# The ALPINE-ALMA [C ii] Survey: Investigation of $10$ Galaxies at $z\sim 4.5$
with [O ii] and [C ii] Line Emission $-$ ISM Properties and [O ii]$-$SFR
Relation

Brittany N. Vanderhoof1, A. L. Faisst2, L. Shen3,4, B. C. Lemaux5, M.
Béthermin6, P. L. Capak7,8, P. Cassata9,10, O. Le Fèvre6, D. Schaerer11,12, J.
Silverman13,14, L. Yan15, M. Boquien16, R. Gal17, J. Kartaltepe1, L. M.
Lubin5, M. Dessauges-Zavadsky11, Y. Fudamoto18,19, M. Ginolfi20, N. P.
Hathi21, G. C. Jones22,23, A. M. Koekemoer21, D. Narayanan7,24,25, M.
Romano9,10, M. Talia26,27, D. Vergani26, G. Zamorani26
1 School of Physics and Astronomy, Rochester Institute of Technology,
Rochester, NY 14623, USA
2 IPAC, California Institute of Technology 1200 E California Boulevard,
Pasadena, CA 91125, USA
3 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China
4 School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei, 230026, China
5 Gemini Observatory, NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawai’i,
96720, USA
6 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de
Marseille) UMR 7326, F-13388, Marseille, France
7 The Cosmic Dawn Center, University of Copenhagen, Vibenshuset, Lyngbyvej 2,
DK-2100 Copenhagen, Denmark
8 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200
Copenhagen, Denmark
9 Dipartimento di Fisica e Astronomia, Università di Padova, vicolo
dell’Osservatorio, 3 I-35122 Padova, Italy
10 INAF, Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5,
I-35122 Padova, Italy
11 Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290
Versoix, Switzerland
12 Institut de Recherche en Astrophysique et Planétologie—IRAP, CNRS,
Université de Toulouse, UPS-OMP, 14, avenue E. Belin, F-31400 Toulouse, France
13 Kavli Institute for the Physics and Mathematics of the Universe, The
University of Tokyo, Kashiwa, 277-8583 (Kavli IPMU, WPI), Japan
14 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033, Japan
15 The Caltech Optical Observatories, California Institute of Technology,
Pasadena, CA 91125, USA
16 Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida Angamos
601, Antofagasta, Chile
17 University of Hawai’i, Institute for Astronomy, 2680 Woodlawn Drive,
Honolulu, HI 96822, USA
18 Research Institute for Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku, Tokyo 169-8555, Japan
19 National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo,
Japan
20 European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748,
Garching, Germany
21 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
22 Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave.,
Cambridge CB3 0HE, UK
23 Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
24 University of Florida Informatics Institute, 432 Newell Drive, CISE Bldg
E251, Gainesville, FL 32611
25 Department of Astronomy, University of Florida, 211 Bryant Space Sciences
Center, Gainesville, FL 32611 USA
26 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via
Gobetti 93/3, I-40129, Bologna, Italy
27University of Bologna—Department of Physics and Astronomy "Augusto Righi"
(DIFA), Via Gobetti 93/2, I-40129 Bologna, Italy
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present $10$ main-sequence ALPINE galaxies ($\log(M/{M}_{\odot})=9.2-11.1$
and ${\rm SFR}=23-190\,{\rm M_{\odot}\,yr^{-1}}$) at $z\sim 4.5$ with optical
[O ii] measurements from Keck/MOSFIRE spectroscopy and Subaru/MOIRCS narrow-
band imaging. This is the largest such multi-wavelength sample at these
redshifts, combining various measurements in the ultra-violet, optical, and
far-infrared including [C ii]158μm line emission and dust continuum from ALMA
and H$\alpha$ emission from Spitzer photometry. For the first time, this
unique sample allows us to analyse the relation between [O ii] and total star-
formation rate (SFR) and the interstellar medium (ISM) properties via [O
ii]/[C ii] and [O ii]/H$\alpha$ luminosity ratios at $z\sim 4.5$. The [O
ii]$-$SFR relation at $z\sim 4.5$ cannot be described using standard local
descriptions, but is consistent with a metal-dependent relation assuming
metallicities around $50\%$ solar. To explain the measured dust-corrected
luminosity ratios of $\log(L_{\rm[OII]}/L_{\rm[CII]})\sim
0.98^{+0.21}_{-0.22}$ and $\log(L_{\rm[OII]}/L_{\rm
H\alpha})\sim-0.22^{+0.13}_{-0.15}$ for our sample, ionisation parameters
$\log(U)<-2$ and electron densities $\log(\rm n_{e}/{\rm[cm^{-3}]})\sim 2.5-3$
are required. The former is consistent with galaxies at $z\sim 2-3$, however
lower than at $z>6$. The latter may be slightly higher than expected given the
galaxies’ specific SFR. The analysis of this pilot sample suggests that
typical $\log(M/{\rm M_{\odot}})>9$ galaxies at $z\sim 4.5$ to have broadly
similar ISM properties as their descendants at $z\sim 2$ and suggest a strong
evolution of ISM properties since the Epoch of Reionisation at $z>6$.

###### keywords:

galaxies: high-redshift – galaxies: evolution – galaxies: ISM

††pubyear: 2021††pagerange: The ALPINE-ALMA [C ii] Survey: Investigation of
$10$ Galaxies at $z\sim 4.5$ with [O ii] and [C ii] Line Emission $-$ ISM
Properties and [O ii]$-$SFR Relation–B

##  1 Introduction

Observing and understanding the interstellar medium (ISM) of galaxies is a key
to understanding galaxy formation and evolution across cosmic time. Next to
the far-infrared providing insights into the abundance of dust and gas, the
rest-frame optical emission lines build an important basis to study the ISM.
Specifically, these lines are sensitive to the instantaneous star formation,
metal content of the gas, hydrogen densities, and ionisation rates among a
variety of other key parameters.

During the recent years, substantial progress have been made in understanding
the ISM properties of galaxies at $z=2-3$ through large spectroscopic survey
such as the Keck Baryonic Structure Survey (KBSS, Steidel et al., 2014) or the
MOSFIRE Deep Evolution Field (MOSDEF) survey (Kriek et al., 2015). Among
others, one goal of such surveys is to pinpoint the properties of the ISM via
three key parameters: the gas-phase metallicity, the electron density $n$ (in
${\rm cm^{-3}}$), and the dimensionless ionisation parameter $U$111The
ionisation parameter is essentially the ratio of ionising photon density to
hydrogen. It is also a measure for the radiation pressure feedback, as it
reflects the radiation-to-gas-pressure ratio (e.g., Yeh & Matzner, 2012)..
Through simultaneous analysis of multiple optical emission lines, it has been
found that these parameters significantly evolve between $z=2$ and local
galaxies, yet there is still some debate on the reasons for this change
(Steidel et al., 2014; Strom et al., 2017; Sanders et al., 2016; Masters et
al., 2016). The evolution of ISM properties questions the reliability (or
validity) of commonly used relations that have been calibrated to local
galaxies. One of those is the relation between [O ii] emission and star
formation rate (SFR) as calibrated to local starburst galaxies (Kennicutt,
1998). Similar to H$\alpha$, [O ii] is related to the current star formation
in galaxies. However, as a forbidden line, the excitation of oxygen is
sensitive to the abundances and ionisation state of the gas and not directly
coupled to the ionisation radiation of young stars. The relation is therefore
expected to change significantly as a function of these properties and hence
likely redshift (Kewley et al., 2004).

This motivates the interesting question of how the ISM changes at early cosmic
times and whether common local relations are still valid at high redshifts.
The latter is important to quantify, as more measurements of optical lines
will be provided soon by the James Webb Space Telescope (JWST). Of specific
interest is the redshift range $z=4-6$. This epoch is fundamental to study the
chemical and structural evolution of galaxies as it connects primordial galaxy
formation during the Epoch of Reionisation at $z>6$ with mature galaxy
evolution at the peak of cosmic SFR density at $z=2-3$. A large number of
galaxies in this redshift range have been robustly identified via Ly$\alpha$
emission and ultra-violet (UV) absorption lines thanks to large spectroscopic
surveys (e.g., Le Fèvre et al., 2015; Hasinger et al., 2018) and are
extensively studied in the UV. Unfortunately, current facilities only allow
the observation of one strong optical line (the [O ii] doublet at rest-frame
$3726\text{\AA}$ and $3729\text{\AA}$) at these redshifts, thus studies with
multiple optical lines as the ones mentioned above are not possible. So far,
[O ii] has only been observed in $4$ galaxies at $z=4-5$, with three of them
lensed (Swinbank et al., 2007; Swinbank et al., 2009; Troncoso et al., 2014;
Shapley et al., 2017). In Shapley et al. (2017), the [O ii] as well as the
detection of [Ne iii]3869 has been used to obtain the first direct measurement
of the gas-phase metallicity at $z\sim 4.5$ via optical lines. However,
further determinations of the ISM properties of these galaxies have not been
possible due to the lack of other emission lines.

In this paper, we investigate for the first time the ISM properties of
high-$z$ galaxy population via the currently largest sample of $10$ typical
main-sequence galaxies at $z\sim 4.5$ with optical [O ii] emission
measurements. We mitigate the lack of optical emission lines by entering the
far-infrared and in addition use spectral information from the rest-frame UV.
Specifically, we are combining measurements of

  * (i)

rest-UV absorption lines (obtained from Keck/DEIMOS spectroscopy).

