.998(\pm 0.004)$ in the framework of weighted
mean and median statistics. Therefore, with ANN algorithm one could expect the
parameter $\eta(z_{i})/\eta(z_{j})$ to be estimated at the precision of
$10^{-3}$, which is more stringent than other results based on currently
available observational data. In order to facilitate comparison between the
inferred values of CDDR parameters obtained from two statistical approaches,
we display the results in Fig. 5. As a final remark, possible violations of
such fundamental relation (cosmic distance duality relation) might have
profound implications for the understanding of fundamental physics and natural
laws. Based on better uv-coverage in the future, we pin our hope on multi-
frequency VLBI observations of more compact radio quasars with higher angular
resolution, smaller statistical and systematic uncertainties. Meanwhile,
considering the variety of different machine learning algorithms, we may also
be optimistic in detecting possible deviation from the CDDR with much higher
precision.

##  IV Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grants Nos. 12203009, 12122504, 12021003, 11875025, 11633001,
11920101003, and 62202469; the Strategic Priority Research Program of the
Chinese Academy of Sciences, Grant No. XDB23000000; Beijing Natural Science
Foundation (Grant No. 4224091); the Interdiscipline Research Funds of Beijing
Normal University; and the China Manned Space Project (Nos. CMS-CSST-2021-B01
and CMS-CSST-2021-A01).

## References

  * [1] I. M. H. Etherington.  On the Definition of Distance in General Relativity.  Philosophical Magazine, 15(18):761, January 1933.
  * [2] I. M. H. Etherington.  Republication of: LX. On the definition of distance in general relativity.  General Relativity and Gravitation, 39(7):1055–1067, July 2007\.
  * [3] Adam G. Riess, Alexei V. Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M. Garnavich, Ron L. Gilliland, Craig J. Hogan, Saurabh Jha, Robert P. Kirshner, B. Leibundgut, M. M. Phillips, David Reiss, Brian P. Schmidt, Robert A. Schommer, R. Chris Smith, J. Spyromilio, Christopher Stubbs, Nicholas B. Suntzeff, and John Tonry.  Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant.  Astron. J., 116(3):1009–1038, September 1998.
  * [4] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and The Supernova Cosmology Project.  Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae.  Astrophys. J. , 517(2):565–586, June 1999.
  * [5] Adam G. Riess, Louis-Gregory Strolger, John Tonry, Stefano Casertano, Henry C. Ferguson, Bahram Mobasher, Peter Challis, Alexei V. Filippenko, Saurabh Jha, Weidong Li, Ryan Chornock, Robert P. Kirshner, Bruno Leibundgut, Mark Dickinson, Mario Livio, Mauro Giavalisco, Charles C. Steidel, Txitxo Benítez, and Zlatan Tsvetanov.  Type Ia Supernova Discoveries at z ¿ 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution.  Astrophys. J. , 607(2):665–687, June 2004.
  * [6] R. A. Knop, G. Aldering, R. Amanullah, P. Astier, G. Blanc, M. S. Burns, A. Conley, S. E. Deustua, M. Doi, R. Ellis, S. Fabbro, G. Folatelli, A. S. Fruchter, G. Garavini, S. Garmond, K. Garton, R. Gibbons, G. Goldhaber, A. Goobar, D. E. Groom, D. Hardin, I. Hook, D. A. Howell, A. G. Kim, B. C. Lee, C. Lidman, J. Mendez, S. Nobili, P. E. Nugent, R. Pain, N. Panagia, C. R. Pennypacker, S. Perlmutter, R. Quimby, J. Raux, N. Regnault, P. Ruiz-Lapuente, G. Sainton, B. Schaefer, K. Schahmaneche, E. Smith, A. L. Spadafora, V. Stanishev, M. Sullivan, N. A. Walton, L. Wang, W. M. Wood-Vasey, and N. Yasuda.  New Constraints on $\Omega$M, $\Omega$Λ, and w from an Independent Set of 11 High-Redshift Supernovae Observed with the Hubble Space Telescope.  Astrophys. J. , 598(1):102–137, November 2003.
