# Covert Transmission with Antenna Selection and Using an External Jammer

Morteza Sarkheil  Department of ECE,
Tarbiat Modares University
Tehran, Iran
<EMAIL_ADDRESS>Paeiz Azmi  Department of ECE,
Tarbiat Modares University
Tehran, Iran
<EMAIL_ADDRESS>Moslem Forouzesh  Department of ECE,
Tarbiat Modares University
Tehran, Iran
<EMAIL_ADDRESS>Ali Kuhestani  Department of ECE,
Sharif University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
###### Abstract

This paper adopts the antenna selection technique to enhance the covert rate
in a wireless communication network comprised of a source, a destination , an
external jammer and an eavesdropper. In the covert communication, the level of
transmit power is low and hence a source with multiple antennas can be adopted
to send the information toward the single antenna destination while
concurrently, the jammer transmits an artificial noise signal. For this system
model, we consider a scenario where the source is forced to select one or
several of its antennas to transmit its confidential information due to its
limited RF chains. Furthermore, we consider two different jamming scenarios to
support our covert communication: 1) The destination is unable to cancel the
jamming signal, 2) The destination can subtract the jamming signal. For such a
communication network, our aim is to maximize the covert rate subject to power
constraint and covert communication requirement. In the first scenario, the
optimization problem is non-convex, and hence, it can be solved through using
Difference of Convex function (DC) method while the optimization problem of
the second scenario is intrinsically convex. Our numerical results show that
the higher the number of selected antennas at the transmitter, the higher the
covert rate will be achieved.

###### Index Terms:

covert communication, antenna selection, external jammer.

##  I Introduction

Security is a critical and important subject in wireless communications
networks. This is because the broadcast nature of wireless networks permits to
the unauthorized nodes to access the contents of the confidential messages
[1]. Physical layer security as a new solution to enhance the confidentiality
of wireless communications has attracted a lot of interest [2]–[4]. Physical
layer secure transmission is provisioned by intelligently exploiting the time
varying properties of fading channels, instead of relying on conventional
cryptographic techniques. This approach uses signal processing and encoding
techniques at the physical layer, to improve the quality of the received
signal at illegitimate receivers compared with the unauthorized users. Toward
this end, the jamming signal can be used to enhance the physical layer
security. Typically, there are two main types of jamming signal to enhance the
security of wireless networks [4], [5]: 1) Friendly jamming (FJ) scenario
where the jamming signal is known at the legitimate receiver, 2) Gaussian
noise jamming (GNJ) where the jamming signal is unknown at the legitimate
receiver[5]-[7]. It should be noted that FJ provides better secrecy
performance compared with GNJ because when FJ is used, legitimate receiver can
cancel the jamming signal. However, GNJ is more simple compared with FJ
scenario. This reveals the trade-off between secrecy performance and
complexity of the network.

In some communication networks, low probability of detection(LPD) or covert
communication is essential for information transmission over electromagnetic
and acoustic channels [8]. For military applications, LPD is interest when the
transmitter wishes to remain undetected, or when the knowledge of
communication may point to the presence of a receiver. As such, in covert
communication only the detection capability of an eavesdropper is considered,
i.e., an eavesdropper need not be able to actually decode the communication
signal. In other words, the covert communication keeps military forces from
possible attacks [9]. In recent years, some few works have investigated the
covert communication in different wireless communication networks [9]–[14].

In this paper, we take into account the power allocation problem of a wireless
communication network, where a multiple antenna source transmits its
confidential message to a single antenna destination in the presence of a
passive eavesdropper. To covert the communication against the eavesdropping
attack, a single antenna external jammer is employed. For this communication
network, we investigate two different jamming scenarios to support covert
communication: 1) The destination is unable to cancel the jamming signal,
i.e., we have FJ, 2) The destination can subtract the jamming signal, i.e., we
have GNJ. We consider a realistic scenario where the number of RF chains at
the source may be fewer than the number of source’s antennas. Instead of
selecting the antennas randomly, we propose to select the best antennas to
transmit the information signal toward the destination. For such a network, we
formulate the power allocation between the source and jammer that maximizes
the instantaneous covert rate while concurrently hiding the communication
against the passive eavesdropping attack. Since the optimization problem of
GNJ scenario is non-convex, we exploit difference of concave (DC) approach to
convert it to a convex optimization problem. For the FJ scenario, we observe
that the optimization problem is convex. We also obtain closed-form solutions
for the optimal power threshold from the eavesdropper’s viewpoint. Our
numerical examples show that by increasing the number of selected antennas at
the transmitter the covert rate is increased. Furthermore, the impact of the
distance between the transmitter and the eavesdropper on the achievable covert
rate is more than the impact of the distance between the transmitter and the
receiver.

##  II System Model

As shown in Fig. 1, the system model under consideration is a multi-input
single-output (MISO) wireless network consisting of an $M_{T}$ antenna
transmitter (Alice), a single antenna jammer, a single antenna receiver (Bob),
and a single antenna eavesdropper (Eve). For this system model, assume Alice
selects several antennas to send information signal and also, the jammer sends
jamming signal in all directions. In this scenario, we assume the
communication channels follow the slow fading. Moreover, the fading
coefficients have Rayleigh distribution. Alice is unaware of Eve’s channel
state information (CSI) and only knows the distance between itself and Eve. We
also assume that the jammer knows the distance between itself and Eve. The
distance between Alice and Bob, Alice and Eve, the jammer and Bob and the
jammer and Eve are defined as ${D_{ab}}$, ${D_{ae}}$, ${D_{jb}}$ and
${D_{je}}$, respectively. The information signal and the jamming signal
denoted by
${\boldsymbol{X_{s}}}=\left[{{x_{s}}^{1},{x_{s}}^{2},..,{x_{s}}^{m}}\right]$
and
${\boldsymbol{X_{j}}}=\left[{{x_{j}}^{1},{x_{j}}^{2},..,{x_{j}}^{m}}\right]$,
respectively, where $m$ shows the number of time slots.

Figure 1: Our MISO system model in the presence of an external jammer and a
passive eavesdropper.

##  III Covert Communication

In the covert communication, our aim is to send a message from Alice to Bob
secretly such that Eve will not be notified of this communication, i.e., based
on the level received energy. The Eve decides about the presence or absence of
signal transmission by Alice. In this scenario, notation ${\Omega_{0}}$ states
that Alice does not transmit the information signal to Bob, while
${\Omega_{1}}$ states that the Alice transmits the information signal to Bob.
If we assume $P({\Omega_{0}})=1-p$ and $P({\Omega_{1}})=p$, the probability of
error at Eve according to its decisions is defined as follows

$\displaystyle{P_{e}}=(1-p){P_{FA}}+p{P_{MD}},$ (1)

where ${P_{MD}}$ is the probability that Alice sends its information signal
but Eve decides on the absence of communication. Furthermore, ${P_{FA}}$ is
the probability that Alice is silent but Eve decides on the presence of
communication between Alice and Bob. The covert communication between Alice
and Bob is established when the following condition holds [12]

$\displaystyle\mathop{{P_{FA}}+{P_{MD}}}\limits_{n\to\infty}\geq
1-\varepsilon.$ (2)

The received signal at node $k\in{Bob,Eve},$ is as follows

$\displaystyle{y_{k}}=\left\\{{\begin{array}[]{*{20}{l}}{\frac{{\sqrt{{p_{j}}}{h_{jk}}{x_{j}}}}{{D_{jk}^{\beta/2}}}+{n_{k}},\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&}{\hfil&{{\Omega_{0},}}&\end{array}}\\\
{\frac{{\sqrt{{p_{j}}}{h_{jk}}{x_{j}}}}{{D_{jk}^{\beta/2}}}+\frac{{\sqrt{{p_{s}}}{\boldsymbol{w}^{H}}\bf{{h_{ak}}}{x_{s}}}}{{D_{ak}^{\beta/2}}}+{n_{k}},\begin{array}[]{*{20}{c}}{}\hfil&{{\Omega_{1}},}\end{array}}\end{array}}\right.$
(7)

where $p_{s}$ and $p_{j}$ are the allocated power for the information signal
and the jamming signal, respectively, such that ${p_{s}}=\alpha{P_{total}}$
and ${p_{j}}=(1-\alpha){P_{total}}$, where $\alpha$ is the power allocation
factor. It is worth noting that the total power consumption constraint
provides a guideline for the power allocation of the Alice and the jammer.
This approach has been widely exploited in the literature for both performance
analysis and network optimization design [7], [11], [14]. Furthermore, in (3),
$\beta$ is the path-loss exponent. Note that
$\boldsymbol{w}=\frac{{{\bf{h_{ab}}}}}{{\left\|{\left.{\bf{{h_{ab}}}}\right\|}\right.}}$represents
the maximum ratio transmission (MRT) beamformer at the source and $n_{k}$ is
the white Gaussian noise received at node $k$ with the distribution of $n\sim
CN(0,\sigma_{k}^{2})$. ${\bf{{h_{ak}}}}$, and ${{\bf{h_{jk}}}}$ are the
channel coefficients of the information signal and the jamming signal,
respectively.

