# Track-based alignment for the BESIII CGEM detector in the cosmic-ray test

A.Q. Guo  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000, China  University of Chinese Academy of Sciences, Beijing 100049,
China  L.H. Wu<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China  L.L. Wang<EMAIL_ADDRESS>Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China
University of Chinese Academy of Sciences, Beijing 100049, China  R.E.
Mitchell  Indiana University, Bloomington, Indiana 47405, USA  A. Amoroso
INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy  Università di
Torino, Dipartimento di Fisica, via P. Giuria 1, 10125 Torino, Italy  R.
Baldini Ferroli  INFN, Laboratori Nazionali di Frascati, via E. Fermi 40,
00044 Frascati (Roma), Italy  I. Balossino  INFN, Sezione di Ferrara, via G.
Saragat 1, 44122 Ferrara, Italy  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  M. Bertani  INFN, Laboratori
Nazionali di Frascati, via E. Fermi 40, 00044 Frascati (Roma), Italy  D.
Bettoni  INFN, Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  F.
Bianchi  INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy
Università di Torino, Dipartimento di Fisica, via P. Giuria 1, 10125 Torino,
Italy  A. Bortone  INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino,
Italy  Università di Torino, Dipartimento di Fisica, via P. Giuria 1, 10125
Torino, Italy  G. Cibinetto  INFN, Sezione di Ferrara, via G. Saragat 1, 44122
Ferrara, Italy  A. Cotta Ramusino  INFN, Sezione di Ferrara, via G. Saragat 1,
44122 Ferrara, Italy  F. Cossio  INFN, Sezione di Torino, via P. Giuria 1,
10125 Torino, Italy  M.Y. Dong  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China  M. Da Rocha Rolo  INFN, Sezione di Torino,
via P. Giuria 1, 10125 Torino, Italy  F. De Mori  INFN, Sezione di Torino, via
P. Giuria 1, 10125 Torino, Italy  Università di Torino, Dipartimento di
Fisica, via P. Giuria 1, 10125 Torino, Italy  M. Destefanis  INFN, Sezione di
Torino, via P. Giuria 1, 10125 Torino, Italy  Università di Torino,
Dipartimento di Fisica, via P. Giuria 1, 10125 Torino, Italy  J. Dong
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing,
100049, China  F. Evangelisti  INFN, Sezione di Ferrara, via G. Saragat 1,
44122 Ferrara, Italy  Università di Ferrara, Dipartimento di Fisica e Scienze
della Terra, via G. Saragat 1, 44122 Ferrara, Italy  R. Farinelli  INFN,
Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  L. Fava  INFN,
Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy  Università di Torino,
Dipartimento di Fisica, via P. Giuria 1, 10125 Torino, Italy  G. Felici  INFN,
Laboratori Nazionali di Frascati, via E. Fermi 40, 00044 Frascati (Roma),
Italy  I. Garzia  INFN, Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara,
Italy  Università di Ferrara, Dipartimento di Fisica e Scienze della Terra,
via G. Saragat 1, 44122 Ferrara, Italy  M. Gatta  INFN, Laboratori Nazionali
di Frascati, via E. Fermi 40, 00044 Frascati (Roma), Italy  G. Giraudo  INFN,
Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy  S. Gramigna  INFN,
Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  Università di
Ferrara, Dipartimento di Fisica e Scienze della Terra, via G. Saragat 1, 44122
Ferrara, Italy  S. Garbolino  INFN, Sezione di Torino, via P. Giuria 1, 10125
Torino, Italy  M. Greco  INFN, Sezione di Torino, via P. Giuria 1, 10125
Torino, Italy  Università di Torino, Dipartimento di Fisica, via P. Giuria 1,
10125 Torino, Italy  Z. Huang  Peking University, Beijing 100871, People’s
Republic of China  Y.R. Hou  University of Chinese Academy of Sciences,
Beijing 100049, China  W. Imoehl  Indiana University, Bloomington, Indiana
47405, USA  L. Lavezzi  INFN, Sezione di Torino, via P. Giuria 1, 10125
Torino, Italy  Università di Torino, Dipartimento di Fisica, via P. Giuria 1,
10125 Torino, Italy  X.L. Lu  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China  M. Maggiora  INFN, Sezione di Torino, via P.
Giuria 1, 10125 Torino, Italy  Università di Torino, Dipartimento di Fisica,
via P. Giuria 1, 10125 Torino, Italy  F. M. Melendi  Università di Ferrara,
Dipartimento di Fisica e Scienze della Terra, via G. Saragat 1, 44122 Ferrara,
Italy  INFN, Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  R.
Malaguti  INFN, Sezione di Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  A.
Mangoni  INFN, Sezione di Perugia, via A. Pascoli, 06123 Perugia, Italy
Università di Perugia, Dipartimento di Fisica e Geologia, via A. Pascoli,
06123 Perugia, Italy  S. Marcello  INFN, Sezione di Torino, via P. Giuria 1,
10125 Torino, Italy  Università di Torino, Dipartimento di Fisica, via P.
Giuria 1, 10125 Torino, Italy  M. Melchiorri  INFN, Sezione di Ferrara, via G.
Saragat 1, 44122 Ferrara, Italy  G. Mezzadri  INFN, Sezione di Ferrara, via G.
Saragat 1, 44122 Ferrara, Italy  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  Q. Ouyang  Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China
University of Chinese Academy of Sciences, Beijing 100049, China  S. Pacetti
INFN, Sezione di Perugia, via A. Pascoli, 06123 Perugia, Italy  Università di
Perugia, Dipartimento di Fisica e Geologia, via A. Pascoli, 06123 Perugia,
Italy  P. Patteri  INFN, Laboratori Nazionali di Frascati, via E. Fermi 40,
00044 Frascati (Roma), Italy  A. Rivetti  INFN, Sezione di Torino, via P.
Giuria 1, 10125 Torino, Italy  R.S. Shi  Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing, 100049, China  University of Chinese
Academy of Sciences, Beijing 100049, China  M. Scodeggio  INFN, Sezione di
Ferrara, via G. Saragat 1, 44122 Ferrara, Italy  Università di Ferrara,
Dipartimento di Fisica e Scienze della Terra, via G. Saragat 1, 44122 Ferrara,
Italy  S. Sosio  INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy
Università di Torino, Dipartimento di Fisica, via P. Giuria 1, 10125 Torino,
Italy  S. Spataro  INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino,
Italy  Università di Torino, Dipartimento di Fisica, via P. Giuria 1, 10125
Torino, Italy  B.L. Wang  University of Chinese Academy of Sciences, Beijing
100049, China  H.P. Wang  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing, 100049, China  University of Chinese Academy of Sciences,
Beijing 100049, China  J.Y. Zhao  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing, 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

The Beijing Electron Spectrometer III (BESIII) is a multipurpose detector
operating on the Beijing Electron Positron Collider II (BEPCII). After more
than ten year’s operation, the efficiency of the inner layers of the Main
Drift Chamber (MDC) decreased significantly. To solve this issue, the BESIII
collaboration is planning to replace the inner part of the MDC with three
layers of Cylindrical triple Gas Electron Multipliers (CGEM).

The transverse plane spatial resolution of CGEM is required to be 120 $\mu$m
or better. To meet this goal, a careful calibration of the detector is
necessary to fully exploit the potential of the CGEM detector. In all the
calibrations, the detector alignment plays an important role to improve the
detector precision. The track-based alignment for the CGEM detector with the
Millepede algorithm is implemented to reduce the uncertainties of the hit
position measurement. Using the cosmic-ray data taken in 2020 with the two
layers setup, the displacement and rotation of the outer layer with respect to
the inner layer is determined by a simultaneous fit applied to more than
160000 tracks. A good alignment precision has been achieved that guarantees
the design request could be satisfied in the future. A further alignment is
going to be performed using the combined information of tracks from cosmic-ray
and collisions after the CGEM is installed into the BESIII detector.

###### Contents

  1. I Introduction
  2. II Detector setup for the cosmic-ray test
  3. III Alignment software
    1. III.1 Cluster reconstruction
    2. III.2 Track finding and fitting for cosmic-ray
    3. III.3 Alignment algorithm
      1. III.3.1 The principle
      2. III.3.2 The implementation at the BESIII
  4. IV Results with the cosmic-ray data
    1. IV.1 The cosmic-ray data
    2. IV.2 The results
  5. V Conclusions and plans
  6. VI Acknowledgments

##  I Introduction

The BESIII spectrometer [1] and the BEPCII $e^{+}e^{-}$ collider is a unique
machine operating in the $\tau$-Charm region and has collected world record
statistics from 2.0 to 5.0 GeV in the last decade. This data set yields
fruitful physics results and its lifespan is likely be extended for another 5
to 10 years with some necessary upgrades both on the accelerator and detector
aspects. Since the Inner Tracker (IT) has a serious aging effect due to the
radiation from beam-related background, replacing the old IT with CGEM
detector [2] is one of the essential upgrades from the detector side. The new
IT has a significant radiation hardness, high rate capability, and excellent
resolutions both in the longitudinal and transverse directions and it is
crucial for the long-term operation of the BESIII detector.

The new IT is composed of 3 layers of CGEM and its geometry can be found in
Fig. 1 (Left). Each layer is a triple-GEM composed of concentric cylindrical
electrodes (Fig. 1 (Right)): the cathode, three GEM foils, and the readout
anode. Each layer plays as an independent tracking detector providing a 3-D
reconstruction of space points along the track with a 2-D readout. Such a
detector will match the requirements for radial resolution ($\sigma_{xy}\sim
120\mu m$) of the existing drift chamber and will improve significantly the
spatial resolution along the beam direction ($\sigma_{z}\sim 220\mu m$) with
very small material budget (less than 1.5% of $\rm X_{0}$) [3]. The new IT
will be installed in the available space left by the removal of the inner MDC.
The new tracker will achieve the requirements for momentum resolution of
$\sigma_{pt}/p_{t}\sim 0.5\%$ at 1 GeV/c in a 1T magnetic field.

