#  Dynamics of GN-z11 Explored by JWST Integral Field Spectroscopy:
Gaseous Rotating Disk at $z=10.60$ Suggestive of Weak Feedback?

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  Masami Ouchi National Astronomical Observatory of Japan,
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  Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Kavli Institute for the Physics and Mathematics
of the Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba, 277-8583, Japan  Hidenobu Yajima Center for Computational
Sciences, University of Tsukuba, Tennodai, 1-1-1 Tsukuba, Ibaraki 305-8577,
Japan  Hajime Fukushima  Center for Computational Sciences, University of
Tsukuba, Tennodai, 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  Kimihiko Nakajima
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  Yoshiaki Ono Institute for Cosmic
Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277-8582, 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  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  Yechi Zhang
National Astronomical Observatory of Japan, 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 Astronomy,
Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba, 277-8583, Japan

(Received —; Revised —; Accepted —)

###### Abstract

We investigate the dynamics of GN-z11, a luminous galaxy at $z=10.60$,
carefully analyzing the public deep integral field spectroscopy (IFS) data
taken with JWST NIRSpec IFU. While the observations of the IFS data originally
targeted a Heii clump near GN-z11, we find that Ciii]$\lambda\lambda
1907,1909$ emission from ionized gas at GN-z11 is bright and spatially
extended significantly beyond the point-spread function (PSF) in contrast with
the GN-z11’s compact UV-continuum morphology. The spatially extended Ciii]
emission of GN-z11 shows a velocity gradient, red- and blue-shifted components
in the NE and SW directions, respectively, which cannot be explained by the
variation of [Ciii]$\lambda$1907/Ciii]$\lambda$1909 line ratios. We perform
forward modeling with GalPak3D, including the effects of PSF smearing and line
blending, and find that the best-fit model is a nearly edge-on disk with a
rotation velocity of $v_{\mathrm{rot}}$=$376_{-151}^{+110}$ km s-1, a velocity
dispersion of $\sigma_{v}=113_{-48}^{+24}$ km s-1, and a ratio of
$v_{\mathrm{rot}}/\sigma_{v}=3.3_{-1.5}^{+1.7}$, indicative of a rotation-
dominated disk at $z=10.6$. Interestingly, the disk rotation velocity is
faster than the circular velocity at the virial radius,
$v_{\mathrm{c}}(r_{200})=205_{-34}^{+28}~{}\mathrm{km~{}s^{-1}}$, estimated
from the stellar mass, suggesting a compact disk produced under weak feedback
such predicted in numerical simulations. In fact, the half-light radius of the
Ciii] emitting gas disk is only $294\pm 45$ pc, while the one of the stellar
UV component is even smaller, $196\pm 12$ pc. While higher S/N data are
necessary for a conclusion, these observational results would suggest that
GN-z11 has a fast-rotating gaseous disk whose center possesses luminous
stellar components or AGN providing weak feedback.

galaxies: evolution — galaxies: kinematics and dynamics

††software: astropy (Astropy Collaboration et al., 2013, 2018), reproject
(Robitaille et al., 2023), galfit (Peng et al., 2002, 2010), GalPak3D (Bouché
et al., 2015)

##  1 Introduction

Primordial galaxies evolve into the various types of galaxies we see today
involving complex interplay between different processes: accretion of cold
gas, minor and major mergers, stellar and active-galactic-nuclei feedback
(e.g., Dekel et al., 2009; Hopkins et al., 2012; Naab & Ostriker, 2017; Nelson
et al., 2019). On the other hand, many of these baryonic processes operate on
physical scales that are well below the resolution of current cosmological
simulations, and are mostly calibrated to match observations in the local
Universe, leaving potential crisis to explain galaxy evolution in the early
universe. Spatially resolved studies of the gas kinematics can be critical to
give a consistent picture of how galaxies grow and evolve across cosmic time.

Ground-based surveys using near-infrared integral field unit (IFU) such as
SINS (Förster Schreiber et al., 2009; Genzel et al., 2011) and KMOS3D
(Wisnioski et al., 2015) provide rich dataset of gas kinematics for star
forming galaxies at $z\sim 2$. Sub-millimeter observations using ALMA show
great potential of investigating gas kinematics at $z>4$. Neeleman et al.
(2020); Rizzo et al. (2020, 2021) analyze the kinematics of $z\sim 4-5$
galaxies and reveal the existence of rotation-dominated disk in the early
universe whose rotation velocity $v_{\mathrm{rot}}$ and be as high as ten
times of the velocity dispersion $\sigma_{v}$. Parlanti et al. (2023) further
report rotating but turbulent disks at $z>5$ while Tokuoka et al. (2022)
identify a potentially rotating disk at $z\sim 9$. After the launch of JWST,
gas kinematics has been investigated for $z\sim 5-8$ galaxies using rest-frame
optical emission observed by JWST NIRSpec MSA (de Graaff et al., 2023) and
NIRCam WFSS (Nelson et al., 2023; Li et al., 2023).

Recent JWST observations provide deep spectroscopic data that reveals the
properties of $z>10$ galaxies (e.g., Arrabal Haro et al., 2023a, b; Curtis-
Lake et al., 2023; Harikane et al., 2024). GN-z11 is a remarkably luminous
high-$z$ galaxies initially identified by Bouwens et al. (2010); Oesch et al.
(2016). Recent JWST observations measure a spectroscopic redshift of $10.603$.
The NIRCam images reveal the compact size of GN-z11 (Tacchella et al., 2023)
while the NIRSpec IFU observations identify extended emission (Maiolino et
al., 2023; Scholtz et al., 2023). Exploiting the spatial and spectral
resolution of 3-dimensional (3d) IFU data, directly analyzing gas kinematics
at $z>10$ becomes possible.