The decision of Eve is based on the energy of the received signal. Eve decides
on the state $\Omega_{1}$ if $\frac{{{Y_{e}}}}{m}\geq V$. Otherwise, Eve
decides on the state $\Omega_{0}$. $V$ is the threshold of decision and
$Y_{e}$ is the energy of the received signal which is
${Y_{e}}={\sum\limits_{l=1}^{m}{\left|{y_{k}^{l}}\right|}^{2}}$. Accordingly,
$P_{FA}$ and $P_{MD}$ can be written, respectively, as

$\displaystyle\begin{array}[]{l}{P_{FA}}=P(({\sigma_{e}}^{2}+\gamma)\frac{{{\chi^{2}}_{2m}}}{m}\geq
V\left|{{\Omega_{0}}}\right.),\end{array}$ (9)
$\displaystyle\begin{array}[]{l}{P_{MD}}=P(({\sigma_{e}}^{2}+\gamma)\frac{{{\chi^{2}}_{2m}}}{m}\leq
V\left|{{\Omega_{1}}}\right.),\end{array}$ (11)

where $\chi^{2}_{2m}$ is a random variable with chi-squared distribution with
$2m$ degrees of freedom. For large number of time slots, $m\to\infty$, we have
$\frac{{\chi_{2m}^{2}}}{m}\to 1$. Therefore, (5) and (6) can be simplified as

$\displaystyle{P_{FA}}=P(({\sigma_{e}}^{2}+\gamma)\geq
V\left|{{\Omega_{0}}}\right.),$ (12)
$\displaystyle{P_{MD}}=P(({\sigma_{e}}^{2}+\gamma)\leq
V\left|{{\Omega_{1}}}\right.).$ (13)

We note that the received signal has the distribution of $y_{k}^{l}\sim
CN(0,({\sigma^{2}}+\gamma))$, where $\gamma$ is defined as follows

$\displaystyle\gamma=\left\\{{\begin{array}[]{*{20}{l}}{\frac{{{P_{j}}}}{{D_{ae}^{\beta}}}{{\left|{{h_{je}}}\right|}^{2}},\begin{array}[]{*{20}{c}}{}\hfil&}{\hfil&{{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}}&{{\Omega_{0}},}\end{array}}\\\
{\frac{{{P_{j}}}}{{D_{je}^{\beta}}}{{\left|{{h_{je}}}\right|}^{2}}+\frac{{{P_{s}}}}{{D_{ae}^{\beta}}}{{\left\|{{\boldsymbol{w}^{H}}\bf{{h_{ae}}}}\right\|}^{2}},\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{{\Omega_{1}}.}\end{array}}\end{array}}\right.$
(18)

In (18), $h_{je}$ and $\bf{h_{ae}}$ are zero-mean Gaussian random variables
with unit variances. Thus, ${\left|{{h_{je}}}\right|^{2}}$ has an exponential
distribution with the parameter $\lambda=1$. Therefore, for the assumption of
${\Omega_{0}}$ , $\gamma$ has an exponential distribution with the parameter
of $\lambda=\frac{1}{{{\phi_{0}}}}$. According to [15], the summation of two
exponential distributions $x$ and $y$ with parameters $\lambda_{1}$ and
$\lambda_{2}$ is given by

$\displaystyle{f_{x+y}}(\gamma)=\frac{{{\lambda_{1}}{\lambda_{2}}}}{{{\lambda_{2}}-{\lambda_{1}}}}[\,{e^{-{\lambda_{1}}\gamma}}-\,{e^{-{\lambda_{2}}\gamma}}]U(\gamma),$
(19)

where $U(\gamma)$ is the step function. Furthermore, $\gamma$ is distributed
as follows

$\displaystyle{f}(\gamma)=\left\\{{\begin{array}[]{*{20}{l}}{\begin{array}[]{*{20}{c}}{\frac{1}{{{\phi_{0}}}}{e^{-\frac{\gamma}{{{\phi_{0}}}}}},}&{}\hfil&{}\hfil&{\begin{array}[]{*{20}{c}}}{\hfil&{\gamma\geq
0}&{{\Omega_{0},}}\end{array}}\end{array}}\\\
{\frac{1}{{{\phi_{0}}-{\phi_{1}}}}({e^{-\frac{\gamma}{{{\phi_{0}}}}}}-{e^{-\frac{\gamma}{{{\phi_{1}}}}}}),\begin{array}[]{*{20}{c}}{}\hfil&{\gamma\geq
0}&{{\Omega_{1},}}\end{array}}\end{array}}\right.$ (25)

where ${\phi_{0}}=\frac{{{P_{j}}}}{{D_{je}^{\beta}}}$ and
${\phi_{1}}=\frac{{{P_{s}}}}{{D_{ae}^{\beta}}}$. By combining (12), (13), (18)
and (25), we obtain

$\displaystyle{P_{FA}}=\left\\{{\begin{array}[]{*{20}{l}}{{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{0}}}}}},\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{V-\sigma_{e}^{2}\geq
0,}\end{array}}\\\
{1,\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{\hfil&{V-\sigma_{e}^{2}\leq
0,}\end{array}}\end{array}}\right.$ (30)

and

$\displaystyle{P_{MD}}=\left\\{{\begin{array}[]{*{20}{l}}{\Gamma,\begin{array}[]{*{20}{c}}{}\hfil&{\begin{array}[]{*{20}{c}}{V-\sigma_{e}^{2}}\end{array}\geq
0,}\end{array}}\\\
{0,\begin{array}[]{*{20}{c}}{}\hfil&{\begin{array}[]{*{20}{c}}{V-\sigma_{e}^{2}}\end{array}\leq
0,}\end{array}}\end{array}}\right.$ (37)

where
${\Gamma}=\frac{1}{{{\phi_{0}}-{\phi_{1}}}}({\phi_{1}}{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{1}}}}}}-{\phi_{0}}{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{0}}}}}}+{\phi_{0}}-{\phi_{1}})$.

###  III-A Eavesdropper’s error

The Eve’s error is defined as ${P_{MD}}+{P_{FA}}$. In this scenario, we
consider the worst case scenario where Eve has an optimal threshold. The
optimal threshold of Eve is minimum decision error. Therefore, we can write

$\displaystyle{P_{MD}}+{P_{FA}}=\left\\{{\begin{array}[]{*{20}{l}}{{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{0}}}}}}+{\Lambda_{1}},\begin{array}[]{*{20}{c}}{}\hfil&{V-\sigma_{e}^{2}\geq
0,}\end{array}}\\\
{1,\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}}&{}\hfil&}{\hfil\end{array}}&{V-\sigma_{e}^{2}\leq
0,}\end{array}}\end{array}}\right.$ (44)

where

$\displaystyle{\Lambda_{1}}=\frac{1}{{{\phi_{0}}-{\phi_{1}}}}({\phi_{1}}{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{1}}}}}}-{\phi_{0}}{e^{-\frac{{(V-\sigma_{e}^{2})}}{{{\phi_{0}}}}}}+{\phi_{0}}-{\phi_{1}}).$
(45)

The minimum Eve’s error will be

$\displaystyle\mathop{\min}\limits_{V}({P_{FA}}+{P_{MD}})={e^{-\frac{{{\phi_{1}A}}}{{{\phi_{1}}-{\phi_{0}}}}}}+{\Lambda_{2}},$
(46)

where

$\displaystyle{\Lambda_{2=}}\frac{1}{{{\phi_{0}}-{\phi_{1}}}}({\phi_{1}}{e^{-\frac{{{\phi_{0}}}}{{{\phi_{1}}-{\phi_{0}}}}A}}-{\phi_{0}}{e^{-\frac{{{\phi_{1}}}}{{{\phi_{1}}-{\phi_{0}}}}A}}+{\phi_{0}}-{\phi_{1}}),$
(47)

where $A=Ln\frac{{{\phi_{1}}}}{{{\phi_{0}}}}$ and the optimal threshold is
given by

$\displaystyle
V^{*}=\frac{{{\phi_{0}}{\phi_{1}}}}{{{\phi_{1}}-{\phi_{0}}}}Ln\frac{{{\phi_{1}}}}{{{\phi_{0}}}}+\sigma_{e}^{2}.$
(48)

Proof: See Appendix A. $\blacksquare$

###  III-B Covert rate

The covert rate at Bob is defined as

$\displaystyle
R=\log(1+\frac{{\alpha{P_{total}}{{\left|{{h_{ab}}}\right|}^{2}}D_{jb}^{\beta}}}{{D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha){P_{total}}{{\left|{{h_{jb}}}\right|}^{2}}D_{ab}^{\beta}}}).$
(49)

Our goal is to increase the covert rate. To this end, from the $M_{T}$
antennas, $N_{D}$ of antennas with the highest channel coefficients between
Alice and Bob are selected to send the information. As such, the corresponding
antennas are selected based on the following criterion

$\displaystyle{\bf{h_{ab}}}={{\rm X}_{{N_{D}}}}(sort({H_{{M_{T}}\times 1}})),$
(50)

where $sort(v)$ arranges the $v$ elements in descending order and ${{\rm
X}_{i}}(u)$ selects first $i$ elements of $u$. According to (50), we can
increase the covert rate by selecting appropriate antennas for sending
information signal. The proposed antenna selection technique, not only
increases the covert rate, but also reduces the hardware equipments at Alice,
including amplifiers and mixers, and also reduces the network complexity and
overhead.

##  IV Optimization Problem

In this paper, our goal is to maximize the covert rate subject to the
limitation of transmit power and the covert communication condition. Moreover,
in the following, we investigate two different scenarios of FJ and GNJ.

###  IV-A Gaussian Noise Jamming Scenario

In this scenario, it is assumed that Bob cannot cancel the jamming signal.
Therefore, we have the following optimization problem

$\displaystyle\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{\alpha}\\{R=\log(1+\frac{{\alpha{P_{total}}{{\left|{{h_{ab}}}\right|}^{2}}D_{jb}^{\beta}}}{{D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha){P_{total}}{{\left|{{h_{jb}}}\right|}^{2}}D_{ab}^{\beta}}})\\},}\\\
{\begin{array}[]{*{20}{c}}{}\hfil\end{array}s.t\begin{array}[]{*{20}{c}}{}\hfil&{{C_{1}}:0<\alpha<1,}\end{array}}\\\
{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{}\hfil&{{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}}\end{array}{C_{2}}}\end{array}:\mathop{\min}\limits_{V}({P_{FA}}+{P_{MD}})\geq
1-\varepsilon.}\end{array}$ (58)

After formulating the constraint $C_{2}$ (See Appendix B), (58) can be
rewritten as follows

$\displaystyle\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{\alpha}\\{R=\log(1+\frac{{\alpha{P_{total}}{{\left|{{h_{ab}}}\right|}^{2}}D_{jb}^{\beta}}}{{D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha){P_{total}}{{\left|{{h_{jb}}}\right|}^{2}}D_{ab}^{\beta}}})\\},}\\\
{\begin{array}[]{*{20}{c}}{}\hfil\end{array}s.t\begin{array}[]{*{20}{c}}{}\hfil&{{C_{1}}:0<\alpha<1,}\end{array}}\\\
{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{}\hfil&}{\hfil\end{array}{C_{2}}}\end{array}:\frac{{(1-\alpha)D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}\ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})<ln\varepsilon.}\end{array}$ (66)

