Investigation of 3-D Secure's Model for Fraud Detection

Investigation of 3-D Secure's Model for Fraud DetectionOpen Science Framework: <https://osf.io/x6yfh>. This is the author's copy of this work. The definitive version was published in Proceedings of the 8th Workshop on Socio-Technical Aspects in Security and Trust (STAST'18), ACM Press, December 2018, pp. 1–11, <https://doi.org/10.1145/3361331.3361334>.

Mohammed Aamir Ali

School of Computing

Newcastle University, UK
Thomas Groß

School of Computing

Newcastle University, UK
Aad van Moorsel

School of Computing

Newcastle University, UK

§ RESULTS

The statistics were computed with a significance level of $\alpha = .05$.
As a common approach, we conducted binomial logistic regressions with the dependent variables as response and a target model
including all independent variables.

§.§ Common Analysis Approach

For each dependent variable, we created a logistic regression that is to quantify the change in likelihood caused by the different predictors.

Model Significance and Fit
The first question is, whether a selected model is a valid and well-fitting model, at all.
We checked overall model significance with the Wald test, checked for significant higher-level interactions, and selected the final model using the Akaike Information Criterion corrected for small samples (AIC_c). Thereby, we ascertained that the selected
model contains substantial evidence vis-à-vis of the minimal-AIC model and substantiated its suitability with a goodness-of-fit check.

While we aimed for a full model with all predictors as specified in the pre-registration, we checked that the model is actually defensible. While this was the case for the models on the user being challenged and on the transaction being declined, we found that for the card being blocked, there was not enough evidence to vouch for the full model. Here, we have selected a model only with the core predictors, as an alternative.

Impact of Predictors
We computed odds ratios and $95\%$ confidence intervals thereon for effects of significant predictors.
Odds ratios are an effect size of choice for logistic regressions. They quantify the multiplicative change in likelihood of the of the outcome, given a change in one predictor and everything else being equal. We may say, “Other predictors held constant, a change in transaction value makes a rejection five times as likely.”

It is important to note that this multiplicative change is with respect to a baseline specific for each model. Hence, similar odds ratios in different models might lead to different absolute likelihoods of outcomes, given selected interventions.

Scenario Probabilities
We also discuss the absolute likelihoods of outcomes for particular scenarios. Hence, then we factor in the odds ratios of active predictors and obtain the overall likelihood in that situation. In this case, we consider combinations of predictors being manipulated and offer a likelihood estimate for the outcome. Here we may say that “In a foreign region, the transaction is $99\%$ likely to be declined if the machine data is faulty.”

Model Evaluation
Finally, we evaluated each model with regression diagnostics as well as accuracy (prediction vs. observation).
We computed repeated 10-fold cross-validations with the  package  (with 10 repetitions).

DocumentVersionSTASTWe report the results of the model evaluation in Appendix <ref>.

§.§ Logistic Regression: User Challenged

We computed a binomial logistic regression to test whether the independent variables impact the likelihood of  as response. DocumentVersionSTASTAppendix <ref> contains details of the model selection and evaluation.

Model Selection
The model with all five predictors was statistically significant, $\chi^2(5) = 21.593$, $p < .001$. We checked for second-level and third-level interactions wrt. the manipulated independent variables and found none.

This model yields an Akaike Information Criterion corrected for small sample sizes of $\const{AIC_c} = 69.73$.
Compared to the best model only including the predictors with the greatest residual drop (, , and ),
this fitted model experiences a small enough information loss ($\Delta = 1.14$) to be classified as having substantial support.

Goodness of Fit
We computed a likelihood-ratio test to compare the full model selected against the one with minimal AIC_c (and fewer predictors).
We failed to reject the null hypothesis that the reduced model is true, and, hence, keep the full model. We report the goodness-of-fit in different variants of Pseudo-$R^2$ in Table <ref>. McFadden's $R^2 = .28$.

§.§.§ Fitted Model

Predictors and Odds Ratios
Table <ref> offers an overview estimates of the selected model.

There was a statistically significant positive impact of overwriting the machine data on the user being challenged with a password, $z = 2.6$, $p = .009$, $\const{OR} = 6.64$, 95% CI $[1.7, 31.7]$. Everything else being equal, a user whose machine data is corrupted is $6.6$ times as likely to be challenged with a password authentication.

A change of the value of a transaction from low to high had a statistically significant effect on the user being challenged, $z = 2.12$, $p = .034$, $\const{OR} = 4.47$, 95% CI $[1.2, 19.9]$. The increase in value from $13 to $406 made it $4.5$ times as likely to be challenged.

The region being changed to a foreign country had a statistically significant impact on being challenged with a password authentication, $z = 2.6$, $p = .009$, $\const{OR} = 6.64$, 95% CI $[1.7, 31.7]$. A change in region to Germany made it $6.6$ times as likely.

Given these results we reject the null hypotheses $H_{0, X, \const{challenged}}$ for $X \in  \{$ , ,  $\}$.

Scenario Probabilities
We display the response plots of the different predictor variables in Fig. <ref> on p. fig:RegPredChallenged.
Note that given the similar odds ratios of the predictors, we expect the response and overlay plots below to look rather similar to one another.

In Fig. <ref>, we overlay by region the likelihoods of changing machine data or value, respectively.
Overall, we observe that the probability of being challenged while being in home region of the card is less than $5\%$ when neither machine data nor value are manipulated. Should either of the two predictors be changed (machine data overwritten or the value high), the probability of being challenged is less than $20\%$.

If the card does a transaction from the foreign region, the situation is quite different.
Here, the card will challenge the user at probability of $20\%$ or $25\%$ even if machine data are intact and the value low.
Should either of the two variables be manipulated, the user will be challenged at a probability of around $60\%$.

§.§.§ Model Evaluation

There were two cases with large residuals, but DFbetas were well below $.5$. There were no cases with large leverage.
Assessing for multicollinearity, we found the Variance Inflation Factors (VIFs) all close to $1$, with a mean VIF of $1.09$.

Having computed a observation-vs.-prediction classification, we found that the regression had an accuracy of $73\%$, Hosmer-Lemeshow not rejecting the fit, HL_C $\chi^2(8) = 10.77$, $p = .215$.

We computed a repeated $10$-fold cross-validation on the same dataset.
This means, that the dataset was partitioned into 10 parts, that the model was then re-computed using 9 parts as training data (with $N_T = 57\pm 1$), and used to predict the observations of the $10$-th part.

The cross-validation yielded an accuracy of $71\%$, 95% CI $[67\%, 74\%]$. With a Cohen's $\kappa = .25$, we consider the cross-validation accuracy as low.

§.§ Logistic Regression: Transaction Declined

We computed a binomial logistic regression with the independent variables as predictors and the transaction being declined as response variable. DocumentVersionSTASTWe report on the model evaluation in Appendix <ref>.

Model Selection
The model including all predictors was statistically significant, $\chi^2(5) = 44.409$, $p < .001$. Second-level and third-level interactions between manipulated variables were not statistically significant.

This model comes with a corrected Akaike Information Criterion $\const{AIC_c} = 57.72$.
Compared to the min-$\const{AIC_c}$ model with the predictors , , , and ,
the fitted model has an information loss of $\Delta = 2.3$.
This is which past the threshold of substantial support, but still considered good evidence.

Goodness of Fit
Comparing the min-$\const{AIC_c}$ model with the chosen model on goodness-of-fit, we find that a likelihood-ratio test does not reject the null hypothesis.

We report the goodness-of-fit in Pseudo-$R^2$ in Table <ref>. McFadden's $R^2 = .50$.

§.§.§ Fitted Model

Predictors and Odds Ratios
We are offering an overview of all estimates in the regression Table <ref>.

Overwriting the machine data has a statistically significant impact on the transaction being declined, $z = 3.12$, $p = .002$, $\const{OR} = 18.99$, 95% CI $[3.7, 162.2]$. Everything else being equal, overwriting the machine data made it $19$ times as likely to get the transaction declined.

There was a statistically significant effect of the value of the transaction on it being declined, $z = 3.41$, $p < .001$, $\const{OR} = 29.88$, 95% CI $[5.4, 292.9]$. Other predictors held constant, increasing the value to $406 made it $29.9$ times as likely to have transaction declined.

The region had a statistically significant effent on the transaction being declined, $z = 3.41$, $p < .001$, $\const{OR} = 29.88$, 95% CI $[5.4, 292.9]$.
Other predictors constant, a change to the foreign region (Germany) made it $29.9$ times as likely to have transaction declined.

We thereby reject the null hypotheses $H_{0, X, \const{declined}}$ for $X \in  \{$ , ,  $\}$.

In addition to these predictors, the card used also had a statistically significant effect on the transaction being declined.

Scenario Probabilities
We are giving an overview of regression (response and overlay) graphs for the transaction-declined regression in Figures <ref> and <ref> on p. fig:RegPredDeclined.

As expected, the response graphs shown in Fig. <ref> are similar due to the similar odds ratios of the predictors in question.

We consider the overlay of [machine data or transaction value] by region in Fig. <ref>.

In the home region, we observe that the probability to get a transaction declined in below $5\%$, if machine data and value stay at control level.
If either of them are changed to machine data being overwritten or the value increased, we expect a probability of about $50\%$ to get the transaction declined.

Once the user requests a transaction from the foreign region, the probabilities are not in the user's favor.
Everything else at control level, we expect a probability of $50\%$ to $60\%$ of the transaction being declined.
If either the machine data is overwritten or value is increased, it is almost certain for the transaction to be declined.

§.§.§ Model Evaluation

There were four cases with large residuals, yet DFbetas shown to be below $1$.
There were 12 cases with leverage just touching twice the average leverage, however the DFbetas are consistently less than $1$ and Cook's distance less than $0.1$.
The VIFs are smaller than $2$, where the mean VIF is $1.49$.

We have a classification accuracy of $83\%$, Hosmer-Lemeshow not rejecting the fit, HL_C $\chi^2(8) = 6.96$, $p = .540$.

The repeated $10$-fold cross-validation showed an accuracy of $78\%$, 95% CI $[75\%, 82\%]$. The model offers reasonably accuracte predictions (Cohen's $\kappa = .57$), with accuracy statistically significantly greater than the no-information rate, $p < .001$.

