# AuthNet: A Deep Learning based Authentication Mechanism using Temporal
Facial Feature Movements

Mohit Raghavendra1*, Pravan Omprakash 2*, Mukesh B R1, Sowmya Kamath1

###### Abstract

Biometric systems based on Machine learning and Deep learning are being
extensively used as authentication mechanisms in resource-constrained
environments like smartphones and other small computing devices. These AI-
powered facial recognition mechanisms have gained enormous popularity in
recent years due to their transparent, contact-less and non-invasive nature.
While they are effective to a large extent, there are ways to gain
unauthorized access using photographs, masks, glasses, etc. In this paper, we
propose an alternative authentication mechanism that uses both facial
recognition and the unique movements of that particular face while uttering a
password, that is, the temporal facial feature movements. The proposed model
is not inhibited by language barriers because a user can set a password in any
language. When evaluated on the standard MIRACL-VC1 dataset, the proposed
model achieved an accuracy of 98.1%, underscoring its effectiveness as an
effective and robust system. The proposed method is also data efficient, since
the model gave good results even when trained with only 10 positive video
samples. The competence of the training of the network is also demonstrated by
benchmarking the proposed system against various compounded Facial recognition
and Lip reading models.

## Introduction

Biometric authentication mechanisms have been long in use in digital systems
for identifying and facilitating access to restricted systems or information
meant only for authorized users. Facial authentication systems have gained
widespread usage in the past few years because of its contact-less non-
invasive nature and ease of use in verifying a person’s identity from a user’s
image. Early image based facial recognition systems were built on hand-
engineered features such as SIFT, LBP, and Fisher vectors. Recently, highly
accurate deep learning methods such as FaceNet (Schroff, Kalenichenko, and
Philbin 2015), Baidu (Liu et al. 2015) and DeepID models (Ouyang et al. 2014;
Sun, Wang, and Tang 2014; Sun et al. 2015) have surpassed human performance.

Figure 1: Data preprocessing Pipeline

Nevertheless, facial authentication systems have often been bypassed by
imposters using a photograph of the authorized user to gain access. To combat
this vulnerability, models that track lip movement patterns of the user
uttering a word are being explored. Deep learning models for lip reading such
as LipNet (Shillingford et al. 2018) have achieved a sentence-level
classification accuracy of 95%. Various other lip reading models have been
built on Hidden Markov models (HMM), Long Short Term Memory (LSTM) Networks,
Convolutional Neural Networks (CNN), etc.

Figure 2: VGGFace Architecture (Adapted from (Parkhi, Vedaldi, and Zisserman
2015))

However, they perform poorly in different lighting conditions and rely solely
on lip movement and lip features. Deep face recognition models combined with
images without the need for audio present a unique approach that are safer
than basic face recognition systems and are immune to noisy environments,
hence addressing the security and privacy concerns. This paper proposes an
authentication model that captures the temporal facial movement patterns of a
person uttering a particular word. A VGGNet model (Liu and Deng 2015) called
VGGFace (Parkhi, Vedaldi, and Zisserman 2015) pre-trained on faces was
employed to capture the facial features and then the sequence of features were
passed through several LSTM layers, and the model learns to predict whether a
valid user is speaking their chosen password or not. The model is not provided
with the actual word of the chosen password, but only the videos of its
utterance. Hence, it is inherently language invariant and can be any
combination of characters and numbers if spoken continuously. The proposed
model was benchmarked on the publicly available standard dataset, MIRACL-VC1
(Rekik, Ben-Hamadou, and Mahdi 2014), and to test the system’s performance on
real-world data and requirements, a separate dataset that consists of videos
from smartphone cameras taken under varying conditions was compiled. The
system was tested on this collated dataset as well.

The key contribution of this paper is a strong video based facial
authentication system, which uses a video of a person uttering a password,
while correctly identifying the as imposter cases, which are 1) same person
uttering a different word (Same Person case) or, 2) another person trying to
gain access, and prevent them from gaining access ( Different person
case).Another key contribution and salient point of this model is that it is
language independent and domain agnostic. To our knowledge such a system has
never been implemented.

The rest of this paper has been structured as follows. Section 2, discusses
relevant works in the field of lip reading and authentication and the observed
gaps that invite further improvements. Section 3 details the proposed
approach, Section 4 describes the process by which the model was evaluated and
validated, followed by conclusion and references.

## Related Work

Several deep learning based face recognition models have been proposed and are
popular currently (Balaban 2015). FaceNet (Schroff, Kalenichenko, and Philbin
2015), Baidu (Liu et al. 2015) and the DeepID models (Ouyang et al. 2014; Sun,
Wang, and Tang 2014; Sun et al. 2015) have shown near perfect results on
benchmark datasets. These face recognition systems have been trained on
millions of images and have surpassed human-level performance. However, they
are still susceptible to attacks if used as authentication systems. In this
section, the state-of-the-art works in this field is discussed and few
observed gaps for further improvement in the performance of these systems are
presented.

Mathulaprangsan et al. (Mathulaprangsan et al. 2015) studied the various
techniques used in lip reading for word classification as well as lip password
authentication. Various localization, segmentation and classification
approaches have been used, including conventional algorithms like FCM and
LCACM that are used for lip localization in an image. They reported that the
most common classifiers used for visual speech recognition at the time were,
namely, Gaussian Mixture Models (GMMs) and Hidden Markov Models (HMMs). Liu et
al. (Liu and Cheung 2014) proposed a lip password verification model using
Multi-boosted HMMs, focusing solely on the behavioral biometrics of lip
movements. They extracted visual sequences of the mouth area region and ran
them through an algorithm that segments lips in order to split the password
into sub units, on which Multi-boosted HMMs were applied to formulate a
decision boundary. They reported an equal error rate (EER) of 4.06%, however,
the passwords that could be used were constrained to only digits. Noda et al.
(Özcan and Basturk 2019) proposed a deep learning based visual speech
recognition system that uses a CNN that is trained on mouth area images to
predict phenomes. The outputs of the CNN were regarded as sequences and a
HMM+GMM observation model was used for the word recognition task. Their work
demonstrated that visual features acquired with a CNN generalized to different
domains outperformed conventional approaches.

Wand et al. (Wand, Koutnik, and Schmidhuber 2016) propose a different approach
for lip reading in the form of LSTM based neural networks. Sequences of mouth
area images, whose features were extracted with a Histogram of Oriented
Gradients (HOG) were fed into the LSTM, which was used for word
classification. Their approach performed significantly better than the
conventional SVM and HMM classifiers. Garg et al. (amit, jnoyola, and sameepb
2016) designed a new approach using a hybrid CNN and LSTM model to classify
words, which was trained on the same dataset that is used in this paper,
MIRACL-VC1 (Rekik, Ben-Hamadou, and Mahdi 2014). They compared different
approaches that included CNN+LSTM, purely CNN and SVM classifiers. They
concluded that the purely CNN approach worked best for visual speech
recognition attaining a 61.2% and also noted that with more training on the
CNN+LSTM model the performance might improve.

Shillingford et al. (Shillingford et al. 2018) introduced a novel approach
called LipNet that performs lip reading classification on sentence level with
an accuracy of 95%. They made use of a combination of spatio-temporal
convolutions and recurrent neural networks to outperform human lip readers
becoming the state of the art model.

From the discussion presented on several previous works and the state-of-the-
art models in the field of facial recognition based biometric systems, several
vulnerabilities that need to be addressed for further improvements were
observed. Facial recognition systems are not immune to attacks and
vulnerabilities in the real world. Existing models for lip password
verification have not achieved high accuracy, and the passwords are
constrained to specific characters, for instance, only digits can be used.
Moreover, most models are restricted to a certain language that they are
trained on. In view of these limitations, an attempt has been made to solve
these issues in this work, by designing an model that is inherently language
and domain agnostic. The details of the proposed model are presented in the
next section.

