# The number of small-degree vertices in matchstick graphs

Jérémy Lavollée  Department of Mathematics, Universitat Politècnica de
Catalunya, Spain.  and  Konrad J. Swanepoel  Department of Mathematics, The
London School of Economics and Political Science, Houghton Street, London WC2A
2AE, U.K.

###### Abstract.

A matchstick graph is a crossing-free unit-distance graph in the plane.
Harborth (1981) proposed the problem of determining whether there exists a
matchstick graph in which every vertex has degree exactly $5$. In 1982,
Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and
Pinchasi (2011) using a discharging method. We combine their method with the
isoperimetric inequality to show that there are $\Omega(\sqrt{n})$ vertices in
a matchstick graph on $n$ vertices that are of degree at most $4$, which is
asymptotically tight.

###### Key words and phrases:

matchstick graph, penny graph, plane unit-distance graph

###### 2020 Mathematics Subject Classification:

Primary 52C10. Secondary 05C10

##  1\. Introduction

Matchstick graphs are graphs which can be drawn in the plane with edges
represented by unit length straight-line segments that intersect only at their
endpoints. Harborth [4, 5] introduced these in 1981 and posed the problem of
finding the least number of vertices for a $k$-regular matchstick graph. In
1982, Blokhuis [2] proved that no 5-regular matchstick graph exists. Kurz and
Pinchasi [6] gave a short proof of the same result by a discharging method.

In this paper we find a stronger result by considering the vertices of degree
at most $4$ in a matchstick graph with no isolated vertices.

######  Theorem 1.

For any matchstick graph on $n$ vertices with no isolated vertices and $n_{i}$
vertices of degree $i$, we have
$4n_{1}+3n_{2}+2n_{3}+n_{4}>\sqrt{\pi\sqrt{3}\cdot n}$.

The proof combines Kurz and Pinchasi’s discharging method with the
isoperimetric inequality. The main result of this paper then follows as a
corollary.

######  Corollary 2.

In a matchstick graph on $n$ vertices, the number of vertices of degree at
most $4$ is greater than $\frac{1}{4}\sqrt{\pi\sqrt{3}\cdot n}$.

######  Remark 3.

Theorem 1 and Corollary 2 are sharp up to the constant. As shown in Figure 1,
it is possible to construct matchstick graphs with $n=3k^{2}+3k$ vertices, of
which all are of degree $5$ except for six of degree $3$ and $6(k-1)$ of
degree $4$. The construction consists of a hexagonal piece of the triangular
lattice with the central vertex removed, as well as removing every second edge
of the boundary of each hexagon except for the smallest and the largest
hexagon. In this case,
$\frac{2n_{3}+n_{4}}{\sqrt{n}}=2\sqrt{3}\sqrt{1+\frac{1}{k}}$ is quite far
from $\sqrt{\pi\sqrt{3}}$ in Theorem 1, partly because for large $k$ the
boundary of this example is a regular hexagon instead of circular, and partly
because in the inequality $b^{2}>\pi\sqrt{3}f_{3}$, we only consider triangles
to bound the area from below. In this example there are many quadrilaterals,
and it is not clear how to get a non-trivial lower bound for the area of many
quadrilaterals in a matchstick graph.

$k=4$$n=60$$n_{3}=6$$n_{4}=18$$2n_{3}+n_{4}=30$ Figure 1. Matchstick graphs
with $n=3k^{2}+3k$ vertices, $n_{3}=6$, $n_{4}=6(k-1)$ and $n_{5}=3k^{2}-3k$.

##  2\. Plane graphs and matchstick graphs

A _plane graph_ $G=(V,E)$ is a drawing of a graph in the plane where all
vertices $v\in V$ are distinct points and all edges $uv\in E$ are Jordan arcs
joining $u$ and $v$, such that edges intersect only at their endpoints. A
_matchstick graph_ is a plane graph with edges drawn as unit length straight-
line segments.

We denote the number of vertices by $n$, the number of vertices of degree $i$
by $n_{i}$, and the number of edges by $e$. Then $n=\sum_{i}n_{i}$ and
$2e=\sum_{i}in_{i}$. We next find similar formulas for the total number of
faces and the number of faces with $k$ sides. We need to be careful in how we
define the number of sides of a face. Note that although $G$ is planar, we do
not assume that $G$ is $2$-connected, nor even that it is connected. This is
to avoid a complicated induction in the proof of Theorem 1. As usual, we
define a _face_ of $G$ to be a connected component of the complement of the
drawing of $G$ in the plane. Thus there is a unique unbounded face, and $0$ or
more bounded faces. We say that a face has $k$ sides or is a $k$-gon if the
face has $k$ edges on its boundary, where we count an edge twice if both sides
of the edge are in the face. For example, the matchstick graph in Figure 2 has
an unbounded face with $4$ sides and a bounded face with $6$ sides, since the
edge $e$ counts twice.

$e$ Figure 2. The bounded face has $6$ sides with $e$ counting twice.

We let $b$ denote the number of sides of the unbounded face, $f_{k}$ the
number of bounded $k$-gons, and $f$ the total number of bounded faces. Then we
have $f=\sum_{k}f_{k}$, and by counting edge-face pairs, we obtain
$2e=b+\sum_{k}kf_{k}$.

Let us consider the possible $k$-gons for some small $k$. This will be
required in the proof of Theorem 1. The only way for a $0$-gon to exist is if
the graph has no edges. Since the graph is simple, a $1$-gon cannot exist and
the only way for a $2$-gon to exist is if the graph has a single edge. The
only possible $3$-gon in a matchstick graph is an equilateral triangle. A
$4$-gon is an equilateral rhombus unless the graph has only two edges with a
common vertex. Similarly, the only possible bounded $5$-gon is an equilateral
pentagon, since the only other combinatorial possibility is a triangle
together with a vertex of degree $1$ joined to one of the vertices of the
triangle. This can only happen if there are exactly $4$ edges, and the fourth
edge is outside the triangle.

An _isolated vertex_ is one with degree $0$. Note that isolated vertices in a
matchstick graph give no structural information about the edges and can simply
be removed from the graph.

##  3\. Proofs

The isoperimetric inequality says that among all simple closed curves of a
fixed length in the plane, the circle is the unique curve that encloses the
largest area [1]. In the proof of Theorem 1 we will need the following
consequence. The $2$-connected case of this lemma is used in [7].

######  Lemma 4.

Let $G$ be a matchstick graph such that the unbounded face has $b$ edges (with
double counting as before) and $f_{3}$ bounded triangular faces. Then
$b^{2}>\pi\sqrt{3}f_{3}$.

###### Proof.

First consider the case where $G$ is 2-connected. The polygon bounding the
unbounded face has $b$ edges and encloses $f_{3}$ equilateral triangles of
area $\sqrt{3}/4$ each. Thus, the polygon has area
$A\geqslant\frac{\sqrt{3}}{4}f_{3}$ and perimeter $b$. By the isoperimetric
inequality, $b^{2}>4\pi A\geqslant\pi\sqrt{3}f_{3}$.

Now assume that $G$ is not 2-connected. If $b=2$ then $f_{3}=0$ and the lemma
is trivially true. We now use induction on $b$. Thus we assume that
$b\geqslant 3$ and that the lemma is true for smaller $b$.

Since $G$ is not $2$-connected there is either a cut-point $v$, since
$b\geqslant 3$, or $G$ is not connected. In either case we can split $G$ into
subgraphs $G^{\prime}$ and $G^{\prime\prime}$ with at most the vertex $v$ in
common and with at least two vertices each. Let $b^{\prime}$ and
$b^{\prime\prime}$ denote the number of edges on their boundaries and
$f_{3}^{\prime}$ and $f_{3}^{\prime\prime}$ the number of triangular faces
respectively. It is clear that $b=b^{\prime}+b^{\prime\prime}$ and
$f_{3}=f_{3}^{\prime}+f_{3}^{\prime\prime}$, so by induction

$b^{2}\geqslant b^{\prime 2}+b^{\prime\prime
2}>\pi\sqrt{3}f^{\prime}_{3}+\pi\sqrt{3}f^{\prime\prime}_{3}=\pi\sqrt{3}f_{3}.\qed$

The proof of Theorem 1 below uses the discharging method from [6]. The
discharging method is used in graph theory and combinatorial geometry to prove
results about planar graphs, geometric graphs and line arrangements [3, 8],
going back to the proof of the Four Colour Theorem. As far as we know, the
charging rule in [6] is the first one that uses Euclidean geometry, as the
redistribution of charges in this rule depends on the sizes of angles. We will
use the exact same charging rule. Compared to the proof in [6], our proof is
slightly more technical not only because we need to consider vertices of
degree other than $5$ too, but also because we make a careful consideration of
the case when the graph is not $2$-regular.

###### Proof of Theorem 1.

Since the theorem is easily checked if $n\leqslant 4$, we assume that
$n\geqslant 5$. By the discussion in Section 2 of the possible $k$-gons for
small $k$, we have $f_{0}=f_{1}=f_{2}=0$, and each $k$-gon with $3\leqslant
k\leqslant 5$ has a $k$-cycle as boundary. The unbounded face cannot be a
triangle, otherwise there are no other edges or vertices. Thus we have
$b\geqslant 4$, and all triangles are bounded faces. We will use the same
discharging method as in [6], which corresponds to the vertex charging version
of [3, Proposition 3.1]. First we assign a charge of $i-6$ to each vertex of
degree $i$ and a charge of $2k-6$ to each face of $k$ sides. As we show next,
the total charge summed over all vertices and faces of the matchstick graph is
at most $-12$. By counting the edge-face and edge-vertex pairs we find

$b+\sum_{k\geqslant 3}kf_{k}=\sum_{i\geqslant 1}in_{i}=2e.$ (1)

Using Euler’s Formula $\sum_{i}n_{i}-e+\sum_{k}f_{k}=c$, where $c\geqslant 1$
is the number of connected components, we have

$-3\geqslant-3\sum_{i\geqslant 1}n_{i}+e+2e-3\sum_{k\geqslant 3}f_{k}$

and using (1) this gives

$\displaystyle-12$ $\displaystyle\geqslant-6\sum_{i\geqslant
1}n_{i}+\sum_{i\geqslant 1}in_{i}+2\sum_{k\geqslant
3}kf_{k}+2(b-3)-6\sum_{k\geqslant 3}f_{k}$ $\displaystyle=\sum_{i\geqslant
1}(i-6)n_{i}+\sum_{k\geqslant 3}(2k-6)f_{k}+2b-6.$ (2)

We now redistribute the charge as follows. For an angle $\alpha$ between a
face and a vertex, transfer from the face to the vertex a charge of

$\begin{cases}0&\text{ if }\alpha\leqslant\frac{\pi}{3},\\\
\frac{3\alpha}{\pi}-1&\text{ if
}\frac{\pi}{3}<\alpha\leqslant\frac{2\pi}{3},\\\ 1&\text{ if
}\alpha>\frac{2\pi}{3}.\end{cases}$

$\alpha$$\frac{\pi}{3}$$\frac{2\pi}{3}$$2\pi$$1$ Figure 3. Charge transfer
from a face to a vertex at an angle of size $\alpha$.

Since this discharging rule was given without motivation in [6], we try to
motivate it now. We want the charges of all faces and all vertices of degree
at least $5$ to end up non-negative. Since triangles start off with a charge
of $0$, we would like there to be no transfer of charge from a triangle to its
vertices. Thus, since all triangles are equilateral, we do not want any
transfer of charge in angles of size $\pi/3$. Next, consider a hexagonal face.
In order for this face to end up with a non-negative charge, we have to move a
charge of on average at most $1$ to each of its vertices. Thus, we cap the
charge transfer at $1$, and since the average degree of an angle of a hexagon
is $2\pi/3$, we let this maximum charge transfer start at this angle. Third, a
vertex of degree $5$ needs to get a charge of at least $1$. If we let the
transferred charge change linearly from $\pi/3$ to $2\pi/3$, then, at least in
the case where all angles at the vertex of degree $5$ lie between $\pi/3$ and
$2\pi/3$, the vertex receives a total charge of $1$, and thus ends up with a
charge of $0$.