In this paper, we analyze the the public data of GN-z11 taken with JWST
NIRSpec IFU. While the observations of the IFS data originally targeted a He
ii clump near GN-z11, we find that Ciii]$\lambda\lambda 1907,1909$ emission at
GN-z11 is bright and spatially extended, which traces the kinematics of
ionized gas. This paper is structured as follows. Section 2 explains our
observations and dataset. Section 3 describes how we analyze the morphology
and kinematics of Ciii] emission. We discuss and summarize our findings in
Sections 4 and 5, respectively. Throughout the paper we adopt a Planck flat
$\Lambda$CDM cosmology with $H_{0}=67.7~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$ and
$\Omega_{\mathrm{m}}=0.310$ (Planck Collaboration et al., 2020).

Figure 1:  Left to right: the moment-0, -1, and -2 maps derived from the Ciii]
emission of GN-z11. The moment-1 and -2 maps are derived by fitting the
profile of Ciii] doublets as shown in Figure 3. The colorbar is plotted in an
arbitrary unit for the moment-0 map while in the unit of
$\mathrm{km~{}s^{-1}}$ for moment-1 and -2 maps. There exists a spatial
extension and a velocity gradient from NE to SW.

##  2 Observations and data reductions

In this work we use the data obtained by the NIRSpec IFU (Jakobsen et al.,
2022; Böker et al., 2022) on JWST. The data was obtained on 22nd – 23rd of May
2023 under the DDT program 4426 (PI: Roberto Maiolino). Details of the
observations are presented in Maiolino et al. (2023). The observational setup
utilised a medium cycling pattern of 10 dithers for a total integration time
on source of 10.6 h with the medium-resolution grating/filter pair
G235M/F170LP, and 3.3 h with the medium-resolution grating/filter pair
G140M/F100LP. These configurations cover the spectral ranges of 0.97–1.89 and
1.66–3.17 $\mu$m, with a nominal spectral resolution of $R\sim 1000$.

We process the data taken with the grating/filter pair G235M/F170LP using the
JWST Science Calibration pipeline version 1.12.5 under CRDS context
jwst_1170.pmap. We make several modifications to the JWST pipeline to increase
data quality, which we summarize here. The raw data files are downloaded from
the MAST archive and processed with calwebb_detector1. The resulting count-
rate frames are corrected for 1/f noise using NSClean (Rauscher, 2024). We
make a customized mask to obtain dark areas that are used to fit a background
in Fourier space. We then perform the default pipeline steps including wcs-
correction, flat-fielding, flux-calibrations, and building 3d data cubes
through calwebb_spec2. We do not apply the calwebb_spec3 process for detecting
outliers and stacking frames as we find the data artefacts cannot be perfectly
removed (see e.g., Perna et al. 2023; D’Eugenio et al. 2023). We utilize a
customized median stacking to combine the 3d data cubes instead of the
“drizzle” method of calwebb_spec3, which is inspired by the procedure taken by
Vayner et al. (2023). The 3d data cube of each exposure frame is firstly
subtracted by background and continuum following the method in Maiolino et al.
(2023). We then use the reproject package (Robitaille et al., 2023) and the
reproject_interp routine to align the different exposure frames for further
median stacking. We choose a pixel scale of $0^{\prime\prime}.06$ for the
spatial information.

We carefully inspect each exposure frame to decide whether they should be
included in the stacking procedure. The observations were not centered
properly as the guide star selected by the observatory was a binary system,
introducing a large offset (1”.4 towards SE; Maiolino et al. 2023). We find in
7 frames (#1, 3, 5, 6, 8, 9, 10) the extended Ciii] emission with a radius of
$\sim 0^{\prime\prime}.3$ is covered within the field-of-view while the other
3 frames (#2, 4, 7) are not used in this study. We identify potential data
artefacts for 3 frames (#3, 8, 10) among the 7 frames. In this study, we
mainly discuss the results based on the stacking data cube for the 4 healthy
frames (#1, 5, 6, 9) while the results of seven-frame stack are also
presented.

In this study, we mainly examine Ciii]$\lambda\lambda 1907,1909$ that is the
emission with highest signal-to-noise ratio (S/N) within the wavelength
coverage of G235M/F170LP and G140M/F100LP grating/filter pairs. We make data
cube for Ciii] using the wavelength from 2.2099$\mu$m to 2.2173$\mu$m and the
pixels in a square region with a radius of 6 pixels from the pixel of Ciii]
intensity peak. The map of the continuum is obtained by sigma-clipping and
collapsing continuum channels around Ciii] emission. We estimate the noise
from the root mean squares (rms) of the fluxes in the IFU field of view with
no visible sources.

It is critical to characterize the point-spread-function (PSF) and line-
spread-function (LSF) of NIRspec IFU in the analysis of kinematics. We utilize
the observation data of star G–191–B2B taken on $15^{\mathrm{th}}$ of
September 2023 under the calibration program 1537 (PI: Karl Gordon). We reduce
the data taken with G235M/F170LP following the procedures mentioned above. The
data cube of G 191–B2B is cropped to have the same dimension and wavelength
range as that of Ciii] to represent the PSF. We also apply the same data
reduction procedure on the NIRSpec IFU observations of planetary nebula
IRAS–05248–7007 conducted on $1^{\mathrm{st}}$ of August 2023 under the
calibration program 1492 (PI: Tracy Beck). We investigate the emission line of
the planetary nebula in the same manner as Isobe et al. (2023a). We obtain
instrumental broadening of $\sigma_{\mathrm{inst}}=8.9~{}\mathrm{\AA}$ that is
consistent with the one of NIRSpec MSA using G235M/F170LP estimated by Isobe
et al. (2023a).