Since the optimization problem (22) is a non-convex one, we rewrite the
objective function as follows

$\displaystyle\Sigma(\alpha)-\Psi(\alpha),$ (67)

where

$\displaystyle\left\\{\begin{array}[]{l}\Sigma(\alpha)=log(D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha){P_{total}}D_{ab}^{\beta}{\left|{{h_{jb}}}\right|^{2}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\alpha{P_{\max}}D_{jb}^{\beta}{\left\|{{h_{ab}}}\right\|^{2}})\\\
\Psi(\alpha)=\log(D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha){P_{total}}D_{ab}^{\beta}{\left|{{h_{jb}}}\right|^{2}})\end{array}\right.$
(71)

To solve the optimization problem, we use the DC method. As such, we
approximate $\Psi(\alpha)$ as follows [16]

$\displaystyle\begin{array}[]{l}\Psi(\alpha)\simeq\tilde{\Psi}(\alpha)=\Psi(\alpha(\eta-1))+\\\
{\nabla^{T}}\Psi(\alpha(\eta-1))(\alpha-\alpha(\eta-1)),\end{array}$ (74)

where $\nabla$ is the gradient operator and ${\nabla^{T}}\Psi(\alpha(\eta-1))$
is calculated as follows

$\displaystyle{\nabla^{T}}\Psi(\alpha(\eta-1))=\frac{{-{P_{total}}D_{ab}^{\beta}{{\left|{{h_{jb}}}\right|}^{2}}}}{{D_{jb}^{\beta}D_{ab}^{\beta}\sigma_{b}^{2}+(1-\alpha(\eta-1)){P_{total}}D_{ab}^{\beta}{{\left|{{h_{jb}}}\right|}^{2}}}}.$

Finally, (49) can be rewritten as $\Sigma(\alpha)-\tilde{\Psi}(\alpha)$ which
is a concave function. $C_{2}$ can be concave by using the change of value
$T=(1-\alpha)D_{ae}^{\beta}-\alpha D_{je}^{\beta}.$ Therefore, we have

$\displaystyle\begin{array}[]{l}\mathop{\max}\limits_{\alpha}\,\,\,\,\,\,\,\Sigma(\alpha)-\tilde{\Psi}(\alpha)\\\
s.t\,\,\,\,{C_{1}}:0<\alpha<1,\\\
\,\,\,\,\,\,\,\,\,\,{C_{2}}:(1-\alpha)D_{ae}^{\beta}\ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})<Tln\varepsilon,\\\
\,\,\,\,\,\,\,\,\,\,{C_{3}}:(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}<T.\end{array}$ (80)

Since the optimization problem in (80) is convex. we can use the available
softwares such as CVX solver to solve it.

###  IV-B Friendly Jamming Scenario

In this scenario, it is assumed that Bob can cancel the jamming signal. Hence,
we have the following optimization problem

$\displaystyle\begin{array}[]{l}\mathop{\max}\limits_{\alpha}\\{R=log(1+\frac{{\alpha{P_{total}}{{\left|{{h_{ab}}}\right|}^{2}}}}{{D_{ab}^{\beta}\sigma_{b}^{2}}})\\},\\\
\,\,\,s.t\,\,\,\,\,\,\,\,\,\,{C_{1}},{C_{2}}.\end{array}$ (83)

In (83), the objective function is convex. Therefore, we can use the CVX
solver to solve it. In addition, we can convert constraints $C_{1}$ and
$C_{2}$ to convex constraints as in the previous scenario.

##  V Simulation Result

In this section, our aim is to evaluate the secrecy performance of the
proposed secure transmission scheme. For simplicity and without loss of
generality, we assume that Alice, Bob, the jammer and the passive Eve are
located at the positions (-2.5,2.5), (2.5,2.5), (2.5,-2.5) and (-2.5,-2.5),
respectively. Unless otherwise stated, the network parameters are: number of
antennas at Alice $M_{T}=10$, number of antenna at Alice that are selected
$N_{D}=6$, $\varepsilon=0.1$, $1-\varepsilon$ is lower bound of error
detection probability, and the path-loss exponent $\beta=2$.

Figure 2: Covert rate versus the total transmit power.

Fig. 2 shows the effect of the transmit power sent by Alice on the covert
rate. According to simulation results, if the transmit power increases, the
covert rate increases. However, as can be seen, the covert rate ceiling is
appeared at high transmit powers. This is because when the transmit power
increases, jammer transmits jamming signal with higher power, hence, Alice can
transmit information signal with higher power which leads to increase covert
rate. Furthermore, we observe that by increasing the number of the best
selected antennas for signal transmission, the achievable covert rate
improves. For example, if the transmit power is constant and the number of
antennas which are selected for sending information signal is doubled, the
covert rate is reduced $27\%$. However, the simulation results shows that the
associated number of antennas should not be larger than 6 in order to balance
the performance gain and implementation cost. If the number of antennas
exceeds 6, in addition to Bob’s condition, the Eve conditions will also
improve for detection, which will make our hidden rate not change
significantly. Since allocating large number of antennas for signal
transmission imposes more hardware equipments (RF chains), higher latency and
more much overhead to the network. Moreover, this figure compares random
antenna selection and proposed antenna selection. It is necessary to note that
random antenna selection with $N_{D}=4$ has more efficiency with respect to
proposed method with $N_{D}=1$ and less efficiency with respect to proposed
method with $N_{D}=2$. The reason is when the number of antennas increase
diversity gain increases. But the proposed method with less diversity gain has
more efficiency with respect to random antenna selection with higher diversity
gain. The Fig. 2 shows proposed method enhances the covert rate approximately
2 times.

Figure 3: Covert rate versus the distance between Alice and Bob. We set
$P_{total}=5W$.

As we can see in Fig. 3, the lower the distance between Alice and Bob, the
higher the covert rate. In addition, the lower the distance between Alice and
Eve, the lower the covert rate. Given Fig. 3, if the distance between Alice
and Bob is increased by 6 meters, the covert rate reduces about $62\%$. Now,
if the distance between Alice and Eve increases the same amount, the cover
rate increases about 4.7 times. Thus, the impact of the distance between Alice
and Eve is more important than the distance between Alice and Bob. Also, as
shown in Figure 3, when the Bob can cancel the jamming signal, the covert rate
increases dramatically. For example, if the distance between Alice and Jammer
is 2 meters and Bob cancels the jamming signal, the covert rate is 5.6 times
more.

Figure 4: Covert rate versus the distance between the jammer and Bob. We set
$P_{total}=5W$.

Also, in Fig. 4, we can see the effect of the distance between the jammer and
Bob and Eve on the covert rate. According to simulation results, when Bob can
not cancel the jamming signal, the lower the distance between the jammer and
Bob, the lower the covert rate. For example, if the the distance between the
jammer and Bob from 3 to 9 meters, the covert rate will increase $40\%$. Since
the jamming signal treats as interference in Bob, by reducing the distance
between the jammer and Bob, the covert rate is reduced. Moreover, the distance
between the jammer and Eve is effective. If their distance is 6 meter high,
the covert rate will decrease $63\%$. Also, when the jamming signal is
canceled, increasing of the distance between Bob and jammer does not change
the covert rate, and this parameter does not affect on the covert rate.

##  VI Conclusion

In the covert communication, the level of transmit power is low which lead to
low SINR hence we aim to exploit from method without increasing the transmit
power increase the covert rate. Our strategy to increase the covert rate is
choosing the appropriate antenna in the transmitter to send the information.
For this purpose, we proposed a system with a multi-antenna transmitter, a
single-antenna jammer , a single-antenna receiver, and a single antenna
eavesdropper. In this scenario, we encountered the optimization problem, which
we use to solve the method of DC. Moreover, we investigated two different
scenarios: 1) GNJ where Bob is unable to cancel the jamming signal, 2) FJ
scenario where Bob is able to cancel the jamming signal. As the simulation
results show, by increasing the number of antennas which are selected in the
transmitter and choosing an appropriate antenna for sending the information,
it causes an increase in the covert rate. Furthermore, the distance between
transmitter and the eavesdropper has more impact on the cover rate compared
with the distance between the transmitter and the receiver.

Appendix A
It is simple to show that ${{P_{FA}}+{P_{MD}}}$ is convex. In order to
calculate the minimum value of ${{P_{FA}}+{P_{MD}}}$, we derivative
${{P_{FA}}+{P_{MD}}}$ with respect to V and then we find the root of it.
Following this, we obtain the optimal $V^{*}$ as follows

$\displaystyle\begin{array}[]{l}{V^{*}}=\frac{{{\phi_{0}}{\phi_{1}}}}{{{\phi_{1}}-{\phi_{0}}}}Ln\frac{{{\phi_{1}}}}{{{\phi_{0}}}}+\sigma_{e}^{2}.\end{array}$
(85)

Appendix B
For the constraint $C_{2}$, we can write

$\displaystyle\begin{array}[]{l}\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}\begin{array}[]{*{20}{c}}\times\end{array}\left[{e\frac{{(1-\alpha)D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})}\right.\\\
\left.{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-e\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})}\right]\end{array}$ (89)
$\displaystyle\begin{array}[]{l}=\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}\times{e^{{{\frac{{(1-\alpha)D_{ae}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}}^{ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})}}}}\\\
\,\,\,\,\,\,\,\,\,\times\,\,\left.{\left[{1-{e^{ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})}}}\right.}\right]>-\varepsilon.\end{array}$
(92)

Therefore, we can rewrite the inequality (92) as

$\displaystyle{{e^{\frac{{(1-\alpha)D_{ae}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}-\alpha
D_{je}^{\beta}}}}}^{ln(\frac{{\alpha
D_{je}^{\beta}}}{{(1-\alpha)D_{ae}^{\beta}}})}<\varepsilon.}$

which is equivalent to the constraint $C_{2}$.