We next show that indeed, in all cases, the charges of all faces and all
vertices of degree at least $5$ will become non-negative, and we obtain the
lower bound (4) for the total charge in terms of $n_{i}$ ($i=1,2,3,4$) and
$b$.

Let us first consider the vertices. A vertex of degree $i=1,2,3$ must have an
angle $\alpha\geqslant\frac{2\pi}{3}$ and hence gains at least a charge of $1$
from the redistribution, ending with a charge of at least $i-5$. This formula
also holds for vertices of degree $4$: this is clear if there is an angle
$\alpha_{j}\geqslant\frac{2\pi}{3}$. Otherwise for each angle $\alpha_{j}$
there is a contribution of at least $\frac{3\alpha_{j}}{\pi}-1$, so the total
contribution is $\frac{3}{\pi}\sum_{j}\alpha_{j}-4=2>1$. For a vertex of
degree at least $6$ it is sufficient to note that the initial charge is non-
negative, and hence so is the final charge. Lastly, for a vertex of degree $5$
let $s>0$ be the number of its angles strictly larger than $\frac{\pi}{3}$.
Since the initial charge is $-1$, if any angle is greater than
$\frac{2\pi}{3}$ then the final charge is non-negative. Otherwise the sum of
these $s$ angles is at least $2\pi-\frac{\pi}{3}(5-s)=\frac{\pi}{3}(s+1)$ and
hence the final charge is at least
$-1+\frac{3}{\pi}\cdot\frac{\pi}{3}(s+1)-s=0$. From these calculations we can
bound the total vertex charge after redistribution from below by

$-4n_{1}-3n_{2}-2n_{3}-n_{4}.$ (3)

Next consider bounded faces of $k$ sides. For $k=3$, a triangular face, the
initial charge is 0 and all angles are $\frac{\pi}{3}$ so the final charge
stays at 0. For $k=4$ the initial charge is $2$. Since the face must be a
rhombus, each angle is equal to at least one other. If exactly two angles are
greater than $\frac{2\pi}{3}$ then the final charge is $0$. Otherwise all
angles must be greater than $\frac{\pi}{3}$ and the final charge is
$2-(\frac{3}{\pi}\cdot 2\pi-4)=0$. For $k=5$ the initial charge is $4$, and as
observed before, the face must be a pentagon. Let $t$ be the number of angles
of the pentagon strictly larger than $\frac{2\pi}{3}$. Then the sum of the
angles less than $\frac{2\pi}{3}$ is at most $3\pi-\frac{2\pi}{3}t$, with a
transferred charge of at most
$\frac{3}{\pi}\left(3\pi-\frac{2\pi}{3}t\right)-(5-t)=4-t$. Adding the charges
of the $t$ angles, totalling to $t$, we get a final transferred charge of at
most $4$.

If the number of sides $k$ is at least 6, the initial charge is $2k-6$ and the
face can lose up to the maximum possible of $k$, since a $k$-gon has at most
$k$ vertices. Hence the final charge is at least $k-6\geqslant 0$. In
particular, the unbounded face with $b$ edges ends up with a charge of at
least $b-6$. We can therefore bound below the total charge of faces after
redistribution by $b-6$. This, together with (3) and (2) gives

$-12\geqslant-4n_{1}-3n_{2}-2n_{3}-n_{4}+b-6,$ (4)

which rearranges to

$4n_{1}+3n_{2}+2n_{3}+n_{4}\geqslant b+6.$ (5)

Now we can write Euler’s Formula as

$4n-2e-2e+4\sum_{k\geqslant 3}f_{k}\geqslant 4,$

and using $2e=b+\sum kf_{k}$ from (1) we get

$4n-2e-b+\sum_{k\geqslant 3}(4-k)f_{k}\geqslant 4.$

Rearranging this gives

$\displaystyle 2e$ $\displaystyle\leqslant 4n-4-b+f_{3}-(f_{5}+2f_{6}+\dots)$
$\displaystyle\leqslant 4n-8+f_{3}.$ (6)

Suppose for contradiction that the theorem is false. Then

$4n_{1}+3n_{2}+2n_{3}+n_{4}\leqslant\sqrt{\pi\sqrt{3}\cdot n},$

hence

$\displaystyle 2e$ $\displaystyle=\sum_{i\geqslant 1}in_{i}$
$\displaystyle\geqslant n_{1}+2n_{2}+3n_{3}+4n_{4}+5\sum_{i\geqslant 5}n_{i}$
$\displaystyle=5n-4n_{1}-3n_{2}-2n_{3}-n_{4}$ $\displaystyle\geqslant
5n-\sqrt{\pi\sqrt{3}\cdot n}.$

Combining this with (6) we get the inequality

$5n-\sqrt{\pi\sqrt{3}\cdot n}\leqslant 4n-8+f_{3}$

and hence

$f_{3}\geqslant n-\sqrt{\pi\sqrt{3}\cdot n}+8.$

Substituting this into the inequality $b^{2}>\pi\sqrt{3}f_{3}$ from Lemma 4
gives

$b>\sqrt{\pi\sqrt{3}(n-\sqrt{\pi\sqrt{3}\cdot n}+8)}.$

Finally, substituting this into (5) gives

$\displaystyle 4n_{1}+3n_{2}+2n_{3}+n_{4}$
$\displaystyle>6+\sqrt{\pi\sqrt{3}\left(n-\sqrt{\pi\sqrt{3}\cdot n}+8\right)}$
$\displaystyle>\sqrt{\pi\sqrt{3}\cdot n},$

which finishes the proof once we verify the last inequality. Suppose to the
contrary that

$\sqrt{\pi\sqrt{3}\left(n-\sqrt{\pi\sqrt{3}\cdot
n}+8\right)}\leqslant\sqrt{\pi\sqrt{3}\cdot n}-6.$

Then squaring both sides gives a contradiction

$\displaystyle\pi\sqrt{3}\cdot n-\pi\sqrt{3}\sqrt{\pi\sqrt{3}\cdot
n}+8\pi\sqrt{3}$ $\displaystyle\leqslant\pi\sqrt{3}\cdot
n-12\sqrt{\pi\sqrt{3}\cdot n}+36,$
$\displaystyle(12-\pi\sqrt{3})\sqrt{\pi\sqrt{3}\cdot n}$
$\displaystyle\leqslant 36-8\pi\sqrt{3}<0.\qed$

###### Proof of Corollary 2.

Consider the matchstick graph $G$ and construct its subgraph $G^{\prime}$ by
removing its isolated vertices, i.e. $n_{0}^{\prime}=0$. Applying the previous
theorem on $G^{\prime}$ we have that

$4n_{1}^{\prime}+3n_{2}^{\prime}+2n_{3}^{\prime}+n_{4}^{\prime}>\sqrt{\pi\sqrt{3}\cdot
n^{\prime}}.$ (7)

Clearly removing the isolated vertices does not change the number of vertices
of each degree $i\geqslant 1$ hence $n_{i}^{\prime}=n_{i}$. We also know that
$n^{\prime}=n-n_{0}$, so (7) gives

$4n_{1}+3n_{2}+2n_{3}+n_{4}>\sqrt{\pi\sqrt{3}(n-n_{0})},$

and hence

$n_{1}+n_{2}+n_{3}+n_{4}>\frac{1}{4}\sqrt{\pi\sqrt{3}(n-n_{0})}.$

Now adding back the isolated vertices to both sides we obtain

$\displaystyle n_{0}+n_{1}+n_{2}+n_{3}+n_{4}$
$\displaystyle>\frac{1}{4}\sqrt{\pi\sqrt{3}(n-n_{0})}+n_{0}$
$\displaystyle\geqslant\frac{1}{4}\sqrt{\pi\sqrt{3}\cdot
n}-\frac{1}{4}\sqrt{\pi\sqrt{3}\cdot n_{0}}+n_{0}$
$\displaystyle\geqslant\frac{1}{4}\sqrt{\pi\sqrt{3}\cdot n}.\qed$

## Acknowledgement

We thank the referees for their careful proof-reading, for providing the
example in Remark 3, and for improving the exposition of the proof of Theorem
1.

## References

  * [1] V. Blåsjö,  The Isoperimetric Problem,  _Amer. Math. Monthly_ 112 (2005), 526–566.
  * [2] A. Blokhuis,  Regular finite planar maps with equal edges, preprint (1982), arXiv:1401.1799.
  * [3] D. W. Cranston and D. B. West,  An introduction to the discharging method via graph coloring,  _Discrete Math._ 340 (2017), 766–793.
  * [4] H. Harborth,  Point sets with equal numbers of unit-distant neighbors (Abstract),  Discrete Geometry, 12–18 July 1981, Oberwolfach,  Tagungsbericht 31/1981, Mathematisches Forschungsinstitut Oberwolfach (1981), pp. 11–12.
  * [5] H. Harborth,  Match sticks in the plane,  In: _The Lighter Side of Mathematics_ , edited by R. K. Guy and R. E. Woodrow (1986), 281–288. Mathematical Association of America, Washington, D.C.
  * [6] S. Kurz and R. Pinchasi,  Regular matchstick graphs,  _Amer. Math. Monthly_ 118 (2011), 264–267.
  * [7] J. Lavollée and K. J. Swanepoel,  Bounding the Number of Edges of Matchstick Graphs,  _SIAM J. Discrete Math._ 36 (2022), 777–785.
  * [8] R. Radoičić and G. Tóth,  The discharging method in combinatorial geometry and the Pach-Sharir conjecture,  Surveys on discrete and computational geometry,  _Contemp. Math._ 453 (2008), Amer. Math. Soc., Providence, RI. pp. 319–342.


11institutetext:  Astrophysics, Department of Physics, Keble Road, Oxford OX1
3RH, United Kingdom
11email<EMAIL_ADDRESS>22institutetext: Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122,
Australia 33institutetext: George P. and Cynthia W. Mitchell Institute for
Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas
A&M University, College Station, TX 77843, USA 44institutetext: Núcleo de
Astronomía, Facultad de Ingeniería y Ciencias. Universidad Diego Portales,
Ejército Libertador 441, Santiago, 8320000, Chile. 55institutetext: AIM-Paris-
Saclay, CEA/DSM/Irfu – CNRS – Université Paris Diderot, CEA-Saclay, pt
courrier 131, 91191 Gif-sur-Yvette, France 66institutetext: Leiden
Observatory, Leiden University, NL-2300 RA Leiden, The Netherlands
77institutetext: Department of Astronomy, University of Geneva, 51 Ch. des
Maillettes, 1290 Versoix, Switzerland 88institutetext: International
Associate, Cosmic Dawn Center (DAWN) 99institutetext: Faculty of Physics,
Ludwig-Maximilians Universität, Scheinerstr. 1, 81679 Munich, Germany
1010institutetext: Research Centre for Astronomy, Astrophysics &
Astrophotonics, Macquarie University, Sydney, NSW 2109, Australia
1111institutetext: Department of Physics & Astronomy, Macquarie University,
Sydney, NSW 2109, Australia 1212institutetext: Max-Planck Institut für
Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany 1313institutetext:
Australia Telescope National Facility, CSIRO Astronomy and Space Science, PO
Box 76, Epping, NSW 1710, Australia 1414institutetext: School of Physics,
University of New South Wales, Sydney, NSW 2052, Australia 1515institutetext:
Institute of Astronomy, The University of Tokyo, Osawa, Mitaka, Tokyo
181-0015, Japan 1616institutetext: National Astronomical Observatory of Japan,
Mitaka, Tokyo 181-8588, Japan

# A low [CII]/[NII] ratio in the center of a massive galaxy at $z=3.7$:
witnessing the transition to quiescence at high-redshift?

