# Bulk Action Growth for Holographic Complexity

Martin Krššák  Department of Theoretical Physics, Faculty of Mathematics,
Physics and Informatics, Comenius University, Bratislava, 842 48, Slovak
Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The action growth proposal relates the holographic complexity to the value of
the action on the Wheeler-de Witt patch. We introduce a new method of
calculating the gravitational action using the “bulk” term, i.e. the part of
the Einstein-Hilbert action quadratic in connection coefficients. We
demonstrate how to address the issue of non-covariance of the bulk action and
evaluate it using the tetrad formalism. Due to the boundary term-free nature
of the bulk action, we can gain further insights into the spatial structure of
the action on the Wheeler-de Witt patch. We then argue that our entire scheme
can be naturally covariantized within the framework of teleparallel geometry.

##  I Introduction

The AdS/CFT correspondence established a relation between a black hole in
asymptotically AdS spacetime and the dual field theory at its boundary [1, 2].
In addition to studying the properties of black holes outside the horizon,
there is currently interest in understanding how the physics is encoded
holographically behind the horizons. One approach is the holographic
complexity conjecture, which suggests that the quantum information complexity
is given by the value of the gravitational action in the interior region of a
black hole [3][4].

The gravitational action in the holographic complexity conjecture is taken to
be the standard Einstein-Hilbert action with the cosmological term and the
Gibbons-Hawking-York boundary term [3][4]

${\mathcal{S}}{}_{\text{grav}}=\frac{1}{2\kappa}\int_{\mathcal{W}}\sqrt{-g}(R-2\Lambda)+\frac{1}{\kappa}\int_{\partial\mathcal{W}}\sqrt{-\gamma}\mathcal{K},$
(1)

where $\kappa=8\pi$ (in $c=G=1$ units), $\gamma_{\mu\nu}$ is the boundary
metric, and $\mathcal{K}$ is its extrinsic curvature.

This action is evaluated in the interior region of a black hole,
$\mathcal{W}$, known as the Wheeler-de Witt (WdW) patch, where the crucial
contribution comes from the boundary term. In the case of a black hole with a
single horizon, the WdW patch represents the whole under the horizon region
and both the horizon and $r=0$ are being effectively treated as boundaries. In
the case of a black hole with two horizons, the WdW patch corresponds to the
region between them and both horizons are treated as boundaries.

We introduce here a new method of evaluating the gravitational action for the
holographic complexity by integrating the bulk gravitational action. Our use
of the term “bulk” comes from the decomposition of the Einstein-Hilbert
Lagrangian [5, 6, 7]

${\mathcal{L}}{}_{\text{EH}}={\mathcal{L}}{}_{\text{bulk}}+{\mathcal{L}}{}_{\text{tot}},$
(2)

where ${\mathcal{L}}{}_{\text{tot}}$ is the total derivative term and the bulk
term, ${\mathcal{L}}{}_{\text{bulk}}$, is the term quadratic in connection
coefficients111Note that the term ”bulk” is commonly used in the literature to
refer to the entire Einstein-Hilbert action, but we refrain from using this
term in that way..

We motivate our approach by an observation that the total derivative term is
not dynamical since it can be transformed into a surface term using the
Stoke’s theorem. The inclusion of the total derivative term introduces second
derivatives of the dynamical gravitational field variables, which leads to a
problem with the variational principle. The standard solution to this problem
is to add the GHY boundary term to cancel out the variations of the total
derivative term [8, 9], resulting in the total gravitational action (1).

However, there is another solution to avoid the issue of second derivatives by
considering the bulk term only. Indeed, it is well-known that both the EH and
bulk actions lead to the same field equations and hence are dynamically
equivalent [5, 6, 7]. Our goal in this paper is to demonstrate that this goes
beyond the dynamical equivalence and that we can evaluate the bulk action to
obtain the same value of the action on the WdW patch as using (1) to study
holographic complexity of black holes. We show that the boundary term-free
nature of the bulk action allows us to study not only the action growth but
also the Lagrangian growth, providing us with further insights into the
structure of the action on the WdW patch.

While the total derivative term in (2) is not dynamical, it ensures the
covariance of the Einstein-Hilbert action. Therefore, if we consider the bulk
term alone, we have to face the issue of non-covariance of the bulk action. In
the standard metric formalism, the bulk term is given by the Einstein
Lagrangian [10] and requires evaluation in a peculiar coordinate system, which
we argue exists only for the exterior solution and not for the interior
solution on the WdW patch that we are interested in. This motivates us then to
consider a tetrad version of the bulk term, where the issue of non-covariance
is not related to the choice of a coordinate system but to the local Lorentz
degrees of freedom of the tetrad. We show that we can find a well-behaved
tetrad on the WdW patch that gives us the correct answer for the action
growth.

We will demonstrate that the issue of the non-covariance can be resolved by
reformulating general relativity using the so-called teleparallel geometry. We
argue that teleparallel geometry is a natural framework for this task due to
the fact that its connection is determined solely by the choice of local
Lorentz degrees of freedom, and covariantizes the action without introducing
second derivatives of the dynamical field variables.

##  II Bulk actions for general relativity

The fundamental variable in the metric formalism is the metric tensor
$g_{\mu\nu}$, from which we define the Christoffel symbols,
$\Gamma^{\rho}{}_{\nu\mu}$, and the Riemannian curvature tensor leading to the
scalar curvature and the Einstein-Hilbert action. The decomposition (2) in the
metric case leads us to the metric bulk term [5, 6, 7]

${\mathcal{L}}{}_{\text{bulk}}=\frac{1}{2\kappa}\sqrt{-g}g^{\mu\nu}(\Gamma^{\rho}{}_{\sigma\mu}\Gamma^{\sigma}{}_{\rho\nu}-\Gamma^{\rho}{}_{\mu\nu}\Gamma^{\sigma}{}_{\rho\sigma}),$
(3)

which is commonly known as the Einstein Lagrangian and defines the Einstein
energy-momentum pseudotensor [10, 11].

The Einstein Lagrangian does not transform as a proper scalar density under
diffeomorphisms and hence its value depends on the choice of a coordinate
system. For instance, in Minkowski spacetime, where gravity is absent and we
expect the action to be zero, the Einstein action (3) indeed vanishes in
Cartesian coordinates, but diverges in the spherical coordinate system.

When including gravity and dealing with a curved spacetime, we must continue
using a quasi-Cartesian coordinate system to avoid the same type of divergent
actions. This is closely related to the well-known problem of various
pseudotensorial definitions of energy, which are required to be evaluated in
such special coordinates [5]. For example, in the case of Schwarzschild, we
use the isotropic Cartesian coordinate system to find the energy. However,
such coordinates are only well-defined in the exterior region of a black hole,
and they do not exist in the interior region that we are interested in.

This problem can be easily avoided by using the tetrad formalism and utilizing
its coordinate-free nature. We select a set of four orthonormal vectors,
$h^{a}{}_{\mu}$, called the tetrad as the fundamental variable, which is
related to the metric by

$g_{\mu\nu}=\eta_{ab}h^{a}_{\ \mu}h^{b}_{\ \nu},$ (4)

where $\eta_{ab}=\text{diag}(-1,1,1,1)$ is the tangent space metric. The
covariant differentiation is done using the Levi-Civita spin connection

$\omega^{a}{}_{b\mu}=\frac{1}{2}h^{\;c}{}_{\mu}\Big{[}f_{b}{}^{a}{}_{c}+f_{c}{}^{a}{}_{b}-f^{a}{}_{bc}\Big{]},$
(5)

where
$f^{c}{}_{ab}=h_{a}{}^{\mu}h_{b}{}^{\nu}(\partial_{\nu}h^{c}{}_{\mu}-\partial_{\mu}h^{c}{}_{\nu})$
are the coefficients of anholonomy. The curvature tensor of the spin
connection is

$\displaystyle
R^{a}_{\,\,\,b\mu\nu}=\partial_{\mu}\omega^{a}_{\,\,\,b\nu}-\partial_{\nu}\omega^{a}_{\,\,\,b\mu}+\omega^{a}_{\,\,\,c\mu}\omega^{c}_{\,\,\,b\nu}-\omega^{a}_{\,\,\,c\nu}\omega^{c}_{\,\,\,b\mu}\,,$
(6)

from where we define the scalar curvature and the tetrad Einstein-Hilbert
action. The decomposition (2) provides us with the tetrad bulk Lagrangian

${\mathcal{L}}{}_{\text{bulk}}=\frac{1}{2\kappa}h\left(\omega^{a}_{\,\,\,ca}\omega^{bc}{}_{b}-\omega^{a}{}_{cb}\omega^{bc}{}_{a}\right),$
(7)

which was introduced originally by Møller and used to define the Møller
energy-momentum complex [12, 13, 14, 15].

To illustrate the problem of non-covariance, we can consider Minkowski
spacetime again. The diagonal Cartesian tetrad gives us the expected zero bulk
action but the diagonal tetrad in the spherical coordinate system
$h^{a}{}_{\mu}=\text{diag}(1,1,r,r\sin\theta)$ leads to a divergent action. We
can solve this problem by considering a local Lorentz transformation of the
tetrad

$\tilde{h}^{a}_{\ \mu}=\tilde{\Lambda}^{a}_{\ b}h^{b}_{\ \mu},$ (8)

with a local Lorentz matrix

$\tilde{\Lambda}^{a}_{\ b}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos\phi\sin\theta&\cos\phi\cos\theta&-\sin\phi\\\
0&-\cos\theta&\sin\theta&0\\\
0&\sin\phi\sin\theta&\sin\phi\cos\theta&\cos\phi\end{array}\right),$ (9)

which will lead to the finite zero action.

Here, we can observe the usefulness of the tetrad formalism, as it allows us
to work both in spherical coordinates and keep the desired properties of
quasi-Cartesian coordinates at the same time. The form of the local Lorentz
transformation (9) can be understood easily from the fact that the tetrad (8)
is obtained from the diagonal Cartesian tetrad by just a coordinate change
[16].

##  III Bulk Action Growth of AdS Black Holes

For the holographic complexity conjecture we are interested in evaluating the
action222The integral here is $\int_{\mathcal{W}}=\int
dt\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int_{r_{-}}^{r_{+}}dr$, where
$r_{+}$ and $r_{-}$ are boundaries of the WdW patch.

${\mathcal{S}}{}=\int_{\mathcal{W}}{\mathcal{L}}{},$ (10)

with the total Lagrangian given by

${\mathcal{L}}{}={\mathcal{L}}{}_{\text{bulk}}+{\mathcal{L}}{}_{\Lambda}+{\mathcal{L}}{}_{\text{EM}},$
(11)

where ${\mathcal{L}}{}_{\Lambda}=\kappa^{-1}h\Lambda=3\kappa^{-1}hl^{-2}$ is
the cosmological term and ${\mathcal{L}}{}_{\text{EM}}$ is the Maxwell term.