Fig. 1: Left: the geometry of CGEM detector. Right: the structure of each CGEM
layer

##  II Detector setup for the cosmic-ray test

Before the installation of CGEM into the BESIII detector, two of the three
CGEM layers have been assembled in the experimental hall of the Institute of
High Energy Physics (IHEP) in Beijing. The whole system, including the
trigger, detector, the readout system, is tested by the cosmic ray as shown in
Fig. 2.

Fig. 2: Left: the setup of the cosmic ray test for the CGEM. Right: a scheme
of the setup with two CGEM (orange box) and the scintillating bars (light dark
boxes).

Two layers of scintillator bars are placed above and below the CGEM detector
and serve as a trigger system. The CGEM layers have an inner radius of 76.9
$mm$ and 121.4 $mm$, while their length is 532 $mm$ and 690 $mm$ in the active
area. The readout layer is segmented by longitudinal (X) and stereo (V)
strips, the length of the strips is fixed for the longitudinal ones, while for
the stereo strips it varies from a few millimeters up to tens of centimeters.
The pitch size of the strips is 650 $\mu m$, therefore the total number of
channels instrumented at present is about five thousand. The gas mixture used
inside the CGEM is $\rm{Ar+iC_{4}H_{10}}$ (90:10) to provide about 55
electrons per m.i.p. in 5 mm. The HV on the electrodes is 835 V which
corresponds to 12 thousand detector gain, while the electric fields between
the electrodes are set to 1.5/3/3/5 kV/cm [4].

The Torino Integrated Gem Electronics for Readout electronics [5] is used to
measure the time and the collected charge. Two ASICs are installed on a Front-
End Board (FEB) to read the signal from 128 strips. The chip has two separate
branches to extract time and charge information. Every signal crossing
threshold on both branches is digitized and transmitted to the off-detector
electronics. The charge can be measured with two methods: Sample and Hold mode
and Time-Over-Threshold mode [6]. The former one was adopted during the
cosmic-ray data taking. Once the trigger is sent to the readout chain, the
data measured by each chip is collected. The GEM Read Out Card (GEMROC)
manages the low voltage of the FEB, the chip configuration, and the data
collection. Since only two layers were used during the cosmic-ray data taking,
the readout electronics have 44 FEB, 11 GEMROC, and two data concentrator
cards to send the whole CGEM-IT outputs to the BESIII DAQ system.

##  III Alignment software

The BESIII Offline Software System (BOSS) is used to manage the geometry
description of the detector, the digitization for the simulation, and the
cluster reconstruction for the data. Within the BOSS framework, the
measurements from the CGEM-IT will be used to create the 1D cluster and 2D
cluster first, then a track-finding algorithm is applied to all the 2D
clusters to choose the best candidate on each layer. The selected clusters are
fitted by a straight line with the least-squares method. The cluster and the
track reconstruction procedures will be explained in the following sections.

###  III.1 Cluster reconstruction

Contiguous firing strips are clustered to form the 1D cluster. Their charge
and time information is used to characterize the signal. There are two methods
to reconstruct the position: the Charge Centroid and the micro-Time Projection
Chamber [7]. The first averages the position of each strip of the cluster with
its charge. The performance of this method is robust, especially for tracks
with small incident angle. Meanwhile, the second associate each strip with a
bi-dimensional point and it uses a linear fit on the points to extrapolate the
position. This method will give better resolution for tracks with large
incident angle. A combined position reconstruction method is being developed
to take advantage of the two methods above. The Charge Centroid method is used
in this study. The clusterization is performed on longitudinal and stereo
strips. The combination of the two measurements gives a 2D cluster. An event
display for both 1D and 2D clusters is shown in Fig. 4 top-right subplot.

###  III.2 Track finding and fitting for cosmic-ray

The trajectory of cosmic-rays is modeled as a straight line and represented by
four parameters: $d\rho$, $\phi_{0}$, $d_{z}$, and $tan\lambda$ as shown in
the Fig. 3. Here, $d\rho$ is the distance of the line projection from the
origin in the x-y plane, $\phi_{0}$ is the angle between the perpendicular
line $\rho$ and the x-axis in the x-y plane, $d_{z}$ is the intercept of the
line projection in the s-z plane, $tan\lambda$ is the slope of the line
projection in the s-z plane. The 2D clusters belonging to one track are fed to
the track fitting algorithm, which is based on the least-squares method, to
obtain the track parameters. For one cosmic-ray, four 2D clusters should to be
collected by the two-layers setup of the CGEM detector as shown in the Fig. 4.
To reject the fake clusters due to the electronic noise, a simple track-
finding algorithm is developed. In the method, at most three clusters with the
largest charge on each half of the layer are selected, then all the possible
combinations of these four sets of clusters are fitted with the linear model,
and at last, the combination with the smallest $\chi^{2}$ is kept as a track
candidate. Figure 4 shows an event display for the track reconstruction.

Fig. 3: Schematic representation of a straight line in the x-y plane (left)
and in the s-z plane (right). Left: the projected line in the x-y plane, where
the dashed line $\rho$ is through the origin $O$ and is perpendicular to the
line with the point $P$ as the intersection (foot of the perpendicular). The
direction $s$ of the projected line in this plane is always defined as
downwards. Right: the projected line in the s-z plane, where the intersect is
$dz$ and the angle between the line and the axis $s$ is $\lambda$.
\begin{overpic}[scale={0.32}]{3D_display.pdf}
\put(65.0,65.0){\includegraphics[scale={0.2}]{local_cluster.png}}
\end{overpic} Fig. 4: A 3D event display for the 2D cluster reconstruction.
The red dots show the collected four 2D cluster when a cosmic-ray pass through
the detector. The blue lines show the fired X strips and the green lines refer
to the fired V strips. The dots are the reconstructed 2D clusters. The red
line is the fitted track. The top right subplot shows the details of the 2D
clusters.

###  III.3 Alignment algorithm

####  III.3.1 The principle

To achieve an optimal track parameter resolution, the position and orientation
of CGEM layers need to be determined precisely. Based on the cosmic-ray data,
the track-based alignment algorithm is implemented [8]. In the track fitting,
the track-hit residual is affected by the displacement of the detector, e.g.
the positions and orientations of the CGEM differ from the true one,
therefore, we can measure the misalignment effect of the detector by fitting a
large number of tracks. In the fit, both the track parameters and the
misalignment parameters, such as the shift and rotation of the detector to its
ideal position, are free. All these parameters are fitted to minimize the
deviation between the measurements and track prediction. Here, the track
parameters are also named local parameters because they are track-independent.
Meanwhile, the parameters to describe the detector displacement are called
global parameters because they are shared by all the tracks. The minimized
function can be defined as:

$\chi^{2}(p,q)=\sum_{j}\sum_{i}(\frac{m_{ij}-f_{ij}(p,q_{j})}{\sigma_{ij}})^{2},$
(1)

where j is the index of track and i is the index of hits in one track.
$f_{ij}$ is the track model prediction at the position of the measurement,
which is a function of global parameters (p) and local parameters ($q_{j}$).
Usually, $f_{ij}$ should be linearized by Taylor expansion, then the
minimization leads to the normal equations $Ca=b$, which is a matrix equation
and its dimension is determined by the number of local and global parameters.
Here, C is a $(n+m)\times(n+m)$-matrix which is constant for a linear problem,
where $n$ and $m$ are the dimensions of global and local parameters. $b$ is
the constant part of the first derivative of the $\chi^{2}(p,q)$ and $a$ is
the parameter vector, including both the global and local part, we want to
know. The complete matrix equation is given by

$\left(\begin{array}[]{c|ccc}\sum C_{k}^{global}&\cdots&H_{j}&\cdots\\\
\hline\cr\vdots&\ddots&0&0\\\ H_{j}^{T}&0&C_{j}^{local}&0\\\
\vdots&0&0&\ddots\end{array}\right)\times\left(\begin{array}[]{c}a_{k}^{global}\\\
\hline\cr\vdots\\\ a_{j}^{local}\\\
\vdots\end{array}\right)=\left(\begin{array}[]{c}\sum b_{k}^{global}\\\
\hline\cr\vdots\\\ b_{j}^{local}\\\ \vdots\end{array}\right).$ (2)

The left side of this equation includes three components. The first part is a
contribution of a symmetric matrix $C_{k}^{global}$ with the dimension of the
global parameters. All the matrices $C_{k}^{global}$ are added up in the upper
left corner of the full matrix of the normal equations. The second
contribution is the symmetric matrix $C_{j}^{local}$, which makes a
contribution to the full matrix on the diagonal and depends only on the
measurement from the j-th track. The third part is a rectangular matrix
$H_{j}$, which has a row number of global parameters dimension and a column
number of the dimension of the local parameters. To achieve desirable
precision, the needed track number could be in the order of $10^{6}$.
Therefore, the $C^{local}$ will be $t*10^{6}$ by $t*10^{6}$ matrix, here, $t$
is the number of the individual track parameters. $H$ will be $t*10^{6}$ by n
matrix. To be able to solve such a large matrix in a reasonable computing
time, a matrix reduction is necessary. We notice that the complete matrix has
a special structure, with many vanishing sub-matrices. Ignoring the global
parameters we could solve the normal equations $C_{j}a_{j}=b_{j}$ for each
partial measurement separately by

$a_{j}^{local}=(C_{j}^{local})^{-1}b_{j}^{local}.$ (3)

The only connection between the local parameters of different partial
measurements is given by the sub-matrices $C_{k}$ and $H$. The special
structure of the full matrix allows a matrix reduction such that the complete
information from each local track fit is transferred to the global parameter-
related matrix. After each track is fitted locally, the corresponding matrices
$C_{k}^{-1}$ and $H_{k}$ are calculated and added to the global matrix and the
vector b in the form of

$C^{\prime}=\sum_{k}C_{k}^{global}-\sum_{k}H_{k}(C_{k}^{local})^{-1}H_{k}^{T},\,b^{\prime}=\sum_{k}b_{k}^{global}-\sum_{k}H_{k}(C_{k}^{local})^{-1}b_{k}^{global}.$
(4)

After the loop over all tracks, the modified normal equations that only
contain the global parameters are obtained:

$C^{\prime}a^{global}=b^{\prime}.$ (5)

The matrix $C^{\prime}$ is reduced to the dimension of the alignment
parameters n therefore the computing time is decreased significantly.

