11institutetext:  Department of Astronomy, University of Geneva, 51 Chemin
Pegasi, 1290 Versoix, Switzerland 22institutetext: CNRS, IRAP, 14 Avenue E.
Belin, 31400 Toulouse, France 33institutetext: Instituto de Investigación
Multidisciplinar en Ciencia y Tecnología, Universidad de La Serena, Raúl
Bitrán 1305, La Serena, Chile 44institutetext: Departamento de Física y
Astronomía, Universidad de La Serena, Avda. Juan Cisternas 1200, La Serena,
Chile 55institutetext: Institut d’astrophysique de Paris, CNRS UMR7095,
Sorbonne Université, 98bis Boulevard Arago, F-75014 Paris, France
66institutetext: School of Earth & Space Exploration, Arizona State
University, Tempe, AZ 85287, USA 77institutetext: Department of Astronomy, The
University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX
78712-1205, USA 88institutetext: Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA 99institutetext: INAF-Osservatorio
Astronomico di Padova, Vicolo dell’ Osservatorio, 5, 35122 Padova, Italy
1010institutetext: Department of Astronomy, Oskar Klein Centre; Stockholm
University; SE-106 91 Stockholm, Sweden 1111institutetext: Department of
Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street,
Baltimore, MD 21218, USA 1212institutetext: Space Telescope Science Institute;
3700 San Martin Drive, Baltimore, MD, 21218, USA 1313institutetext: Bogolyubov
Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
14b Metrolohichna str., Kyiv, 03143, Ukraine 1414institutetext: Astronomy
Department, Williams College, Williamstown, MA 01267, USA 1515institutetext:
University of Michigan, Department of Astronomy, 323 West Hall, 1085 S.
University Ave, Ann Arbor, MI 48109, USA 1616institutetext: Department of
Physics and Astronomy, Minnesota State University, Mankato, MN, 56001, USA
1717institutetext: Infrared Processing and Analysis Center, California
Institute of Technology, Pasadena, CA 91125, USA 1818institutetext: Astronomy
Department, University of Virginia, P.O. Box 400325, Charlottesville, VA
22904-4325, USA 1919institutetext: Institut für Physik und Astronomie,
Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany

# No correlation of the Lyman continuum escape fraction with spectral hardness

R. Marques-Chaves No correlation of the Lyman continuum escape fraction with
spectral hardnessNo correlation of the Lyman continuum escape fraction with
spectral hardness D. Schaerer No correlation of the Lyman continuum escape
fraction with spectral hardnessNo correlation of the Lyman continuum escape
fraction with spectral hardnessNo correlation of the Lyman continuum escape
fraction with spectral hardnessNo correlation of the Lyman continuum escape
fraction with spectral hardness R. O. Amorín No correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardness H. Atek No correlation of the
Lyman continuum escape fraction with spectral hardnessNo correlation of the
Lyman continuum escape fraction with spectral hardness S. Borthakur No
correlation of the Lyman continuum escape fraction with spectral hardnessNo
correlation of the Lyman continuum escape fraction with spectral hardness J.
Chisholm No correlation of the Lyman continuum escape fraction with spectral
hardnessNo correlation of the Lyman continuum escape fraction with spectral
hardness V. Fernández No correlation of the Lyman continuum escape fraction
with spectral hardnessNo correlation of the Lyman continuum escape fraction
with spectral hardness S. R. Flury No correlation of the Lyman continuum
escape fraction with spectral hardnessNo correlation of the Lyman continuum
escape fraction with spectral hardness M. Giavalisco No correlation of the
Lyman continuum escape fraction with spectral hardnessNo correlation of the
Lyman continuum escape fraction with spectral hardness A. Grazian No
correlation of the Lyman continuum escape fraction with spectral hardnessNo
correlation of the Lyman continuum escape fraction with spectral hardness M.
J. Hayes No correlation of the Lyman continuum escape fraction with spectral
hardnessNo correlation of the Lyman continuum escape fraction with spectral
hardness T. M. Heckman No correlation of the Lyman continuum escape fraction
with spectral hardnessNo correlation of the Lyman continuum escape fraction
with spectral hardness A. Henry No correlation of the Lyman continuum escape
fraction with spectral hardnessNo correlation of the Lyman continuum escape
fraction with spectral hardness Y. I. Izotov No correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardness A. E. Jaskot No correlation
of the Lyman continuum escape fraction with spectral hardnessNo correlation of
the Lyman continuum escape fraction with spectral hardness Z. Ji No
correlation of the Lyman continuum escape fraction with spectral hardnessNo
correlation of the Lyman continuum escape fraction with spectral hardness S.
R. McCandliss No correlation of the Lyman continuum escape fraction with
spectral hardnessNo correlation of the Lyman continuum escape fraction with
spectral hardness M. S. Oey No correlation of the Lyman continuum escape
fraction with spectral hardnessNo correlation of the Lyman continuum escape
fraction with spectral hardness G. Östlin No correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardness S. Ravindranath No
correlation of the Lyman continuum escape fraction with spectral hardnessNo
correlation of the Lyman continuum escape fraction with spectral hardness M.
J. Rutkowski No correlation of the Lyman continuum escape fraction with
spectral hardnessNo correlation of the Lyman continuum escape fraction with
spectral hardness A. Saldana-Lopez No correlation of the Lyman continuum
escape fraction with spectral hardnessNo correlation of the Lyman continuum
escape fraction with spectral hardness H. Teplitz No correlation of the Lyman
continuum escape fraction with spectral hardnessNo correlation of the Lyman
continuum escape fraction with spectral hardness T. X. Thuan No correlation of
the Lyman continuum escape fraction with spectral hardnessNo correlation of
the Lyman continuum escape fraction with spectral hardness A. Verhamme No
correlation of the Lyman continuum escape fraction with spectral hardnessNo
correlation of the Lyman continuum escape fraction with spectral hardness B.
Wang No correlation of the Lyman continuum escape fraction with spectral
hardnessNo correlation of the Lyman continuum escape fraction with spectral
hardness G. Worseck No correlation of the Lyman continuum escape fraction with
spectral hardnessNo correlation of the Lyman continuum escape fraction with
spectral hardness X. Xu No correlation of the Lyman continuum escape fraction
with spectral hardnessNo correlation of the Lyman continuum escape fraction
with spectral hardness