The spherically symmetric spacetime is given by the metric

$ds^{2}=-f^{2}dt^{2}+f^{-2}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ (12)

that can be represented by a diagonal tetrad

$h^{a}{}_{\mu}=\text{diag}(f,f^{-1},r,r\sin\theta).$ (13)

We encounter a similar issue with this diagonal tetrad, resulting in the same
kind of divergence as discussed in the Minkowski case. Fortunately, it has the
exact same solution even in the curved case, i.e. to use a tetrad (8) with the
same local Lorentz transformation matrix (9) but with $h^{a}{}_{\mu}$ given by
(13).

###  III.1 Uncharged case

In the case of the uncharged AdS black hole, we consider the action without
the Maxwell term and the $f$-function given by

$f^{2}=1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}.$ (14)

We find the total Lagrangian (11) to be

${\mathcal{L}}{}=\frac{2\sin\theta}{\kappa
l^{2}rf}[Ml^{2}+r^{3}+r(l^{2}+3r^{2})(f-1)].$ (15)

By integrating, we obtain the action

${\mathcal{S}}{}=\int_{\mathcal{W}}{\mathcal{L}}{}=\int
dt\left.\frac{r^{3}+rl^{2}(1-f)}{l^{2}}\right|_{0}^{r_{h}},$ (16)

leading to the action growth

$\frac{d{\mathcal{S}}{}}{dt}=2M,$ (17)

which exactly coincides with the result obtained in [4, 3] using the standard
action (1).

###  III.2 Charged case

The Reissner-Nordstrom solution is given by the same metric (12) and the
tetrad (8) as in the uncharged case, but with the $f$-function given by

$f^{2}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{r^{2}}{l^{2}},$ (18)

from where two horizons $r_{\pm}$ are found as the real solutions to
$f(r_{\pm})=0$

The total Lagrangian, including the Maxwell term for a point charge
${\mathcal{L}}{}_{\text{EM}}=\kappa^{-1}h\,Q^{2}r^{-4}$, after evaluation
takes the same form as in the uncharged case (15) but with the $f$-function
given by (18). By integrating, we obtain the bulk action

$\displaystyle{\mathcal{S}}{}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{W}}{\mathcal{L}}{}=\int
dt\left.\frac{r^{3}+rl^{2}(1-f)}{l^{2}}\right|_{r_{-}}^{r_{+}}.$ (19)

Using $f(r_{\pm})=0$, we find the action growth to be

$\frac{d{\mathcal{S}}{}}{dt}=Q^{2}\left(\frac{1}{r_{+}}-\frac{1}{r_{-}}\right),$
(20)

what is again exactly the result obtained in [4, 3] using the standard action
(1).

##  IV Covariantization and teleparallel geometry

We would like to demonstrate that the teleparallel geometry is a natural
framework where the bulk action can be covariantized. The basic idea
underlying the teleparallel approach to gravity is to replace the Riemannian
torsion-free connection (5) by the curvature-free teleparallel spin connection
[17, 18]333We use a common convention that geometric objects with respect to
the teleparallel connection have $``\bullet"$ above them [17], to distinguish
them from the Riemannian geometry quantities.

${\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}_{\ b\mu}=\Lambda^{a}_{\
c}\partial_{\mu}(\Lambda^{-1})^{c}_{\ b},$ (21)

where $\Lambda^{a}_{\ c}$ is a local Lorentz transformation.

The torsion tensor ${\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{a}_{\
\mu\nu}=2\partial_{[\mu}h^{a}_{\
\nu]}+2{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}_{\ b[\mu}h^{b}_{\
\nu]}$ is generally non-zero and can be used to formulate a gravity theory by
considering the torsion-based Lagrangian [17]444We follow [3] and use
$(-,+,+,+)$ convention, unlike [17].

$\displaystyle{\stackrel{{\scriptstyle\bullet}}{{\mathcal{L}}}}{}_{\text{TG}}=-\frac{h}{2\kappa}\left[{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{abc}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}_{cba}-{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{ac}{}_{a}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{b}{}_{cb}\right],$
(22)

where ${\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{a}_{\
bc}=\frac{1}{2}\left({\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{\ a}_{c\
b}+{\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{\ a}_{b\
c}-{\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{a}_{\ bc}\right)$ is the
contortion tensor. Using the Ricci theorem

${\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}_{\ b\mu}=\omega^{a}_{\
b\mu}+{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{a}_{\ b\mu},$ (23)

it is possible to show that (22) is equivalent to the EH action up to a
surface term [17]

${\mathcal{L}}{}_{EH}={\stackrel{{\scriptstyle\bullet}}{{\mathcal{L}}}}{}_{\text{TG}}+\partial_{\mu}\left(\frac{h}{\kappa}\,{\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{\nu\mu}{}_{\nu}\right),$
(24)

from where follows that the theory given by (22) is dynamically equivalent to
general relativity [17, 18].

Moreover, we can relate the teleparallel (22) and bulk (7) Lagrangians by
using our previous result [16], where we have derived a relation between
${\stackrel{{\scriptstyle\bullet}}{{\mathcal{L}}}}{}_{\text{TG}}(h^{a}_{\
\mu},{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}_{\ b\mu})$ and
${\stackrel{{\scriptstyle\bullet}}{{\mathcal{L}}}}{}_{\text{TG}}(h^{a}_{\
\mu},0)$. Using the Ricci theorem (23), it is straightforward to see that the
latter is equivalent to the bulk Lagrangian (7). This leads to the relation

${\stackrel{{\scriptstyle\bullet}}{{\mathcal{L}}}}{}_{\text{TG}}={\mathcal{L}}{}_{\text{bulk}}+\frac{1}{\kappa}\partial_{\mu}\left(h{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{\nu\mu}{}_{\nu}\right).$
(25)

The teleparallel Lagrangian (22) is invariant under simultaneous local Lorentz
transformation of the tetrad and spin connection

$h^{a}_{\ \mu}\rightarrow\Lambda^{a}{}_{b}h^{b}_{\
\mu},\quad{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}{}^{a}_{\
b\mu}\rightarrow\Lambda^{a}{}_{c}{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}{}^{c}_{\
d\mu}\Lambda_{b}^{\ d}+\Lambda^{a}_{\ c}\partial_{\mu}\Lambda_{b}{}^{c},$ (26)

where $\Lambda_{b}{}^{c}=(\Lambda^{-1})^{c}_{\ b}$, and can be viewed as a
covariantization of the bulk Lagrangian (7).

Note that while the total derivative term in (2) covariantizes the bulk action
by introducing the second derivatives of the dynamical field variables, the
total derivative term in (25) contains only second derivatives of the local
Lorentz degrees of freedom $\Lambda^{a}{}_{b}$, which are not dynamical.

Figure 1:  The real (red) and imaginary (blue) parts of the Lagrangian growth
$\mathcal{G}=\kappa\frac{d{\mathcal{L}}{}}{dt}$ ( for $\theta=\pi/2$), where
${\mathcal{L}}{}$ is given by (15), for: AdS (upper left), charged AdS (upper
right), Schwarzschild (lower left), Reissner-Nordström (lower right) black
holes.

##  V Discussion and Conclusions

We have introduced a new method of evaluating the gravitational action on the
WdW patch, which instead of the standard action (1) uses the tetrad bulk
action (7), and tested it on the asymptotically AdS black holes. In addition
to being a useful new tool for calculating the action, our results are
interesting from several perspectives.

The boundary term-free nature of the bulk action allows us to not only
calculate the action’s value but study the Lagrangian as well. This helps us
to gain a deeper understanding of the action’s structure on the WdW patch. We
plot the “Lagrangian growth” as a function of distance $r$ in Figure 1. The
upper two plots corresponds to the results for AdS black holes from Section
III.

We observe a peculiar feature that the Lagrangian becomes complex on the WdW
patch. Although the total action is real since the imaginary part is
antisymmetric around the center of the WdW patch, the presence of a complex
phase is intriguing and may play a role, for example, in situations where we
consider a gravitational path integral [19] on a portion of the WdW patch.

It is also interesting to plot the Lagrangian growth for the asymptoticaly
flat black holes, which can be seen in the lower two plots in Figure 1. We
observe that on the WdW patch, the real part of the Lagrangian is constant. As
a result, the growth of the bulk action is determined just by the radial
length of the WdW patch.

Our results are interesting not only in relation to the holographic complexity
proposal but also from the perspective of general relativity alone. While the
dynamical equivalence of the EH and bulk actions is well-established, the bulk
action is considered to be a rather obscure object due to its non-covariance.
Similarly, various pseudotensorial definitions of energy-momentum derived from
the bulk action, despite being known to give correct results [20], have fallen
out of favor recently.

We have demonstrated that this is mostly unjustified and the bulk action can
provide meaningful results as long as we learn how to deal with its non-
covariance properly. We have demonstrated this in the tetrad case (7), where
we can use spherical coordinates and eliminate divergences through local
Lorentz transformation (9). It remains an open question whether we can use the
metric bulk Lagrangian (3) in the same fashion. The isotropic Cartesian
coordinates were used successfully in the exterior region [21], but it is not
clear how to proceed on the WdW patch.

We have then explored an alternative framework where instead of finding a
well-behaved tetrad by applying (9) on (13), we used (9) to define a
teleparallel spin connection (21). The torsion tensor of this connection
behaves covariantly under transformations (26) and defines a covariant
teleparallel gravity [22, 18, 23]. Since the Lagrangian of teleparallel
gravity is equivalent to the bulk Lagrangian (25), and it is covariant,
teleparallel gravity can be viewed as a natural covariant framework for the
use of the bulk action.

From another perspective, we can view our results as that the physics is given
solely by the bulk action. In the standard approach, the bulk term is
covariantized by addition of the total derivative term (2), but this
introduces second derivatives of dynamical field variables that need to be
removed by the GHY term. Teleparallel gravity achieves covariance of the bulk
action by using the teleparallel connection (21), which only contains non-
dynamical local Lorentz degrees of freedom that fully remove divergences of
the bulk action caused by its non-covariance.

Furthermore, this is useful also from a rather practical perspective as it
allows us to utilize many results already existing in the literature on
teleparallel gravity in a new way. In fact, we have here just used our
previous results about the teleparallel action in the exterior region of a
black hole [22, 16] and applied them to the interior geometry. In recent
years, there has been significant interest in various modified teleparallel
theories of gravity, such as $f(T)$ gravity [24, 25], where one of the central
problems is finding well-behaved tetrads [26, 27, 28, 29, 30] (See reviews
[31, 18, 32] as well). While these modified theories are rather speculative in
their nature, the results regarding these well-behaved tetrads are valid on
their own and can now be utilized towards studying holographic complexity
using the bulk action growth.