  * (ii)

optical emission of [O ii] (obtained from Subaru/MOIRCS narrow-band imaging
and Keck/MOSFIRE spectroscopy) and H$\alpha$ (obtained from Spitzer/IRAC
colours), and

  * (iii)

singly ionised Carbon ([C ii]158μm at $158\,{\rm\mu m}$, hereafter denoted as
[C ii]) and $150\,{\rm\mu m}$ dust continuum from ALMA.

The galaxies, located in the COSMOS field (Scoville et al., 2007), are all
part of the ALPINE survey (Béthermin et al., 2020; Faisst et al., 2020a; Le
Fèvre et al., 2020), which currently provides the largest multi-wavelength
post-reionisation dataset with observations from UV to far-infrared. Thanks to
the indirect measurement of metallicity from UV absorption lines as well as
additional emission lines such as H$\alpha$ and [C ii], we can derive strong
constraints on key ISM parameters such as metallicity, electron density, and
ionisation parameter. Furthermore, the knowledge of the total SFR of the
galaxies enables a first calibration of the relation between [O ii] and SFR at
$z\sim 4.5$.

This paper is organised as follows: In Section 2 we present the spectroscopic
and narrow-band observations of [O ii], constraints on the metal content from
rest-UV absorption lines, and the measurement of H$\alpha$ emission from
Spitzer. In the following section, we discuss the relation between total SFR
and [O ii] emission (Section 3.1) and the [O ii]/[C ii] and [O ii]/H$\alpha$
luminosity ratio in the context of the ISM properties of our galaxies (Section
3.2). We conclude in Section 4.

Throughout this work, we assume a $\Lambda$CDM cosmology with $H_{0}=70\,{\rm
km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{\Lambda}=0.70$, and $\Omega_{\rm m}=0.30$.
All magnitudes are given in the AB system (Oke, 1974) and stellar masses and
SFRs are normalised to a Chabrier (2003) initial mass function (IMF) unless
noted otherwise.

##  2 Sample and Data

The $10$ galaxies at $z\sim 4.5$ reside in the Cosmic Evolution Survey
(COSMOS, Scoville et al., 2007) field and are a subset of the ALPINE survey
(Béthermin et al., 2020; Faisst et al., 2020a; Le Fèvre et al., 2020), which
covers in total $118$ galaxies in COSMOS and the Extended Chandra Deep Field
South (ECDFS, Grogin et al., 2011; Giacconi et al., 2002). The galaxies are
covered by a wealth of observations at rest-frame UV (obtained by ground-based
telescopes and the Hubble space telescope), optical (obtained by the Spitzer
space telescope), and far-infrared (obtained by ALMA) wavelengths. The
measurement of the $\sim 150\,{\rm\mu m}$ dust continuum and [C ii] line are
detailed in Béthermin et al. (2020). Physical properties including stellar
mass, SFR, dust attenuation (E(B-V)), and UV continuum slope ($\beta$) are
derived from the rest-frame UV to optical photometry by using the spectral
energy distribution (SED) fitting code LePhare (Arnouts et al., 1999; Ilbert
et al., 2006). Bruzual & Charlot (2003) composite stellar population models of
different ages, metallicities, star-formation histories, and dust attenuation
are assumed for the fitting as described in detail in Faisst et al. (2020a).
The galaxies are typical galaxies for their cosmic times and lie on the
expected $z\sim 4.5$ main-sequence (Figure 1). The total SFR in this case is
computed from the UV and far-infrared continuum or alternatively from the [C
ii]$-$SFR relation (Schaerer et al., 2020), which was derived from the
IRX$-\beta$ relation for galaxies without far-infrared continuum detection
(Fudamoto et al., 2020). The measurements coincide closely with two
parameterisations of the main-sequence at $z\sim 4.5$ from Speagle et al.
(2014) and Schreiber et al. (2015). The latter is extrapolated from Herschel
observations at $z\sim 3.5$.

###  2.1 A Unique Sample of $z\sim 4.5$ Galaxies with [O ii] Measurements

In this study, we present the largest galaxy sample with [O ii] measurement at
$z\sim 4.5$. For one of the $10$ galaxies, we obtain the [O ii] measurement
from spectroscopy while for the others [O ii] is obtained from narrow-band
photometry. In the following, we detail the measurements in both cases.

Figure 1: Relation between stellar mass and total SFR for the $10$ galaxies
with [O ii] and [C ii] measurements studied in this work (red with far-
infrared continuum detection; blue without). The total SFR in this case is
computed by the UV$+$far-infrared continuum (if available) or [C ii] alone
according to the Schaerer et al. (2020) relation (see Section 3.1.1).
DC_881725, for which a spectroscopic measurement of [O ii] exists, is shown as
a yellow star. Other ALPINE galaxies with [C ii] measurements are shown in
grey. We show the main-sequence (MS) parameterisations at $z\sim 4.5$ from
Speagle et al. (2014) and Schreiber et al. (2015), the latter extrapolated
from Herschel observations at $z\sim 3.5$, as well as from ALPINE at $z\sim
4.5$ and $z\sim 5.5$ (Khusanova et al., 2020).  Figure 2: Keck/MOSFIRE slit
layout on DC_881725. Upper panel: ACS/F814W image of DC_881725 overlaid with
contours (1, 3, 5, and 10$\sigma$) of far-infrared continuum (red) and [C ii]
emission (blue) observed by ALMA. The MOSFIRE slit is indicated as the cyan
box. The F814W image (Koekemoer et al., 2007) is smoothed by a Gaussian to
represent the seeing (FWHM$\,=0.4\arcsec$) during the Keck observations.
Bottom panel: Two-dimensional Keck/MOSFIRE spectrum in K-band with [O ii]
emission indicated. The orange box visualises the extraction window ($6\,{\rm
pixels}$ or $1.08\arcsec$).

####  2.1.1 DC_881725: Spectroscopic [O ii] Measurement at $z=4.58$

DC_881725 was observed at rest-frame optical wavelengths with the Multi-Object
Spectrometer for Infrared Exploration (MOSFIRE; McLean et al., 2010, 2012) on
the Keck I telescope. This galaxy was originally a filler target part of a
MOSFIRE program to observe a $z\sim 2.2$ overdensity (PI: Nick Scoville).
Hence the galaxy was basically randomly picked from the ALPINE sample with the
only requirement of having an [O ii] flux (estimated from the local SFR vs. [O
ii] relation; Kennicutt, 1998) that is bright enough to be detected by the
parent observations. The observations cover the optical [O ii] doublet at
rest-frame wavelengths of $3726\,{\rm\text{\AA}}$ and $3729\,{\rm\text{\AA}}$.
This is the first main-sequence galaxy at $z\sim 4.5$ with spectroscopic [O
ii] measurement as well as observations of the far-infrared [C ii] line
emission and continuum at rest-frame $150\,{\rm\mu m}$.

The observations with MOSFIRE were carried out on 2019 January 14 in $K$-band
($1.93-2.40\,{\rm\mu m}$) under clear sky conditions with an average seeing of
0.4$\arcsec$ and airmass of $1.1-1.2$. The galaxy was observed in MCSD mode
with 16 reads, ABBA dither pattern, and a slit width of $0.7\arcsec$ for a
total on-target exposure time of $1.2\,{\rm h}$ ($24\times 3\,{\rm min}$). The
resolution of the spectrum corresponds to $R\sim 3600$, which would allow us
theoretically to barely resolve the [O ii] doublet if it were at higher
signal-to-noise. Figure 2 shows DC_881725 in the Hubble ACS/F814W COSMOS
mosaics (Koekemoer et al., 2007) (convolved to match the resolution of the
ground-based seeing at the time of the Keck observations) with the MOSFIRE
slit overlaid. The far-infrared and [C ii] emission observed with ALMA are
superimposed as contours. The lower panel shows the two-dimensional spectrum
and the window used to extract the one-dimensional spectrum.

The data was reduced and wavelength calibrated using the current MOSFIRE data
reduction pipeline222https://keck-
datareductionpipelines.github.io/MosfireDRP/. Sky lines were used by the
pipeline to perform a wavelength calibration. The absolute flux calibration of
the spectrum is detailed in Appendix A. In brief, a magnitude $17$ standard
star from the 2MASS star catalogue was included in the mask to derive a
wavelength-dependent absolute flux calibration of the extracted one-
dimensional spectrum.

The one-dimensional spectrum of DC_881725 is extracted in a spatial window of
$6$ pixels (corresponding to $1.08\arcsec$ at a pixel scale of
$0.18\arcsec/{\rm px}$) around the centre of the identified [O ii] emission
line. The extracted spectrum in $\rm e^{-}\,s^{-1}$ is converted to $\rm
erg\,s^{-1}\,cm^{-2}\,\text{\AA}^{-1}$ using the wavelength dependent
normalisation described in Appendix A. The extracted spectrum is shown in
Figure 3 (blue dashed line) around the expected location of the [O ii]
doublet.