## References

  * [1] Y. Liang, H. V. Poor, and L. Ying, ”Wireless broadcast networks: reliability, security, and stability,”  Inf. Theory and App. Work., USA, pp. 249-255, Feb. 2008.
  * [2] A. Mukherjee, S. A. A. Fakoorian, J. Huang, and A. L. Swindlehurst,” Principles of physical layer security in multiuser wireless networks: A survey,” IEEE Commun. Surveys Tuts. , Vol. 16, no. 3, pp. 1550-1573, Feb. 2014.
  * [3] A. Kuhestani, A. Mohammadi and M. Mohammadi,” Joint Relay Selection and Power Allocation in Large-Scale MIMO Systems With Untrusted Relays and Passive Eavesdroppers,” IEEE Trans. Inf. Foren. Sec., vol. 13, no. 2, pp. 341-355, Feb. 2018.
  * [4] M. Tatar Mamaghani, A. Kuhestani and K.-K. Wong,” Secure Two-Way Transmission via Wireless-Powered Untrusted Relay and External Jammer,” IEEE Trans. Veh. Tech. ,pp. 1-1, Jul. 2018.
  * [5] A. Kuhestani, A. Mohammadi and P. L. Yeoh, ”Optimal power allocation and secrecy sum rate in two-way untrusted relaying networks with an external jammer,” IEEE Trans. Commun ,Vol. 66, no.6, pp. 2671-2684, Jun. 2018 .
  * [6] A. Yener and S. Ulukus, ”Wireless physical-layer security: Lessons learned from information theory ,” Proc. IEEE ,Vol. 103, no. 10, pp. 1814-1825, Oct. 2015 .
  * [7] E. Tekin and A. Yener, ”The general Gaussian multiple-access and two-way wiretap channels: Achievable rates and cooperative jamming,” IEEE Trans. Inf. Theory ,Vol. 54, no. 6, pp. 2735-2751, Jun. 2008 .
  * [8] R. Diamant and L. Lampe, ”Low Probability of Detection for Underwater Acoustic Communication: A Review,” IEEE Access ,Vol. 6, pp. 19099-19112, Mar. 2018 .
  * [9] K. Shahzad, X. Zhou, and S. Yan, ”Covert Communication in Fading Channels under Channel Uncertainty,” . IEEE 85th Veh. Tech.Conf. (VTC Spring), Sydney, NSW, Australia, pp. 1-5, Nov. 2017.
  * [10] A. Abdelaziz and C. Emre Koksal, ”Fundamental Limits of Covert Communication over MIMO AWGN Channel,” . IEEE Conf. on Commun. and Net. Security (CNS), Las Vegas, NV, USA, pp. 1-9, Dec. 2017.
  * [11] M. Forouzesh, P. Azmi, N. Mokari, and K.-K. Wong,” Covert Communications Versus Physical Layer Security,” arXiv:1803.06608.
  * [12] J. Zhang and M. C. Gursoy, ”Collaborative relay beamforming for secure broadcasting,” IEEE Wireless Commun. and Net. Conf. (WCNC), Sydney, NSW, Australia, pp. 1-6, Apr. 2010.
  * [13] J. Hu, S. Yan, X. Zhou ,F. and J. Wang, ”Covert Communication in Wireless Relay Networks,” IEEE GLOBECOM 2017, Singapore, pp. 1-6, Jan. 2018.
  * [14] B. A. Bash, D. Goeckel, D. Towsley, and S. Guha, ” Hiding Information in Noise: Fundamental Limits of Covert Wireless Communication,” IEEE Commun. Magazine, Vol. 53, no. 12, pp. 26-31, Dec. 2015.
  * [15] P.E. Oguntunde, O.A. Odetunmibi and A.O. Adejumo, ”On the Sum of Exponentially Distributed Random Variables: A Convolution Approach,” European Journal of Statistics and Probability, Vol. 1, no. 2, pp. 1-8, Dec 2013.
  * [16] S. Boyd and L. Vandenberghe, ”Convex Optimization”, Cambridge University Press, 2004\.


# A Reliable Calibration of HII Galaxies Hubble Diagram with Cosmic
Chronometers and Artificial Neural Network

Jian-Chen Zhang(张建臣) Institute for Frontiers in Astronomy and Astrophysics of
Beijing Normal University, Beijing 100875, China<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University, Beijing 100875, China
College of Computer and Information Engineering, Dezhou University, Dezhou
253023, China  Institute for Astronomical Science, Dezhou University, Dezhou
253023, China  Kang Jiao(焦康) Institute for Frontiers in Astronomy and
Astrophysics of Beijing Normal University, Beijing 100875, China;
<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University,
Beijing 100875, China  Dipartimento di Fisica e Astronomia “Augusto Righi”,
Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, I-40129
Bologna, Italy  Tingting Zhang  College of Command and Control Engineering,
PLA Army Engineering University, Nanjing 210000, China<EMAIL_ADDRESS>Tong-Jie Zhang(张同杰) Institute for Frontiers in Astronomy and Astrophysics of
Beijing Normal University, Beijing 100875, China<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University, Beijing 100875, China
Institute for Astronomical Science, Dezhou University, Dezhou 253023, China
Bo Yu(于波) Institute for Astronomical Science, Dezhou University, Dezhou
253023, China  School of Mathematics and Big Data, Dezhou University, Dezhou
253023, China  Shandong Provincial Key Laboratory of Biophysics, Institute of
Biophysics, Dezhou University, Dezhou 253023, China

###### Abstract

The $L-\sigma$ relation of HII galaxies (HIIGx) calibrated by a distance
indicator is a reliable standard candle for measuring the Hubble constant
$H_{0}$. The most straightforward calibration technique anchors them with the
first tier of distance ladders from the same galaxies. Recently another
promising method that uses the cosmological model-independent Cosmic
Chronometers (CC) as a calibrator has been proposed. We promote this technique
by removing the assumptions about the cosmic flatness and using a non-
parametric Artificial Neural Network for the data reconstruction process. We
observe a correlation between the cosmic curvature density parameter and the
slope of the $L-\sigma$ relation, thereby improving the reliability of the
calibration. Using the calibrated HIIGx Hubble diagram, we obtain a Type Ia
Supernovae Hubble diagram free of the conventional assumption about $H_{0}$.
Finally we get a value of
$H_{0}=65.9_{-2.9}^{+3.0}~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, which is
compatible with latest Planck18 measurement.

Hubble diagram(759); H II regions(694); Standard candles(1563); Hubble
constant(758); Astronomy data analysis(1858); Neural networks(1933); Type Ia
supernovae(1728)

††software: GaPP (Seikel et al., 2012), ReFANN (Wang et al., 2020b), emcee
(Foreman-Mackey et al., 2013), ChainConsumer (Hinton, 2016), Numpy (Harris et
al., 2020), Scipy (Virtanen et al., 2020), Matplotlib (Hunter, 2007).

##  1 Introduction

Hubble diagrams contain a wealth of information on the mass-energy content and
evolutionary history of the Universe via the distance redshift relation they
present. A Hubble diagram can be constructed using any astronomical source
with standardizable intrinsic properties that are calibrated using known
credible distance indicators. For example, Type Ia Supernovae (SNe Ia), which
share the same explosion mechanism, thus have an intrinsic luminosity that can
be used to measure the luminosity distance (Riess et al., 1998; Perlmutter et
al., 1999). The $E_{p,i}-E_{iso}$ correlation of Gamma-ray bursts (GRBs) has
no significant redshift dependence and thus can be used to measure distance
(Demianski, Marek et al., 2017). The non-linear relationship between X-ray and
UV emission in quasars enables a novel, highly accurate method for estimating
the absolute luminosity, converting it to standard candles (Lusso, E. &
Risaliti, G., 2017; Salvestrini, F. et al., 2019; Lusso, E. et al., 2020).
Fast radio bursts (FRBs) dispersion is a convolution of the cosmic distance
element and the electron density, which enables the use of these events as
cosmological distance measurements (Kumar & Linder, 2019).

HII galaxies (HIIGx) are massive and compact aggregates of star formation. The
starburst nearly entirely dominates the total luminosity of an HIIGx. The
strong correlation between the luminosity $L(H\beta)$ in $H\beta$ lines and
the ionized gas velocity dispersion $\sigma_{v}$ (Terlevich & Melnick, 1981;
Chávez et al., 2014; Melnick, J. et al., 2017) provides a tool for
constructing the Hubble diagram and is then used to estimate $H_{0}$ after
calibration with the first tier of distance ladders (Chávez et al., 2012;
Fernández Arenas et al., 2017). Apart from this type of local calibration, a
new method was recently presented that extends to high redshifts by
reconstructing Cosmic Chronometers (CC) to obtain distance information (Wu et
al., 2020). They applied the prevalent Gaussian Process (GP) (Seikel et al.,
2012) reconstruction in their approach and obtained an accurate calibration of
the HIIGx Hubble diagram. However, their result is based on the assumptions
that the Universe is flat and the observational data are correlated by a
kernel function. Removing the assumptions will further improve the reliability
of this calibration method, especially for the concern that the last
assumption is relatively strong.

Artificial Neural Network (ANN) is a machine learning technique that excels at
regression. It has recently been broadly utilized in astronomical studies.
Such as distinguishing standard and modified gravity cosmologies (Peel et al.,
2019), estimating cosmological parameters (Wang et al., 2020b), simulating
Cosmic Microwave background anisotropy maps(Mishra et al., 2019), creating
weak lensing convergence map (Mustafa et al., 2019), discriminating
cosmological models (Schmelzle et al., 2017), likelihood-free cosmological
constraints (Wang et al., 2021) Etc. Wang et al. (2020a) make use of ANN in
reconstructing functions of the $H(z)$ and the $D_{L}(z)$ using OHD and SNe Ia
data. They prove its reliability and superiority in characterizing data
uncertainties.

In this article, we propose using the ANN and CC to obtain a reliable
calibration of the HIIGx Hubble diagram without any assumptions. In section 2,
we illustrate the principle of calibrating the HIIGx Hubble diagram. In
section 3, we describe the principle and methodology in the local measurement
of $H_{0}$, specifically about using the calibrated HIIGx Hubble diagram in
section 3.1 and the cross-calibrated SNe Hubble diagram in section 3.2. In
section 4, we present the data used in this article and present our results.
In section 5, we draw our conclusions and engage in some discussion.