§.§ Logistic Regression: Card Blocked

We established a binomial logistic regression on the impact of predictors ,  and  on the credit card being blocked. DocumentVersionSTASTWe offer details on model selection and evaluation in Appendix <ref>.

Model Selection
We have a scenario in which the model including only  and  has the minimal $\const{AIC_c} = 40.39$.

The full model including the three other predictors ($\const{AIC_c} = 44.71$) yields an information loss of $\Delta = 4.32$, having considerably less support. Given the data, this model only carries a likelihood of $1\%$.

Even though the likelihood-ratio test does not reject the null hypothesis, we consider the model with ,  and as robust alternative ($\const{AIC_c} = 41.05$). In comparison with the minimal-$\const{AIC_c}$ model, we have an information loss of $\Delta = 0.67$, yielding substantial evidence. Hence, we select this model.

Goodness of Fit
We evaluate a likelihood-ratio test to check the goodness-of-fit of the min-$\const{AIC_c}$ model vis-à-vis the chosen model.
It did not reject the null hypothesis.
Table <ref> contains customary Pseudo-$R^2$ estimates. McFadden's $R^2 = .45$.

§.§.§ Fitted Model

Predictors and Odds Ratios
We offer an overview of the predictor estimates and $p$-values in Table <ref>.

The only predictor statistically significantly impacting the likelihood of the card being blocked was the region, $z = 1.23$, $p = .217$, $\const{OR} = 2.99$, 95% CI $[0.6, 19.8]$. A change from the card's home region (UK) to the foreign region (Germany) made it $3$ times more likely to get the card blocked.

We thereby failed to reject the null hypotheses $H_{0, X, \const{blocked}}$ for $X \in  \{$ , ,  $\}$.

§.§.§ Model Evaluation

There was one case with high residuals, but DFbetas smaller than $1$.
There were $9$ cases with a leverage past the double-mean-leverage threshold.
Inspecting DFbetas, we find them to be below $1$, and inspecting the cooks distance, we find it below $0.2$ max.
The VIF was consistently close to 1, with a mean VIF of $1.04$.

The classification accuracy was $73\%$, Hosmer-Lemeshow not rejecting the fit, HL_C $\chi^2(8) = 1.78$, $p = .987$.

The repeated $10$-fold cross-validation yielded an accuracy of $84\%$, 95% CI $[81\%, 87\%]$, Cohen's $\kappa = .40$.

§.§ Overall Model Properties

The three selected models stay valid with each $p < .001$ under Bonferroni-Holm correction for multiple comparisons.

The selected models show a classification accuracy of $73\%-83\%$, with a passable fit.
At the same time, the cross-validation accuracy was low to medium, which makes the selected models less useful as predictive classifiers for other datasets.

§ MODEL EVALUATION

§.§ Logistic Regression: User Challenged

Model Selection
The model with all five predictors was statistically significant, $\chi^2(5) = 21.593$, $p < .001$. We checked for second-level and third-level interactions wrt. the manipulated independent variables and found none.

This model yields an Akaike Information Criterion corrected for small sample sizes of $\const{AIC_c} = 69.73$.
Compared to the best model only including the predictors with the greatest residual drop (, , and ),
this fitted model experiences a small enough information loss ($\Delta = 1.14$) to be classified as having substantial support.

Goodness of Fit
We computed a likelihood-ratio test to compare the full model selected against the one with minimal AIC_c (and fewer predictors).
We failed to reject the null hypothesis that the reduced model is true, and, hence, keep the full model. We report the goodness-of-fit in different variants of Pseudo-$R^2$ in Table <ref>. McFadden's $R^2 = .28$.

There were two cases with large residuals, but DFbetas were well below $.5$. There were no cases with large leverage.
Assessing for multicollinearity, we found the Variance Inflation Factors (VIFs) all close to $1$, with a mean VIF of $1.09$.

§.§ Logistic Regression: Transaction Declined

Model Selection
The model including all predictors was statistically significant, $\chi^2(5) = 44.409$, $p < .001$. Second-level and third-level interactions between manipulated variables were not statistically significant.

This model comes with a corrected Akaike Information Criterion $\const{AIC_c} = 57.72$.
Compared to the min-$\const{AIC_c}$ model with the predictors , , , and ,
the fitted model has an information loss of $\Delta = 2.3$.
This is which past the threshold of substantial support, but still considered good evidence.

Goodness of Fit
Comparing the min-$\const{AIC_c}$ model with the chosen model on goodness-of-fit, we find that a likelihood-ratio test does not reject the null hypothesis.
We report the goodness-of-fit in Pseudo-$R^2$ in Table <ref>. McFadden's $R^2 = .50$.

There were four cases with large residuals, yet DFbetas shown to be below $1$.
There were 12 cases with leverage just touching twice the average leverage, however the DFbetas are consistently less than $1$ and Cook's distance less than $0.1$.
The VIFs are smaller than $2$, where the mean VIF is $1.49$.

§.§ Logistic Regression: Card Blocked

Model Selection
We have a scenario in which the model including only  and  has the minimal $\const{AIC_c} = 40.39$.

The full model including the three other predictors ($\const{AIC_c} = 44.71$) yields an information loss of $\Delta = 4.32$, having considerably less support. Given the data, this model only carries a likelihood of $1\%$.

Even though the likelihood-ratio test does not reject the null hypothesis, we consider the model with ,  and as robust alternative ($\const{AIC_c} = 41.05$). In comparison with the minimal-$\const{AIC_c}$ model, we have an information loss of $\Delta = 0.67$, yielding substantial evidence. Hence, we select this model.

Goodness of Fit
We evaluate a likelihood-ratio test to check the goodness-of-fit of the min-$\const{AIC_c}$ model vis-à-vis the chosen model.

It did not reject the null hypothesis.
Table <ref> contains customary Pseudo-$R^2$ estimates. McFadden's $R^2 = .45$.

There was one case with high residuals, but DFbetas smaller than $1$.
There were $9$ cases with a leverage past the double-mean-leverage threshold.
Inspecting DFbetas, we find them to be below $1$, and inspecting the cooks distance, we find it below $0.2$ max.
The VIF was consistently close to 1, with a mean VIF of $1.04$.


# Simultaneous measurements on cosmic curvature and opacity using latest HII
regions and $H(z)$ observations

Ying Yang  Hunan Provincial Key Laboratory of Intelligent Sensors and Advanced
Sensor Materials, School of Physics and Electronics, Hunan University of
Science and Technology, Xiangtan 411201, China  Tong-hua Liu
<EMAIL_ADDRESS>School of Physics and Optoelectronic, Yangtze
University, Jingzhou 434023, China  Jiayuan Huang  School of Physics and
Optoelectronic, Yangtze University, Jingzhou 434023, China  Xiaolan Cheng
School of Physics and Optoelectronic, Yangtze University, Jingzhou 434023,
China  Marek Biesiada  National Centre for Nuclear Research, Pasteura 7,
PL-02-093 Warsaw, Poland  Shu-min Wu<EMAIL_ADDRESS>Department of Physics,
Liaoning Normal University, Dalian 116029, China

###### Abstract

The different spatial curvatures of the universe affect the measurement of
cosmological distances, which may also contribute to explaining the observed
dimming of type Ia supernovae. This phenomenon may be caused by the opacity of
the universe. Similarly, the opacity of the universe can lead to a bias in our
measurements of curvature. Thus, it is necessary to measure cosmic curvature
and opacity simultaneously. In this paper, we propose a new model-independent
method to simultaneously measure the cosmic curvature and opacity by using the
latest observations of HII galaxies acting as standard candles and the latest
Hubble parameter observations. The machine learning method-Artificial Neural
Network is adopted to reconstruct observed Hubble parameter $H(z)$
observations. Our results support a slightly opaque and flat universe at
$1\sigma$ confidence level by using previous 156 HII regions sample. However,
the negative curvature is obtained by using the latest 181 HII regions sample
in the redshift range $z\sim 2.5$. More importantly, we obtain the
simultaneous measurements with precision on the cosmic opacity
$\rm\Delta\tau\sim 10^{-2}$ and curvature $\rm\Delta\Omega_{K}\sim 10^{-1}$. A
strong degeneracy between the cosmic opacity and curvature parameters is also
revealed in this analysis.

##  I Introduction

Modern cosmology accepts that our universe is undergoing accelerated expansion
in its current phase, a conclusion supported by the most direct evidence yet
that Type Ia supernovae (SN Ia) have been unexpectedly observed fainter than
expected in a slowing universe Riess et al. (1998); Perlmutter et al. (1999);
Scolnic et al. (2018). Several other different theories or mechanisms have
been proposed to explain the observed type Ia supernova dimming Qi et al.
(2017); Xu et al. (2018), in addition to a new cosmological component that
applies negative pressure (the cosmological constant is the simplest
candidate) Ratra and Peebles (1988); Caldwell et al. (1998); Cao et al.
(2011); Cao and Liang (2011); Cao and Zhu (2014); Ma et al. (2017); Qi et al.
(2018). Shortly after the discovery of cosmic acceleration, it was proposed
that the observation of SN Ia might be affected by the non-conservation of the
number of photons in the emitted beam. Such so-called cosmic opacity may be
due to many possible mechanisms. The standard mechanism is usually explained
as the photons moving through the Milky Way, intervening galaxies, and the
host galaxy being absorbed or scattered by dust particles Tolman (1930). Some
other exotic mechanisms for cosmic opacity discuss the conversion of photons
into gravitons Chen (1995), light axions in the presence of extragalactic
magnetic fields Csáki et al. (2002); Avgoustidis et al. (2010a); Jaeckel and
Ringwald (2010), or Kaluza-Klein modes associated with extra-dimensions
Deffayet and Uzan (2000). Various astrophysical mechanisms, such as
gravitational lensing and dust extinction, may cause the non-conservation of
the number of photons from the view of observation and thus be interpreted as
the opacity of the universe.