####  III.3.2 The implementation at the BESIII

Since the alignment method is based on the reconstructed track, usually, the
alignment procedure will take place after the data taking. To obtain an
unbiased and reliable result, both the tracks from cosmic rays and the
collision data will be used to determine the displacement between the
different parts of the detector. Before the CGEM detector is installed into
the BESIII, it will be tested by the cosmic ray. It is necessary to obtain the
displacements between the layers of CGEM detector. Firstly, the unexpected
displacement or distortion could be found in time and fixed before the
installation. Secondly, we can obtain the misalignment parameters between the
layers and use them as initial values in the following alignment procedure.
Therefore, a dedicated alignment study using the cosmic-ray data is performed.

As shown in the Fig. 5 left, the cosmic-ray test setup consists of two CGEM
layers. The innermost layer is made of one single sheet, we treat it as a
rigid body and use it as a reference object to study the displacement of the
outer layer. The outer layer includes two sheets covering the top and bottom
parts of the detector. So each sheet is considered as an independent component
in the alignment study. For each sheet, three translation parameters Dx, Dy,
and Dz corresponding to the shifts in the local reference system, and three
rotation angles Rx, Ry, and Rz corresponding to small rotations around the
respective local axes are taken into account. All these alignment parameters
are illustrated in the left plot of Fig. 5.

Fig. 5: Left: the alignment parameters for outer layer. Right: the hit
position distribution.

Several constraints are applied to consider the relations between the
alignment parameters. Due to the specific setup for the cosmic-ray data
taking, most of the hits are collected in the center part of the detector as
shown in the right plot of Fig. 5. In this case, the Dy parameter is
insensitive to the track, therefore, we constrain the Dy values to be zero. In
addition, the correlation between Dx and Rz is quite strong since the rotation
along the Z axis is equivalent to a shift to opposite directions on the X
dimension of the two sheets. To reduce this uncertainty, we require the Dx
values of the two sheets to be the same.

##  IV Results with the cosmic-ray data

###  IV.1 The cosmic-ray data

The data sets used in the study were collected in 2020. The run numbers are
from 10 to 17 and the event numbers are shown in the Table 1. The position of
the trigger system is slightly different in each run. The setup of the last
run, which has the largest statistics, is shown in Fig. 2 right.

Table 1: The event numbers in each data set.  Data set | run10 | run11 | run12 | run13 | run14 | run15 | run16 | run17
---|---|---|---|---|---|---|---|---
Event Number | 18039 | 12831 | 4957 | 2928 | 3318 | 1083 | 5000 | 112405

In the alignment procedure, the initial global parameters are set to zero and
the global fit will be performed iteratively until the fit results are
converged. During the iteration, the output of the previous fit will be used
as the initial parameters of the next iteration.

###  IV.2 The results

The inner layer is labeled as ”Layer1” and the outer one is labeled as
”Layer2”. On each layer, the residuals in $X$ and $V$ dimensions (named as
$\delta X$ and $\delta V$), as illustrated in Fig. 6, represent the
discrepancies between the expected positions and the measurements in the two
different dimensions.

Fig. 6: The definitions of $\delta X$ and $\delta V$

The $\delta X$ and $\delta V$ are investigated before and after alignment
correction to check the algorithm as shown in Fig. 7. In these plots, clear
shifts to zero are found in the residual distributions for both X and V
measurements. These shifts due to displacement are corrected after taking the
misalignment parameters into account.

Fig. 7: The residuals of X and V comparison before and after alignment
correction. Fig. 8: Distributions of the residuals as a function of z before
alignment correction.

We also investigate the residual distribution with respect to z. As shown in
Fig. 8, we can find that the $\delta X$s are obviously z dependent. For
example, the $\delta X$ is increased from 3 mrad to 6 mrad in the range of 400
mm along z. This behavior can be interpreted as a rotation of the detector
around the Y axis. The scale of the residual and the z-dependent behaviors on
the top and bottom part of the same layer are different, which proves that it
is necessary to separate the two sheets on the same layer as an individual
component in the alignment study. The different layer shows the opposite shift
direction and it is also consistent with our expectation. After applying the
alignment correction, both the displacement and z-dependent variation are
eliminated as shown in Fig. 9.

Fig. 9: Distributions of the residuals as a function of z after alignment
correction.

We check the the fit converges after seven iterations from different data sets
and the results are shown in Fig. 10. We find the fit will converge after 7
times iterations. The fitted global parameters from run10, run11, and run17,
which have large statistics, are consistent with each other. The fitted global
parameters from the largest data set run17 are shown in the Table 2.

Fig. 10: The fitted alignment parameter $Dz$ in each iteration from different
data sets. Table 2: The fitted global parameters for the outer layer of the
CGEM.

Position | Dx(mm) | Dz(mm) | Rx(mrad) | Ry(mrad) | Rz(mrad)
---|---|---|---|---|---
Bottom | $0.377\pm 0.008$ | $-1.635\pm 0.005$ | $-0.512\pm 0.033$ | $0.999\pm 0.006$ | $6.528\pm 0.064$
Top | $0.377\pm 0.008$ | $-1.984\pm 0.005$ | $-2.127\pm 0.040$ | $0.302\pm 0.007$ | $15.572\pm 0.064$

##  V Conclusions and plans

To summarize, the alignment parameters for the BESIII CGEM detector have been
determined using the global fit approach of track-based alignment with the
Millepede program. In this study, the cosmic-ray data taken in 2020 with two
layers setup is used and about 160000 tracks are fitted simultaneously. Ten
alignment parameters are measured with high precision. The results from
different data sets are consistent with each other. Clear shift and rotation
effects between the two layers are observed. After performing the alignment
correction, the consistency between the intersection point determined by the
track and the measurement is significantly improved.

This is the first step of the whole alignment procedure for the CGEM project.
The inclusion of cosmic ray tracks in the alignment procedure is not
sufficient to control all the weak modes biasing the curvatures of high
momentum tracks. After all the three layers will be assembled and installed
into the BESIII, further alignment study will be performed using the electron-
position collision data together with the cosmic-ray data to get a more
precise and reliable result. In addition, the displacement between the CGEM
and the outer drift chamber also will be measured with the same software
framework.

##  VI Acknowledgments

We would like to express our great appreciation to the BEPCII accelerator team
for their enlightening discussion and helpful suggestions. This work is
supported in part by National Natural Science Foundation of China (NSFC) under
Contracts Nos. 12275296, 12275297, U1832204, 12175256; National Key R&D
Program of China under Contract No. 2020YFA0406304.

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11institutetext:  LERMA, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Université, UPMC, 75014 Paris,France 22institutetext: Univ. Grenoble
Alpes, CNRS, IPAG, 38000 Grenoble, France 33institutetext: Aix Marseille Univ,
CNRS, CNES, LAM (Laboratoire d’Astrophysique de Marseille), Marseille, France
44institutetext: LLR (Laboratoire Leprince-Ringuet), CNRS, École
Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
55institutetext: School of Physics and Astronomy, Cardiff University, Queen’s
Buildings, The Parade, Cardiff, CF24 3AA, UK 66institutetext: AIM, CEA, CNRS,
Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191
Gif-sur-Yvette, France 77institutetext: Univ. Grenoble Alpes, CNRS, Grenoble
INP, LPSC-IN2P3, 53, avenue des Martyrs, 38000 Grenoble, France
88institutetext: Institut Néel, CNRS, Université Grenoble Alpes, France
99institutetext: Institut de RadioAstronomie Millimétrique (IRAM), Grenoble,
France 1010institutetext: Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale Aldo Moro 5, I-00185 Roma, Italy 1111institutetext: Centro de
Astrobiología (CSIC-INTA), Torrejón de Ardoz, 28850 Madrid, Spain
1212institutetext: High Energy Physics Division, Argonne National Laboratory,
9700 South Cass Avenue, Lemont, IL 60439, USA 1313institutetext: Instituto de
Radioastronomía Milimétrica (IRAM), Granada, Spain 1414institutetext:
Laboratoire de Physique de l’École Normale Supérieure, ENS, PSL Research
University, CNRS, Sorbonne Université, Université de Paris, 75005 Paris,
France 1515institutetext: INAF-Osservatorio Astronomico di Cagliari, Via della
Scienza 5, 09047 Selargius, IT 1616institutetext: Department of Physics and
Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia,
PA, 19104, USA 1717institutetext: Institut d’Astrophysique de Paris, CNRS
(UMR7095), 98 bis boulevard Arago, 75014 Paris, France 1818institutetext:
University of Lyon, UCB Lyon 1, CNRS/IN2P3, IP2I, 69622 Villeurbanne, France
1919institutetext: School of Earth and Space Exploration and Department of
Physics, Arizona State University, Tempe, AZ 85287, USA 2020institutetext:
Caltech, Pasadena, CA 91125, USA