Figure 3: Proposed Architecture built on LSTM Network

## Proposed Approach

The proposed model uses a 2 step CNN-RNN architecture depicted in Fig. 3. A
pre-trained VGGFace model (Parkhi, Vedaldi, and Zisserman 2015) is used to
extract the essential facial features at each timestep. To identify and learn
the movement patterns of these features, a small network of stacked LSTM
layers is employed. After the training is complete, this network can determine
if a particular test case’s person-word combination matches the trained
person-word combination.

The MIRACL-VC1 Words dataset consists of 10 speakers speaking 10 different
words, 10 times each. This has been split into different sections for
different forms of testing and validation. It was randomly chosen to set aside
5 speakers i.e 5 X 10 X 10 = 500 utterances as unseen data for the model to
exhibit its robustness against the Different Person Case (mentioned above).
Among the remaining 5 speakers, we again set aside 3 words each, i.e 5 X 3 X
10 = 150 utterances to test the effectiveness of the model for the Same Person
Case. Thus, a total of 650 examples were set aside. The remaining set has 5 X
7 X 10 = 350 utterances, in which 10 examples are positive(from the right
speaker saying right word), 60 are negative samples because of the right
speaker saying wrong word and 280 examples of wrong speakers. The 10 positive
samples are oversampled to generate 100 new positive samples, giving a total
of 350 + 100 = 450 examples. This is split into a 70 : 30 train-test split,
and the the 650 examples that were set aside before are then added to this
test set as negative samples,to cover all the aforementioned imposter
cases.This approach is adopted to ensure that we are dealing with completely
new data that model can possibly have never seen before.

In summary, for any given person-word combination, there are 10 samples of
correct person correct word (oversampled to 110), 90 samples of correct person
saying wrong word, 90 samples of wrong person saying correct word and 810
samples of wrong person saying wrong word, making the total of 1000 words i.e
10 speakers uttering 10 words, 10 times each.

To provide a comprehensive analysis of the performance and demonstrate the
generality and robustness of the system, the model was trained in an iterative
manner on all such possible combinations and tested against unseen examples.
So, the training is repeated for the 5 randomly chosen speakers, each speaking
7 different words, generating 35 possible person-word combinations.

Data preprocessing.  Each training and testing sample in the MIRACL-VC1
dataset is a sequence of images of the person uttering a password, typically
consisting of 5 to 15 images, as depicted in Fig. 1. Only the color images for
the words have been utilised in training. The images are first passed through
a Haar cascade face detector (Viola and Jones 2004) to crop out the region
containing faces from the images. The Haar cascade detector is commonly used
to detect faces and face features like eyes, lips, nose, etc., and is
available in the OpenCV library. It can be used to detect faces or objects in
general. The algorithm works in four steps; starting with selecting Haar
features, creating a set of images for easier processing, training with
Adaboost and passing them through a set of cascading classifiers. To ensure
uniformity in number of timesteps, the images are padded with white images.
Each image is then resized into 224 X 224 pixels to feed it into the pre-
trained VGGNet model. For the manually compiled dataset, a similar approach
was followed. This dataset represents a subset of possible real world
conditions. They are first segmented into images using a constant frame rate,
padded to ensure uniformity in number of timesteps and resized into 224 X 224
pixels to be fed into the VGGFace model.

VGGFace.  Each image is passed through the pre-trained VGGFace model, which
consists of a long sequence of convolutional layers that are trained on
hundreds of thousands of images of celebrity faces. We obtain a 2,622
dimensional feature vector output for each image. This process is repeated for
each word and for each person (as illustrated in Fig. 4), and reshaped to make
it suitable for feeding into the LSTM layer. Thus each sample has 20 timesteps
and 2622 features. The samples are labelled 1 for the correct person-word
combination, and 0 for all other combinations, making this a binary
classification problem. The resulting class imbalance was solved by using the
technique of oversampling.

Figure 4: Variations in Features Extracted using VGGFace

LSTM network.  LSTM networks have reported astounding results across a wide
variety of application domains where data is sequential. So, a network of 4
LSTM layers is employed, each consisting of 20 timesteps and an output sigmoid
layer for predicting probabilities. Once trained, this model can determine if
the input test video contains the correct person-word combination or not, so
that an authenticated user can be granted access.

## Experimental Evaluation & Metrics

Several experiments were performed to validate the proposed approach. The
model was trained on a system with an i7-8640U CPU and 8GB of RAM. The model
was bench marked on the standard MIRACL-VC1 dataset and also on a collated
dataset, which consists of videos, each of 2 seconds duration. The videos were
obtained from a OnePlus 6 smartphone with 1080p resolution and 16MP camera, a
OnePlus 7 smartphone with 1080p resolution and 16MP camera, and an Apple
iPhone XR smartphone with 720p resolution and 7MP camera. They were captured
under different lighting conditions and also against varying backgrounds. The
detailed statistics of each dataset used is described in Table 1. The proposed
model took 154.2 seconds per person-word combination.

Table 1: Dataset statistics Dataset | MIRACL-VC1 | Collated Dataset
---|---|---
# Speakers | 10 | 3
# Words | 10 | 3
# Utterances | 10 | 6
Image Resolution | $640\times 480\times 3$ | $350\times 640\times 3$
Language used | English | English and Hindi

The proposed pipeline has an binary classification task as the objective, and
is designed to take a series of images of size $640\times 480$, ordered
according to timesteps, as input. The images were then passed through the
pretrained VGGFace model as described in Figure 2. The resultant feature
vectors were then used for training and evaluating the LSTM network as
described in Figure 3. The proposed model was trained using the binary
crossentropy (BCE) loss function (calculated as per Eq. (1)) and the Adam
optimizer, with an initial learning rate of 0.001 for 60 epochs and a batch
size of 75 .

$\mathbf{BCE}=-{(y\log(p)+(1-y)\log(1-p))}$ (1)

### Evaluation Metrics

Various standard metrics were employed for assessing the performance of the
proposed model, and to demonstrate its effectiveness. Sensitivity (measure as
per Eq. (2)) evaluates the acceptance ratio of the system i.e., the
probability that the authorised user gains rightful access. Specificity
measures the rejection ratio of the system i.e., probability that an
unauthorised attacker is thwarted from gaining access, given by Eq. (3).
Accuracy is used to measure the overall performance of the model (given by Eq.
(4)) and it demonstrates the efficacy of the authentication mechanism as a
whole.

$\mathbf{Sensitivity}=\frac{TP}{TP+FN}$ (2)
$\mathbf{Specificity}=\frac{TN}{TN+FP}$ (3)
$\mathbf{Accuracy}=\frac{TP+TN}{TP+TN+FP+FP}$ (4)

TP and TN stand for True Positive samples, whereas FP and FN stand for False
Positive and False Negative samples. Two other metrics were also used to
evaluate the other aspects of the proposed model. The Equal Error Rate (EER)
metric is defined as a point where the the probability of an intruder gaining
access is equal to the probability that an authorised user is refused access.
Hence, it is also the point where False Acceptance Ratio (FAR) is equal to the
False Rejection Rate (FRR). FAR is calculated as per Eq. 5 and FRR is
calculated as per Eq. 6. EER is a standard metric used for comparison across
biometric systems. The EER was calculated from the Receiver Operating
Characteristic curve (ROC) as this is less dependent on scaling. The ROC Curve
is plotted and depicted in Fig. 5.

$\mathbf{FAR}=\frac{FP}{TP+TN+FP+FN}$ (5)
$\mathbf{FRR}=\frac{FN}{TP+TN+FP+FN}$ (6)

The ROC curve is a curve of True positive rate (TPR) against False positive
rate (FPR or FAR) as the threshold is varied. The ROC curve acts as a cost-
benefit analysis while making decisions. It is also used to determine the
threshold to ensure most true positives are gained for each false positives
that the pipeline incurs.This ensures that allow as much authorised accesses
are allowed while limiting the amount of unauthorised access that’s allowed by
the system. From the ROC curve intersecting with the line $x+y=1$ , point of
equal error is found out. The Area Under the Receiver Operating Characteristic
curve (AUC) is calculated using the ROC curve. The probability that the model
scores a random positive example higher than a random negative example is
called AUC. AUC is a threshold-independent metric and hence measures the
confidence that the model has in its decision boundary.