Figure 2:  Radial profile of the enclosed light of Ciii] and continuum compared to the shape of PSF. Shaded areas indicate the uncertainties. The Ciii] is more extended than continuum and PSF enabling the analysis of kinematics.  Table 1: Dynamical Properties of GN-z11 Property | Value
---|---
Morphology Fitting
Half-light radius $R_{\mathrm{e}}$ [pc] | $294\pm 45$
Position angle $\theta$ [deg] | $247\pm 12$
Axial ratio $b/a$ | $0.35\pm 0.30$
Observed Kinematic Ratio
Observed velocity difference $\Delta v_{\mathrm{obs}}$ [km s-1] | $366_{-68}^{+62}$
Median velocity dispersion $\sigma_{\mathrm{med}}$ [km s-1] | $122_{-83}^{+3}$
$\Delta v_{\mathrm{obs}}/2\sigma_{\mathrm{med}}$ | $1.50_{-0.28}^{+1.05}$
Forward Modeling
Half-light radius $R_{\mathrm{e}}$ [pc] | $342_{-32}^{+31}$
Position angle $\theta$ [deg] | $220_{-14}^{+18}$
Inclination $i$ [deg] | $69\pm 16$
Rotational velocity $v_{\mathrm{rot}}$ [km s-1] | $376_{-151}^{+110}$
Velocity dispersion $\sigma_{v}$ [km s-1] | $113_{-48}^{+24}$
$v_{\mathrm{rot}}/\sigma_{v}$ | $3.33_{-1.50}^{+1.72}$

##  3 Analysis and Results

###  3.1 Morphology

We make the narrow band image or the so-called moment-0 map of Ciii] by
summing up the data cube in the wavelength direction within the velocity range
of $-500~{}\mathrm{km~{}s^{-1}}$ to $800~{}\mathrm{km~{}s^{-1}}$ around the
centroid of [Ciii]$\lambda 1907$. The moment-0 map in the left panel of Figure
1 shows a slight spatial extension from NE to SW, which is consistent with the
descriptions in Maiolino et al. (2023). However, the funnel-shaped geometry
extended towards the SW is not clearly seen in our median stack results after
removing the 3 frame affected by the guiding problem. We fit the moment-0 map
with a Sersic model using galfit (Peng et al., 2002, 2010). The shape of PSF
we measure in Section 2 is convolved with the model during the fitting. We
obtain a Sersic index of $n<0.5$ but with large uncertainties. We thus adopt
the fiducial fitting result with a $n=1$ Sersic profile and obtain a half-
light radius of $R_{\mathrm{e}}=294\pm 45~{}\mathrm{pc}$ as shown in Table 1.
Tacchella et al. (2023) find the NIRCam images of GN-z11 are composed of a
central unresolved component and a extended exponential component. Effective
radius of Ciii] is even larger than the extended component
($R_{\mathrm{e}}=196\pm 12~{}\mathrm{pc}$) that is composed of more matured
stellar population than the central region. In Figure 2, we show the radial
profile of Ciii] emission that is more extended than those of the continuum
and PSF. The enclosed light profile is derived by integrating the fluxes
within a series of circular apertures that oversample the moment-0 map. From
the enclosed light profile, we derive the half-light radius of
$R_{\mathrm{e}}\sim 285$ pc after subtracting the size of PSF, which is
consistent with the once derived from galfit fitting. The Ciii] morphology is
clearly extended and resolved, which enables further kinematic analysis.

Figure 3: Spectra of Ciii] in the central $49=7\times 7$ pixels of Figure 1.
In each pixel with $S/N>6$, we overplot the best-fit profile (red solid curve)
composed of two Gaussian components (red dashed curves) that is used to derive
the line-of-sight velocity and velocity dispersion. The red vertical line
indicates the central velocity of [Ciii]$\lambda 1907$. Figure 4:  Moment maps
compared with the best-fit rotation disk model. Left to right: observed data,
GalPak3D model, and the residuals.

###  3.2 Kinematics

We examine the Ciii] kinematics by calculating the moment-1 and -2 maps that
represent the line-of-sight velocity and velocity dispersion, respectively.
Instead of the method by summing up the velocity weighted by the fluxes (e.g.,
Walter et al., 2008), we adopt a method of profile fitting to the spectrum in
each pixel as shown in Figure 3. The profile fitting method on high S/N pixels
is more robust against the fluctuations of flux especially at the wavelength
channels that correspond to large velocity offset. We select spaxels with a
S/N$>6$ that can be fitted reliably. Because Ciii] emission is a doublet, we
fit the emission line with two Gaussian components with the constraints of: 1)
flux ratio of [Ciii]$\lambda 1907/$Ciii]$\lambda 1909=1.48$ (Kewley et al.
2019; see Section 3.3 for further discussions), 2) velocity offset between the
two centroids fixed to the one at rest-frame, 3) identical line widths larger
than the LSF dispersion of $\sigma_{\mathrm{inst}}=8.9~{}\mathrm{\AA}$. By
conducting the profile fitting to the integrated spectrum of S/N$>6$ spaxels,
we find the central wavelength of [Ciii]$\lambda 1907$ to be $2.2127\mu$m
corresponding to the redshift of $10.605$ that is defined as the systematic
velocity. As shown in the middle panel of Figure 1, there is a clear velocity
gradient from NE to SW consistent with the Ciii] morphology. We measure a
maximum observed velocity difference of $\Delta
v_{\mathrm{obs}}=366_{-68}^{+62}~{}\mathrm{km~{}s^{-1}}$ and a median velocity
dispersion of $\sigma_{\mathrm{med}}=122_{-83}^{+3}~{}\mathrm{km~{}s^{-1}}$.
The velocity dispersion are the line width $\sigma$ quadratically subtracted
by the $\sigma_{\mathrm{inst}}$. This value is likely an upper limit because
the emission line only marginally resolved compared to the LSF dispersion of
$\sim 120~{}\mathrm{km~{}s^{-1}}$. Here we report the observed kinematic ratio
of $\Delta v_{\mathrm{obs}}/2\sigma_{\mathrm{med}}=1.50$ that is possibly
given by a rotational dominated disk.