# Candidate cosmic filament in the GJ526 field, mapped with the NIKA2 camera

J.-F. Lestrade Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera<EMAIL_ADDRESS>F.-X. Désert Candidate cosmic
filament in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic
filament in the GJ526 field, mapped with the NIKA2 camera G. Lagache Candidate
cosmic filament in the GJ526 field, mapped with the NIKA2 cameraCandidate
cosmic filament in the GJ526 field, mapped with the NIKA2 camera R. Adam
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera P. Ade Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera H. Ajeddig Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera P. André Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera E. Artis Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera H. Aussel Candidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 camera A. Beelen Candidate cosmic filament in the GJ526
field, mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526
field, mapped with the NIKA2 camera A. Benoît Candidate cosmic filament in the
GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in the
GJ526 field, mapped with the NIKA2 camera S. Berta Candidate cosmic filament
in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in
the GJ526 field, mapped with the NIKA2 camera M. Béthermin Candidate cosmic
filament in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic
filament in the GJ526 field, mapped with the NIKA2 camera L. Bing Candidate
cosmic filament in the GJ526 field, mapped with the NIKA2 cameraCandidate
cosmic filament in the GJ526 field, mapped with the NIKA2 camera O. Bourrion
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera M. Calvo Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera A. Catalano Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera A. Coulais Candidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 camera M. De Petris Candidate cosmic filament in the
GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in the
GJ526 field, mapped with the NIKA2 camera S. Doyle Candidate cosmic filament
in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in
the GJ526 field, mapped with the NIKA2 camera E. F. C. Driessen Candidate
cosmic filament in the GJ526 field, mapped with the NIKA2 cameraCandidate
cosmic filament in the GJ526 field, mapped with the NIKA2 camera A. Gomez
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera J. Goupy Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera F. Kéruzoré Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera C. Kramer Candidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 camera B. Ladjelate Candidate cosmic filament in the
GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in the
GJ526 field, mapped with the NIKA2 camera S. Leclercq Candidate cosmic
filament in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic
filament in the GJ526 field, mapped with the NIKA2 camera J. F. Macías-Pérez
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera A. Maury Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera P. Mauskopf Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera F. Mayet Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera A. Monfardini Candidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 camera M. Muñoz-Echeverría Candidate cosmic filament in
the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in the
GJ526 field, mapped with the NIKA2 camera L. Perotto Candidate cosmic filament
in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in
the GJ526 field, mapped with the NIKA2 camera G. Pisano Candidate cosmic
filament in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic
filament in the GJ526 field, mapped with the NIKA2 camera N. Ponthieu
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera V. Revéret Candidate cosmic filament in the GJ526 field, mapped with
the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera A. J. Rigby Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera A. Ritacco Candidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526 field,
mapped with the NIKA2 camera C. Romero Candidate cosmic filament in the GJ526
field, mapped with the NIKA2 cameraCandidate cosmic filament in the GJ526
field, mapped with the NIKA2 camera H. Roussel Candidate cosmic filament in
the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in the
GJ526 field, mapped with the NIKA2 camera F. Ruppin Candidate cosmic filament
in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic filament in
the GJ526 field, mapped with the NIKA2 camera K. Schuster Candidate cosmic
filament in the GJ526 field, mapped with the NIKA2 cameraCandidate cosmic
filament in the GJ526 field, mapped with the NIKA2 camera S. Shu Candidate
cosmic filament in the GJ526 field, mapped with the NIKA2 cameraCandidate
cosmic filament in the GJ526 field, mapped with the NIKA2 camera A. Sievers
Candidate cosmic filament in the GJ526 field, mapped with the NIKA2
cameraCandidate cosmic filament in the GJ526 field, mapped with the NIKA2
camera C. Tucker Candidate cosmic filament in the GJ526 field, mapped with the
NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with the
NIKA2 camera R. Zylka Candidate cosmic filament in the GJ526 field, mapped
with the NIKA2 cameraCandidate cosmic filament in the GJ526 field, mapped with
the NIKA2 camera

(Received January 15, 2022; accepted January 15 , 2022)

Distinctive large-scale structures have been identified in the spatial
distribution of optical galaxies up to redshift $z\sim 1$. In the more distant
universe, the relationship between the dust-obscured population of star-
forming galaxies observed at millimetre wavelengths and the network of cosmic
filaments of dark matter apparent in all cosmological hydrodynamical
simulations is still under study. Using the NIKA2 dual-band millimetre camera,
we mapped a field of $\sim 90$ arcminutes2 in the direction of the star GJ526
simultaneously in its 1.15-mm and 2.0-mm continuum wavebands to investigate
the nature of the quasi-alignment of five sources found ten years earlier with
the MAMBO camera at 1.2 mm. We find that these sources are not clumps of a
circumstellar debris disc around this star as initially hypothesized. Rather,
they must be dust-obscured star-forming galaxies, or sub-millimetre galaxies
(SMGs), in the distant background. The new NIKA2 map at 1.15 mm reveals a
total of seven SMGs distributed in projection on the sky along a filament-like
structure crossing the whole observed field. Furthermore, we show that the
NIKA2 and supplemental Herschel photometric data are compatible with a model
of the spectral energy distributions (SEDs) of these sources when a common
redshift of 2.5 and typical values of the dust parameters for SMGs are
adopted. Hence, we speculate that these SMGs might be located in a filament of
the distant ‘cosmic web’. The length of this candidate cosmic filament
crossing the whole map is at least 4 cMpc (comoving), and the separations
between sources are between 0.25 cMpc and 1.25 cMpc at this redshift, in line
with expectations from cosmological simulations. Nonetheless, further
observations to determine the precise spectroscopic redshifts of these sources
are required to definitively support this hypothesis of SMGs embedded in a
cosmic filament of dark matter.

###### Key Words.:

galaxies: evolution – Galaxy: formation – galaxies: groups – large-scale
structure of Universe – galaxies: clusters: intracluster medium

##  1 Introduction

Sub-millimetre Galaxies (SMGs) found in submillimetre- and millimetre-
wavelength surveys are forming stars at prodigious rates, sometimes exceeding
1000 $M_{\odot}$/yr, and are enshrouded in dust with bright far-infrared
luminosities $>10^{12}\,L_{\odot}$ (Hodge & da Cunha, 2020, e.g.). Thus, SMGs
have only faint or undetectable optical counterparts but probe the most active
cosmic epoch of galaxy formation at $z\cong 2$.

Theory predicts that galaxy formation preferentially occurs along large-scale
dark matter filamentary or sheetlike overdense regions that are related to
initial fluctuations of the density field in the primordial universe (Bond et
al., 1996) and formed the ‘cosmic web’ as depicted by cosmological
hydrodynamical simulations. It is thought that galaxy growth and high star
formation at an early stage of cosmic evolution is critically tied to massive
haloes of dark matter at the knots of this network of filaments (Béthermin et
al., 2013). A large fraction of the neutral hydrogen from the inter galactic
medium falls into these filaments and constantly replenishes the gas
reservoirs of galaxies through cold gas accretion; the cold gas streams along
the filaments, survives the shocks at the virial radii of the dark matter
haloes and infalls onto the galaxies to fuel their star formation and growth
(Kereš et al., 2005; Dekel et al., 2009).

Definitive observational confirmation of this mechanism $-$ cold gas accretion
along filaments of the cosmic web $-$ is actively sought. Photoionized
Lyman-$\alpha$ nebulae around quasars have been observed with morphologies and
kinematics suggestive of intergalactic gas filaments and gas infall (Cantalupo
et al., 2014; Hennawi et al., 2015; Martin et al., 2015, 2019) but alternative
scenarios based on outflows cannot be ruled out. Recently, extended
Ly-$\alpha$ emission plausibly powered by collisional ionization along
filamentary structures has been observed. Umehata et al. (2019) have
discovered two Ly-$\alpha$ filaments more than 1 Mpc in extent within the
protocluster SSA22 at $z=3.1$ and, remarkably, also detected embedded SMGs and
X-ray sources that are distributed all along these two filaments. Bacon et al.
(2021) report the discovery of diffuse extended Ly-$\alpha$ emission from
redshift 3.1 to 4.5, tracing cosmic web filaments on scales of $2.5-4$ Mpc in
the MUSE Extremely Deep Field, which partially overlaps the Hubble Ultra Deep
Field. Also, Daddi et al. (2021) report on a plausible observation of cold gas
accretion streams towards the massive potential well of the protocluster
RO-1001 at $z=2.91$ and on the detection of three filaments apparently
connecting to the associated 300 kpc-wide Ly-$\alpha$ nebula. In cosmological
simulations, the relationship between SMGs and the network of cosmic filaments
of dark matter is debated (Miller et al., 2015).

Nowadays, a region of the sky several hundreds of arcseconds in size,
corresponding to several cMpc (comoving) at redshift $z\leavevmode\nobreak\
>\leavevmode\nobreak\ 1$, can be efficiently mapped at millimetre wavelengths
using a modern camera such as NIKA2 installed on the IRAM 30-metre telescope
(Adam et al., 2018). In maps covering such a scale, the spatial distribution
of SMGs can be compared to the cosmic web in cosmological simulations. We
present NIKA2 data that are a serendipitous contribution to this endeavour.

In 2007, as part of a survey designed to search for new circumstellar discs
around 50 stars, the field in the direction of the nearby star GJ526 was
mapped in the 1.2-mm continuum waveband using the MAMBO camera installed on
the IRAM 30-metre telescope and a quasi-alignment of five sources along with
this star was found in projection on the sky. At the time, it was not possible
to conclude whether these sources were background SMGs or part of a large,
clumpy circumstellar disc seen edge-on and centred on this star (Lestrade et
al., 2009). In 2017, we again observed the same field with the NIKA2 camera to
test whether these sources had remained fixed and were in the background, or
whether they had moved at the same rate and in the same direction as the star,
and were clumps in a circumstellar disc. Based on the proper motion of the
star, this possible displacement was estimated to be as large as
$22.6^{\prime\prime}$ in the south-west direction after ten years.

In this paper, we report our finding with NIKA2 that the MAMBO sources did not
move over this ten-year period and so they must be SMGs in the distant
background. In Sect. 2 we present the NIKA2 observations, and in Sect. 3 we
present the NIKA2 maps at 1.15 mm and 2.0 mm, and discuss the overdensity of
sources in the field. In Sect. 4 we discuss the nature of the sources
characterized by their spectral energy distributions (SEDs) constrained with
our NIKA2 photometry and Herschel supplemental data. Finally, in Sect. 5 we
discuss the remarkable spatial distribution of these sources in the 1.15-mm
NIKA2 map along a filament-like structure and a possible association with the
cosmic web of the cosmological simulations.