C. Schreiber 11 K. Glazebrook 22 C. Papovich 33 T. Díaz-Santos 44 A. Verma 11
D. Elbaz 55 G. G. Kacprzak 22 T. Nanayakkara 66 P. Oesch 7788 M. Pannella 99
L. Spitler 10101111 C. Straatman 1212 K.-V. Tran 13131414 T. Wang 15151616

(Received 5 August 2019; accepted 26 November 2020)

Understanding the process of quenching is one of the major open questions in
galaxy evolution, and crucial insights may be obtained by studying quenched
galaxies at high redshifts, at epochs when the Universe and the galaxies were
younger and simpler to model. However, establishing the degree of quiescence
in high redshift galaxies is a challenging task. One notable example is Hyde,
a recently discovered galaxy at $z_{\rm spec}=3.709$. As compact ($r_{1/2}\sim
0.5\,{\rm kpc}$) and massive ($M_{\ast}\sim 10^{11}\,{\rm M}_{\odot}$) as its
quenched neighbor Jekyll, it is also extremely obscured yet only moderately
luminous in the sub-millimeter. Panchromatic modeling suggested it could be
the first galaxy found in transition to quenching at $z>3$, however the data
were also consistent with a broad range of star-formation activity, from fully
quenched to moderate star formation rates (SFR) in the lower scatter of the
galaxy main-sequence. Here, we describe Atacama Large Millimeter Array (ALMA)
observations of the ${\rm[{C\textsc{II}}]}$ $157\,\mu{\rm m}$ and
${\rm[{N\textsc{II}}]}$ $205\,\mu{\rm m}$ far-infrared emission lines. The [C
II] emission within the half-light radius is dominated by ionized gas, while
the outskirts are dominated by photo-dissociation regions or neutral gas. This
suggests that the ionization in the center is not primarily powered by on-
going star formation, and could come instead from remnant stellar populations
formed in an older burst, or from a moderate active galactic nucleus (AGN).
Accounting for this information in the multi-wavelength modeling provides a
tighter constraint on the star formation rate of ${\rm
SFR}=50^{+24}_{-18}\,{\rm M}_{\odot}/{\rm yr}$. This rules out fully quenched
solutions, and favors ${\rm SFR}$s more than factor of two lower than expected
for a main-sequence galaxy, confirming the nature of Hyde as a transition
galaxy. Theses results suggest that quenching happens from inside-out, and
starts before the galaxy expels or consumes all its gas reservoirs. Similar
observations of a sample of massive and obscured galaxies would determine
whether this is an isolated case or the norm for quenching at high-redshift.

###### Key Words.:

Galaxies: evolution – galaxies: high-redshift – galaxies: star formation –
sub-millimeter: galaxies

##  1 Introduction

Massive galaxies in the low-redshift Universe are observed to be mostly
quiescent, with current star formation rates less then 1% of their past
average (e.g., Pasquali et al. 2006). But how galaxies quench their star
formation and turn into massive, red early-type galaxies is one of the key
unresolved questions of galaxy evolution.

While a number of mechanisms have been proposed in the literature to cause
galaxies to stop, reduce, or prevent star formation – including black hole
feedback, strong outflows, or gas stripping (e.g., Silk & Rees 1998; Birnboim
& Dekel 2003; Croton et al. 2006; Gabor & Davé 2012; Martig et al. 2009;
Förster Schreiber et al. 2014; Genzel et al. 2014; Peng et al. 2015), it is
still unknown which of these mechanism is most important to explain emergence
of quenched galaxies. Crucial insight can be gained by studying quiescent
galaxies at higher redshift, where the available time for feedback processes
to act is shorter, limiting the range of possible mechanisms. Recent work has
shown that massive quiescent galaxies already existed as early as $z=4$ (e.g.
Labbé et al. 2005; Kriek et al. 2009; Gobat et al. 2012; Merlin et al. 2018;
Schreiber et al. 2018b; Belli et al. 2019), at an epoch where state-of-the-art
numerical simulations predict all galaxies to be forming stars (e.g., Wellons
et al., 2015; Davé et al., 2016). These galaxies must have had massive star-
forming progenitors at even higher redshifts, and because the age of the
Universe was then comparable to their estimated stellar ages (of order a
billion years or less), they must have quenched shortly before being observed.
The actor of this abrupt quiescence may therefore be easier to identify than
in lower redshift objects, which are seen after several billion years of
passive evolution.

An interesting case is that of the most distant known quiescent galaxy to
date, at $z_{\rm spec}=3.715$ (Glazebrook et al., 2017). Detection of sub-
millimeter emission toward this object (Simpson et al., 2017) was later found
to arise from to a neighboring extremely obscured galaxy at the same redshift
(Schreiber et al. 2018c, hereafter S18). This was demonstrated by the
detection with ALMA of the [C II] line blueshifted by $550\,{\rm km/s}$
compared to the Balmer absorption lines of the quiescent galaxy, and most
convincingly by the improved spatial resolution ($0.4\arcsec$) and depth of
the sub-millimeter imaging, showing that both the [C II] and dust emissions
are produced by a separate, rotating galaxy, mostly unresolved ($0.1\arcsec$
radius) and located $0.5\arcsec$ away from the quiescent galaxy. The pair was
dubbed “Jekyll and Hyde”, with Jekyll being the quiescent galaxy, and Hyde the
obscured galaxy.

In S18, we performed an extensive analysis of the rich multi-wavelength data
at hand to understand the physical conditions in the obscured galaxy Hyde. We
showed that, despite its extreme obscuration and sub-mm detection, this galaxy
appears to form stars at a relatively slow pace, with a star-formation rate
${\rm SFR}<100\,{\rm M}_{\odot}/{\rm yr}$ and a stellar mass $M_{\ast}\simeq
10^{11}\,{\rm M}_{\odot}$. Given the strong obscuration ($A_{V}\simeq 3$) and
the galaxy’s large stellar mass, the infrared luminosity is in fact low enough
($L_{\rm IR}\simeq 10^{12}\,L_{\odot}$) that it may be entirely powered by
intermediate-age stars, such that its current ${\rm SFR}$ (averaged over the
last $10\,{\rm Myr}$) could be as low as zero. Although these data could not
exclude that Hyde is simply a normal galaxy in the lower envelope of the main
sequence, the possibility of it being quenched or in transition to quenching
is particularly interesting. Indeed, given that this galaxy is still obscured
and is therefore likely to contain substantial gas reservoirs, this would be
at odds with a number of proposed quenching mechanisms which require full
removal or consumption of the gas reservoirs prior to quenching.

This surprising conclusion is nevertheless independently supported by a number
of pieces of evidence, first reported in S18 and summarized here for the
reader’s convenience: a) its dust temperature ($T_{\rm dust}\simeq 30\,{\rm
K}$) is almost $10\,{\rm K}$ lower than the average for $z\sim 4$ galaxies
(Schreiber et al., 2018a), which suggests a softer-than-average radiation
field, b) its compact dust continuum size of about $0.5\,{\rm kpc}$ is smaller
than the rest-ultraviolet size of all $z\sim 4$ star-forming galaxies in the
same field, and is instead similar to that of quiescent galaxies (Straatman et
al., 2015), and c) its $L_{[{C\textsc{ii}}]}/L_{\rm FIR}$ ratio is abnormally
low for a galaxy of this luminosity, which we showed could be explained by a
recent truncation of star formation, although other explanations for the
observed ${\rm[{C\textsc{II}}]}$ deficit could not be excluded. Considered
independently, none of these facts is conclusive or extremely unusual, and
indeed there are other examples of galaxies which share at least one of these
properties (e.g., similar compactness, dust temperature, or
$L_{[{C\textsc{ii}}]}/L_{\rm FIR}$ ratio). It is, however, the sum of these
facts which suggests an abnormal process is at play in this galaxy; to our
knowledge, such a combination of observables is unique among distant massive
galaxies. Although all these signs may point toward Hyde being caught in a
very specific phase, possibly in transition to quiescence, none of the data
available at the time allowed us to prove this conclusively. In light of the
results presented in this paper, we describe these points further in our
discussion.

In this paper, we exploit new spectroscopic data from the Atacama Large
Millimeter Array (ALMA) to address this question, combining observations of
the [C II] and [N II] ($205.178\,\mu{\rm m}$) emission lines. [N II], owing to
its high ionization potential ($14.5\,{\rm eV}$), is exclusively found in
ionized gas regions, where it can be found alongside [C II] with an almost
constant line ratio
${\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}\sim 3$ (see Oberst et
al. 2006). In contrast, photo-dissociation regions (PDRs) surrounding stellar
birthclouds are a neutral medium that emits [C II] but no [N II]. The observed
${\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}$ ratio can therefore be used as a
probe of the fraction of the [C II]-emitting gas which is associated with
ionized gas. While star-forming regions will typically be comprised of both
neutral and ionized gas, the former is found to dominate the [C II] emission
in star-forming galaxies (Pavesi et al., 2016; Díaz-Santos et al., 2017). In
contrast, nearby early-type galaxies, which are forming stars at much lower
rates, were shown to have a significantly larger fraction of their [C II]
associated with ionized gas (Lapham et al., 2017). This implies that the
ionized component of [C II] is mostly not associated with star-formation,
hence that the ${\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}$ ratio can be used
as an independent tracer of star formation activity, which we apply here to
the galaxy Hyde.

The structure of this paper is as follows. In section 2 we describe the new
ALMA observations and detections, in section 3 we describe the method used to
analyze them, in section 4 we describe our results and the ionization state of
the gas, and use these observations to refine our estimate of the current
${\rm SFR}$ in Hyde. In section 5 we discuss how other observables support
these results, and we finally conclude in section 6.

In the following, we assumed a $\Lambda$CDM cosmology with $H_{0}=70\ {\rm
km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$, $\Omega_{\rm M}=0.3$, $\Omega_{\Lambda}=0.7$
and a Chabrier (2003) initial mass function (IMF), to derive both star-
formation rates and stellar masses.

##  2 Data

The data we use in this paper consist of ALMA observations of Hyde obtained in
two different programs. The first data set was obtained in the Director’s
Discretionary Time (DDT) program 2015.A.00026.S (PI: Schreiber) to measure the
[C II] emission; the galaxy was observed in band 8 (TDM correlator, covering
$401.05$–$416.68\,{\rm GHz}$ with four $1.875\,{\rm GHz}$ spectral windows at
$31.25\,{\rm MHz}$ resolution) for 1.2 hours (on-source), with a synthesized
beam size of $0.52$$\times$$0.42\arcsec$ (natural weighting). These
observations were already presented in S18, and were reprocessed here for
homogeneity. The second data set was obtained in the regular call program
2018.1.00216.S (PI: Schreiber) to measure the [N II] emission; the galaxy was
observed in band 7 (TDM correlator, covering $295.34$–$311.20\,{\rm GHz}$ with
four $1.875\,{\rm GHz}$ spectral windows at $31.25\,{\rm MHz}$ resolution) for
0.8 hours (on-source), with a synthesized beam size of
$0.29$$\times$$0.25\arcsec$ (natural weighting).

Both data sets were reduced with the same procedure, using the ALMA pipeline
to produce dirty images with natural weighting (to maximize $S/N$) and a
spectral averaging of three elements (so that the spectral response function
is effectively one channel, see Section A.6.1 in the ALMA Proposer’s Guide).
The pixel size was left to its default value of $0.05\arcsec$ and
$0.085\arcsec$ in band 7 and band 8, respectively (corresponding to $0.36$ and
$0.61\,{\rm kpc}$), which generously samples the core of the dirty beam; finer
pixel sizes would not benefit the quality of the analysis. Since both data
sets were observed with the TDM correlators, the final cubes have a spectral
resolution of $35$ and $45\,{\rm km/s}$, respectively, which is sufficient to
resolve the broad line profiles expected in massive compact galaxies.