### Results and Discussion

To quantify AuthNet’s performance, a cross validation dataset was prepared,
that accounted for the various imposter cases available for the person-word
combination, namely, the same person saying a different word, a different
person saying the same word and a different person saying a different word.
The different speakers chosen for testing were that of new faces completely
unseen by the model during the training phase. The different words spoken by
the same person were also chosen such that they are completely new and hence
the capability of the model to generalize to new person-word combinations is
thoroughly tested. The performance of the proposed model with other metrics
methods are shown in Table 3 and the confusion matrix is shown in Fig. 7.

Figure 5: ROC Curve for the MIRACL-V1 dataset Figure 6: Confusion Matrix of
proposed AuthNet model for the MIRACL-VC1 dataset Figure 7: Performance
analysis of the Proposed Approach

$\cos\\_diff({\bf t},{\bf e})=1-\frac{\sum_{i=1}^{n}{{\bf t}_{i}{\bf
e}_{i}}}{\sqrt{\sum_{i=1}^{n}{({\bf t}_{i})^{2}}}\sqrt{\sum_{i=1}^{n}{({\bf
e}_{i})^{2}}}}$ (7)

Cosine Difference, measured as per Eq. (7), was used to compute the
differences between images. The purpose of measuring cosine difference between
the feature vectors across speakers and words is to show that there is a
significant difference between the feature vectors obtained through the
VGGFace model, among different people and even different words spoken by the
same person as time progresses. This difference in feature vectors between
different people and different words provides an empirical justification as to
how facial features differ when people utter their passwords.Hence, when the
pre-trained VGGFace model extracts a feature vector for each image, the
considerable differences across images can be then be captured by the LSTM
network. This is visualised in Fig. 4), This underscores the fact that the
proposed model is capable of handling imposters uttering the same password,
thus allowing it to be an open password system.

Table 2: Benchmarking the proposed AuthNet model against State-of-the-art models Models | Test Accuracy
---|---
AuthNet (Proposed) | 0.980
FaceNet + Garg et al. |
(amit, jnoyola, and sameepb 2016) | 0.611
FaceNet + Chung & Zisserman |
(Chung et al. 2017) | 0.652
FaceNet + Gergen et al. |
(Gergen et al. 2016) | 0.864
FaceNet + LipNet |
(Wand, Koutnik, and Schmidhuber 2016) | 0.951

In Table 3, the high sensitivity value of 0.998 indicates that the system
incorrectly classifies an input user-password combination 2 times out of 1000
only, to recognize the authorised user-password combination accurately. It is
a good measure of AuthNet’s ability to capture true positives. The high
specificity value demonstrates the model’s ability to correctly classify
imposter attacks. The individual specificity errors for the different imposter
cases are shown in Table 4. The values show that the system denies the attacks
from a different person around 98% of the time, while it correctly identifies
a wrong password being spoken 94% of the time. This indicates the model’s
ability to act as an excellent defense mechanism against attacks by malicious
entities, thereby ensuring a high level of privacy. From the ROC curve in Fig.
5 the point where the TPR is equal to 0.977 and FPR to be 0.023 is calculated,
thus the EER value is 0.023. A low equal error rate signifies the pipeline’s
ability to reduce both false acceptances and false rejections to a minimum.
The AUC was found to be 0.990, which signifies the pipeline’s confidence in
its prediction.

On the collated dataset obtained from a capturing smartphone camera videos, we
find that the performance of the model is on par with the performance on the
MIRACL-VC1 dataset developed in a controlled lab setting. This signifies that
the system generalises well to real world cases, even in the presence of image
variance and varied lighting conditions. So, the system has an inherent
capability to capture patterns unique to the user, and the results are not
just a product of overfitting to the data from a simulated environment with
identical background and lighting conditions.

Table 3: Performance of proposed AuthNet model w.r.t different datasets Metric | MIRACL-VC1 | Collated Dataset
---|---|---
Sensitivity | 0.998 | 0.987
Specificity | 0.969 | 0.976
Accuracy | 0.980 | 0.981
AUC Score | 0.990 | 0.989
Equal Error Rate | 0.023 | 0.037

To demonstrate the effectiveness of the training, AuthNet model is benchmarked
against a two-level system - several state-of-the-art models in face
recognition combined with lip reading. Since FaceNet (Schroff, Kalenichenko,
and Philbin 2015) is the current best-performing face recognition model
reporting an accuracy of 99.76%, it was used for comparison. Garg et al.
(amit, jnoyola, and sameepb 2016) proposed a lip reading model that achieved
an accuracy of 61.2%. The models of Chung & Zisserman (Chung et al. 2017) and
Gergen et al. (Gergen et al. 2016) were the previous state-of-the-art models
in lip reading on word level with their reported accuracy values of 65.4% and
86.6% respectively. Lipnet (Shillingford et al. 2018) has been included to
benchmark the performance of the proposed AuthNet model. The scores and
comparisons are reported in Table 2, which shows that the performance of the
trained model matches the combined performance of state-of-the-art models
while overcoming errors or imposter attacks in these models.

Table 4: Specificity for different types of errors (MIRACL-VC1 dataset) Imposter types/Error cases | Samples | Specificity
---|---|---
Same Person-Different Words | 210 | 0.9420
Different Person-Same Words | 150 | 0.9893
Different Person-Different Words | 100 | 0.9730

## Conclusion and Future Work

In this paper, AuthNet, a temporal facial movement based authentication
mechanism using 2-level trained CNN-RNN deep neural models was presented. When
evaluated on the standard MIRACL-VC1 dataset, AuthNet achieved an accuracy of
98.1%, underscoring its effectiveness. The results obtained from cross-
validation on a collated dataset proved its inherent language and domain
agnostic nature. It also demonstrated the real-world application of such a
system in mobile phones and other resource constrained smart devices,
involving varying backgrounds, different lighting contents, varying video
resolutions, etc.

AuthNet can function as an open password system, as the model effectively
classified the same password being spoken by a different person. Hence,
disclosing the password does not cause any security concerns. It also has no
language barrier, since the model is not dependent on knowledge of the
language in which the password/phrase is being spoken, but only the temporal
facial movements of the speaker extracted from the videos of its utterance.
The proposed method is also data efficient, since the model gave good results
even when trained with only 10 positive video samples. As part of future work,
the proposed model could be optimised in its time complexity, as it takes
approximately 2 seconds for testing per sample currently. Since fast
processing is a critical requirement for deployment as an authentication
system for smartphones and computers, optimising the testing time will be a
primary concern to be addressed, which will be explored.

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# Observations favor the redshift-evolutionary $L_{X}$-$L_{UV}$ relation of
quasars from copula

Bao Wang Department of Physics and Synergetic Innovation Center for Quantum
Effects and Applications, Hunan Normal University, Changsha, Hunan 410081,
China  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, China  School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China  Yang Liu Department of Physics
and Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha, Hunan 410081, China  Hongwei Yu  Department of
Physics and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha, Hunan 410081, China  Institute of
Interdisciplinary Studies, Hunan Normal University, Changsha, Hunan 410081,
China  Puxun Wu  Department of Physics and Synergetic Innovation Center for
Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Institute of Interdisciplinary Studies, Hunan Normal
University, Changsha, Hunan 410081, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compare, with data from the quasars, the Hubble parameter measurements, and
the Pantheon+ type Ia supernova, three different relations between X-ray
luminosity ($L_{X}$) and ultraviolet luminosity ($L_{UV}$) of quasars. These
three relations consist of the standard and two redshift-evolutionary
$L_{X}$-$L_{UV}$ relations which are constructed respectively by considering a
redshift dependent correction to the luminosities of quasars and using the
statistical tool called copula. By employing the PAge approximation for a
cosmological-model-independent description of the cosmic background evolution
and dividing the quasar data into the low-redshift and high-redshift parts, we
find that the constraints on the PAge parameters from the low-redshift and
high-redshift data, which are obtained with the redshift-evolutionary
relations, are consistent with each other, while they are not when the
standard relation is considered. If the data are used to constrain the
coefficients of the relations and the PAge parameters simultaneously, then the
observations support the redshift-evolutionary relations at more than
$3\sigma$. The Akaike and Bayes information criteria indicate that there is
strong evidence against the standard relation and mild evidence against the
redshift-evolutionary relation constructed by considering a redshift dependent
correction to the luminosities of quasars. This suggests that the redshift-
evolutionary $L_{X}$-$L_{UV}$ relation of quasars constructed from copula is
favored by the observations.