The [O ii] line flux is derived by fitting a single Gaussian with variable
mean, $\sigma$, and total flux to the one-dimensional spectrum (Figure 3,
black solid line). We also tried to fit a double Gaussian to account for the
doublet nature of the line. However, given the low signal-to-noise and
resolution, no robust fit was obtained. The uncertainties of the total flux
are determined using $500$ Monte Carlo samples for each of which the fluxes
are changed according to their uncertainties (assumed Gaussian) obtained from
the variance of the spectrum. We find a total [O ii] line flux of
$3.00^{+0.47}_{-0.46}\times 10^{-17}\,{\rm
erg\,s^{-1}\,cm^{-1}\,\text{\AA}^{-1}}$ and measure an [O ii] redshift of
z${}_{[\text{O\,{ii}}]}~{}=~{}4.5793$. The latter matches closely the redshift
determined from [C ii] ($\sim 123\pm 30\,{\rm km\,s^{-1}}$ blue-shifted w.r.t.
[O ii]) but is significantly red-shifted with respect to Ly$\alpha$ by $328\pm
30\,{\rm km\,s^{-1}}$.333The velocity shifts quoted assume equal flux of the
two blended [O ii] lines. We estimate an uncertainty of $\pm 30\,{\rm
km\,s^{-1}}$ assuming [O ii]λ3729/[O ii]λ3726 ratios between $0.35$ and $1.5$,
corresponding to the high and low electron density regime (Osterbrock, 1974).
These quantities as well as the [O ii] luminosity, L${}_{[\text{O\,{ii}}]}$,
are listed in Table 1 along with other properties measured from the ancillary
data.

Figure 3: Extracted 1D MOSFIRE spectrum (blue) of DC_881725 at the location of
the [O ii] emission line doublet (indicated by the vertical dot-dashed lines).
The Gaussian fit (assuming blended lines) is shown in black.

####  2.1.2 [O ii] Measurements from Narrow-Band Imaging

We complement DC_881725 with $9$ additional galaxies at $z\sim 4.5$ which have
[O ii] luminosities derived from narrow-band imaging taken with the Multi-
Object Infrared Camera and Spectrograph (MOIRCS; Ichikawa et al., 2006; Suzuki
et al., 2008) on the Subaru Telescope. The observations were part of the
Charting Cluster Construction with VUDS (Le Fèvre et al., 2015) and ORELSE
(Lubin et al., 2009) survey (C3VO, Lemaux et al., 2020; Shen et al., 2021). In
brief, the $9$ ALPINE galaxies fall in the footprint of three pointings
targeting the massive proto-cluster PCl J1001+0220 at $z\sim 4.57$ (Lemaux et
al., 2018) and its surrounding in the COSMOS field. Similar to DC_881725, they
are therefore randomly selected from the ALPINE sample but without imposed
restrictions on the [O ii] flux. The $9$ galaxies may be part of the proto-
cluster structure, however, as shown by studies at lower redshifts (e.g.,
Darvish et al., 2015), we do not expect significant differences in their
properties compared to field galaxies.

The observations were executed between February 2 2020 and January 31 2021,
using NB2071 ($2.043-2.097\,{\rm\mu m}$) and NB2083 ($2.056-2.110\,{\rm\mu
m}$) narrow-band filters, under seeing that ranged from $\sim 0.4-0.8\arcsec$
and conditions that varied from light cirrus to photometric. The total
effective integration time across all three pointings and two filters was
approximately $18$ hours, which was split into $150{\rm s}$ individual
exposures. A standard circular dither pattern with a set of 10 dithers was
adopted.

All raw data were reduced with the IRAF based reduction pipeline MCSRED2
(Tanaka et al., 2011), which performed flat-fielding, masking objects, and sky
subtraction. Astrometry calibration was performed by SCAMP (Bertin, 2006) for
every individual exposure. For each filter and chip, a final narrow-band image
was stacked using SWarp (Bertin et al., 2002). The $5\sigma$ limiting
magnitudes of the images have a median of $23.1\,{\rm mag}$ and in a range of
$22.7-23.3\,{\rm mag}$, where $\sigma$ is measured from the flux scatter of
randomly distributed $2\arcsec$ aperture. Note that some exposures were
shallower due to the filter wheel occultation. Finally, photometric
calibration was performed to the Ks$-$band image from the UltraVISTA DR4
(McCracken et al., 2012a) by selecting a set of bright but unsaturated point
sources as the reference stars.

For the source detection and extraction on each narrow-band image, SExtractor
(Bertin & Arnouts, 1996) was used in dual-image mode using the much deeper
($24.8\,{\rm mag}$ at $5\sigma$) Ks$-$band image as the detection image.444We
set detection parameters of DETECT_MINAREA $=5$, DETECT_THRESH $=1.5$, and
ANALYSIS_THRESH $=1.5$. The latter two are relative ($\sigma$) thresholds. The
point-spread functions (PSFs) of the narrow-band images were degraded to match
the Ks-band image. In details, the PSFs for the narrow-band and Ks-image were
derived using the PSFex code (Bertin, 2011). Subsequently, the Photutil
package with a Split Cosine Bell window was used to generate a matching kernel
between two PSFs. The uncertainties on the fluxes estimated by SExtractor were
scaled to account for correlated pixel noise following the method of Pelliccia
et al. (2021).

The following system of two equations involving the narrow-band and the
underlying Ks-band photometry was solved for the continuum ($f^{\lambda}_{\rm
cont}$) and [O ii] line flux ($f^{\lambda}_{\rm line}$) at wavelength
$\lambda$ of the [O ii] line (c.f. Hu et al., 2019):

$\bar{f}^{\rm\lambda}_{\rm NB/Ks}=\frac{\int\left(f^{\lambda}_{\rm
line}+f^{\lambda}_{\rm cont}\right)T^{\lambda}_{\rm NB/Ks}d\lambda}{\int
T^{\lambda}_{\rm NB/Ks}d\lambda}.$ (1)

In the above equation, $\bar{f}^{\lambda}_{\rm NB}$ and
$\bar{f}^{\lambda}_{\rm Ks}$ denote the detected flux densities in each
narrow-band and the Ks-band, and $T^{\lambda}_{\rm NB}$ and $T^{\lambda}_{\rm
Ks}$ are the corresponding filter transmission functions. In the calculation,
we assumed an [O ii] line profile resembling a $\delta$-function in each
narrow-band filter and a flat continuum profile. As the observed line width of
the [O ii] doublet ($29.01_{-4.64}^{+5.33}$Å, see Figure 3) is much narrower
than the width of the narrow-band filter ($\sim 270\,{\rm\text{\AA}}$), so the
[O ii] line profile does not affect the result of this calculation. The
uncertainties of the [O ii] fluxes were obtained using Monte-Carlo iterations
for each of which the narrow-band and Ks-band fluxes were drawn from a
Gaussian distribution with a width corresponding to their associated
uncertainties. The [O ii] flux uncertainty is then defined as the 16th and
84th quantiles of the resulting [O ii] flux distribution. Finally, an inverse-
variance weighting scheme was adopted to combine fluxes obtained from the two
narrow-band filters, with weights defined as $W_{i}=T_{i}^{2}/\sigma_{i}^{2}$,
where $T_{i}$ is the narrow-band transmission curve at the [C ii] redshift and
$\sigma_{i}$ is the narrow-band flux uncertainty.

Table 2 lists the measured [O ii] luminosities as well as other properties
derived from the ancillary data for these $9$ galaxies.

###  2.2 Estimate of Metal-Enrichment from Rest-Frame UV Absorption Lines

Commonly, the gas-phase metal content of galaxies is measured by flux ratios
of bright optical lines such as [O ii], [O iii], and H$\alpha$ (e.g., Pettini
& Pagel, 2004; Maiolino & Mannucci, 2019). Recently, thanks to ALMA, also far-
infrared emission lines such as [N ii]205μm, [O ii]88μm, and [C ii] are used
to estimate the gas-phase metal content of high-redshift galaxies (e.g.,
Vallini et al., 2021; Yang & Lidz, 2020; Jones et al., 2020; Pavesi et al.,
2019; Croxall et al., 2017). The former are not accessible at these redshifts
currently (except [O ii]), while the latter are not available for our galaxies
as their observations are costly (except [C ii]) or only possible for galaxies
at $z>6$ (in the case of [O ii]88μm).

Nonetheless, we can estimate the metal content of our galaxies from the
exquisite rest-frame UV spectra in an indirect way. Specifically, the
strength, i.e., equivalent-width (EW; note that EWs of absorption lines are
defined positive), of rest-frame UV absorption lines correlates with metal
content (e.g., Leitherer et al., 2011). This correlation has different origins
including the evolutionary stage of stellar populations, metal-dependent winds
impacting the width of absorption lines, or dust extinction. The technique has
already been applied to similar main-sequence galaxies at $z\sim 4-6$ (Faisst
et al., 2016b; Ando et al., 2007) and has been tested empirically at lower
redshifts (Faisst et al., 2016b). The S/N of individual rest-UV spectra of our
galaxies is too low to estimate their metal content. We therefore perform
median stacking the $10$ spectra before we compute the EWs of several
absorption complexes around Si iii ($\sim 1300\,{\rm\text{\AA}}$), C ii ($\sim
1335\,{\rm\text{\AA}}$), Si iv ($\sim 1400\,{\rm\text{\AA}}$), and C iv ($\sim
1550\,{\rm\text{\AA}}$). The uncertainties are computed via Monte-Carlo
iterations including the spectral variances. We then use the relations between
absorption line EW and metallicity as calibrated via galaxies in the local
universe as well as $z\sim 2$ and $z\sim 3$ in Faisst et al. (2016b).