##  2 Observations

The observations were conducted with the NIKA2 dual-band millimetre camera
installed on the IRAM 30-metre telescope at Pico Veleta (altitude 2850m) in
Spain (Monfardini et al., 2011; Catalano et al., 2014; Calvo et al., 2016;
Bourrion et al., 2016; Adam et al., 2018; Perotto et al., 2020). NIKA2
operates with kinetic inductance detectors (KID) arranged into three arrays :
two with a maximum transmission around 260 GHz (1.15 mm) with 1140 detectors
in each polarization, and one at 150 GHz (2.0 mm) with 616 detectors at a
single polarization. The transmission bands of the detectors are broad with a
$\sim 50$ GHz full width at half maximum (FWHM) transmission at both
wavelengths. The combination of its large field of view (6.5 arcminutes) and
high sensitivity provides the NIKA2 camera with a high mapping speed (111 and
1388 arcminutes2 mJy-2 h-1 at 260 GHz and 150 GHz, respectively, at zero
atmospheric opacity ; Perotto et al., 2020). The angular resolutions (main
beam FWHM) are 17.7 and 11.2 arcseconds at 150 and 260 GHz, respectively.

The data presented in this study were obtained during the science verification
observations of the NIKA2 camera on 24 April 2017, on 14, 17, 18, 19, and 20
February 2018, and on 14 March 2018, in average weather conditions
(atmospheric opacity between 0.10 and 0.25 at 2 mm, and between 0.20 and 0.45
at 1 mm) and at elevations between 33∘ and 67∘. The observing strategy
consisted of a series of $9^{\prime}\times 4.5^{\prime}$ on-the-fly scans with
a scanning speed of 40”/sec along two orthogonal orientations ($-60^{\circ}$,
$+30^{\circ}$) in the AZ-EL coordinate system of the telescope to minimize
residual stripping patterns in the maps. We defined the map centre at the
J2000 coordinates of the star GJ526 as in the initial MAMBO map (RA :
13h45m44.52s and DEC : $+14^{\circ}53^{\prime}20.6^{\prime\prime}$). These
observations are split in 167 scans that correspond to a total of 9.3 hours of
observations on source. For calibration purposes, skydips were conducted to
determine the coupling of each KID to the sky emission. An opacity correction
could then be computed along the line of sight of each observation scan.

The time-ordered information (TOI) of each detector was processed in order to
remove the contribution of atmospheric and electronic correlated noises and to
obtain the maps at both wavelengths following the procedure initially
described in Adam et al. (2014), and modified in Ponthieu et al. (in
preparation). In short, an iterative process between maps and TOI was carried
out until the maps did not change anymore (less than $0.1\sigma$) after three
iterations. For each sub-scan, the first four harmonics at 2 mm (five at 1
mm), each made of one sine and one cosine, were fit to the TOI with a least-
square method and removed. In more detail, there was no masking of the source
but rather a downweighting of large signal-to-noise excursions. This
downweighting becomes more accurate when the map is close to the final map
after three iterations. On the first iteration, the initial map was set to
zero. The process includes some inhomogeneous weighing of the data samples and
so leads to some large spatial scale filtering. Thus, the procedure is
designed for optimized detection of point-sources and not extended sources.
Nonetheless, the more aggressive filtering and the avoidance of masks in this
procedure induce a filtering effect that has to be determined by simulations
of point sources injected into the data. Thus, we established that a
photometric correction of 25 % that is constant for the range of S/Ns
displayed by the detected sources in our study must be applied. It should be
noted that this iterative part of the pipeline used to process the data is an
addition to the standard pipeline used by Perotto et al. (2020). Finally, we
add that the noise of each sample was propagated to the map of each scan with
a standard inverse variance weighing. A global rescaling of the noise for each
map was done to make the standard deviation of S/Ns in noise map unity (see
Sect. 3), in order to measure an accurate S/N for point sources.

Point sources were detected by fitting the map with a Gaussian point spread
function, in other words effectively applying a noise-weighted matched-
filtering of the map with a fixed width Gaussian. Position accuracy is better
than 3 arcseconds (for each coordinate) for sources detected above $4\sigma$.
The photometric correction of 25 % described above was applied to the fit
amplitude. The flux density scale was based on specific observations of the
primary calibrator Uranus to calibrate all individual detectors of each array
and on frequent standard observations of the secondary calibrator MWC349 to
check the stability of this calibration over a NIKA2 run of several days. The
resulting absolute photometric accuracy is of the order of 15 %. Although the
weather conditions were not optimal, the expected NIKA2 sensitivity at zero
opacity ($30\pm 3$ and $9\pm 1$ mJy$.\sqrt{s}$ at 1.15 mm and 2.0 mm,
respectively ; Perotto et al., 2020) was recovered once the opacity was taken
into account.

Figure 1:  GJ526 field, mapped with NIKA2 in the 1.15-mm continuum waveband
(colour scale : mJy/beam, contours : S/N). Source IDs are those in Table 1.
The curve traces the filament-like structure identified across the map. S/N
contours are $-3,+3,+4,+5,+6,....$ (the negative contours are dashed white
lines and can be seen better when the map is zoomed in the electronic version.
The S/N=$-4$ level was processed in plotting the map but no such negative
feature was apparent. For the noise rms per pixel, we refer the reader to Fig.
5. Pixel size is 2 arcseconds. J2000 coordinates of the map centre are
13h45m44.52s and $+14^{\circ}53^{\prime}20.6^{\prime\prime}$. The filled black
circle in the bottom right corner shows the HPBW.

Figure 2: Same as for Fig. 2, but observed at 2.0 mm.

Figure 3:  S/N histogram of the jack-knifed map at 1.15 mm. The mean $\mu$ and
standard deviation $\sigma$, as well as the minimum and maximum of the S/Ns,
are indicated in the figure. In red, a Gaussian not fit but set with zero
mean, standard deviation unity, and its integral summing up to the total
number of S/Ns, is overplotted to show that the theoretical distribution of
S/Ns for Gaussain noise satisfactorily matches the histogram of the data in
the map.

Figure 4: Same as for Fig. 5, but observed at 2 mm. The slight excess found at
the positive tail is discussed in the text.

Figure 5: Radial variation in the noise rms per pixel in the 1.15-mm map
(pixel size = $2^{\prime\prime}$). At 2mm, the variation is the same but
reduced by a factor of roughly three. The variation is isotropic to a good
approximation over the maps.

##  3 NIKA2 maps

The NIKA2 maps of the GJ526 field $560^{\prime\prime}\times
560^{\prime\prime}$ in size are shown in Fig. 2 at 1.15 mm and in Fig. 2 at
2.0 mm. Millijansky point-like sources are apparent at both wavelengths. In
order to determine the detection threshold, we analysed the noise distribution
in computing S/Ns at all pixels of the jack-knifed map at each wavelength. In
such a map of the noise, the sources were eliminated after subtraction between
individual maps of consecutive pairs of scans with the two orthogonal scanning
orientations and finally the co-addition of all pairs. At 1.15 mm, we found
that the S/N distribution is Gaussian, centred almost on zero and has a
standard deviation of almost unity, as expected for normally distributed noise
(see Fig. 5). However, at 2 mm, although the distribution is also centred
close to zero and has a standard deviation close to unity, there is an excess
at the positive tail that makes it deviate slightly from Gaussian, despite
careful reprocessing (see Fig. 5). This was unexpected since the atmospheric
fluctuations are less severe at this wavelength. Close inspection of the data
showed that electronic correlated noise in the 2.0 mm sampling modules of the
instrument was higher than usual during the observations. With this caveat in
mind, source extraction and photometry were carried out with caution.

To set the source detection threshold, we searched for any possible detections
in the jack-knifed maps using the same fitting procedure of a Gaussian point
spread function as described in Sect. 2. Clearly, any statistically
significant detection in this noise-only map would be spurious. At 1.15 mm, we
find that there are nine spurious detections with S/N $>\leavevmode\nobreak\
+3$ and ten with S/N $<\leavevmode\nobreak\ -3$, evenly distributed over the
jack-knifed map, and none with —S/N— $\geq\leavevmode\nobreak\ 4$. Similarly,
at 2.0 mm, there are eight spurious detections with S/N $>\leavevmode\nobreak\
+3$ and nine with S/N $<\leavevmode\nobreak\ -3$, and none with —S/N—
$\geq\leavevmode\nobreak\ 4$. We conclude that the threshold S/N = 4 is a
reliable criterion for a robust source detection at both wavelengths.

In Table 1 we list the 21 sources with S/N $\geq\leavevmode\nobreak\ 3$ found
in the 1.15-mm NIKA2 map, among which ten are unambiguously detected according
to our criterion (S/N $\geq\leavevmode\nobreak\ 4$) and have flux densities
between $\sim$2 and $\sim$5 mJy. Statistically, about two additional sources
should be real among the remaining eleven sources ($3\leavevmode\nobreak\
\leq\leavevmode\nobreak\ {\rm S/N}\leavevmode\nobreak\ <\leavevmode\nobreak\
4$) since only nine positive detections are expected to be spurious above
$+3\sigma$, as explained above. Evidently, these potential sources at 1.15 mm
cannot be identified unless additional inputs are included. For instance, ID 8
is a $3.6\sigma$ source in the 1.15-mm NIKA2 map but can also be identified in
the MAMBO map at slightly above $3\sigma$ (see map in Fig. 1 of Lestrade et
al., 2009) ; thus ID 8 is likely a real source at the level of 4.7$\sigma$ in
quadratically combining the two S/Ns. Among the unambiguously detected sources
at 1.15 mm, only two are also unambiguously detected at 2 mm (ID 16 and ID 9),
and four additional sources have S/Ns of between 3 and 4 at 2.0 mm (see Table
1). This deficit of detections at 2 mm can be explained by the dust emissivity
index $\beta$ of SMGs that is between 1 and 2, or sometimes steeper (see also
Sect. 4). Finally, there is a source which is slightly above $4\sigma$ at 2.0
mm (north-east of map and see Table 1) but has surprisingly no 1.15 mm
counterpart. As already mentioned, electronic correlated noise higher than
standard during the observations must explain this deviation from normal
statistics in the 2.0 mm map, which we consider not as reliable as the 1.15-mm
map.