Compact continuum emission was clearly detected in both data sets ($S_{\rm
peak}/N$ of $46$ and $41$, respectively), which had to be removed prior to
analyzing the line emission. To optimize the $S/N$, this was done in the image
domain by fitting the best continuum spectral model obtained in S18 to the
spectral data at each pixel of the image, excluding spectral elements with
expected line emission. The resulting spectra, extracted at the peak pixel,
are shown on Fig. 1. Line maps were finally created by spectrally-averaging
the continuum-subtracted cubes over $800\,{\rm km/s}$ around the expected line
frequencies at $z=3.7087$ (see Fig. 1), the redshift obtained from the [C II]
emission in S18. The lines were detected with a $S_{\rm peak}/N$ of $17$ and
$5$, respectively, and the maps are shown in Fig. 2. Despite these moderate
$S_{\rm peak}/N$, the detection of [N II] is still highly significant: since
we did not fit for the position of the line emission in neither the spatial
not the frequency domains, the null hypothesis of a non-detection can be
safely rejected.

##  3 Spatial profile modeling

###  3.1 Description of the modeling

Figure 1: [C II] (green) and [N II] (red) spectra extracted at the peak pixel
of the line emission, after continuum subtraction. The [N II] spectrum was
rescaled upward by a factor five for easier comparison to the [C II] spectrum.
The integration window used to create the line maps is shown with vertical
dotted lines.  Figure 2: Dirty beam (top), continuum (center) and line maps
(bottom) of Hyde in band 8 (left) and 7 (right) produced by the ALMA pipeline,
without any cleaning applied (“dirty” images). Line maps are continuum-
subtracted and were produced by summing the flux in a $800\,{\rm km/s}$
velocity window centered on the [C II] mean velocity. The half-intensity area
of the corresponding dirty beams are shown in the bottom-right corner of each
panel. Contours shown are $3\sigma$ (dotted line) and $10\sigma$ (solid line).

Using a custom-build software, we then modeled each line map independently in
the image domain. We did not attempt to model the maps directly in the
visibility domain for two reasons; first, fitting complex profiles other than
Gaussians and point sources in the visibility domain is particularly
challenging and time-consuming, and second, it has been shown that image-based
and visibility-based methods actually provide similar results for ALMA data
(Hodge et al., 2016). Nevertheless, we double check in section 3.2 the result
of our modeling against the observed visibilities.

Our model consists of a large grid of two-component exponential profiles. The
first “central” component was given a small scale length of $0.1$ to
$0.5\,{\rm kpc}$ to represent an unresolved central region (S18 showed the
continuum emission extends over a $\sim 0.7\,{\rm kpc}$ half-light radius),
while the scale length of the second “extended” component was allowed to vary
between $0.15$ and $3\,{\rm kpc}$, with the constraint that its scale length
must be at least $0.05\,{\rm kpc}$ larger than that of the central
component111We tried extending this grid to a maximum of $1.5\,{\rm kpc}$ for
the “central” component and $5\,{\rm kpc}$ for the “extended” component; this
had no significant impact on our flux measurements, suggesting the original
grid was large enough.. We let the flux of each component vary independently
and freely, with the sole constraint that the combined light profile must
remain positive at all positions. We also varied the axis ratio and the
position angle, however to save on computation time we forced the values of
both components to match. Finally, the position of both components was fixed
to the centroid of the continuum emission. Since the size of the galaxy is
small compared to the image pixel size, we generated models with a $\times$$9$
oversampling factor, and a further $\times$$3$ oversampling for the central
$3$$\times$$3$ pixels to properly sample the exponential core. This
corresponds to an over-sampled pixel size of $0.01$ (band 6) and $0.02\,{\rm
kpc}$ (band 8) at the core. We then convolved the model with the ALMA dirty
beam, and selected the best-fit model using maximum likelihood estimation,
accounting for spatially-correlated noise in the computation of the likelihood
(see Appendix C).

To estimate the uncertainty on the model, we used Monte Carlo simulations: the
whole procedure was repeated $200$ times on mock images that were created by
perturbing the observed images with realistic noise (same amplitude and same
covariance as in the real data, see Appendix A), leading to $200$ other
“acceptable” models from which we estimated confidence intervals using the
$16$th and $84$th percentiles (see Appendix E). Because the distributions of
the observables derived from this modeling (e.g., the ionized gas fraction)
can be strongly non-Gaussian, we carried out all our calculations on each of
the $200$ Monte Carlo realizations, and derived confidence intervals for all
derived quantities in the same fashion throughout the paper. With $200$ Monte
Carlo simulations, $1\sigma$ asymmetric error bars have an accuracy of $10\%$.

###  3.2 Validation of the image-domain modeling

Figure 3: Binned amplitude of the ALMA visibilities as a function of the UV
distance ($u^{2}+v^{2}$). The [C II] visibilities are shown on the left
(green), and the [N II] visibilities are shown on the right (red). Observed
visibilities are displayed as filled circles with error bars; for clarity, the
visibilities on long baselines, which have large uncertainties, are not
displayed. The best fit models obtained in this paper (from the image-domain
analysis) are shown as solid lines (green for [C II], red for [N II]). For
comparison purposes, on each panel, the model of the other line is also
displayed, but re-normalized to fit the observed visibilities. The 1$\sigma$
confidence interval of the model (determined from our Monte Carlo simulations)
is shown as a shaded region in the background.

Since we performed all our analysis in the image-domain, we checked that the
same trends we detect in this analysis are present in the raw ALMA
visibilities. To do so, we used CASA to perform continuum subtraction in the
visibility domain using uvcontsub (first order polynomial), excluding channels
containing the line in the fit. From the resulting measurement set, we then
extracted and averaged the frequency channels covered by the line using split.
We note that this data set is not strictly equivalent to the image-domain data
we used in our analysis, where the continuum subtraction and frequency
averaging were performed in the image domain (with slightly different weights,
computed from the image RMS in each channel). The images created from these
alternative measurement sets have $10$-$20\%$ larger noise RSM; the $S/N$ will
therefore be lower than in our main analysis, but this will be sufficient for
our purpose.

We finally extracted the visibilities using ms.getdata() and manually
performed the remaining averaging (polarization) and the binning by $(u,v)$
distance. The real and imaginary parts of the visibilities were averaged
separately in each bin, using the weights in the measurement set, and combined
to form the binned amplitude. The uncertainties on the binned amplitudes were
estimated from the weighted standard deviation of the data in the bin, and
scaled down by the square root of the number of points.

In parallel, we also produced mock visibilities for our best fit and Monte
Carlo models and binned them in a similar way. The corresponding visibilities
were obtained from the $\times$9 oversampled model images created by our
fitting procedure. These visibilities were computed and injected into the real
measurement set using the CASA task ft, which generates mock measurement sets
with the exact same $(u,v)$ coverage and weights as the real data. To correct
for any discrepancy the data reduction between these visibilities and the
images from which we derived the models, we applied a global rescaling factor
to our best-fit model amplitudes to best match the observed visibilities.

The binned amplitudes are shown in Fig. 3. These figures show the [C II]
emission is clearly resolved. The [N II] emission is more noisy, but appears
more compact and consistent with being unresolved. The best-fit models
obtained from the image-domain analysis provide good matches to the visibility
data, but seem mutually inconsistent between the two lines. In fact, trying to
fit the [C II] data with the [N II] model, we obtained a worse $\chi^{2}$
($22.4$, vs $20.9$ for the best fit [C II] model), and similarly for the [N
II] data and the [C II] model ($15.0$, vs $8.0$ for the best fit [N II]
model). This implies that the two emission lines have different spatial
distributions, a point we analyze in greater detail in section 4.

###  3.3 Aperture fluxes

We first applied our model to the new band 7 continuum image, which has a
higher $S/N$ and sharper angular resolution than the band 8 data used in S18,
and updated the dust continuum half-light radius to $0.51\pm 0.07\,{\rm kpc}$
(major axis), which is consistent with the value obtained from the band 8
data. This value was used to separate the emission into a “center” and
“outskirts” components, as we now describe.

Coming back to the other images (line and continuum maps), we summed the flux
of every model (both for the best fit and the $200$ Monte Carlo simulations)
in two circular annuli: from $0$ to $0.5\,{\rm kpc}$ for the “central”
annulus, and from $0.5$ to $3.5\,{\rm kpc}$ for the “outskirts” annulus. We
note that the central annulus actually contains more than half ($\sim 75\%$)
of the continuum emission since the model axis ratio is lower than one.

The obtained “central” and “outskirts” fluxes (for both lines and continua)
are the main measurements we use in the rest of this analysis. The other
parameters of the model, such as the respective size and flux of both
exponential components, are effectively marginalized over and considered
unimportant. Aperture fluxes are typically simpler to constrain than other
shape-related model parameters, because they tend to be less model-dependent.
For example, the flux inside our “central” aperture depends only slightly on
the adopted size for the central component; if the aperture is larger or
comparable in size, it then contains all the flux from that component, no
matter how that flux is distributed internally. Furthermore, the total flux in
that aperture has a natural upper bound from the data. To illustrate this, we
can consider the peak pixel of the image, which for [C II] entirely contains
our aperture. Because of the convolution with the dirty beam, this pixel
contains flux from both inside and outside of our aperture. Yet, even if the
relative contribution of one versus the other is uncertain, neither can exceed
the observed peak flux.

In this light, it may seem overly complex to introduce a rich two-component
model to, in the end, simply measure two aperture fluxes. We stress that this
model complexity is in fact crucial to assess the reliability of our
measurement. Indeed, the constraining power of our data (owing both to $S/N$
and angular resolution) is not sufficient to allow us to determine the exact
shape of the intensity profile at all radii. Therefore, by exploring as wide a
range of models as possible – including numerous models which are barely
distinguishable on the ALMA images – we make sure our uncertainties on the
aperture fluxes encompass all credible scenarios allowed by the data.

To test our method and the accuracy of our uncertainties, we applied our full
measurement procedure to simulated images with injected sources, as described
in Appendix D. We found that our method can recover the total, central, and
outskirts fluxes in all our simulated images with no detectable systematic
bias. The estimated uncertainties from Monte Carlo simulations were found to
correctly capture the noise in our measurements, with at most a $10\%$ under-
estimation.

##  4 Results

###  4.1 Spatial distribution of the lines

In Fig. 4 (left) we show the surface brightness profile of Hyde inferred from
our two-component model for the dust continuum and the two emission lines, [C
II] and [N II]. Although the $S/N$ of the [N II] data is low, the angular
resolution is sufficient to show that most of the emission is centrally-
concentrated, with a profile that is similar to that of the dust continuum.
The [C II] emission, however, appears more extended (as is commonly found in
the literature, see, e.g., Gullberg et al. 2018; Rybak et al. 2019; Tadaki et
al. 2019). The intrinsic half-light radii we obtained for the continuum, [C
II], and [N II] are (respectively) $r_{1/2}=0.51\pm 0.07$, $1.2\pm 0.3$, and
$0.3^{+0.3}_{-0.1}\,{\rm kpc}$ (along the major axis). The lower error bar on
the [N II] size is limited by the minimum source size in our model grid,
$0.1\,{\rm kpc}$; the data are consistent with the [N II] emission being
point-like.

Martí-Vidal et al. (2012) quantify what we should be able to measure in our
data given the angular resolution and $S/N$: based on their Eq. (7), our data
should be able to distinguish, with more than $95\%$ reliability ($2\sigma$),
between a point source and an extended profile of size $0.30$, $0.96$, and
$0.94\,{\rm kpc}$, respectively. The sizes of the continuum and [C II] are
both larger than these values and are indeed “measured” ($r/\sigma_{r}=7$ and
$4$, respectively), and the size we report for [N II] indeed has a $2\sigma$
upper limit of $0.9\,{\rm kpc}$. Our measurements are thus consistent with
these theoretical expectations.

