# Prospects of Detecting the Global HI-21cm Signal at uGMRT through the
Gravitational Lensing by an isolated Neutron Star

Rupa Basu  Siddhartha Bhattacharyya  National Centre for Radio Astrophysics,
Pune, India, Pin-411007  Anjan Kumar Sarkar  National Centre for Radio
Astrophysics, Pune, India, Pin-411007  Shibaji Banerjee  Department of
Physics, St. Xavier’s College, Kolkata, India, Pin-700016  Debasis Majumder
Department of Physics, Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur Math, Howrah, India, Pin-711202

###### Abstract

The strength of the global HI-21cm signal is several orders of magnitude lower
than the foreground and background noise and hence it is difficult to observe
this signal at a given radio telescope. However, a few recent studies reported
the detection of that signal at the radio band suggests the strength of this
signal is somehow magnified. In this analysis, we study the prospects of
detecting this global signal at different frequency bands of uGMRT where this
global signal is supposed to be amplified through the strong gravitational
lensing by an isolated neutron star located in a cosmological distance. Our
study shows the effects of the lensing parameters on the observables of that
amplified global signal and discusses its variation with the frequency bands
considered here. We present a method to estimate the position and size of an
isolated neutron star using the signal-to-noise ratio of that global signal
supposed to be detected at different frequency bands of uGMRT. We discuss the
scope of multi-messenger astronomy in the era of HI-21cm observation where the
estimated lensing parameters can be cross-validated using the pulsar detection
at the X-ray band from the same location in the sky. Our analysis is equally
applicable to any radio telescope with given specifications.

##  1 Introduction

The redshifted HI-21cm signal is an important tool in modern cosmology. It is
believed that in the post-reionization era ($z<6$), the HI-21cm signal
originates from the self-shielded dense pockets of neutral hydrogen. From
observing the damped Lyman-alpha absorption lines, it has been noticed that
these systems contain the bulk of neutral hydrogen. However, the HI-21cm
observation does not need to resolve the individual neutral hydrogen sources
because in low-frequency observations (below $1420\,{\rm MHz}$) emission from
the individual hydrogen clouds is very faint and diffuse background radiation
where the source clustering is imprinted on the fluctuation of the background
emission. In this work, we consider the ”global” HI-21cm signal which is the
spatially averaged HI-21cm signal, which gives us the essential information
about the mean properties of the IGM for a given redshift $z$. In general, the
brightness temperature of the global HI-21cm signal (e.g. Zaldarriaga et al.
2004) depends on the cosmic microwave background (CMB) temperature, neutral
hydrogen fraction in the IGM and the spin temperature of neutral hydrogen
which quantifies the relative abundance of hydrogen atoms in low and high
energy states. There are many theoretical models for the evolution of the IGM
and the global HI-21cm signal. However, the amplitude, duration and timing of
this signal differ for the different proposed models since the exact
properties of stars and their evolution are still unknown. The models proposed
to date can be roughly divided into two groups; (1) the global HI-21cm line in
emission where reionization occurs after the sufficient heating of the IGM
such that its temperature is well above the CMB temperature, and (2) the
global HI-21cm line in absorption where reionization occurs before the
substantial heating of the IGM. However, in both cases, the model-based
stellar and galactic properties have been extrapolated from the existing
observations which are mostly low-redshift observations.

There are several upcoming experiments which are planned to detect the global
HI-21cm signal a few of them are, the Shaped Antenna measurement of the
background RAdio Spectrum (SARAS) (e.g. Singh et al. 2017), the Broadband
Instrument for Global Hydrogen Reionisation Signal (BIGHORNS) (e.g. Sokolowski
et al. 2015), the Large-Aperture Experiment to Detect the Dark Ages (LEDA)
(e.g. Price et al. 2018), etc. Apart from the earth-based telescopes, there is
a proposal to go to the far side of moon, Dark Ages Radio Explorer (DARE)
(e.g. Burns et al. 2017) to avoid the man-made radio frequency interference
(RFI) and ionospheric effects on the ground-based observations. In recent
Bowman et al. (2018) reported the detection of a deep absorption trough in the
global HI-21cm signal using the Experiment to Detect the Global Epoch of
reionization Signature (EDGES). The reported amplitude of this absorption deep
is $\sim 500\,{\rm mK}$ which is roughly $2.5$ times larger than the
theoretical estimated amplitude of $150-200\,{\rm mK}$. This detection will
give us some new inputs in the physical mechanism of the evolution of the
universe, however, there are a few controversies regarding this detection and
the reader is referred to Singh et al. (2022) for more details.

The upgraded Giant Metrewave Radio Telescope (uGMRT) (e.g. Gupta et al. 2017)
is an excellent instrument to detect the global HI-21cm signal. The uGMRT
contains a total of $30$ parabolic dishes of $45\,{\rm m}$ diameter each
distributed in a ”Y” shaped pattern with the longest baseline of $25\,{\rm
km}$. The uGMRT is operating in four frequency bands; Band-2 $(125-250\,{\rm
MHz})$, Band-3 $(250-500\,{\rm MHz})$, Band-4 $(550-850\,{\rm MHz})$ and
Band-5 $(1060-1460\,{\rm MHz})$. In this paper, we denote Band-2, Band-3,
Band-4 and Band-5 of the uGMRT as B2, B3, B4 and B5 respectively. We note that
the information of observational frequency is reflected in the redshift of the
neutral hydrogen source. Each frequency band has a system noise which depends
on the system temperature, antenna gain, frequency bandwidth, integration
time, etc. On top of that, the foreground radio sources of the global HI-21cm
signal create additional noise which is a few orders of magnitude higher than
the signal. In general, extragalactic point sources, galactic synchrotron
emission, and galactic and extra-galactic free-free emissions are considered
the continuum foreground sources (e.g. Ghosh et al. 2011a; Di Matteo et al.
2002; Ali et al. 2008). However, to tackle these foreground sources, various
techniques have already been discussed in the literature (e.g. Ghosh et al.
2011b; Datta et al. 2010; Wang et al. 2006; Morales et al. 2006; McQuinn et
al. 2006). The removal of these foreground sources and detection of the faint
global HI-21cm signal is very challenging for radio astronomers. Recently,
Chakraborty, Roy (2023) reported the detection of the HI-21cm signal from a
neutral hydrogen source in a galaxy at redshift $z\sim 1.3$ using B4. This
observation suggests that the amplitude of the signal is somehow magnified for
which the signal strength is greater than the foreground noise as well as the
system noise of the telescope and the HI-21cm signal was able to detect. There
are many ways to magnify the strength of the weak HI-21cm signal and the
strong gravitational lensing is one of them. Chakraborty, Roy (2023) also
mentioned the same and quoted the nature and the redshift of the lensing
medium without giving any further details.

In the current analysis, we present a method to quantify the size and position
of the lensing object using the lensed global HI-21cm signal which is supposed
to be detected at different frequency bands of uGMRT. Many astrophysical
massive objects such as MACHOs (e.g. Banerjee et al. 2003), black holes
(Hawking, Israel 1989) can lens the global HI-21cm signal, however here we
consider the neutron star as the probable lensing object for that purpose. Our
proposed method is equally applicable to any radio telescope with a given
specification. A brief outline of the paper is as follows: Section 2 presents
how the lensed global HI-21cm signal can be used to estimate the size and
position of the lensing object. Section 3 discusses the results of this
analysis and finally, we summarize and conclude our findings in Section 4.

##  2 Formalism

In this section, we discuss the model of the strong gravitational lensing by
which a neutron star (as the lensing object) can amplify the global $\rm
HI-21cm$ signal such that it can be detected at different frequency bands of
uGMRT. This section is divided into several sub-sections where we discuss (1)
the flux density of the global $\rm HI-21cm$ signal, (2) the telescope and
background noise which suppress that global signal, (3) the gravitational
lensing model considered here for the amplification of that global signal such
that it can cross noise level and be detected at uGMRT, (4) the variation of
the mass of a neutron star with its radius which gives a major impact on the
amplification of the global $\rm HI-21cm$ signal, (5) the optical depth which
signifies the probability of detecting that global signal, and (6) the chance
coincidence which is an essential scenario for the amplification of the global
$\rm HI-21cm$ signal supposed to be detected at uGMRT.

###  2.1 Flux Density

The brightness of any radio signal is generally quantified by the average flux
density $\overline{S}_{\nu}=\int\overline{I}_{\nu}\,d\Omega$, where $d\Omega$
is the solid angle subtended by the source to the telescope and
$\overline{I}_{\nu}$ is the average specific intensity of the source. The
solid angle $d\Omega=A(\theta)\sin\theta\,d\theta\,d\phi$ depends on the
antenna beam pattern $A(\theta)$. For the uGMRT, the antenna beam pattern is
approximated as the Gaussian function
$A(\theta)=\exp(-\theta^{2}/\theta_{0}^{2})$, where
$\theta_{0}=0.6\,\theta_{FWHM}$ and $\theta_{FWHM}$ is the angular extent of
the telescope beam pattern for which the power drops by $50\%$ of its maximum
value. For a neutral hydrogen source located at a redshift $z$, the spatially
averaged specific intensity of the global HI-21cm signal can be estimated
using the model given by Bharadwaj, Saiyad Ali 2005, is given by

$\overline{I}_{\nu}(z)=\frac{2.5\times
10^{2}}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}\left(\frac{\Omega_{b}h^{2}}{0.02}\right)\left(\frac{0.7}{h}\right)\quad{\rm
Jy\,rad^{-2}}$ (1)

where, $\Omega_{b}$, $\Omega_{m}$ and $\Omega_{\Lambda}$ are the density
parameters for baryon, dark matter and dark energy respectively, and $h$ is
the Hubble parameter. We estimate the average flux density
$\overline{S}_{\nu}$ for the future detected global HI-21cm signal at
different frequency bands of the uGMRT using equation 1. The redshift of the
neutral hydrogen source is governed by the observational frequency $\nu_{obs}$
of the telescope using the relation $z=\nu_{HI}/\nu_{obs}-1$, where
$\nu_{HI}=1420\,{\rm MHz}$ is the emission frequency of the HI-21cm signal.
Each frequency band of the uGMRT correspond to a redshift range of the neutral
hydrogen source, i.e. $4.68\leq z\leq 10.83$ for B2, $1.84\leq z\leq 4.46$ for
B3, $0.67\leq z\leq 1.58$ for B4 and $0\leq z\leq 0.34$ for B5 respectively.
However, here we define a redshift called the characteristic redshift $z_{c}$
which is governed by the central frequency ($\nu_{0}$) of each frequency band
of the uGMRT to keep the analysis straightforward. We note that
$z_{c}=6.18,\,2.73,\,1.03\,{\rm and}\,0.13$ for B2, B3, B4 and B5
respectively. In the present analysis, we consider the scenario where the
global HI-21cm signal originates from the self-shielded dense pockets of
neutral hydrogen, which can be treated as the point source of neutral
hydrogen. It is believed that this scenario happened in the post-reionization
era of the evolving universe for which the redshift of neutral hydrogen source
should be $z<6$. Considering the characteristic redshift $z_{c}$ of each
frequency band of the uGMRT, we simply restrict our analysis to B3, B4 and B5
only for which the value of average flux density is
$\overline{S}_{\nu}(z_{c})=0.38$ mJy, $0.84$ mJy and $1.40$ mJy respectively.
We consider the $\Lambda CDM$ cosmology (Planck Collaboration et al., 2020) to
estimate the values of $\overline{S}_{\nu}(z_{c})$.