  * [7] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. F. Cardoso, J. Carron, A. Challinor, H. C. Chiang, J. Chluba, L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, J. Delabrouille, J. M. Delouis, E. Di Valentino, J. M. Diego, O. Doré, M. Douspis, A. Ducout, X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, Y. Fantaye, M. Farhang, J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Génova-Santos, M. Gerbino, T. Ghosh, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, F. K. Hansen, D. Herranz, S. R. Hildebrandt, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, A. Karakci, E. Keihänen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J. M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, P. Lemos, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, M. López-Caniego, P. M. Lubin, Y. Z. Ma, J. F. Macías-Pérez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, M. Maris, P. G. Martin, M. Martinelli, E. Martínez-González, S. Matarrese, N. Mauri, J. D. McEwen, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M. A. Miville-Deschênes, D. Molinari, L. Montier, G. Morgante, A. Moss, P. Natoli, H. U. Nørgaard-Nielsen, L. Pagano, D. Paoletti, B. Partridge, G. Patanchon, H. V. Peiris, F. Perrotta, V. Pettorino, F. Piacentini, L. Polastri, G. Polenta, J. L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubiño-Martín, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard, C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev, A. S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, L. Toffolatti, M. Tomasi, T. Trombetti, L. Valenziano, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Zacchei, and A. Zonca.  Planck 2018 results. VI. Cosmological parameters.  Astron. Astrophys., 641:A6, September 2020.
  * [8] Shuo Cao and Zonghong Zhu.  The distance duality relation and the temperature profile of galaxy clusters.  Science China Physics, Mechanics, and Astronomy, 54(12):2260–2264, December 2011.
  * [9] Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.  Testing the gas mass density profile of galaxy clusters with distance duality relation.  Mon. Not. Roy. Astron. Soc., 457(1):281–287, March 2016.
  * [10] Tonghua Liu, Shuo Cao, Jia Zhang, Marek Biesiada, Yuting Liu, and Yujie Lian.  Testing the cosmic curvature at high redshifts: the combination of LSST strong lensing systems and quasars as new standard candles.  Mon. Not. Roy. Astron. Soc., 496(1):708–717, July 2020.
  * [11] Jun-Qing Xia, Hai Yu, Guo-Jian Wang, Shu-Xun Tian, Zheng-Xiang Li, Shuo Cao, and Zong-Hong Zhu.  Revisiting Studies of the Statistical Property of a Strong Gravitational Lens System and Model-Independent Constraint on the Curvature of the Universe.  Astrophys. J. , 834(1):75, January 2017.
  * [12] Jing-Zhao Qi, Shuo Cao, Sixuan Zhang, Marek Biesiada, Yan Wu, and Zong-Hong Zhu.  The distance sum rule from strong lensing systems and quasars - test of cosmic curvature and beyond.  Mon. Not. Roy. Astron. Soc., 483(1):1104–1113, February 2019.
  * [13] Puxun Wu, Zhengxiang Li, Xiaoliang Liu, and Hongwei Yu.  Cosmic distance-duality relation test using type Ia supernovae and the baryon acoustic oscillation.  Phys. Rev. D, 92(2):023520, July 2015.
  * [14] R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro.  Testing the Distance-Duality Relation with Galaxy Clusters and Type Ia Supernovae.  Astrophys. J. Lett., 722(2):L233–L237, October 2010.
  * [15] Shuo Cao and Nan Liang.  Testing the distance-duality relation with a combination of cosmological distance observations.  Research in Astronomy and Astrophysics, 11(10):1199–1208, October 2011.
  * [16] Kamal Bora and Shantanu Desai.  A test of cosmic distance duality relation using SPT-SZ galaxy clusters, Type Ia supernovae, and cosmic chronometers.  JCAP, 2021(6):052, June 2021.
  * [17] Xiao-Lei Meng, Tong-Jie Zhang, Hu Zhan, and Xin Wang.  Morphology of Galaxy Clusters: A Cosmological Model-independent Test of the Cosmic Distance-Duality Relation.  Astrophys. J. , 745(1):98, January 2012.
  * [18] Cheng-Zong Ruan, Fulvio Melia, and Tong-Jie Zhang.  Model-independent Test of the Cosmic Distance Duality Relation.  Astrophys. J. , 866(1):31, October 2018.