##  VI Acknowledgements

This work was funded through SASPRO2 project AGE of Gravity: Alternative
Geometries of Gravity, which has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No. 945478.

## References

  * [1] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252, [hep-th/9711200].
  * [2] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150].
  * [3] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93 (2016), no. 8 086006, [arXiv:1512.04993].
  * [4] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016), no. 19 191301, [arXiv:1509.07876].
  * [5] L. D. Landau and E. M. Lifschits, The Classical Theory of Fields.  Pergamon Press, 1975.
  * [6] R. d’Inverno, Introducing Einstein’s relativity.  Oxford University Press, 1992.
  * [7] T. Padmanabhan, Gravitation: Foundations and frontiers.  Cambridge University Press, 2014.
  * [8] J. W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082–1085.
  * [9] G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D15 (1977) 2752–2756.
  * [10] A. Einstein, Hamilton’s Principle and the General Theory of Relativity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916 (1916) 1111–1116.
  * [11] A. Einstein, The foundation of the general theory of relativity., Annalen Phys. 49 (1916), no. 7 769–822.
  * [12] C. Møller, Conservation Laws and Absolute Parallelism in General Relativity, K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 1 (1961), no. 10 1–50.
  * [13] C. Pellegrini and J. Plebanski, Tetrad Fields and Gravitational Fields, K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 2 (1963), no. 4 1–39.
  * [14] C. Møller, Survey of Investigations on the Energy-Momentum Complex in General Relativity, K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 35 (1966), no. 3 1–14.
  * [15] C. Møller, On the Crisis in the Theory of Gravitation and a Possible Solution, K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 89 (1978), no. 13 1–32.
  * [16] M. Krššák, Holographic Renormalization in Teleparallel Gravity, Eur. Phys. J. C77 (2017), no. 1 44, [arXiv:1510.06676].
  * [17] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity: An Introduction.  Springer, 2013.
  * [18] M. Krššák, R. J. van den Hoogen, J. G. Pereira, C. G. Böhmer, and A. A. Coley, Teleparallel theories of gravity: illuminating a fully invariant approach, Class. Quant. Grav. 36 (2019), no. 18 183001, [arXiv:1810.12932].
  * [19] S. W. Hawking, Quantum Gravity and Path Integrals, Phys. Rev. D 18 (1978) 1747–1753.
  * [20] C.-C. Chang, J. M. Nester, and C.-M. Chen, Pseudotensors and quasilocal gravitational energy momentum, Phys. Rev. Lett. 83 (1999) 1897–1901, [gr-qc/9809040].
  * [21] J. Beltrán Jiménez, L. Heisenberg, and T. S. Koivisto, Teleparallel Palatini theories, JCAP 08 (2018) 039, [arXiv:1803.10185].
  * [22] M. Krššák and J. G. Pereira, Spin Connection and Renormalization of Teleparallel Action, Eur. Phys. J. C75 (2015), no. 11 519, [arXiv:1504.07683].
  * [23] M. Krššák and E. N. Saridakis, The covariant formulation of f(T) gravity, Class. Quant. Grav. 33 (2016), no. 11 115009, [arXiv:1510.08432].
  * [24] R. Ferraro and F. Fiorini, Modified teleparallel gravity: Inflation without inflaton, Phys. Rev. D75 (2007) 084031, [gr-qc/0610067].
  * [25] R. Ferraro and F. Fiorini, On Born-Infeld Gravity in Weitzenbock spacetime, Phys. Rev. D78 (2008) 124019, [arXiv:0812.1981].
  * [26] R. Ferraro and F. Fiorini, Spherically symmetric static spacetimes in vacuum f(T) gravity, Phys. Rev. D84 (2011) 083518, [arXiv:1109.4209].
  * [27] N. Tamanini and C. G. Boehmer, Good and bad tetrads in f(T) gravity, Phys. Rev. D86 (2012) 044009, [arXiv:1204.4593].
  * [28] M. Hohmann, L. Järv, M. Krššák, and C. Pfeifer, Modified teleparallel theories of gravity in symmetric spacetimes, Phys. Rev. D 100 (2019), no. 8 084002, [arXiv:1901.05472].
  * [29] A. A. Coley, R. J. Van Den Hoogen, and D. D. McNutt, Symmetry and Equivalence in Teleparallel Gravity, J. Math. Phys. 61 (2020), no. 7 072503, [arXiv:1911.03893].
  * [30] E. D. Emtsova, M. Krššák, A. N. Petrov, and A. V. Toporensky, On conserved quantities for the Schwarzschild black hole in teleparallel gravity, Eur. Phys. J. C 81 (2021), no. 8 743, [arXiv:2105.13312].
  * [31] Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79 (2016), no. 10 106901, [arXiv:1511.07586].
  * [32] S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Teleparallel gravity: from theory to cosmology, Rept. Prog. Phys. 86 (2023), no. 2 026901, [arXiv:2106.13793].


#  Strong He i Emission Lines in High N/O Galaxies at $z\sim 6$ Identified in
JWST Spectra:
High He/H Abundance Ratios or High Electron Densities?

Hiroto Yanagisawa  Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Department of Physics,
Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033, Japan  Masami Ouchi National Astronomical Observatory of
Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Department of Astronomical
Science, SOKENDAI (The Graduate University for Advanced Studies), Osawa
2-21-1, Mitaka, Tokyo, 181-8588, Japan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba
277-8583, Japan  Kuria Watanabe  Department of Astronomical Science, SOKENDAI
(The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo,
181-8588, Japan  National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo, 181-8588, Japan  Akinori Matsumoto  Institute for Cosmic Ray
Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582,
Japan  Department of Physics, Graduate School of Science, The University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Kimihiko Nakajima National
Astronomical Observatory of Japan, National Institutes of Natural Sciences,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Hidenobu Yajima Center for
Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba,
Ibaraki 305-8577, Japan  Kentaro Nagamine Theoretical Astrophysics, Department
of Earth and Space Science, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba
277-8583, Japan  Department of Physics & Astronomy, University of Nevada, Las
Vegas, 4505 S. Maryland Pkwy, Las Vegas, NV 89154-4002, USA  Theoretical Joint
Research, Forefront Research Center, Graduate School of Science, Osaka
University, Toyonaka, Osaka, 560-0043, Japan  Koh Takahashi National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588,
Japan  Minami Nakane Institute for Cosmic Ray Research, The University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Department of
Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan  Nozomu Tominaga National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan  Department
of Physics, Faculty of Science and Engineering, Konan University, 8-9-1
Okamoto, Kobe, Hyogo 658-8501, Japan  Department of Astronomical Science,
SOKENDAI (The Graduate University for Advanced Studies), 2-21-1 Osawa, Mitaka,
Tokyo, 181-8588, Japan  Hiroya Umeda Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
Department of Physics, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Hajime Fukushima Center for
Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba,
Ibaraki 305-8577, Japan  Yuichi Harikane Institute for Cosmic Ray Research,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
Yuki Isobe Waseda Research Institute for Science and Engineering, Faculty of
Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo
169-8555, Japan  Yoshiaki Ono Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Yi Xu
Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8582, Japan  Department of Astronomy, Graduate School of
Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
Yechi Zhang National Astronomical Observatory of Japan, National Institutes of
Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

###### Abstract

We present Hei/H$\beta$-flux and He/H-abundance ratios in three JWST galaxies
with significant constraints on N/O-abundance ratios, GS-NDG-9422,
RXCJ2248-ID, and GLASS150008 at $z\sim 6$ mostly with the spectroscopic
coverage from Hei$\lambda$4471 and Heii$\lambda$4686 to Hei$\lambda$7065,
comparing with 68 local-dwarf galaxies. We find that these high-$z$ galaxies
present strong Hei emission with Hei/H$\beta$ flux ratios generally larger
than those of local-dwarf galaxies. We derive He/H with all of the detected
Hei, Heii, and $2-3$ hydrogen Balmer lines in the same manner as the local
He/H determination conducted for cosmology studies. These high-$z$ galaxies
show He overabundance He/H$\gtrsim 0.10$ or high electron density $n_{\rm
e}\sim 10^{3-4}$ cm-3 much larger than local values at low O/H,
$12+\log{\rm(O/H)}=7-8$. In contrast, we obtain low He/H and $n_{\rm e}$
values for our local-dwarf galaxies by the same technique with the same helium
and hydrogen lines, and confirm that the difference between the high-$z$ and
local-dwarf galaxies are not mimicked by systematics. While two scenarios of
1) He overabundance and 2) high electron density are not clearly concluded, we
find that there is a positive correlation on the He/H-N/O or $n_{\rm e}$-N/O
plane by the comparison of the high-$z$ and local-dwarf galaxies. The scenario
1) suggests that the overabundant helium and nitrogen are not explained by the
standard chemical enrichment of core-collapse supernovae, but the CNO-cycle
products and equilibrium ratios, respectively. The scenario 2) indicates that
the strong helium lines are originated from the central dense clouds of the
high-$z$ galaxies by excessive collisional excitation.

Galaxy chemical evolution (580), Galaxy evolution (594), High-redshift
galaxies (734), Chemical abundances (224), Chemical enrichment (225), Galaxy
formation (595)

††software: astropy (Astropy Collaboration et al., 2013), NumPy (Harris et
al., 2020), matplotlib (Hunter, 2007), SciPy (Virtanen et al., 2020), corner
(Foreman-Mackey, 2016), emcee (Foreman-Mackey et al., 2013), and YMCMC (Hsyu
et al., 2020)

##  1 Introduction

The James Webb Space Telescope (JWST) provides high-sensitivity near-infrared
(NIR) spectroscopy and allows us to detect various elemental emission lines of
high-$z$ galaxies. The high N/O values in high-$z$ galaxies such as GN-z11 are
reported in recent studies (Cameron et al., 2023a; Senchyna et al., 2023;
Isobe et al., 2023b). Isobe et al. (2023b) investigate abundances of C, N, and
O in high-$z$ galaxies and suggest that the low C/N and high N/O ratios are
explained by the chemical composition of CNO-cycle equilibrium. Isobe et al.
(2023b) also find that the low C/N and high N/O ratios found in the high-$z$
galaxies coincide with those of globular cluster stars, suggesting that these
galaxies may be forming globular clusters. Topping et al. (2024) present
RXCJ2248-ID at $z\sim 6$ showing not only high N/O but also high electron
densities, and suggest that CNO-cycled gas coincide with the formation of
globular clusters under the very dense conditions. This may be connected with
a trend of high electron densities in high-$z$ galaxies, as suggested by Isobe
et al. (2023a).