##  2 Calibrating the HIIGx Hubble Diagram

The luminosity of Balmer $\mathrm{H}\beta$ lines that emitted in HIIGx is
strongly correlated with the ionized gas velocity dispersion $\sigma_{v}$
(Terlevich & Melnick, 1981), since both the intensity of ionizing radiation
and the value of $\sigma_{v}$ grow as the starbust mass increases (Siegel et
al., 2005). This correlation can be approximated by the expression in Chávez
et al. (2012),

$\lg\left[\frac{L(\mathrm{H}\beta)}{\mathrm{erg}\,\mathrm{s}^{-1}}\right]=\alpha\,\lg\left[\frac{\sigma_{v}(\mathrm{H}\beta)}{\mathrm{km}\,\mathrm{s}^{-1}}\right]+\kappa\
,$ (1)

where $\alpha$ and $\kappa$ are constant coefficients in this linear relation.
Due to the low dispersion of this relation observed between
$L(\mathrm{H}\beta)$ and $\sigma_{v}$, these galaxies and local HII regions
can be utilized as standard candles (Terlevich et al., 2015; Wei et al., 2016;
Yennapureddy & Melia, 2017; Leaf & Melia, 2018).

The selection criteria ensure that the selected HIIGx are comprised of systems
in which the luminosity is dominated by single and very young starbursts (less
than $5\,\mathrm{Myr}$ in age) (Terlevich et al., 2015). Accordingly, the
bolometric flux of the HIIGx can thus be regarded as primarily constituting
the $\mathrm{H}\beta$ line. Therefore the luminosity distance-$D_{L}(z)$ of an
HIIGx can be approximated using the luminosity $L(\mathrm{H}\beta)$ and flux
$F(\mathrm{H}\beta)$ associated to the $\mathrm{H}\beta$ line,

$D_{L}=\left[\frac{L(\mathrm{H}\beta)}{4\pi
F(\mathrm{H}\beta)}\right]^{1/2}\;,$ (2)

where $F(\mathrm{H}\beta)$ denotes the reddening corrected $\mathrm{H}\beta$
flux. By the definition of distance modulus

$\mu=5\lg D_{L}+25\ ,$ (3)

we can obtain

$\mu_{\mathrm{HII}}=2.5[\alpha\lg\sigma_{v}(\mathrm{H}\beta)+\kappa-\lg
F(\mathrm{H}\beta)]-100.2,$ (4)

while the error propagation is

$\sigma^{2}_{\mu_{\mathrm{HII}}}(\alpha,z_{i})=2.5^{2}[(\alpha\sigma_{\lg\sigma_{v}})^{2}+\sigma^{2}_{\lg
F}],$ (5)

where $\sigma_{\lg\sigma_{v}}$ and $\sigma_{\lg F}$ are the measurement error
${\lg\sigma_{v}}$ and $\lg F$ respectively.

If we can measure the $D_{L}$ at the same redshift as the observed HIIGx, we
are able to calibrate the slop ($\alpha$) and intercept ($\kappa$) of the
$L-\sigma$ relation. Locally, it is not difficult to anchor the HIIGx with
reliable distance modulus determined by other distance indicators within the
same galaxy, such as the period-luminosity relation of Cepheid. For a higher
redshift range that the first tier of the distance ladder is not applicable,
we might obtain the anchors by integrating the inverse of the Observation
$H(z)$ Data (OHD, also known as CC) across redshift,

$\frac{D_{L}}{(1+z)}=\left\\{\begin{array}[]{ll}\frac{D_{H}}{\sqrt{\Omega_{k0}}}\sinh\left[\sqrt{\Omega_{k0}}D_{C}/D_{H}\right]&\Omega_{k0}>0\\\
D_{C}&\Omega_{k0}=0\\\
\frac{D_{H}}{\sqrt{\left|\Omega_{k0}\right|}}\sin\left[\sqrt{\left|\Omega_{k0}\right|}D_{C}/D_{H}\right]&\Omega_{k0}<0\end{array}\right.$
(6)

where $\Omega_{k0}$ is the curvature energy desity parameter, the Hubble
radius $D_{H}=c/H_{0}$, and the comoving distance

$D_{C}=c\int_{0}^{z}\frac{dz^{\prime}}{H\left(z^{\prime}\right)},$ (7)

$c$ denotes the speed of light, and $H(z^{\prime})$ denotes the Hubble
parameter at redshift $z^{\prime}$. If we want to get the $D_{C}$ at any
redshift of interest, a reconstruction based on the OHD is a prerequisite.
Reconstructing $H(z)$ would complicate the covariance propagation, therefore
we choose the same method as Wu et al. (2020) that reconstruct $c/H(z)$. We do
the integration using

$D_{C}\simeq\sum\frac{c}{H(z_{i})}\Delta z_{i},$ (8)

which strictly holds if the redshift interval $\Delta
z_{i}=\frac{1}{2}(z_{i-1}-z_{i+1})$ is smaller enough. The covariance of the
reconstructed data will propagate to the $D_{C}$ in an approximate form of

$\operatorname{Cov}_{D_{C}}(z_{i},z_{j})=\sum_{l}^{i}\sum_{k}^{j}\operatorname{Cov}\big{(}\frac{c}{H_{k}},\frac{c}{H_{l}}\big{)}\Delta
z_{k}\Delta z_{l},$ (9)

which further propagate to the covariance of $\mu$ as

$\operatorname{Cov}_{\mu}(H_{0},\Omega_{k0};z_{i},z_{j})=\mathcal{C}_{i}\mathcal{C}_{j}\operatorname{Cov}_{D_{C}}(z_{i},z_{j}),$
(10)

where the coefficient $\mathcal{C}_{i}$ in error propagation is a function of
$\Omega_{k}$ and $H_{0}$,

$\mathcal{C}_{i}\frac{\ln
10}{5}=\left\\{\begin{array}[]{ll}\frac{\sqrt{\Omega_{k0}}}{D_{H}}\coth\left[\sqrt{\Omega_{k0}}D_{Ci}/D_{H}\right]&\Omega_{k0}>0\\\
\frac{1}{D_{C}}&\Omega_{k0}=0\\\
\frac{\sqrt{\left|\Omega_{k0}\right|}}{D_{H}}\cot\left[\sqrt{\left|\Omega_{k0}\right|}D_{Ci}/D_{H}\right]&\Omega_{k0}<0\end{array}\right..$
(11)

Figure 1: Left: reconstructions of $c/H(z)$ using Gaussian Process (blue) and
Artificial Neural Network (orange). The solid central lines represent the
reconstructed values, whereas the shaded regions represent the 1 $\sigma$
confidence intervals. The black error bars are the 32 OHD that are used as the
basis in our reconstructions. Right: the corresponding reconstructions of the
comoving distances, while 1 $\sigma$ regions are calculated based on the upper
and lower bounds of the $c/H(z)$ in the left panel.

The HIIGx Hubble diagram can be calibrated without any assumption of the
cosmological model or the geometry of the Universe, using the Bayesian
inference method by defining the log-likelihood function

$\displaystyle\ln\mathcal{L}=$
$\displaystyle-\frac{1}{2}\ln\Big{(}(2\pi)^{N}\det\boldsymbol{\operatorname{Cov}}\Big{)}-\frac{1}{2}\Delta\boldsymbol{\mu}^{\top}\boldsymbol{\operatorname{Cov}}^{-1}\Delta\boldsymbol{\mu},$
(12)

where the distance modulus difference vector is

$\Delta\boldsymbol{\mu}=\boldsymbol{\mu_{\text{HII}}}(\alpha,\kappa;\boldsymbol{z}_{\text{HII}})-\boldsymbol{\mu^{\text{rec}}_{\text{OHD}}}(H_{0},\Omega_{k0};\boldsymbol{z}_{\text{HII}}),$
(13)

and the corresponding total covariance matrix is

$\boldsymbol{\operatorname{Cov}}=\boldsymbol{\operatorname{Cov}^{\text{rec}}_{\mu_{\text{OHD}}}}+\operatorname{diag}(\sigma^{2}_{\mu_{\mathrm{HII}}}),$
(14)

where $\boldsymbol{\operatorname{Cov}^{\text{rec}}_{\mu_{\text{OHD}}}}$ is
covariance matrix of the reconstructed $\mu_{\text{OHD}}$ that described in
eq. 10, while the covariance matrix of $\mu_{\mathrm{HII}}$ is diagonal that
described in eq. 5.

##  3 Local Measurement the $H_{0}$

In cosmography, the $D_{L}(z)$, as a function of redshift, can be expanded
into the Taylor series as

$D_{L}(z)=\frac{cz}{H_{0}}\left[\mathcal{D}_{L}^{0}+\mathcal{D}_{L}^{1}z+\mathcal{D}_{L}^{2}z^{2}+\mathcal{O}(z^{3})\right]\;,$
(15)

where

$\displaystyle\mathcal{D}_{L}^{0}=1$ (16)
$\displaystyle\mathcal{D}_{L}^{1}=-\frac{1}{2}(1-q_{0})$
$\displaystyle\mathcal{D}_{L}^{2}=-\frac{1}{6}(1-q_{0}-3q_{0}^{2}+j_{0})\;.$

The $q_{0}$ and $j_{0}$ in above equation are the deceleration parameter the
jerk parameter respectively, which are the dimensionless second and third
derivative of the scale factor $a(t)$ with respect to cosmic time $t_{0}$. And
the first-order approximation at low redshifts in eq. 15 is the famous
Hubble’s law,

$D_{L}(z)=\frac{cz}{H_{0}}\ .$ (17)

It is independent of the mass-energy content of the Universe but only valid
locally ($z\lesssim 0.1$).