  * [19] Kai Liao.  The Cosmic Distance Duality Relation with Strong Lensing and Gravitational Waves: An Opacity-free Test.  Astrophys. J. , 885(1):70, November 2019.
  * [20] Yu-Bo Ma, Shuo Cao, Jia Zhang, Jingzhao Qi, Tonghua Liu, Yuting Liu, and Shuaibo Geng.  Testing Cosmic Opacity with the Combination of Strongly Lensed and Unlensed Supernova Ia.  Astrophys. J. , 887(2):163, December 2019.
  * [21] Kai Liao, Marek Biesiada, and Zong-Hong Zhu.  Strongly Lensed Transient Sources: A Review.  Chinese Physics Letters, 39(11):119801, November 2022.
  * [22] G. F. R. Ellis and W. Stoeger.  The ’fitting problem’ in cosmology.  Classical and Quantum Gravity, 4(6):1697–1729, November 1987.
  * [23] Zhengxiang Li, Guo-Jian Wang, Kai Liao, and Zong-Hong Zhu.  Model-independent Estimations for the Curvature from Standard Candles and Clocks.  Astrophys. J. , 833(2):240, December 2016.
  * [24] Jun-Jie Wei and Xue-Feng Wu.  An Improved Method to Measure the Cosmic Curvature.  Astrophys. J. , 838(2):160, April 2017.
  * [25] Rubén Arjona, Hai-Nan Lin, Savvas Nesseris, and Li Tang.  Machine learning forecasts of the cosmic distance duality relation with strongly lensed gravitational wave events.  Phys. Rev. D, 103(10):103513, May 2021.
  * [26] N. B. Hogg, M. Martinelli, and S. Nesseris.  Constraints on the distance duality relation with standard sirens.  JCAP, 2020(12):019, December 2020.
  * [27] Purba Mukherjee and Ankan Mukherjee.  Assessment of the cosmic distance duality relation using Gaussian process.  Mon. Not. Roy. Astron. Soc., 504(3):3938–3946, June 2021.
  * [28] Ariadna Montiel, Ruth Lazkoz, Irene Sendra, Celia Escamilla-Rivera, and Vincenzo Salzano.  Nonparametric reconstruction of the cosmic expansion with local regression smoothing and simulation extrapolation.  Phys. Rev. D, 89(4):043007, February 2014.
  * [29] Jack A. Baldwin.  Luminosity Indicators in the Spectra of Quasi-Stellar Objects.  Astrophys. J. , 214:679–684, June 1977.
  * [30] D. Watson, K. D. Denney, M. Vestergaard, and T. M. Davis.  A New Cosmological Distance Measure Using Active Galactic Nuclei.  Astrophys. J. Lett., 740(2):L49, October 2011.
  * [31] Jian-Min Wang, Pu Du, David Valls-Gabaud, Chen Hu, and Hagai Netzer.  Super-Eddington Accreting Massive Black Holes as Long-Lived Cosmological Standards.  Phys. Rev. Lett. , 110(8):081301, February 2013.
  * [32] G. Risaliti and E. Lusso.  Cosmological Constraints from the Hubble Diagram of Quasars at High Redshifts.  Nature Astronomy, 3:272–277, January 2019.
  * [33] G. Risaliti and E. Lusso.  A Hubble Diagram for Quasars.  Astrophys. J. , 815(1):33, December 2015.
  * [34] G. Risaliti and E. Lusso.  Cosmology with AGN: can we use quasars as standard candles?  Astronomische Nachrichten, 338(329):329–333, March 2017.
  * [35] Tonghua Liu, Shuo Cao, Marek Biesiada, Yuting Liu, Shuaibo Geng, and Yujie Lian.  Testing the Cosmic Opacity at Higher Redshifts: Implication from Quasars with Available UV and X-Ray Observations.  Astrophys. J. , 899(1):71, August 2020.
  * [36] K. I. Kellermann.  The cosmological deceleration parameter estimated from the angular-size/redshift relation for compact radio sources.  Nature (London), 361(6408):134–136, January 1993.