###  2.2 Telescope and Background Noise

The faint Global HI-21cm signal is attenuated by the huge telescope noise
which is mainly dominated by the foreground and background noise of
extragalactic point sources, galactic synchrotron emission, galactic and
extra-galactic free-free emissions, etc (e.g. Ghosh et al. 2011b, Di Matteo et
al. 2002, Ali et al. 2008). The noise of any radio telescope with given
parameters can be estimated using the radiometer equation (Lorimer, Kramer,
2005)

$\Delta
S=\frac{T_{sys}+T_{BG}}{N_{A}\,G\,\sqrt{(n_{P}\,T_{samp}\,\Delta\nu)}}$ (2)

where, $n_{P}$, $N_{A}$ and $T_{samp}$ are the number of polarization, number
of antennae and sampling time for an observation, which are frequency band
independent parameters, i.e. the value of $N_{A}=14$, $n_{p}=2$ and
$T_{samp}=0.671$ sec are same for B3, B4 and B5 respectively. The antenna gain
$G$ varies across the frequency range of each frequency band of uGMRT.
However, in this analysis, we consider the minimum value of $G$ in each
frequency band to maximise the telescope noise, where $G=0.38$ K Jy-1, $0.35$
K Jy-1 and $0.22$ K Jy-1 for B3, B4 and B5 respectively (Gupta et al., 2017).
The system temperature $T_{sys}$ is also changing across the frequency range
of each uGMRT band. However, we consider the maximum value of $T_{sys}$ in
each frequency band to maximise the telescope noise, where we consider
$T_{sys}=165$ K, $100$ K and $75$ K for B3, B4 and B5 respectively (Gupta et
al., 2017). The frequency bandwidth $\Delta\nu$, which is often called the
usable bandwidth, is also different for the different frequency bands of
uGMRT, where $\Delta\nu=120$ MHz, $200$ MHz and $280$ MHz for B3, B4 and B5
respectively (Gupta et al., 2017).

The noise temperature ($T_{BG}$) due to foreground and background
astrophysical sources also makes a significant contribution to the telescope
noise estimated using eq. 2. The value of $T_{BG}$ varies with $\nu_{obs}$,
where it is believed that $T_{BG}$ decreases with increasing $\nu_{obs}$ of
the telescope. Bowman et al. 2018 mentioned that the value of sky temperature
for the foreground and background noise of the global HI-21cm signal is $\sim
6000$ K at the observational frequency of $78\pm 1$ MHz using EDGES telescope,
which will be even smaller if we consider B3, B4 and B5 as the observing band
to detect that signal. However, in this analysis, we consider a typical large
value of $T_{BG}=10^{4}$ K same for all B3, B4 and B5 to maximise the
telescope noise. Considering the value of noise parameters (eq. 2) discussed
above, the value of telescope noise $\Delta S=150.56$ mJy, $125.82$ mJy and
$118.75$ mJy for B3, B4 and B5 respectively which are roughly $400$, $150$ and
$85$ times larger than the flux density of the global HI-21cm signal. For any
radio astronomical observation, we generally define a quantity called the
signal-to-noise ratio, $(SNR)_{\nu}$, which determines how much the strength
of the signal is greater than the telescope noise, which is defined as
$(SNR)_{\nu}=\overline{S}_{\nu}/\Delta S$. The values of $(SNR)_{\nu}$ for the
non-amplified global HI-21cm signal are $2.5\times 10^{-3}$, $6.6\times
10^{-3}$ and $1.2\times 10^{-2}$ at B3, B4 and B5 respectively. The estimated
values of $(SNR)_{\nu}$ mentioned above suggest that it is challenging to
detect the global HI-21cm signal at different frequency bands of the uGMRT
without considering any signal amplification. However, recently Chakraborty,
Roy 2023 reported the $5\sigma$ (i.e. $(SNR)_{\nu}=5$) detection of HI-21cm
signal from a star-forming galaxy at redshift of $z\sim 1.3$ using B4. They
also mentioned that the HI-21cm signal is boosted by the strong gravitational
lens of an early-type elliptical galaxy at redshift $z\sim 0.13$. This
observation (Chakraborty, Roy, 2023) motivates us to think about whether the
global HI-21cm signal can also be magnified by the strong gravitational
lensing of any compact object along the line-of-sight of the source. In the
present analysis, we consider the global HI-21cm signal to be amplified by the
neutron star for which the signal has to be magnified by a factor $\sim
10^{4}$ to detect that signal with $(SNR)_{\nu}=10$.

###  2.3 Gravitational Lensing Model

Figure 1: A toy diagram for the gravitational lensing of the global $\rm
HI-21cm$ signal by an isolated neutron star located in a cosmological
distance.

Figure 1 shows a diagram that exhibits the lensing of the global HI-21cm
signal by a compact object which is considered to be a neutron star for this
analysis. The source ”S” emits the global HI-21cm signal which initially
propagates through the path ”SP” and is then lensed at the point ”P” due to
the presence of the lensing object ”L”. The observer ”O” receives the signal
along the path ”PO” and the image of ”S” is formed at ”I”. The line ”ON”
joining the observer ”O” and compact object ”L” is considered the line-of-
sight (LOS) of the observation. The angular position of source ”S” and image
”I” concerning ”ON” are $\beta$ and $\theta$ respectively. It is also assumed
that both the image and source lie on the same plane which is perpendicular to
”ON”. The amount of bending in the signal path is generally quantified by the
parameter $\xi$ called the impact parameter which determines the closest
distance of the signal path to the lensing object and $\alpha$ is the angle of
bending concerning ”PS”.

Both source and lensing objects are located at cosmological distances and we
associate the angular diameter distance with them instead of their physical
distance. For a cosmological source located at redshift $z$, the angular
diameter distance
$D(z)=(1+z)^{-1}\,\int_{0}^{z}dz^{\prime}\,(c/H(z^{\prime}))$ where $H(z)$ is
the Hubble parameter and $c$ is the speed of light in vacuum. Referring to
figure 1, we consider the angular diameter distance of ”L” from ”O” and ”S”
from ”O” are $D_{L}$ and $D_{S}$ respectively. The angular diameter distance
of ”S” from ”L” is $D_{LS}$, where $D_{LS}\neq D_{S}-D_{L}$ since both ”S” and
”L” are located at a cosmological distance. The distance of the image and the
source from ”ON”, where ”ON” is perpendicular to the plane of ”I” and ”S”, is
$\beta D_{S}$ and $\theta D_{S}$ respectively whereas the distance between ”I”
and ”S” is $\alpha D_{LS}$. We introduce a parameter $\gamma$, defined as
$D_{L}=\gamma D_{S}$, which determines the position of the lensing object,
where $\gamma\to 0$ implies the lensing object is closed to the observer and
$\gamma\to 1$ implies the lensing object is closed to the source. We use the
relation $D_{L}=\gamma D_{S}$ to determine the redshift of the lensing object
$z_{L}$ as a function of both $\gamma$ and the redshift of the source
($z_{S}$), i.e. for a fixed value of $\gamma$ and $z_{S}$, the value of
$z_{L}$ is constant. This implies that both $D_{LS}$ and $D_{L}$ are the
functions of $\gamma$ and $z_{S}$ whereas $D_{S}$ is the function of $z_{S}$
only.

Now we discuss the angular position ($\theta$) of the image ”I” (figure 1)
which is formed due to gravitational lensing of the signal from ”S” (figure 1)
by a massive compact object ”L” (figure 1), where the value of $\theta$ can be
estimated using the relation

$\theta_{\pm}=\frac{\beta}{2}\left[1\pm\sqrt{1+\left(\frac{2\theta_{E}}{\beta}\right)^{2}}\right]$
(3)

where, $\beta$ is the angular position of the source and $\theta_{E}$ is
called the Einstein’s angle. From eq. 3, it is clear that two images (of the
source) are formed with angular positions $\theta_{+}$ and $\theta_{-}$
concerning ”ON” (figure 1) respectively. The image with angular position
$\theta_{+}$ is formed outside of Einstein’s circle centred at ”S” (figure 1)
with angular radius $\theta_{E}$ whereas the other image with angular position
$\theta_{-}$ is formed inside of Einstein’s circle. However, both images do
not have the same intensity where the image with angular position $\theta_{+}$
has more intensity in comparison to the other image. This has been discussed
later in this paper. Einstein’s angle $\theta_{E}$ strongly depends on the
mass of the lensing object, and the angular diameter distance of the source
and lensing object from the observer, which is given by

$\theta_{E}=\left[\frac{4GM_{L}(R)}{c^{2}}\left(\frac{D_{LS}(\gamma,z_{S})}{D_{L}(\gamma,z_{S})\,\,D_{S}(z_{S})}\right)\right]^{1/2}$
(4)

where, $G$ is the universal gravitational constant, $c$ is the speed of light
in a vacuum and $M_{L}(R)$ is the total mass of the lensing object depending
on the radius of the object which is discussed in detail in the later part of
this paper. The parameters $D_{LS}$, $D_{L}$ and $D_{S}$ have already been
discussed earlier. Eq. 4 implies that the Einstein’s angle $\theta_{E}$
depends on three parameters $R$, $\gamma$ and $z_{S}$.