Because the CNO-cycle produces not only a large amount of nitrogen but also
helium, the high-$z$ galaxies that have the high N/O abundances may show high
He/H abundances. Strong He i emission lines are detected in some of the
high-$z$ galaxies (Cameron et al., 2023b; Topping et al., 2024). In this work,
we clarify whether there is a difference in helium abundance between local and
high-$z$ galaxies by comparing intensities of the He i emission lines. We also
investigate the connection between He/H, N/O and the electron density. We
derive helium abundances with high accuracy using a method developed for Big-
Bang nucleosynthesis studies (Izotov et al., 2014; Aver et al., 2015; Hsyu et
al., 2020; Matsumoto et al., 2022). This work aims to understand the process
of chemical enrichment in the high-$z$ universe by comparing the observed
helium abundances with local galaxies and chemical evolution models.

This paper is organized as follows. In Section 2, we present our sample of
galaxies. In Section 3, we derive the helium abundances of the galaxies in our
sample. Section 4 shows our results of He/H measurements and the correlation
between the He/H and N/O ratios. We also present the correlation between the
electron densities and N/O in Section 4. We compare our observational results
with several chemical evolution models in Section 5. We summarize our results
in Section 6.

##  2 Data and Sample

At present there are 5 galaxies at $z\gtrsim 6$ whose N/O values are
significantly constrained (GN-z11; Cameron et al. 2023a; Charbonnel et al.
2023; CEERS01019, GLASS150008; Isobe et al. 2023b, RXCJ2248-ID; Topping et al.
2024, GS-NDG-9422; Cameron et al. 2023b). To derive He/H, hydrogen and helium
emission lines within the rest-frame wavelength range approximately from
$4000\,\mathrm{\AA}$ to $7000\,\mathrm{\AA}$ are needed, which requires
galaxies to be in the redshift range of $0.5\lesssim z\lesssim 6.5$. We select
3 galaxies (GS-NDG-9422, RXCJ2248-ID, and GLASS150008) that satisfy these
criteria, which are presented in Sections 2.1, 2.2, and 2.3. Note that we do
not require large N/O values in order to obtain the unbiased sample. Local
dwarf galaxies are also presented as a control sample in Section 2.4.

Table 1: Observed emission line flux ratios

Emission line | GS-NDG-9422 | RXCJ2248-ID | GLASS150008 | J2115-1734†
---|---|---|---|---
H$\delta$ | $25.0\pm 2.9$ | $28.3\pm 1.2$ | $25.4\pm 2.5$ | $27.0\pm 0.2$
H$\gamma$ | $48.3\pm 2.3$ | $51.2\pm 0.8$ | $\dots$ | $47.1\pm 0.3$
He i $\lambda$4471 | $4.7\pm 0.6$ | $9.4\pm 0.8$ | $\dots$ | $3.8\pm 0.1$
He ii $\lambda$4686 | $6.4\pm 1.2$ | $\dots$ | $\dots$ | $2.6\pm 0.2$
H$\beta$ | $100.0\pm 1.7$ | $100.0\pm 0.8$ | $100.0\pm 3.7$ | $100.0\pm 0.4$
He i $\lambda$5876 | $12.8\pm 5.8$ | $20.5\pm 0.8$ | $18.7\pm 3.2$ | $10.9\pm 0.1$
H$\alpha$ | $264.5\pm 3.5$ | $254.7\pm 0.8$ | $\dots$ | $288.9\pm 0.1$
He i $\lambda$7065 | $7.0\pm 0.6$ | $12.2\pm 2.0$ | $\dots$ | $3.7\pm 0.2$

Note. — Observed flux ratios relative to H$\beta$ ($\times 100$). For GS-
NDG-9422 and RXCJ2248-ID, the flux values are taken from Topping et al. (2024)
and Cameron et al. (2023b), respectively. ($\dagger$) One out of the 68 local
galaxies is presented as an example.

Figure 1: 2D and 1D spectra of GS-NDG-9422 (top) and GLASS150008 (bottom). The
black solid lines and gray shaded regions represent the spectra and 1$\sigma$
errors, respectively. From left to right, H$\delta$, H$\gamma$, He i 4472, He
ii $\lambda$4686, H$\beta$, He i $\lambda$5876, and He i $\lambda$7065 are
shown. The undetected emission lines are flagged as ‘Not detected’. The
emission lines that are out of the wavelength range of the detector are
flagged as ‘Not covered’.

###  2.1 GS-NDG-9422

Spectroscopic data of GS-NDG-9422 at $z=5.94$ was obtained with JWST NIRSpec
via the JWST Advanced Deep Extragalactic Survey (JADES; PID: 1210, PI:
Luetzgendorf; Bunker et al. 2023; Eisenstein et al. 2023). Cameron et al.
(2023b) have investigated the spectra and measured the rest-UV and optical
emission line fluxes. We calculate the flux ratios relative to H$\beta$ using
the flux values of prism spectrum reported in Cameron et al. (2023b). The flux
ratios are shown in Table 1. Since the prism spectrum covers all of the
emission lines used in this study, systematic uncertainty caused by slit-loss
is negligibly small. The prism spectrum of GS-NDG-9422 is shown in Figure 1.

Since no nitrogen line is detected in GS-NDG-9422, Cameron et al. (2023b)
derived 3$\sigma$ upper limits of $\log(\mathrm{N/O})<-0.85$ and $-1.01$ using
N iii]$\lambda$1750 / [O iii]$\lambda$5007 and N iii]$\lambda$1750 / [O
iii]$\lambda$1666, respectively. These N/O ratios are consistent with those of
local galaxies. In this study, we adopt the more conservative upper limit of
$\log(\mathrm{N/O})<-0.85$.

###  2.2 RXCJ2248-ID

JWST NIRSpec observation of RXCJ2248-ID at $z=6.11$ was conducted in General
Observers (GO) program (PID: 2478, PI: Stark). Topping et al. (2024) have
analysed the spectra and measured the rest-UV and optical emission line
fluxes. The flux ratios relative to H$\beta$ are calculated with the flux
values reported in Topping et al. (2024). These flux ratios are shown in Table
1.

###  2.3 GLASS150008

GLASS150008 at $z=6.23$ was observed with the JWST NIRSpec in the GLASS JWST
Early Release Science (GLASS; Proposal ID: 1324, PI: T. Treu; Treu et al.
2022). The GLASS data were acquired by using high resolution ($R\sim 2700$)
filter-grating pairs of F100LP-G140H, F170LP-G235H, and F290LP-G395H, which
cover the wavelength ranges of $1.0-1.6$, $1.7-3.1$ and $2.9-5.1$ $\mu$m,
respectively. The total exposure time of the GLASS data is 4.9 hours for each
filter-grating pair. We use the spectroscopic data that were reduced by
Nakajima et al. (2023). Nakajima et al. (2023) conducted the level-2 and -3
calibrations on the raw data using the JWST Science Calibration Pipeline
version 1.8.5 with the Calibration Reference Data System (CRDS) context file
of $\tt jwst\\_1028.pmap$ or $\tt jwst\\_1027.pmap$. Nakajima et al. (2023)
also performed additional processes that improve the flux calibration, noise
estimate, and composition. See Nakajima et al. (2023) for the details of the
data reduction. The reduced spectrum of GLASS150008 is shown in Figure 1.

Isobe et al. (2023b) have derived the N/O ratio using the N iii]$\lambda$1750
/ O iii]$\lambda\lambda$1661,1666 and obtained
$\log(\mathrm{N/O})=-0.40^{+0.05}_{-0.07}$, which is significantly higher than
those of typical local galaxies.

We measure flux values of the hydrogen and helium emission lines that are used
for helium abundance estimation by fitting a Gaussian function. We detect
H$\delta$, H$\beta$, and He i $\lambda$ 5876 at more than the 6$\sigma$
levels. We do not use He i $\lambda$3889 and 5015 because these lines are
blended with H8 and [O iii] $\lambda$5007, respectively. The errors of fluxes
are obtained from uncertainties of the Gaussian fit. The measured emission
line fluxes are listed in Table 1.

Table 2: Summary of our sample

Name | $z$ | $\log M_{*}/M_{\odot}$ | $\log\mathrm{SFR}/M_{\odot}\,\mathrm{yr^{-1}}$ | Ref.
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
GS-NDG-9422 | 5.94 | $7.80^{+0.01}_{-0.01}$ | $8.2^{+2.8}_{-2.8}$ | Terp et al. (2024)
RXCJ2248-ID | 6.11 | $8.05^{+0.17}_{-0.15}$ | $1.8^{+0.2}_{-0.2}$ | Topping et al. (2024)
GLASS150008 | 6.23 | $8.39^{+0.35}_{-0.19}$ | $1.0^{+0.9}_{-0.7}$ | Jones et al. (2023)
J2115-1734† | 0.02 | $6.56^{+0.02}_{-0.02}$ | $0.27^{+0.01}_{-0.01}$ | Kojima et al. (2020)

Note. — (1) Name. (2) Redshift. (3) Stellar mass. For GS-NDG-9422, a stellar
mass is not provided in the literature. (4) Star-formation rate. (5) Reference
for the stellar mass and star-formation rate. ($\dagger$) One out of the 68
local galaxies is presented as an example.

###  2.4 Local Dwarf Galaxies

As a control sample, we use a total of 68 local dwarf galaxies reported by
Hsyu et al. (2020), Matsumoto et al. (2022) and Yanagisawa et al. (in prep.).
For the galaxies reported by Hsyu et al. (2020) and Matsumoto et al. (2022),
we use the flux values that are presented by Hsyu et al. (2020) and Matsumoto
et al. (2022), respectively. For the other galaxies that will be reported by
Yanagisawa et al. (in prep.), optical spectra are obtained from SDSS and by
Magellan/MagE observation (PI: M. Rauch). For detail of the MagE observation,
see Xu et al. (2022). Emission line fluxes and errors are measured in the same
manner as explained in Section 2.3. The calculated flux ratios for one out of
the 68 local galaxies, J2115-1734, are presented in Table 1.

###  2.5 Our Sample

Our sample comprises of a total of 3 high-$z$ galaxies and 68 local galaxies.
Hereafter these 71 ($=3+68$) galaxies are referred to as our sample. We
summarize the 3 high-$z$ galaxies and 1 representative local galaxy in Table
2. Figure 2 presents the He i $\lambda$5876/H$\beta$ ratios of these galaxies.
RXCJ2248-ID and GLASS150008 show significantly intense He i emission compared
to those of the local galaxies, while GS-NDG-9422 shows the He i emission
comparable to (but moderately stronger than) those of the local galaxies.

Figure 2: Flux ratios of He i $\lambda$5876/H$\beta$ as a function of O/H. The
red square, diamond, and hexagon represent GS-NDG-9422, RXCJ2248-ID, and
GLASS150008, respectively. The blue and gray dots denote the local galaxies,
where the blue dots are the galaxies that have N/O constraints.