###  3.1 Measurement with the Calibrated HIIGx Hubble Diagram

Once the Hubble diagram of HIIGx had been properly calibrated, it could be
widely used to constrain cosmological model parameters. For instance, the
$H_{0}$ could be constrained directly using only the local HIIGx measurements,
or it could be constrained together with energy density parameters using the
entire sample and assuming a cosmological model. We will concentrate on the
direct measurement of $H_{0}$ in this article, which can be achieved by
performing a Markov Chain Monte Carlo (MCMC) to maximize the log-likelihood
function that described in eq. 12 with

$\Delta\boldsymbol{\mu}=\boldsymbol{\mu^{\text{local}}_{\text{HII}}}(\alpha,\kappa;\boldsymbol{z}^{\text{local}}_{\text{HII}})-\boldsymbol{\mu_{\text{th}}}(H_{0};\boldsymbol{z}^{\text{local}}_{\text{HII}}),$
(18)

where the theoretical $H_{0}$-dependent distance module
$\boldsymbol{\mu_{\text{th}}}(H_{0},\boldsymbol{z}^{\text{local}}_{\text{HII}})$
is determined by substituting eq. 17 into eq. 3 at local redshifts ($z<0.1$)
of HIIGx, and the corresponding covariance matrix is sub-matrix of the full
$\boldsymbol{\operatorname{Cov}^{\text{rec}}_{\mu_{\text{OHD}}}}$ ,

$\boldsymbol{\operatorname{Cov}}=\boldsymbol{\operatorname{Cov}^{\text{rec}}_{\mu_{\text{OHD}}}}(\boldsymbol{z}^{\text{local}}_{\text{HII}},\boldsymbol{z}^{\text{local}}_{\text{HII}}).$
(19)

To achieve a better constraint on the $H_{0}$ and the calibration of the HIIGx
Hubble diagram, we make a joint constraint by maximizing the summation of the
log-likelihood function

$\displaystyle\ln\mathcal{L}=$
$\displaystyle\ln\mathcal{L}(\boldsymbol{\mu_{\text{HII}}},\boldsymbol{\mu^{\text{rec}}_{\text{OHD}}}|H_{0},\Omega_{k0},\alpha,\kappa)$
(20)
$\displaystyle+\ln\mathcal{L}(\boldsymbol{\mu^{\text{local}}_{\text{HII}}}|H_{0},\alpha,\kappa)$

which are described in section 2 and here above.

Figure 2: 68% and 95% confidence contours for the parameters involved in
calibrating the HIIGx Hubble diagram with OHD. Results from the fit of not
assuming flat Universe are show in blue and orange contours for using the GP
and ANN reconstructed $\mu_{\mathrm{OHD}}$ data respectively, whereas the red
and violet contours are that from the assuming a flat Universe.

###  3.2 Measurement with the Cross-Calibrated SNe Hubble Diagram

SNe Ia is widely used as secondary standard candles to measure luminosity
distances $D_{L}$ because the peak luminosities of light curves of all SNe Ia
are nearly identical. By identifying an SNe Ia that shares a host galaxy with
a Cepheid variable, one can determine the host galaxy’s $D_{L}$ via its
period-luminosity relation. Principally, by combining the $D_{L}$ with the
peak apparent magnitude $m_{\max}$ of the SNe Ia, we can obtain the peak
absolute magnitude $M_{\max}$ of all the SNe Ia using

$\mu=5\lg\frac{D_{L}}{\mathrm{Mpc}}+25=m_{\max}-M_{\max}.$ (21)

Then the $D_{L}$ of an arbitrary SNe Ia can be easily obtained by measuring
its $m_{\text{max}}$.

However, the $M_{\max}$ of SNe Ia are not identical but also related to the
shapes can colors of the light curves. Taking this into account, the formula
for SNe Ia distance modulus should be modified by including perturbations of
shapes $x$ and color $c$ as shown in Guy et al. (2007),

$\mu_{B}(\alpha_{X},\beta,M_{B})=m_{B}^{\rm max}-M_{B}+\alpha_{X}X-\beta C,$
(22)

where the subscript $B$ stands for B band, while $\alpha$ and $\beta$ are
nuisance parameters for modification. It is well known that the $M_{B}$ is
strongly correlated with $H_{0}$. As a convention in SNe Ia cosmology, it is
safe to constrain parameters that are not very much dependent on $M_{B}$,
based on an assumed value of $H_{0}=70\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$.
However, if the objective is to precisely measure the $H_{0}$, we must first
obtain an accurate measurement of the $M_{B}$, which should also be modified
according to the stellar mass ($M_{\mathrm{stellar}}$), in a step function
form of

$M_{B}=\left\\{\begin{array}[]{ll}M_{B}^{1}&\text{ if }\ M_{\text{stellar
}}<10^{10}M_{\odot}\\\ M_{B}^{1}+\Delta_{M}&\text{ otherwise.
}\end{array}\right.,$ (23)

where the modified term $\Delta_{M}$ only takes effect for massive objects.
The eq. 22 can be written in matrix form of

$\boldsymbol{\mu}_{B}=\mathbf{A}\boldsymbol{\eta}-\boldsymbol{M}_{B},$ (24)

where the light-curve parameters vector is

$\boldsymbol{\eta}=\left(m_{B1},X_{1},C_{1},\cdots,m_{Bn},X_{n},C_{n}\right),$
(25)

and the coefficient matrix is

$\mathrm{A}=\mathrm{A}_{0}+\alpha\mathrm{A}_{1}-\beta\mathrm{A}_{2},$ (26)

whose elements can be decide according to

$\left(\mathbf{A}_{k}\right)_{i,j}=\delta_{3i,j+k}.$ (27)

The covariance matrix of the $\eta$ consists of statistical uncertainties of
light-curve fitting and seven sources of systematic uncertainties, including
the covariance of calibration, light curve model, bias correction, mass step
uncertainty, Milky Way dust extinction correction, peculiar velocity
corrections, contamination of Hubble diagram by non-Ia. For detail
descriptions please referred to Betoule et al. (2014) . The covariance further
propagate to $\boldsymbol{\mu_{B}}$ in the form of

$\displaystyle\operatorname{Cov}_{\boldsymbol{\mu_{B}}}=\mathbf{A}\operatorname{Cov}_{\eta}\mathbf{A}^{\top}$
$\displaystyle+\operatorname{diag}\left(\frac{5\sigma_{z}}{z\ln
10}\right)^{2}$ (28)
$\displaystyle+\operatorname{diag}\left(\sigma_{\text{lens
}}^{2}\right)+\operatorname{diag}\left(\sigma_{\mathrm{coh}}^{2}\right),$

taking account of the uncertainty in cosmological redshift due to peculiar
velocities ($\sigma_{z}$), the variation of magnitudes caused by gravitational
lensing ($\sigma_{\text{lens }}$), and the intrinsic variation in SN magnitude
not described by the other terms ($\sigma_{\mathrm{coh}}$).

Once we get the HIIGx Hubble diagram which is precisely calibrated using the
method described in section 2, we can take distance anchors from it to cross
calibrate the SNe Ia Hubble diagram. We define the log-likelihood function in
eq. 12 with the distance modulus difference vector in the form of

$\Delta\boldsymbol{\mu}=\boldsymbol{\mu_{B}}\left(\alpha_{x},\beta,M_{B}^{1},\Delta
M;\boldsymbol{z}_{\text{SNe}}\right)-{\boldsymbol{\mu}^{\text{rec}}_{\mathrm{HII}}}(\boldsymbol{z}_{\text{SNe}}),$
(29)

where
${\boldsymbol{\mu}^{\text{rec}}_{\mathrm{HII}}}(\boldsymbol{z}_{\text{SNe}})$
is reconstructed based on the calibrated
$\boldsymbol{\mu}^{\text{calib}}_{\mathrm{HII}}$ described in section 2, whose
error propagation is given by

$\displaystyle\begin{aligned}
\sigma^{2}_{\boldsymbol{\mu}^{\text{calib}}_{\mathrm{HII}}}=2.5^{2}[\alpha^{2}\sigma^{2}_{\lg\sigma}+\sigma_{\alpha}^{2}\lg^{2}\sigma+\sigma^{2}_{\kappa}+\sigma^{2}_{\lg
F}],\end{aligned}$ (30)

where $\sigma_{\alpha}$, $\sigma_{\kappa}$, are the calibration errors of
$\alpha$ and $\kappa$, while $\sigma_{\lg\sigma_{v}}$, $\sigma_{\lg F}$ are
measurement errors of $\lg\sigma(\rm H\beta)$ and $\lg F(\rm H\beta)$
respectively. The total covariance matrix corresponding to eq. 29 is

$\boldsymbol{\operatorname{Cov}}=\boldsymbol{\operatorname{Cov}}_{\boldsymbol{\mu}_{B}}+\boldsymbol{\operatorname{Cov}}^{\text{rec}}_{\boldsymbol{\mu}_{\mathrm{HII}}}.$
(31)

We take the same procedure in section 3.1 to determine the value of $H_{0}$
using this cross calibrated SNe Hubble diagram, therefore we define

$\Delta\boldsymbol{\mu}=\boldsymbol{\mu}^{\text{local}}_{B}\left(\alpha_{x},\beta,M_{B}^{1},\Delta
M;\boldsymbol{z}^{\text{local}}_{\text{SNe}}\right)-\boldsymbol{\mu_{\text{th}}}(H_{0};\boldsymbol{z}^{\text{local}}_{\text{SNe}})$
(32)

only using the local samples and get the covariance matrix

$\boldsymbol{\operatorname{Cov}}=\boldsymbol{\operatorname{Cov}^{\text{rec}}_{\mu_{\text{HII}}}}(\boldsymbol{z}^{\text{local}}_{\text{SNe}},\boldsymbol{z}^{\text{local}}_{\text{SNe}})$
(33)

correspondingly. We perform a MCMC to maximize the summation of the log-
likelihood function

$\displaystyle\ln\mathcal{L}=$
$\displaystyle\ln\mathcal{L}(\boldsymbol{\mu}_{B},\boldsymbol{\mu^{\text{rec}}_{\text{HII}}}|\alpha_{x},\beta,M_{B}^{1},\Delta
M)$ (34)
$\displaystyle+\ln\mathcal{L}(\boldsymbol{\mu_{\textit{B}}^{\text{local}}}|H_{0},\alpha_{x},\beta,M_{B}^{1},\Delta
M)$

and get the $H_{0}$ and calibrated SNe Hubble diagram simultaneously.