Assuming the velocity gradient is given by disk rotation, we conduct forward
modelling using a software named GalPak3D (Bouché et al., 2015). GalPak3D
allows us to fit 3d data cube of doublets assuming a fixed flux ratio and
convolve the dynamical model with the PSF shape we derive in Section 2. We
choose a thick disk model with the disk height equals to $30\%$ of the
effective radius. The surface brightness is an exponential function of radius,
i.e., Sersic profile with a Sersic index of $n=1$. The choice of the surface
brightness profile and disk height is consistent with the modelling result of
Ciii] morphology. We choose the arctan rotation curve with two parameters,
maximum rotation velocity ($v_{\mathrm{rot}}$) and turn-over radius
($r_{\mathrm{v}}$):

$v(r)=v_{\mathrm{rot}}\frac{2}{\pi}\arctan(r/r_{\mathrm{v}}).$ (1)

The disk model consists of 10 parameters in total that are x- and
y-coordinates of the center, total flux, effective radius ($r_{\mathrm{e}}$),
$r_{\mathrm{v}}$, inclination ($i$), position angle, systemic velocity,
$v_{\mathrm{rot}}$, and intrinsic dispersion ($\sigma_{v}$). We fix the prior
to the uniform distributions whose lower and upper boundaries are given by the
galfit modeling of Ciii] morphology. The intrinsic dispersion is free from the
broadening given by instrument, the local isotropic velocity dispersion driven
by disk self-gravity, and the mixture of the line-of-sight velocities due to
the disk thickness Bouché et al. (2015), and the blending of Ciii] doublets.
The best-fit model and the residuals of moment maps are presented in Figure 4.
While the velocity gradient can be explained by the best-fit model, there
still remain residuals on the moment-1 maps possible given by complex
dynamical component that is not resolved by the current data. The dynamical
properties are shown in Table 1 with the uncertainties obtained by applying
forward modelling above on 200 data cubes generated by Monte-Carlo method. We
obtain $v_{\mathrm{rot}}/\sigma_{v}\gtrsim 1$ featuring a rotation dominated
disk in GN-z11.

The results above are based on the stacked data cube for 4 healthy frames.
Although we find potential data artefacts in the other 3 frames (see Section
2), we also conduct the same analysis on the seven-frame stack and obtain
similar results, such as $\Delta
v_{\mathrm{obs}}=238_{-67}^{+78}~{}\mathrm{km~{}s^{-1}}$,
$\sigma_{\mathrm{med}}=77_{-77}^{+68}~{}\mathrm{km~{}s^{-1}}$,
$v_{\mathrm{rot}}=361_{-179}^{+184}~{}\mathrm{km~{}s^{-1}}$, and
$\sigma_{v}=75_{-52}^{+22}~{}\mathrm{km~{}s^{-1}}$. We adopt the results
derived from the four-frame stack as the fiducial values for further
discussions.

Figure 5:  Integrated Ciii] spectra of S/N$>6$ pixels on the NE (top), SW
(middle), and the whole area (bottom) of GN-z11. In the same manner as Figure
3, we show the best-fit profile but with flux ratio
[Ciii]$\lambda$1907/Ciii]$\lambda$1909 as a free parameter. The solid vertical
lines indicate the centroids of [Ciii]$\lambda 1907$ and Ciii]$\lambda 1909$
obtained from the best-fit profile. The left and right boundaries of the
shaded areas are obtained from the best-fit profiles with flux ratio fixed to
$0.588$ ($n_{\mathrm{e}}=10^{5}~{}\mathrm{cm^{-3}}$) and $1.48$
($n_{\mathrm{e}}=100~{}\mathrm{cm^{-3}}$), respectively. The velocity offset
between the NE and SW pixels cannot be explained by the variation of flux
ratio.

###  3.3 Velocity gradient explained by flux ratio variation?

The flux ratio of [Ciii]$\lambda$1907/Ciii]$\lambda$1909 is sensitive to the
electron density between $n_{\mathrm{e}}=10^{2}-10^{5}~{}\mathrm{cm^{-3}}$. We
examine the flux ratio based on the ionization models of (Kewley et al., 2019)
with electron temperature $T_{\mathrm{e}}=10^{4}~{}\mathrm{K}$. In Figure 5,
we presents spectra of Ciii] integrated for all S/N$>6$ pixels, those on the
NE, and those on the SW together with the best-fit profiles with flux ratio
[Ciii]$\lambda$1907/Ciii]$\lambda$1909 as a free parameter. To examine the
influence of flux ratio on the central wavelengths, we also conduct fitting
with flux ratio [Ciii]$\lambda$1907/Ciii]$\lambda$1909 fixed to $0.588$
($n_{\mathrm{e}}=10^{5}~{}\mathrm{cm^{-3}}$) and $1.48$
($n_{\mathrm{e}}<100~{}\mathrm{cm^{-3}}$). Changing the flux ratio results in
a velocity difference of $\lesssim 100~{}\mathrm{km~{}s^{-1}}$. The velocity
difference between the NE to SW pixels cannot be explained by the variation of
flux ratios as shown in Figure 5, which confirms the velocity gradient
presented in Figure 1.

We obtain [Ciii]$\lambda 1907$/Ciii]$\lambda 1909=1.12$ corresponding to
$n_{\mathrm{e}}\sim 10^{4}~{}\mathrm{cm^{-3}}$ from the best-fit profile of
the integrated spectrum. Relatively high electron density of $\gtrsim
10^{3}~{}\mathrm{cm^{-3}}$ is consistent with reshift evolution of
$n_{\mathrm{e}}$ up to $z\sim 9$ discussed in Isobe et al. (2023a).
Particularly, the peak of [Ciii]$\lambda 1907$ appears to be lower than that
of Ciii]$\lambda 1909$ (Figure 3), which is a hint of high electron density
possibly given by the central black hole (Maiolino et al., 2023, 2024).
However, we obtain large $1\sigma$ uncertainty $=0.54$ on the flux ratio based
on the integrated spectrum. The models with
$n_{\mathrm{e}}=10^{5}~{}\mathrm{cm^{-3}}$ or
$n_{\mathrm{e}}=100~{}\mathrm{cm^{-3}}$ cannot be ruled out with $\chi^{2}$
statistics or Bayesian evidence. We thus choose a modest flux ratio of $1.48$
that is stable for $n_{\mathrm{e}}\lesssim 10^{3}~{}\mathrm{cm^{-3}}$ to
measure the velocities and conduct dynamical modelling.