To further confirm this difference in half-light radii with a simpler
independent method, we used the simulated noise maps described in the previous
section and injected sources of sizes matching the above values, and convolved
with the dirty beam. For each source, we computed its observed radial profile
on the noisy image, and located the radius at which the emission falls below
half of the peak. Comparing this simulation to the values observed on the real
images, we found that the [C II] emission is significantly ($3.6\sigma$)
larger than the continuum, and that [N II] is significantly ($2.4\sigma$)
smaller than [C II], while the difference between [N II] and the continuum is
only marginal ($1.2\sigma$).

This confirms that the difference in size between the [C II] and [N II]
emission is not simply due to noise, and implies the presence of a strong
gradient in the ${\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}$ ratio, with
inner regions having lower ${\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}$ and
therefore a higher proportion of ionized gas.

Table 1:  Measured line fluxes from the image analysis, corresponding ionized
gas fractions and infrared luminosities.

| $I_{\rm[{C\textsc{II}}]}$ | $I_{\rm[{N\textsc{II}}]}$ | [C II]/[N II] | $f_{\rm{\rm[{C\textsc{II}}]},ion}$ | $f_{\rm IR,ISM}$ | $\log_{10}(L_{\rm IR}/L_{\odot})$ | $\log_{10}(L_{\rm IR,BC}/L_{\odot})$ | $\log_{10}(L_{[{C\textsc{ii}}]}/L_{\rm FIR})$
---|---|---|---|---|---|---|---|---
Flux origin | ${\rm Jy.km/s}$ | ${\rm Jy.km/s}$ | e | % | % | f |  |
Totala | $1.73^{+0.13}_{-0.10}$ | $0.227^{+0.053}_{-0.046}$ | $9.8^{+2.6}_{-1.8}$ | $28^{+8}_{-6}$ | $60^{+8}_{-10}$ | $12.03^{+0.14}_{-0.14}$ |  $11.52^{+0.17}_{-0.24}$ g | $-2.89^{+0.23}_{-0.26}$
Centralb | $0.69^{+0.25}_{-0.19}$ | $0.187^{+0.049}_{-0.039}$ | $4.8^{+2.4}_{-1.7}$ | $58^{+32}_{-20}$ | $84^{+12}_{-15}$ | $11.93^{+0.15}_{-0.16}$ | $11.12^{+0.30}_{-0.69}$ | $-3.17^{+0.26}_{-0.28}$
Outskirtsc | $1.04^{+0.25}_{-0.27}$ | $0.022^{+0.072}_{-0.022}$ | $60^{+\rm inf}_{-46}$ | $5^{+14}_{-5}$ | $13^{+33}_{-13}$ | $11.39^{+0.13}_{-0.19}$ | $11.26^{+0.21}_{-0.21}$ | $-2.46^{+0.33}_{-0.29}$
Peak pixeld | $1.24^{+0.07}_{-0.07}$ | $0.190^{+0.038}_{-0.038}$ | $8.6^{+2.4}_{-1.8}$ | $32^{+11}_{-9}$ | $64^{+12}_{-11}$ | $11.95^{+0.14}_{-0.14}$ | $11.37^{+0.17}_{-0.17}$ | $-2.96^{+0.22}_{-0.23}$

a Summed flux inside a $3.5\,{\rm kpc}$ radius. b Summed flux inside a
$0.5\,{\rm kpc}$ radius. c Summed flux between $0.5$ and $3.5\,{\rm kpc}$
radius. d Flux of the peak pixels on the ALMA image (warning: because of the
poorer resolution, this method measures flux on a larger scale for [C II] than
for [N II]). e Ratio of the [C II] to [N II] flux, with fluxes expressed in
${\rm W}/{\rm m}^{2}$. f Total infrared luminosity ($8$–$1000\,\mu{\rm m}$),
computed from the total $L_{\rm IR}$ estimated in Schreiber et al. (2018c)
scaled by the fraction of [N II] continuum flux in each region. g Computed as
the sum of the central and outskirts luminosities.

To quantify the gradient in the line ratio, we now turn to the measured
fluxes. We list the measured values and their uncertainties in Table 1, and
provide in Appendix E the full observed distributions in the Monte Carlo
simulations, which we use to determine and propagate uncertainties throughout
this paper. We found no significant flux in ${\rm[{N\textsc{II}}]}$ beyond our
fiducial $0.5\,{\rm kpc}$ radius, and conversely, we found less than half of
the ${\rm[{C\textsc{II}}]}$ flux is located inside this radius. This leads to
a line ratio in the central region of ${\rm[{C\textsc{II}}]}_{\rm
center}/{\rm[{N\textsc{II}}]}_{\rm center}=4.8^{+2.4}_{-1.7}$, and a lower
limit on the line ratio in the outskirts of ${\rm[{C\textsc{II}}]}_{\rm
outskirts}/{\rm[{N\textsc{II}}]}_{\rm outskirts}>15$ ($1\sigma$ limit).

###  4.2 Ionized gas distribution

Figure 4: Left: modeled intrinsic surface brightness profiles of Hyde in the
dust continuum (black, horizontal stripes), [C II] (green, +45 degrees
stripes), and [N II] (red, -45 degrees stripes). The dust continuum half-light
radius is indicated with a vertical dotted black line. The PSF HWHM of the [C
II] (resp. [N II]) line map is indicated with vertical dot dashed (resp.
dashed) green (resp. red) line. The hashed regions show the $1\sigma$
confidence intervals obtained from the $200$ mock noise realizations, and are
centered on the best-fit model. Right: Ionized gas fraction
($f_{\rm{\rm[{C\textsc{II}}]},ion}$) determined from the [C II]/[N II] line
ratio in different regions of Hyde (green circles; center, outskirts, and
total), compared to other values reported in the literature. For all galaxies,
the ionized gas fraction was computed assuming a fixed
${\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}=2.80\pm 0.18$ (see
text). We show literature values for local ETGs (red circles) and KINGFISH
galaxies (purple circles) from Lapham et al. (2017), local infrared luminous
galaxies in GOALS (black circles) from Zhao et al. (2016) and Díaz-Santos et
al. (2017) and HERCULES from Rosenberg et al. (2015) and Kamenetzky et al.
(2016), and high-redshift SMGs from Pavesi et al. (2016). We also show the
upper limit of Zhang et al. (2018) obtained by stacking high-redshift lensed
SMGs (downward-pointing orange triangle).

Using the fluxes estimated in the previous section, we estimated the fraction
of [C II] emission associated with ionized gas (or “ionized gas fraction” for
short), $f_{\rm{\rm[{C\textsc{II}}]},ion}$, following a method similar to
Díaz-Santos et al. (2017) and Lapham et al. (2017):

$\displaystyle
f_{\rm{\rm[{C\textsc{II}}]},ion}=\frac{{\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}}{{\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}}\,.$
(1)

Here we adopt a fixed
${\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}=2.80\pm 0.18$, which
is the mean value found by Lapham et al. (2017). In principle this ratio has a
weak dependence on the electron density, which can be measured using the
${\rm[{N\textsc{II}}]}_{122}/{\rm[{N\textsc{II}}]}_{205}$ ratio, but the
${\rm[{N\textsc{II}}]}_{122}$ line is not observable for Hyde due to poor
atmospheric transmission at that redshifted wavelength. If Hyde turns out to
have an unusual electron density, based on the modeling of Lapham et al.
(2017) its ${\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}$ could
only be higher, up to
${\rm[{C\textsc{II}}]_{ion}}/{\rm[{N\textsc{II}}]_{ion}}=4$, which would only
increase the value of $f_{\rm{\rm[{C\textsc{II}}]},ion}$ we infer from the
data.

In Fig. 4 (right), we display the ionized gas fraction of Hyde in the
“central” and “outskirts” annuli (which we recall correspond to inside and
outside of the dust half-light radius, respectively). We find a clear
difference between the two regions. The outskirts of the galaxy has a low
$f_{\rm{\rm[{C\textsc{II}}]},ion}$ of $0$–$19\%$, which is typical of
infrared-luminous local galaxies and distant SMGs (Pavesi et al., 2016; Díaz-
Santos et al., 2017). Likewise, the $L_{[{C\textsc{ii}}]}/L_{\rm FIR}$ ratio
in the outskirts, $\log_{10}(L_{[{C\textsc{ii}}]}/L_{\rm
FIR})=-2.46^{+0.33}_{-0.29}$, is similar to that observed in other distant
galaxies of this luminosity (e.g., Capak et al., 2015). This suggests that the
majority of the [C II] emission in the outskirts is associated with cold star-
forming gas.

On the other hand, the center of the galaxy has a high222As a sanity check, we
also computed the $f_{\rm{\rm[{C\textsc{II}}]},ion}$ using the peak fluxes
only; these peak fluxes are not model-dependent, but they will contain flux
from both inside and outside of our central aperture; we still found a high
$f_{\rm{\rm[{C\textsc{II}}]},ion}$ of $23$–$43$%, see Table 1.
$f_{\rm{\rm[{C\textsc{II}}]},ion}$ of $38$–$90\%$, which has never been
observed in a distant SMG and is rarely observed in local infrared-luminous
galaxies. Finding such high $f_{\rm{\rm[{C\textsc{II}}]},ion}$ is instead not
uncommon in more normal or quiescent local galaxies, such as main-sequence
galaxies and ETGs (Lapham et al., 2017). It should be noted, however, that
$f_{\rm{\rm[{C\textsc{II}}]},ion}$ or
${\rm[{C\textsc{II}}]}/{\rm[{N\textsc{II}}]}$ values from the literature are
typically only quoted for the galaxies as a whole, and few studies prior to
this one have attempted to separate the emission from the center and outskirts
(see, e.g., Parkin et al. 2013 where a $f_{\rm{\rm[{C\textsc{II}}]},ion}$
gradient was found in M51). It is possible that higher
$f_{\rm{\rm[{C\textsc{II}}]},ion}$ values could also be found in the center of
some SMGs and other IR-luminous galaxies if they were observed with a
sufficient angular resolution. This would be an interesting avenue for future
observations, in particular as it could allow finding analogs to Hyde in the
local Universe, which would provide invaluable insights on the physical
processes at play.

The $\log_{10}(L_{[{C\textsc{ii}}]}/L_{\rm FIR})=-3.17^{+0.26}_{-0.28}$ in the
center of Hyde is also a factor five lower than in the outskirts, which is
expected if the gas is predominantly ionized (Díaz-Santos et al., 2017). This
supports the hypothesis that the majority of the [C II] emitting gas in the
galaxy center is ionized and not star-forming, and therefore that an energy
source other than on-going star formation is significantly contributing to the
energy budget in the center of the galaxy. If Hyde is indeed on the path to
quenching, this would strongly suggest an inside-out quenching channel.

We cannot determine with certainty the nature of this central ionizing source
with the available data, but we can propose two likely candidates. First, the
source could be a compact population of intermediate-age stars born in a
recent past. Indeed, Hyde is believed to contain about $4\times
10^{10}\,M_{\ast}$ of stars within its half light radius (S18), and a Bruzual
& Charlot (2003) single stellar population of this mass should generate an [N
II]-ionizing flux with total energy $\gtrsim 10^{8}\,L_{\odot}$, even at
advanced ages of several hundred million years, which is larger than the
observed central [N II] luminosity, $(6.5^{+1.7}_{-1.4})\times
10^{7}\,L_{\odot}$. However, hydrogen and carbon and will also consume part of
this ionizing flux; determining whether this is indeed a viable hypothesis
would require dedicated modeling, a measure of the gas metallicity, and a
better understanding of the geometry of the system, which are all presently
lacking.