(Received –; accepted –)

The properties that govern the production and escape of hydrogen-ionizing
photons (Lyman continuum, LyC; with energies ¿13.6 eV) in star-forming
galaxies are still poorly understood, but they are key to identifying and
characterizing the sources that reionized the Universe. Here we empirically
explore the relation between the hardness of ionizing radiation and the LyC
leakage in a large sample of low-$z$ star-forming galaxies from the recent
Hubble Space Telescope Low-$z$ Lyman Continuum Survey. Using Sloan Digital Sky
Survey stacks and deep XShooter observations, we investigate the hardness of
the ionizing spectra ($Q_{\rm He^{+}}/Q_{\rm H}$) between 54.4 eV (He+) and
13.6 eV (H) from the optical recombination lines He ii 4686Å and H$\beta$
4861Å for galaxies with LyC escape fractions spanning a wide range, $f_{\rm
esc}\rm(LyC)\simeq 0-90\%$. We find that the observed intensity of He
ii/H$\beta$ is primarily driven by variations in the metallicity, but is not
correlated with LyC leakage. Both very strong (¡$f_{\rm esc}\rm(LyC)$¿ $\simeq
0.5$) and nonleakers (¡$f_{\rm esc}\rm(LyC)$¿ $\simeq 0$) present similar
observed intensities of He ii and H$\beta$ at comparable metallicity, between
$\simeq 0.01$ and $\simeq 0.02$ for $12+\log({\rm O/H})$ $>8.0$ and $<8.0$,
respectively. Our results demonstrate that $Q_{\rm He^{+}}/Q_{\rm H}$ does not
correlate with $f_{\rm esc}\rm(LyC)$, which implies that strong LyC emitters
do not show harder ionizing spectra than nonleakers at similar metallicity.

###### Key Words.:

Galaxies: starburst – Galaxies: high-redshift – Cosmology: dark ages,
reionization, first stars

##  1 Introduction

Numerous studies of star-forming galaxies at $z\sim 1-3$ have shown
differences with low-$z$ galaxies in the Sloan Digital Sky Survey (SDSS),
based on shifts in the classical, optical BPT emission-line diagrams toward
higher excitation in the nebular gas phase (see the review of Kewley et al.,
2019, and references therein). Such shifts can be due to different physical
effects causing harder ionizing spectra of their stellar populations (e.g.,
lower metallicities, younger ages), interstellar medium (ISM) properties
(higher ionization parameter, different gas pressure and/or density),
$\alpha$-element enhancements, selection effects, or others (cf. Steidel et
al., 2016; Kewley et al., 2019; Izotov et al., 2021a).