Figure 4: Top panel: Flux-normalized stacked UV spectrum of the $10$ galaxies
studied in this work. Prominent spectral features are indicated. The light
area shows the $1\sigma$ scatter determined from individual variances of the
spectra. Bottom panels: The EWs of rest-frame UV absorption lines provide an
indirect measure of the metal content. The blue swaths show the measured EWs
of each absorption complex from a stack of all $10$ galaxies in our sample.
The grey swath shows the calibration between EW and metallicity from Faisst et
al. (2016b). The vertical dashed lines mark $0.1$, $0.5$, and $1\,{\rm
Z_{\odot}}$.

The stacked spectrum and the results from this analysis are shown in Figure 4.
The blue horizontal swaths show the measured EWs of each absorption complex
with uncertainty and the grey swaths show the calibration derived in Faisst et
al. (2016b). Looking at the individual absorption complexes suggests average
metallicites between $10\%$ and $50\%$ of solar, except for Si iii, which is
consistent with solar metallicity. We note that even for the stack the S/N is
relatively low, which is likely the cause for the apparent differences between
the individual absorption complexes. Combining the probability distribution
functions of all absorption complexes, we find an average metallicity of
$12+\log{(\rm O/H})=8.41^{+0.31}_{-0.54}$. This corresponds to roughly $\sim
50\%$ of the solar metallicity555We assume a solar oxygen abundance of
${\rm\log(O/H)+12}=8.76$ according to the “galactic concordance abundances”
(Nieva & Przybilla, 2012; Nicholls et al., 2017), which is close to the
primordial solar abundance (Asplund et al., 2009).. Such values are consistent
with the expected metallicity based on the stellar masses of our galaxies at
these redshifts (Faisst et al., 2016b; Ando et al., 2007).

###  2.3 Photometric Measurement of H$\alpha$ from Spitzer

The coverage of our galaxies by Spitzer allows us to photometrically estimate
their H$\alpha$ emissions via the [$3.6\,{\rm\mu m}$]$-$[$4.5\,{\rm\mu m}$]
colours. This method has been successfully used by several studies very early
on (e.g., Shim et al., 2011; Stark et al., 2013; Mármol-Queraltó et al., 2016;
Rasappu et al., 2016) and it is shown to result in photometric H$\alpha$
measurements that are statistically consistent with spectroscopy (e.g., Faisst
et al., 2016a). The detailed derivation of this measurement for the ALPINE
galaxies is described in Faisst et al. (2020a). The luminosities are corrected
for dust in the same way as [O ii]. As an additional check, we compare our
H$\alpha$ measurement to the EL-COSMOS catalogue (Saito et al., 2020), which
contains H$\alpha$ emission line predictions from SED fitting for all galaxies
from the Laigle et al. (2016) COSMOS catalogue. We find good agreement within
a factor of $2-5$ (see Appendix B). We find a similar agreement when comparing
the [O ii] luminosities provided in the EL-COSMOS catalogue to our
spectroscopic and narrow-band imaging measurements, which verifies the
reliability of the catalogue.

##  3 Discussion

Figure 5:  Left panels: Histograms of the $L_{{\rm OII}}/L_{{\rm IR}}$ (left)
and $L_{{\rm OII}}/L_{{\rm CII}}$ (right) ratios for the $10$ galaxies. The
observed ratios for DC_881725 are indicated with arrows. The filled histograms
show observed ratios (not corrected for dust) and the dashed histograms show
dust-corrected ratios (see text for details). Right panel: Relation between [O
ii] emission and total SFR. The red (blue) symbols show galaxies with
(without) far-infrared detections from ALMA. In the former case, the total SFR
is ${\rm SFR_{UV+IR}}$; in the latter case, the total SFR is derived from [C
ii] emission. The filled lighter symbols show observed [O ii] luminosities
(not corrected for dust) while the empty darker symbols show dust corrected [O
ii] luminosities. For far-infrared detected galaxies, the dust correction is
derived from the IRX value directly. For the others, the relation between
$\beta$ and IRX is used (see Section 3.1.2). Note that some galaxies have
little dust, in which cases the filled and empty symbols coincide. DC_881725
is shown as star. The purple “plus” denotes the median dust corrected [O ii]
luminosity. We also show different empirical [O ii]$-$SFR relations from
Kennicutt (1998, long dashed line) and the updated version from Kewley et al.
(2004, solid line), as well as the metallicity dependent models from Kewley et
al. (2004, dotted, short dashed, and dot-dashed lines for solar, $50\%$ solar,
and $20\%$ solar metallicity).

###  3.1 The [O ii]$-$SFR relation at $z\sim 4.5$

We first investigate whether the local relation between [O ii] emission and
total star formation is still valid at $z\sim 4.5$. For this we make use of
our multi-wavelength measurements of the total SFR from UV and far-infrared
data.

####  3.1.1 Total SFRs Derived from UV$+$Far-Infrared Continuum and [C ii]
Emission

In order to relate [O ii] emission to total star formation, we have to provide
measurements of the latter that are as robust as possible. Our sample allows
us to do this via the combination of measurement of UV and far-infrared
continuum as well as [C ii] emission.

For galaxies detected in far-infrared continuum ($5$ out of $10$), we compute
the total SFR as the sum of UV and dust obscured star formation. Specifically,
we make use of the relations detailed in Kennicutt (1998) (see also Kennicutt
& De Los Reyes, 2021),

${\rm SFR_{UV}\,(M_{\odot}\,yr^{-1})}=0.79\times 10^{-28}L_{\rm\nu}({\rm
erg\,s^{-1}\,Hz^{-1}})$ (2)

and

${\rm SFR_{IR}\,(M_{\odot}\,yr^{-1})}=2.54\times 10^{-44}L_{\rm IR}({\rm
erg\,s^{-1}}).$ (3)

These SFR values have already been converted from a Salpeter (1955) to a
Chabrier (2003) IMF by division of a factor of $1.77$ (in linear scale). For
details on the derivation of the UV and far-infrared luminosity we refer to
Faisst et al. (2020a) and Béthermin et al. (2020), respectively. Note that the
derivation of $L_{\rm IR}$ depends strongly on the shape of the far-infrared
SED, with specifically the dust temperature playing an important role (e.g.,
Faisst et al., 2017). Here, we make use of the stacked far-infrared SED
derived from Herschel, SCUBA, and ALMA photometry of $z=4-6$ galaxies in
COSMOS with similar SFR and stellar masses as the ALPINE galaxies (see
Béthermin et al., 2020). The dust temperature of this stack is $43\pm 5\,{\rm
K}$, which is consistent with the individual dust temperatures measured for
$4$ ALPINE galaxies with sufficient far-infrared coverage from ALMA (Faisst et
al., 2020b).

For the remaining five galaxies without far-infrared continuum detection, we
use the relation between [C ii] line emission and total SFR as presented by
Schaerer et al. (2020),

$\log({\rm SFR_{[CII]}/[M_{\odot}\,yr^{-1}]})=\frac{\log({\rm
L_{[CII]}/L_{\odot}})-6.61}{1.17}.$ (4)

The above relation is based on total SFRs of all ALPINE galaxies (with limits
from [C ii] non-detection properly taken into account). For galaxies without
far-infrared detection, the IRX$-\beta$ relation (Fudamoto et al., 2020) is
used to derive their far-infrared luminosities and total SFRs.

The total SFR for the $9$ narrow-band detected galaxies are listed in Table 2.
The total SFRs of DC_881725 derived from UV and far-infrared as well as [C ii]
for comparison are listed in Table 3. Note that the [C ii] and far-infrared
(if available) derived SFRs are very comparable as the former have been
calibrated by the latter in Schaerer et al. (2020).

We can also compare different [C ii]$-$SFR relations derived in De Looze et
al. (2014) with our total SFRs derived for the $5$ galaxies with far-infrared
continuum detection. These relations have been calibrated using low-
metallicity dwarfs ($12+\log({\rm O/H})\sim 7.1-8.4$), starbursts (and H ii
regions), galaxies with an active galactic nucleus (AGN), ultra-luminous
infrared galaxies (ULIRGs), and high-redshift galaxies (mostly at $z\sim
1-3$). We find that (i) the [C ii]$-$SFR relation of local metal-poor dwarfs
underestimates the total SFR by $0.43\pm 0.13\,{\rm dex}$, while (ii) the
relation derived from ULIRGs overestimates the total SFRs by $0.77\pm
0.15\,{\rm dex}$. The other calibrations are roughly consistent with our
derived values within $1\sigma$ uncertainties. Similar results are found for
the remaining $5$ galaxies with SFRs derived from [C ii] using the Schaerer et
al. (2020) relation. This SFR comparison suggests that our galaxies are not
comparable to either local low-metallicity dwarfs or local ULIRGs. This is not
surprising as rest-frame UV absorption line spectroscopy suggests that our
galaxies are rather metal enriched (see Section 2.2). Furthermore, their far-
infrared luminosities are about an order of magnitude lower than those of
local ULIRGs included in De Looze et al. (2014) ($L_{\rm IR}\sim 3\times
10^{12}\,{\rm L_{\odot}}$). Table 3 shows a detailed list of SFRs derived by
the different [C ii]$-$SFR relations for the example of DC_881725.

####  3.1.2 The [O ii]$-$SFR relation at $z\sim 4.5$

With measured robust total SFRs for our galaxies, we can now relate these
values to the [O ii] emission.