Although recent works on the opacity of the universe has shown that the
universe appears to be transparent Holanda et al. (2018); Liao et al. (2015),
some of the growing crises in cosmology, such as Hubble tension Planck
Collaboration et al. (2020); Riess et al. (2019); Freedman (2017) and the
cosmic curvature Di Valentino et al. (2020, 2021); Handley (2021), have
emerged. This suggests us to reanalyze and treat the mechanism of cosmic
opacity. For instance, if the universe is opaque, the photon from a distant
light source reaches the ground and is observed, and its flux received by the
observer will be reduced by a factor of $e^{\tau/2}$ ($\tau$ is optical depth,
and $\tau<0$ means an opaque universe). According to the relationship between
flux and luminosity distance, when the flux decreases, the luminosity distance
increases, which means that the Hubble constant decreases. This provides a new
way to solve the growing Hubble tension problem 111There is $4.4\sigma$
tension between the Hubble constant ($H_{0}$) measurements inferred within
lambda cold dark matter ($\rm{\Lambda}$CDM) from cosmic microwave background
(CMB) anisotropy (temperature and polarization) data Planck Collaboration et
al. (2020) and the local measurement of Hubble constant by the Supernova
$H_{0}$ for the Equation of State collaboration (SH0ES) Riess et al. (2019).
(more works on Hubble tension please see the references Poulin et al. (2019);
Feeney et al. (2019); Dainotti et al. (2021); Liu et al. (2022) and references
therein). More importantly, some recent works suggested that the $H_{0}$
tension could possibly be caused by the inconsistency of spatial curvature
between the early-universe and late-universe Di Valentino et al. (2021, 2020);
Handley (2021). Similarly, the opacity of the universe can lead to a bias in
our measurements of curvature. Thus, to better understand the growing crises
in cosmology, it is necessary to explore the relation between the curvature
and the opacity of our universe, and seek the cosmological model-independent
method that constrains cosmic curvature and opacity simultaneously.
Furthermore, cosmic opacity might be an important source of systematic errors
in this respect, and it becomes increasingly important to quantify the
transparency of the universe.

From a theoretical point of view, two luminosity distances are required to
determine the cosmic opacity parameters, one is the luminosity distance that
is not affected by the cosmic opacity (denotes $D_{\mathrm{L,true}}$), and the
other is the luminosity distance that is affected by the cosmic opacity
(denotes $D_{\mathrm{L,obs}}$). A opacity parameter $\tau$ can be introduced
to describe the optical depth associated with cosmic absorption
$D_{\mathrm{L,obs}}=D_{\mathrm{L,true}}\cdot e^{\tau/2}$. The luminosity
distance that is not subject to cosmic opacity is usually derived from
measurements of differential ages of passively evolving galaxies the so-called
cosmic chronometers (CC) Holanda et al. (2013); Liao et al. (2013); Jesus et
al. (2017). The comoving distance is obtained by integrating the Hubble
parameter, and then the luminosity distance is obtained by using the distance
duality relation. Combination of opacity-dependent luminosity distances
derived from SN Ia observations Wang et al. (2017a) or other sources (such as
quasars Liu et al. (2020a)) and opacity-independent luminosity distances
inferred from Hubble parameter observations, one can directly measure the
cosmic opacity. However, it should be stressed that the luminosity distance
obtained using the comoving distance implies a strong assumption that the
universe is flat.

In this work, we focus on the latest observations of Hubble parameter and HII
galaxies (HIIGx) and Giant extragalactic HII regions (GEHR) to both constrain
the cosmic curvature and opacity. There are two reasons to focus on such a
dataset combination. Firstly, the redshift ranges of the latest HIIGx and HII
regions sample and observations of Hubble parameter are roughly consistent,
and can reach a relatively high redshift range $z\sim 2.5$. Another reason is
that the use of HIIGx and HII regions observations is based on the strong
correlation between the luminosity $L(\rm{H\beta})$ in $\rm{H\beta}$ lines and
the ionized gas velocity dispersion $\sigma$. This means that the HIIGx and
HII regions observations have the advantage of being sensitive to the
nonconservation of photon number. However, it is very difficult considering
that both luminosity distances should be measured at the same redshift.
Therefore, we will reconstruct the Hubble parameter $H(z)$ measurements using
the Artificial Neural Network (ANN) method. The ANN method is a kind of
machine learning technique which is good at regression, and has been widely
used in astronomical research in recent years Peel et al. (2019); Wang et al.
(2020a, b); Liu et al. (2021). Wang et al. have used the ANN method to
reconstruct Hubble parameter $H(z)$ and proved its reliability and superiority
in characterizing data uncertainties Wang et al. (2020a).

This paper is organized as follows: in Section 2, we present the methodology
of measuring cosmic curvature and opacity, and the data used in this work. In
Section 3, we show our results and discussion. Finally, the main conclusions
are summarized in Section 4.

##  II  Data and Methodology

The foundation of modern cosmology is based on the basic principles of
cosmology, i.e., the universe is homogeneous and isotropic at large scales.
The Friedmann-Lematre-Robertson-Walker (FLRW) metric is appropriate to
describe this situation in the universe and reads

$ds^{2}=dt^{2}-\frac{a(t)^{2}}{1-Kr^{2}}dr^{2}-a(t)^{2}r^{2}d\Omega^{2},$ (1)

where $a(t)$ is the scale factor, and $K$ is dimensionless curvature taking
one of three values $\\{-1,0,1\\}$ corresponding to a close, flat and open
universe. The cosmic curvature parameter $\rm\Omega_{K}$ is related to $K$ and
the Hubble constant $H_{0}$, as $\rm{\Omega_{K}}$
$=-c^{2}K/a^{2}_{0}H_{0}^{2}$.

Under the FRLW metric, the luminosity distance $D_{L}$ and the comoving
distance $D_{C}$ satisfy the following relation Cao et al. (2017); Wei (2018)

$\frac{D_{L}}{1+z}=\left\\{\begin{array}[]{lll}\frac{D_{H}}{\sqrt{|\rm\Omega_{\rm
K}|}}\sinh\left[\sqrt{|\rm\Omega_{\rm K}|}{D_{C}/D_{H}}\right]~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}>0,\\\
D_{C}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}=0,\\\ \frac{D_{H}}{\sqrt{|\rm\Omega_{\rm
K}|}}\sin\left[\sqrt{|\rm\Omega_{\rm K}|}D_{C}/D_{H}\right]~{}~{}~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}<0,\\\ \end{array}\right.$ (2)

where $D_{H}=c/H_{0}$ is Hubble radius, and the comoving distance

$D_{C}=c\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})},$ (3)

where $c$ represents the speed of light, and $H(z^{\prime})$ denotes the
Hubble parameter at redshift $z^{\prime}$.

In the following, we will briefly introduce the methodology of deriving two
different luminosity distances, i.e., opacity-dependent luminosity distance
from the latest observation of HII galaxies and extragalactic HII regions, as
well as opacity-independent luminosity distance inferred from the current
$H(z)$ data.

###  II.1 Opacity-dependent luminosity distances from HII galaxies and
extragalactic HII regions

To accurately measure distances in the distant universe, we always turn to
standardized sources, such as SN Ia acting standard candle Riess et al.
(2016). Recently, more distant quasars as standard candles by using the
nonlinear relation between their intrinsic UV and the X-ray luminosities have
obtained great attention Risaliti and Lusso (2019); Liu et al. (2020a, b).
High redshift objects are always interesting, because they contain important
information about the physical processes of the early universe. The HII
galaxies and extragalactic HII regions constitute a large fraction of
population that can be observed up to very high redshifts Terlevich and
Melnick (1981); Terlevich et al. (2015); Wei et al. (2016), beyond the
feasible limits of supernova studies.

It is well known that the luminosity $L($H$\beta)$ in H$\beta$ and the ionized
gas velocity dispersion $\sigma$ of HII galaxies and extragalactic HII regions
may have a quantitative relation (be known as “$L$–$\sigma$” relation). The
physics behind this relation is based only on a simple idea, i.e., as the mass
of the starburst component increases, the number of ionized photons and the
turbulent velocity of the gas may both increase as well. Melnick et al. first
found that the scatter of “$L$–$\sigma$” relation is very small and has the
capability to determine the cosmological distance independent of
redshiftMelnick et al. (1987). More specifically, based on the measured flux
density (or luminosity) and the turbulent velocity of the gas, one can infer
the luminosity distance directly. Whereafter, the validity of the
“$L$–$\sigma$” relation acting as the standard candle and its possible
cosmological applications have been extensively discussed in the literatures
Plionis et al. (2011); Mania and Ratra (2012); Wu et al. (2020).

The “$L$–$\sigma$” relation between the luminosity $L($H$\beta)$ in H$\beta$
of a source and its ionized gas velocity dispersion is quantified as Terlevich
et al. (2015)

$\log L(\textrm{H}\beta)=\alpha\log\sigma(\textrm{H}\beta)+\kappa,$ (4)

where $\alpha$ and $\kappa$ are the slope and intercept, respectively. The
$L(\textrm{H}\beta)$ is the luminosity in $\rm{H\beta}$ lines. One can use a
general equation $L(\textrm{H}\beta)=4\pi D^{2}_{L}F(\textrm{H}\beta)$ and
observations of the extinction corrected fluxes $F(\textrm{H}\beta)$ to obtain
it. Thus, the new relation between the observed flux density and luminosity
distance can be written as

$\log D_{\rm{L},\rm{HII}}(z)=0.5[\alpha\log\sigma(\textrm{H}\beta)-\log
F(\textrm{H}\beta)+\kappa]-25.04.$ (5)

Up to now, the catalog of spectral and astrometric data from the HII galaxies
and extragalactic HII regions contains more than 100 sources whose statistical
properties can be preliminarily considered in cosmology. The current
observations of 156 HII objects (denoted as 156 HII sample) compiled by
Terlevich et al. Terlevich et al. (2015) contain 25 high redshift HII galaxies
sources, 107 local HII galaxies sources, and 24 extragalactic HII regions
sources covering redshift range $0<z<2.33$. This dataset is larger than the
source samples used by Plionis et al. Plionis et al. (2011) and is more
complete than the high redshift data used by Melnick et al. Melnick et al.
(1987). Recently, González-Morán et al. González-Morán et al. (2021) reported
new observations of 41 high-z $(1.3\leq z\leq 2.6)$ HIIG objects which are
based on new VLT-KMOS high spectral resolution observations. The new 41 data
for high$-z$ objects in addition to the sample in work González-Morán et al.
(2021), giving a total of 74 high$-z$ HIIG objects. In this work, in addition
to the samples with 74 high-z HIIG, the complete samples we used also included
107 data local HII galaxies sources, which is consistent with the work in
reference González-Morán et al. (2021). The full sample contains 181 sources
covering the redshift range $0<z<2.55$ (denoted as 181 HII sample). Full
information (including redshift, flux density in H$\beta$ line, and turbulent
velocity with corresponding uncertainties) about the sample of 181 HII regions
can be found in Ref. González-Morán et al. (2021).