  * [37] L. I. Gurvits.  Apparent Milliarcsecond Sizes of Active Galactic Nuclei and the Geometry of the Universe.  Astrophys. J. , 425:442, April 1994.
  * [38] R. G. Vishwakarma.  Consequences on variable $\Lambda$-models from distant type Ia supernovae and compact radio sources.  Classical and Quantum Gravity, 18(7):1159–1172, April 2001.
  * [39] Shuo Cao, Marek Biesiada, John Jackson, Xiaogang Zheng, Yuhang Zhao, and Zong-Hong Zhu.  Measuring the speed of light with ultra-compact radio quasars.  JCAP, 2017(2):012, February 2017.
  * [40] Shuo Cao, Xiaogang Zheng, Marek Biesiada, Jingzhao Qi, Yun Chen, and Zong-Hong Zhu.  Ultra-compact structure in intermediate-luminosity radio quasars: building a sample of standard cosmological rulers and improving the dark energy constraints up to z 3.  Astron. Astrophys., 606:A15, September 2017.
  * [41] R. A. Preston, D. D. Morabito, J. G. Williams, J. Faulkner, D. L. Jauncey, and G. Nicolson.  A VLBI survey at 2.29 GHz.  Astron. J., 90:1599–1603, September 1985.
  * [42] J. C. Jackson.  Tight cosmological constraints from the angular-size/redshift relation for ultra-compact radio sources.  JCAP, 2004(11):007, November 2004.
  * [43] Xiaolei Li, Shuo Cao, Xiaogang Zheng, Jingzhao Qi, Marek Biesiada, and Zong-Hong Zhu.  Testing and selecting cosmological models with ultra-compact radio quasars.  European Physical Journal C, 77(8):677, August 2017.
  * [44] Yubo Ma, Jia Zhang, Shuo Cao, Xiaogang Zheng, Tengpeng Xu, and Jingzhao Qi.  The generalized cosmic equation of state: a revised study with cosmological standard rulers.  European Physical Journal C, 77(12):891, December 2017.
  * [45] Jing-Zhao Qi, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.  New observational constraints on f( T) cosmology from radio quasars.  European Physical Journal C, 77(8):502, August 2017.
  * [46] Xiaogang Zheng, Kai Liao, Marek Biesiada, Shuo Cao, Tong-Hua Liu, and Zong-Hong Zhu.  Multiple Measurements of Quasars Acting as Standard Probes: Exploring the Cosmic Distance Duality Relation at Higher Redshift.  Astrophys. J. , 892(2):103, April 2020.
  * [47] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, R. Chornock, R. J. Foley, M. E. Huber, R. Kessler, G. Narayan, A. G. Riess, S. Rodney, E. Berger, D. J. Brout, P. J. Challis, M. Drout, D. Finkbeiner, R. Lunnan, R. P. Kirshner, N. E. Sanders, E. Schlafly, S. Smartt, C. W. Stubbs, J. Tonry, W. M. Wood-Vasey, M. Foley, J. Hand, E. Johnson, W. S. Burgett, K. C. Chambers, P. W. Draper, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, F. Bresolin, E. Gall, R. Kotak, M. McCrum, and K. W. Smith.  The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample.  Astrophys. J. , 859(2):101, June 2018.
  * [48] A. Rest, D. Scolnic, R. J. Foley, M. E. Huber, R. Chornock, G. Narayan, J. L. Tonry, E. Berger, A. M. Soderberg, C. W. Stubbs, A. Riess, R. P. Kirshner, S. J. Smartt, E. Schlafly, S. Rodney, M. T. Botticella, D. Brout, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters.  Cosmological Constraints from Measurements of Type Ia Supernovae Discovered during the First 1.5 yr of the Pan-STARRS1 Survey.  Astrophys. J. , 795(1):44, November 2014.
  * [49] D. Scolnic, A. Rest, A. Riess, M. E. Huber, R. J. Foley, D. Brout, R. Chornock, G. Narayan, J. L. Tonry, E. Berger, A. M. Soderberg, C. W. Stubbs, R. P. Kirshner, S. Rodney, S. J. Smartt, E. Schlafly, M. T. Botticella, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters.  Systematic Uncertainties Associated with the Cosmological Analysis of the First Pan-STARRS1 Type Ia Supernova Sample.  Astrophys. J. , 795(1):45, November 2014.