The left panels of Figure 5 show the histograms of the $L_{{\rm OII}}/L_{{\rm
IR}}$ and $L_{{\rm OII}}/L_{{\rm CII}}$ luminosity ratios. We find observed
median ratios of $\log(L_{\rm[OII]}/L_{\rm[CII]})~{}=~{}0.39^{+0.35}_{-0.32}$
and $\log(L_{\rm[OII]}/L_{\rm IR})=-2.45^{+0.23}_{-0.09}$. The corresponding
dust-corrected ratios are
$\log(L_{\rm[OII]}/L_{\rm[CII]})~{}=~{}0.98^{+0.21}_{-0.22}$ and
$\log(L_{\rm[OII]}/L_{\rm IR})=~{}-1.64^{+0.23}_{-0.13}$, respectively.

With [O ii] being an optical line, it has to be corrected for the effect of
dust attenuation. We compute [O ii] dust correction factors using the stellar
continuum dust attenuation $E_{s}{(B-V)}$ values via

$f_{\rm corr}=10^{0.4E_{s}{(B-V)}k_{\lambda}/f},$ (5)

where $k_{\rm\lambda}$ is the reddening curve with
$\lambda=3727\,{\rm\text{\AA}}$ and we assume a differential dust attenuation
factor between stellar continuum and nebular emission of $f=0.44$ (Calzetti et
al., 2000)666The “$f-$factor” is the differential dust attenuation between
nebular emission and stellar continuum and is defined as
$f={E_{s}(B-V)}/{E_{n}(B-V)}$.. The $E_{s}{(B-V)}$ values are estimated by the
following procedure. For galaxies that are detected in far-infrared continuum,
we used their IRX ratio ($\equiv\log(L_{\rm IR}/L_{\rm UV})$) to estimate the
dust attenuation at rest-frame $1600\,{\rm\text{\AA}}$ ($A_{\rm 1600}$) via
the relation given in Hao et al. (2011)777Note that this parameterization is
consistent with earlier derivations (e.g., from Buat et al., 2005; Burgarella
et al., 2005; Kong et al., 2004; Meurer et al., 1999).,

$A_{\rm 1600}=2.5\log(1+0.46\times 10^{\rm IRX}).$ (6)

The nebular $E(B-V)$ values, $E_{n}{(B-V)}$, are then estimated using the
relation $A_{\rm 1600}=4.39\times E_{n}{(B-V)}$ (Calzetti et al., 2000). Other
parameterisations (e.g., Reddy et al., 2015) lead to values that are different
by less than $20\%$. For galaxies that are not detected in far-infrared
continuum, we derive the stellar $E_{s}{(B-V)}$ values from the UV continuum
photometry. Specifically, we explore the values from SED fitting directly as
well as derived from the UV continuum slope ($\beta$) using the dust
parameterisations by Calzetti et al. (2000) and Reddy et al. (2015). As an
additional comparison, we also derive IRX values directly from the $\beta$
slopes using the IRX$-\beta$ relation fit for ALPINE galaxies at $z\sim 4.5$
in Fudamoto et al. (2020). We find that all four methods agree within
$0.05\,{\rm mag}$, which results in $<30\%$ differences in the dust factors
for [O ii]. All in all, we find [O ii] dust attenuation factors (linear)
between $1-5$, except in one case where we find a value of $f_{\rm corr}\sim
10$. The different $k_{\rm\lambda}$ parameterisations used above result in
variations of $<40\%$ (leading to uncertainties of $<0.14\,{\rm dex}$ in [O
ii] luminosity). The differential dust attenuation between nebular emission
and stellar continuum is by far the largest contribution to the total
uncertainty. There is observational evidence that the $f-$factor of $z>2$
galaxies is closer to $0.7$ than the locally measured value of $0.44$ (Kashino
et al., 2017; Faisst et al., 2019; Rodriguez-Munoz et al., prep). In the case
of $f=0.7$, the dust attenuation factors would decrease by a factor of $1.5$
($0.18\,{\rm dex}$ change in luminosity) on average for our sample.

Finally, the right panel of Figure 5 relates the observed (solid symbols) and
intrinsic (dust-corrected, empty symbols) [O ii] luminosities to the total
star formation (as described in Section 3.1.1). The far-infrared detected
galaxies (dust correction from the IRX ratio) are shown as triangles, the
remaining galaxies are shown as circles. DC_881725 is denoted with a star and
the median of the dust-corrected [O ii] luminosity and total SFR is marked as
a purple cross. The measurement uncertainties in [O ii] luminosities are large
compared to the measurement uncertainties in total SFR (on the order of
$0.1\,{\rm dex}$). However, we expect systematic errors in the total SFRs to
be a factor of $2-3$ larger given the uncertainties in the relation between [C
ii] and SFR.

Along with our data, we show different parameterisations of the [O ii]$-$SFR
relation from the literature. We include the original relation by Kennicutt
(1998)888Converted to a Chabrier (2003) IMF.,

${\rm SFR_{[\text{O\,{ii}}]}^{\rm K98}\,(M_{\odot}\,yr^{-1})}=(0.79\pm
0.23)\times 10^{-41}L_{\rm[OII]}({\rm erg\,s^{-1}}),$ (7)

as well as an updated version provided by equation 4 in Kewley et al.
(2004)999Converted to a Chabrier (2003) IMF.

${\rm SFR_{[\text{O\,{ii}}]}^{\rm K04}\,(M_{\odot}\,yr^{-1})}=(3.72\pm
0.93)\times 10^{-42}L_{\rm[OII]}({\rm erg\,s^{-1}}).$ (8)

As pointed out by these authors, this relation may depend significantly on the
gas-phase metallicity of the galaxies. We therefore also show their
theoretical relations for solar, half-solar, and one-fifth solar metallicity
obtained by equation 10 in Kewley et al. (2004)101010Converted to a Chabrier
(2003) IMF.

${\rm SFR_{[\text{O\,{ii}}]}^{\rm
K04}}(Z)\,{\rm(M_{\odot}\,yr^{-1})}=\frac{4.46\times 10^{-42}L_{\rm[OII]}({\rm
erg\,s^{-1}})}{(-1.75\pm 0.25)[\log({\rm O/H})+12]+(16.73\pm 2.23)}.$ (9)

We find that the original Kennicutt (1998) relation would significantly
overestimate [O ii]-derived SFRs by factors of $3-5$ if applied to dust-
corrected [O ii] luminosities. Note that the relation in Equation 7 required
dust corrections of [O ii] at the H$\alpha$ rest-frame wavelength (see
description in Kennicutt, 1998). However, at the dust attenuation values of
our galaxies, this would reduce the dust correction factors by less than
$40\%$ (and similarly the overestimation), hence cannot account for the
discrepancy. A much better estimate of the total SFR is provided by the
updated and metal-dependent [O ii]$-$SFR relations by Kewley et al. (2004)
(detailed numbers for the case of DC_881725 are listed in Table 3). The large
uncertainties, mainly due to the unknown differential dust attenuation factors
but also the uncertain total SFR, do not allow us to distinguish relations for
different metallicities at significance. As shown by the purple cross denoting
the median value of the dust corrected [O ii] luminosities and total SFRs,
sub-solar metallicities are the most likely choice but solar metallicites
cannot be excluded given the combination of measurement and systematic
uncertainties. Also, note that an increase of the $f-$factor from local $0.44$
to $\sim 0.7$ would reduce the dust correction for the [O ii] luminosity by
$<0.2\,{\rm dex}$ (see above), which would keep our observations consistent
with the (Kewley et al., 2004) relation at sub-solar metallicity.

Figure 6: Predictions of the emission line ratio $\log(L_{\rm[OII]}/L_{\rm
H\alpha})$ (left) and $\log(L_{\rm[OII]}/L_{\rm[CII]})$ (right) from Cloudy
simulations as a function of ionisation parameter ($U$) for $0.1\,{\rm
Z_{\odot}}$ (dotted lines), $0.5\,{\rm Z_{\odot}}$ (dashed lines), and
$1\,{\rm Z_{\odot}}$ (solid lines) metallicities as well as hydrogen densities
of $\log(n/[{\rm cm^{3}}])=0.5$ (blue) and $\log(n/[{\rm cm^{3}}])=2$ (red).
The different panels show different assumptions for the background stellar
population (see text): burst of star formation (panels $A$), constant star
formation ($B$), and best-fit SED for DC_881725 ($C$). Our observed (dust-
corrected) line ratios are shown as horizontal lines. In the case of
$\log(L_{\rm[OII]}/L_{\rm[CII]})$ (right panels), we show DC_881725 in black
(with the $1\sigma$ uncertainty shown as grey swath) and the $9$ narrow-band
detected galaxies as grey lines. For $\log(L_{\rm[OII]}/L_{\rm H\alpha})$
(left panels), we only show the median line ratio marginalised over all the
uncertainties of the individual galaxies ($1\sigma$ uncertainties shown by the
grey swath). Note that the $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ ratio is
largely insensitive to the hydrogen density, hence provides a good constraint
on $U$.

###  3.2 Constraints on ISM Properties from Cloudy Analysis

In this section, we compare the $\log(L_{\rm[OII]}/L_{\rm[CII]})$ and
$\log(L_{\rm[OII]}/L_{\rm H\alpha})$ luminosity ratios from measurements
presented in Sections 2.1 and 2.3 to theoretical values obtained by Cloudy
(C17.02; Ferland et al., 2017) to study key parameters of the ISM of our
galaxies.