##  1 Introduction

Quasars are extremely luminous and persistent energy sources powered by
supermassive black holes. The luminosities of quasars are so great that the
maximum redshift of quasars can reach $z>7$ (Mortlock et al., 2011; Bañados et
al., 2018; Lyke et al., 2020; Wang et al., 2021). If quasars can be regarded
as standard candles, they will cover the redshift desert of cosmological data,
and may play an important role in understanding the property of dark energy
and the possible origin of the Hubble constant ($H_{0}$) tension. To use data
from quasars for cosmological purposes, one needs to construct a luminosity
relation to determine the distance of quasars. Several empirical relations
have been proposed (Baldwin, 1977; Paragi et al., 1999; Chen & Ratra, 2003;
Watson et al., 2011; La Franca et al., 2014; Wang et al., 2014; Cao et al.,
2021). Among them, the nonlinear relation between the X-ray luminosity
($L_{X}$) and the ultraviolet (UV) luminosity ($L_{UV}$) (Risaliti & Lusso,
2015, 2019; Lusso & Risaliti, 2016; Lusso et al., 2020) is a very popular one.
The $L_{X}$-$L_{UV}$ relation has been utilized to construct the Hubble
diagram of the quasars up to $z\sim 7.5$, and has been applied in the quasar
cosmology widely (Khadka & Ratra, 2020a, b; Wei & Melia, 2020; Khadka & Ratra,
2021; Li et al., 2021; Lian et al., 2021; Bargiacchi et al., 2022).

With the $L_{X}$-$L_{UV}$ relation and the logarithmic polynomial expansion of
the cosmic distance, it has been found that the distance modulus/redshift
relation of the quasars at $z>1.4$ has a more than 4$\sigma$ deviation from
the prediction of the cosmological constant plus the cold dark matter
($\Lambda$CDM) model with $\Omega_{m0}=0.3$ (Risaliti & Lusso, 2019; Lusso et
al., 2019) 111It should be pointed out that controversies have arisen
regarding this deviation (Yang et al., 2020; Banerjee et al., 2021; Bargiacchi
et al., 2021; Khadka & Ratra, 2022; Sacchi et al., 2022; Khadka et al., 2022;
Dainotti et al., 2022; Lenart et al., 2023; Wang et al., 2022; Mehrabi &
Basilakos, 2020; Velten & Gomes, 2020; Petrosian et al., 2022). Some
researches indicated that this deviation may originate from the divergence of
the logarithmic polynomial expansion at the high-redshift regions (Yang et
al., 2020; Banerjee et al., 2021)., where $\Omega_{m0}$ is the present
dimensionless matter density parameter. This deviation may indicate that the
standard $L_{X}$-$L_{UV}$ relation of quasars may not be accurate. Recently,
Khadka & Ratra (2022) found that part of the quasar data shows evidence of
redshift evolution of the $L_{X}$-$L_{UV}$ relation, suggesting a possible
redshift-evolutionary $L_{X}$-$L_{UV}$ relation. Dainotti et al. (2022) first
obtained a three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$
relation by considering a redshift dependent correction to the luminosities of
quasars. We also constructed a three-dimensional and redshift-evolutionary
$L_{X}$-$L_{UV}$ relation by using the powerful statistical tool called copula
(Wang et al., 2022). For the standard and two redshift-evolutionary
$L_{X}$-$L_{UV}$ relations, it remains interesting to determine which one is
favored by observations.

To compare three different $L_{X}$-$L_{UV}$ relations with observations, the
theoretical value of the luminosity distance $d_{L}$ must be given. One way to
achieve this goal is to use a cosmological model to derive the luminosity
distance. The obtained results will then be model-dependent. The other one is
to resort to the so-called cosmography, which is cosmological-model-
independent. However, the usual cosmographic method, which is based on the
Taylor expansion or the Padé approximation of the Hubble expansion rate,
suffers the divergence problem in the high-redshift regions. Recently, Huang
(2020) proposed the PAge approximation and found that it can describe the
global expansion history of our universe with high accuracy. The PAge
approximation can be considered as a cosmological-model-independent method for
the high-redshift cosmography (Huang, 2020; Huang et al., 2021; Cai et al.,
2022a) and has been applied in cosmology widely. For example, the PAge
approximation has been used to explore the supernova magnitude evolution
(Huang, 2020), test the high redshift gamma-ray burst luminosity correlations
(Huang et al., 2021), reaffirm the cosmic expansion history (Luo et al.,
2020), and investigate the $S_{8}$ tension (Huang et al., 2022) and the
$H_{0}$ tension (Cai et al., 2022b, c). More recently, by dividing the quasar
data into the low-redshift and high-redshift parts and using them to constrain
the coefficients of the $L_{X}$-$L_{UV}$ relation and the PAge parameters, Li
et al. (2022) found that there are apparent inconsistencies between the
results from the low-redshift data and the high-redshift ones respectively,
and concluded that the deviation between the distance modulus/redshift
relation of the high-redshift quasars and the prediction from the $\Lambda$CDM
model found in Ref. (Risaliti & Lusso, 2019) probably originates from the
redshift-evolutionary effects of the $L_{X}$-$L_{UV}$ relation.

In this paper, we check whether the $L_{X}$-$L_{UV}$ relation is redshift-
evolutionary and compare three different $L_{X}$-$L_{UV}$ relations (Dainotti
et al., 2022; Wang et al., 2022; Risaliti & Lusso, 2015) by using the PAge
approximation with the observational data comprising of the 2421 quasar sample
(Lusso et al., 2020), the Hubble parameter ($H(z)$) measurements (Moresco et
al., 2020) and the Pantheon+ type Ia supernova (SNe Ia) data (Brout et al.,
2022). We find that the three-dimensional and redshift-evolutionary
$L_{X}$-$L_{UV}$ relation from copula (Wang et al., 2022) is favored by the
observational data.

The rest of the paper is organized as follows. In section 2, we introduce the
PAge approximation, the three different $L_{X}$-$L_{UV}$ relations and the
data sets. The redshift-evolutionary effects of $L_{X}$-$L_{UV}$ relations are
examined in section 3. A comparison of the three different relations is given
in section 4 and the selection effects are discussed in section 5. We
summarize our conclusions in section 6.

##  2 Models and Data

###  2.1 PAge Approximation

The PAge approximation (Huang, 2020) is a cosmographic method proposed to
describe the cosmic expansion history. The detailed process to construct the
PAge approximation is given in the Appendix (A). Compared with the Taylor
expansion and the Padé approximation, the PAge approximation is accurate
enough to perform well in the high-redshift cosmography (Huang et al., 2021;
Cai et al., 2022a). In the PAge approximation, the Hubble expansion rate $H$
is taken as a function of the cosmic time $t$:

$\frac{H(t)}{H_{0}}=1+\frac{2}{3}\left(1-\eta\frac{H_{0}t}{p_{\rm
age}}\right)\left(\frac{1}{H_{0}t}-\frac{1}{p_{\rm age}}\right),$ (1)

where parameters $p_{\rm age}$ and $\eta$ are defined, respectively, as

$p_{\rm age}=H_{0}t_{0},$ (2)

and

$\eta=1-\frac{3}{2}p_{\rm age}^{2}(1+q_{0}).$ (3)

Here $t_{0}$ is the cosmic age and $q_{0}$ is the present deceleration
parameter. According to the definition of $H(t)$:
$H(t)=\frac{1}{a(t)}\frac{{\rm d}a(t)}{{\rm d}t}$ with $a(t)$ being the cosmic
scale factor, Eq. (1) can be rewritten as a differential equation. Solving
this differential equation, one can obtain the relation between $a(t)$ and
$t$:

$a(t)=\left(\frac{t}{t_{0}}\right)^{\frac{2}{3}}\exp\left[\frac{1}{3t_{0}^{2}}(t-t_{0})\left(\eta
t+3H_{0}t_{0}^{2}-\eta t_{0}-2t_{0}\right)\right].$ (4)

Apparently, the expansion of the universe can be described by three
parameters: $t_{0}$, $H_{0}$ and $q_{0}$. Thus, the PAge approximation
provides a cosmological-model-independent method to describe the cosmic
background evolution. With the PAge approximation, the luminosity distance
$d_{L}$ has the form

$d_{L}(z)=(1+z)\int^{t_{0}}_{t_{z}}{\frac{{\rm d}t}{a(t)}},$ (5)

where $t_{z}$ is the time of light emission, and $z=\frac{1}{a(t)}-1$ is the
redshift. The relation between $t_{z}$ and $z$ can be inferred from Eq. (4).