##  3 Chemical Abundances

###  3.1 Helium Abundance

The number abundance ratio of helium to hydrogen $y\equiv{\rm He/H}$ is given
by

$y=\frac{\rm He^{+}}{\rm H^{+}}+\frac{\rm He^{++}}{\rm H^{+}}=y^{+}+y^{++},$
(1)

where $y^{+}$ and $y^{++}$ are the number abundance ratios of singly and
doubly ionized helium to ionized hydrogen, respectively. We ignore the neutral
hydrogen and helium because the observed emission lines come from H ii
regions, where most of the hydrogen and helium atoms are ionized.

We present the methods to obtain $y^{+}$ and $y^{++}$ in sections 3.1.1 and
3.1.2, respectively.

Figure 3: Probability distribution functions (PDFs) obtained with modified
YMCMC. The diagonal and off-diagonal panels show 1D and 2D PDFs, respectively.
The blue solid lines and black dashed lines represent the median and 68%
confidence levels, respectively. On top of each column, the median and 68%
confidence level values of the PDF are presented. Table 3: Best recovered
parameters from MCMC analysis

Name | $y^{+}$ | $T_{\mathrm{e}}$ | $\log(n_{\mathrm{e}}/\mathrm{cm^{-3}})$ |  $c$(H$\beta$) | $\tau_{\rm He}$ | $\log\xi$ | $\log N_{2}$
---|---|---|---|---|---|---|---
|  | [K] |  |  |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
GS-NDG-9422 | $0.098^{+0.007}_{-0.007}$ | $17379^{+1006}_{-955}$ | $1.36^{+0.91}_{-0.89}$ | $0.05^{+0.07}_{-0.04}$ | $8.86^{+1.89}_{-1.77}$ | $-4.3^{+1.5}_{-1.2}$ | $12.5^{+0.3}_{-0.2}$
RXCJ2248-ID | $0.166^{+0.018}_{-0.014}$ | $23241^{+1711}_{-1686}$ | $0.93^{+0.74}_{-0.65}$ | $0.03^{+0.06}_{-0.03}$ | $8.48^{+2.96}_{-2.75}$ | $-4.5^{+1.2}_{-1.0}$ | $13.4^{+0.1}_{-0.1}$
GLASS150008 | $0.142^{+0.066}_{-0.039}$ | $21889^{+4320}_{-400}$ | $2.17^{+1.58}_{-1.49}$ | $0.70^{+0.19}_{-0.22}$ | $\dots{}^{\dagger}$ | $-3.7^{+1.2}_{-1.7}$ | $11.2^{+1.5}_{-1.5}$
J2115-1734‡ | $0.084^{+0.005}_{-0.005}$ | $16238^{+2703}_{-2553}$ | $2.15^{+0.49}_{-1.22}$ | $0.21^{+0.04}_{-0.03}$ | $2.71^{+1.32}_{-1.19}$ | $-4.0^{+1.4}_{-1.5}$ | $11.4^{+1.4}_{-1.5}$

Note. — (1) Name. (2) Number abundance ratio of singly ionized helium to
hydrogen. (3) Electron temperature. (4) Electron density. (5) Reddening
correction coefficient. (6) Helium optical depth parameter. ($\dagger$) For
GLASS150008, best-fit $\tau_{\mathrm{He}}$ is not shown because the obtained
probability distribution is almost constant. (7) Number abundance ratio of
neutral to ionized hydrogen. (8) Column density of the hydrogen atom at the
state with the principal quantum number 2. ($\ddagger$) One out of the 68
local galaxies is presented as an example.

Table 4: Chemical abundances

Name | $12+\log(\rm O/\rm H)$ | $\log(\mathrm{N/O})$ | $y^{+}$ | $y^{++}$ | $y$ | Ref.
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
GS-NDG-9422 | $7.59^{+0.01}_{-0.01}$ | $<-0.85$ | $0.098^{+0.007}_{-0.007}$ | $0.006^{+0.001}_{-0.001}$ | $0.104^{+0.007}_{-0.007}$ | Cameron et al. (2023b)
RXCJ2248-ID | $7.43^{+0.17}_{-0.09}$ | $-0.39^{+0.10}_{-0.08}$ | $0.166^{+0.018}_{-0.014}$ | $<0.005$ | $0.166^{+0.018}_{-0.014}$ | Topping et al. (2024)
GLASS150008 | $7.65^{+0.14}_{-0.08}$ | $-0.40^{+0.05}_{-0.07}$ | $0.142^{+0.066}_{-0.039}$ | $<0.002$ | $0.142^{+0.066}_{-0.039}$ | Isobe et al. (2023b)
J2115-1734† | $4.27^{+0.10}_{-0.10}$ | $-1.56^{+0.01}_{-0.01}$ | $0.084^{+0.005}_{-0.005}$ | $0.0023^{+0.0002}_{-0.0002}$ | $0.086^{+0.005}_{-0.005}$ | Kojima et al. (2021)

Note. — (1) Name. (2) Oxygen abundance. (3) N/O abundance ratio. (4) Singly
ionized helium abundance. (5) Doubly ionized helium abundance. (6) Total
helium abundance. (7) Reference for the O/H and N/O ratios. ($\dagger$) One
out of the 68 local galaxies is presented as an example.

####  3.1.1 Singly Ionized Helium

The $y^{+}$ value is estimated with modified YMCMC, which is based on YMCMC
(Hsyu et al., 2020) and modified to model the optical depth for the Balmer
lines by Yanagisawa et al. (2024). Modified YMCMC uses the Markov Chain Monte
Carlo (MCMC) algorithm and searches for the best-fit parameters that reproduce
the observed hydrogen and helium emission lines (for detail, see Hsyu et al.
2020 and Yanagisawa et al. 2024). We constrain 7 free parameters of $y^{+}$,
the electron temperature $T_{\mathrm{e}}$, electron density $n_{\rm e}$,
reddening correction parameter $c({\rm H\beta})$, helium optical depth
$\tau_{\rm He}$, neutral to ionized hydrogen fraction $\xi$, and column
density of the hydrogen atom with a principal quantum number 2 $N_{2}$. We use
the flat priors of

$\begin{split}&0.01\leq y^{+}\leq 0.30,\\\
&1\leq\log_{10}(n_{\mathrm{e}}/\mathrm{cm^{-3}})\leq 5,\\\ &0\leq c({\rm
H\beta})\leq 1,\\\ &0\leq\tau_{\mathrm{He}}\leq 20,\\\
&-6\leq\log_{10}(\xi)\leq-1,\\\ &9\leq\log_{10}(N_{2}/\mathrm{cm^{-2}})\leq
15.\end{split}$ (2)

We also use a Gaussian prior for $T_{\mathrm{e}}$ to prevent the walkers from
falling in an unphysical solution, following the method presented by (Aver et
al., 2010). We use $T_{\rm e}(\textrm{He}\,\textsc{ii})=0.95\,T_{\rm
e}(\textrm{O}\,\textsc{iii})$ as a central value of a Gaussian prior, where
$T_{\rm e}(\textrm{He}\,\textsc{ii})$ and $T_{\rm
e}(\textrm{O}\,\textsc{iii})$ is the electron temperature of a He ii and an [O
iii] region, respectively. This assumption is based on Peimbert et al. (2002),
who claim that $T_{\mathrm{e}}(\mathrm{He}\,\textsc{ii})$ is 0.9-1.0 times
lower than $T_{\rm e}(\textrm{O}\,\textsc{iii})$ in typical H ii regions. The
obtained probability distributions for GS-NDG-9422, RXCJ2248-ID, and
GLASS150008 are presented in Figure 3. These high-$z$ galaxies show
$y^{+}\gtrsim 0.10$, which is remarkably larger than those of the local
galaxies ($y^{+}\sim 0.08$; Hsyu et al. 2020; Matsumoto et al. 2022). The
probability distributions for GS-NDG-9422 and RXCJ2248-ID present
$\tau_{\mathrm{He}}\sim 8-9$, which are significantly larger than those of
typical local galaxies ($\tau_{\mathrm{He}}\lesssim 3$; Hsyu et al. 2020;
Matsumoto et al. 2022). This is likely to coincide with the optically-thick
hydrogen gas suggested by Yanagisawa et al. (2024). We also derive He/H for
the control sample of the local galaxies in the same manner, using the same
helium and hydrogen lines as the input. The probability distribution of one of
the local galaxies, J2115-1734, is also shown in Figure 3. The $y^{+}$ value
for J2115-1734 is comparable to the typical values of 0.08, indicating that
the limited number of the input emission lines is not the cause of the large
$y^{+}$ in the high-$z$ galaxies.

####  3.1.2 Doubly Ionized Helium

We derive the $y^{++}$ values from Equation (17) of Pagel et al. (1992):

$y^{++}=0.084\left[\frac{T_{\textrm{e}}(\textrm{O\,{iii}})}{10^{4}}\right]^{0.14}\frac{F\left(\textrm{He\,{ii}}\,\lambda
4686\right)}{F\left(\textrm{H}\beta\right)},$ (3)

where $F(\lambda)$ represents the flux value of the emission line $\lambda$.
Measurements and errors of $T_{\rm e}(\textrm{O}\,\textsc{iii})$ are taken
from Cameron et al. (2023b), Topping et al. (2024), and Jones et al. (2023)
for GS-NDG9422, RXCJ2248-ID, and GLASS150008, respectively. For the local
galaxies presented by Hsyu et al. (2020) and Matsumoto et al. (2022),
$T_{\mathrm{e}}(\mathrm{O\,\textsc{iii}})$ values are taken from Hsyu et al.
(2020). The $T_{\mathrm{e}}(\mathrm{O\,\textsc{iii}})$ values for the galaxies
that will be presented by Yanagisawa et al. (in prep.) are taken from Kojima
et al. (2020), Izotov et al. (2020, 2021), and Sánchez Almeida et al. (2015).
If the He ii $\lambda$4686 line is not detected, we put the 2$\sigma$ upper
limit on the $y^{++}$ value. We present the $y^{++}$ values of our sample
galaxies in Table 4.

###  3.2 Oxygen and Nitrogen Abundances

The O/H and N/O abundance ratios of GS-NDG-9422, RXCJ2248-ID, and GLASS150008
are taken from Cameron et al. (2023b), Topping et al. (2024), and Isobe et al.
(2023b), respectively. The O/H abundance ratios of the local galaxies that are
presented by Hsyu et al. (2020) and Matsumoto et al. (2022) are taken from
Hsyu et al. (2020) and Matsumoto et al. (2022), respectively. The O/H
abundance ratios for the galaxies that will be presented by Yanagisawa et al.
(in prep.) are taken from Kojima et al. (2020), Izotov et al. (2020, 2021),
and Sánchez Almeida et al. (2015). The N/O abundance ratios of the local dwarf
galaxies are taken from Hägele et al. (2011), Duarte Puertas et al. (2022),
and Kojima et al. (2021). The O/H and N/O abundance ratios of the 3 high-$z$
galaxies and 1 representative local galaxy in our sample are presented in
Table 4.