Figure 3: Hubble diagrams of HIIGx and JLA SNe Ia samples. Error-bars
represent the calibrated distance modulus of the HIIGx and JLA sample and
their corresponding 1 $\sigma$ uncertainties, labeled by the sample names and
calibration methods, with calib1 and calib2 stand for using Gaussian Process
(GP) and Artificial Neural Network (ANN) reconstructed data respectively. The
blue (orange) solid curves and shaded regions are the GP (ANN) reconstructed
$\mu^{\text{rec}}_{\mathrm{HII}}$ and their corresponding 1 $\sigma$
confidence regions at the redshift of JLA sample. Figure 4: 68% and 95%
confidence contours for the parameters involved in calibrating the SNe Hubble
diagram with calibrated HIIGx data. Blue (orange) contours result from the fit
of using the GP (ANN) reconstructed $\mu_{\mathrm{HII}}$ data and the full JLA
sample.

##  4 Data and Results

We applied the methods that are described in section 2 and 3 to the following
list of data in detail.

  1. (1)

OHD. We take the updated full sample of cosmological model-independent OHD
which are measured according the equation

$H(z)=-\frac{1}{1+z}\frac{\mathrm{d}z}{\mathrm{d}t}\simeq-\frac{1}{1+z}\frac{\Delta
z}{\Delta t}$ (35)

by measuring the time derivative of redshift $\mathrm{d}z/\mathrm{d}t$ without
assuming any cosmological model. The derivative is measured by taking the
differential age of two massive and passively evolving galaxies at different
redshifts as an approximation. A total number of 32 OHD (listed in table 1)
are given by Jimenez et al. (2003); Simon et al. (2005); Stern et al. (2010);
Moresco et al. (2012); Zhang et al. (2014); Moresco (2015); Moresco et al.
(2016); Ratsimbazafy et al. (2017); Borghi et al. (2022); Jiao et al. (2022).
Considering the last two measurements are not fully independent because they
are derived from the same sample of passive galaxies sample, and their
covariance is not clear yet, we decide only use the latest Jiao et al. (2022)
measurement taking advantage of $\sim 1/\sqrt{2}$ fraction of systematic
uncertainty that is improved by approximately double number galaxies were
used. One could either use the Borghi et al. (2022) measurement or both of
them if the covariance is cleverly considered.

  2. (2)

HIIGx. We use the full sample of HIIGx composed by González-Morán et al.
(2021), which includes 107 local HII galaxies (Chávez et al., 2014)
recalibrated by (González-Morán et al., 2019) in the redshift range of
$0.0088\leq z\leq 0.16417$, and 74 high$\text{-}z$ HII galaxies in the
redshift range of $0.63427\leq z\leq 2.545$. The fluxes and gas velocity
dispersions (along with their uncertainties) of HIIGx and GEHR that we use in
this article are all referred to from this catalogue.

  3. (3)

JLA. This SNe Ia catalog is composed by Betoule et al. (2014) from
observations obtained by the SDSS-II and SNLS collaborations. The data set
includes several low-redshift samples ($z<0.1$), all three seasons from the
SDSS-II ($0.05<z<0.4$), and three years from SNLS ($0.2<z<1$), and it totals
740 spectroscopically confirmed SNe Ia with high quality light curves. The
light curve parameters
$\left(m_{B1},X_{1},C_{1},\cdots,m_{Bn},X_{n},C_{n}\right)$ and the
corresponding covariance matrix are all obtained from this catalogue.

Table 1: All the Observational $H(z)$ Data derived from the Cosmic Chronomters
(CC) method

_z_ | $H(z)$ | $\sigma_{H}$ | References
---|---|---|---
0.09 | $69$ | $12$ | Jimenez et al. (2003)
0.17 | $83$ | $8$ |
0.27 | $77$ | $14$ |
0.4 | $95$ | $17$ |
0.9 | $117$ | $23$ | Simon et al. (2005)
1.3 | $168$ | $17$ |
1.43 | $177$ | $18$ |
1.53 | $140$ | $14$ |
1.75 | $202$ | $40$ |
0.48 | $97$ | $62$ | Stern et al. (2010)
0.88 | $90$ | $40$ |
0.1791 | $75$ | $4$ |
0.1993 | $75$ | $5$ |
0.3519 | $83$ | $14$ |
0.5929 | $104$ | $13$ | Moresco et al. (2012)
0.6797 | $92$ | $8$ |
0.7812 | $105$ | $12$ |
0.8754 | $125$ | $17$ |
1.037 | $154$ | $20$ |
0.07 | $69$ | $19.6$ |
0.12 | $68.6$ | $26.2$ | Zhang et al. (2014)
0.2 | $72.9$ | $29.6$ |
0.28 | $88.8$ | $36.6$ |
1.363 | $160$ | $33.6$ | Moresco (2015)
1.965 | $186.5$ | $50.4$ |
0.3802 | $83$ | $13.5$ |
0.4004 | $77$ | $10.2$ |
0.4247 | $87.1$ | $11.2$ | Moresco et al. (2016)
0.4497 | $92.8$ | $12.9$ |
0.4783 | $80.9$ | $9$ |
0.47 | $89$ | $34$ | Ratsimbazafy et al. (2017)
0.75 | $98.8$ | $33.6$ | Borghi et al. (2022)
0.80 | $113.1$ | $28.5$ | Jiao et al. (2022)

* $H(z)$ and $\sigma_{H}$ are in the unit of $\rm km\ s^{-1}\ Mpc^{-1}$

For the calibration of the HIIGx Hubble diagram, we reconstruct $D_{C}$ at the
redshifts of HIIGx sample based on the OHD, whereas for the calibration of the
JLA SNe Ia Hubble diagram we reconstruct ${\boldsymbol{\mu}_{\mathrm{HII}}}$
at redshifts of JLA sample based on the pre-calibrated HIIGx Hubble diagram.
We perform the following two methods to test the reliability of
reconstructions. GP reconstructs values that are connected in each pair of
data points by assuming a covariance function $k(z,\tilde{z})$. We use GaPP
(Seikel et al., 2012) and adopt the squared exponential kernal function

$k(z,\tilde{z})=\sigma_{f}^{2}\exp\left(-\frac{(z-\tilde{z})^{2}}{2\ell^{2}}\right),$
(36)

where the signal variance $\sigma_{f}$ and length scale $\ell$ are two
hyperparameters that should be optimized. The covariance matrix

$\boldsymbol{\operatorname{Cov}}=\boldsymbol{K}_{**}-\boldsymbol{K}_{o*}^{\top}\boldsymbol{K}_{oo}^{-1}\boldsymbol{K}_{o*},$
(37)

where the subscripts o and $*$ stand for calculating eq. 36 at the redshifts
of the observational data and the reconstructed data, with the first and
second position stand for the rows and columns respectively. ANN is an
alternative non-parametrical reconstruction method free of assumptions that
can improve reliability. We use ReFANN as our second tool, which is validated
in reconstructing OHD and $D_{L}(z)$ (Wang et al., 2020b). The covariance
matrix of can be estimated by the method described in section 6.2 of Wang et
al. (2020b),

$\begin{array}[]{l}\operatorname{Cov}\left(\frac{c}{H_{i}},\frac{c}{H_{j}}\right)=\frac{1}{N}\sum_{k=1}^{N}\left[\left(\frac{c}{H_{i}}_{k}-\bar{\frac{c}{H_{i}}}\right)\left(\frac{c}{H_{j}}_{k}-\bar{\frac{c}{H_{j}}}\right)\right],\end{array}$
(38)

which averaged over a large number of reconstructions (i.e., $N=1000$) based
on Gaussian re-sampling of the observational data. We reconstruct solely
within the intercept redshift interval shared by the basis data and the data
for comparison in our Bayesian analysis, considering that interpolation is
more trustworthy than extrapolation.

In Figure 1 we show the reconstructed $c/H(z)$ and $D_{C}(z)$ which are based
on the OHD. The GP and ANN behave different on $c/H(z)$ reconstruction at both
lower and higher end of the redshift range, which result to the difference of
the $D_{C}(z)$. Based on the reconstruction, we calibrate the HIIGx Hubble
diagram without assuming the flatness of the Universe and show the involved
parameters in Figure 2 and their summary in Table 2. We observe inverse
correlation ($r\sim-0.49$) between $H_{0}$ and $\Omega_{k}$, which is known as
the geometry degeneracy. We also observe strong inverse correlation
($r\sim-0.98$) between $\alpha$ and $\kappa$. The correlation between the
cosmic parameters and the calibration coefficients should not be overlooked,
especially for the correlation between $\Omega_{k0}$ and $\alpha$ that shows
correlation coefficient $r\sim 0.36$. As a comparison result, we also show the
calibration coefficients from assuming a flat Universe and find 1.07 $\sigma$
and 1.14 $\sigma$ difference between the two set of $\alpha$ and $\kappa$
respectively.

We show the OHD calibrated HIIGx Hubble diagrams and their reconstructions in
Figure 3, companion with the calibrated JLA SNe Ia Hubble diagrams based on
the reconstructed data. The calibration parameters and the jointly constrained
$H_{0}$ are shown in Figure 4 and summarised in Table 2. We observe strong
correlation ($r\sim 0.92$) between $H_{0}$ and $M^{1}_{B}$.

Table 2: Calibration Parameters of Hubble Diagrams. HIIGx | $H_{0}$ | $\Omega_{k0}$ | $\alpha$ | $\kappa$ |
---|---|---|---|---|---
OHD+GP | $67.0\pm 2.4$ | $-0.04^{+0.29}_{-0.24}$ | $4.877^{+0.103}_{-0.098}$ | $33.51\pm 0.16$ |
OHD+ANN | $64.4^{+3.9}_{-3.4}$ | $0.17^{+0.42}_{-0.37}$ | $4.861^{+0.101}_{-0.099}$ | $33.57\pm 0.17$ |
SNe | $H_{0}$ | $\alpha_{x}$ | $\beta$ | $M_{B}^{1}$ | $\Delta M$
HII+GP | $68.7^{+2.6}_{-2.5}$ | $0.1419\pm 0.0058$ | $3.071^{+0.074}_{-0.070}$ | $-18.987^{+0.073}_{-0.074}$ | $-0.071\pm 0.021$
HII+ANN | $65.9^{+3.0}_{-2.9}$ | $0.1422\pm 0.0058$ | $3.068\pm 0.073$ | $-19.078^{+0.094}_{-0.093}$ | $-0.071\pm 0.021$

##  5 Conclusion and Discussion

Cosmic Chronometers (also known as OHD) can be used as distance anchors to
constrain the $L-\sigma$ relation coefficients of the HII Galaxies, therefore
provide us an alternative cosmological-model independent method to calibrate
the HIIGx Hubble diagram. We promote this method by removing the flat Universe
assumption and observe weak correlations between the cosmic parameters
($H_{0}$ and $\Omega_{k0}$) and the coefficients ($\alpha$ and $\kappa$). We
observe approximately 1 $\sigma$ difference between the calibrations assuming
or not the flatness of the Universe. This indicates the necessity of
assumption-free for improving the reliability of the calibration.