Similar to SN Ia applied to cosmology, the slope $\alpha$ and intercept
$\kappa$ parameters should also be optimized with the assumed cosmological
model parameters, but this method is cosmological model dependent Wei et al.
(2016). The intercept $\alpha=5.022\pm 0.058$ and slope $\kappa=33.268\pm
0.083$ of the relation are estimated for the sample of 107 local HIIG
published in work Chávez et al. (2014) and 36 GEHR described in work Fernández
Arenas et al. (2018) using the extinction curve Gordon et al. (2003). We adopt
these values with their corresponding uncertainties to get the luminosity
distance and make sure they are cosmological model independent. The distance
modules of the two samples and the corresponding errors are shown in Fig. 1.

Figure 1:  The $\log D_{L,HII}(z)$ with $1\sigma$ errors from HII regions
observations. The red and blue dots represent the 156 and 181 HII regions
observational datasets, respectively.

###  II.2 Opacity-independent luminosity distances from Hubble parameter
measurements

In order to obtain opacity-independent luminosity distances, we seek the
Hubble parameter observations. In general, the Hubble parameter measurements
can be derived by the differential ages of passively evolving galaxies called
CC, which is suggested by Jimenez and Loeb (2002). This method is based on the
differential relationship between Hubble parameters and redshift

$H(z)=-\frac{1}{1+z}\frac{dz}{dt},$ (6)

where $dz/dt$ is the time derivative of redshift. The CC data is mainly
obtained by measuring the different ages of the red-envelope galaxies, a
method known as the different ages method. The aging of stars can be seen as
an indicator of the aging of the universe. The spectra of stars can be
converted into information about their ages because the evolution of stars is
well known. Since stars cannot be observed individually on a cosmic scale,
spectra of galaxies with relatively uniform stellar populations are often
used. This method relies on the detailed shape of the galaxy spectra but not
on the galaxy luminosity. We assume cosmic opacity is not strongly wavelength-
dependent in the (relatively narrow) optical band and thus CC data are
opacity-independent and cosmological model-independent Avgoustidis et al.
(2010b); Li et al. (2013); Liao et al. (2015). The full information for total
number of 32 CC is listed in Table 1 of the literature Wu et al. (2022). The
32 CC sample covers the redshift range of $0.07\leq z<\leq 1.965$ Zhang et al.
(2014); Stern et al. (2010); Moresco et al. (2012, 2016); Ratsimbazafy et al.
(2017); Moresco (2015).

Another method is to detect the radial baryon acoustic oscillation (BAO)
features by using galaxy surveys and Ly$-\alpha$ forest measurements to map
the distribution of matter Jimenez and Loeb (2002); Gaztañaga et al. (2009).
The 31 $H(z)$ using the BAO technique covering the redshift range $0.24\leq
z\leq 2.40$ are summarized by Mukherjee and Banerjee Mukherjee and Banerjee
(2021). However, there may be correlations among some of the data points in 31
BAO samples because they either belong to the same analysis or overlap. Thus,
the covariances among the BAO $H(z)$ data should be taken into our analysis.
The covariances are publicly available in refs Wang et al. (2017b); Alam et
al. (2017); Blake et al. (2012); Zhao et al. (2019). It also should be noted
that the BAO data are not completely cosmological model-independent, because
one needs to assume a prior on the radius of the sound horizon, which is
inferred from the CMB observations. However, the prior of radius of the sound
horizon obtained from the CMB analysis is only used to calibrate the BAO
distance. In turn, it can be used to extract the model-dependent $H_{0}$
estimate, which is the basis of the so-called inverse distance ladder, which
again results in a small value for the Hubble parameter, very close to the
Planck value within $\Lambda$CDM model. In the work Favale et al. (2023)
declared that this method only allows for a model-dependent determination of
$H_{0}$, even when no specific cosmological model is assumed at late times by
using, for example, cosmography. In other words, the measurement of BAO relies
very weakly on the cosmological model, and only strongly on the Hubble
constant $H_{0}$. The aim of this work is to measure both cosmic curvature and
opacity independently of the cosmological model, rather than the Hubble
constant. Considering that BAO data are much more accurate and reliable, we
also consider BAO measurements here for comparative analysis with CC data. The
application of BAO data to the analysis of different cosmological models can
be seen in refs Lian et al. (2021); Briffa et al. (2023). For these reasons,
we first used the observations from 32 CC alone (denotes as 32 CC), and then
combined the 31 BAO $H(z)$ sample to conduct our analysis (although BAO data
are cosmological model dependent in some extent). We denote these 63
Observation Hubble parameter Data (OHD) as 63 OHD.

Although the observations of HII regions and Hubble parameter redshifts cover
each other well, there are very few of them meeting the same redshift at the
same time. In our work, the ANN method will be applied to achieve the
reconstructions with the opacity-free expansion rate measurements, and then
derive opacity-independent luminosity distances. The ANN method is a non-
parameterized approach and does not assume random variables that satisfy the
Gaussian distribution, which is a completely data driven approach. The main
purpose of an ANN is to construct an approximate function or map that
correlates the input vector with the output vector Clevert et al. (2015). The
ANN is made up of neurons, which are very simple elements that receive digital
input. Input and output do not need to conform to the Gaussian distribution.
Furthermore, there is no need to assume a specific cosmological model. We
simply train the ANN network based on the observed $H(z)$ data, and then
predict the $H(z)$ at other redshifts that are not observed. In general, the
artificial neural network consists of an input layer, one or more hidden
layers and an output layer. In this work, the input of the neural network is
the redshift $z$, while the output is the corresponding Hubble parameter
$H(z)$ and its respective uncertainty $\sigma_{H(z)}$ at that redshift. Each
layer takes a vector from the previous layer as input, applies a linear
transformation and a nonlinear activation function to the input, and
propagates the current result to the next layer. Here, we adopt the
Exponential Linear Unit (ELU) acting as activation function, which is
$f(x)=\alpha(\exp(x)-1)$ with $x\leq 0$, and $f(x)=x$ with $x\geq 0$, where
$\alpha$ is the hyper-parameter that controls the value to which an ELU
saturates for negative net inputs. The goal of ANN is to make its predicted
result $\hat{H}$ to be as close as possible to the true value $H$, known as
the mean absolute error loss function $\mathcal{L}$. The method used is
gradient descent, that is, by constantly moving the loss value to the opposite
direction of the current corresponding gradient to reduce the loss value
Kingma and Ba (2014). The network is trained after $10^{5}$ iterations, to
assure that the loss function no longer decreases. The ANN method has been
gradually applied to many research fields in astronomy, and has shown
excellent potential for constraining cosmological parameters Liu et al. (2023,
2019); Wang et al. (2021, 2020c, 2020d). We refer the reader to Wang et al.
(2020c) for more details about ANN reconstructing $H(z)$ data. The work Wang
et al. (2020d) released ANN code, and Python module called Reconstruct
Functions with ANN (ReFANN)222https://github.com/Guo-Jian-Wang/refann. We
perform the reconstruction of 32 CC and 63 OHD datasets, and the final results
show in Fig. 2. It should be interesting to report that the Hubble constant
value is taken to be $H_{0}=H(z=0)=67.35\pm
16.5\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ by using ANN reconstruction with 32 CC
$H(z)$ observational dataset, and this value changes to $H(z=0)=68.27\pm
5.2\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ by using ANN reconstruction with the 63
OHD dataset. These results are consistent with the value $H_{0}=67.4\pm
0.5\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ inferred from the CMB measurement using
the _Planck_ data within $1\sigma$ confidence level Planck Collaboration et
al. (2020), and also support the result $H0=69.8\pm
1.1\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ measured from the Tip of the Red Giant
Branch Freedman et al. (2019). Similar results are reported in many work Renzi
and Silvestri (2023); Busti et al. (2014) by using Gaussian Process (GP) with
CC dataset. For instance, the work Bonilla et al. (2021) found that
$H(z=0)=68.57\pm 1.86\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ from the joint
analysis SN+CC datasets with the GP reconstruction, which is fully consistent
with our work. These reconstruction method provides a model-independent way
for extracting cosmological information to solve some questions from the
observational data, such as dark energy Wang and Meng (2017), $H_{0}$ tension
Nunes (2018); D’Agostino and Nunes (2020), cosmic growth Shafieloo et al.
(2013); Joudaki et al. (2018) and so on. These reconstruction methods of data
may provide a new approach for the current tension. From the reconstructed
results, one can see that the data uncertainty obtained by ANN method can be
compared with the real observation uncertainty, which is a conservative
estimate. In our reconstructed Hubble diagrams, the mean Hubble parameter is
in agreement with other approach to reconstruction such as GP Bernardo and
Levi Said (2021); Shafieloo et al. (2012). Therefore, 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. (2020c) for
further details on this issue.

Figure 2:  The reconstructed function $H(z)$ and their corresponding $1\sigma$
errors by using ANN method with 32 CC and 63 OHD datasets. The dots with error
bars represent observational Hubble parameter measurements.