We have discussed earlier that the strong gravitational lensing magnifies the
intensity of the global $\rm HI-21cm$ signal due to the massive compact object
”L” (figure 1) where we scale the amplified intensity of that signal through
the relation $\overline{I}_{\mu}=\mu\times\overline{I}_{\nu}$ and the value of
$\overline{I}_{\nu}$ can be estimated using the eq. 1. The parameter $\mu$ is
called the magnification factor. As a result, the observed flux from a neutral
hydrogen source is also scaled similarly, i.e. the amplified flux density can
be written as $\overline{S}_{\mu}=\mu\times\overline{S}_{\nu}$. The parameter
$(SNR)_{\nu}$ is the ratio of $\overline{S}_{\nu}$ and $\Delta S$, where
$\Delta S$ does not depend on the magnification of the global $\rm HI-21cm$
signal by the massive compact object ”L” (figure 1). This further implies that
the parameter $(SNR)_{\nu}$ is also scaled similarly as we have seen for
$\overline{I}_{\nu}$ and $\overline{S}_{\nu}$, i.e. the amplified signal-to-
noise ratio will be $(SNR)_{\mu}=\mu\times(SNR)_{\nu}$. We have discussed
earlier that two images of ”S” (figure 1) will be formed by the strong
gravitation lensing due to the massive compact object ”L” (figure 1), where
one image will be formed inside and another will be formed outside of the
Einstein’s circle respectively. This implies that the magnification factor
$\mu$ will also be different for the different images mentioned above. For a
given angular position of the source ”S” (figure 1), the value of $\mu$ can be
estimated using the relation

$\mu_{\pm}=\frac{\beta^{2}+2\theta_{E}^{2}}{2\beta\sqrt{\beta^{2}+4\theta_{E}^{2}}}\pm\frac{1}{2}$
(5)

where $\mu_{+}$ and $\mu_{-}$ are the magnification factors of the images
formed outside and inside of Einstein’s circle respectively. Here we can
conclude that the value $(SNR)_{\mu}$ is larger for the image formed outside
of Einstein’s circle in comparison to another image formed there due to the
strong gravitational lensing of the global $\rm HI-21cm$ signal. We have seen
earlier that the value of $\theta_{E}$ (eq. 4) depends on $R$, $\gamma$ and
$z_{S}$, and as a consequent $\mu_{\pm}$ also depends on that three parameters
mentioned above. However, from now onwards we only consider the image of ”S”
(figure 1) formed outside of Einstein’s circle for two reasons, viz. (A) the
value of $\theta_{-}$ is very close to $\beta$ and it may be difficult to
distinguish both images simultaneously due to the small field-of-view of the
uGMRT and (B) the strength of that image ($\overline{S}_{\mu}$), determined
through the parameter $\mu_{-}$, may be insufficient to detect. We request
readers to have a quick look at the review article on gravitational lenses by
Narayan, Bartelmann (1996) for more details.

###  2.4 Mass of a Neutron Star

We have already discussed that Einstein’s angle $\theta_{E}$ (eq. 4) depends
on the mass of the lensing object. As a consequence, both the parameters
$\theta_{+}$ (eq. 3) and $\mu_{+}$ (eq. 5) also vary with the variation of the
mass of the lensing object. Here we consider a neutron star, a compact massive
object, as the lensing medium for which the mass is a non-linear function of
its radius. The variation of mass $M_{L}(R)$ of a compact neutron star can be
numerically evaluated by solving the Tolman-Oppenheimer-Volkov equations (TOV)
(Oppenheimer, Volkoff, 1939), which is given by

$\frac{dP}{dR}=-\frac{GM_{L}(R)}{R^{2}}\,\rho\,\left(1+\frac{P}{\rho
c^{2}}\right)\left(1+\frac{4\pi
R^{3}P}{M_{L}(R)c^{2}}\right)\left(1-\frac{2GM_{L}(R)}{c^{2}R}\right)^{-1}$
(6)

where $P$, $\rho$ and $R$ are the pressure, density and radius of a
spherically symmetric compact neutron star. The parameter $P$ is related with
$\rho$ through a polytropic relation $P=K\rho^{\Gamma}$, where $K$ is a
constant and $\Gamma$ is the polytropic exponent given by $\Gamma=1+(1/n)$
with $n$ being the polytropic index. The hydrostatic equilibrium inside the
neutron star (Shapiro, Teukolsky, 2008) can be described by the variation of
mass $M_{L}$ as a function of the radial distance $R$, which is given by

$\frac{dM_{L}}{dR}=4\pi R^{2}P$ (7)

We estimate the value of $M_{L}(R)$ by solving the coupled differential
equations 6 and 7 numerically keeping with some suitable parameters. For a
neutron star, which is considered here as the lensing object, we assume the
value of core density $\rho_{c}=2.5\times 10^{17}$ kg m-3 and the polytropic
exponent $\Gamma=5/3$. The value of $M_{L}(R)$ increases with increasing $R$
attains a maximum value and then starts decreasing. For the neutron star with
the combination of $\rho_{c}$ and $\Gamma$ value mentioned above, we have seen
that $M_{L}(R)$ is maximum to the value of $\approx 2M_{\odot}$ for $R\approx
10$ km and this is denoted here as $R_{\rm peak}$.

###  2.5 Optical Depth

Now we shift our attention to another parameter called optical depth $\tau$
which determines the detection probability of the lensed global $\rm HI-21cm$
signal using any radio telescope with a given observational frequency. The
parameter $\tau$ (detection probability) depends on the density and the
position of the lensing object along the LOS of the observation and this is
given by

$\tau=\int_{0}^{z_{S}}d\chi(z)\,(1+z)^{2}\,n_{L}\,\sigma(z)$ (8)

where, $\chi(z)$ is the co-moving distance of an object located at redshift
$z$, $n_{L}$ is the co-moving number density of the lensing medium and
$\sigma(z)$ is the lensing cross section of the incoming global $\rm HI-21cm$
signal for a lensing medium of mass $M_{L}$ located at redshift $z$. Using a
flat $\Lambda CDM$ cosmology and some parameter arrangements the eq. 8 can be
written as

$\tau(z_{S},\gamma)=\frac{3}{2}\,\Omega_{CO}\,f\int_{0}^{z_{S}}dz\,\,\frac{(1+z)^{2}\,H_{0}^{2}\,D_{LS}(z,\gamma)\,D_{L}(z)}{c\,H(z)\,D_{S}(z)}$
(9)

where, $H(z)=H_{0}\,\sqrt{\Omega_{M}\,(1+z)^{3}\,+\,\Omega_{\Lambda}}$ is the
Hubble parameter with $H_{0}$, $\Omega_{M}$ and $\Omega_{\Lambda}$ are Hubble
constant, dark matter density parameter and dark energy density parameter
respectively and the value of those parameters are taken from Planck
Collaboration et al. 2020. The dimensionless parameter $f$ is closed to unity
and the density parameter of compact objects is found to be $\Omega_{CO}\leq
0.1$ (e.g. Schneider 1993). Here we introduced $\Omega_{CO}$ because we study
the detection probability of the lensed global $\rm HI-21cm$ signal by a
compact neutron star. From the above equation, it is clear that the value of
$\tau$ solely depends on the redshift of the neutral hydrogen source ($z_{S}$)
and the position of the lensing object which is determined by the parameter
$\gamma$.

###  2.6 Chance Coincidence

Till now we have discussed how one can predict the $SNR$ of the amplified
global $\rm HI-21cm$ signal supposed to be detected at different frequency
bands of uGMRT. We considered the neutral hydrogen source located at redshift
$z_{S}$ emits the signal further amplified through gravitational lensing by a
compact neutron star located at redshift $z_{L}$. We have seen that the mass
of the neutron star, which is a function of its radius, makes a significant
contribution to this amplification and the angular position of the image,
where the mass of a neutron star can be estimated using the TOV equation
(Oppenheimer, Volkoff, 1939). Apart from the mass of the lensing medium, the
angular position of the neutral hydrogen source ($\beta$) also makes a
significant contribution to the amplification of the global $\rm HI-21cm$
signal, where $\beta$ is a free parameter for this analysis. We mentioned
earlier that the angle $\beta$ is taken concerning ”ON” as the LOS of the
observation (figure 1). From eq. 5, we have seen that the amplification, as
well as the $SNR$ of the observation, increases sharply with decreasing
$\beta$, however, this is not shown in this paper. We have found a maximum
allowed value of $\beta$, denoted as $\beta_{\rm max}$, above which we will
not have sufficient amplification of the global $\rm HI-21cm$ signal such that
it will be detected at the different frequency bands of uGMRT. We have seen
that the value of $\beta_{\rm max}$ is different for different frequency bands
of uGMRT, size and position of the lensing object. For a given size and
position of the lensing object, i.e. $R=10$ km and $\gamma=0.5$, the value of
maximum allowed the angular position of the neutral hydrogen source is
$\beta_{\rm max}=9.23\times 10^{-26}$ rad, $9.86\times 10^{-26}$ rad and
$1.06\times 10^{-25}$ rad such that the global $\rm HI-21cm$ signal can be
detected at B3, B4 and B5 respectively. However, for different values of $R$
and $\gamma$, the value of $\beta_{\rm max}$ does not change significantly.
All the values of $\beta_{\rm max}$ mentioned above are very small and we may
conclude that $\beta_{\rm max}\rightarrow 0$ for all the frequency bands
considered here. This implies the source has to be located close to the LOS
(”ON” of figure 1) such that the global $\rm HI-21cm$ signal can be detected
at different frequency bands of uGMRT. This is called ”chance coincidence”
which was already mentioned in different studies (e.g. Saini et al. 2001)
earlier.