We implement a non-parametric Artificial Neural Network method for
reconstructing data in the intermediate process of calibration and obtain
$\alpha=4.861_{-0.099}^{+0.101}$ and $\kappa=33.57\pm 0.17$ with its
reliability affirmed by highly compatible Gaussian Process involved result. We
obtain Hubble constant with a value of
$H_{0}=64.4_{-3.4}^{+3.9}~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, which is lower
but still compatible with the Planck18 measurement ($H_{0}=67.4\pm
0.5~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, Planck Collaboration et al. (2020)).

We calibrate the JLA SNe Ia Hubble diagram using the calibrated HIIGx Hubble
diagram as an anchor, making it possible to be free of the conventional
assumption of $H_{0}=70~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$. The newly
calibrated JLA Hubble diagram can be used to constrain any cosmological
parameters including those degenerated with the $H_{0}$. We obtain a jointly
constrained value of
$H_{0}=65.9_{-2.9}^{+3.0}~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, which is more
compatible with the Planck18 measurement than what we get from the HIIGx
Hubble diagram.

In conclusion, our method further improves the reliability of the HIIGx Hubble
diagram calibrated with OHD by removing the assumption of cosmological
flatness in the method of Wu et al. (2020) and the assumption of kernel
functions in the reconstruction. This method can be used to obtain
cosmological-model independent calibrations of other Hubble diagrams.

We appreciate the referee’s several useful suggestions, as well as Guo-Jian
Wang’s kindly instruction about the ReFANN. We also thank Yu-Chen Wang, Yun-
Long Li and Cheng-Zong Ruan for valuable discussions. This work is supported
by the National Science Foundation of China(Grants Nos. 61802428, 11929301).
Kang Jiao is also funded by the China Scholarship Council (CSC) from the
Ministry of Education of P.R. China.

## References

  * Betoule et al. (2014) Betoule, M. e. a., Kessler, R., Guy, J., et al. 2014, A&A, 568, A22, doi: 10.1051/0004-6361/201423413
  * Borghi et al. (2022) Borghi, N., Moresco, M., & Cimatti, A. 2022, ApJL, 928, L4, doi: 10.3847/2041-8213/ac3fb2
  * Chávez et al. (2012) Chávez, R., Terlevich, E., Terlevich, R., et al. 2012, MNRASL, 425, L56, doi: 10.1111/j.1745-3933.2012.01299.x
  * Chávez et al. (2014) Chávez, R., Terlevich, R., Terlevich, E., et al. 2014, MNRAS, 442, 3565, doi: 10.1093/mnras/stu987
  * Demianski, Marek et al. (2017) Demianski, Marek, Piedipalumbo, Ester, Sawant, Disha, & Amati, Lorenzo. 2017, A&A, 598, A112, doi: 10.1051/0004-6361/201628909
  * Fernández Arenas et al. (2017) Fernández Arenas, D., Terlevich, E., Terlevich, R., et al. 2017, MNRAS, 474, 1250, doi: 10.1093/mnras/stx2710
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * González-Morán et al. (2019) González-Morán, A. L., Chávez, R., Terlevich, R., et al. 2019, MNRAS, 487, 4669, doi: 10.1093/mnras/stz1577
  * González-Morán et al. (2021) González-Morán, A. L., Chávez, R., Terlevich, E., et al. 2021, MNRAS, 505, 1441, doi: 10.1093/mnras/stab1385
  * Guy et al. (2007) Guy, J., Astier, P., Baumont, S., et al. 2007, A&A, 466, 11, doi: 10.1051/0004-6361:20066930
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hinton (2016) Hinton, S. R. 2016, The Journal of Open Source Software, 1, 00045, doi: 10.21105/joss.00045
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Jiao et al. (2022) Jiao, K., Borghi, N., Moresco, M., & Zhang, T.-J. 2022, arXiv e-prints, arXiv:2205.05701.  https://arxiv.org/abs/2205.05701
  * Jimenez et al. (2003) Jimenez, R., Verde, L., Treu, T., & Stern, D. 2003, ApJ, 593, 622, doi: 10.1086/376595
  * Kumar & Linder (2019) Kumar, P., & Linder, E. V. 2019, PRD, 100, 083533, doi: 10.1103/PhysRevD.100.083533
  * Leaf & Melia (2018) Leaf, K., & Melia, F. 2018, MNRAS, 474, 4507, doi: 10.1093/mnras/stx3109
  * Lusso, E. & Risaliti, G. (2017) Lusso, E., & Risaliti, G. 2017, A&A, 602, A79, doi: 10.1051/0004-6361/201630079
  * Lusso, E. et al. (2020) Lusso, E., Risaliti, G., Nardini, E., et al. 2020, A&A, 642, A150, doi: 10.1051/0004-6361/202038899
  * Melnick, J. et al. (2017) Melnick, J., Telles, E., Bordalo, V., et al. 2017, A&A, 599, A76, doi: 10.1051/0004-6361/201629728
  * Mishra et al. (2019) Mishra, A., Reddy, P., & Nigam, R. 2019, CMB-GAN: Fast Simulations of Cosmic Microwave background anisotropy maps using Deep Learning
  * Moresco (2015) Moresco, M. 2015, MNRAS, 450, L16, doi: 10.1093/mnrasl/slv037
  * Moresco et al. (2012) Moresco, M., Verde, L., Pozzetti, L., Jimenez, R., & Cimatti, A. 2012, JCAP, 7, 053, doi: 10.1088/1475-7516/2012/07/053
  * Moresco et al. (2016) Moresco, M., Pozzetti, L., Cimatti, A., et al. 2016, JCAP, 5, 014, doi: 10.1088/1475-7516/2016/05/014
  * Mustafa et al. (2019) Mustafa, M., Bard, D., Bhimji, W., et al. 2019, CompAC, 6, 1, doi: 10.1186/s40668-019-0029-9
  * Peel et al. (2019) Peel, A., Lalande, F., Starck, J.-L., et al. 2019, PRD, 100, 023508, doi: 10.1103/PhysRevD.100.023508
  * Perlmutter et al. (1999) Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565, doi: 10.1086/307221
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910
  * Ratsimbazafy et al. (2017) Ratsimbazafy, A. L., Loubser, S. I., Crawford, S. M., et al. 2017, MNRAS, 467, 3239, doi: 10.1093/mnras/stx301
  * Riess et al. (1998) Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009, doi: 10.1086/300499
  * Salvestrini, F. et al. (2019) Salvestrini, F., Risaliti, G., Bisogni, S., Lusso, E., & Vignali, C. 2019, A&A, 631, A120, doi: 10.1051/0004-6361/201935491
  * Schmelzle et al. (2017) Schmelzle, J., Lucchi, A., Kacprzak, T., et al. 2017, Cosmological model discrimination with Deep Learning
  * Seikel et al. (2012) Seikel, M., Clarkson, C., & Smith, M. 2012, JCAP, 2012, 036, doi: 10.1088/1475-7516/2012/06/036
  * Siegel et al. (2005) Siegel, E. R., Guzmán, R., Gallego, J. P., López, M. O., & Hidalgo, P. R. 2005, MNRAS, 356, 1117, doi: 10.1111/j.1365-2966.2004.08539.x
  * Simon et al. (2005) Simon, J., Verde, L., & Jimenez, R. 2005, PRD, 71, 123001, doi: 10.1103/PhysRevD.71.123001
  * Stern et al. (2010) Stern, D., Jimenez, R., Verde, L., Kamionkowski, M., & Stanford, S. A. 2010, JCAP, 2, 008, doi: 10.1088/1475-7516/2010/02/008
  * Terlevich & Melnick (1981) Terlevich, R., & Melnick, J. 1981, MNRAS, 195, 839, doi: 10.1093/mnras/195.4.839
  * Terlevich et al. (2015) Terlevich, R., Terlevich, E., Melnick, J., et al. 2015, MNRAS, 451, 3001, doi: 10.1093/mnras/stv1128
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wang et al. (2020a) Wang, G.-J., Li, S.-Y., & Xia, J.-Q. 2020a, ApJS, 249, 25, doi: 10.3847/1538-4365/aba190
  * Wang et al. (2020b) Wang, G.-J., Ma, X.-J., Li, S.-Y., & Xia, J.-Q. 2020b, ApJS, 246, 13, doi: 10.3847/1538-4365/ab620b
  * Wang et al. (2021) Wang, Y.-C., Xie, Y.-B., Zhang, T.-J., et al. 2021, ApJS, 254, 43, doi: 10.3847/1538-4365/abf8aa
  * Wei et al. (2016) Wei, J.-J., Wu, X.-F., & Melia, F. 2016, MNRAS, 463, 1144, doi: 10.1093/mnras/stw2057
  * Wu et al. (2020) Wu, Y., Cao, S., Zhang, J., et al. 2020, ApJ, 888, 113, doi: 10.3847/1538-4357/ab5b94
  * Yennapureddy & Melia (2017) Yennapureddy, M. K., & Melia, F. 2017, JCAP, 2017, 029, doi: 10.1088/1475-7516/2017/11/029
  * Zhang et al. (2014) Zhang, C., Zhang, H., Yuan, S., et al. 2014, RAA, 14, 1221, doi: 10.1088/1674-4527/14/10/002