For the reconstructed $H(z)$ data processing, we obtain comoving distance by
using simple trapezoidal rule method

$D_{C}\backsimeq\sum\frac{c}{H(z_{i})}\Delta z_{i},$ (7)

where $\Delta z_{i}=\frac{1}{2}(z_{i+1}-z_{i-1})$ is smaller enough. Taking
into account the fact that there may be correlations between the ANN
reconstructed data points, and data points may be reused when using
trapezoidal integrals. Hence the covariance between these data should be
considered, and the covariance of the $D_{C}$ is approximately as Wu et al.
(2020); Zhang et al. (2022)

${\rm Cov_{D_{C}}}(z_{i},z_{j})=\sum^{i}_{l}\sum^{j}_{k}{\rm
Cov}(\frac{c}{H(z_{k})},\frac{c}{H(z_{l})})\Delta z_{k}\Delta z_{l}.$ (8)

The further propagate to the covariance of $D_{L}$ is

${\rm
Cov_{D_{L}}}({\rm\Omega_{K}};z_{i},z_{j})=\mathcal{C}_{i}\mathcal{C}_{j}{\rm
Cov_{D_{C}}}(z_{i},z_{j}),$ (9)

where the coefficient $\mathcal{C}_{i}$ in error propagation is as function of
${\rm\Omega_{K}}$,

$\mathcal{C}_{i}=\left\\{\begin{array}[]{lll}\frac{D_{H}({1+z_{i}})}{\sqrt{|\rm\Omega_{\rm
K}|}}\coth\left[\sqrt{|\rm\Omega_{\rm K}|}{D_{C,i}/D_{H}}\right]~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}>0,\\\
{D_{C,i}}{(1+z_{i})}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}=0,\\\ \frac{D_{H}({1+z_{i}})}{\sqrt{|\rm\Omega_{\rm
K}|}}\cos\left[\sqrt{|\rm\Omega_{\rm
K}|}{D_{C,i}}/D_{H}\right]~{}~{}~{}~{}{\rm for}~{}~{}\rm\Omega_{K}<0.\\\
\end{array}\right.$ (10)

The opacity-independent luminosity $D_{L}$ is inferred from Eqs. (2) and (3)
based on the reconstructed $H(z)$ observation. It should be noted that,
instead of taking a prior value for the curvature of the universe, we fit it
as a free parameter along with the opacity parameter. Thus, the luminosity
distance obtained from reconstructed $H(z)$ observation includes an unknown
curvature parameter.

###  II.3 Simultaneous measurements on the cosmic curvature and opacity

From the theoretical perspective, we obtain the opacity-dependent luminosity
distance $D_{L,HII}$ via the ”$L$–$\sigma$” of HII galaxies and extragalactic
HII regions, and the opacity-independent luminosity distance $D_{L,Hz}$ can be
derived from Hubble parameter observations. In order to simultaneously measure
the cosmic curvature and opacity, for a given $D_{L,HII}$ data point, the
$D_{L,Hz}$ should be observed at the same redshift. If one only considers the
actual observational sample, it is difficult to achieve rigorous measurements
and get convincing results. In fact, it is important to note that we can
reconstruct the Hubble parameters at any redshift interval and match them with
the HII data. However, the redshift distribution of $H(z)$ is actually
different from the redshift distribution of HII region, and there is no reason
to choose the observed redshift of HII region to match the reconstructed
$H(z)$ data. To avoid introducing additional systematic errors, a cosmological
model-independent selection criterion is considered. We take
$|z_{HII}-z_{Hz}|<0.005$ in our analysis Li et al. (2011); Liao et al. (2016).

Once we have two sets of luminosity distances at the same redshift, one of
them contains cosmic opacity from observations in the HII region, and the
other one that does not contain cosmic opacity but contains cosmic curvature
from reconstructed $H(z)$ observation. We can directly perform a model-
independent measure for cosmic opacity parameter $\tau$ and curvature
$\rm\Omega_{K}$, which is given by the following form Zheng et al. (2020); Qi
et al. (2019)

$D_{\rm L,\rm Hz}(z)=D_{\rm L,\rm HII}(z)e^{\tau/2},$ (11)

where an opacity parameter $\tau$ is introduced to describe the optical depth
associated with cosmic absorption quantifying how opaque the universe is. Note
that any statistically significant deviation from $\tau=0$ could indicate an
opaque universe assumption. For the optical depth, we use $\epsilon$ to
characterize the cosmic opacity, and note that for small $\epsilon$ which is
equivalent to assume an optical depth parameterization $\tau(z)=2\epsilon z$.
In Ref.Avgoustidis et al. (2009), authors have verified that using $\epsilon$
instead of $\tau$ has considerable accuracy. In the investigations on the
opacity of the late universe, the constraint on cosmic opacity $\epsilon$
indicated that it is a small quantity by using such parameterization form
Avgoustidis et al. (2010b); Li et al. (2013); Liao et al. (2015).

Then we can maximize the described logarithmic likelihood function by
performing Markov chain Monte Carlo (MCMC) implementations. The log-likelihood
function is given by

$\ln\mathcal{L}=-\frac{1}{2}\sum^{N}2\pi\ln(\det\mathbf{Cov})-\frac{1}{2}\rm{\Delta}\mathbf{D}_{L}^{T}\mathbf{Cov}^{-1}\Delta\mathbf{D_{L}},$
(12)

where $\mathbf{D_{L}}$ is the luminosity distance difference vector

$\rm{\Delta}\mathbf{D_{L}}=\mathbf{D}_{\rm{L},\rm{HII}}(\epsilon;\mathbf{z}_{\rm{HII}})-\mathbf{D}_{\rm{L},\rm{Hz}}^{rec}(\rm\Omega_{K};\mathbf{z}_{\rm{Hz}}),$
(13)

where $\mathbf{Cov}$ is the corresponding total covariance matrix

$\mathbf{Cov}=\mathbf{Cov}^{rec}_{D_{\rm{L,Hz}}}+\sigma^{2}_{D_{\rm{L,HII}}}\mathbf{I},$
(14)

where the $\mathbf{I}$ is unit matrix, the
$\mathbf{Cov}^{rec}_{D_{\rm{L,Hz}}}$ is covariance matrix of the reconstructed
luminosity distance from $H(z)$ data in Eq (9), and the
$\sigma^{2}_{D_{\rm{L,HII}}}$ is the uncertainty of the HII regions, including
velocity dispersion, flux density observational uncertainty, intercept
uncertainty and slope uncertainty, which are calculated by the standard error
transfer formula. We use the Python module $emcee$ Foreman-Mackey et al.
(2013) to perform the MCMC analysis.

##  III Results and Discussion

Figure 3:  The 1D and 2D marginalized probability distributions for the cosmic opacity parameter $\epsilon$ and curvature parameter $\rm\Omega_{K}$ by using ANN reconstructed different observation datasets. Table 1:  Summary of the constraints on the spatial curvature parameter $\rm\Omega_{K}$ and cosmic opacity parameter $\epsilon$ by using different data combination. Data Combination | $\rm\Omega_{K}$ | $\epsilon$
---|---|---
CC+156 HII | $0.016^{+0.762}_{-0.884}$ | $-0.145^{+0.036}_{-0.036}$
CC+181 HII | $-0.800^{+0.384}_{-0.389}$ | $-0.028^{+0.040}_{-0.040}$
OHD+156 HII | $-0.007^{+0.793}_{-0.845}$ | $-0.146^{+0.036}_{-0.035}$
OHD+181 HII | $-0.811^{+0.389}_{-0.391}$ | $-0.029^{+0.040}_{-0.038}$
CC+SN Ia | $0.043^{+0.446}_{-0.380}$ | $-0.007^{+0.026}_{-0.026}$
OHD+SN Ia | $0.094^{+0.395}_{-0.393}$ | $-0.008^{+0.024}_{-0.022}$

Let’s start with the ANN reconstructed 32 CC $H(z)$ observational data. It
shoud be stressed that the $H_{0}$ value is adopted from the the ANN
reconstruction at $z=0$ with the 32 CC $H(z)$ observation in this framework,
i.e., $H_{0}=67.35\pm 16.5\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$. Combined with
156 HII regions sample, the 1D marginalized probability distributions and 2D
regions with $1\sigma$ and $2\sigma$ contours corresponding to cosmic opacity
and curvature parameters are shown in Fig. 3. The best-fitting values with the
corresponding $1\sigma$ uncertainties are
$\rm\Omega_{K}=0.016^{+0.762}_{-0.884}$ and
$\epsilon=-0.145^{+0.036}_{-0.036}$. Meanwhile, for the latest 181 HII regions
sample, we obtain the best-fitting values $\rm\Omega_{K}$ and $\epsilon$ with
1$\sigma$ confidence level are $\rm\Omega_{K}=-0.800^{+0.384}_{-0.389}$ and
$\epsilon=-0.028^{+0.040}_{-0.040}$. In Fig. 4, the $1\sigma$ and $2\sigma$
confidence level contours for parameter estimations represent the constraint
results. The numerical results of the constraints on $\rm\Omega_{K}$ and
$\epsilon$ are shown in Table 1. The best-fitting values for both the cosmic
curvature and opacity are negative, and our results support a transparent and
closed universe at $1\sigma$ confidence level.

Working on the ANN reconstructed 63 OHD $H(z)$ sample, the Hubble constant
value is taken to be $H_{0}=68.27\pm 5.2\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$
from the ANN reconstruction at $z=0$, and we adopt this value for a prior. In
this case, the result seems to support a slight preference for the nonzero
cosmic opacity ($\epsilon=-0.146^{+0.036}_{-0.035}$) within 1$\sigma$
confidence level by combining 151 HII regions sample, while supporting a flat
universe in this case. However, combining 181 HII regions sample, we obtain a
closed universe ($\rm\Omega_{K}=-0.811^{+0.389}_{-0.391}$) within $1\sigma$
confidence level. In this case, the result indicates that a transparent
universe is supported. The corresponding numerical results are displayed in
lines 3 and 4 of Table 1. Recent reports suggest that there may be
inconsistencies in the spatial curvature between the early and late universe,
i.e., combining the Planck temperature and polarization power spectra data,
the work showed that a closed universe
($\rm\Omega_{K}=-0.044^{+0.018}_{-0.015}$) was supported. However, with the
combination of the Planck lensing data and low redshift baryon acoustic
oscillations observation, a flat universe was precisely constrained within
$\rm\Omega_{K}=0.0007\pm 0.0019$ Di Valentino et al. (2021, 2020); Handley
(2021).