##  3 Result

In this section, we discuss the variation of several lensing parameters and
scenarios and their physical implications on the detection of the global $\rm
HI-21cm$ signal at different frequency bands (B3, B4 and B5) of uGMRT. We
start our discussion with the variation of the neutron star’s (lensing object)
redshift ($z_{L}$) with the parameter $\gamma$, which determines the position
of the lensing object, assuming the amplified global $\rm HI-21cm$ signal is
detected at different frequency bands (B3, B4 and B5) of uGMRT. The left panel
of figure 2 shows the variation of $z_{L}$ with $\gamma$ ($0.1\leq\gamma\leq
0.9$) for different neutral hydrogen source redshift. Here, we use the
characteristic redshift $z_{c}$ corresponding to different observing bands
(showing in different colour lines) of uGMRT, i.e. $z_{c}=2.73,\,1.03\,{\rm
and}\,0.13$ for B3, B4 and B5 respectively. We see that the value of $z_{L}$
increases with increasing $\gamma$ for all the frequency bands of GMRT. For a
fixed value of $\gamma$, $z_{L}$ is maximum for B4 (blue line) and minimum for
B5 (green line), however, the variation of $z_{L}$ with $\gamma$ for B3 (red
line) and B4 (blue line) is almost similar. Considering all the values of
$z_{L}$ corresponding to the complete range of $\gamma$, we find that the
maximum value of $z_{L}$ is around $1$ for $\gamma=0.9$ which is nearly the
same for both B3 and B4, i.e. $z_{c}=2.73$ and $1.03$, and minimum value of
$z_{L}$ is around $0.01$ for $\gamma=0.1$ with B5, i.e. $z_{c}=0.13$. This
implies that for detection of the lensed global HI-21cm signal at uGMRT, the
redshift of the neutron star as the lensing object lies within the range
$0.01<z_{L}<1$ corresponding to the different frequency bands.

Figure 2: The variation of the neutron star’s redshift (left panel) and the
optical depth fraction (right panel) with the neutron star’s position
considering the detection of the global $\rm HI-21cm$ signal at Band-3 (red
line), Band-4 (blue line) and Band-5 (green line) of uGMRT.

Next, we discuss the variation of the optical depth with the position of the
neutron star (lensing object). The parameter optical depth ($\tau$) determines
the detection probability of the amplified global $\rm HI-21cm$ signal, where
$\tau$ depends on several factors, viz. redshift ($z_{S}$) of neutral hydrogen
source, position ($\gamma$) of neutron star along LOS and density parameter
($\Omega_{CO}$) of compact objects. We do not have a clear picture of the
value of $\Omega_{CO}$, however, Schneider 1993 put a constraint on the upper
limit of $\Omega_{CO}$ ($\leq 0.1$). Here we study the variation of
$\tau/\Omega_{CO}$ with the variation of $\gamma$ for different source
redshifts where these are assumed to be the characteristic redshifts
corresponding to different frequency bands of uGMRT. The right panel of figure
2 shows the variation of $\tau/\Omega_{CO}$ with $\gamma$ ($0.1\leq\gamma\leq
0.9$) for different redshifts of the neutral hydrogen source. The red, blue
and green lines show the variation of $\tau/\Omega_{CO}$ for the detection of
amplified global $\rm HI-21cm$ signal at B3 ($z_{c}=2.73$), B4 ($z_{c}=1.03$)
and B5 ($z_{c}=0.13$) respectively. For all the frequency bands considered
here, we see $\tau/\Omega_{CO}$ decreases with increasing $\gamma$. This
implies that the probability of detecting amplified global $\rm HI-21cm$
signal at different frequency bands is large if the neutron star is close to
the observer. For a particular value of $\gamma$, we find the value of
$\tau/\Omega_{CO}$ is highest for B3 and lowest for B5. This points to the
fact that we have a good amount of detection probability of the amplified
global $\rm HI-21cm$ signal if the neutral hydrogen source is far away from
the observer and this is relatively poor for the scenario where the neutral
hydrogen source is nearby. Considering all the values of $\tau/\Omega_{CO}$
corresponding to the complete range of $\gamma$, we find that the maximum
value of $\tau/\Omega_{CO}$ is $7.72$ for $\gamma=0.1$ with B3 ($z_{c}=2.73$),
and minimum value of $\tau/\Omega_{CO}$ is $1.78\times 10^{-3}$ for
$\gamma=0.9$ with B5 ($z_{c}=0.13$). Considering the upper limit of
$\Omega_{CO}$ (Schneider, 1993), we find the maximum and minimum values of
optical depth are $\tau_{\rm max}=0.77$ and $\tau_{\rm min}=1.78\times
10^{-4}$. Combining all the information mentioned above, we may conclude that
the probability of detecting amplified global $\rm HI-21cm$ signal is maximum
at B3 ($z_{c}=2.73$) if the neutron star (lensing object) is close to the
observer.

Figure 3: The effect of radius (upper panels) and position (lower panels) of
an isolated neutron star on the signal-to-noise ratio ($SNR$) of the global
$\rm HI-21cm$ signal supposed to be detected at Band-3 (left column), Band-4
(middle column) and Band-5 (right column) of uGMRT. The black horizontal
dashed line shows the threshold value of $SNR$, i.e. $SNR_{\rm Th}=5$.

The amplification strength of the global $\rm HI-21cm$ signal through the
gravitational lensing by an isolated neutron star mainly depends on two
physical parameters of the lensing object, viz. its mass and position along
LOS. Previously, we have discussed that the neutral hydrogen source has to be
located close to LOS, i.e. $\beta\rightarrow 0$, such that the amplified
global $\rm HI-21cm$ signal can be detected at different frequency bands of
uGMRT which is called chance coincidence. For the fact stated above, we
exclude the parameter $\beta$ in the phenomena of the amplification of global
$\rm HI-21cm$ signal. Further, we have also seen that the mass of an isolated
neutron star strongly depends on its radius and that can be estimated using
the TOV equation (eq. 6). We have converted the amplification strength into
observed $SNR$ of the global $\rm HI-21cm$ signal which is supposed to be
detected at different frequency bands of uGMRT. Therefore, we may conclude
that the observed $SNR$ strongly depends on the size and position of an
isolated neutron star along LOS.

Figure 3 shows the variation of $SNR$ with the variation of neutron star
radius and position along LOS assuming the amplified global $\rm HI-21cm$
signal is detected at B3 (left column), B4 (middle column) and B5 (right
column). The upper panels show the variations of $SNR$ with neutron star
radius ($R$) for its different positions ($\gamma$) along LOS where we
consider three distinct values of $\gamma$, i.e. $0.1$, $0.5$ and $0.9$, which
spans the entire range of $\gamma$. The lower panels show the variations of
$SNR$ with neutron star positions ($\gamma$) for its different radius ($R$)
values where we consider three distinct values of $R$, i.e. $8$ km, $10$ km
and $16$ km, which spans the entire range of radius. Here we introduce a
threshold value for $SNR$, i.e. $SNR_{\rm Th}=5$, such that any detection of
the global $\rm HI-21cm$ signal with $SNR<SNR_{\rm Th}$ will not be considered
as a genuine detection which is shown in the black horizontal dashed lines in
the figure 3.

First, we discuss the variation of $SNR$ with $R$ for the detection of
amplified global $\rm HI-21cm$ signal at different frequency bands of uGMRT.
For B3 ($z_{c}=2.73$) with all the values of $\gamma$ shown in the left upper
panel, $SNR$ increases with increasing $R$ attains a maximum value near
$R_{\rm max}\approx 10$ km and then starts decreasing with further increasing
of $R$. However, the sharpness of the plot changes with different values of
$\gamma$, where this is steeper for $\gamma=0.1$ and relatively flatter for
$\gamma=0.9$. Further, the peak value of $SNR$ also changes with different
values of $\gamma$ where this is maximum, i.e. $SNR_{\rm max}=34.47$, for
$\gamma=0.1$ and this is minimum, i.e. $SNR_{\rm min}=3.68$, for $\gamma=0.9$.
For different values of $\gamma$ with $8\,{\rm km}\leq R\leq 16\,{\rm km}$,
the values of predicted $SNR$ have not been always above $SNR_{\rm Th}$,
rather for some of the values of $\gamma$, it goes below the threshold value.
Considering $8\,{\rm km}\leq R\leq 16\,{\rm km}$, we find that $SNR>SNR_{\rm
Th}$ for $\gamma=0.1$ and $SNR<SNR_{\rm Th}$ for $\gamma=0.9$, whereas for
$\gamma=0.5$, $SNR>SNR_{\rm Th}$ is possible only if $R<12.5$ km. The similar
qualitative natures of $SNR$ vs $R$ plots have been seen for the other two
frequency bands of uGMRT. However the allowed range of $R$ for different
values of $\gamma$, and the values of $SNR_{\rm max}$ and $SNR_{\rm min}$ are
different for different frequency bands. For the detection of amplified global
$\rm HI-21cm$ signal at B4 with characteristic redshift $z_{c}=1.03$ shown in
the middle upper panel of figure 3, we find that $SNR>SNR_{\rm Th}$ for
$\gamma=0.1$ with $8\,{\rm km}\leq R\leq 16\,{\rm km}$ and for $\gamma=0.5$
with $R<14\,{\rm km}$, whereas the predicted values of $SNR$ are always below
$SNR_{\rm Th}$ for $\gamma=0.9$ with the full range of $R$ considered here.
The value of $SNR_{\rm max}$ and $SNR_{\rm min}$ are $75.03$ and $4.02$
respectively. For the detection of amplified global $\rm HI-21cm$ signal
coming from a neutral hydrogen source located at $z_{c}=0.13$ (B5) shown in
the right upper panel of figure 3, we find that $SNR>SNR_{\rm Th}$ for both
$\gamma=0.1$ and $0.5$ with $8\,{\rm km}\leq R\leq 16\,{\rm km}$. However for
$\gamma=0.9$, $SNR>SNR_{\rm Th}$ is possible only if $R<11\,{\rm km}$. The
value of $SNR_{\rm max}$ and $SNR_{\rm min}$ are $340.42$ and $5.22$
respectively.