Figure 6:  $v_{\mathrm{rot}}/\sigma_{v}$ of GN-z11 compared to those obtained
by previous studies at different redshifts. Our results of GN-z11 are shown
with the red star and square that are $v_{\mathrm{rot}}/\sigma_{v}$ obtained
with GalPak3D modelling and the observed kinematic ratio $\Delta
v_{\mathrm{obs}}/\sigma_{\mathrm{med}}$, respectively. The blue circles are
galaxies whose [Cii] kinematics is measured with ALMA taken from Neeleman et
al. (2020); Rizzo et al. (2020, 2021); Herrera-Camus et al. (2022); Fraternali
et al. (2021); Parlanti et al. (2023). The blue squares are recent JWST
results using NIRSpec MSA taken from de Graaff et al. (2023), while the blue
triangles and diamonds are taken from Nelson et al. (2023) and Li et al.
(2023), respetively, who use NIRCam WFSS.

##  4 Discussions

###  4.1 Rotation disk at $z>10$

The ratio between ordered rotation and turbulent motion is often used to
indicate a rotation disk. Our estimations of $v_{\mathrm{rot}}/\sigma_{v}$ are
displayed in Figure 6 with a comparison to previous results at different
redshifts. Both the observed kinematic ratio and the intrinsic
$v_{\mathrm{rot}}/\sigma_{v}$ agree with the scenario of a rotation dominated
disk. Our results are comparable to previous rotating galaxies identified by
ground-based sub-millimeter observations (Rizzo et al., 2020, 2021; Fraternali
et al., 2021; Herrera-Camus et al., 2022; Tokuoka et al., 2022; Parlanti et
al., 2023), rest-frame optical spectroscopy using JWST NIRSpec MSA (de Graaff
et al., 2023), and NIRCam WFSS (Nelson et al., 2023; Li et al., 2023).
Nevertheless, it is surprising to find a rotation disk at $z>10$ because
galaxies are more likely to be turbulent with increasing redshift due to
feedback from massive stars and supernovae (e.g., Pillepich et al., 2019;
Yajima et al., 2022, 2023). Simulations of Yajima et al. (2017) show that
galactic disks are destroyed due to supernova feedback, while galaxies in
simulations with no feedback or lower SF efficiency models can sustain a
galactic disk. Recent JWST results of massive luminous galaxies found at
$z>10$ (e.g., Donnan et al., 2023; Harikane et al., 2023, 2024) suggest lack
of suppression on star formation. High electron density such found by Isobe et
al. (2023a) could be an evidence of efficient cooling in which case feedback
becomes inefficient. Dekel et al. (2023) also discuss that high densities and
low metallicities in galaxies at $z\gtrsim 10$ result in a high star formation
efficiency with feedback-free starbursts. The scenario of weak feedback is
consistent with GN-z11 being luminous and rotational dominated.

Figure 7:  Rotation velocity and velocity dispersion as a function of stellar
mass. The symbols are the same as those in Figure 6. The grey crosses are
$z\sim 2$ galaxies taken from Förster Schreiber et al. (2018). The grey line
on the left panel indicates the Tully-Fisher relation calculated by Di Teodoro
et al. (2021).

In Figure 7, we examine mass dependence of the rotation velocity and velocity
dispersion combining previous observations and our results of GN-z11. We adopt
the stellar mass $\log(M_{*}/\mathrm{M_{\odot}})=9.1_{-0.4}^{+0.3}$ of GN-z11
derived by Tacchella et al. (2023). GN-z11 shows comparable rotation velocity
and velocity dispersion to some previous reported high-$z$ galaxies, but
possibly larger than those with similar stellar mass of
$10^{9}~{}\mathrm{M_{\odot}}$. Since the Ciii] doublets are only marginally
resolved with moderate S/N and spectral resolution, we cannot rule out the
possibility that the $v_{\mathrm{rot}}$ and $\sigma_{v}$ of GN-z11 are
overestimated even with our best efforts. Fortunately, GN-z11 will be observed
with the $R\sim 2700$ JWST NIRspec high resolution gratings in the coming
cycle-3 missions. Nevertheless, it is possible that GN-z11 has
$v_{\mathrm{rot}}$ and $\sigma_{v}$ larger than those of $z\sim 2$ star
forming galaxies (Förster Schreiber et al., 2018) and offsetted from the local
Tully-Fisher relation (e.g., Di Teodoro et al., 2021), which can be explained
by the compactness of GNz-11.

Figure 8:  Circular velocities of GN-z11 at $2R_{\mathrm{e}}$ (red star) and
virial radius of the dark matter halo (red circle). The black lines are the
simulation predictions of Kimm et al. (2015) with a velocity offset for
presentation purpose. The peaked $v_{\mathrm{c}}$ in inner radius is
consistent with the model of no thermal/mechanical feedback.