###  2.2 X-ray and UV Luminosity Relations

Risaliti & Lusso (2015) discovered that there is a nonlinear relation between
the X-ray luminosity and the UV luminosity of quasars, which takes the form

$\displaystyle\log(L_{X})=\beta+\gamma\log(L_{UV}),$ (6)

where $\beta$ and $\gamma$ are two constants.

Recently, by using the Gaussian copula, we have constructed a three-
dimensional and redshift-evolutionary X-ray and UV luminosity relation (Wang
et al., 2022)

$\log(L_{X})=\beta+\gamma\log(L_{UV})+\alpha\ln(\bar{a}+z)$ (7)

with $\bar{a}=5$, where $\alpha$ is a new parameter that characterizes the
redshift-evolution of the relation and $\ln\equiv\log_{e}$. In deriving the
above three-dimensional luminosity relation, we have assumed that
$\log(L_{UV})$ and $\log(L_{X})$ satisfy, respectively, the following Gaussian
distributions

$\displaystyle
f(x)=\frac{1}{\sqrt{2\pi}\sigma_{x}}e^{-\frac{(x-\bar{a}_{x})^{2}}{2\sigma_{x}^{2}}},\quad
f(y)=\frac{1}{\sqrt{2\pi}\sigma_{y}}e^{-\frac{(y-\bar{a}_{y})^{2}}{2\sigma_{y}^{2}}},$
(8)

where $x=\log(L_{UV})$, $y=\log(L_{X})$, $\bar{a}_{x}$ and $\bar{a}_{y}$
represent the mean value, and $\sigma_{x}$ and $\sigma_{y}$ are the standard
deviations. Furthermore, the probability density distribution of the quasars
needs to be known to construct the three-dimensional luminosity relation.
Lusso et al. (2020) have pointed out that the 2421 quasar data points satisfy
the Gamma distribution in the $z$ space. Since the Gaussian distribution will
give a simple expression of the X-ray and UV luminosity relation and the log-
transformation is a common way to transform the non-Gaussian distribution into
the Gaussian one, we consider the $z_{*}$ space, where
$z_{*}\equiv\ln(\bar{a}+z)$ with $\bar{a}$ being a constant, and find that

$\displaystyle
f(z_{*})=\frac{1}{\sqrt{2\pi}\sigma_{z_{*}}}e^{-\frac{(z_{*}-\bar{a})^{2}}{2\sigma_{z_{*}}^{2}}}$
(9)

can describe approximately the redshift distribution of the quasars. Utilizing
the Gaussian copula and Eqs. (8, 9), Wang et al. (2022) constructed the
relation shown in Eq. (7).

In Eq. (7) , the standard relation is recovered when $\alpha=0$. If
$\bar{a}=1$, the relation given in Eq. (7) reduces to the one proposed in
(Dainotti et al., 2022) by assuming that the luminosities of quasars are
corrected by a redshift-dependent function $(1+z)^{\alpha}$. In (Dainotti et
al., 2022), the value of $\alpha$ is determined by using the EP method (Efron
& Petrosian, 1992), while in our analysis, $\alpha$ is treated as a free
parameter. To avoid confusion, we name the relations with $\bar{a}=5$ and
$\bar{a}=1$ as Type I and Type II, respectively. Converting luminosity to
flux, we can define a function $\Phi$ via

$\displaystyle\log(F_{X})$ $\displaystyle=$
$\displaystyle\Phi(\log(F_{UV}),d_{L})$ (10) $\displaystyle=$
$\displaystyle\beta+\gamma\log(F_{UV})+\alpha\ln(\bar{a}+z)+(\gamma-1)\log(4\pi
d_{L}^{2}).$

Here $F_{X}=\frac{L_{X}}{4\pi d_{L}^{2}}$ and $F_{UV}=\frac{L_{UV}}{4\pi
d_{L}^{2}}$ are the fluxes of the X-ray and UV respectively.

###  2.3 Data Sets

To compare three different $L_{X}$-$L_{UV}$ relations, we will use the latest
quasar data. Furthermore, we also consider the Hubble parameter measurements
and Pantheon+ SNe Ia sample in order to constrain the PAge parameters and the
coefficients of the relations tightly and break the degeneracy between
different parameters. The data sets are as follows.

  * •

Quasars

The data of quasars in our analysis comprise of 2421 X-ray and UV flux
measurements (Lusso et al., 2020), which cover a redshift range of
$z\in[0.009,7.541]$. The relation coefficients and the PAge parameters can be
obtained by maximizing $-\ln\mathcal{L}$, where $\mathcal{L}$ is the
D’Agostinis likelihood function (D’Agostini, 2005)

$\mathcal{L}_{Q}(\delta,\beta,\gamma,\alpha,{\bf
p})\propto\prod_{i}\frac{1}{\sqrt{2\pi(\delta^{2}+\sigma_{{\rm
tot},i}^{2})}}\exp\left\\{-\frac{[\log(F_{X})_{i}-\Phi(\log(F_{UV}),d_{L}({\bf
p}))_{i}]^{2}}{2\left(\delta^{2}+\sigma_{{\rm tot},i}^{2}\right)}\right\\}.$
(11)

Here $\sigma_{{\rm tot},i}=\sigma_{X,i}^{2}+\gamma^{2}\sigma_{UV,i}^{2}$
represents the total measurement error in $\log(F_{X})$ and $\log(F_{UV})$,
$\delta$ is the intrinsic dispersion, function $\Phi$ is defined in Eq. (10),
$d_{L}$ is the luminosity distance predicted by the PAge approximation in Eq.
(5) and $\bf p$ represent the PAge parameters.

  * •

Hubble parameter measurements

The updated 32 $H(z)$ measurements have a redshift range of $z\in[0.07,1.965]$
(Moresco et al., 2020; Liu et al., 2023), which contain 17 uncorrelated and 15
correlated measurements. The 15 correlated measurements are from Refs.
(Moresco et al., 2012; Moresco, 2015; Moresco et al., 2016) and their
covariance matrices are given in (Moresco et al., 2020)222All $H(z)$
measurements and their errors can be found at
https://gitlab.com/mmoresco/CCcovariance/.. The $\chi^{2}$ of the $H(z)$ data
is

$\chi^{2}_{H}=\sum_{i=1}^{17}{\frac{[H_{{\rm th},i}(t)-H_{{\rm
obs},i}(t)]^{2}}{\sigma_{H,i}^{2}}}+\Delta\hat{H}^{T}{\bf
C^{-1}_{H}}\Delta\hat{H},$ (12)

where $H_{\rm obs}(t)$ is the observed value of the Hubble parameter, $H_{\rm
th}(t)$ represents the theoretical value from the PAge approximation, which is
given in Eq. (1), $\sigma_{H}$ is the observed uncertainty,
$\Delta\hat{H}=H_{\rm th}(t)-H_{\rm obs}(t)$ is the data vector of the 15
correlated measurements, and $\bf C^{-1}_{H}$ is the inverse of the covariance
matrix.