Now, we discuss the variation of $SNR$ with $\gamma$ for the detection of
amplified global $\rm HI-21cm$ signal at different frequency bands of uGMRT.
For B3 ($z_{c}=2.73$) with all the values of $R$ shown in the left lower
panel, $SNR$ decreases with increasing $\gamma$. The qualitative nature of
$SNR$ vs $\gamma$ plot is almost similar for both $R=10$ km $8$ km, however,
this is largely differed for $R=16$ km. However, for a fixed value of
$\gamma$, the predicted $SNR$ is maximum for $R=10$ km and minimum for $R=16$
km and for that reason, $R\approx 10$ km is denoted here as $R_{\rm peak}$,
i.e. a particular value of neutron star’s radius for which the signal
amplification is maximum. We find the predicted values of $SNR$ are not always
above $SNR_{\rm Th}$ for the range of $\gamma$ considered here which further
gives us the upper limit of $\gamma$, denoted by $\gamma_{\rm cut}$, and this
varies with different values of $R$ mentioned in the figure. For both $R=10$
km and $8$ km, we see $\gamma_{\rm cut}\approx 0.71$ whereas for $R=16$ km, we
find $\gamma_{\rm cut}\approx 0.18$. The similar qualitative natures of $SNR$
vs $\gamma$ plots have been seen for the other two frequency bands of uGMRT.
However the value of $\gamma_{\rm cut}$ differs for different values of $R$
and frequency bands considered here. For the detection of amplified global
$\rm HI-21cm$ signal at B4, we find $\gamma_{\rm cut}\approx 0.86$ for both
$R=10$ km and $8$ km whereas $\gamma_{\rm cut}\approx 0.37$ for $R=16$ km.
Similarly for B5, we see $\gamma_{\rm cut}\approx 0.91$ for both $R=10$ km and
$8$ km whereas $\gamma_{\rm cut}\approx 0.69$ for $R=16$ km. In short for a
fixed value of $R$, the value of $\gamma_{\rm cut}$ increases with decreasing
redshift of the neutral hydrogen source.

Combining the information discussed above we may conclude that a neutron star
with radius $8\,{\rm km}\leq R\leq 16\,{\rm km}$ located close to the observer
can amplify the global $\rm HI-21cm$ signal originating from a neutral
hydrogen source with redshift within the range $0.13\leq z_{S}\leq 2.73$ such
that it can be detected at different frequency bands of uGMRT. This can also
happen for the other position of the neutron star along LOS with an upper
limit on its radius. Previously we have discussed the variation of a neutron
star’s mass with its radius governed by the TOV equation (eq. 6) and the same
feature has been replicated in the variation of $SNR$ with $R$ because the
mass of a neutron star as a whole gives a major contribution to the
amplification of any astrophysical signal through gravitational lensing. The
value of $SNR_{\rm max}$ increases with the decreasing redshift of the neutral
hydrogen source because the flux density ($\overline{S}_{\nu}$) from that
source is large if the source is located nearby in comparison to the source
which is relatively away from the observer and also the telescope noise is
minimum at B5 and maximum at B3. Further, the value of $\gamma_{\rm cut}$
decreases with the increasing of neutral hydrogen source redshift which
implies if the source is away from the observer, a neutron star with a fixed
radius (say $10$ km) has to be located close to the observer to get sufficient
amplification of the global $\rm HI-21cm$ signal such that it can be detected
at uGMRT. For all of the results discussed above we have seen that a neutron
star with a radius $10$ km, denoted by $R_{\rm peak}$, amplifies the global
$\rm HI-21cm$ signal in such a way that the predicted $SNR$ is always the
maximum, which is irrespective of the neutral hydrogen source redshift. From
now on, we consider $R_{\rm peak}$ as the standard radius of a neutron star
for further discussion.

Figure 4: The estimation of the radius and position of an isolated neutron
star using the signal-to-noise ratio ($SNR$) of the global $\rm HI-21cm$
signal supposed to be detected at Band-3 (left column), Band-4 (middle column)
and Band-5 (right column) of uGMRT. The white vertical dashed line shows the
standard radius of that neutron star, i.e. $R=R_{\rm peak}=10\,{\rm km}$.

Till now we have analysed how the lensing parameters, such as the position and
radius of a neutron star, control the amplification of the global $\rm
HI-21cm$ signal which is supposed to be detected at different frequency bands
of uGMRT. Now we discuss if an amplified global $\rm HI-21cm$ signal is
detected at a particular frequency band of uGMRT with a given $SNR$ then how
we can provide constraints on the radius and position of a neutron star which
amplifies that signal. Figure 4 shows the variation of $SNR$ on $\gamma-R$
plane for different frequency bands of uGMRT which are supposed to be used to
detect that global $\rm HI-21cm$ signal. Previously we have discussed the
parameter $\gamma_{\rm cut}$ which varies with $R$ and different frequency
bands and the same feature is also exhibited here. Further, the parameter
$R_{\rm peak}$, which is the standard radius of a neutron star, has been shown
in the figure by a white vertical dashed line. Let us first consider B3 (left
panel), if an amplified global $\rm HI-21cm$ signal ($z_{c}=2.73$) is detected
with $SNR=10$, we may conclude that a neutron star with a radius $R_{\rm
peak}\approx 10$ km located at a position $\gamma=0.37$ amplifies that signal.
However, the value of $\gamma$ may change if that neutron star has a different
radius rather than $R_{\rm peak}$. For a given $SNR$, the value of $\gamma$
increases with increasing $R$, attains a maximum value and then starts
decreasing with further increase of $R$. Therefore it may be concluded that we
have a maximum allowed value of $\gamma$ which in general corresponds to
$R_{\rm peak}$ for a given $SNR$ of the global signal. We can make the same
kind of prediction about the position and radius of a neutron star if that
amplified global $\rm HI-21cm$ signal is detected at B4 (middle panel) and B5
(right panel) respectively. Suppose a global $\rm HI-21cm$ signal is detected
at B4 ($z_{c}=1.03$) with $SNR=15$, then that signal can be amplified by a
neutron star with $\gamma=0.42$ and $R=R_{\rm peak}\approx 10$ km. Similarly,
if that global signal is detected at B5 ($z_{c}=0.13$) with $SNR=20$, then
that signal can be amplified by a neutron star with $\gamma=0.68$ and
$R=R_{\rm peak}\approx 10$ km. Briefly, we may conclude that if a global $\rm
HI-21cm$ signal is detected at a particular frequency band of uGMRT with a
given $SNR$, we can predict the position and radius of an isolated neutron
star which may amplify that global signal. However, the position ($\gamma$)
and radius ($R$) of that neutron star are highly correlated, but we can put a
tight constraint on the value of $\gamma$ if we consider the standard radius
$R=R_{\rm peak}$ for that neutron star.

##  4 Discussion

The detection of the global $\rm HI-21cm$ signal at a given telescope is a
challenging part of radio astronomy since the strength of that global signal
is several orders of magnitude lower than the foreground/background noises.
However, recently, several studies (e.g. Chakraborty, Roy 2023) reported the
detection of that global signal at different radio telescopes, which implies
that the strength of that signal is somehow magnified. There are several ways
to magnify that global signal and strong gravitational lensing is one of them,
where the amount of magnification of that global signal is generally
quantified by the parameter signal-to-noise ratio ($SNR$). In this scenario,
we have presented a method to estimate the position and size of the lensing
medium using the $SNR$ of the global $\rm HI-21cm$ signal which is supposed to
be detected at different frequency bands of uGMRT considering the fact an
isolated neutron star amplifies the strength of that global signal, where the
Band-3 (B3), Band-4 (B4) and Band-5 (B5) of uGMRT have been considered for
this analysis. In this paper, we have studied the effects of several lensing
parameters, viz. size, mass and position of the neutron star, detection
probability, etc., on the $SNR$ of that global signal and its variation with
frequency bands. Our analysis shows that there is a possibility of detecting
the amplified global $\rm HI-21cm$ signal at different frequency bands of
uGMRT with a sufficiently large amount of $SNR$, where the important findings
are listed below.

  1. 1.

The neutral hydrogen source has to be located close to the line (line-of-
sight) which connects the observer and the lensing medium such that the
amplification of the global $\rm HI-21cm$ signal will be sufficient to detect
at different frequency bands of uGMRT. This phenomenon is called the ”chance
coincidence”.

  2. 2.

Considering the detection of that amplified global signal at different
frequency bands of uGMRT, the redshift of an isolated neutron star, which
amplifies that global signal, should lie within the range $0.01<z_{L}<1$
corresponds to different redshifts of the neutral hydrogen source.

  3. 3.

The probability of detecting global $\rm HI-21cm$ signal is maximum at Band-3
if the isolated neutron star is located close to the observed and this is
minimum at Band-5 if that lensing medium is located close to the source.

  4. 4.

In contrast, the predicted $SNR$ of that amplified global signal is relatively
high for Band-5 in comparison to the other two frequency bands considered
here.

  5. 5.

An isolated neutron star with radius $8\,{\rm km}\leq R\leq 16\,{\rm km}$
located close to the observer can amplify that global signal such that it can
be detected at different frequency bands of uGMRT, however, this can also
happen for other position of that neutron star with an upper limit on its
radius. However, there is also an upper limit on the neutron star’s position
irrespective of its radius above which it can not perform the required
amplification of that global signal.

  6. 6.

The amplification of the global $\rm HI-21cm$ signal is maximum by an isolated
neutron star with radius $R=R_{\rm peak}=10\,{\rm km}$ and it is minimum for
$R=16\,{\rm km}$ irrespective of its position.

  7. 7.

One can estimate the position of an isolated neutron star using the $SNR$ of
that amplified global signal, which is supposed to be detected at different
frequency bands of uGMRT, by considering the value of its radius $R_{\rm
peak}=10\,{\rm km}$.

  8. 8.

On the other way, one may also determine the radius of that neutron star
independently if its position can be located by some other surveys.