For the Cloudy simulation, we assume gas with plain-parallel geometry
including Orion-type grains as well as Polycyclic Aromatic Hydrocarbons
(PAHs). We ran models for gas clouds with electron densities of $\log(n_{\rm
H}/{\rm cm^{-3}})\,=\,[0.5,2]$, gas-phase metallicities of
$Z=[0.1,0.5,1.0]\,{\rm Z_{\odot}}$, and ionisation parameters between
$-3.0<\log(U)<-0.5$ in steps of $0.5\,{\rm dex}$. The stopping criteria of the
simulation is crucial, especially for computing the [C ii] emission, which can
originate from low-density warm ISM. We use a stopping point at $A_{\rm
v}=100\,{\rm mag}$ and we found that lower values such as $A_{\rm v}=10\,{\rm
mag}$ do not have a significant impact on the our results. We also
investigated the dependence of our results on different star formation
histories for the underlying stellar population. For this, we assumed three
different simple models: (A) a starburst with an age of $100\,{\rm Myr}$; (B)
a constant star formation history; (C) the best-fit SED of DC_881725. The
first two are produced by the BPASS v2.0 models (Eldridge et al., 2017;
Stanway & Eldridge, 2018), assume $10\%$ solar stellar metallicity, and
include binary-star evolution. The latter is derived from broad-band
photometry using CIGALE (Burgarella et al., 2005; Noll et al., 2009; Boquien
et al., 2019). The results are shown in Figure 6. The panel labels indicate
the assumption of the underlying stellar population (A, B, or C,
respectively). The left and right columns show the result for
$\log(L_{\rm[OII]}/L_{\rm H\alpha})$ and $\log(L_{\rm[OII]}/L_{\rm[CII]})$,
respectively. In the following, we compare the models to the observations in
more detail.

####  3.2.1 The $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ luminosity ratio

Let us first have a look at the $\log(L_{\rm[OII]}/L_{\rm H\alpha})$
luminosity ratio. As seen on the left panels in Figure 6, this ratio shows a
strong negative correlation with the ionisation parameter (radiation pressure)
$U$, however is largely independent of the electron density and the changes
with metallicity are small compared to the accuracy of our observations. Thus
the $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ luminosity ratio allows us to put
constraints on the ionisation parameter $U$. The trends are slightly flatter
for a constant star formation history or using the best-fit SED. However,
overall the impact of the three different star formation histories is
negligible at the current accuracy of our measurements.

The observed median dust-corrected $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ ratio
($-0.22_{-0.15}^{+0.13}$) and population scatter is indicated as black line
with grey area, respectively. Note that we are only using $7$ out of the $10$
galaxies whose Spitzer photometry is not contaminated by nearby sources.
Spitzer-derived H$\alpha$ luminosities are mainly uncertain because of the
unknown dust correction (specifically the unknown $f-$factor). However, the [O
ii] to H$\alpha$ luminosity ratio is expected to be rather stable as the [O
ii] and H$\alpha$ lines are affected by dust in a similar way (and have likely
similar $f-$factors). For example, a change in $f$ from $0.44$ to $0.7$
affects the luminosity ratio by less than $0.1\,{\rm dex}$, which is smaller
than sample scatter and measurement uncertainties. Different reddening curves
contribute even less to the uncertainties.

Comparing our observations to the Cloudy models, we find that low ionisation
parameters ($\log(U)<-2$ or $\log(q/{\rm cm\,s^{-1}})<8.5$)111111Note that
$U=q/c$, where $c=3\times 10^{10}\,{\rm cm\,s^{-1}}$ is the speed of light.
are favored independent of the assumed electron density and metallicity. Such
ionisation parameter values are very consistent with typical measurements at
$z=2-3$ (via the [O iii]/[O ii] line ratio) finding $\log(U)\sim-2.5$ or
$\log(q/{\rm cm\,s^{-1}})\sim 8$ (Nakajima & Ouchi, 2014; Sanders et al.,
2016, 2020). On the other hand, studies at $z>6$ suggest $\log(U)\sim-1.7$,
which would indicate an increase in ionisation parameter in the Epoch of
Reionisation (Stark et al., 2014, 2015; Hutchison et al., 2019; Harikane et
al., 2020). Note that observations of local H ii regions suggest an upper cut-
off due to radiation confinement of $\log(U)~{}\sim~{}-1$ (e.g., Yeh &
Matzner, 2012).

####  3.2.2 The $\log(L_{\rm[OII]}/L_{\rm[CII]})$ luminosity ratio

Next, we focus on the $\log(L_{\rm[OII]}/L_{\rm[CII]})$ luminosity ratio. The
models from Cloudy are shown in the right panels of Figure 6. Increasing
ionisation parameter as well as increasing electron density result in a higher
luminosity ratio. On the other hand, more metal enriched environments are
expected to have lower $\log(L_{\rm[OII]}/L_{\rm[CII]})$ ratios. The trends
are not significantly affected by the different assumptions on the underlying
stellar population.

The black and grey horizontal lines show the measured ratios for DC_881725 and
the $9$ narrow-band detected galaxies, respectively. The [O ii] luminosities
have been corrected for dust attenuation using the methods described in
Section 3.1.2. We find a log-ratio of $1.11_{-0.08}^{+0.08}$ for DC_881725 and
a median ratio of $0.98^{+0.21}_{-0.22}$ for all $10$ galaxies. Note that the
true stellar reddening curve plays a sub-dominant role in this rather
quantitative analysis.

The models show that the intrinsic $\log(L_{\rm[OII]}/L_{\rm[CII]})$
luminosity ratio is significantly degenerate with metallicity, ionisation
parameter, and electron density. From Section 3.2.1 we expect an ionisation
parameter $\log(U)<-2$. Metallicity seems to have the least impact on the
luminosity ratio assuming a reasonable range between $10\%$ solar and solar
metallicity. An analysis of the rest-frame UV absorption line strengths (see
Section 2.2) suggests that our galaxies have $\sim 50\%$ of the solar gas-
phase metal enrichment on average. Fixing the metallicity to half-solar is
therefore a reasonable assumption. Within these assumptions our measurements
would argue for electron densities of $\log(n/{\rm[cm^{-3}]})\sim 2.5-3$,

Typical electron densities are measured to be around $100-200\,{\rm cm^{-3}}$
in galaxies with similar rates of star formation at $z\sim 2$. This is about a
factor of $5-10$ higher than observed on average in local galaxies (Steidel et
al., 2014; Masters et al., 2014; Sanders et al., 2016; Davies et al., 2021).
However, as pointed out by Kaasinen et al. (2017) (see also Shimakawa et al.,
2015; Davies et al., 2021), the electron density is positively correlated with
(specific) SFR, hence the higher $\log(n)$ found in high-$z$ galaxies are
likely due to a higher normalization of the main-sequence (i.e., higher
average SFR). In line with this, it was found that $\log(n)$ does not
significantly evolve with redshift for SFR-matched samples. Taking the
measurements in Davies et al. (2021) at face value suggests at least
$\log(n/{\rm cm^{-3}})\sim 2.5$ ($300\,{\rm cm^{-3}}$) for the average
specific SFR of our sample ($\sim 0.9\,{\rm Gyr^{-1}}$) according to $z\sim 2$
galaxies. Our measurements ($\log(n/{\rm[cm^{-3}]})\sim 2.5-3$) are generally
consistent with these expectations given their high SFRs, although at the high
end. Lower electron densities (e.g., $\sim 100-200\,{\rm cm^{-3}}$, typical at
$z\sim 2$) would indicate ionisation parameters of $\log(U)\gtrsim-1.7$, which
is clearly inconsistent with our measured $\log(L_{\rm[OII]}/L_{\rm H\alpha})$
luminosity ratio.

Combining the results of this section with Section 3.2.1, we conclude that our
$z\sim 4.5$ galaxies have ionisation parameters $\log(U)<-2$ and electron
densities $\log(n/{\rm[cm^{-3}]})\sim 2.5-3.0$. While the former is consistent
with measurements in galaxies at $z=2-3$, the latter is consistent with or
slightly higher than the expected electron densities given the SFRs of our
galaxies and the relation between $\log(n)$ and star formation measured in
lower redshift galaxies. Significantly higher ionisation parameters are
suggested for galaxies in the Epoch of Reionisation at $z>6$, which would
argue for a fast evolution of the ISM properties at times earlier than $z\sim
4.5$. Furthermore, we find that these typical main-sequence galaxies are much
less extreme than sub-millimetre galaxies at similar redshifts showing
electron densities of $\log(n/{\rm[cm^{-3}]})=4$ or more (e.g., Bothwell et
al., 2017; De Breuck et al., 2019).

We note that nine of the galaxies lie in a greater structure associated with a
proto-cluster at $z\sim 4.57$ (Lemaux et al., 2018). It is possible that the
proto-cluster environment has altered their properties in some way that could
be relevant for this study. For example, Darvish et al. (2015) argues for a
lower electron density in galaxies residing in dense environments
($\Sigma/\Sigma_{\rm mean}>4$) based on a comprehensive study of an over dense
region at $z\sim 0.5$. The translation of this result to (less) over-dense
regions at $z\sim 5$ and hence the impact of the environment on the properties
of our galaxies remains unclear. As far as we can tell, the nine galaxies have
statistically indistinguishable properties with respect to the other ALPINE
(hence field) galaxies using the metrics that we are able to compare (e.g.,
Figure 1). As such, it is at least likely that any effect of environment on
the ISM is subtle at best.

####  3.2.3 Caveats

Several other parameters could change the modeled line ratios. Here, we
investigate the impact of the stellar metallicity of the stellar population
producing the incident spectrum as well as the geometrical covering factor.