  * •

SNe Ia

The updated Pantheon+ SNe Ia sample contains 1701 data points (Brout et al.,
2022). The nearby ($z<0.01$) supernova data will not be used in our work since
they are sensitive to the peculiar speed (Brout et al., 2022). Thus, our
analysis only includes 1590 data points in the redshift range of
$z\in[0.01,2.2613]$. The distance modulus $\mu_{\rm SN}$ of SNe Ia is defined
as

$\mu_{\rm SN}=m-M_{B},$ (13)

where $m$ is the apparent magnitude and $M_{B}$ is the absolute magnitude. The
theoretical value of the distance modulus relates to the luminosity distance
$d_{L}$ through

$\mu_{\rm th}=5\log\left(\frac{d_{L}}{\mathrm{Mpc}}\right)+25.$ (14)

Then the $\chi^{2}_{\rm SN}$ of SNe Ia has the form

$\chi^{2}_{\rm SN}=\Delta\mu^{T}{\bf C^{~{}~{}-1}_{SN}}\Delta\mu,$ (15)

where $\Delta\mu=\mu_{{\rm SN}}-\mu_{{\rm th}}$ and $\bf C^{~{}~{}-1}_{SN}$ is
the inverse of the covariance matrix which contains the systematic and
statistical uncertainties.

The constraints on the relation coefficients, the PAge parameters, the
intrinsic dispersion and the absolute magnitude from all observational data
can be obtained by maximizing
$\mathcal{L}=e^{-\frac{1}{2}(\chi^{2}_{H}+\chi^{2}_{\rm SN})}\mathcal{L}_{Q}$.

##  3 Tests the redshift-evolutionary effects of relations

After dividing all the quasar data into the high-redshift part and the low-
redshift one, Li et al. (2022) used them to constrain the standard
$L_{X}$-$L_{UV}$ relation with the PAge approximation and found that there
exists tension between the results from the high-redshift and low-redshift
data respectively. They concluded that this tension may originate from the
redshift-evolutionary effects of the $L_{X}$-$L_{UV}$ relation. In this
section, we discuss the constraints on the two redshift-evolutionary
$L_{X}$-$L_{UV}$ relations with the PAge approximation using data from the
low-redshift ($z<1.4$) and high-redshift ($z>1.4$) quasars respectively. For
comparison, the standard $L_{X}$-$L_{UV}$ relation is also considered in our
analysis. Since the Hubble constant $H_{0}$ is degenerate with parameter
$\beta$ in the relations and the quasars have large dispersions, we also
include the $H(z)$ measurements and the SNe Ia data.

Table 1: Marginalized constraints on the PAge parameters and the coefficients
of the relations with 1$\sigma$ confidence level (CL) from the $H(z)$+SNe
Ia+quasar data in the PAge approximation.

| Type I relation |  | Type II relation |  | Standard relation
---|---|---|---|---|---
| Low-redshift | High-redshift |  | Low-redshift | High-redshift |  | Low-redshift | High-redshift
$t_{0}$ | $13.84^{+0.50}_{-0.62}$ | $13.83^{+0.49}_{-0.55}$ |  | $13.89^{+0.51}_{-0.65}$ | $13.91^{+0.47}_{-0.57}$ |  | $13.76\pm 0.52$ | $13.49^{+0.44}_{-0.54}$
$H_{0}$ | $69.2\pm 2.8$ | $69.1\pm 2.8$ |  | $69.1\pm 2.8$ | $69.0\pm 2.4$ |  | $68.2\pm 2.6$ | $67.1^{+2.9}_{-2.5}$
$q_{0}$ | $-0.407\pm 0.059$ | $-0.408^{+0.053}_{-0.060}$ |  | $-0.404\pm 0.056$ | $-0.398\pm 0.059$ |  | $-0.419\pm 0.059$ | $-0.488\pm 0.053$
$M_{B}$ | $-19.357\pm 0.086$ | $-19.362\pm 0.0.89$ |  | $-19.361\pm 0.087$ | $-19.364\pm 0.074$ |  | $-19.389\pm 0.084$ | $-19.432^{+0.098}_{-0.078}$
$\delta$ | $0.2357\pm 0.0048$ | $0.2065\pm 0.0049$ |  | $0.2357\pm 0.0048$ | $0.2058\pm 0.0054$ |  | $0.2368\pm 0.0048$ | $0.2124^{+0.0050}_{-0.0056}$
$\beta$ | $7.33^{+0.37}_{-0.41}$ | $8.42\pm 0.41$ |  | $8.06\pm 0.45$ | $9.59\pm 0.41$ |  | $7.27\pm 0.39$ | $8.24\pm 0.40$
$\gamma$ | $0.606\pm 0.016$ | $0.551\pm 0.015$ |  | $0.604\pm 0.016$ | $0.548\pm 0.014$ |  | $0.632^{+0.014}_{-0.012}$ | $0.604\pm 0.013$
$\alpha$ | $0.420\pm 0.150$ | $0.739^{+0.096}_{-0.087}$ |  | $0.131\pm 0.047$ | $0.352\pm 0.045$ |  | $-$ | $-$

Note. — The unit of $t_{0}$ is Gyr and $H_{0}$ is ${\rm
km\;s^{-1}\;Mpc^{-1}}$.

Figure 1: One-dimensional marginalized distributions and two-dimensional
contours at 1$\sigma$ and 2$\sigma$ CLs of the Type I and Type II relations
from low-redshift and high-redshift quasars.

The best-fitting values with uncertainties are obtained by using the Markov
Chain Monte Carlo (MCMC) method, which is available via the $emcee$ package in
Python 3.7 (Foreman-Mackey et al., 2013). Table 1 and Figure 1 show the
marginalized results and the probability contour plots, respectively. From
them, we obtain the following main results:

  * •

For the standard relation, the constraints from the low-redshift and high-
redshift data on all the parameters are inconsistent. This agrees with the
conclusion obtained in (Li et al., 2022).

  * •

For the redshift-evolutionary relations, the cosmographic parameters ($t_{0}$,
$H_{0}$ and $q_{0}$) and the absolute magnitude ($M_{B}$) are consistent in
two different redshift ranges. However, the values of the relation
coefficients ($\beta$, $\gamma$ and $\alpha$) and the intrinsic dispersion
($\delta$) from the low-redshift and high-redshift data respectively are
discrepant. This is similar to what obtained in the standard relation case.

  * •

The redshift-evolutionary coefficient $\alpha$ deviates from zero at more than
$2\sigma$, which implies that the observational data support a redshift-
dependent relation. Furthermore, the redshift-evolutionary character is more
apparent in the Type I relation than that in the Type II case.

Therefore, the redshift-evolutionary $L_{X}$-$L_{UV}$ relations are favored as
opposed to the standard one since consistent PAge parameters can be obtained
in the redshift-evolutionary relations and the value of $\alpha$ deviates from
zero at more than $2\sigma$.