Figure 4: Helium abundance as a function of oxygen abundance. The squares,
diamonds, and hexagons represent GS-NDG-9422, RXCJ2248-ID, and GLASS150008,
respectively. The open square denotes the He/H derived by Cameron et al.
(2023b). To avoid overlapping the data points, the open square is shifted to
the left by 0.2 dex. The blue and gray points denote the local dwarf galaxies
(Hsyu et al. 2020, Matsumoto et al. 2022, and Yanagisawa et al. in prep.) with
and without N/O measurement, respectively. The black line and blue shaded
region are the linear relation of local galaxies and its 3$\sigma$ error,
respectively (Matsumoto et al., 2022). Figure 5: He/H as a function of N/O.
The symbols are the same as Figure 4. To avoid overlapping the data points,
the point for GLASS150008 is shifted to the left by 0.1 dex.

##  4 Result

In Figure 4, we show the helium and oxygen abundances of our sample.
RXCJ2248-ID and GLASS150008 show the helium abundances that are significantly
higher than those of the local galaxies. These galaxies show the large He
i/H$\beta$ flux ratios as presented in Figure 2. In Figure 4, He/H of GS-
NDG-9422 derived in this work and by Cameron et al. (2023b) are presented as
the red and open squares, respectively. The result of Cameron et al. (2023b)
are consistent with our result within the 1$\sigma$ level. Note that some
local galaxies show He/H lower than the primordial value, although these
values are consistent with those derived by Matsumoto et al. (2022), who
conducted YMCMC analyses with smaller systematics using $\sim$12 emission
lines. In Figure 5, the He/H and N/O values of our sample galaxies are
presented. GLASS150008 and RXCJ2248-ID, which show the helium overabundance in
Figure 4, also have large N/O values. We can see a positive correlation
between He/H and N/O. These He-rich galaxies introduce significant uncertainty
on the primordial helium abundance, which is determined by estimating an
intercept of linear fitting in He/H-O/H plane. We suggest that galaxies with
large N/O ratios should not be used for the determination of the primordial
helium abundance because the N-rich galaxies are likely to have the high He/H
ratios (see also Izotov et al. 2007, who suggest the positive correlation
between He/H and N/H).

There may exist the neutral helium in the outer layer of the ionized hydrogen
region, which we do not consider at present. If the neutral helium exists, the
He/H ratio still increases, which strengthens the result of helium
overabundance.

The electron densities of GS-NDG-9422 and GLASS150008 derived from YMCMC are
consistent with the upper limits of $\log(n_{\mathrm{e}}/\mathrm{cm^{-3})}<3$
and $<3.9$ estimated by Cameron et al. (2023b) and Jones et al. (2023) with
the nebular continuum and the C iii]$\lambda\lambda$1907,1909 doublet ratio,
respectively. However, the electron density of RXCJ2248-ID derived from YMCMC
is lower than $\log(n_{\mathrm{e}}/\mathrm{cm^{-3})}=4-5$, which are derived
with the line ratios of doubly and triply ionized metal lines including C
iii]$\lambda\lambda$1907,1909 (Topping et al., 2024). There is a possibility
that the strong He i emission is produced by collisional excitation due to the
high electron density.

Including the high electron density scenario, we adopt a Gaussian prior on
$y^{+}$ with a central value of 0.08 and standard deviation of 0.01 in the
modified YMCMC analyses to examine whether the strong He i emission can be
reproduced with the low $y^{+}$. The best-fit parameters and probability
distributions are shown in Table 5 and Figure 6, respectively. For GS-
NDG-9422, the best-estimate electron density is $\sim 100\,\mathrm{cm^{-3}}$,
while those of RXCJ2248-ID and GLASS150008 are as high as
$1,000\,\mathrm{cm^{-3}}$. For GS-NDG-9422 and GLASS150008, these electron
densities are consistent with the upper limits of those derived by Cameron et
al. (2023b) and Jones et al. (2023), respectively. For RXCJ2248-ID,
$\log(n_{\mathrm{e}}/\mathrm{cm^{-3}})\sim 3$ is still lower than that derived
from C iii] doublet by Topping et al. (2024) by $\sim$2 dex. This may be
explained as the distribution of the metal is different from that of helium.
If the metal enrichment occurs only within the central region, the metal
emission lines suggest the very dense condition. In this case, the helium gas
exists more extended region, indicating the lower density. We also conduct the
modified MCMC analyses in the same manner for one of the local dwarf galaxies,
J2115-1734 (Figure 6). For the local galaxy, the result is unchanged from the
one without the Gaussian prior on $y^{+}$ because the $y^{+}$ value is $\sim
0.08$ even without the Gaussian prior.

In Figure 7, the relation between N/O and $n_{e}$ is presented. Under the
assumption of the low $y^{+}$, we can see the positive correlation between N/O
and $n_{e}$, which may suggest that the dense situation leads to the excessive
enhancement of nitrogen. The difference in $n_{\mathrm{e}}$ between the
high-$z$ and local galaxies may be originated from the compact morphology in
high-$z$ galaxies, as suggested by Isobe et al. (2023a).

Table 5: Best recovered parameters from MCMC analysis with the Gaussian priors
on $y^{+}$

Name | $y^{+}$ | $T_{\mathrm{e}}$ | $\log(n_{\mathrm{e}}/\mathrm{cm^{-3}})$ |  $c$(H$\beta$) | $\tau_{\rm He}$ | $\log\xi$ | $\log N_{2}$
---|---|---|---|---|---|---|---
|  | [K] |  |  |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
GS-NDG-9422 | $0.087^{+0.005}_{-0.006}$ | $16339^{+2568}_{-2559}$ | $2.02^{+0.69}_{-1.32}$ | $0.11^{+0.12}_{-0.07}$ | $8.89^{+2.80}_{-3.15}$ | $-4.29^{+1.32}_{-1.16}$ | $12.1^{+1.3}_{-2.0}$
RXCJ2248-ID | $0.094^{+0.008}_{-0.006}$ | $28012^{+1428}_{-6060}$ | $3.06^{+0.30}_{-0.19}$ | $0.04^{+0.08}_{-0.03}$ | $1.44^{+3.43}_{-1.13}$ | $-5.1^{+0.8}_{-0.6}$ | $13.4^{+0.1}_{-0.1}$
GLASS150008 | $0.082^{+0.007}_{-0.007}$ | $22757^{+3408}_{-2931}$ | $3.65^{+0.49}_{-0.49}$ | $0.80^{+0.14}_{-0.19}$ | $7.30^{+7.30}_{-5.26}$ | $-4.5^{+1.0}_{-1.0}$ | $11.3^{+1.6}_{-1.6}$
J2115-1734† | $0.082^{+0.003}_{-0.004}$ | $15992^{+2655}_{-2625}$ | $2.22^{+0.42}_{-1.30}$ | $0.21^{+0.04}_{-0.02}$ | $2.74^{+1.46}_{-1.25}$ | $-4.2^{+1.3}_{-1.26}$ | $11.3^{+1.4}_{-1.6}$

Note. — Same as Table 3, but with the Gaussian priors on $y^{+}$ (see Section
4). ($\dagger$) One out of the 68 local galaxies is presented as an example.

Hereafter the case where the density is low and He/H is high is referred to as
helium overabundance case, and the case where the density is high and He/H is
low is referred to as high density case. In either cases, the high-$z$
galaxies in our sample are quite different from the local galaxies, whose He/H
and $n_{\mathrm{e}}$ values derived in the same manner show significantly
lower than those of the high-$z$ galaxies.

Figure 6: Same as Figure 3, but with the Gaussian prior on $y^{+}$. Figure 7:
Electron density as a function of N/O. The symbols are the same as Figure 4.
To avoid overlapping the data points, the red hexagon are shifted to the left
by 0.1 dex.

##  5 Discussion

In the previous section, we find that the strong He i emission is achieved by
either the high He/H ratios or the high electron densities. In the following
sections, assuming the helium overabundance case, we compare the observed
chemical compositions of He/H, O/H, and N/O with the chemical evolution
models.

###  5.1 He/H and O/H

####  5.1.1 Comparison with the CCSNe Model

We first compare the observed He/H and O/H ratios with a standard core-
collapse supernovae (CCSNe) model. We use the CCSNe model developed by
Watanabe et al. (2024), who calculate chemical evolution of galaxies by
utilizing the explosive nucleosynthesis code developed by Tominaga et al.
(2007). In this model, stars in a mass range of $9\,M_{\odot}\leq M\leq
40\,M_{\odot}$ are instantaneously created according to Kroupa IMF (Kroupa,
2001). At the end of the lifetime, these stars cause CCSNe and eject gas. The
yield of CCSNe is taken from Nomoto et al. (2013). For the details of the
model, see Watanabe et al. (2024). In this model, the ejecta have abundances
of $\mathrm{He/H}\sim 0.20$ and $12+\log\mathrm{(O/H)}\sim 10$. Because the
high-$z$ galaxies in our sample have the oxygen abundances of
$12+\log\mathrm{(O/H)}\sim 7.4-7.7$, the ejecta need to be diluted by
primordial gas in the interstellar medium (ISM). The observed emission lines
come from this mixture of ejecta and primordial gas. We thus calculate the
He/H and O/H abundance ratios in the mixed gas that are given by

$\begin{split}&\textrm{He/H}=r_{\mathrm{mix}}(\textrm{He/H})_{\textrm{e}}+(1-r_{\mathrm{mix}})(\textrm{He/H})_{\textrm{p}}\\\
&\textrm{O/H}=r_{\mathrm{mix}}(\textrm{O/H})_{\textrm{e}}+(1-r_{\mathrm{mix}})(\textrm{O/H})_{\textrm{p}},\\\
\end{split}$ (4)

where the subscripts e and p stand for ‘ejecta’ and ‘primordial gas’,
respectively, and $r_{\mathrm{mix}}$ is a mixing parameter which satisfies
$0<r_{\mathrm{mix}}\leq 1$. $r_{\mathrm{mix}}$ is defined so that
$r_{\mathrm{mix}}=0$ corresponds to the primordial gas and
$r_{\mathrm{mix}}=1$ corresponds to the pure ejecta.

In Figure 8(a), we compare our observational results with the CCSNe model.
Because the oxygen abundance is rapidly enhanced by the CCSNe ejecta,
$r_{\mathrm{mix}}$ needs to be smaller than 0.01 to fall within the observed
O/H range. However, the small $r_{\mathrm{mix}}$ reduces the He/H to almost
the primordial value of 0.08, which indicates that the CCSN model is unlikely
to reproduce the observed high He/H and low O/H. The same conclusion was also
reached by Fukushima et al. (2024) using a different one-zone chemical
evolution model.