Possibly larger differences have been observed in rest-UV spectra of distant
galaxies, where strong nebular emission from C iv $\lambda$1550 and He ii
$\lambda$1640 has been found (e.g., Stark et al., 2015; Mainali et al., 2017;
Vanzella et al., 2021). These lines have also been detected in confirmed or
suspected Lyman-continuum (LyC) emitters at $z\sim 2-4$ (see Vanzella et al.,
2018, 2020; Naidu et al., 2022). These observations suggest hard ionizing
spectra in particular because C iv and He ii probe energies above 47.8 and
54.5 eV, respectively, and are thus sensitive to higher-energy radiation than
the classical strong optical emission lines of [O iii], [O ii], [N ii], and [S
ii] (e.g., Kewley et al., 2019; Feltre et al., 2016).

Nebular He ii emission provides the best measure of the hardness of the
ionizing radiation field because its recombination lines111The strongest and
most commonly detected lines are He ii $\lambda$1640 and He ii $\lambda$4686.
are basically direct photons counters for energies $>54$ eV, whereas the
forbidden metal lines depend on many parameters (ionization parameter and
others). Naidu et al. (2022) stacked rest-UV spectra of Ly$\alpha$ emitters
(LAEs) at $z\sim 2,$ for which they estimated LyC escape fractions using
indirect methods, finding narrow He ii $\lambda$1640, C iv $\lambda$1550, and
other lines in sources with high LyC escape, whereas low escape sources only
show C iii] and O iii] emission. Based on this, they suggested a possible
relation between the hardness of the ionizing spectra and LyC escape. He ii is
also seen in the stacks of the LyC candidates of Marchi et al. (2018) selected
as Lyman break galaxies (LBGs), although the poor resolution makes it
difficult to exclude a significant contribution from stellar emission.
Moreover, nebular He ii emission has not been reported in the LBG stacked LyC
emitter spectra of Steidel et al. (2018). These findings call for a
clarification of the possible link between LyC escape, nebular He ii emission,
and the hardness of the radiation between energies 13.6 and 54 eV.

At low redshift, Jaskot & Oey (2013) have examined how hard ionizing radiation
can influence emission line diagnostics of the optical depth of LyC radiation
in green pea galaxies, a class of strong emission line galaxies now known to
contain LyC emitters. Schaerer et al. (2022) have recently discovered intense
C iv and He ii $\lambda$1640 emission lines in three low-$z$ strong LyC
emitters, ($f_{\rm esc}>0.1$), with UV properties similar to the high-$z$
galaxies mentioned above. They proposed that strong C iv $\lambda$1550
emission indicates high LyC escape fractions. Furthermore, they also estimated
that strong LyC leakers do not have harder ionizing spectra than nonleakers,
and that the presence of strong C iv and He ii in the spectra of LyC leakers
could be primarily due to a high ionizing photon production (Schaerer et al.,
2022).

On the other hand, Pérez-Montero et al. (2020) suggested that the observed He
ii $\lambda$4686 emission in metal-poor low-$z$ galaxies could be explained by
significant photon leakage, without the need for additional sources of hard
ionizing photons, while it is generally accepted that normal stellar
populations cannot produce sufficient photons above 54 eV (see, e.g.,
Stasińska et al., 2015; Schaerer et al., 2019; Olivier et al., 2021; Simmonds
et al., 2021). These results again call for a systematic approach to determine
whether LyC emitting galaxies show a different He+-ionizing radiation than
comparable galaxies with negligible Lyman continuum escape.

In this Letter we determine empirically and in a differential manner the
hardness of the ionizing spectra between 54.4 and 13.6 eV (i.e., the
ionization potential of He+ and that of H) as a function of the LyC escape
fraction for the first time. To achieve this goal, we use the data from 89
low-redshift star-forming galaxies for which Lyman continuum and nonionizing
UV spectra are available from the recent Hubble Space Telescope (HST) Low-$z$
Lyman Continuum Survey and other observations (see Sect. 2). Using optical
spectra and stacking, we determine the hardness of ionizing radiation field
from the optical recombination lines He ii $\lambda$4686 and H$\beta$ in a
straightforward manner, in contrast to methods using forbidden optical lines
and to methods relying on UV emission lines. Our first constraints on the
hardness of LyC emitters and a control sample thus contribute to a better
knowledge of the analogue sources of cosmic reionization, their ionizing
radiation field and ISM.