Since the large of deviations of the results with the recent work Liao et al.
(2016, 2013); Liu et al. (2021), this prompted one thought about the above
results: whether the HII region samples are robust and reliable enough to
support us to infer phenomenology? In the subsequent analysis, we combine the
ANN reconstructed $H(z)$ data and latest SN Ia observational sample released
by the SH0ES and Pantheon$+$ collaboration to constrain on cosmic opacity and
curvature parameters Scolnic et al. (2022). The Pantheon$+$ dataset includes
1701 light curves, spectroscopically confirmed SN Ia and covering the redshift
range $0.001<z<2.26$ which comes from 18 different surveys. For SN Ia dataset,
the value of absolute magnitude $M_{B}$ is important. The $M_{B}$ is usually
calibrated by local measurement such as Cepheid variable stars. However, there
is strong degeneracy between the $M_{B}$ and $5\log_{10}H_{0}$ term inside the
distance modulus. Considering the consistency of the work, we convert the
Hubble constant used to absolute magnitudes, i.e., $M_{B}=-19.435$ $mag$
corresponds to $H_{0}=67.35\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ with the ANN
reconstructed 32 CC dataset, and $M_{B}=-19.406$ $mag$ corresponds to
$H_{0}=68.27\mathrm{~{}km~{}s^{-1}~{}Mpc^{-1}}$ with the ANN reconstructed 63
OHD dataset. The $1\sigma$ and $2\sigma$ confidence level contours for $H(z)+$
SN Ia constraint is shown in the right panel of the Fig. 3. We obtain the
best-fitting values are $\rm\Omega_{K}=0.043^{+0.446}_{-0.380}$ and
$\epsilon=-0.007^{+0.026}_{-0.026}$ by using the ANN reconstructed CC and the
Pantheon$+$ datasets. Working on 63 OHD sample, the best-fitting values are
$\rm\Omega_{K}=0.094^{+0.395}_{-0.393}$ and
$\epsilon=-0.008^{+0.024}_{-0.022}$. This result suggests that, combined with
the most popular astronomical observations of SN Ia plus $H(z)$, our universe
is a flat and transparent Universe. Compared with results from the HII regions
datasets, we find that the deviation of the cosmic curvature and opacity
parameters from zero values is largely due to the HII samples. Although the
samples of HII region are not sufficiently reliable (it may be that the small
sample size and large uncertainty lead to the bias of the inferred
phenomenology), this improved model-independent approach proposed in this work
to infer phenomenology may have a place in the future.

The second question is whether the degeneracy between curvature and opacity
reinforces the negative curvature results obtained from the HII region
datasets. In order to better analyze the degeneracy between curvature and
opacity, we first assume a flat universe ($\rm\Omega_{K}=0$). Combining the
results given by CC+156 HII and OHD+156 HII datasets, we are able to conclude
that the universe is opaque at $1\sigma$ confidence level, and the numerical
results are given in Table 2. The results given by CC+181 HII and OHD+181 HII
are shown a more opaque universe at $1\sigma$ confidence level compared to the
case of freeing the cosmic curvature parameter. We plot the probability
density function of the cosmic opacity parameter in Fig. 5. This result is
perfectly consistent with our hypothesis. When true HII galaxies emit photons
travel through the universe to the observer, the flux received by the observer
decreases because the universe may be opaque, which leads to an increase in
the actual observed luminosity distance. However, the observations of SN Ia
plus $H(z)$ give a almost transparency Universe. This results is in perfect
agreement with the recent work Liao et al. (2016); Liu et al. (2021); Liao et
al. (2015). Thus, we are able to conclude that this phenomenological deviation
($\epsilon\neq 0$) is mainly due to the unreliability or bias of the HII
region datasets, rather than the CC and OHD samples.

Assuming a transparent universe ($\epsilon=0$), we also plot the probability
density as the function of the cosmic curvature parameter in Fig. 6, and the
numerical results are shown in Table 2. We find that if we increase the degree
of freedom of the opacity parameter of the universe, the curvature of the
universe has a larger negative value. From above results, there is little
difference between the OHD and CC samples, however, the difference between the
156 HII and 181 HII samples is very large, which will lead to different cosmic
opacity and curvature. Obviously, the difference between the two HII samples
has a significant effect on the curvature constraint, and the curvature value
of the sample in the 181 HII region is greater than the negative curvature
value of the sample in the 156 HII region. Working on the Pantheon+ with CC
and OHD datasets, a flat Universe is both supported within the 1$\sigma$
uncertainties, but the center value is a little negative. This result is
consistent with the value inferred from Planck data Planck Collaboration et
al. (2020). These results strengthen our argument in answer to the first
question that the HII region samples are not particularly reliable. However,
although the degeneracy between the cosmic opacity and the spatial curvature
still exists, our method provides a new approach for constraining both the
cosmic opacity and the spatial curvature, and alleviates this situation to
some extent compared to considering cosmic curvature or opacity alone.

Figure 4: The probability density function of the cosmic opacity parameter with a flat universe ($\rm\Omega_{K}=0$). Figure 5: The probability density function of the cosmic curvature with a transparent universe ($\epsilon=0$). Table 2:  Summary of the constraints on the spatial curvature parameter $\rm\Omega_{K}$ and cosmic opacity parameter $\epsilon$ by using different data combination, with the assumption of a flat universe ($\rm\Omega_{K}=0$) or a transparent universe ($\epsilon=0$). Data Combination | $\rm\Omega_{K}$ | $\epsilon$
---|---|---
CC+156 HII (fixed $\rm\Omega_{K}$) | $-$ | $-0.143^{+0.039}_{-0.034}$
CC+156 HII (fixed $\rm\epsilon$) | $-0.988^{+0.847}_{-0.985}$ | $-$
CC+181 HII (fixed $\rm\Omega_{K}$) | $-$ | $-0.075^{+0.034}_{-0.032}$
CC+181 HII (fixed $\rm\epsilon$) | $-0.937^{+0.332}_{-0.368}$ | $-$
OHD+156 HII (fixed $\rm\Omega_{K}$) | $-$ | $-0.148^{+0.036}_{-0.033}$
OHD+156 HII (fixed $\rm\epsilon$) | $-1.003^{+0.857}_{-0.995}$ | $-$
OHD+181 HII (fixed $\rm\Omega_{K}$) | $-$ | $-0.072^{+0.033}_{-0.031}$
OHD+181 HII (fixed $\rm\epsilon$) | $-0.908^{+0.342}_{-0.382}$ | $-$
CC+SN Ia (fixed $\rm\Omega_{K}$) | $-0.012^{+0.269}_{-0.292}$ | $-$
CC+SN Ia (fixed $\rm\epsilon$) | $-$ | $-0.003^{+0.019}_{-0.018}$
OHD+SN Ia (fixed $\rm\Omega_{K}$) | $-0.056^{+0.266}_{-0.280}$ | $-$
OHD+SN Ia (fixed $\rm\epsilon$) | $-$ | $-0.003^{+0.018}_{-0.019}$

Benefiting from the ANN reconstructed technology, the $\rm{Hz/HII}$ data pairs
satisfying the redshift selection criteria have a considerable growth.
Therefore, a considerable amount of high-redshift samples (beyond the redshift
limit of SN Ia $z>1.4$) have been included in our analysis. Actually, the
combination of Hubble parameter and the HII regions observation allows us to
use ANN reconstructed technology to get more precise measurements on the
cosmic opacity parameter at level $\rm\Delta\epsilon\sim 10^{-2}$, although
the measurement precision of the curvature is at the level of
$\rm\Delta\Omega_{K}\sim 10^{-1}$. Our method provides a model-independent
constraint both on cosmic opacity and curvature, more stringent than other
current results based on real observational data. In addition, our results
show that there is a very strong degeneracy between cosmic opacity and
curvature, and this degeneracy is negatively correlated.

In order to highlight the potential of our method, it is necessary to compare
our results with previous works. Many works turned to luminous sources with
known (or standardizable) intrinsic luminosity in the universe, like SN Ia,
quasars and so on, to infer observations of opacity-dependent luminosity
distances. For example, some works Li et al. (2013); Liao et al. (2013, 2015)
used Union 2.1 + galaxies cluster sample, Union2.1 + $H(z)$, JLA + $H(z)$ to
constrain cosmic opacity, and obtained the precision on cosmic opacity
parameter at level $\rm\Delta\epsilon\sim 10^{-1}$. The recent works Qi et al.
(2019); Fu et al. (2021); Geng et al. (2020); Wei (2019) used simulated
gravitational wave observations to replace Hubble parameter observations to
constrain cosmic opacity. These works showed the opacity parameter
$\Delta\epsilon\sim 10^{-3}$ at 68.3% confidence level. However, we should
seek other methods and technologies until the observed gravitational wave
events will be sufficient to get statistical results in the future.

##  IV Conclusion

The cosmic opacity and curvature parameter both play the important roles in
modern cosmology. The cosmic opacity may indicate that photon numbers are not
conserved from source to the observer due to some new physical phenomenon.
Cosmic opacity may be caused by absorption/scattering due to matter in the
universe, or by extragalactic magnetic fields that can turn photons into
unobserved particles (e.g. light axions, chameleons, gravitons, Kaluza-Klein
modes), and it thus is crucial to correctly interpret cosmic opacity for
astronomical photometric measurements, like SN Ia, HII regions and quasar
observations. According to general relativity, photons move along geodesics in
the universe, and curvature determines the spatial structure of the universe,
which means that the path of photons from the source to the observer will be
affected by the curvature of the universe. Different curvature of the universe
will result in different measured cosmological distances, which may also cause
the supernova to faint. Therefore, it is necessary to measure cosmic curvature
and opacity simultaneously.

In this paper, we have proposed a new model-independent method to
simultaneously measure the cosmic curvature and opacity by using the latest
observations of HII galaxies acting as standard candles and the latest Hubble
parameter observations. We adopt the non-parameterized method Artificial
Neural Network to reconstruct observed Hubble parameter. Our results support a
slightly opaque and closed universe at $1\sigma$ confidence level. However, it
should be emphasized that the cosmic absorption (caused by the opaque of the
Universe) affects the luminosity distances derived from “$L$–$\sigma$”
relation of HII regions, but also generates influences on the measured
luminosity distances of SN Ia. In particular, considering the fact that a
opaque Universe will lead an increase in the actual observed luminosity
distance, which means that the sources we observe are farther away and less
bright than they really are. Therefore, this is also a mechanism to explain
the dimming of distant SN Ia. In the subsequent analysis, we suggest that
there could be two possible reasons for this results. Firstly, the increase in
the degree of freedom of the opacity parameter makes curvature more negative,
that is, the negative correlation between the opacity and the curvature
parameter leads to this result. Secondly, unreliability of the two HII region
samples rather than the $H(z)$ samples, cosmic curvature obtained using the
HII sample has the large negative values. This conclusion is supported by
jointly analysis of the most popular astronomical observations of SN Ia plus
$H(z)$ dataset, such combination supports a flat and transparent Universe
within 1$\sigma$ uncertainty. More importantly, we obtain the measurement
precision on the cosmic opacity parameter $\rm\Delta\epsilon\sim 10^{-2}$, and
the measurement precision on curvature $\rm\Delta\Omega_{K}\sim 10^{-1}$,
simultaneously. In the case of assuming a transparent universe, our results
indicate in some ways that our universe has a spatial structure of negative
curvature. A strong degeneracy between the cosmic opacity and curvature
parameters is also revealed in this analysis. Note that such negative
correlation could potentially affect our constraint on cosmic opacity or
curvature alone. Meanwhile, we need to emphasize that our work suggests a new
approach to constrain the cosmic opacity and the spatial curvature,
simultaneously. Although the degeneracy between the cosmic opacity and the
spatial curvature still exists, our method alleviates this situation to some
extent compared to considering cosmic curvature or opacity alone.