To examine the mass concentration of GN-z11, we calculate the circular
velocity from the rotation curve of Equation (1) considering the asymmetric
drift (see Equation (12) of Burkert et al. 2016):

$v_{\mathrm{c}}^{2}(r)=v(r)^{2}+2\sigma_{v}^{2}\times\left(\frac{r}{R_{\mathrm{d}}}\right),$
(2)

where $R_{\mathrm{d}}=R_{\mathrm{e}}/1.68$ is the disk scale height. We obtain
$v_{\mathrm{c}}(2R_{\mathrm{e}})=385_{-101}^{+86}~{}\mathrm{km~{}s^{-1}}$ at
the radius that roughly corresponds to the extension of Ciii] investigated by
this study. This circular velocity is compared with the circular velocity at
the virial radius ($r_{200}$) of the dark matter halo that can be calculated
from the halo mass ($M_{\mathrm{h}}$) using the following equations (Mo &
White, 2002):

$v_{\mathrm{c}}(r_{200})=\left(\frac{GM_{\mathrm{h}}}{r_{200}}\right)^{1/2},$
(3)
$r_{200}=\left(\frac{GM_{\mathrm{h}}}{100\Omega_{\mathrm{m}}H_{0}^{2}}\right)^{1/3}(1+z)^{-1}.$
(4)

We adopt the $z\sim 8$ stellar mass – halo mass relation from Behroozi et al.
(2019) and scale it to match the $z\sim 10$ relation, as the stellar mass of
GN-z11 is not well probed by the $z\sim 10$ relation (see Figure 9 of Behroozi
et al. 2019). We obtain $M_{\mathrm{h}}=1.35_{-0.67}^{+0.56}\times
10^{11}~{}\mathrm{M_{\odot}}$ (c.f., Scholtz et al. 2023; Ferrara 2023) and
$v_{\mathrm{c}}(r_{200})=205_{-34}^{+28}~{}\mathrm{km~{}s^{-1}}$. As shown in
Figure 8, $v_{\mathrm{c}}(2R_{\mathrm{e}})$ is surprisingly larger than
$v_{\mathrm{c}}(r_{200})$ with at least $1\sigma$ significance. The peaked
velocity in the inner radius is consistent with the model shown in Kimm et al.
(2015) with no thermal or mechanical feedback, instead of the strong feedback
model that would introduce a smoothly rising velocity curve. Such a centrally
concentrated mass profile is a favorable explanation for disk formation and
efficient star formation as predicted by simulations (e.g., Hopkins et al.,
2023).

We further investigate the dynamical state of GN-z11 by firstly estimating the
dynamical mass with the following equation:

$\displaystyle M_{\mathrm{dyn}}=$ $\displaystyle 2.33\times
10^{5}\left(\frac{r}{\mathrm{kpc}}\right)\left(\frac{v_{\mathrm{c}}(r)}{\mathrm{km~{}s^{-1}}}\right)^{2}.$
(5)

We compare dynamical mass within the effective radius
$M_{\mathrm{dyn}}(r<R_{\mathrm{e}})=4.3_{-1.7}^{+3.1}\times
10^{9}~{}\mathrm{M_{\odot}}$ with the stellar mass and find that the gas mass
takes up $\sim 70\%$ of the dynamical mass, assuming negligible mass budget
from dark matter. A similar gas mass fraction ($f_{\mathrm{gas}}\sim 0.6$) can
be obtained if we adopt the SFR reported by Tacchella et al. (2023) and
estimate the gas mass from the star formation rate surface density following
the Kenicutt-Schmidt law (Kennicutt, 1998). Here we give a rough evaluation of
disk instability using the Toomre-Q value:

$Q=\frac{\sigma_{v}}{v_{\mathrm{rot}}}\frac{a}{f_{\mathrm{gas}}},$ (6)

where the parameter $a$ ranges from 1 to 2 depending on the gas distribution.
We assume $a=\sqrt{2}$ which corresponds to a disk with constant rotational
velocity (Genzel et al., 2011). We find $Q\lesssim 1$ consistent with picture
of a rotation disk experiencing gas accretion though disk instability. There
is a surprising agreement with the observational results of local dwarf
galaxies (e.g., Green et al., 2014; Isobe et al., 2023b; Xu et al., 2024).
Green et al. (2014) measure the gas mass for local star forming galaxies and
find $M_{*}\sim 10^{9}~{}\mathrm{M_{\odot}}$ galaxies typically have
$f_{\mathrm{gas}}\sim 0.5$, which implies $Q\sim 1$ following the calculation
of this study. A dynamically unstable rotation disk in GN-z11 is consistent
with the picture of efficient gas accretion and star formation, which could be
related to weak feedback from central stellar component or AGN.

###  4.2 Alternative interpretations

We identify a clear velocity gradient using the NIRSpec IFU data of GN-z11
that is well explained by a rotating disk. However, there could be alternative
interpretations of the Ciii] morphology and kinematics:

  * •

Ionization cone of central black hole (BH). Maiolino et al. (2023) explains
the spatial extension of Ciii] as the ionization cone of central BH. It is
consistent with the scenario of disk rotation, although the Ciii] emission may
only trace part of the gaseous disk illuminated by the ionization cone.

  * •

Merger. Distinguishing between rotation disk and galaxy mergers has been a
longstanding problem for high-$z$ galaxies due to limited spatial resolution.
GN-z11 is very compact and may not be well resolved until next-generation
facilities such as extremely large telescopes with adaptive optics. With the
current observational evidence, the scenario of major merger is not favored
for GN-z11 and similar high$-z$ galaxies with compact sizes (see e.g., Ono et
al., 2023).

  * •

Outflow. Bunker et al. (2023) find suggestions of galactic outflows from the
velocity shift of Ly$\alpha$. Ferrara (2023) obtain a fiducial terminal
velocity of $555~{}\mathrm{km~{}s^{-1}}$ for the outflows that composed of
highly ionized gas. However, outflow signatures are not seen in the relatively
bright Ciii] or He ii$\lambda 1640$ emission. Contributions of outflow to the
observed Ciii] kinematics is likely weak.

##  5 Summary

In this paper we investigate the dynamics of GN-z11, a luminous galaxy at
$z=10.60$. We carefully analyze the public deep integral field spectroscopy
(IFS) data taken with JWST NIRSpec IFU and exploit the spatial and spectral
resolution to determine the dynamical properties of GN-z11. Our findings are
summarized below:

  * •

We carefully investigate the Ciii]$\lambda\lambda 1907,1909$ emission that
traces ionized gas. The half-light radius of the Ciii] emitting gas disk is
only $294\pm 45$ pc but spatially extended significantly beyond the PSF in
contrast with the GN-z11’s compact UV-continuum morphology.