Second, the ionizing source could be an AGN. Given that none of the available
data indicate the presence of an AGN in this galaxy ($L_{X}<10^{44}\,{\rm
erg/s}$; Civano et al. 2016, $L_{1.4\,\rm GHz}<10^{24}\,{\rm W/Hz}$; Smolčić
et al. 2017), this may seem less likely, yet a weak or obscured AGN cannot be
ruled out. The presence of an AGN would make the interpretation of the
observed line ratios more difficult, and could invalidate some of the
assumptions and finer calculations presented in the following section, which
only account for stellar ionization flux. Yet, the implication on the galaxy’s
${\rm SFR}$ estimate would follow a similar path: if an AGN exists in the
center of this galaxy, with a luminosity large enough to affect the continuum
and line emission, it would necessarily contribute to the observed $L_{\rm
IR}$. This, in turn, would imply that ${\rm SFR}$ estimates using $L_{\rm IR}$
would be biased high. This would include our initial estimate from S18, which
already tentatively placed the galaxy below the galaxy main sequence. If an
AGN is indeed present, and given the large amount of dust and the compactness
of the galaxy, the AGN radiation could be trapped (e.g., Costa et al., 2018)
and couple efficiently with the gas to suppress star formation.

###  4.3 Revised star formation rate

Figure 5: Likelihood of the photometric data for Hyde as a function of ${\rm
sSFR}={\rm SFR}/M_{\ast}$. The likelihood is shown without any constrain on
the dust luminosity (gray), using the total infrared luminosity $L_{\rm IR}$
(orange), or using the birthcloud infrared luminosity $L_{\rm IR,BC}$ (red).
The horizontal red lines show the $68\%$ (thick) and $90\%$ (thin) confidence
intervals. The vertical lines indicate the locus of the galaxy main-sequence
and its scatter from Schreiber et al. (2017).  Figure 6: Updated location of
Hyde with respect to the galaxy main-sequence. The dashed (resp., dotted) line
indicate the locus of the galaxy main-sequence (resp., its scatter) from
Schreiber et al. (2017). The dark (resp., light) red region shows the final
68% (resp., 90%) confidence region for Hyde obtained in this paper. The red
cross indicates the location of our best-fit solution.

We now use the characterization of the ionizing emission presented in the
previous section to provide a refined estimate of the galaxy’s ${\rm SFR}$. We
do so by estimating the infrared luminosity produced in birth clouds by young
stars, and feed this luminosity into our SED modeling as a prior.

The total infrared luminosity, $L_{\rm IR}$, is a very useful observable in
SED modeling, as it offers an independent constraint on the obscured
luminosity of a galaxy. This is particularly crucial to constrain the ${\rm
SFR}$. However, like [C II], the $L_{\rm IR}$ can originate from different
regions in the galaxy. Here we will consider two main contributors:
birthclouds (BC), which are heated exclusively by stars younger than $10\,{\rm
Myr}$ (Charlot & Fall, 2000), and the inter-stellar medium (ISM), which is
heated by the older stars (e.g., da Cunha et al., 2008). We note the
corresponding infrared luminosities $L_{\rm IR,BC}$ and $L_{\rm IR,ISM}$,
respectively. $L_{\rm IR,BC}$ is the quantity we aim to estimate, as by
definition it is most directly tracing the recent ${\rm SFR}$.

Next, we need to estimate the fraction of the infrared luminosity produced in
the ISM, $f_{\rm IR,ISM}=L_{\rm IR,ISM}/L_{\rm IR}$. Intuitively, we can
expect this fraction to be closely related to
$f_{\rm{\rm[{C\textsc{II}}]},ion}$, which we computed earlier, since PDRs are
co-located with birthclouds, and since ionized gas is part of the ISM. In what
follows we will therefore assume
$L_{\rm{\rm[{C\textsc{II}}]},ion}=L_{\rm{\rm[{C\textsc{II}}]},ISM}$ and
$L_{\rm{\rm[{C\textsc{II}}]},PDR}=L_{\rm{\rm[{C\textsc{II}}]},BC}$.

With these assumptions, we can relate $f_{\rm IR,ISM}=L_{\rm IR,ISM}/L_{\rm
IR}$ to $f_{\rm{\rm[{C\textsc{II}}]},ion}$, using the following relation:

$\displaystyle\frac{1}{f_{\rm IR,ISM}}-1=\frac{L_{\rm IR,BC}}{L_{\rm IR,ISM}}$
$\displaystyle=\frac{L_{\rm IR,BC}}{L_{{\rm[{C\textsc{II}}]},\rm
BC}}\,\frac{L_{{\rm[{C\textsc{II}}]},\rm ISM}}{L_{\rm
IR,ISM}}\,\frac{L_{{\rm[{C\textsc{II}}]},\rm BC}}{L_{{\rm[{C\textsc{II}}]},\rm
ISM}}$ $\displaystyle=\frac{({\rm[{C\textsc{II}}]}/{\rm IR})_{\rm
ISM}}{({\rm[{C\textsc{II}}]}/{\rm IR})_{\rm
BC}}\,\left(\frac{1}{f_{\rm{\rm[{C\textsc{II}}]},ion}}-1\right)$
$\displaystyle=\alpha\,\left(\frac{1}{f_{\rm{\rm[{C\textsc{II}}]},ion}}-1\right)$
(2)

We can determine $\alpha$ empirically using a set of reference galaxies for
which we can estimate both $f_{\rm{\rm[{C\textsc{II}}]},ion}$ and $f_{\rm
IR,ISM}$. Here we used the local infrared-luminous galaxies from GOALS (Díaz-
Santos et al., 2017), which have direct measurements of [C II] and [N II] from
Herschel and an average of $f_{\rm{\rm[{C\textsc{II}}]},ion}=12.6\pm 0.7\%$,
similar to high-z SMGs. Based on the short gas depletion times typically
observed for starbursting galaxies (e.g., Béthermin et al. 2015), we assumed
that the GOALS galaxies were mostly formed in a brief burst lasting $50$ to
$200\,{\rm Myr}$333As shown in da Cunha et al. (2010), starbursting galaxies
can also contain a component of older stars, which would also contribute to
the IR emission. If true, this would increase our estimate of $\alpha$ from
the GOALS galaxies, which would in turn decrease the final value of $L_{\rm
IR,BC}$ and ${\rm SFR}$ for Hyde. Our assumption is therefore conservative..
We then used Bruzual & Charlot (2003) stellar populations and the uniform
Calzetti et al. (2000) dust screen to compute the expected fraction of their
bolometric luminosity produced by young stars for such star formation
histories, $f_{\rm bol,BC}=65\pm 5\%$. Young stars were defined as stars
younger than $10\,{\rm Myr}$, as above. Given the strong attenuation in these
galaxies (Howell et al., 2010), the bolometric luminosity is practically equal
to the infrared luminosity, and this latter fraction can thus be identified as
$1-f_{\rm IR,ISM}$ for GOALS galaxies. Feeding these estimates back to Eq. 2
gives $\log_{10}(\alpha)=-0.57\pm 0.10$ (in other words, the
${\rm[{C\textsc{II}}]}/{\rm IR}$ ratio is a factor of four times lower in the
ambient ISM than in birthclouds).

With the knowledge of $f_{\rm IR,ISM}$, which we can compute independently in
the center ($f_{\rm IR,ISM}=69$–$96\%$) and outskirts ($f_{\rm
IR,ISM}=0$–$46\%$) of Hyde, we can estimate the summed infrared luminosity
produced in birth clouds for both regions: $\log_{10}(L_{\rm
IR,BC}/L_{\odot})=11.52\pm 0.20$. We then ran
FAST++444https://github.com/cschreib/fastpp v1.3 to model the multi-wavelength
photometry with the same setup as in S18, but now using $L_{\rm IR,BC}$ as a
prior on the obscured luminosity of stars younger than $10\,{\rm Myr}$.
Briefly, in S18 we used a flexible model for the star-formation history,
formulated as an exponentially rising ${\rm SFR}$ followed by an exponential
decline, with a variable time of transition between the two phases and
variable exponential timescales.

The outcome is illustrated in Figs. 5 and 6. The stellar mass was essentially
unaffected, but the updated model produced an ${\rm SFR}=50^{+24}_{-18}\,{\rm
M}_{\odot}/{\rm yr}$ (averaged over the last $10\,{\rm Myr}$), which is non-
zero. This leads to ${\rm sSFR}=0.71^{+0.58}_{-0.34}\,{\rm Gyr}^{-1}.$ With a
main-sequence locus at ${\rm sSFR}=2.8\,{\rm Gyr}^{-1}$ (Schreiber et al.,
2017), this places the galaxy a factor $4.0$ below the main sequence, with a
lower limit of $>$$2.3$ at 68% confidence (and just $>$$1.3$ at 90%
confidence).

The previous estimate of the ${\rm SFR}$ from S18, using instead the total
$L_{\rm IR}$ as a constraint, only allowed us to obtain an upper limit on the
${\rm SFR}$ (see Fig. 5). The new data, however, exclude the possibility that
Hyde has fully quenched (${\rm SFR}\ll{\rm SFR}_{\rm MS}$). While the $90\%$
upper limit on the ${\rm SFR}$ is actually unchanged compared to our earlier
estimates, we are now able to constrain both sides of the ${\rm SFR}$
probability distribution, and we can therefore give more credit to the maximum
probability solution and the standard $68\%$ confidence interval. In this
light, we can claim that the ${\rm SFR}$ is now constrained to intermediate
values, a factor of $2$–$10$ lower than the main-sequence level. Accounting
for the observational uncertainty and the log-normal main-sequence scatter,
the probably of observing ${\rm sSFR}<0.71\,{\rm Gyr}^{-1}$ when randomly
drawing from the main-sequence distribution is $4\%$.

A factor limiting the strength of this conclusion is that the location of the
galaxy main sequence is not yet perfectly determined at $z\sim 4$. Here we
based our comparison on the estimate from Schreiber et al. (2017) since it is
also based on ALMA-derived ${\rm SFR}$s and with stellar masses derived in a
similar way; this mitigates the impact of systematic biases on both $M_{\ast}$
and ${\rm SFR}$ estimations, and makes the relative ${\rm SFR}$ difference
between Hyde and the main sequence more robust. In fact, the most robust
comparison would be obtained by applying the exact same method to determine
the ${\rm SFR}$ for Hyde and main-sequence galaxies. Unfortunately, this
cannot be achieved until high-quality [C II] and [N II] data are available for
a representative sample of other normal galaxies. Yet, even setting aside all
our ${\rm SFR}$ modeling, it is still true that Hyde has a high overall
${\rm[{N\textsc{II}}]}/{\rm[{C\textsc{II}}]}$ ratio compared to high-z SMGs,
and that it is “special” in a number of ways (high attenuation, compact size,
etc.). Based on this, we argue that most galaxy are not like Hyde, and
therefore that on average the ${\rm SFR}$ of typical main-sequence galaxies
should be correct.

Nevertheless, we can quantify how the above result depends on the estimated
locus and scatter of the main-sequence. If the main-sequence scatter is
increased to $0.4\,{\rm dex}$ (resp. $0.5\,{\rm dex}$), we find that the
probably $P$ of observing Hyde’s ${\rm sSFR}$ when drawing from the main-
sequence distribution increases to $9\%$ (resp. $13\%$). This does not appear
to be supported by observations however, as other references in the literature
typically report a lower scatter. For example, if the scatter is decreased to
$0.25\,{\rm dex}$ (resp. $0.15\,{\rm dex}$; Speagle et al. 2014; Pearson et
al. 2018), $P$ drops to $2.5\%$ (resp. $0.5\%$). Similarly, if the main-
sequence mean ${\rm sSFR}$ is decreased by $0.1\,{\rm dex}$ (Pearson et al.,
2018), $P$ increases to $8\%$. However, most references in the literature
using FIR, sub-mm, or radio-based ${\rm SFR}$ estimates actually report a
similar or higher mean ${\rm sSFR}$ at $z\sim 4$ and $\log_{10}(M_{\ast})\sim
10.8$; with a difference of $+0.1\,{\rm dex}$ (Speagle et al., 2014),
$+0.02\,{\rm dex}$ (Tomczak et al., 2016), $+0.03\,{\rm dex}$ (Bourne et al.,
2017), $+0.07\,{\rm dex}$ (Leslie et al., 2020). Considering an increase of
the main-sequence ${\rm sSFR}$ by $0.1\,{\rm dex}$ would decrease $P$ to
$2.3\%$.