First, we ran the simulation with a covering factor of $0.1$ instead of $1$
and found that it does not have a significant impact ($<0.05\,{\rm dex}$) on
either line ratio. Second, we implemented a solar stellar metallicity (instead
of $10\%$ solar) for the incident spectrum produced by the BPASS models. We
find that this increases both $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ and
$\log(L_{\rm[OII]}/L_{\rm[CII]})$ ratios by $0.15\,{\rm dex}$. This seems to
be because a factor $8$ decrease in [O ii], which is over-balanced by a factor
$11$ and $12$ decrease in H$\alpha$ and [C ii], respectively.

An increase in the $\log(L_{\rm[OII]}/L_{\rm[CII]})$ luminosity ratio would
slightly reduce the likelihood of high electron densities for a given
ionisation parameter. And increased $\log(L_{\rm[OII]}/L_{\rm H\alpha})$ ratio
would further allow slightly higher ionisation parameters (especially if
assuming a gas-phase metallicity of $50\%$ solar). All in all, a higher
stellar metallicity would argue for $\sim 0.2-0.3\,{\rm dex}$ higher
ionisation parameters and roughly the same amount lower electron densities,
making our galaxies consistent with their counterparts at $z\sim 2-3$ with
similar star formation properties.

Finally, our simple model assumes that the galaxy is a single H ii region.
More complicated “multi-sector” models could be investigated, however, we do
not think that given the uncertainties in our (photometric) measurements of [O
ii] and H$\alpha$ this would improve the robustness of our results.

##  4 Conclusions

We have assembled the best-studied sample of main-sequence galaxies at $z\sim
4.5$ to date, to investigate key parameters of the ISM as well as the relation
between [O ii] and total SFR of galaxies in the early Universe right after the
Epoch of Reionisation.

The $10$ main-sequence galaxies at $z\sim 4.5$ have measurements of rest-frame
UV absorption lines, optical [O ii] (from spectroscopy and narrow-band
imaging) and H$\alpha$ emission (from Spitzer), as well as far-infrared
observations of [C ii] and dust continuum from ALMA. We use the total SFRs
derived from [C ii] emission as well as far-infrared continuum to calibrate
the relation between SFR and [O ii] emission for the first time at these
redshifts (Section 3.1). Furthermore, we constrain key parameters of their ISM
(metallicity, electron density, ionisation parameter) via the UV absorption
lines and [O ii]/[C ii] and [O ii]/H$\alpha$ luminosity ratios, which we
compare to Cloudy simulations (Sections 2.2 and 3.2). Our findings can be
summarised as follows:

  * •

The relation between dust-corrected [O ii] luminosities and total SFR is best
described by sub-solar metallicity models from Kewley et al. (2004). For the
median of our sample, we find a total SFR of $53.57^{+23.48}_{-31.30}\,{\rm
M_{\odot}\,yr^{-1}}$ and $\log(L_{\rm[OII]}/{\rm
L_{\odot}})=9.88^{+0.29}_{-0.51}$. The original Kennicutt (1998) [O ii]$-$SFR
relation would overestimate the SFR by a factor of $\sim 3$ (Section 3.1.2).

  * •

By comparing the $\log(L_{\rm[OII]}/L_{\rm[CII]})$ and
$\log(L_{\rm[OII]}/L_{\rm H\alpha})$ luminosity ratio to Cloudy models, we
find that our galaxies have ionisation parameters $\log(U)<-2$ and electron
densities of $\log(n/[{\rm cm^{3}}])\sim 2.5-3$. The former is consistent with
$z\sim 2-3$ galaxies, the latter may be slightly higher than expected based on
our galaxies’ specific SFR. However, these results depend on the input
parameters for the Cloudy simulation. Specifically, increasing the stellar
metallicity of the incident spectrum from $10\%$ solar to solar would make our
observations more consistent with electron densities measured in $z=2-3$
galaxies.

All in all, we find that the ISM properties of this representative sample of
$z\sim 4.5$ galaxies are similar to their descendants at $z\sim 2-3$ when
matched by their specific SFRs. There are indications that the ionisation
parameter of more primordial galaxies in the Epoch of Reionisation at $z>6$ is
increased (see, e.g., Harikane et al., 2020) compared to our findings. This
suggests a rapid evolution of the ISM in the few $100\,{\rm Myrs}$ spanning
these epochs. The relation between [O ii] emission and SFR is consistent if
assuming a gas-phase metallicity of $\sim 50\%$ solar, which is in line with
the estimates from rest-frame UV absorption spectroscopy. This result is an
important step towards using optical emission lines as total SFR indicators in
the era of JWST.

In this work, we demonstrated the necessity the combination of data from many
facilities covering the rest-frame UV to far-infrared to decipher the physical
properties of high-redshift galaxies. Although this is the largest and best-
studied sample of typical $z\sim 4.5$ galaxies to-date, larger samples with
similar multi-wavelength observations are crucial for a better statistical
study of the dependence of the reported parameters on other galaxy properties.
Larger surveys with current facilities (e.g., Keck) and later JWST will
provide these samples.

Table 1: Summary of measured properties for DC_881725 from broad-band photometry, ALMA observations, and new Keck/MOSFIRE spectroscopy. Broad-band | ALMA | Keck/DEIMOS | Keck/MOSFIRE
---|---|---|---
————————————————————————— | ——————————————– | ————————— | —————————
$\log({\rm M_{*}})$ | $\rm SFR_{\rm SED}$ | $\rm E(B-V)_{SED}$ | $\rm\log(L_{UV})$ | $\beta$ | $z_{\rm[CII]}$ | $\log(\rm L_{IR})$ | $\log(\rm L_{[CII]})$ | $z_{\rm Ly\alpha}$ | EW(Ly$\rm\alpha$) | $z_{\rm[OII]}$ | $\log(\rm L_{[OII]})$
$\left[\rm{M_{\odot}}\right]$ | $\left[\rm{M_{\odot}\,yr^{-1}}\right]$ | $\left[\rm{mag}\right]$ | $\left[\rm{L_{\odot}}\right]$ |  |  | $\left[\rm{L_{\odot}}\right]$ | $\left[\rm{L_{\odot}}\right]$ |  | [Å] |  | $\left[\rm{L_{\odot}}\right]$
9.96${}^{+0.16}_{-0.11}$ | 88.0${}^{+61.1}_{-43.3}$ | 0.25${}^{+0.05}_{-0.05}$ | 10.97${}^{+0.06}_{-0.06}$ | -1.20${}^{+0.42}_{-0.18}$ | 4.5777 | 11.67${}^{+0.10}_{-0.13}$ | 8.84${}^{+0.04}_{-0.04}$ | 4.5854 | 57.4${}^{+18.0}_{-18.0}$ | 4.5793 | 9.22${}^{+0.05}_{-0.06}$

Table 2: Summary of properties derived from the ancillary data for the $9$
galaxies with [O ii] detection from narrow-band imaging.

ID | z[CII] | log(LOII) | log(L[CII]) | log(LFIR) | log(M) | SFR[CII] | SFRFIR | SFRUV
---|---|---|---|---|---|---|---|---
|  | $\left[\rm{L_{\odot}}\right]$ | $\left[\rm{L_{\odot}}\right]$ | $\left[\rm{L_{\odot}}\right]$ | $\left[\rm{M_{\odot}}\right]$ | $\left[\rm{M_{\odot}\,yr^{-1}}\right]$ | $\left[\rm{M_{\odot}\,yr^{-1}}\right]$ | $\left[\rm{M_{\odot}\,yr^{-1}}\right]$
DC_665626 | 4.5830 | 9.36${}^{+0.08}_{-0.09}$ | 8.21${}^{+0.10}_{-0.13}$ | – | 9.21${}^{+0.16}_{-0.18}$ | 23.15${}^{+1.22}_{-1.30}$ | – | 5.77${}^{+1.24}_{-1.23}$
DC_680104 | 4.5320 | 9.51${}^{+0.10}_{-0.13}$ | 8.74${}^{+0.12}_{-0.16}$ | – | 9.23${}^{+0.18}_{-0.12}$ | 66.30${}^{+1.25}_{-1.36}$ | – | 14.55${}^{+1.12}_{-1.10}$
VC_5100969402 | 4.5869 | 9.34${}^{+0.08}_{-0.10}$ | 8.72${}^{+0.04}_{-0.05}$ | 11.65${}^{+0.11}_{-0.16}$ | 10.00${}^{+0.14}_{-0.12}$ | 63.42${}^{+1.09}_{-1.10}$ | 43.23${}^{+13.06}_{-13.06}$ | 12.50${}^{+1.14}_{-1.13}$
VC_5100994794 | 4.5783 | 9.14${}^{+0.11}_{-0.15}$ | 8.75${}^{+0.04}_{-0.04}$ | 11.20${}^{+0.12}_{-0.16}$ | 9.73${}^{+0.13}_{-0.15}$ | 66.93${}^{+1.08}_{-1.09}$ | 15.50${}^{+4.71}_{-4.71}$ | 10.69${}^{+1.19}_{-1.23}$
VC_5101209780 | 4.5700 | 8.99${}^{+0.30}_{-0.38}$ | 8.86${}^{+0.08}_{-0.10}$ | 11.62${}^{+0.13}_{-0.19}$ | 10.05${}^{+0.12}_{-0.12}$ | 84.45${}^{+1.18}_{-1.23}$ | 40.65${}^{+14.64}_{-14.64}$ | 22.07${}^{+1.09}_{-1.10}$
VC_5101210235 | 4.5733 | 9.06${}^{+0.16}_{-0.30}$ | 8.35${}^{+0.11}_{-0.14}$ | – | 9.78${}^{+0.15}_{-0.12}$ | 30.71${}^{+1.24}_{-1.33}$ | – | 24.08${}^{+1.08}_{-1.07}$
VC_5101218326 | 4.5678 | 9.32${}^{+0.08}_{-0.10}$ | 9.26${}^{+0.02}_{-0.02}$ | 11.79${}^{+0.07}_{-0.08}$ | 11.01${}^{+0.05}_{-0.07}$ | 184.45${}^{+1.04}_{-1.04}$ | 60.38${}^{+10.30}_{-10.30}$ | 27.94${}^{+1.06}_{-1.07}$
VC_5101244930 | 4.5769 | 9.09${}^{+0.30}_{-0.19}$ | 8.70${}^{+0.08}_{-0.10}$ | – | 9.67${}^{+0.13}_{-0.16}$ | 61.73${}^{+1.18}_{-1.23}$ | – | 17.45${}^{+1.12}_{-1.12}$
VC_5110377875 | 4.5441 | 9.21${}^{+0.18}_{-0.32}$ | 9.23${}^{+0.03}_{-0.03}$ | – | 10.17${}^{+0.21}_{-0.12}$ | 174.87${}^{+1.05}_{-1.06}$ | – | 24.67${}^{+1.09}_{-1.08}$