Finally, we address flux boosting that alters the measured flux densities
provided in Table 1. In a flux-limited survey, it is well known that sources
with a low S/N are detected at flux densities systematically higher than their
true flux densities and this effect is exacerbated when the source population
increases in number with decreasing flux (e.g. Coppin et al., 2005). We
estimated flux boosting with 1000 simulations of the sky in our NIKA2 maps
following the prescriptions by Scott et al. (2008) with their semi-Bayesian
approach. We find that it is statistically less than 15% for a S/N above 4,
and steadly increases up to 50% for a S/N of 3, similarly at 1 mm and 2 mm. We
chose not to correct downwards the quoted flux densities in Table 1.

Table 1: NIKA2 astrometry and photometry. Source IDs assigned in column 1 are used throughout the text.  ID | Name | $\alpha_{2000}$ | $\delta_{2000}$ | $S_{260GHz}$ | $S/N$ | $S_{150GHz}$ | $S/N$
---|---|---|---|---|---|---|---
|  | $(h\leavevmode\nobreak\ \leavevmode\nobreak\ m\leavevmode\nobreak\ \leavevmode\nobreak\ s)$ |  $(^{\circ}\leavevmode\nobreak\ \leavevmode\nobreak\ {}^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\ {}^{\prime\prime})$ | (mJy) | 260GHz | (mJy) | 150GHz
Unambiguously detected NIKA2 sources (S/N $\geq$ 4 at 1.15 mm)
16 | NIKA2$\\_$J134546.6+145239 | 13:45:46.63 | 14:52:39.7 | 4.11$\pm$ 0.43 | 9.6 | 0.98$\pm$ 0.09 | 10.3
1 | NIKA2$\\_$J134533.4+144950 | 13:45:33.48 | 14:49:50.5 | 3.96$\pm$ 0.75 | 5.3 | 0.41$\pm$ 0.17 | 2.4
3 | NIKA2$\\_$J134537.1+145631 | 13:45:37.18 | 14:56:31.9 | 3.67$\pm$ 0.60 | 6.1 | 0.39$\pm$ 0.13 | 3.0
9 | NIKA2$\\_$J134540.6+145442 | 13:45:40.61 | 14:54:42.8 | 3.13$\pm$ 0.45 | 6.9 | 0.42$\pm$ 0.10 | 4.3
10 | NIKA2$\\_$J134541.0+145414 | 13:45:41.09 | 14:54:14.8 | 2.97$\pm$ 0.44 | 6.8 | 0.23$\pm$ 0.09 | 2.4
6 | NIKA2$\\_$J134538.3+145542 | 13:45:38.36 | 14:55:42.2 | 2.95$\pm$ 0.52 | 5.7 | 0.37$\pm$ 0.11 | 3.3
7 | NIKA2$\\_$J134538.9+145145 | 13:45:38.97 | 14:51:45.4 | 2.87$\pm$ 0.47 | 6.1 | 0.41$\pm$ 0.11 | 3.9
22 | NIKA2$\\_$J134555.5+145401 | 13:45:55.51 | 14:54: 1.1 | 2.40$\pm$ 0.52 | 4.6 | 0.02$\pm$ 0.11 | 0.2
18 | NIKA2$\\_$J134551.5+145159 | 13:45:51.51 | 14:51:59.0 | 2.15$\pm$ 0.48 | 4.5 | 0.34$\pm$ 0.11 | 3.1
13 | NIKA2$\\_$J134544.6+145159 | 13:45:44.69 | 14:51:59.4 | 2.08$\pm$ 0.44 | 4.7 | 0.06$\pm$ 0.10 | 0.6
Possibly detected NIKA2 sources (3 $\leq$ S/N $<$ 4 at 1.15 mm)
0 |  | 13:45:27.41 | 14:51: 6.8 | 2.68$\pm$ 0.79 | 3.4 | 0.20$\pm$ 0.18 | 1.1
25 |  | 13:45:59.81 | 14:50:29.8 | 2.61$\pm$ 0.79 | 3.3 | 0.38$\pm$ 0.18 | 2.1
5 |  | 13:45:38.33 | 14:49:24.6 | 2.57$\pm$ 0.71 | 3.6 | 0.28$\pm$ 0.16 | 1.7
17 |  | 13:45:49.08 | 14:57:36.4 | 2.25$\pm$ 0.72 | 3.1 | 0.14$\pm$ 0.15 | 0.9
14 |  | 13:45:45.51 | 14:57: 5.0 | 2.20$\pm$ 0.61 | 3.6 | 0.27$\pm$ 0.13 | 2.0
21 |  | 13:45:55.42 | 14:50:19.7 | 2.20$\pm$ 0.67 | 3.3 | 0.48$\pm$ 0.15 | 3.1
12 |  | 13:45:44.66 | 14:50:15.4 | 2.15$\pm$ 0.56 | 3.8 | 0.41$\pm$ 0.13 | 3.2
2 |  | 13:45:34.13 | 14:52:18.1 | 1.67$\pm$ 0.52 | 3.2 | 0.16$\pm$ 0.11 | 1.4
19 |  | 13:45:51.63 | 14:55: 1.6 | 1.57$\pm$ 0.49 | 3.2 | 0.15$\pm$ 0.11 | 1.4
4 |  | 13:45:37.30 | 14:53:43.1 | 1.55$\pm$ 0.45 | 3.4 | 0.08$\pm$ 0.10 | 0.8
8 |  | 13:45:40.15 | 14:53:36.2 | 1.52$\pm$ 0.43 | 3.6 | 0.39$\pm$ 0.09 | 4.1
Table 2: Comparison between NIKA2 and MAMBO ID | $NIKA2$ | $MAMBO$ | Differences
---|---|---|---
| $S_{260GHz}$ | $S_{250GHz}$ | $\Delta\alpha$ | $\Delta\delta$
| (mJy) | (mJy) | $(^{\prime\prime})$ | $(^{\prime\prime})$
7 | 2.87$\pm$ 0.47 | 3.2$\pm$0.7 | $0$ | $+6$
9 | 3.13$\pm$ 0.45 | 4.3$\pm$0.8 | $-3$ | $-2$
10 | 2.97$\pm$ 0.45 | 6.3$\pm$1.0 | $-3$ | $-2$
16 | 4.11$\pm$ 0.43 | 5.6$\pm$0.7 | $0$ | $-1$
MM134543+145317 | 1.3$\pm$ 0.45 | 3.0$\pm$0.7 | $+3$ | $-2$

###  3.1 Nature of the MAMBO sources

As already mentioned, the field around GJ526 was mapped in 2007 with MAMBO.
Five sources were found quasi-aligned and within $100^{\prime\prime}$ from the
star GJ526 coordinates, and they were reported in Table 2 of Lestrade et al.
(2009). They were hypothesized to be the clumps of a large, broken disc seen
almost edge-on surrounding this star. With our new NIKA2 observations, four of
these five MAMBO sources were detected anew in the NIKA2 map at 1.15 mm, and
are IDs 7, 9, 10, and 16 in Table 1 herein. The fifth MAMBO source $-$
MM134543+145317, a 4.3$\sigma$ MAMBO detection in Table 2 of Lestrade et al.
(2009) $-$ was also identified in the 1.15-mm NIKA2 map but only at the level
of 2.8$\sigma$ and so was not retained in our final NIKA2 source list. Such a
difference in detection levels would arise from a S/N underestimate and
overestimate of $0.75\sigma$ by NIKA2 and MAMBO, respectively.

Astrometric and photometric comparisons between the MAMBO map and the NIKA2
map, which is 40% deeper, are given in Table 2 herein. The coordinate
differences between the four MAMBO sources at epoch 2007 (Table 2 of Lestrade
et al. 2009) and their NIKA2 counterparts at epoch 2017 in Table 1 herein are
only a few arcseconds, and so much less than the displacement of
$22.6^{\prime\prime}$ in the south-east direction inferred from the star
motion. Rather, these coordinates differences are consistent with telescope
pointing errors (rms$<3^{\prime\prime}$, see Table 18 of Perotto et al., 2020)
combined with astrometric errors for sources with low S/Ns. Thus, we conclude
that the MAMBO sources have not moved in concert with the star and are not
clumps of a disc associated with GJ526. They must instead be SMGs in the
background, probably unlensed owing to their low flux density level.

The MAMBO flux densities in Table 2 of Lestrade et al. (2009) are comparable
with the NIKA2 flux densities at 1.15 mm in Table 1 herein to better than 1.5
times the quadratically combined uncertainties of the two measurements, except
for source ID 10 for which the difference is 3.1 times the combined
uncertainty. This source would have faded from $6.3\pm 1$mJy at epoch 2007 to
$2.97\pm 0.62$mJy ten years later if this difference were real. This is
physically unrealistic for an SMG and we favour calibration difficulties with
MAMBO and/or NIKA2, inherent to low S/N sources. It should be noted that the
differences in flux densities due to differences in reference frequencies of
the two cameras, 250 GHz for MAMBO and 260 GHz for NIKA2, are negligible.