Now we present the cross-validation of the position and size of an isolated
neutron star estimated using the $SNR$ of that amplified global signal. It is
believed that an isolated neutron star is one of the sources of pulsar
emission. In this scenario, we propose to cross-validate the position and size
of that isolated neutron star through the pulsar observation in the same
location in the sky from where that amplified global signal is supposed to be
detected. The required luminosity of that extragalactic pulsar emission is the
order of $\sim 10^{32}\,{\rm erg\,s^{-1}}$ such that it can be detected at
different frequency bands of uGMRT, which is several orders of magnitude
higher than the brightest pulsar (e.g. villa, Crab, etc.) detected at the
radio band to date. This implies that it is difficult to detect the pulsar
signal and the amplified global $\rm HI-21cm$ signal simultaneously at any
frequency band of uGMRT. However, a pulsar signal with that required
luminosity can be detected at the X-ray band (e.g. Walter et al. 1996). In
this scenario, we discuss the scope of multi-messenger astronomy, where that
amplified global signal may be detected at the radio band and the
corresponding pulsar signal can be detected at the X-ray band from the same
location in the sky. Several studies (e.g. Pons et al. 2002; Kaplan et al.
2007) show how one can estimate the redshift and radius of an isolated neutron
star which emits an extragalactic pulsar signal at the X-ray band. In this
regard, we propose to detect the amplified global $\rm HI-21cm$ signal at the
radio band and cross-validate the estimated position and size of an isolated
neutron star using the pulsar observation at the X-ray band from the same
location in the sky.

We have discussed the cross-validation of the estimated lensing parameters
using the pulsar observation but if that pulsar signal has not been detected
along with that amplified global signal from the same location in the sky,
this implies that the global $\rm HI-21cm$ signal may not be amplified by an
isolated neutron star where some other massive compact objects can perform
this amplification. Here we have considered an isolated neutron star as a
lensing medium, however, an accreting neutron star in a binary system can also
perform the same kind of amplification which is beyond the scope of this
analysis and we have a plan to study this prospect in our subsequent paper.

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# J0107a: A Barred Spiral Dusty Star-forming Galaxy at $z=2.467$

Shuo Huang Institute of Astronomy, Graduate School of Science, The University
of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Ryohei Kawabe National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Department of Astronomy, School of Science, The Graduate University for
Advanced Studies (SOKENDAI), Osawa, Mitaka, Tokyo 181-8588, Japan  Kotaro
Kohno Institute of Astronomy, Graduate School of Science, The University of
Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Research Center for the
Early Universe, Graduate School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Toshiki Saito National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Shoichiro
Mizukoshi Institute of Astronomy, Graduate School of Science, The University
of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Daisuke Iono National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Department of Astronomy, School of Science, The Graduate University for
Advanced Studies (SOKENDAI), Osawa, Mitaka, Tokyo 181-8588, Japan  Tomonari
Michiyama Faculty of Welfare and Information, Shunan University, 43-4-2,
Gakuendai, Shunan, Yamaguchi, 745-8566, Japan  Yoichi Tamura Department of
Physics, Graduate School of Science, Nagoya University, Furocho, Chikusa,
Nagoya 464-8602, Japan  Christopher C. Hayward Center for Computational
Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
Hideki Umehata Institute for Advanced Research, Nagoya University, Furocho,
Chikusa, Nagoya 464-8602, Japan  Department of Physics, Graduate School of
Science, Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  Cahill
Center for Astronomy and Astrophysics, California Institute of Technology, MS
249-17, Pasadena, CA 91125, USA

###### Abstract

Dusty Star-Forming Galaxies (DSFGs) are amongst the most massive and active
star-forming galaxies during the cosmic noon. Theoretical studies have
proposed various formation mechanisms of DSFGs, including major merger-driven
starbursts and secular star-forming disks. Here, we report J0107a, a bright
($\sim 8$ mJy at observed-frame 888 $\mu$m) DSFG at $z=2.467$ that appears to
be a gas-rich massive disk and might be an extreme case of the secular disk
scenario. J0107a has a stellar mass $M_{\star}\sim 5\times 10^{11}M_{\odot}$,
molecular gas mass $M_{\mathrm{mol}}\sim(1\textendash 6)\times
10^{11}M_{\odot}$, and a star formation rate (SFR) of $\sim 500M_{\odot}$
yr-1. J0107a does not have a gas-rich companion. The rest-frame 1.28 $\mu$m
JWST NIRCam image of J0107a shows a grand-design spiral with a prominent
stellar bar extending $\sim 15$ kpc. ALMA band 7 continuum map reveals that
the dust emission originates from both the central starburst and the stellar
bar. 3D disk modeling of the CO(4-3) emission line indicates a dynamically
cold disk with rotation-to-dispersion ratio $V_{\mathrm{max}}/\sigma\sim 8$.
The results suggest a bright DSFG may have a non-merger origin, and its
vigorous star formation may be triggered by bar and/or rapid gas inflow.

Barred spiral galaxies(136); CO line emission(262); Galaxy formation (595);
High-redshift galaxies(734); Ultraluminous infrared galaxies (1735)

††software: numpy (Harris et al., 2020), scipy (Virtanen et al., 2020),
matplotlib (Hunter, 2007), astropy (Astropy Collaboration et al., 2018),
photutils (Bradley et al., 2022), dynesty (Speagle, 2020)

##  1 Introduction

The cosmic star formation rate density increases from early times to its peak
at $z\sim 2-3$, called cosmic noon, and then decreases steadily to the present
day (for a review, see Madau & Dickinson, 2014). During cosmic noon, dusty
star-forming galaxies (DSFGs) populate the massive and high-SFR end of star-
forming galaxies, with $M_{\star}\sim 10^{11}M_{\odot}$ and SFR$\sim
100-1000M_{\odot}$ yr-1(e.g., da Cunha et al., 2015; Miettinen et al., 2017;
Dudzevičiūtė et al., 2020). The large infrared luminosity of DSFGs makes them
easy to be detected with submillimeter/millimeter observations, and they are
often referred to as submillimeter galaxies (SMGs; e.g., Smail et al., 1997;
Hughes et al., 1998; Casey et al., 2014)

The origin of DSFGs has been debated since their discovery. Based on insights
from studying local ultraluminous infrared galaxies (ULIRGs), which are
exclusively late-stage major mergers (e.g., Sanders & Mirabel, 1996), and
observations of high-z DSFGs, many authors have argued that bright DSFGs (or
classical SMGs with $S_{\mathrm{850}\mu\mathrm{m}}\gtrsim 5$ mJy) are mainly
major mergers (e.g., Tacconi et al., 2008; Engel et al., 2010; Chen et al.,
2022a). Early theoretical models demonstrated that major mergers can indeed
power bright DSFGs (e.g., Narayanan et al., 2010; Hayward et al., 2011), but
it was unclear whether there were enough major mergers to account for the
observed number counts of bright DSFGs. Hayward et al. (2011, 2012, 2013)
argued that the high-z bright DSFG population is heterogeneous, with merger-
driven starbursts dominating at the bright end and ‘quiescently star-forming’
disks dominating at fainter submm fluxes. Others have argued that the DSFG
population is predominantly gas-rich massive disks (i.e. not ongoing major
mergers) that sustain high SFRs due to high gas accretion rates from the
cosmic web (e.g., Dekel et al., 2009; Davé et al., 2010; Narayanan et al.,
2015; Lovell et al., 2021).

Morphological information can thus place critical constraints on the formation
channels of DSFGs by distinguishing disks and mergers. Previous studies using
deep Hubble Space Telescope (HST) imaging have revealed a prevalence of
disturbed morphology (e.g., Chen et al., 2015), inline with interaction or
merger. However, due to the high redshift and heavy dust obscuration of DSFGs,
the rest-frame UV/optical emission observed with HST might not be an ideal
tracer of stellar distribution.

The advent of James Webb Space Telescope (JWST) makes rest-frame optical-to-
NIR imaging with unprecedented sensitivity and spatial resolution available
for DSFGs. Recent JWST studies of submm-selected galaxies have found a large
fraction of isolated disks (Chen et al., 2022b; Cheng et al., 2023), in
contrast to more clumpy and irregular shapes in HST bands. JWST rest-frame NIR
imaging will provide a more robust view of the stellar distribution of DSFGs
and hints for their possibly diverse formation channels.

In this paper, we report J0107a, a $z=2.467$ DSFG with a clear barred spiral
structure in JWST/NIRCam images and bright dust continuum and CO/[CI](1-0)
line emission. We describe the observation in Section 2 and data analyses and
results in Section 3. We discuss the results and summarize in Section 4.
Throughout this paper we assume the Chabrier (2003) IMF and Planck 2018
cosmology ($H_{0}=67.4$ km s-1 Mpc-1, $\Omega_{m}=0.315$ and
$\Omega_{\mathrm{\Lambda}}=0.685$; Planck Collaboration et al., 2020).

##  2 DATA

###  2.1 The Target

The target of this study, J0107a was first identified by Tamura et al. (2014)
as a serendipitously detected line emitter at the observed frequency of 99.75
GHz in ALMA band 3 observation towards the local galaxy merger VV114 at
$z=0.0205$. Based on photometric redshift, the line is identified as CO(3-2)
at $z=2.467$. The redshift is confirmed by Mizukoshi et al. (2021) with
CO(3-2), CO(4-3), and [CI](1-0) emission lines (Figure 1a). J0107a is also
detected in continuum with flux densities of $3.4\pm 0.3$ mJy at 1.3 mm and
$7.9\pm 1.0$ mJy at $888$ $\mu$m, which satisfy the selection criteria of a
classical SMG.

###  2.2 ALMA Data

J0107a has been covered by the ALMA campaigns toward VV114. In this study, we
include ALMA band 3 (Project code 2013.1.00469.S and 2013.1.01057.S), band 4
(2013.1.01057.S), band 6 (2015.1.00973.S and 2015.1.00902.S), and band 7
(2013.1.00740.S) data. Data reduction and imaging are conducted with CASA
(CASA Team et al., 2022) in a standard manner following Mizukoshi et al.
(2021), except for band 3 and 4, additional spectral cubes with natural
weighting are made to maximize sensitivity. The integrated 1D ALMA spectra of
J0107a are shown in Figure 1a.

Figure 1: A multiwavelength view of J0107a: (a) the integrated 1D spectra of
J0107a from ALMA Band 3/4 data. The channel widths are 50 km s-1. (b) Left:
JWST/NIRCam color composite image (red: F356W; green: F200W; blue: F150W) of
J0107a. Middle: F356W image with ALMA band 7 continuum of J0107a (dark orange)
and the zeroth-moment map of the band 3 emission line from the $z=1.186$
southwest (SW) spiral galaxy (magenta) overlaid. The insert panel shows the
band 3 spectrum of the SW spiral galaxy, with the line’s central frequency of
105.458 GHz marked with the gray dashed line. Right: Chandra ACIS-S 0.5-10 keV
count image smoothed with a $0\farcs 5$ FWHM circular Gaussian kernel.
Contours show the zeroth-moment maps of J0107a’s emission lines, with white,
red, and blue for CO(3-2), CO(4-3), and [CI](1-0), respectively. Each panel is
$5\arcsec\times 5\arcsec$ in size. ALMA beam sizes are shown with the same
colors as the contours. Contour levels are $-2$, 2, 4, 6, 8, 10… in the middle
panel and $-2$, 2, 4, 8, 16… in the right panel.