##  5 Our result in context

###  5.1 The transition to quiescence?

The modeling in the previous section allowed us to obtain a revised ${\rm
SFR}$ estimate for Hyde, which confirms and refines the earlier estimate from
S18. This ${\rm SFR}$, together with the estimated large stellar mass, would
place the galaxy significantly outside of the standard galaxy main-sequence
and its scatter (e.g., Schreiber et al. 2017). This supports the hypothesis
that the galaxy does not belong to the main-sequence, and is instead in
transition to quiescence.

As in S18, we are still not able to determine the future of this galaxy with
certainty; although its ${\rm SFR}$ at the time of observation does appear to
be low, we cannot exclude that this only corresponds to a temporary pause in
its activity. At best, we can bring forward two arguments which disfavor (but
not disprove) this possibility. Firstly, the galaxy is already among the most
massive individual known at high redshift, and sits beyond the knee of the
stellar mass function. This implies that, if its ${\rm SFR}$ does increase in
the future, it cannot do so for very long. Secondly, if high-redshift galaxies
commonly experience large (but temporary) fluctuations in their ${\rm SFR}$,
this would be detectable as an increase in the scatter of the galaxy-main
sequence. This does not appear to be the case at least at $z\sim 4$ (e.g.,
Schreiber et al. 2017).

###  5.2 Other evidence

As pointed out in our introduction, Hyde compiles an unusual combination of
observables, when compared to other massive galaxies (or SMGs) at a similar
epoch. We expand on these aspects in the following paragraphs, and discuss
what additional observations could help confirm or contradict our results.

One particularly unusual combination is that of a lower-than-average dust
temperature and a compact geometry. Hyde is indeed unusually compact; in
recent sub-millimeter surveys, only 4 to $6\%$ of SMGs turn out to have sizes
as small as Hyde (e.g., Ikarashi et al. 2017; Gullberg et al. 2019). For
massive star-forming galaxies in general, compact sizes are typically observed
in starburst galaxies located above the galaxy main-sequence (e.g., Elbaz et
al. 2011). These galaxies also tend to have a higher dust temperature (e.g.,
Elbaz et al. 2011; Magnelli et al. 2014; Béthermin et al. 2015), which can be
seen as a natural consequence of their high luminosity and compact geometry.
Based on the very compact size of Hyde, we could have therefore expected to
see an enhanced dust temperature, but the opposite is observed.

Indeed, when comparing to the average dust temperature of massive $z\sim 4$
galaxies from Schreiber et al. (2018a), Hyde displays a temperature about
$10\,{\rm K}$ lower than average. Although dust temperatures are notoriously
difficult to measure (e.g., Casey 2012), in this case we can compare Hyde’s
temperature to a reference which has been measured with the same method (SED
fitting and SED templates) and the same wavelength coverage (from Herschel to
ALMA band 7). This eliminates most of the systematics, and allows us to
robustly quantity the relative difference.

In an attempt to understand these possibly conflicting observations, we can
try to draw a comparison to other known galaxies. Here, we have selected the
few known massive galaxies at high-redshift with a similar dust temperature of
$\sim$$30\,{\rm K}$, and with a known spectroscopic redshift; the latter being
required to measure the dust temperature accurately. The two most famous
examples include GN20 (Daddi et al. 2009; Hodge et al. 2013; Tan et al. 2014)
and HDF850.1 (Walter et al., 2012). These two galaxies have a half-light
radius of $6$–$7\,{\rm kpc}$, which is an order of magnitude larger than Hyde.
This may on its own explain their lower-than-average dust temperature,
although recent results suggest this could also be caused by optically-thick
dust, at least for GN20 (Cortzen et al., 2020). To our knowledge, the only
other reported instance of a low dust temperature combined with a compact
geometry can be found in the interacting pair SGP 38326 (Oteo et al., 2016);
unfortunately this system only has poor wavelength coverage, which renders the
temperature uncertain ($T_{\rm dust}=33$–$55\,{\rm K}$). Although comparably
massive to Hyde, these galaxies are also an order of magnitude brighter, hence
would fit in the traditional picture of a starburst galaxy.

To date, the case of Hyde therefore appears to be unique. Its low dust
temperature could correspond to a softer radiation field (or, equivalently, to
a low star formation efficiency), which would match the results in this paper.
However, as for GN20, it could also be caused by optically-thick dust.
Disentangling the two possibilities would require an alternative measurement
of the temperature (e.g., using the CO or [C I] line ratios; Cortzen et al.
2020).

##  6 Conclusions

We have obtained new ALMA observations toward the galaxy Hyde to observe its
${\rm[{N\textsc{II}}]}_{205}$ emission. We showed that the line emission is
more concentrated in [N II] than in [C II], which implies a gradient in the
ionized gas fraction. We found the center of the galaxy to be predominantly
ionized, which suggests that young stars in their birth-clouds are not the
dominant ionization source in the galaxy center, leaving the room for other
sources such as intermediate-age-stars, or an AGN. In contrast, the outskirts
are dominated by neutral gas which suggests the [C II] emission there is
mostly associated with star-forming regions. Using these new insights, we
obtained an updated estimate of the galaxy’s star formation rate, placing it
securely below the galaxy main sequence, hence possibly in transition to
quiescence.

These results point toward an ongoing inside-out quenching mechanism, and show
that this process may start before the galaxy has fully expelled or consumed
its gas reservoirs, as demonstrated by the strong obscuration of this galaxy.
This confirms the importance of studying the properties of this galaxy in
greater details, to better understand the process of quenching at high
redshifts.

Further observations of this system would enable a better understanding of the
state of the gas, and of the mechanism responsible for quenching, in
particular whether the galaxy hosts an AGN or not. New observations with ALMA
are already scheduled, including high-resolution imaging of the dust continuum
emission to establish the morphology and geometry of the galaxy, as well as [C
I] for measuring the cold gas mass to study the interplay between the ionized
and neutral gas reservoirs. Otherwise, the James Webb Space Telescope (JWST)
could be used to look at the rest-frame optical lines, in particular ${\rm
H}_{\alpha}$, to better constrain the ${\rm SFR}$ of the galaxy and obtain an
estimate of the gas-phase metallicity. Unfortunately, even JWST may not have a
sharp enough resolution to resolve the optical emission line profiles, which
would then require larger telescopes, such as the European Extremely Large
Telescope (E-ELT). Finally, deep low-frequency radio observations, for example
with the LOw Frequency ARray (LOFAR), would reveal even moderate AGN activity
and thus determine the role of AGNs in the high-redshift quenching process.

To go beyond the case study of this single object and determine the rate of
occurrence of these features, similar observations of a sample of massive,
obscured galaxies would be required. As illustrated here, finding
transitioning galaxies is not an easy task, especially if most of them are
strongly obscured and would not be included in the $H$-band selected catalogs
produced from Hubble imaging. Samples of Spizter-IRAC-selected galaxies may be
more adequate (see Caputi et al. 2015; Wang et al. 2016, 2019), and in the
near future JWST will hopefully open this search to fainter, more numerous
galaxies.

###### Acknowledgements.

The authors want to thank the two anonymous referees for their comments that
clearly improved the consistency, thoroughness, and overall quality of this
paper. CS would like to thank Leah Morabito and Ian Heywood for sharing their
expertise on interferometric imaging. Most of the numerical analysis conducted
in this work has been performed using vif, a free and open source C++ library
for fast and robust numerical astrophysics (http://cschreib.github.io/vif/).
This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.A.00026.S
and ADS/JAO.ALMA#2018.1.00216.S. ALMA is a partnership of ESO (representing
its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and
NSC 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. GGK acknowledges the support of the Australian Research Council through
the Discovery Project DP170103470.

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##  Appendix A Simulated noise maps

To estimate parameter uncertainties, the method we used in this paper is to
repeat our measurement on mock images, created by duplicating the true image
and adding different realizations of noise to it. In this experiment, the
observed noisy image and our best-fit model become the “truth”, and we can
determine how far away our fitting procedure is from this “truth” on each mock
image. The strength of this method is that it requires no assumption on the
model, and no analytical calculations; the parameter probability distribution
can be extracted straight away by gathering the fits to all mock images.
Furthermore, it automatically takes care of parameters that are strongly
correlated with one another, as well as correlated noise. The difficulty is
that it strongly depends on the quality of the noise that is injected on top
of the “truth” image. For uncertainties to be accurate, the noise needs to
have the exact same amplitude and covariance matrix as in the real data.

Figure 7: Observed and simulated noise maps (top) and their respective auto-
correlation functions (bottom). The real [C II] map is displayed in the top
left corner, with the source masked in the center. For comparison, we also
display the [C II] dirty beam in the middle of the bottom panel. All the
images on each row are displayed with the same color bar. The auto-correlation
functions were rescaled prior to display to a peak value of unity. For easier
comparison, we also show a circular average of the covariance in shown in the
central column of the first row, with the observed (black) and simulated
(green dashed) auto-correlation functions, and the dirty-beam profile (blue).

While in non-interferometric data sets the noise can usually be assumed
uncorrelated, this is not the case with our ALMA data; the noise observed in
the [C II] line map, and its auto-correlation function (ACF), are shown in
Fig. 7 (left). One can observe that the ACF has a structure very similar to
the dirty beam (see Fig. 7, center), and we demonstrate in Appendix B that
this is indeed expected when the data is imaged with natural weighting.

This makes it easy to reproduce noise with a similar covariance. Indeed, if a
uniform uncorrelated random noise is convolved with a two-dimensional kernel
$K$, its ACF will be $C=K\otimes K$, or in the Fourier domain,
$\hat{C}=\hat{K}^{2}$ (where “hat” symbolizes the Fourier transform). If we
set $C=P$, where $P$ is the dirty beam (see Appendix B), we have:

$\displaystyle\hat{K}=\sqrt{\hat{P}}.$ (3)

Since the Fourier transform introduces aliasing artifacts on the edges of $K$,
it is desirable to use a dirty beam image $P$ that is significantly larger
than the final dimensions of the noise map. This can be achieved by padding.
To stabilize this further, and since we are mostly interested in preserving
the “core” of the covariance, we smoothed out the transition to the edges of
the image by multiplying the dirty beam image with a broad Gaussian of FWHM
$4\arcsec$.

We used this empirical convolution kernel $K$ to generate new correlated noise
realizations, shown in Fig. 7 (right). The resulting noise maps are visually
similar to the real noise map, and the core of the covariance is well
reproduced. We note that, since the dirty beam side lobes in our data have a
relatively low amplitude, very similar results could have been achieved by
simply using a Gaussian kernel of FWHM equal to half that of the dirty beam.

Finally, to set the noise RMS, we simply measured the RMS on the real image
and on the simulated noise images, and rescaled the simulated images to match
the real observed RMS. Because our target is at the phase center, we did not
correct for the primary beam attenuation in the real (or simulated) images,
hence the RMS was constant across the entire image.

##  Appendix B Auto-correlation function of image-domain noise

For a generic interferometer, the observed dirty image is the Fourier
transform of the weighted and sampled sky visibility:

$\displaystyle I_{D}(x,y)$ $\displaystyle\equiv\int
A(u,v)\,W(u,v)\,V(u,v)\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv\,,$
(4)

where $A$ is the visibility sampling of the interferometer (a sum of delta
functions), $W$ is the imaging weight, and $V$ is the complex visibility.
Without loss of generality and for simplicity of notation, we drop the time,
frequency, and polarization dependence in all quantities.