The ANN method we used here is not unique. It is still worth exploring whether
these conclusions will change with different machine learning reconstruction
methods. However, both GP and ANN method have their own advantages and
disadvantages Wang et al. (2020d), and show great potential in the studies of
precision cosmology. For instance, the GP reconstruction greatly reduces the
uncertainty of data, and the reconstructed data points are related to each
other. The ANN method is more like a black box, and we don’t know what’s going
on inside the box. Therefore, we emphasize here that in the era before the
wealth of available data, it is also very necessary to carefully choose the
method of data reconstruction.

As a final remark, although the simultaneous constraint on cosmic opacity and
curvature by using HII regions and Hubble parameter measurements do not
significantly improve precision, yet it helps us to gain a deeper
understanding relation between cosmic opacity and curvature in the early
universe ($z\sim 2.5$). We also look forward to a large amount of future data,
not only from the Hubble parameters, but also from the HII regions, allowing
us to further improve the precision of the cosmic opacity and curvature
constraints through various machine learning techniques in the future.

## Acknowledgments

The authors are grateful to the referee for constructive comments, which
allowed to improve the paper substantially. Liu. T.-H was supported by
National Natural Science Foundation of China under Grant No. 12203009; Chutian
Scholars Program in Hubei Province (X2023007); Hubei Province Foreign Expert
Project (2023DJC040). Y. Y. was supported by the National Natural Science
Foundation of China under Grant No. 12105097, and Scientific Research Fund of
Hunan Provincial Education Department(No. 20C0787). Wu. S.-M was supported by
National Natural Science Foundation of China under Grant No. 12205133.