  * •

The spatially extended Ciii] emission of GN-z11 shows a velocity gradient,
red- and blue-shifted components in the NE and SW directions, respectively,
which cannot be explained by the variation of
[Ciii]$\lambda$1907/Ciii]$\lambda$1909 line ratios. We perform forward
modeling with GalPak3D, including the effects of PSF smearing and line
blending, and find that the best-fit model is a nearly edge-on disk with a
rotation velocity of $v_{\mathrm{rot}}$=$376_{-151}^{+110}$ km s-1, a velocity
dispersion of $\sigma_{v}=113_{-48}^{+24}$ km s-1. Both the observed kinematic
ratio $\Delta v_{\mathrm{obs}}/2\sigma_{\mathrm{med}}=1.50$ and the intrinsic
$v_{\mathrm{rot}}/\sigma_{v}=3.3_{-1.5}^{+1.7}$ would indicate a rotation-
dominated disk.

  * •

The rotation velocity and velocity dispersion of GN-z11 is comparable to
$4<z<9$ galaxies previously reported by ground-based sub-millimeter
observations and JWST, but possibly larger than those with similar stellar
mass of $10^{9}~{}\mathrm{M_{\odot}}$. Observations with better S/N and
resolution are required to confirm this result.

  * •

The disk rotation velocity is faster than the circular velocity at the virial
radius, $v_{\mathrm{c}}(r_{200})=205_{-34}^{+28}~{}\mathrm{km~{}s^{-1}}$,
suggesting a compact disk produced under weak feedback such predicted in
numerical simulations. Our results would suggest a fast-rotating gaseous disk
at $z>10$ whose center possesses luminous stellar components or AGN providing
weak feedback.

This work is based on observations made with the NASA/ESA/CSA James Webb Space
Telescope. The data were obtained from the Mikulski Archive for Space
Telescopes at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127 for JWST. These observations are associated with programs
1429, 1537, and 4426. We thank the research team led by Roberto Maiolino for
developing their observing program of GN-z11. We thank Yuzo Ishikawa for the
discussions on data analysis of JWST NIRspec IFU. We are grateful to staff of
the James Webb Space Telescope Help Desk for letting us know useful
information. This work was supported by the joint research program of the
Institute for Cosmic Ray Research (ICRR), University of Tokyo. This
publication is based upon work supported by the World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan. M.O., H.Y., H.F.,
K.N., Y.H., and Y.I. are supported by JSPS KAKENHI Grant Nos.
20H00180/21H04467, 21H04489, 3K13139, 20K22373, 21K13953, and 24KJ0202
respectively. H.Y. is supported by JST FOREST Program, Grant Number JP-
MJFR202Z.

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11institutetext: Department of Language and Literature
Max Planck Institute for Empirical Aesthetics, Frankfurt
22institutetext: Institut für Maschinelle Sprachverarbeitung (IMS)
University of Stuttgart
22email<EMAIL_ADDRESS>
# Diachronic Topics in New High German Poetry

Thomas N. Haider 1122

###### Abstract

Statistical topic models are increasingly and popularly used by Digital
Humanities scholars to perform distant reading tasks on literary data
(Navarro-Colorado, 2018), (Hettinger et al., 2016). It allows us to estimate
what people talk about. Especially Latent Dirichlet Allocation (LDA), see
(Blei et al., 2003), has shown its usefulness, as it is unsupervised, robust,
easy to use, scalable, and it offers interpretable results. In a preliminary
study, we apply LDA to a corpus of New High German poetry (textgrid, with 51k
poems, 8m token) and interpret salient topics, their trend over time
(1575–1925 A.D.), and use the distribution of topics over documents for a
classification of poems into time periods and for authorship
attribution.00footnotetext: Proceedings of the International Digital
Humanities Conference DH2019, Utrecht
https://dev.clariah.nl/files/dh2019/boa/1031.html

##  1 Corpus

The Digital Library in the TextGrid Repository represents an extensive
collection of German texts in digital form (Vanscheidt et al., 2016). It was
mined from http://zeno.org and covers a time period from the mid 16th century
up to the first decades of the 20th century. It contains many important texts
that can be considered as part of the literary canon, even though it is far
from complete (e.g. it contains only half of Rilke’s work). We find that
around 51k texts are annotated with the label ’verse’ (TGRID-V), not
distinguishing between ’lyric verse’ and ’epic verse’. However, the average
length of these texts is around 150 token, dismissing most epic verse tales.
Also, the poems are distributed over 229 authors, where the average author
contributed 240 poems (median 131 poems). A drawback of TGRID-V is the
circumstance that it contains a noticeable amount of French, Dutch and Latin
(over 400 texts). To constrain our dataset to German, we filter foreign
language material with a stopword list, as training a dedicated language
identification classifier is far beyond the scope of this work.

Figure 1: 25 year Time Slices of Textgrid Poetry (1575–1925)

##  2 Experiments

We approach diachronic variation of poetry from two perspectives. First, as
distant reading task to visualize the development of clearly interpretable
topics over time. Second, as a downstream task, i.e. supervised machine
learning task to determine the year (the time-slot) of publication for a given
poem. We infer topic distributions over documents as features and pit them
against a simple style baseline.

We use the implementation of LDA as it is provided in genism (Rehurek and
Sojka, 2011). LDA assumes that a particular document contains a mixture of few
salient topics, where words are semantically related. We transform our
documents (of wordforms) to a bag of words representation, filter stopwords
(function words), and set the desired number of topics=100 and train for 50
epochs to attain a reasonable distinctness of topics. We choose 100 topics
(rather than a lower number that might be more straightforward to interpret)
as we want to later use these topics as features for downstream tasks. We find
that wordforms (instead of lemma) are more useful for poetry topic models, as
these capture style features (rhyme), orthographic variations (’hertz’ instead
of ’herz’), and generally offer more interpretable results.