##  4 Simultaneous Constraints

To further compare the three different $L_{X}$-$L_{UV}$ relations, we will use
all $H(z)$, SNe Ia and quasar data to constrain the PAge parameters ($t_{0}$,
$H_{0}$ and $q_{0}$), the absolute magnitude ($M_{B}$), the coefficients of
the relations ($\beta$, $\gamma$ and $\alpha$) and the intrinsic dispersion
($\delta$) simultaneously. We will utilize the Akaike information criterion
(AIC) (Akaike, 1974, 1981) and the Bayes information criterion (BIC) (Schwarz,
1978) to find the relation favored by the observations. The AIC and BIC are
defined, respectively, as

$\displaystyle\mathrm{AIC}$ $\displaystyle=$ $\displaystyle
2p-2\ln\mathcal{L}_{\rm max}\,,$ (16) $\displaystyle\mathrm{BIC}$
$\displaystyle=$ $\displaystyle p\ln N-2\ln\mathcal{L}_{\rm max}\,,$ (17)

where $\mathcal{L}_{\rm max}$ is the maximum value of the likelihood function,
$p$ represents the number of free parameters and $N$ is the number of data. We
calculate $\Delta$AIC(BIC) of the three relations, which denotes the
difference in the AIC(BIC) of a given relation relative to the reference
relation (the Type I relation ), i.e., $\rm\Delta
AIC(BIC)=AIC(BIC)-AIC_{ref}(BIC_{ref})$, for the relation comparison. If
$0<\Delta\mathrm{AIC(BIC)}\leq 2$, it is difficult to single out a better
relation. If $2<\Delta\mathrm{AIC(BIC)}\leq 6$ we have mild evidence against
the given relation, and we have strong evidence against the given relation if
$\Delta\mathrm{AIC(BIC)}>6$. The parameter values and $\Delta$AIC(BIC)
obtained from MCMC are summarized in Table 2, and the posterior distribution
contours are shown in Figure 2. In the last column of Table 2, the constraints
on the PAge parameters and the absolute magnitude obtained from data without
the quasars are also displayed in order to assess the influence of the
quasars.

Figure 2: The marginalized posterior distributions and two-dimensional
contours at 1$\sigma$ and 2$\sigma$ CLs from $H(z)$+SNe Ia+quasar data in the
PAge approximation.  Table 2: The marginalized results with 1$\sigma$ CL for
three different luminosity relations from $H(z)$+SNe Ia+quasar data in the
PAge approximation. The results from data without the quasars are also
displayed for a comparison.

| Type I Three-Dimensional | Type II Three-Dimensional | Standard | Without quasars
---|---|---|---|---
$t_{0}$ | $13.72\pm 0.50$ | $13.65^{+0.44}_{-0.56}$ | $13.30\pm 0.43$ | $13.67^{+0.44}_{-0.53}$
$H_{0}$ | $69.1\pm 2.8$ | $68.6\pm 2.5$ | $65.0\pm 2.5$ | $69.4\pm 2.7$
$q_{0}$ | $-0.471\pm 0.059$ | $-0.496\pm 0.055$ | $-0.606\pm 0.054$ | $-0.471\pm 0.057$
$M_{B}$ | $-19.377^{+0.092}_{-0.079}$ | $-19.395\pm 0.078$ | $-19.521\pm 0.084$ | $-19.369^{+0.091}_{-0.077}$
$\delta$ | $0.2248\pm 0.0033$ | $0.2245^{+0.0032}_{-0.0037}$ | $0.2289\pm 0.0035$ | $-$
$\beta$ | $7.68\pm 0.25$ | $8.80\pm 0.32$ | $6.62\pm 0.24$ | $-$
$\gamma$ | $0.579\pm 0.010$ | $0.577\pm 0.011$ | $0.655\pm 0.008$ | $-$
$\alpha$ | $0.672^{+0.063}_{-0.073}$ | $0.252\pm 0.025$ | $-$ | $-$
$-2\ln\mathcal{L}_{\rm max}$ | $1246.550$ | $1250.415$ | $1350.201$ | $-$
$\Delta$AIC | $-$ | $3.865$ | $101.651$ | $-$
$\Delta$BIC | $-$ | $3.865$ | $95.346$ |  $-$

Note. — The unit of $t_{0}$ is Gyr and $H_{0}$ is ${\rm
km\;s^{-1}\;Mpc^{-1}}$.

Figure 3: Probability density of Hubble constant $H_{0}$ for different
luminosity relations. The gray shades represent the results from the Planck
2018 CMB observations ($H_{0}=67.4\pm 0.5\;{\rm km\;s^{-1}\;Mpc^{-1}}$ (Planck
Collaboration et al., 2020)) and the SH0ES ($H_{0}=73.04\pm 1.04\;{\rm
km\;s^{-1}\;Mpc^{-1}}$) (Riess et al., 2022) at 68% CL (dark gray) and 95% CL
(light gray).

One can see that the constraints on the PAge parameters ($H_{0}$, $t_{0}$ and
$q_{0}$) and the absolute magnitude $M_{B}$ from $H(z)$+SNe Ia+quasars with
the Type I relation are well consistent with those from data without the
quasars. The values of the PAge parameters and the absolute magnitude are
compatible with each other for the Type I and Type II relations, while they
apparently deviate for the standard relation. The values of $t_{0}$ for the
Type I and Type II relations are more comparable with the result from the
Planck 2018 CMB observations ($t_{0}=13.801\pm 0.024$ Gyr) (Planck
Collaboration et al., 2020) than those for the standard relation. The
deceleration parameter $q_{0}$ for the Type I and Type II relations is
slightly larger than that in the $\Lambda$CDM model
($q_{0}=3\Omega_{m0}/2-1=-0.528\pm 0.011$) from the Planck 2018 CMB
observations (Planck Collaboration et al., 2020), while its value for the
standard relation is smaller than $q_{0}=-0.528\pm 0.011$. Furthermore, we
find that the value of the Hubble constant $H_{0}$ for the Type I and Type II
relations is lager than that for the standard relation, and locates in the
region between the Planck 2018 CMB observations (Planck Collaboration et al.,
2020) and the local distance ladders from SH0ES (Riess et al., 2022), which
can be seen in Figure 3, where the values of $H_{0}$ from the
quasars+$H(z)$+SNe Ia, the Planck 2018, and the SH0ES are shown.

The intrinsic dispersion $\delta$ and the relation coefficient $\gamma$ have
almost the same values in the Type I and Type II relations, which are larger
than those obtained in the standard relation. The value of $\gamma$ in the two
three-dimensional relations is consistent with $\gamma=0.586\pm 0.061$
obtained through the narrow redshift bin method (Lusso et al., 2020). For the
relation coefficient $\beta$, the Type II relation has the maximum value and
the standard relation has the minimum one. The deviations of $\beta$ among
three different relations are of more than $2\sigma$. The observations favor
strongly the redshift-evolutionary relations since $\alpha=0$ is ruled out at
more than $3\sigma$. The value of $\alpha$ in the Type I relation is
apparently larger than that in the Type II relation, which indicates that the
redshift-evolutionary character in the Type I relation is more apparent than
that in the Type II relation. The $\alpha$ value in the Type I relation is
comparable with $\alpha=0.580^{+0.084}_{-0.099}$ obtained in the $\Lambda$CDM
model (Wang et al., 2022). From Table 2, we find that the Type I three-
dimensional relation has the minimum AIC(BIC). The $\Delta$AIC(BIC) of the
standard relation is much larger than 6, which indicates that we have strong
evidence against it. The $\Delta$AIC(BIC) of the Type II three-dimensional
relation is 3.865. Thus, the Type I three-dimensional relation has a slight
advantage.

##  5 Selection effects

To check whether our result that the redshift-evolutionary relation is favored
by the observations is caused by the selection effects involved in the
measurement process, we follow the method given in (Li, 2007; Singh et al.,
2022) and use firstly Monte Carlo simulations to simulate a realistic
population of quasars as a function of redshift. Since the 2421 quasar data
points satisfy the Gamma distribution in the $z$ space (Lusso et al., 2020),
we use this Gamma distribution to generate the redshift distribution of the
simulated data. At redshift $z$, the value of $\log(F_{UV})$ is assumed to
satisfy the Gaussian distribution shown in Eq. (8) with the mean value and the
standard deviation being $-27.503$ and $0.488$, respectively, which are
obtained from the real data. We further assume that the error
($\sigma_{{UV}}$) of $\log(F_{UV})$ satisfies the lognormal distribution and
find, by using the real data, the mean value and the standard deviation being
$-4.342$ and $0.566$ respectively. Since the real data show a correlation
between $z$ and $\log(F_{UV})$ with the correlation coefficient being
$\rho=-0.255$, we also consider this correlation when generating the
($z,\log(F_{UV})$) pair. Then, we remove the simulated data below the UV flux
limit, i.e., the minimum observable flux $\log(F_{UV})_{min}=-28.759$. The
flux limit reflects the detection capability of the instrument, and quasars
are unobservable by current detectors if their $\log(F_{UV})$ are less than
$\log({F_{UV}})_{min}$. For any pair of ($z,\log(F_{UV})$), we generate
$\log(F_{X})$ by assuming its distribution function to be
$f(\log(F_{X}))=\frac{1}{\sqrt{2\pi}\delta}\exp\left\\{-\frac{[\log(F_{X})-\Phi(\log(F_{UV}),d_{L}({\bf
p}))]^{2}}{2\delta^{2}}\right\\}$ and the luminosity relation to be the
standard $L_{X}$-$L_{UV}$ relation, where the intrinsic dispersion $\delta$,
the relation coefficients and the PAge parameters ${\bf p}$ are taken as fixed
values shown in the third column of Table 2.