  * •

Notes: The UV SFRs have not been corrected for dust attenuation. [C ii] SFRs
derived using the Schaerer et al. (2020) relation (see Section 3.1.1). Far-
infrared SFRs derived using the Kennicutt (1998) relation.

Table 3: Summary of SFRs measured from [O ii] and ancillary ALPINE data
available for DC_881725. Note that SFRs from UV and optical indicators have
not been corrected for dust attenuation.

| SFR $[\rm M_{\astrosun}\,yr^{-1}]$
---|---
| ———————————————————–
Reference | [O ii] | [C ii] | IR | UV
K98 | 50.0${}^{+6.7}_{-6.1}$ |  | 46.0${}^{+11.9}_{-11.9}$ | 13.3${}^{+2.0}_{-1.8}$
K04† |  |  |  |
\- no met. dep. | 23.6${}^{+3.2}_{-2.7}$ |  |  |
\- solar | 18.6${}^{+2.5}_{-2.3}$ |  |  |
\- half-solar | 13.8${}^{+1.9}_{-1.7}$ |  |  |
\- $1/5^{\rm th}$-solar | 10.3${}^{+1.4}_{-1.5}$ |  |  |
L14 |  |  |  |
\- metal-poor dwarfs |  | 22.0${}^{+1.6}_{-1.6}$ |  |
\- starburst |  | 59.9${}^{+5.5}_{-5.5}$ |  |
\- AGN |  | 73.0${}^{+6.0}_{-6.1}$ |  |
\- high-$z$ |  | 80.9${}^{+8.9}_{-8.7}$ |  |
\- ULIRGs |  | 360.7${}^{+4.3}_{-4.3}$ |  |
S20 |  | 64.1${}^{+6.2}_{-5.7}$ |  |
dust correction factor‡ | 5.2 (4.3) | $-$ | $-$ | 3.4 (3.5)

  * •

† The first value is derived using their equation 4. The other values are
derived for different gas-phase metallicities and their equation 10.

  * •

‡ Dust correction factor for [O ii] and UV continuum emission assuming a
reddening curve from Calzetti et al. (2000) and Reddy et al. (2015) (in
parenthesis), respectively (see text for more details). The [O ii] correction
factor assumes a differential dust attenuation $f=0.44$. For $f=0.7$, decrease
the value by a factor of $1.8$.

  * •

References: K98 $-$ Kennicutt (1998); K04 $-$ Kewley et al. (2004); L14 $-$ De
Looze et al. (2014); S20 $-$ Schaerer et al. (2020).

## Acknowledgements

We thank Yuichi Harikane for the helpful inputs on running Cloudy and the
anonymous referee for the comments which improved this manuscript. We also
thank Behnam Darvish and Nick Scoville for letting us add DC_881725 to their
MOSFIRE mask. G.C.J. acknowledges ERC Advanced Grant 695671 “QUENCH” and
support by the Science and Technology Facilities Council (STFC). This work
made use of v2.2.1 of the Binary Population and Spectral Synthesis (BPASS)
models as described in Eldridge et al. (2017) and Stanway & Eldridge (2018).
This work was supported by the Programme National Cosmology et Galaxies (PNCG)
of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. This paper uses
data obtained with the ALMA Observatory, under Large Program 2017.1.00428.L.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. This work is based on
observations and archival data made with the Spitzer Space Telescope, which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under a contract with NASA. Some of the material presented in this paper is
based upon work supported by the National Science Foundation under Grant No.
1908422. MT acknowledges the support from grant PRIN MIUR 2017 20173ML3WW 001.
Furthermore, this work is based on data from the W. M. Keck Observatory and
the Subaru Telescope. The authors wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Mauna Kea has
always had within the indigenous Hawaiian community. We are most fortunate to
have the opportunity to conduct observations from this mountain. Based on data
products from observations made with ESO Telescopes at the La Silla Paranal
Observatory under ESO programme ID 179.A-2005 and on data products produced by
CALET and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA
consortium. Finally, we would also like to recognize the contributions from
all of the members of the COSMOS Team who helped in obtaining and reducing the
large amount of multi-wavelength data that are now publicly available through
IRSA at http://irsa.ipac.caltech.edu/Missions/cosmos.html.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Absolute Flux Calibration of the MOSFIRE Spectrum

For the absolute calibration of the MOSFIRE spectrum, we use a $17^{\rm th}$
magnitude standard star from the 2MASS star catalogue, which was included for
this purpose in the mask. The one-dimensional spectrum of the star is
extracted within $10\,{\rm pixels}$ ($1.8\arcsec$) around the centre of the
continuum emission on the two-dimensional spectrum in ${\rm e^{-}/s}$. The
wavelength-dependent calibration is then derived by comparing the observed
spectrum with models of different stellar spectral types.

In detail, we use the spectra from the Pickles (1998) stellar model library,
which offers a wide range in spectral types from O to M and covers a
wavelength from $0.115-2.5\,{\rm\mu m}$ in steps of $5\,{\rm\text{\AA}}$. This
is sufficiently red to match the MOSFIRE $K-$band. We find that the spectrum
at $>2\,{\rm\mu m}$ only changes slowly as a function of stellar type, hence
we only focus on a coarse grid including O5 V, B3 V, A5 V, F5 V, G5 V, K5 V,
and M5 V type stars. The model spectrum of each of these is normalised to the
UltraVista $K-$band flux of the observed 2MASS star, obtained from the
COSMOS2015 catalogue (Laigle et al., 2016; McCracken et al., 2012b). A
wavelength-dependent conversion from units of $\rm e^{-}\,s^{-1}$ to $\rm
erg\,s^{-1}\,cm^{-2}\,\text{\AA}^{-1}$ is then derived by comparing the
normalised model spectrum to the observed spectrum of the 2MASS star. The
resulting normalisations are shown in Figure 7. The arrow indicates the
wavelength of [O ii]. The computed normalization is within $5\%$ for all
spectral types except the coolest M-dwarf. By comparing the $K-$band
normalised stellar spectra to the UltraVISTA $Y$, $J$, and $H$ photometry, we
find that our standard star fits best a G$-$type star. We therefore use the G5
V normalisation through out the paper.

Figure 7: Wavelength-dependent normalisation of the MOSFIRE/K band spectrum
derived for various stellar types from the Pickles (1998) library. Our
standard star is close to a G5 V type. The wavelength of [O ii] for DC_881725
is indicated by an arrow. The dashed line represents the wavelength range of
the MOSFIRE K-band.

##  Appendix B Comparison of Optical Emission Line Measurements to EL-COSMOS

We derived the H$\alpha$ luminosities for our $10$ galaxies from their Spitzer
[$3.6\,{\rm\mu m}$]$-$[$4.5\,{\rm\mu m}$] colours (see Faisst et al. (2020a)
and Faisst et al. (2016a) for a detailed description of the methods used).
Here we compare our measurement to the recent EL-COSMOS catalogue (Saito et
al., 2020), which provides predictions of the intrinsic (dust-corrected) [O
ii] and H$\alpha$ optical lines from SED fitting calibrated to spectroscopic
measurements for all galaxies in the Laigle et al. (2016) COSMOS2015
catalogue. Our luminosity measurements have been dust corrected using the
method described in the this paper. Figure 8 shows the result of the
comparison. EL-COSMOS agrees very well with our narrow-band and spectroscopic
[O ii] measurements. Comparing the H$\alpha$ measurements, we find a good
agreement within a factor of $2$. Note that we marked galaxies that have
uncertain H$\alpha$ measurements due to contaminated Spitzer photometry (see
also Faisst et al., 2020a).

Figure 8: Comparison between our measurements of H$\alpha$ (blue circles,
derived from Spitzer colours) and [O ii] (orange squares, derived from narrow-
band imaging or spectroscopy) and the derivations in the EL-COSMOS catalog.
Galaxies whose H$\alpha$ measurements are uncertain due to contaminated
Spitzer photometry are shown as open circles. The dark (light) grey areas
denote deviations of a factor of $2$ ($5$) from the 1$-$to$-$1 relation
(dashed line).