###  3.2 Counterparts in the optical, infrared and radio domains

To search for counterparts to the NIKA2 sources in the optical, mid-infrared,
far-infrared, and radio domains, we inspected images of the Sloan Digital Sky
Survey (SDSS), the Wide-field Infrared Survey Explorer (WISE) (Wright et al.,
2010), the Herschel Observatory (Griffin et al., 2010; Poglitsch et al., 2010)
and the NRAO VLA Sky Survey (NVSS) (Condon et al., 1998). Owing to the NIKA2
position uncertainties, we found no convincing counterpart in SDSS, WISE, or
NVSS. In NVSS, with a 3$\sigma$ limit of 1.5 mJy at 1.4 GHz, this is not
surprising unless it is from an AGN

More interesting were our findings in the Herschel archive. Seven NIKA2
sources have supplemental photometric data from SPIRE (250 $\mu$m, 350 $\mu$m
and 500 $\mu$m) and two sources have PACS data (100 $\mu$m and 160 $\mu$m).
For SPIRE, the flux densities of three sources (IDs 7, 10, and 18) are taken
from the Herschel/SPIRE Point Source Catalogue (HSPSC) (Schulz et al., 2017)
111https://irsa.ipac.caltech.edu/Missions/herschel.html and are reported in
Table 3 herein, along with their HSPSC identifiers. We also uploaded the SPIRE
observation ID$\\_$obs 1342234791 of the Herschel Open-Time Key project DEBRIS
(Matthews et al., 2010) to determine the flux densities of three additional
sources in Table 3 (IDs 14, 17, and 21). Their coordinate differences between
SPIRE and NIKA2 are, at most, 1.5 times the quadratically combined astrometric
uncertainties of the two instruments ($\sim 5^{\prime\prime}$). Additionally,
we determined SPIRE upper limits for the other NIKA2 sources since they are
within the SPIRE maps of this programme and not close to the map border of
lesser quality. We measured a representative rms in an area where there is no
apparent source in the maps to estimate the SPIRE $3\sigma$ upper limits of 34
mJy, 33 mJy, and 31 mJy at $250\mu$m, $350\mu$m, and $500\mu$m, respectively.
For PACS, we uploaded the observation ID$\\_$obs 1342213079 of the same Open-
Time Key project and found that ID 7 is detected and ID 10 is not. We
estimated the flux densities of ID 7 using aperture photometry (radius=30′′)
and found $S_{\nu}=10\pm 2$ mJy at 100 $\mu$m and $S_{\nu}=52\pm 3$ mJy at 160
$\leavevmode\nobreak\ \mu$m. The $3\sigma$ upper limits of ID 10 from this
observation are 6 mJy at 100 $\mu$m and 9 mJy at 160 $\mu$m. The other NIKA2
sources are not part of the PACS maps (ID 18 is at the border of the maps and
the data are unusable).

Table 3:  Herschel/SPIRE photometry  ID | $SPIRE$
---|---
| 250$\mu$m | 350$\mu$m | 500$\mu$m | HSPSC ID
| (mJy) | (mJy) | (mJy) |
7 | 65.6$\pm$7.0 | 59.9$\pm$7.4 | 30.9$\pm$6.9 | J1345.65+1451.7
10 | 32.3$\pm$6.0 | 35.8$\pm$7.8 | 31.6$\pm$7.6 | J1345.68+1454.3
14 | 49.4$\pm$9.3 | 40.5$\pm$9.1 | 27.4$\pm$6.5 |
17 | 33.1$\pm$6.5 | 29.8$\pm$5.4 | $<23$ |
18 | 77.9$\pm$6.9 | 62.2$\pm$7.2 | 29.1$\pm$7.5 | J1345.85+1451.9
21 | 45.8$\pm$9.6 | 31.2$\pm$7.2 | $<23$ |

###  3.3 Background source number count in the GJ526 field

In Sect. 3.1, we concluded that there are ten sources unambiguously detected
in the 1.15-mm NIKA2 map with flux densities larger than 2 mJy. This can be
usefully compared to the cumulative source number counts in other fields at a
similar wavelength. The COSMOS field and the Lockman Hole North field have
been surveyed at 1.2 mm with MAMBO/IRAM 30m (Greve et al., 2004; Bertoldi et
al., 2007; Lindner et al., 2011) and at 1.1 mm with AzTEC/ASTE (Scott et al.,
2010; Hatsukade et al., 2011). Also, part of the GOODS field has been surveyed
at 1.1mm with ALMA (Franco et al., 2018).

The noise rms in the NIKA2 maps is not uniform, and increases radially from
the centre, as shown in Fig. 5, because of the scanning strategy used for the
observations. However, as it can be seen in this figure, the noise rms
increases by less than 10% within a radial distance of $150^{\prime\prime}$
from the map centre and so we first consider this central part of the map as
the region of interest with a quasi-uniform noise rms. Practically, the
statistical uncertainty of the flux density of a source in this central region
is $\sim 0.50$mJy/beam at 1.15 mm. Thus, the detection threshold of $4\sigma$
corresponds to a flux density of 2 mJy at this wavelength. Lindner et al.
(2011) combined their observations with other single dish studies to report
the cumulative source number count $N(S_{1.2mm}\geq 2.0{\rm mJy})=500\pm 200$
sources/deg2. Franco et al. (2018) report the lower formal value
$N(S_{1.1mm}\geq 1.8{\rm mJy})=209_{-119}^{+178}$ sources/deg2 with ALMA at a
similar wavelength. It is known that the number counts derived from
interferometric data are lower than the number counts derived from single-dish
data because of source blending (Karim et al., 2013; Béthermin et al., 2017).
Therefore, consistently with our NIKA2 observations, we adopted the single
dish count of Lindner et al. (2011) for our probability calculation. There are
six sources (IDs 7, 9, 10, 13, 16, and 18) that are unambiguously detected
(S/N $>$ 4) in the central region of the 1.15-mm map with quasi-uniform rms
($r<150^{\prime\prime}$), while 2.7 sources are expected with the density 500
sources/deg2. The Poisson probability of exactly six sources with
$S_{1.2mm}>2$ mJy being found in this region, given this mean surface density,
is as low as 3.7%. It may be argued that the likelihood of finding an excess
of sources in the GJ526 field is actually enhanced because this field is one
in a collection of 50 other similar-sized fields that were mapped in the
initial MAMBO survey (Lestrade et al., 2006, 2009). Although it is relevant to
recall this context, a clear-cut conclusion cannot be drawn because the other
fields were observed with significantly less integration times than GJ526, and
so are less deep. We discuss now the more intriguing spatial distribution of
the sources of this overdensity along a filament-like structure.

###  3.4 Filament-like structure

Now using the whole map, it is remarkable that seven sources (IDs 3, 6, 9, 10,
16, 18, and 25) are spatially distributed along an arc in projection on the
sky across the whole map, as highlighted in Fig. 2. The first six sources are
unambiguously detected with flux densities larger than 2 mJy at 1.15 mm, while
the last one (ID 25) is a $3.3\sigma$ possible detection with a flux density
of $2.61\pm 0.79$ mJy at this wavelength (Table 1). We can estimate the
probability that such a quasi-alignment of sources occurs by chance using
small, consecutive sectors all along the arc with the small opening angle
$2\delta$, as sketched in Fig. 6, and in which the averaged source number
surface density $\mu$ is $N(S_{1.2mm}\geq 2$mJy) $=500$ sources/degree2, as
referenced in Sect. 3.3. For any triplet of sources along the arc, the
probability that the third source does not deviate more than $\pm\delta$ in
degrees from alignment with the first two sources is the Poisson probability
$\mu\times\pi r^{2}\times 2\delta/360^{\circ})\exp(\mu\times\pi r^{2}\times
2\delta/360^{\circ}$ that exactly one source is found within the sector
centred on the second source, and having a radius $r$, which is the distance
between the second and third sources. In applying this formula for the first
triplet in the NIKA2 map (source IDs 3, 6, and 9), the probability of a chance
occurrence of quasi-alignment within $\pm 10^{\circ}$ is
$\sim\rm{500\leavevmode\nobreak\
sources/deg}^{2}\times\pi(70^{\prime\prime}/3600^{\prime\prime})^{2}\times
20^{\circ}/360^{\circ}=3.3\%$ with $r_{1}=70^{\prime\prime}$ for the distance
between source IDs 6 and 9. Similarly, the probability is 0.6% for the second
triplet (IDs 6, 9, and 10) with $r_{2}=30^{\prime\prime}$ between source IDs 9
and 10, it is 10.1% for the third triplet (IDs 9, 10, and 16) with
$r_{3}=130^{\prime\prime}$, it is 4.1% for the fourth triplet (IDs 10, 16, and
18) with $r_{4}=80^{\prime\prime}$, and it is 13.0% for the fifth triplet (IDs
16, 18, and 25) with $r_{5}=150^{\prime\prime}$. These triplets defined five
consecutive sectors (only two are represented in Fig. 6) that are independant,
and so the resulting probability is the product of the five probabilities
estimated above, that is $\sim\prod\limits_{i=1}^{5}\mu\pi
r_{i}^{2}2\delta/360^{\circ}$ (the exponential is approximatively unity in our
application), and this combined probability is as low as $1.1\times 10^{-7}$.
Therefore, it is very unlikely that this filament-like structure has occurred
by chance. This is true whatever is our concern that 50 similar-sized fields
have been observed in the initial MAMBO survey. In fact, to account for this,
the probability can be boosted 50 times and is $5.5\times 10^{-6}$ (i.e. still
very low).

Figure 6: Estimation of the probability for the chance alignment of sources.

##  4 Spectral energy distribution

Physical quantities of an SMG can be estimated from a model of its SED fit to
photometric data. In an SMG, copious amounts of diffuse small dust grains
absorb the intense UV radiation field of massive stars and re-radiate in the
far-infrared. Its intrinsic SED peaks at a wavelength corresponding to dust
temperature typically between 20 K and 60 K, and has a slope steeper than the
Rayleigh-Jeans regime because of the dust emissivity in the (sub-)millimetre
domain. For an SMG with bolometric luminosity $L_{bol}$ at redshift $z$, its
intrinsic SED $f_{\nu}$ is shifted towards longer wavelengths and the
detectable flux density at an observed frequency $\nu$ is (Blain et al., 2002,
eq. 3) :

$S_{\nu}={{1+z}\over{4\pi D_{L}^{2}}}L_{bol}{{f_{\nu(1+z)}}\over{\int
f_{\nu^{\prime}}d\nu^{\prime}}},$ (1)

where $D_{L}$ is the luminosity distance to redshift $z$ (Weinberg, 1972, eq.
15.3.24) and can be computed using the on-line calculator by Wright (2006)
222https://ned.ipac.caltech.edu/help/cosmology-calc.html in standard
$\Lambda$CDM cosmology 333$H_{0}=67.4$km/s/Mpc and $\Omega_{m}=0.315$. In this
equation, the intrinsic SED $f_{\nu}$ is from a modified black-body to account
for dust emissivity in the far-infrared and sub-millimetre. In other words, it
is a Planck function at dust temperature $T_{dust}$ multiplied by the
attenuation factor $1-exp(-(\nu/\nu_{c})^{\beta})$ with the critical frequency
$\nu_{c}$ at which the dust opacity reaches unity. In our study, we fixed
$\nu_{c}$ to the standard value 3000 GHz (100 $\mu$m) (Blain et al., 2003).