With the above approach, we generate 2500 quasars. Using these simulated data,
we can fit the intrinsic dispersion and the Type I relation coefficients with
the fixed PAge parameters. This process is repeated twenty times, resulting in
the average values of the intrinsic dispersion and the relation coefficients
to be: $\bar{\delta}=0.2254\pm 0.0032$, $\bar{\beta}=6.46\pm 0.21$,
$\bar{\gamma}=0.657\pm 0.009$, and $\bar{\alpha}=0.050\pm 0.054$. Apparently,
they have slightly smaller errors than those shown in the first column of
Table 2. This is because the errors of the PAge parameters are ignored when
fitting the intrinsic dispersion and the relation coefficients from the
simulated data. In Figure 4, the twenty values of the relation coefficient
$\alpha$ from the simulated data are displayed to compare with $\alpha=0$ used
in the simulation. It is easy to see that $\alpha=0$ is included in the
1$\sigma$ CL of the average value of twenty $\alpha$. Comparing
$\bar{\alpha}=0.050\pm 0.054$ with $\alpha=0.672^{+0.063}_{-0.073}$ shown in
Table 2, we find that the selection effects are ruled out as the possible
cause of redshift-evolution of the $L_{X}$-$L_{UV}$ relation.

Figure 4:  The twenty results of relation coefficient $\alpha$ from the
simulated data. The orange base line shows the $\alpha=0$ line. The blue
dotted line represents the best fitting value of the average value
$\bar{\alpha}$ and the blue shaded area corresponds to the $1\sigma$ CL of
$\bar{\alpha}$.

##  6 Conclusions

Recently, a three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$
relation of quasars was constructed from the Gaussian copula (Wang et al.,
2022). In this paper, we compare, by using the observational data from
quasars, the Hubble parameter measurements and the Pantheon+ SNe Ia sample,
this redshift-evolutionary $L_{X}$-$L_{UV}$ relation with another redshift-
evolutionary $L_{X}$-$L_{UV}$ relation constructed by considering a redshift
dependent correction to the luminosities of quasars and the standard
$L_{X}$-$L_{UV}$ relation. We do not assume any cosmological model but use the
PAge approximation to describe the cosmic background evolution. The quasar
data are divided into the low-redshift part and the high-redshift one. The
low-redshift and high-redshift data are then utilized respectively to
constrain the relation coefficients, the SNe Ia absolute magnitude, the
intrinsic dispersion of quasars and the PAge parameters. For the two redshift-
evolutionary relations, we find that the low-redshift and high-redshift data
can give consistent results on the PAge parameters and the absolute magnitude
of the SNe Ia. While, the values of the relation coefficients respectively
from the low-redshift and high-redshift data are discrepant. If the standard
relation is considered, all the results from the low-redshift and high-
redshift data are incomparable. Furthermore, we find that the observations
favor the redshift-evolutionary relations at more than $2\sigma$. Thus, the
redshift-evolutionary relations of quasars are favored by the observations as
opposed to the standard one.

To further compare three different relations, we use the quasar, $H(z)$ and
SNe Ia data to simultaneously constrain the PAge parameters, the intrinsic
dispersion, the relation coefficients and the absolute magnitude of the SNe
Ia. We find that the constraints obtained with the two redshift-evolutionary
relations on the PAge parameters, the absolute magnitude, the intrinsic
dispersion and the relation coefficient $\gamma$ are well consistent with each
other, but they deviate significantly from those obtained in the standard
relation. For the relation coefficient $\beta$, different relations lead to
different values and the deviations between them are more than $2\sigma$. The
observations favor strongly the redshift-evolutionary relations since
$\alpha=0$ is ruled out more than $3\sigma$. According to the AIC and BIC, we
find that there is strong evidence against the standard relation and mild
evidence against the redshift-evolutionary relation obtained by considering a
correction to luminosities of quasars. We also confirm that the redshift-
evolution of the $L_{X}$-$L_{UV}$ relation can not be caused by the selection
effects. Therefore, we can conclude that among the three different
$L_{X}$-$L_{UV}$ relations the redshift-evolutionary relation from copula is
the most favored one.

We appreciate very much the insightful comments and helpful suggestions by the
anonymous referee. Bao Wang thanks Zhuoyang Li for the helpful discussions
about the PAge approximation. This work was supported in part by the NSFC
under Grants No. 12275080 and No. 12075084.

##  Appendix A PAge Approximation

When constructing the PAge approximation, two underlying assumptions are used
(Huang, 2020):

  * •

The product of the Hubble expansion rate $H(t)$ and the cosmic time $t$ can be
approximated as a quadratic function of $t$.

  * •

The universe is matter-dominated at redshift $z\gg 1$ with the radiation-
dominant epoch, which occurs shortly before the matter domination, being
disregarded.

Thus, expanding $H(t)t$ at the cosmic age $t_{0}$ to the second order, we can
obtain

$\displaystyle H(t)t$ $\displaystyle=$ $\displaystyle p_{\rm age}-(1-p_{\rm
age}-p_{\rm age}q_{0})(t_{0}-t)H_{0}$ (A1) $\displaystyle-$
$\displaystyle\left(1+q_{0}-\frac{p_{\rm
age}}{2}(2+3q_{0}+j_{0})\right)(t_{0}-t)^{2}H_{0}^{2}+\mathcal{O}\left((t_{0}-t)^{3}\right),$

where $p_{\rm age}\equiv H_{0}t_{0}$, $q_{0}$ is the present deceleration
parameter, $j_{0}$ is the present jerk parameter, and

$a(t)=1-H_{0}(t_{0}-t)-\frac{1}{2}H_{0}^{2}q_{0}(t_{0}-t)^{2}-\frac{1}{6}H_{0}^{3}j_{0}(t_{0}-t)^{3}+\mathcal{O}\left((t_{0}-t)^{4}\right)$
(A2)

has been used.

In order to achieve a highly precise approximation at high redshift, we now
consider the second assumption. When the redshift becomes very high ($z\gg 1$)
as the time tends to zero ($t\to 0$), the universe is dominated by matter.
Thus, the Hubble parameter can be expressed as $H(t)=\frac{2}{3t}$ at $t\to
0$, which means $\lim\limits_{t\to 0}H(t)t=\frac{2}{3}$. To ensure that Eq.
(A1) satisfies the condition $\lim\limits_{t\to 0}H(t)t=\frac{2}{3}$, we
modify the coefficient of the $(t_{0}-t)^{2}$ term and then obtain

$\displaystyle H(t)t$ $\displaystyle=$ $\displaystyle p_{age}-(1-p_{\rm
age}-p_{\rm age}q_{0})(t_{0}-t)H_{0}$ (A3)
$\displaystyle-\left(1+q_{0}-\frac{2}{3p_{\rm
age}^{2}}\right)(t_{0}-t)^{2}H_{0}^{2}.$

Performing a mathematical transformation, one has

$\frac{H(t)}{H_{0}}=1+\frac{2}{3}\left(1-\eta\frac{H_{0}t}{p_{\rm
age}}\right)\left(\frac{1}{H_{0}t}-\frac{1}{p_{\rm age}}\right),$ (A4)

which is the common form of the PAge approximation. Here $\eta\equiv
1-\frac{3}{2}p_{\rm age}^{2}(1+q_{0})$.

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