  * [50] M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas, P. Astier, P. El-Hage, M. Konig, S. Kuhlmann, J. Marriner, R. Pain, N. Regnault, C. Balland, B. A. Bassett, P. J. Brown, H. Campbell, R. G. Carlberg, F. Cellier-Holzem, D. Cinabro, A. Conley, C. B. D’Andrea, D. L. DePoy, M. Doi, R. S. Ellis, S. Fabbro, A. V. Filippenko, R. J. Foley, J. A. Frieman, D. Fouchez, L. Galbany, A. Goobar, R. R. Gupta, G. J. Hill, R. Hlozek, C. J. Hogan, I. M. Hook, D. A. Howell, S. W. Jha, L. Le Guillou, G. Leloudas, C. Lidman, J. L. Marshall, A. Möller, A. M. Mourão, J. Neveu, R. Nichol, M. D. Olmstead, N. Palanque-Delabrouille, S. Perlmutter, J. L. Prieto, C. J. Pritchet, M. Richmond, A. G. Riess, V. Ruhlmann-Kleider, M. Sako, K. Schahmaneche, D. P. Schneider, M. Smith, J. Sollerman, M. Sullivan, N. A. Walton, and C. J. Wheeler.  Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples.  Astron. Astrophys., 568:A22, August 2014.
  * [51] R. Kessler and D. Scolnic.  Correcting Type Ia Supernova Distances for Selection Biases and Contamination in Photometrically Identified Samples.  Astrophys. J. , 836(1):56, February 2017.
  * [52] Zhengxiang Li, Puxun Wu, and Hongwei Yu.  Cosmological-model-independent Tests for the Distance-Duality Relation from Galaxy Clusters and Type Ia Supernova.  Astrophys. J. Lett., 729(1):L14, March 2011.
  * [53] Kai Liao, Zhengxiang Li, Shuo Cao, Marek Biesiada, Xiaogang Zheng, and Zong-Hong Zhu.  The Distance Duality Relation from Strong Gravitational Lensing.  Astrophys. J. , 822(2):74, May 2016.
  * [54] Tonghua Liu, Shuo Cao, Sixuan Zhang, Xiaolong Gong, Wuzheng Guo, and Chenfa Zheng.  Revisiting the cosmic distance duality relation with machine learning reconstruction methods: the combination of HII galaxies and ultra-compact radio quasars.  European Physical Journal C, 81(10):903, October 2021.
  * [55] Philip R. Bevington, D. Keith Robinson, J. Morris Blair, A. John Mallinckrodt, and Susan McKay.  Data Reduction and Error Analysis for the Physical Sciences.  Computers in Physics, 7(4):415, January 1993.
  * [56] Shuo Cao, Jingzhao Qi, Zhoujian Cao, Marek Biesiada, Jin Li, Yu Pan, and Zong-Hong Zhu.  Direct test of the FLRW metric from strongly lensed gravitational wave observations.  Scientific Reports, 9:11608, August 2019.
  * [57] Eric D. Feigelson and G. Jogesh Babu.  Modern Statistical Methods for Astronomy.  2012\.
  * [58] III Gott, J. Richard, Michael S. Vogeley, Silviu Podariu, and Bharat Ratra.  Median Statistics, H0, and the Accelerating Universe.  Astrophys. J. , 549(1):1–17, March 2001.
  * [59] Arman Shafieloo, Varun Sahni, and Alexei A. Starobinsky.  New null diagnostic customized for reconstructing the properties of dark energy from baryon acoustic oscillations data.  Phys. Rev. D, 86(10):103527, November 2012.
  * [60] Xiaogang Zheng, Xuheng Ding, Marek Biesiada, Shuo Cao, and Zong-Hong Zhu.  What Are the Omh2̂ (z1, z2) and Om (z1, z2) Diagnostics Telling us in Light of H(z) Data?  Astrophys. J. , 825(1):17, July 2016.
  * [61] Xiaogang Zheng, Marek Biesiada, Xuheng Ding, Shuo Cao, Sixuan Zhang, and Zong-Hong Zhu.  Statistical analysis with cosmic-expansion-rate measurements and two-point diagnostics.  European Physical Journal C, 78(10):274, July 2018.