###  2.1 Topic Trends

We retrieve the most important (likely) words for all 100 topics and interpret
these (sorted) word lists as aggregated topics, e.g. topic 27 (figure 2)
contains: Tugend (virtue), Kunst (art), Ruhm (fame), Geist (spirit), Verstand
(mind) and Lob (praise). This topic as a whole describes the concept of
’artistic virtue’.

In certain clusters (topics) we find poetic residuals, such that rhyme words
often cluster together (as they stand in proximity), e.g. topic 52 with: Mund
(mouth), Grund (cause, ground), rund (round).

To discover trends of topics over time, we bin our documents into time slots
of 25 years width each. See figure 1 for a plot of the number of documents per
bin. The chosen binning slots offer enough documents per slot for our
experiments. To visualize trends of singular topics over time, we aggregate
all documents d in slot s and add the probabilities of topic t given d and
divide by the number of all d in s. This gives us the average probability of a
topic per timeslot. We then plot the trajectories for each single topic. See
figures 2–6 for a selection of interpretable topic trends. Please note that
the scaling on the y-axis differ for each topic, as some topics are more
pronounced in the whole dataset overall.

Figure 2: left: Topic 27 ’Virtue, Arts’ (Period: Enlightenment), right: Topic
55 ’Flowers, Spring, Garden’ (Period: Early Romanticism) Figure 3: left: Topic
63 ’Song’ (Period: Romanticism), right: Topic 33 ’German Nation’ (Period:
Vormärz, Young Germany)) Figure 4: left: Topic 28 ’Beautiful Girls’ (Period:
Omnipresent, Romanticism), right: Topic 77 ’Life & Death’ (Period:
Omnipresent, Barock Figure 5:  left: Topic 60 ’Fire’ (Period: Modernity),
right: Topic 42 ’Family’ (no period, fluctuating) Figure 6:  Most informative
topics for classification; left: Topic 11 ’World, Power, Lust, Time’ (Period:
Barock), right: Topic 19 ’Heaven, Depth, Silence’ (Period: Romanticism,
Modernity)

Some topic plots are already very revealing. The topic ‘artistic virtue’
(figure 2, left) shows a sharp peak around 1700—1750, outlining the period of
Enlightenment. Several topics indicate Romanticism, such as ‘flowers’ (figure
2, right), ‘song’ (figure 3, left) or ‘dust, ghosts, depths’ (not shown). The
period of ’Vormärz’ or ’Young Germany’ is quite clear with the topic ‘German
Nation’ (figure 3, right). It is however hardly distinguishable from romantic
topics.

We find that the topics ’Beautiful Girls’ (figure 4, left) and ’Life & Death’
(figure 4, right) are always quite present over time, while ’Girls’ is more
prounounced in Romanticism, and ’Death’ in Barock.

We find that the topic ’Fire’ (figure 5, left) is a fairly modern concept,
that steadily rises into modernity, possibly because of the trope ’love is
fire’. Next to it, the topic ’Family’ (figure 5, right) shows wild fluctuation
over time.

Finally, figure 6 shows topics that are most informative for the downstream
classification task: Topic 11 ’World, Power, Time’ (left) is very clearly a
Barock topic, ending at 1750, while topic 19 ’Heaven, Depth, Silence’ is a
topic that rises from Romanticism into Modernity.

###  2.2 Classification of Time Periods and Authorship

To test whether topic models can be used for dating poetry or attributing
authorship, we perform supervised classification experiments with Random
Forest Ensemble classifiers. We find that we obtain better results by training
and testing on stanzas instead of full poems, as we have more data available.
Also, we use 50 year slots (instead of 25) to ease the task.

For each document we determine a class label for a time slot. The slot
1575–1624 receives the label 0, the slot 1625–1674 the label 1, etc.. In
total, we have 7 classes (time slots).

As a baseline, we implement rather straightforward style features, such as
line length, poem length (in token, syllables, lines), cadence (number of
syllables of last word in line), soundscape (ratio of closed to open
syllables, see (Hench, 2017)), and a proxy for metre, the number of syllables
of the first word in the line.

We split the data randomly 70:30 training:testing, where a 50:50 shows (5
points) worse performance. We then train Random Forest Ensemble classifiers
and perform a grid search over their parameters to determine the best
classifier. Please note that our class sizes are quite imbalanced.

The Style baseline achieves an Accuracy of 83%, LDA features 89% and a
combination of the two gets 90%. However, training on full poems reduces this
to 42—52%.

The most informative features (by information gain) are: Topic11 (.067), Topic
37 (.055), Syllables Per Line (.046), Length of poem in syllables (.031),
Topic19 (.029), Topic98 (.025), Topic27 (’virtue’) (.023), and Soundscape
(.023).

For authorship attribution, we also use a 70:30 random train:test split and
use the author name as class label. We only choose the most frequent 180
authors. We find that training on stanzas gives us 71% Accuracy, but when
trained on full poems, we only get 13% Accuracy. It should be further
investigated is this is only because of a surplus of data.

###  2.3 Conclusion & Future Work

We have shown the viability of Latent Dirichlet Allocation for a visualization
of topic trends (the evolution of what people talk about in poetry). While
most topics are easily interpretable and show a clear trend, others are quite
noisy. For an exploratory experiment, the classification into time slots and
for authors attribution is very promising, however far from perfect. It should
be investigated whether using stanzas instead of whole poems only improves
results because of more available data. Also, it needs to be determined if
better topic models can deliver a better baseline for diachronic change in
poetry, and if better style features will outperform semantics. Finally, only
selecting clear trending and peaking topics (through co-variance) might
further improve the results.

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