###  2.3 JWST Data

JWST observations of the VV114 field have been conducted as part of the GOALS-
JWST survey (Evans et al., 2022; Linden et al., 2023), with NIRCam imaging in
the F150W, F200W, F356W, and F444W filters and MIRI imaging in the F560W,
F770W, and F1500W filters. 111The JWST data presented in this paper were
obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space
Telescope Science Institute. The specific observations analyzed can be
accessed via https://doi.org/10.17909/g4k8-3922 (catalog DOI:
10.17909/g4k8-3922). We obtain images from the MAST archive and align the
astrometry to GAIA DR3 (Gaia Collaboration et al., 2016, 2023). To remove the
foreground emission from VV114, we first apply standard background
subtraction, then identify point sources with a flux ratio
$\log(S_{\mathrm{F356W}}/S_{\mathrm{F200W}})<-0.05$ as bright stars in VV114
and subtract them using the point spread function (PSF) model built with the
WebbPSF222https://github.com/spacetelescope/webbpsf package before performing
photometry.

##  3 Analysis and Results

###  3.1 JWST and ALMA Look at J0107a

The left panel of Figure 1b shows the NIRCam image of J0107a. J0107a is
clearly resolved, showing a grand-design barred spiral structure with two arms
and a blue central point-like source, in line with an active galactic nucleus
(AGN) as indicated by its X-ray detection in Chandra ACIS-S data (Grimes et
al., 2006). The bar structure is also visible in the ALMA band 7 continuum
image with $0\farcs 18\times 0\farcs 15$ beam size, implying star formation in
the bar (Figure 1b, the middle panel). To quantify the bar structure, we fit
ellipse isophotes to the F444W image (Figure 2). The projected bar size is
determined as the semimajor axis (SMA) at maximum ellipticity
$e_{\mathrm{max}}$. For J0107a’s stellar bar, we find an SMA of 7.5 kpc and
$e_{\mathrm{max}}=0.50$. The $e_{\mathrm{max}}$ is similar to that measured by
Guo et al. (2023) for $z>1$ barred galaxies in the CEERS survey, but the SMA
is longer. We also perform parametric modeling of the F444W image, assuming a
PSF model for the AGN, three Sérsic profiles for the bar, bulge, and disk, and
two logarithmic curves for the arms (Kennicutt, 1981). We implement the models
using the numpy, scipy, and astropy packages, then use the dynesty code
(Speagle, 2020) to sample the joint posterior probability density function
(PDF) of model parameters. The best-fit model and residual are shown in panels
c and d of Figure 2, respectively. The rest-frame NIR light is dominated by
the three Sérsic components with light fraction of $11.8_{-3.0}^{+2.3}\%$,
$80.5_{-5.0}^{+4.6}\%$ and $7.9_{-3.5}^{+4.3}\%$ for the AGN, sum of Sérsic
components, and spiral arms, respectively.

Figure 2: Left: a) $3\arcsec\times 3\arcsec$ F444W cutout of J0107a displayed
in $-2\textendash 80\sigma$ range; b) the ellipse isophote fit to J0107a
(cyan); c) the best-fit 2D parametric model of J0107a; d) the residual of the
best-fit model. During modeling processes, the two neighboring galaxies are
masked. Right: radial surface brightness (SB, top), ellipticity (middle), and
position angle (PA, bottom) distributions of J0107a. The vertical dashed line
marks the spatial resolution in the F444W band.

J0107a is detected with three different molecular gas tracers, namely CO,
[CI], and dust continuum. The derived $M_{\mathrm{mol}}$ ranges from $(9.2\pm
1.7)\times 10^{10}M_{\odot}$ using the CO(3-2) line to $(3.2\pm 1.6)\times
10^{11}M_{\odot}$ using the dust continuum, assuming SMG/ULIRG-like CO
excitation and conversion factors (Mizukoshi et al., 2021). If normal star-
forming galaxies-like conversions (Daddi et al., 2010, 2015; Decarli et al.,
2016) are adopted, the $M_{\mathrm{mol}}$ will be $\sim 6\times
10^{11}M_{\odot}$. Therefore, J0107a is a very gas-rich DSFG with gas fraction
$\sim 20\textendash 60\%$. No companion of J0107a is found in the band 3 cube.
Assuming $\alpha_{\mathrm{CO}}=3.6M_{\odot}$/(K km s-1 pc2), $r_{31}=0.42\pm
0.07$ and a Gaussian line profile with FWHM of 300 km s-1, the $3\sigma$
sensitivity of 1.45 mJy beam-1 in 50 km s-1 channels translates to an upper
limit of $M_{\mathrm{mol}}\lesssim 10^{11}M_{\odot}$, implying the absence of
any companion with molecular gas mass comparable with that of J0107a.

###  3.2 Photometry and Spectral Energy Distribution Modeling

Figure 3: Left: EAZY fit of the JWST photometry (black circles and error
bars). The median and 16th-84th percentile ranges of the SED at the best-fit
redshift $z_{\mathrm{best}}$ are shown with the solid lines and shaded
regions, respectively. The insert panels show the PDFs of photometric
redshift. The photometric redshifts discussed in Section 3.3 are
16th-50th-84th percentile values of the PDF. The vertical dashed lines mark
the spectroscopic redshift identification. Right: X-ray-to-millimeter SED fit
of J0107a at $z_{\mathrm{spec}}=2.467$. For the total SED model and individual
components, we show median and 16th-84th percentile ranges with the solid
lines and shaded regions, respectively.

To derive the photometry in JWST bands, we first combine the F356W and F444W
images and then run source detection routines in the photutils package to
generate Kron apertures for J0107a and the two neighboring galaxies in the
combined $5\arcsec\times 5\arcsec$ cutout image. The fluxes are measured with
the Kron apertures and corrected for aperture loss using the PSF in each band.
Flux uncertainties are estimated from sky by taking the standard deviations of
fluxes in 1000 apertures randomly placed at blank positions within $5\arcsec$
radius from J0107a.

We fit the observed spectral energy distribution (SED) of J0107a using the
CIGALE code (Boquien et al., 2019). In addition to the JWST data, we include
ALMA band 6/7 continuum and Chandra X-ray fluxes taken from Tamura et al.
(2014) and Mizukoshi et al. (2021). We assume a nonparametric star formation
history (Leja et al., 2019) and adopt Noll et al. (2009) dust attenuation
laws. The AGN emission is modeled with a power-law X-ray component plus the
SKIRTOR library (Stalevski et al., 2016). For dust emission, we use the Draine
et al. (2014) templates. We fix the redshift of J0107a at
$z_{\mathrm{spec}}=2.467$ in CIGALE SED fitting.

The SED fit of J0107a is shown in the right panel of Figure 3. From SED
fitting we derive the basic physical parameters of J0107a as follows:
$M_{\star}=4.5_{-1.4}^{+1.7}\times 10^{11}M_{\odot}$,
SFR=$499_{-179}^{+357}M_{\odot}$ yr-1, dust mass
$M_{\mathrm{dust}}=4.3_{-1.0}^{+1.0}\times 10^{9}M_{\odot}$, and rest-frame
2-10 keV luminosity of $4.9_{-0.9}^{+0.9}\times 10^{43}$ erg s-1. The SFR of
J0107a is $0.8_{-0.4}^{+0.9}$ times of the main sequence value of star-forming
galaxies with the same redshift and $M_{\star}$ (Speagle et al., 2014).

###  3.3 Properties of the Two Neighboring galaxies and Association with
J0107a

Two neighboring galaxies are visible within $2\farcs 5$ from J0107a. The
spiral galaxy $1\farcs 7$ southwest of J0107a exhibits a single-peaked
redshift PDF (Figure 3, the top-left panel). This PDF, derived from the SED
fit of the JWST photometry using the EAZY333https://github.com/gbrammer/eazy-
py (Brammer et al., 2008) code, yields a photometric redshift of
$z_{\mathrm{phot}}=1.12_{-0.03}^{+0.05}$. An emission line at the central
frequency of 105.458 GHz is detected at the position of the southwest spiral
galaxy in the ALMA band 3 spectral cube (Figure 1b, the middle panel), with
the direction of velocity gradient aligning with the NIR major axis.
Therefore, we are able to identify the emission line as redshifted CO(2-1) and
derive spectroscopic redshift $z_{\mathrm{spec}}=1.186$ for this galaxy. The
neighboring spheroid galaxy $1\farcs 6$ southeast to J0107a has
$z_{\mathrm{phot}}=1.69_{-0.26}^{+0.75}$ from the redshift PDF (Figure 3, the
bottom-left panel) and no line or continuum detection in the ALMA data, so its
relation with J0107a remains uncertain. If this spheroid galaxy is at the same
redshift of J0107a, the $M_{\star}$ and SFR derived from SED fit of the JWST
data will be $6.9_{-1.3}^{+1.2}\times 10^{10}M_{\odot}$ and
$4_{-4}^{+18}M_{\odot}$ yr-1, respectively, and the stellar mass ratio of 1:4
does not satisfy the definition of a major merger.

###  3.4 Dynamical Modeling

A clear velocity gradient is seen in the velocity fields of all three spectral
lines (Figure 4, see also Mizukoshi et al. (2021)). We choose CO(4-3) to
perform kinematic modeling because it has the highest S/N. We create a
$9\farcs 3\times 9\farcs 3$ cutout cube centered at J0107a with $20$ km s-1
channel width and model the data with the GalPak3D code (Bouché et al., 2015),
assuming an exponential disk profile and arctan rotation curve. This gives an
inclination of $9\fdg 7\pm 0\fdg 8$, half-light radius $r_{\mathrm{e}}=0\farcs
54\pm 0\farcs 01$, maximum rotation velocity $v_{\mathrm{MAX}}=281\pm 39$ km
s-1, and intrinsic velocity dispersion $\sigma=35\pm 3$ km s-1. In addition,
we fit the data with the 3D-Barolo code (Di Teodoro & Fraternali, 2015) and
find consistent values of an inclination of $10\fdg 4$, maximum rotation
velocity of $283$ km s-1 and intrinsic velocity dispersion of 31 km s-1. The
best-fit models and residual maps are shown in Figure 4. These results show
J0107a has a dynamically cold gas disk with $v_{\mathrm{MAX}}/\sigma\sim 8$.