Figure 7: SEDs of all NIKA2 sources of interest. Each SED is from a single-
component modified black-body adjusted to the Herschel and NIKA2 data by
varying the dust temperature $T_{dust}$, the bolometric luminosity $L_{bol}$,
and the dust emissivity index $\beta$, and keeping the redshift fixed at
$z=2.5$ for all sources. Empty markers mark the upper limits. The seven
sources in the candidate cosmic filament discussed in this study are labelled
in red. $\chi_{\nu}^{2}$ is reduced with the number of degrees of freedom
$\nu$ (see details in Sect. 4).

For our sources, photometry data were limited to the flux densities from NIKA2
and Herschel in Tables 1 and 3, and only to the SPIRE upper limits for some of
them. Fitting the SED of each source to such a limited amount of data cannot
yield a reliable photometric redshift because of the known degeneracy between
redshift and dust temperature when only far-infrared and sub-millimetre data
are available (e.g. Sect. 2.10.1 in Blain et al. (2002)). Instead, we made the
most of the data at our disposal in testing whether or not they are compatible
with the hypothesis that all sources are at the same redshift. To this end, we
constrained $T_{dust}$ to the range from 20 K to 60 K (e.g. Chapman et al.,
2005; Strandet et al., 2016), and constrained all sources to be at the same
median redshift $z=2.5$ of the phenomenological model of SMGs by Béthermin et
al. (2015) for a survey depth similar to our NIKA2 observations at 1.15 mm.
Figure 7 shows that the photometric data are compatible with the hypothesis
that all sources are at the same redshift, and thus are in the same cosmic
structure. This remains true whether or not flux boosting is applied.
Evidently, it is only when their spectroscopic redshifts are measured that
this hypothesis could be confirmed. As far as the three parameters varied in
the model, we provide their acceptable ranges for each source in Fig. 7 ; it
is from 26 K to 60 K for $T_{dust}$ (keeping in mind our initial constraint),
it is from $2.5\times 10^{12}$ L⊙ to $23\times 10^{12}$ L⊙ for $L_{bol}$, and
it is from 1.0 to 2.0 for $\beta$. These luminosities are in the upper range
of star-forming ultra-luminous infrared galaxies (ULIRG) in the local universe
($10^{12}\leq L_{IR}/L_{\odot}\leq 10^{13}$ L⊙). The brightest source, ID 18,
is even formally in the range of hyper-luminous infrared galaxies (HyLIG). In
addition, in Fig. 7 we provide the star formation rates (SFRs) between 330 and
2255 M⊙/yr, as derived from the luminosity Lbol using SFR(M⊙/yr)=$1.1\times
10^{-10}\times$ Lbol(L⊙) (Hayward et al., 2014).

##  5 Discussion

The distinctive quasi-alignment of the seven sources, IDs 3, 6, 9, 10, 16, 18,
and 25, across the 1.15-mm NIKA2 map is qualitatively reminiscent of the
imprint of a cosmic filament in cosmological hydrodynamical simulations (e.g.
Illustris (Springel et al., 2018), SIMBA (Davé et al., 2019), NewHorizon
(Dubois et al., 2021), and Uchuu (Ishiyama et al., 2021)). Quantitatively,
these simulations provide spatial gauges that characterize the distribution of
dark matter haloes that host galaxies. These gauges can be usefully compared
to our observations. First, it is noticeable that the candidate filament
stretches across the whole NIKA2 maps $560^{\prime\prime}$ in size and so
extends at least over $\sim$4 Mpc (comoving) using the scaling factor of
8.3”/kpc, which depends only weakly on redshift for $1<z<6$
444https://ned.ipac.caltech.edu/help/cosmology-calc.html. Such an observed
extent is larger than the size of a typical cluster of galaxies and is
comparable to cosmic filaments in cosmological simulations where they extend
from 3 Mpc to 15 Mpc (see e.g. NewHorizon snapshot at $z=2$ in Figure 1 of
Dubois et al. (2021)). Second, it is noticeable that the five angular
separations $r_{i}$ between our NIKA2 sources given in Sect. 3.4 (completing
with $r_{0}=20^{\prime\prime}$ between IDs 3 and 8) correspond to linear
separations between $\sim$0.25 Mpc and $\sim$1.25 Mpc, using the same scaling
factor. Such observed separations are also comparable to cosmological
simulations where haloes hosting galaxies are typically separated by 0.1 to
1.5 Mpc (see e.g. the NewHorizon snapshot mentioned above). This match of
spatial gauges between our NIKA2 observations and cosmological simulations
supports our suggestion that the seven SMGs, which are quasi-aligned, bright
(L${}_{FIR}\leavevmode\nobreak\ >\leavevmode\nobreak\ 10^{12}$L⊙) , and
possibly at the same redshift, may trace a cosmic filament.

A more detailed comparison between our observations and cosmological
simulations is not an easy task. The first simulation that has been
specifically motivated to study SMGs is the Gadget-2 simulation by Davé et al.
(2010) based on a scenario of gas-rich satellite infall rather than
intergalactic gas infall along cosmic filaments. However, the spatial
distribution of SMGs of this simulation is not presented. In a more recent
study derived from the SIMBA simulation (Davé et al., 2019), Lovell et al.
(2021) also focus on SMGs and in their Fig. 4, they provide a 140 cMpc sized
map of the sky projection of all simulated galaxies with SFR $>$ 20 M⊙/yr and
within $0.1<z<10$. This simulated map is as deep as 0.25 mJy at 850 $\mu$m
(corresponding to SFR $=$ 20 M⊙/yr) and is convolved with the SCUBA2 beam.
Their resulting spatial distribution of SMGs is clearly non-uniform and
structured along filaments. Further comparison will require a simulation
zoomed to match the smaller scale of our NIKA2 maps ($\sim$ 5cMpc) and evolved
to match our NIKA2 source redshift, when measured.

##  6 Conclusions

With the NIKA2 dual-band camera at the 30-metre IRAM telescope, we made two
deep maps of a relatively large field of $\sim$90 arcminutes2 in the direction
of the star GJ526, simultaneously in its 1.15-mm and 2.0-mm continuum
wavebands with a $1\sigma$ sensitivity of 0.45 mJy/beam and 0.15 mJy/beam,
respectively, and with 9.3 hours of on-source observations owing to the high
mapping speed of the NIKA2/IRAM 30-metre system. In total, ten background
millijansky sources are unambiguously detected with $S/N>4$ and must be SMGs.

In projection on the sky, the spatial distribution of seven of these SMGs
along a filamentary structure that completely crosses both maps is remarkable
and has only a low probability of chance occurrence ($1.1\times 10^{-7}$). We
show that all available photometric data, though limited, are compatible with
a model of their SEDs in which they are at the same redshift $z=2.5$. Of
course, this must be confirmed with spectroscopic redshift measurements. At
this stage, we can only speculate that this structure could trace a cosmic
filament of dark matter, as is predicted by theory and apparent in
cosmological hydrodynamical simulations. In assuming the scaling factor of 8.3
kpc$/^{\prime\prime}$ at high redshift with standard cosmology for our maps,
the extent of this candidate cosmic filament is at least 4 Mpc and the
separations between SMGs are a fraction of Mpc comparable to distances between
dark matter haloes in high-resolution cosmological simulations. Again, our
interpretation is speculative at this stage and a full assessment requires
measurements of the spectroscopic redshifts of the SMGs in this candidate
filament. Deep spectroscopic follow-up observations of their molecular (CO) or
atomic (CI) lines redshifted into the ALMA or NOEMA bands can be envisioned
(Inoue et al., 2016; Hashimoto et al., 2018; Solimano et al., 2021; Bing et
al., 2022; Birkin et al., 2021). An attempt to optically identify them would
require both deeper optical observations and interferometric millimetre-
wavelength continuum observations to improve their positions to sub-arcsecond
accuracy Finally, the NIKA2 Cosmology Legacy Survey underway at the IRAM
30-metre telescope (Bing et al., 2022), designed to map fields more than 30
times larger than the GJ526 field, will be of great interest to further probe
the spatial distribution of SMGs in the millimetre-wavelength domain and
address their relationship with the network of cosmic filaments of dark matter
in simulations. In particular, establishing this relationship would support
the cold gas accretion mechanism to fuel galaxy growth. This endeavour is also
pursued with imaging of diffuse Ly $\alpha$ emission along filamentary
structures (e.g Umehata et al. (2019)).

###### Acknowledgements.

This work is based on observations carried out under project number 133-17
with the IRAM 30m telescope. IRAM is supported by INSU/CNRS (France), MPG
(Germany) and IGN (Spain). We would like to thank the IRAM staff for their
support during the campaigns. The NIKA2 dilution cryostat has been designed
and built at the Institut Néel. In particular, we acknowledge the crucial
contribution of the Cryogenics Group, and in particular Gregory Garde, Henri
Rodenas, Jean Paul Leggeri, Philippe Camus. This work has been partially
funded by the Foundation Nanoscience Grenoble and the LabEx FOCUS
ANR-11-LABX-0013. This work is supported by the French National Research
Agency under the contracts “MKIDS”, “NIKA” and ANR-15-CE31-0017 and in the
framework of the “Investissements d’avenir” program (ANR-15-IDEX-02). This
work has benefited from the support of the European Research Council Advanced
Grant ORISTARS under the European Union’s Seventh Framework Program (Grant
Agreement no. 291294). This research has made use of Herschel data ; Herschel
is an ESA space observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from NASA.
This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France. This research has made use of the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration.

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