The dirty beam is the response to a point source, which has constant
visibilities in the Fourier domain:

$\displaystyle P(x,y)$ $\displaystyle\propto\int
A(u,v)\,W(u,v)\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv\,.$ (5)

If we define the inverse Fourier transform of the dirty image

$\displaystyle V_{D}(u,v)$ $\displaystyle\equiv\int
I_{D}(x,y)\,\exp\large[2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv$ (6)
$\displaystyle=A(u,v)\,W(u,v)\,V(u,v)\,,$ (7)

then by definition, the image auto-correlation function (ACF) is

$\displaystyle C(x,y)$ $\displaystyle\equiv\int
V_{D}(u,v)\,\bar{V}_{D}(u,v)\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv$
(8) $\displaystyle=\int
A(u,v)\,W(u,v)^{2}\,V(u,v)\,\bar{V}(u,v)\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv\,,$
(9)

where we used $A^{2}=A$ since it is a sum of delta functions.

To study the ACF of the noise, we now assume that the sky contains no source
and the visibilities are therefore only made of noise of amplitude
$\sigma(u,v)$, which we assume is uncorrelated. Thus,
$\left<V\,\bar{V}\right>=\sigma^{2}$. If we now compute the expectation value
of the ACF, we get

$\displaystyle\left<C(x,y)\right>$ $\displaystyle=\int
A(u,v)\,W(u,v)^{2}\,\sigma(u,v)^{2}\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv\,.$
(10)

In the case of natural weighting, the weights are chosen as $W\propto
1/\sigma^{2}$. Injecting this into the above equation, we finally get

$\displaystyle\left<C(x,y)\right>$ $\displaystyle\propto\int
A(u,v)\,W(u,v)\,\exp\large[-2\pi\,i\,(u\,x+v\,y)\large]\,du\,dv$ (11)
$\displaystyle\propto P(x,y)\,.$ (12)

Therefore, for images produced with natural weighting, the expectation value
of the noise ACF is the dirty beam.

In practice the match is not perfect (see Fig. 7), and this can be explained
by at least three possible causes: a) we can only measure the ACF on a finite-
sized image, so our estimate of the image ACF is noisy; b) the $\sigma$ that
enters in the definition of $W$ is only an estimate of the true visibility
uncertainty; and c) the noise in the visibilities can itself be correlated.
These potential issues seem to have only a moderate impact however; as
demonstrated in Fig. 7 (middle, top), in our data the core of the ACF is
extremely well reproduced by the dirty beam.

##  Appendix C Likelihood for correlated Gaussian noise

The general formulation of the likelihood for Gaussian correlated noise is:

$\displaystyle\mathcal{L}\propto\exp(-\frac{\chi^{2}}{2})\,,\quad\text{with}\quad\chi^{2}=(\mathbf{d}-\mathbf{m})^{t}\,\Sigma^{-1}\,(\mathbf{d}-\mathbf{m})\,,$
(13)

where $\mathbf{d}$ is a vector of observed data, $\mathbf{m}$ is a vector of
model data, and $\Sigma$ is the data covariance matrix (symmetric and positive
definite). For our data (see Appendix B), we have:

$\displaystyle\Sigma_{ij}=\sigma^{2}\,P(x_{i}-x_{j},y_{i}-y_{j})\,,$ (14)

where $\sigma$ is the image RMS, $(x_{i},y_{i})$ are the image coordinates of
pixel $i$, and $P(\delta x,\delta y)$ is the value of the dirty beam at an
offset position $(\delta x,\delta y)$ from the peak.

In practice, inverting $\Sigma$ is numerically unstable. The best way to
evaluate it is to perform a Singular Value Decomposition (SVD) such that
$\Sigma=U\lambda\,U^{t}$, where $U$ is a unitary matrix ($U^{-1}=U^{t}$) and
$\lambda$ is a diagonal matrix whose entries are the singular values
$\lambda_{i}$ sorted by decreasing value. Then we have
$\Sigma^{-1}=U\lambda^{-1}\,U^{t}$. If we define
$L^{-1}=(1/\\!\sqrt{\lambda})\,U^{t}$, then Eq. 13 can be rewritten in a
simpler form:

$\displaystyle\chi^{2}=(\tilde{\mathbf{d}}-\tilde{\mathbf{m}})^{t}\,(\tilde{\mathbf{d}}-\tilde{\mathbf{m}})$
(15)

where $\tilde{\mathbf{d}}=L^{-1}\,\mathbf{d}$ and
$\tilde{\mathbf{m}}=L^{-1}\,\mathbf{m}$. These can be seen as the “de-
correlated” observation and model vectors, respectively.

The SVD is not sufficient to make the computation of $\Sigma^{-1}$ or $L^{-1}$
stable. In all cases, the matrix $\lambda$ contains small entries that cannot
be inverted safely. A workaround is to truncate the matrix
$(1/\\!\sqrt{\lambda})$ by setting to zero the inverse of those singular
values that are smaller than some chosen threshold. The resulting matrix
inverse is then a “pseudo-inverse” (i.e., an approximation of the true
inverse), but it is usually better behaved.

The method we adopted to choose the threshold is the following. We defined the
normalized singular values $\bar{\lambda}_{i}=\lambda_{i}/\lambda_{0}\leq 1$,
and we created a logarithmic grid of $100$ threshold values $v$ ranging from
${\rm min}(\bar{\lambda})$ to $0.1$, such that all $\bar{\lambda}_{i}<v$ were
removed from the inverse. For each value of $v$, we evaluated the
corresponding $L^{-1}$, and used it to “de-correlate” one of our noise
realization from the simulations (this noisy image did not contain any
source). We computed the covariance matrix of the resulting image, normalized
it to unit diagonal, and defined the metric $k$ as the sum the absolute value
of entries with $|x_{i}-x_{j}|\leq 1$ and $|y_{i}-y_{j}|\leq 1$. We then
picked the value of $v$ which minimized $k$, or in other words, the value
which produced an image with the lowest noise covariance. Experiments showed
that $k$ is large when $v$ is too small, as the matrix inverse is unstable and
the resulting images are degraded. On the other hand, $k$ is also large when
$v$ is too large, as this leads to a poorer approximation of the inverse which
leaves more correlated noise.

We note that, while Eq. 13 is formally the correct expression to compute the
likelihood for our data, we obtained very similar results when we used the
simpler, standard expression for uncorrelated noise,
$\chi^{2}=\sum_{i}\,[(d_{i}-m_{i})/\sigma]^{2}$. This is because we used Monte
Carlo simulations to determine the confidence intervals rather than relying on
the shape of the likelihood.

##  Appendix D Input/output simulations

Figure 8: Outcome of the input/output analysis of our fitting procedure,
modeling sources of known profiles in mock images. The simulations matching
the [C II] and [N II] maps are shown at the top and bottom, respectively. From
left to right, we show how our method recovers the total flux, the flux inside
the center, the flux in the outskirts, and the ratio of center-to-total (with
“center” and “outskirts” as defined in the main text). The red line is the
line of perfect agreement.

To test the accuracy of our profile-fitting method, we performed an
input/output analysis, where we placed sources of known light profiles in
simulated noise maps devoid of sources, and tried to recover their profile
with our method. The input sources were modeled as two-component exponential
disks, with a small “core” component and an “extended” component, as is
assumed in our model. The size of the “core” component was chosen randomly in
the range $r_{\rm core}=0.1$–$0.5\,{\rm kpc}$, and the size of the extended
component was chosen randomly in the range $r_{\rm extended}=r_{\rm
core}+0.2$–$2.2\,{\rm kpc}$. The relative flux of these components was chosen
uniformly between 0% (all flux in extended component) to 100% (all flux in
core component). The axis ratio was chosen randomly between $0.4$ to $1.0$,
and the position angle could take any value. Finally, the peak flux of each
source was matched to the observed peak flux in our real images. As for our
real data, we then computed the true fluxes inside and outside of a
$0.5\,{{\rm kpc}}$ aperture to obtain the “central” and “outskirts” fluxes. We
note here that, since the [N II] image has a sharp angular resolution, the
step of fixing the peak flux in the simulations has a strong impact on the
true flux in the central aperture, as can be seen in Fig. 8 where the range of
$F_{\rm center}$ is limited. This is less true for [C II], where the coarser
PSF means that the central pixel can contain more flux from the extended
component.

The mock sources were injected directly in the image domain, by convolving the
source’s known light profile with the image dirty beam at $\times$$9$
oversampling. To test the accuracy of this source injection method, we also
used CASA to create simulated visibilities for each mock source using
simobserve (without thermal noise), and produced corresponding dirty images
using clean with the same (oversampled) cell size. The resulting mock images
looked identical to those produced by convolution with the dirty beam. To
quantify this, for each mock source we computed the second moment ($R^{2}$) of
the images produced by the two methods (visibilities imaging, and image
convolution). We found a maximum relative error on $R^{2}$ of only $3\times
10^{-5}$, i.e., close to numerical noise, with no dependence on the size of
the mock sources. This implies that injecting directly in the image domain is
an accurate source injection method, which is also much faster than creating
mock visibilities and imaging them. We caution that this property only holds
because we work with dirty images; such images can be described as a Fourier
transform of an incomplete $(u,v)$ plane, but cleaned images cannot.

We repeated this procedure with 200 different mock sources, and for each
source we replicated the noise level, noise covariance matrix, peak flux, and
dirty beam of the [C II], [N II], and continuum maps separately, to test the
impact of the varying $S/N$ and angular resolutions encountered in this work.
Our fitting procedure was then applied to each of these mock sources, and the
recovered profiles were compared to the truth. Since these simulations are
accurate mocks of the real images, they allow us to reproduce exactly the
measurements we perform in this paper, so we can study any bias arising from
our method.

The outcome of this analysis, illustrated in Fig. 8 for [C II] and [N II],
shows that our modeling method recovers the total, central, and outskirts
fluxes with no detectable bias. The measurements are noisy, particularly for
the central flux, however the observed scatter is correctly captured by the
error bars, which were estimated exactly the same way as for the real data. To
quantify this, for each observed quantity $F$ shown in Fig. 8, we computed the
normalized residual $\delta_{F}=(F_{\rm obs}-F_{\rm true})/\sigma_{\rm obs}$,
where $\sigma_{\rm obs}$ is our estimated uncertainty. We then computed the
fraction of simulated points with $|\delta_{F}|<1$. If our uncertainties are
perfectly accurate, and if the noise on the quantity $F$ is roughly Gaussian,
this fraction should be equal to $68$%. Since we only use $200$ points to
compute these fractions, the expected statistical error on this number is
about $3$%. We find fractions for [C II] (resp. [N II]) of $67$% (resp. $66$%)
for the total flux, $62$% (resp. $61$%) for the flux in the inner $0.5\,{\rm
kpc}$, and $61$% (resp. $62$%) for the flux in the outskirts. These values are
exactly in the expected range for the total flux, but the fractions for the
central and outskirts fluxes are slightly lower than $68$%. For Gaussian
noise, this would imply that the uncertainties are under-estimated by at most
$10\%$; taking this small correction into account would not affect our
conclusions.

##  Appendix E Distribution in Monte Carlo simulation

Figure 9: Joint distribution of the fluxes, flux ratios, and ionized gas
fraction in the Monte Carlo simulations executed for the real images. As is
customary, on the diagonal we show the histogram of each quantity; the
$y$-axis for these plots is scaled automatically to include the maximum counts
in a bin. The green horizontal bars correspond the $16$th and $84$th
percentiles, which we report as uncertainties, and the horizontal green bar is
the bests-fit value.