The dynamical mass within a radius of $2.2r_{e}=9.8$ kpc at $z=2.467$ can be
calculated as:

$M_{\mathrm{dyn}}=\frac{2.2r_{\mathrm{e}}v^{2}_{\mathrm{circ}}}{G}$ (1)

where $G$ is the gravitational constant and
$v_{\mathrm{circ}}=\sqrt{v_{\mathrm{MAX}}^{2}+4.4\sigma^{2}}$ is the
circularized velocity (Burkert et al., 2010) computed using the results from
GalPak3D modelling. The resulting $M_{\mathrm{dyn}}$ of $\sim 2\times
10^{11}M_{\odot}$ is less than the sum of $M_{\star}$ and $M_{\mathrm{mol}}$,
indicating the total mass within a radius of 10 kpc is dominated by baryon.

Previous analysis in Mizukoshi et al. (2021) reported a dynamical mass of
$\sim 10^{10}M_{\odot}$ based on similar velocity dispersion but a much lower
$v_{\mathrm{MAX}}=69$ km s-1, and their $M_{\mathrm{dyn}}$ conflicts with
$M_{\star}$ and $M_{\mathrm{mol}}$, mainly because of their initial guess of
inclination of $60\arcdeg$. We have tested initial guesses of $10\arcdeg$,
$30\arcdeg$, and $60\arcdeg$, and let the value vary from $0\arcdeg$ to
$90\arcdeg$. For initial guesses of $10\arcdeg$, $30\arcdeg$, the fit
converges to the same results presented above, while for the initial guess of
$60\arcdeg$, the fit converges to $\sim 60\arcdeg$. Apparently, the
inclination cannot be constrained alone with the $1\farcs 2$ resolution CO
data even the peak S/N reaches $\sim 50$, and the new low inclination solution
seems more appropriate. The outermost isophote of J0107a has an ellipticity of
$\sim 0.3$ which translates to an inclination of $<45\arcdeg$. The inclination
from isophote fitting should be treated as an upper limit because of
distortion of the isophotes by strong spiral arms (Yu & Ho, 2020) and
insufficient depth of the F444W image to detect the underlying smooth stellar
disk. We caution that the current $M_{\mathrm{dyn}}$ estimate can still have
large uncertainty due to the low inclination, beam smearing effect, and a
possible more complex rotation pattern due to the bar. Future high-resolution
CO observations are needed to better constrain the dynamics and mass
distribution.

Figure 4: The zeroth, first, and second moment maps of the CO(4-3) emission
line of J0107a. Also shown are the GalPak3D/3D-Barolo models and corresponding
residual maps. The black ellipse in the top left panel indicates the beam
size. The residual in the first moment map with redshifted velocity on the
northeast edge overlaps with a horn-like structure in the zeroth moment map
(Figure 1b, the right panel) and may originate from gas inflow. However, the
S/N is low so further observation is needed to confirm it.

##  4 Discussion and Conclusion

A bright DSFG with a flux density of $\sim 8$ mJy at the observed wavelength
of 888 $\mu$m is expected to be the product of a gas-rich major merger
according to the conventional wisdom. Nonetheless, J0107a’s symmetric barred
spiral morphology, lack of massive and gas-rich companion, and large
dynamically cold gas disk all support a non-merger origin. Instead of merger-
induced starburst, a more plausible formation channel would be cold mode
accretion via inflowing gas streams from the cosmic web which penetrate the
hot medium in the host halo and reach the central galaxy to fuel star
formation and AGN activity (e.g., Kereš et al., 2009). Assuming the stellar-
to-halo mass ratio $\log(M_{\star}/M_{\mathrm{halo}})$ ranges from $-1.5$ dex
to $-1.8$ dex (Stach et al., 2021), J0107a has a halo mass of
$M_{\mathrm{halo}}=(1\textendash 3)\times 10^{13}M_{\odot}$, which lies at the
boundary between hot and cold in hot modes of accretion (Dekel & Birnboim,
2006) and represents the most efficient normal star-forming galaxy in
converting gas into stars at this redshift. The predicted baryon accretion
rate is $1500\textendash 5400M_{\odot}$ yr-1 (Dekel et al., 2009), which is
sufficient to supply J0107a’s star formation. If J0107a is indeed fed by such
streams, it will be an extreme case with the maximum allowed accretion rate,
thus an ideal target to directly detect such streams onto a bright DSFG and
demonstrate their heterogeneous origins.

The large stellar bar of J0107a is of particular interest in the context of
galaxy assembly. J0107a has the highest redshift among known $z>1$ galaxies
with a clear stellar bar identified (e.g., Simmons et al., 2014; Guo et al.,
2023), but also see a lensed case at $z=4.26$ (Smail et al., 2023) and
potential gas/dust bars at $z=1.5\textendash 4.4$ (Hodge et al., 2019; Tsukui
et al., 2023). In local strongly barred galaxies, the global SFR is suppressed
compared to unbarred spiral galaxies with similar mass (e.g., Gavazzi et al.,
2015; Géron et al., 2021), while J0107a lies above the $z\sim 2.5$ star-
forming main sequence. This may be interpreted as the bar-accelerated stellar
mass build-up in the past (e.g., Kim et al., 2017; Fraser-McKelvie et al.,
2020) due to the effective gas redistribution as we will discuss below, and
J0107a is caught in the early starburst phase of a massive barred galaxy.

In addition to the stellar bar, the 888 $\mu$m dust continuum map suggests the
presence of a dust/gas bar (Figure 1b, the middle panel), which is the site of
star formation. The dust emission shows three peaks and is reminiscent of the
enhancement of SFR at the galactic center and the bar ends in local barred
galaxies (Maeda et al., 2023). Elevated central star formation is common in
local barred galaxies (e.g., Chown et al., 2019; Lin et al., 2020) and
explained by theoretical studies as a consequence of gas being funneled by the
bar toward the center and feeds the star formation and supermassive black hole
(e.g., Carles et al., 2016). The higher central molecular gas concentration
observed in local barred spiral galaxies (Kuno et al., 2007; Yu et al., 2022b)
provides direct evidence for bar-driven gas inflow, and Yu et al. (2022b) also
suggest the spiral arms can playing a role in transporting gas to the center.
On the other hand, barred galaxies with suppressed central SFR often exhibit
weak to nearly vanished spiral arms (Wang et al., 2020; Yu et al., 2022a).
J0107a possesses a strong stellar bar and pronounced spiral arms, and the high
SFR may be triggered by gas inflow driven by stellar structures in its gas-
rich disk.

Numerical simulations have shown that an isolated baryon-dominated and
dynamically cold disk allows bar formation at early times (Rosas-Guevara et
al., 2022; Bland-Hawthorn et al., 2023), though it has also been suggested
that a high gas fraction can weaken the bar and slows down its formation
(e.g., Athanassoula et al., 2013). Alternative formation channels of barred
spiral structures include interaction with a companion galaxy (e.g., Elmegreen
et al., 1990; Oh et al., 2008). The photometric redshift of the southeast
spheroid galaxy does not exclude the possibility of it being a companion of
J0107a. From the analysis of the JWST data, the spheroid galaxy will have
$M_{\star}$ of $\sim 1/4\times$ J0107a and a distance of 13 kpc to J0107a if
it is at $z=2.467$, which is sufficient to affect J0107a’s disk. While the
southeast spiral arm indeed seems brighter, tidal features were not found.
Furthermore, J0107a’s CO disk is dynamically cold and dominated by ordered
rotation ($V_{\mathrm{max}}/\sigma\sim 8$). Considering that cold accretion
can also bring a low $V_{\mathrm{max}}/\sigma$ component (Ho, 2007), secular
bar formation is thus favored and consistent with the first massive and well-
ordered disks emerging in the cosmic noon (Simons et al., 2017).

Previous simulations of bar formation have neither covered the parameter space
with $M_{\star}$, SFR, and gas fraction as high as J0107a, nor gas accretion
from the surrounding environment. Refined simulations reproducing the
properties of J0107a are needed to understand the formation mechanisms of
stellar and gas bars in the earliest massive disks. In terms of observations,
deep high-resolution spectroscopy is needed to confirm the isolated disk
nature of J0107a and to investigate the interplay between bar, gas, and star
formation.

Finally, J0107a has been mis-identified as a strongly lensed SMG magnified by
approximately ten times in Mizukoshi et al. (2021) to explain the bright
CO(4-3) emission with narrow line FWHM (FWHM = $193\pm 11$ km s-1; $S\Delta
v=3.0\pm 0.2$ Jy km s-1). However, it is now clear that this is caused by a
low inclination of $\sim 10\arcdeg$, as the regular morphology and absence of
a nearby massive source can rule out strong gravitational lensing effect.
While the empirical relation between CO line FWHM and velocity-integrated
intensity (e.g., Harris et al., 2012; Bothwell et al., 2013) is a natural
consequence of the virial theorem for large samples of unlensed galaxies, the
use of such relations to identify individual lensed source should be careful
as the result is sensitive to the unknown inclination.

We are grateful to the anonymous reviewer for the helpful and constructive
comments that improved the quality of this manuscript. SM is supported by
Japan Society for the Promotion of Science (JSPS) KAKENHI with the Grant
number of 23KJ0450. YT is supported by JSPS KAKENHI (No. 22H04939). This paper
makes use of the following ALMA data: ADS/JAO.ALMA #2013.1.00469.S,
#2013.1.01057.S, #2015.1.00973.S, #2015.1.00902.S, and #2013.1.00740.S. ALMA
is a partnership of ESO (representing its member states), NSF (USA), and NINS
(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. The Flatiron Institute is
supported by the Simons Foundation.

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