####  5.1.2 Comparison with the Other Models

We then explore another physical origin of the helium overabundances. The high
He/H may be explained by gas produced in the CNO cycle, which occurs in an
outer layer of massive stars and produce a large amount of helium from
hydrogen. If the hydrogen burning layer is ejected into the ISM, the He/H
value may become high. In CNO cycle equilibrium, a large amount of nitrogen is
accumulated compared to oxygen, which also enhances N/O (Isobe et al., 2023b).
In previous studies, models including Wolf-Rayet (WR) stars, supermassive
stars (SMSs), and tidal disruption events (TDEs) are presented to explain the
anomalous chemical abundances of high-$z$ galaxies such as GN-z11 (Isobe et
al., 2023b; Charbonnel et al., 2023; Nagele & Umeda, 2023; Watanabe et al.,
2024; Senchyna et al., 2023; Cameron et al., 2023a). We thus incorporate the
yields of the WR star, SMS, and TDE into the model. We use chemical evolution
models of WR stars developed by Watanabe et al. (2024) and of SMSs and TDEs
developed by Watanabe et al. in (prep.). In these models, stars in a mass
range of $9\,M_{\odot}\leq M\leq 25\,M_{\odot}$ cause CCSNe at the end of the
lifetimes. On the other hand, stars in a mass range of $M>25\,M_{\odot}$ end
up in CCSNe at a probability $f_{\mathrm{SN}}$ and direct collapses (DCs) at a
probability $1-f_{\mathrm{SN}}$. The yields of each model are shown in Table
6. We allow the mixing that is described as $r_{\mathrm{mix}}$ in Equation
(4). In Figure 8(b), (c), and (d), we compare the observational results with
the WR star, SMS, and TDE models, respectively. The WR star model explains the
high He/H of the ISM in the three high-$z$ galaxies within the $\sim 1\sigma$
levels. The SMS model also reproduces the He/H and O/H of the ISM in all of
the three high-$z$ galaxies. The TDE model explains the chemical compositions
of GS-NDG-9422 and GLASS150008 within the 1$\sigma$ level, while that of
RXCJ2248-ID is not reproduced by the TDE model.

Figure 8: Comparison with chemical evolution models calculated by Watanabe et
al. (2024) and Watanabe et al. (in prep.). The symbols are the same as Figure
4. The blue shaded regions show the regions that are reproduced by the models
with $0\leq r_{\mathrm{mix}}\leq 1$. (a): CCSN model. The black lines indicate
the models with $r_{\mathrm{mix}}=0.002$ and $0.1$. Models with
$r_{\mathrm{mix}}\gtrsim 0.1$ extend beyond the range of the figure due to the
rapid enhancement of oxygen. (b)-(d): WR star, SMS, and TDE models. The black
solid lines indicate the model tracks with fixed $r_{\mathrm{mix}}$ values.
The black dashed lines present the models with the fixed values of the log of
the age (yr). Here, $f_{\mathrm{SN}}=0.005$ is assumed. Note that
$r_{\mathrm{mix}}=0$ corresponds to the point $(\mathrm{O/H})\times 10^{5}=0$,
$\mathrm{He/H}=0.08$.

In Figure 9, we show the He/H and O/H of the models as a function of age. For
RXCJ2248-ID, only the SMS and WR star model is presented since the TDE model
does not explain the He/H and O/H values of RXCJ2248-ID. We calculate the
chemical evolution at $r_{\mathrm{mix}}=0.1-0.5$ and
$f_{\mathrm{SN}}=0.0001-0.005$. At $\log\mathrm{(Age/yr)}\sim 6.9$, O/H
rapidly increases because the stars with $M<25M_{\odot}$ start to cause CCSNe.
For GS-NDG-9422, the SMS model reproduces the He/H and O/H ratios within the
1$\sigma$ levels for $r_{\mathrm{mix}}=0.1$ and $f_{\mathrm{SN}}=0.005$, while
the O/H value is reproduced within the very short timescale because of the
rapid oxygen enhancement by CCSNe. On the other hand, the TDE model with
$r_{\mathrm{mix}}=0.5$ and $f_{\mathrm{SN}}=0.0001$ reproduces He/H and O/H
within the longer timescale ($\log(\mathrm{Age/yr})=6.45-6.9$) due to the
moderate oxygen enrichment by the TDE. The WR model with
$r_{\mathrm{mix}}=0.3$ and $f_{\mathrm{SN}}$ value between 0.0001 and 0.005 is
likely to reproduce the He/H and O/H of GS-NDG-9422. For GLASS150008, the SMS
model reproduces the He/H and O/H values with all of the parameter sets shown
in Figure 9 except for $r_{\mathrm{mix}}=0.1$ and $f_{\mathrm{SN}}=0.0001$,
while the TDE models with $r_{\mathrm{mix}}\sim 0.5$ explain the He/H and O/H
within the $1\sigma$ level. The WR models with $r_{\mathrm{mix}}=0.3-0.5$ and
$f_{\mathrm{SN}}\sim 0.005$ are likely to reproduce the He/H and O/H on
GLASS150008. For RXCJ2248-ID, the SMS model reproduces the observed He/H and
O/H within the $\sim 1\sigma$ levels for both of $f_{\mathrm{SN}}=0.0001$ and
0.005 if $r_{\mathrm{mix}}=0.3$ is assumed, while the WR model requires larger
$r_{\mathrm{mix}}$ values of $\sim 1$.

In all cases explained above, the large fraction of massive stars need to
cause DCs ($f_{\mathrm{SN}}\lesssim 0.01$) to explain the observed O/H within
the sufficiently long timescale. This is consistent with the result of
Watanabe et al. (2024), who suggest that most of WR stars need to cause DCs to
explain the observed large N/O in GN-z11. However, this result of low
$f_{\mathrm{SN}}$ depends on the assumed IMF and star-formation timescale,
which need to be further investigated

Figure 9: He/H and O/H as a function of age. The black, red and orange lines
represent the models with mixing parameters of $r_{\mathrm{mix}}=0.1$, $0.3$,
and $0.5$, respectively. The solid and dashed lines denote the models with
$f_{\mathrm{SN}}$ values of $0.0001$ and $0.005$, respectively. The blue
shaded regions in the top, middle, and bottom panels indicate the 1$\sigma$
areas of observed abundance ratios of GS-NDG-9422, RXCJ2248-ID, and
GLASS150008, respectively.  Table 6: Yields of the models

Model | Mass [$M_{\odot}$] |  1H [$M_{\odot}$] |  4He [$M_{\odot}$] |  14N [$M_{\odot}$] |  16O [$M_{\odot}$] | Reference
---|---|---|---|---|---|---
WR† | 13 | 4.52 | 3.89 | 0.13 | 1.51 |
| 15 | 5.02 | 4.51 | 0.15 | 1.82 |
| 20 | 6.73 | 6.20 | $1.18\times 10^{-2}$ | 2.94 |
| 25 | 7.81 | 5.29 | $1.66\times 10^{-2}$ | 5.93 |
| 30 | 8.94 | 4.00 | $1.69\times 10^{-4}$ | $1.55\times 10^{-4}$ | Limongi & Chieffi (2018)
| 40 | 11.1 | 4.38 | $1.90\times 10^{-4}$ | $1.99\times 10^{-4}$ |
| 60 | 15.5 | 6.37 | $2.91\times 10^{-4}$ | $2.53\times 10^{4}$ |
| 80 | 17.5 | 9.71 | $2.14\times 10^{-2}$ | $5.80\times 10^{-3}$ |
| 120 | 22.6 | 14.7 | $1.17\times 10^{-3}$ | $2.66\times 10^{-4}$ |
SMS | $10^{5}$ | $4.19\times 10^{4}$ | $5.8\times 10^{4}$ | 81.4 | 0.884 | Nagele & Umeda (2023)
TDE | 13 | 4.61 | 1.79 | $5.33\times 10^{-3}$ | $1.12\times 10^{-2}$ |
| 15 | 5.21 | 2.20 | $1.63\times 10^{-3}$ | $3.13\times 10^{-3}$ |
| 18 | 5.66 | 3.15 | $2.53\times 10^{-3}$ | $3.17\times 10^{-3}$ |
| 20 | 6.41 | 3.28 | $2.66\times 10^{-3}$ | $3.50\times 10^{-3}$ | Watanabe et al. (in prep.)
| 25 | 7.61 | 4.23 | $3.68\times 10^{-3}$ | $4.02\times 10^{-3}$ |
| 30 | 8.75 | 5.27 | $4.71\times 10^{-3}$ | $4.35\times 10^{-3}$ |
| 40 | 10.5 | 7.30 | $6.60\times 10^{-3}$ | $4.88\times 10^{-3}$ |

Note. — ($\dagger$) Metallicity of $\mathrm{[Fe/H]}=-3$ and an initial
rotation velocity of $v=300\,\mathrm{km\,s^{-1}}$ is assumed.

###  5.2 He/H and N/O

In Figure 10, we compare the N/O ratios with the same models as presented in
Section 5.1.2. In the SMS model, $r_{\mathrm{mix}}\sim 0.3$ (0.1) reproduces
the chemical abundances of RXCJ2248-ID and GLASS150008 (GS-NDG-9422), which
are consistent with the result presented in Figure 8. In the TDE and WR star
models, chemical abundances of GS-NDG-9422 and GLASS150008 are reproduced at
various values of $r_{\mathrm{mix}}$, while those of RXCJ2248-ID are not
explained by any $r_{\mathrm{mix}}$ within the 1$\sigma$ level. These results
suggest that the high He/H and N/O is achieved by the gas produced by the CNO-
cycle, which supplies a large amount of both He and N.

These He- and N-rich situations are not found in the local dwarf galaxies.
This may be because the CNO-cycled gas is sufficiently diluted by the
primordial gas and ejecta of the CCSNe in the local dwarf galaxies, which
leads to the low He/H and N/O ratios.

Figure 10: Same as Figure 8, but for He/H as a function of N/O. From left to
right, SMS, TDE, and WR star models are shown. The symbols are the same as
Figure 4. The blue shaded regions represent the regions that are reproduced by
the models. The black solid lines indicate the trajectories of each model with
fixed mixing parameters shown beside the lines. The black dashed lines
represent the model tracks with the fixed values of the log of the age. Here,
$f_{\mathrm{SN}}=0.005$ is assumed. Since the grids in the three panels are
drawn at the same mixing parameters and stellar ages, the values are only
shown in the left panel.

##  6 Summary

In this paper, using the MCMC algorithm, we derive the He/H abundance ratios
of 3 high-$z$ galaxies and compare them with the chemical evolution models.
Our main results are summarized below.

  * •

Among the 5 galaxies at $z\simeq 6$ that have the N/O constraints, we select 3
galaxies whose optical helium and hydrogen lines necessary for the He/H
estimates are covered by JWST NIRSpec. We also select the local dwarf galaxies
as the control sample. We find the flux ratios of He i to H$\beta$ in the 3
high-$z$ galaxies are comparable to or significantly larger than those in the
local galaxies (Figure 2).