## References

  * Riess et al. (1998) A. G. Riess, A. V. Filippenko, P. Challis, et al., Astron. J. 116, 1009 (1998), eprint astro-ph/9805201.
  * Perlmutter et al. (1999) S. Perlmutter, G. Aldering, G. Goldhaber, T. S. C. Project, et al., Astrophys. J. 517, 565 (1999), eprint astro-ph/9812133.
  * Scolnic et al. (2018) D. M. Scolnic, D. O. Jones, A. Rest, et al., Astrophys. J. 859, 101 (2018), eprint 1710.00845.
  * Qi et al. (2017) J.-Z. Qi, S. Cao, M. Biesiada, X. Zheng, and Z.-H. Zhu, European Physical Journal C 77, 502 (2017), eprint 1708.08603.
  * Xu et al. (2018) T. Xu, S. Cao, J. Qi, M. Biesiada, X. Zheng, and Z.-H. Zhu, JCAP 2018, 042 (2018), eprint 1708.08631.
  * Ratra and Peebles (1988) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988).
  * Caldwell et al. (1998) R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998), eprint astro-ph/9708069.
  * Cao et al. (2011) S. Cao, N. Liang, and Z.-H. Zhu, Mon. Not. Roy. Astron. Soc. 416, 1099 (2011), eprint 1012.4879.
  * Cao and Liang (2011) S. Cao and N. Liang, arXiv e-prints arXiv:1105.6274 (2011), eprint 1105.6274.
  * Cao and Zhu (2014) S. Cao and Z.-H. Zhu, Phys. Rev. D 90, 083006 (2014), eprint 1410.6567.
  * Ma et al. (2017) Y. Ma, J. Zhang, S. Cao, X. Zheng, T. Xu, and J. Qi, European Physical Journal C 77, 891 (2017).
  * Qi et al. (2018) J.-Z. Qi, S. Cao, M. Biesiada, T.-P. Xu, Y. Wu, S.-X. Zhang, and Z.-H. Zhu, Research in Astronomy and Astrophysics 18, 066 (2018), eprint 1803.04109.
  * Tolman (1930) R. C. Tolman, Proceedings of the National Academy of Science 16, 511 (1930).
  * Chen (1995) P. Chen, Phys. Rev. Lett. 74, 634 (1995).
  * Csáki et al. (2002) C. Csáki, N. Kaloper, and J. Terning, Phys. Rev. Lett. 88, 161302 (2002), eprint hep-ph/0111311.
  * Avgoustidis et al. (2010a) A. Avgoustidis, C. Burrage, J. Redondo, L. Verde, and R. Jimenez, JCAP 2010, 024 (2010a), eprint 1004.2053.
  * Jaeckel and Ringwald (2010) J. Jaeckel and A. Ringwald, Annual Review of Nuclear and Particle Science 60, 405 (2010), eprint 1002.0329.
  * Deffayet and Uzan (2000) C. Deffayet and J.-P. Uzan, Phys. Rev. D 62, 063507 (2000), eprint hep-ph/0002129.
  * Holanda et al. (2018) R. F. L. Holanda, S. H. Pereira, and D. Jain, Phys. Rev. D 97, 023538 (2018), eprint 1801.04344.
  * Liao et al. (2015) K. Liao, A. Avgoustidis, and Z. Li, Phys. Rev. D 92, 123539 (2015), eprint 1512.01861.
  * Planck Collaboration et al. (2020) Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, et al., Astron. Astrophys. 641, A6 (2020), eprint 1807.06209.
  * Riess et al. (2019) A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. 876, 85 (2019), eprint 1903.07603.
  * Freedman (2017) W. L. Freedman, Nature Astronomy 1, 0169 (2017), eprint 1706.02739.
  * Di Valentino et al. (2020) E. Di Valentino, A. Melchiorri, and J. Silk, Nature Astronomy 4, 196 (2020), eprint 1911.02087.
  * Di Valentino et al. (2021) E. Di Valentino, L. A. Anchordoqui, Ö. Akarsu, et al., Astroparticle Physics 131, 102605 (2021), eprint 2008.11284.
  * Handley (2021) W. Handley, Phys. Rev. D 103, L041301 (2021), eprint 1908.09139.
  * Poulin et al. (2019) V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Phys. Rev. Lett. 122, 221301 (2019), eprint 1811.04083.
  * Feeney et al. (2019) S. M. Feeney, H. V. Peiris, A. R. Williamson, S. M. Nissanke, D. J. Mortlock, J. Alsing, and D. Scolnic, Phys. Rev. Lett. 122, 061105 (2019), eprint 1802.03404.
  * Dainotti et al. (2021) M. G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, and G. Lambiase, Astrophys. J. 912, 150 (2021), eprint 2103.02117.
  * Liu et al. (2022) T. Liu, S. Cao, X. Li, H. Zheng, Y. Liu, W. Guo, and C. Zheng, Astron. Astrophys. 668, A51 (2022), eprint 2210.02765.
  * Holanda et al. (2013) R. F. L. Holanda, J. C. Carvalho, and J. S. Alcaniz, JCAP 2013, 027 (2013), eprint 1207.1694.
  * Liao et al. (2013) K. Liao, Z. Li, J. Ming, and Z.-H. Zhu, Physics Letters B 718, 1166 (2013), eprint 1212.6612.
  * Jesus et al. (2017) J. F. Jesus, R. F. L. Holanda, and M. A. Dantas, General Relativity and Gravitation 49, 150 (2017), eprint 1605.01342.
  * Wang et al. (2017a) G.-J. Wang, J.-J. Wei, Z.-X. Li, J.-Q. Xia, and Z.-H. Zhu, Astrophys. J. 847, 45 (2017a), eprint 1709.07258.
  * Liu et al. (2020a) T. Liu, S. Cao, M. Biesiada, Y. Liu, S. Geng, and Y. Lian, Astrophys. J. 899, 71 (2020a), eprint 2006.14776.
  * Peel et al. (2019) A. Peel, F. Lalande, J.-L. Starck, et al., Phys. Rev. D 100, 023508 (2019), eprint 1810.11030.
  * Wang et al. (2020a) G.-J. Wang, S.-Y. Li, and J.-Q. Xia, Astrophys. J.s 249, 25 (2020a), eprint 2005.07089.
  * Wang et al. (2020b) G.-J. Wang, X.-J. Ma, S.-Y. Li, and J.-Q. Xia, Astrophys. J.s 246, 13 (2020b), eprint 1910.03636.
  * Liu et al. (2021) T. Liu, S. Cao, S. Zhang, X. Gong, W. Guo, and C. Zheng, European Physical Journal C 81, 903 (2021), eprint 2110.00927.
  * Cao et al. (2017) S. Cao, J. Qi, M. Biesiada, X. Zheng, T. Xu, Y. Pan, and Z.-H. Zhu, arXiv e-prints arXiv:1708.08608 (2017), eprint 1708.08608.
  * Wei (2018) J.-J. Wei, Astrophys. J. 868, 29 (2018), eprint 1806.09781.
  * Riess et al. (2016) A. G. Riess, L. M. Macri, S. L. Hoffmann, et al., Astrophys. J. 826, 56 (2016), eprint 1604.01424.
  * Risaliti and Lusso (2019) G. Risaliti and E. Lusso, Nature Astronomy 3, 272 (2019), eprint 1811.02590.
  * Liu et al. (2020b) T. Liu, S. Cao, J. Zhang, M. Biesiada, Y. Liu, and Y. Lian, Mon. Not. Roy. Astron. Soc. 496, 708 (2020b), eprint 2005.13990.
  * Terlevich and Melnick (1981) R. Terlevich and J. Melnick, Mon. Not. Roy. Astron. Soc. 195, 839 (1981).
  * Terlevich et al. (2015) R. Terlevich, E. Terlevich, J. Melnick, R. Chávez, M. Plionis, F. Bresolin, and S. Basilakos, Mon. Not. Roy. Astron. Soc. 451, 3001 (2015), eprint 1505.04376.
  * Wei et al. (2016) J.-J. Wei, X.-F. Wu, and F. Melia, Mon. Not. Roy. Astron. Soc. 463, 1144 (2016), eprint 1608.02070.
  * Melnick et al. (1987) J. Melnick, M. Moles, R. Terlevich, and J.-M. Garcia-Pelayo, Mon. Not. Roy. Astron. Soc. 226, 849 (1987).
  * Plionis et al. (2011) M. Plionis, R. Terlevich, S. Basilakos, F. Bresolin, E. Terlevich, J. Melnick, and R. Chavez, Mon. Not. Roy. Astron. Soc. 416, 2981 (2011), eprint 1106.4558.
  * Mania and Ratra (2012) D. Mania and B. Ratra, Physics Letters B 715, 9 (2012), eprint 1110.5626.
  * Wu et al. (2020) Y. Wu, S. Cao, J. Zhang, T. Liu, Y. Liu, S. Geng, and Y. Lian, Astrophys. J. 888, 113 (2020), eprint 1911.10959.
  * González-Morán et al. (2021) A. L. González-Morán, R. Chávez, E. Terlevich, et al., Mon. Not. Roy. Astron. Soc. 505, 1441 (2021), eprint 2105.04025.
  * Chávez et al. (2014) R. Chávez, R. Terlevich, E. Terlevich, et al., Mon. Not. Roy. Astron. Soc. 442, 3565 (2014), eprint 1405.4010.
  * Fernández Arenas et al. (2018) D. Fernández Arenas, E. Terlevich, R. Terlevich, et al., Mon. Not. Roy. Astron. Soc. 474, 1250 (2018), eprint 1710.05951.
  * Gordon et al. (2003) K. D. Gordon, G. C. Clayton, K. A. Misselt, A. U. Landolt, and M. J. Wolff, Astrophys. J. 594, 279 (2003), eprint astro-ph/0305257.
  * Jimenez and Loeb (2002) R. Jimenez and A. Loeb, Astrophys. J. 573, 37 (2002), eprint astro-ph/0106145.
  * Avgoustidis et al. (2010b) A. Avgoustidis, C. Burrage, J. Redondo, L. Verde, and R. Jimenez, JCAP 2010, 024 (2010b), eprint 1004.2053.
  * Li et al. (2013) Z. Li, P. Wu, H. Yu, and Z.-H. Zhu, Phys. Rev. D 87, 103013 (2013), eprint 1304.7317.
  * Wu et al. (2022) P.-J. Wu, J.-Z. Qi, and X. Zhang, arXiv e-prints arXiv:2209.08502 (2022), eprint 2209.08502.
  * Zhang et al. (2014) C. Zhang, H. Zhang, S. Yuan, S. Liu, T.-J. Zhang, and Y.-C. Sun, Research in Astronomy and Astrophysics 14, 1221-1233 (2014), eprint 1207.4541.
  * Stern et al. (2010) D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, and S. A. Stanford, JCAP 2010, 008 (2010), eprint 0907.3149.
  * Moresco et al. (2012) M. Moresco, A. Cimatti, R. Jimenez, et al., JCAP 2012, 006 (2012), eprint 1201.3609.
  * Moresco et al. (2016) M. Moresco, L. Pozzetti, A. Cimatti, et al., JCAP 2016, 014 (2016), eprint 1601.01701.
  * Ratsimbazafy et al. (2017) A. L. Ratsimbazafy, S. I. Loubser, S. M. Crawford, et al., Mon. Not. Roy. Astron. Soc. 467, 3239 (2017), eprint 1702.00418.
  * Moresco (2015) M. Moresco, Mon. Not. Roy. Astron. Soc. 450, L16 (2015), eprint 1503.01116.
  * Gaztañaga et al. (2009) E. Gaztañaga, A. Cabré, and L. Hui, Mon. Not. Roy. Astron. Soc. 399, 1663 (2009), eprint 0807.3551.
  * Mukherjee and Banerjee (2021) P. Mukherjee and N. Banerjee, Phys. Rev. D 103, 123530 (2021), eprint 2105.09995.
  * Wang et al. (2017b) Y. Wang, G.-B. Zhao, C.-H. Chuang, et al., Mon. Not. Roy. Astron. Soc. 469, 3762 (2017b), eprint 1607.03154.
  * Alam et al. (2017) S. Alam, M. Ata, S. Bailey, et al., Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), eprint 1607.03155.
  * Blake et al. (2012) C. Blake, S. Brough, M. Colless, et al., Mon. Not. Roy. Astron. Soc. 425, 405 (2012), eprint 1204.3674.
  * Zhao et al. (2019) G.-B. Zhao, Y. Wang, S. Saito, et al., Mon. Not. Roy. Astron. Soc. 482, 3497 (2019), eprint 1801.03043.
  * Favale et al. (2023) A. Favale, A. Gómez-Valent, and M. Migliaccio, Mon. Not. Roy. Astron. Soc. (2023), eprint 2301.09591.
  * Lian et al. (2021) Y. Lian, S. Cao, M. Biesiada, Y. Chen, Y. Zhang, and W. Guo, Mon. Not. Roy. Astron. Soc. 505, 2111 (2021), eprint 2105.04992.
  * Briffa et al. (2023) R. Briffa, C. Escamilla-Rivera, J. L. Said, and J. Mifsud, Mon. Not. Roy. Astron. Soc. 522, 6024 (2023), eprint 2303.13840.
  * Clevert et al. (2015) D.-A. Clevert, T. Unterthiner, and S. Hochreiter, arXiv e-prints arXiv:1511.07289 (2015), eprint 1511.07289.
  * Kingma and Ba (2014) D. P. Kingma and J. Ba, arXiv e-prints arXiv:1412.6980 (2014), eprint 1412.6980.
  * Liu et al. (2023) T. Liu, S. Cao, S. Ma, Y. Liu, C. Zheng, and J. Wang, Physics Letters B 838, 137687 (2023).
  * Liu et al. (2019) T. Liu, S. Cao, J. Zhang, S. Geng, Y. Liu, X. Ji, and Z.-H. Zhu, Astrophys. J. 886, 94 (2019), eprint 1910.02592.
  * Wang et al. (2021) G.-J. Wang, X.-J. Ma, and J.-Q. Xia, Mon. Not. Roy. Astron. Soc. 501, 5714 (2021), eprint 2004.13913.
  * Wang et al. (2020c) G.-J. Wang, S.-Y. Li, and J.-Q. Xia, Astrophys. J.s 249, 25 (2020c), eprint 2005.07089.
  * Wang et al. (2020d) G.-J. Wang, X.-J. Ma, S.-Y. Li, and J.-Q. Xia, Astrophys. J.s 246, 13 (2020d), eprint 1910.03636.
  * Freedman et al. (2019) W. L. Freedman, B. F. Madore, D. Hatt, T. J. Hoyt, I. S. Jang, R. L. Beaton, C. R. Burns, M. G. Lee, A. J. Monson, J. R. Neeley, et al., Astrophys. J. 882, 34 (2019), eprint 1907.05922.
  * Renzi and Silvestri (2023) F. Renzi and A. Silvestri, Phys. Rev. D 107, 023520 (2023).
  * Busti et al. (2014) V. C. Busti, C. Clarkson, and M. Seikel, Mon. Not. Roy. Astron. Soc. 441, L11 (2014), eprint 1402.5429.
  * Bonilla et al. (2021) A. Bonilla, S. Kumar, and R. C. Nunes, European Physical Journal C 81, 127 (2021), eprint 2011.07140.
  * Wang and Meng (2017) D. Wang and X.-H. Meng, Phys. Rev. D 95, 023508 (2017), eprint 1708.07750.
  * Nunes (2018) R. C. Nunes, JCAP 2018, 052 (2018), eprint 1802.02281.
  * D’Agostino and Nunes (2020) R. D’Agostino and R. C. Nunes, Phys. Rev. D 101, 103505 (2020), eprint 2002.06381.
  * Shafieloo et al. (2013) A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D 87, 023520 (2013), eprint 1211.6128.
  * Joudaki et al. (2018) S. Joudaki, M. Kaplinghat, R. Keeley, and D. Kirkby, Phys. Rev. D 97, 123501 (2018), eprint 1710.04236.
  * Bernardo and Levi Said (2021) R. C. Bernardo and J. Levi Said, JCAP 2021, 014 (2021), eprint 2105.12970.
  * Shafieloo et al. (2012) A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D 85, 123530 (2012), eprint 1204.2272.
  * Zhang et al. (2022) J.-C. Zhang, K. Jiao, T. Zhang, T.-J. Zhang, and B. Yu, Astrophys. J. 936, 21 (2022), eprint 2208.03960.
  * Li et al. (2011) Z. Li, P. Wu, and H. Yu, Astrophys. J. Lett. 729, L14 (2011), eprint 1101.5255.
  * Liao et al. (2016) K. Liao, Z. Li, S. Cao, M. Biesiada, X. Zheng, and Z.-H. Zhu, Astrophys. J. 822, 74 (2016), eprint 1511.01318.
  * Zheng et al. (2020) X. Zheng, K. Liao, M. Biesiada, S. Cao, T.-H. Liu, and Z.-H. Zhu, Astrophys. J. 892, 103 (2020), eprint 2002.09909.
  * Qi et al. (2019) J.-Z. Qi, S. Cao, C. Zheng, Y. Pan, Z. Li, J. Li, and T. Liu, Phys. Rev. D 99, 063507 (2019), eprint 1902.01988.
  * Avgoustidis et al. (2009) A. Avgoustidis, L. Verde, and R. Jimenez, JCAP 2009, 012 (2009), eprint 0902.2006.
  * Foreman-Mackey et al. (2013) D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Publications of the Astronomical Society of the Pacific 125, 306 (2013), eprint 1202.3665.
  * Scolnic et al. (2022) D. Scolnic, D. Brout, A. Carr, A. G. Riess, T. M. Davis, A. Dwomoh, D. O. Jones, N. Ali, P. Charvu, R. Chen, et al., Astrophys. J. 938, 113 (2022), eprint 2112.03863.
  * Fu et al. (2021) X. Fu, L. Zhou, J. Yang, Z.-Y. Lu, Y. Yang, and G. Tang, Chinese Physics C 45, 065104 (2021).
  * Geng et al. (2020) S. Geng, S. Cao, T. Liu, M. Biesiada, J. Qi, Y. Liu, and Z.-H. Zhu, Astrophys. J. 905, 54 (2020), eprint 2010.05151.
  * Wei (2019) J.-J. Wei, Astrophys. J. 876, 66 (2019), eprint 1902.00223.