  * [62] Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, and Jonathan Goodman.  emcee: The MCMC Hammer.  Publications of the Astronomical Society of the Pacific, 125(925):306, March 2013.
  * [63] R. F. L. Holanda, F. S. Lima, Akshay Rana, and Deepak Jain.  Strong lensing systems and galaxy cluster observations as probe to the cosmic distance duality relation.  European Physical Journal C, 82(2):115, February 2022.
  * [64] Janis Fluri, Tomasz Kacprzak, Alexandre Refregier, Adam Amara, Aurelien Lucchi, and Thomas Hofmann.  Cosmological constraints from noisy convergence maps through deep learning.  Phys. Rev. D, 98(12):123518, December 2018.
  * [65] Janis Fluri, Tomasz Kacprzak, Aurelien Lucchi, Alexandre Refregier, Adam Amara, Thomas Hofmann, and Aurel Schneider.  Cosmological constraints with deep learning from KiDS-450 weak lensing maps.  Phys. Rev. D, 100(6):063514, September 2019.
  * [66] Dezső Ribli, Bálint Ármin Pataki, José Manuel Zorrilla Matilla, Daniel Hsu, Zoltán Haiman, and István Csabai.  Weak lensing cosmology with convolutional neural networks on noisy data.  Mon. Not. Roy. Astron. Soc., 490(2):1843–1860, December 2019.
  * [67] Guo-Jian Wang, Si-Yao Li, and Jun-Qing Xia.  ECoPANN: A Framework for Estimating Cosmological Parameters Using Artificial Neural Networks.  Astrophys. J., Suppl. Ser., 249(2):25, August 2020.
  * [68] Djork-Arné Clevert, Thomas Unterthiner, and Sepp Hochreiter.  Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs).  arXiv e-prints, page arXiv:1511.07289, November 2015.
  * [69] Diederik P. Kingma and Jimmy Ba.  Adam: A Method for Stochastic Optimization.  arXiv e-prints, page arXiv:1412.6980, December 2014.
  * [70] Guo-Jian Wang, Xiao-Jiao Ma, and Jun-Qing Xia.  Machine learning the cosmic curvature in a model-independent way.  Mon. Not. Roy. Astron. Soc., 501(4):5714–5722, March 2021.
  * [71] M. Martinelli, C. J. A. P. Martins, S. Nesseris, D. Sapone, I. Tutusaus, A. Avgoustidis, S. Camera, C. Carbone, S. Casas, S. Ilić, Z. Sakr, V. Yankelevich, N. Auricchio, A. Balestra, C. Bodendorf, D. Bonino, E. Branchini, M. Brescia, J. Brinchmann, V. Capobianco, J. Carretero, M. Castellano, S. Cavuoti, R. Cledassou, G. Congedo, L. Conversi, L. Corcione, F. Dubath, A. Ealet, M. Frailis, E. Franceschi, M. Fumana, B. Garilli, B. Gillis, C. Giocoli, F. Grupp, S. V. H. Haugan, W. Holmes, F. Hormuth, K. Jahnke, S. Kermiche, M. Kilbinger, T. D. Kitching, B. Kubik, M. Kunz, H. Kurki-Suonio, S. Ligori, P. B. Lilje, I. Lloro, O. Marggraf, K. Markovic, R. Massey, S. Mei, M. Meneghetti, G. Meylan, L. Moscardini, S. Niemi, C. Padilla, S. Paltani, F. Pasian, V. Pettorino, S. Pires, G. Polenta, M. Poncet, L. Popa, L. Pozzetti, F. Raison, J. Rhodes, M. Roncarelli, R. Saglia, P. Schneider, A. Secroun, S. Serrano, C. Sirignano, G. Sirri, F. Sureau, A. N. Taylor, I. Tereno, R. Toledo-Moreo, L. Valenziano, T. Vassallo, Y. Wang, N. Welikala, J. Weller, and A. Zacchei.  Euclid: Forecast constraints on the cosmic distance duality relation with complementary external probes.  Astron. Astrophys., 644:A80, December 2020.