  * •

We obtain the He/H with modified YMCMC code, which is the modified version of
the YMCMC code (Hsyu et al., 2020) developed for the primordial helium
abundance determination. The He/H abundance ratio of the high-$z$ galaxies are
higher than those of the local galaxies (Figure 4). We also derive He/H for
the local galaxies as a control sample using the same helium and hydrogen
lines. These local galaxies show the low helium abundances, meaning that the
limited number of input emission lines is not the cause of the helium
overabundances. We also compare our sample galaxies in the He/H-N/O plane
(Figure 5). We find that He/H has the positive correlation with N/O. This may
be caused by the gas produced by the CNO-cycle, which enhances both He and N.

  * •

There is the possibility that the strong He i emission is achieved by the
excessive collisional excitation caused by the high electron density, not by
the high He/H ratio. We rerun modified YMCMC with the Gaussian prior on
$y^{+}$ with the central value of 0.08, which is nearly the primordial helium
abundance. We find that if the high-$z$ galaxies in our sample have the He/H
ratios comparable to the local galaxies, these high-$z$ galaxies have the
electron densities significantly higher than those of the local galaxies
(Figure 7). In this case, we also find the positive correlation between N/O
and $n_{\mathrm{e}}$, implying that the nitrogen enrichment may be connected
with the dense situation.

  * •

We also conduct analyses with the modified YMCMC for the control sample of 68
local dwarf galaxies in the same manner. These local galaxies show the low
He/H and $n_{e}$ values, which indicate that in either of the helium
overabundance and high density case, the high-$z$ galaxies in our sample are
different from the local galaxies and these differences are not caused by
systematics.

  * •

Assuming the high-$z$ galaxies in our sample have the high helium abundance,
we compare the He/H and O/H abundance ratios with the standard chemical
enrichment by the CCSNe (Figure 8 (a)). The helium overabundance in the
high-$z$ galaxies are unlikely to be explained by the abundance of the mixed
gas of the CCSNe ejecta and primordial gas. We then introduce the WR star,
TDE, and SMS models, which eject He-rich CNO-cycled gas into the ISM, to
explain the high He/H values in the high-$z$ galaxies (Figure 8 (b), (c), and
(d), respectively). The WR star and SMS models reproduce the He/H and O/H in
the three high-$z$ galaxies. The TDE model explains the He/H and O/H values
observed in GS-NDG-9422 and GLASS150008, while those of RXCJ2248-ID are not
reproduced in this model. Comparing the time evolution of He/H and O/H of the
models with the observational results (Figure 9), we find that low
$f_{\mathrm{SN}}$ values ($\lesssim$ 1 %) are needed to explain the observed
abundances in sufficiently long timescales.

  * •

We also compare the He/H and N/O values with the models (Figure 10). We find
that the He/H and N/O ratios of GS-NDG-9422 and GLASS150008 are reproduced by
the SMS, TDE, and WR star models, while those of RXCJ2248-ID are reproduced
simultaneously only by the SMS model with $r_{\mathrm{mix}}\sim 0.3$ within
the $1\sigma$ level.

## Acknowledgement

We are thankful to Daniel P. Stark, Ricardo Amorin, and Siwei Zou for helpful
discussions. This work is based on the observations with the NASA/ESA/CSA
James Webb Space Telescope associated with programs of GTO-1210 (JADES) and
GO-2478. We are grateful to GTO-1210 and GO-2478 teams led by Nora
Luetzgendorf and Daniel P. Stark, respectively. This publication is based upon
work supported by the World Premier International Research Center Initiative
(WPI Initiative), MEXT, Japan, KAKENHI (20H00180, 21H04467, and 21H04489)
through Japan Society for the Promotion of Science, and JST FOREST Program
(JP-MJFR202Z). This work was supported by the joint research program of the
Institute for Cosmic Ray Research (ICRR), University of Tokyo.

## References

  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Aver et al. (2010) Aver, E., Olive, K. A., & Skillman, E. D. 2010, J. Cosmology Astropart. Phys, 2010, 003, doi: 10.1088/1475-7516/2010/05/003
  * Aver et al. (2015) —. 2015, J. Cosmology Astropart. Phys, 2015, 011, doi: 10.1088/1475-7516/2015/07/011
  * Bunker et al. (2023) Bunker, A. J., Cameron, A. J., Curtis-Lake, E., et al. 2023, arXiv e-prints, arXiv:2306.02467, doi: 10.48550/arXiv.2306.02467
  * Cameron et al. (2023a) Cameron, A. J., Katz, H., Rey, M. P., & Saxena, A. 2023a, MNRAS, 523, 3516, doi: 10.1093/mnras/stad1579
  * Cameron et al. (2023b) Cameron, A. J., Katz, H., Witten, C., et al. 2023b, arXiv e-prints, arXiv:2311.02051, doi: 10.48550/arXiv.2311.02051
  * Charbonnel et al. (2023) Charbonnel, C., Schaerer, D., Prantzos, N., et al. 2023, A&A, 673, L7, doi: 10.1051/0004-6361/202346410
  * Duarte Puertas et al. (2022) Duarte Puertas, S., Vilchez, J. M., Iglesias-Páramo, J., et al. 2022, A&A, 666, A186, doi: 10.1051/0004-6361/202141571
  * Eisenstein et al. (2023) Eisenstein, D. J., Willott, C., Alberts, S., et al. 2023, arXiv e-prints, arXiv:2306.02465, doi: 10.48550/arXiv.2306.02465
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24, doi: 10.21105/joss.00024
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fukushima et al. (2024) Fukushima, K., Nagamine, K., Matsumoto, A., et al. 2024, arXiv e-prints, arXiv:2401.06450, doi: 10.48550/arXiv.2401.06450
  * Hägele et al. (2011) Hägele, G. F., García-Benito, R., Pérez-Montero, E., et al. 2011, MNRAS, 414, 272, doi: 10.1111/j.1365-2966.2011.18389.x
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hsyu et al. (2020) Hsyu, T., Cooke, R. J., Prochaska, J. X., & Bolte, M. 2020, ApJ, 896, 77, doi: 10.3847/1538-4357/ab91af
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Isobe et al. (2023a) Isobe, Y., Ouchi, M., Nakajima, K., et al. 2023a, ApJ, 956, 139, doi: 10.3847/1538-4357/acf376
  * Isobe et al. (2023b) Isobe, Y., Ouchi, M., Tominaga, N., et al. 2023b, ApJ, 959, 100, doi: 10.3847/1538-4357/ad09be
  * Izotov et al. (2020) Izotov, Y. I., Schaerer, D., Worseck, G., et al. 2020, MNRAS, 491, 468, doi: 10.1093/mnras/stz3041
  * Izotov et al. (2014) Izotov, Y. I., Thuan, T. X., & Guseva, N. G. 2014, MNRAS, 445, 778, doi: 10.1093/mnras/stu1771
  * Izotov et al. (2021) —. 2021, MNRAS, 504, 3996, doi: 10.1093/mnras/stab1099
  * Izotov et al. (2007) Izotov, Y. I., Thuan, T. X., & Stasińska, G. 2007, ApJ, 662, 15, doi: 10.1086/513601
  * Jones et al. (2023) Jones, T., Sanders, R., Chen, Y., et al. 2023, ApJ, 951, L17, doi: 10.3847/2041-8213/acd938
  * Kojima et al. (2020) Kojima, T., Ouchi, M., Rauch, M., et al. 2020, ApJ, 898, 142, doi: 10.3847/1538-4357/aba047
  * Kojima et al. (2021) —. 2021, ApJ, 913, 22, doi: 10.3847/1538-4357/abec3d
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Limongi & Chieffi (2018) Limongi, M., & Chieffi, A. 2018, ApJS, 237, 13, doi: 10.3847/1538-4365/aacb24
  * Matsumoto et al. (2022) Matsumoto, A., Ouchi, M., Nakajima, K., et al. 2022, ApJ, 941, 167, doi: 10.3847/1538-4357/ac9ea1
  * Nagele & Umeda (2023) Nagele, C., & Umeda, H. 2023, ApJ, 949, L16, doi: 10.3847/2041-8213/acd550
  * Nakajima et al. (2023) Nakajima, K., Ouchi, M., Isobe, Y., et al. 2023, ApJS, 269, 33, doi: 10.3847/1538-4365/acd556
  * Nomoto et al. (2013) Nomoto, K., Kobayashi, C., & Tominaga, N. 2013, ARA&A, 51, 457, doi: 10.1146/annurev-astro-082812-140956
  * Pagel et al. (1992) Pagel, B. E. J., Simonson, E. A., Terlevich, R. J., & Edmunds, M. G. 1992, MNRAS, 255, 325, doi: 10.1093/mnras/255.2.325
  * Peimbert et al. (2002) Peimbert, A., Peimbert, M., & Luridiana, V. 2002, ApJ, 565, 668, doi: 10.1086/324601
  * Sánchez Almeida et al. (2015) Sánchez Almeida, J., Elmegreen, B. G., Muñoz-Tuñón, C., et al. 2015, ApJ, 810, L15, doi: 10.1088/2041-8205/810/2/L15
  * Senchyna et al. (2023) Senchyna, P., Plat, A., Stark, D. P., & Rudie, G. C. 2023, arXiv e-prints, arXiv:2303.04179, doi: 10.48550/arXiv.2303.04179
  * Terp et al. (2024) Terp, C., Heintz, K. E., Watson, D., et al. 2024, arXiv e-prints, arXiv:2404.06543, doi: 10.48550/arXiv.2404.06543
  * Tominaga et al. (2007) Tominaga, N., Umeda, H., & Nomoto, K. 2007, ApJ, 660, 516, doi: 10.1086/513063
  * Topping et al. (2024) Topping, M. W., Stark, D. P., Senchyna, P., et al. 2024, arXiv e-prints, arXiv:2401.08764, doi: 10.48550/arXiv.2401.08764
  * Treu et al. (2022) Treu, T., Roberts-Borsani, G., Bradac, M., et al. 2022, ApJ, 935, 110, doi: 10.3847/1538-4357/ac8158
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Watanabe et al. (2024) Watanabe, K., Ouchi, M., Nakajima, K., et al. 2024, ApJ, 962, 50, doi: 10.3847/1538-4357/ad13ff
  * Xu et al. (2022) Xu, Y., Ouchi, M., Rauch, M., et al. 2022, ApJ, 929, 134, doi: 10.3847/1538-4357/ac5e32
  * Yanagisawa et al. (2024) Yanagisawa, H., Ouchi, M., Nakajima, K., et al. 2024, arXiv e-prints, arXiv:2403.20118, doi: 10.48550/arXiv.2403.20118