##  2 Observations

###  2.1 Lyman continuum and UV observations

We used the recent LzLCS, which consists of 66 star-forming galaxies at
$z\simeq 0.2-0.4$ (Flury et al., 2022a, b; Saldana-Lopez et al., 2022) and 23
additional sources previously studied by Izotov et al. (2016b, c, 2018a,
2021b) and Wang et al. (2019). This unique dataset includes 37 LyC emitters
detected with high signal-to-noise ratio (S/N¿3), with escape fractions
spanning a wide range ($f_{\rm esc}$$\sim$ 1–90 %), and strong upper limits
for the remaining sources. The galaxies span a wide range of physical
properties with stellar masses $\log(M_{*}/M_{\odot})=7.2-10.8$, star
formation rates SFR$=(3-80)$ $M_{\odot}$ yr-1, and nebular abundances
$12+\log({\rm O/H})=7.5-8.6$, derived using the direct method (Flury et al.,
2022a).

###  2.2 SDSS optical spectra

Optical spectra are available for all sources from SDSS and BOSS surveys
(Eisenstein et al., 2011). We retrieved optical spectra from the SDSS Science
Archive Server. SDSS spectra have a spectral resolution $R\sim 2000$. The
extinction curve of Fitzpatrick (1999) and the Galactic $E(B-V)$ values from
the dust maps of Green et al. (2018) were used to correct the reddening effect
of the Galaxy. Flux measurements were derived using Gaussian profiles and the
Python nonlinear least-squares function curve_fit and corresponding
uncertainties using a Monte Carlo method. The Cardelli et al. (1989) reddening
law ($R_{V}=3.1$) was adopted to correct for the internal extinction using the
ratios of well-detected Balmer emission lines (following Izotov et al., 1994).

In the lines of interest in this work, H$\beta$ is detected with high
significance in all galaxies, but not He ii, yielding $3\sigma$ limits of He
ii/H$\beta\lesssim 0.04$ on average, which are fairly above the typical
intensities He ii/H$\beta\lesssim 0.02$ found in other compact star-forming
galaxies (e.g., Izotov et al., 2016a), including LyC emitters (Guseva et al.,
2020). Therefore, we performed a stacking analysis of the SDSS spectra to
improve the He ii/H$\beta$ limits.

Table 1: Summary of the results of SDSS stacks

SDSS Bins | Denomination | Definition | N | $f_{\rm esc}$ | $12+\log({\rm O/H})$ | He ii/H$\beta$ | $Q_{\rm He^{+}}/Q_{\rm H}$
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
$f_{\rm esc}$ | very strong leakers | $f_{\rm esc}\geq 0.2$ | 8 | $0.497_{-0.140}^{+0.242}$ | $7.849_{-0.165}^{+0.265}$ | $0.016\pm 0.007$ | $0.005\pm 0.002$
| strong leakers | $0.05\leq f_{\rm esc}\ <0.2$ | 13 | $0.117_{-0.042}^{+0.053}$ | $8.107_{-0.221}^{+0.296}$ | $0.013\pm 0.006$ | $0.007\pm 0.003$
| weak leakers | $f_{\rm esc}\ <0.05$ | 29 | $0.025_{-0.011}^{+0.014}$ | $8.115_{-0.174}^{+0.221}$ | $0.012\pm 0.004$ | $0.007\pm 0.002$
| non-leakers | $\rm S/N(LyC)<2$ | 39 | $<0.011$ | $8.175_{-0.186}^{+0.245}$ | $0.015\pm 0.005$ | $0.009\pm 0.003$
$f_{\rm esc}$ and metallicity |  |  $f_{\rm esc}\geq 0.05$ and $12+\log({\rm O/H})$ ¡ 8.0 | 11 | $0.346_{-0.228}^{+0.322}$ | $7.811_{-0.109}^{+0.161}$ | $0.020\pm 0.005$ | $0.008\pm 0.002$
|  |  $f_{\rm esc}\geq 0.05$ and $12+\log({\rm O/H})$ $=8.0-8.3$ | 6 | $0.203_{-0.113}^{+0.159}$ | $8.095_{-0.086}^{+0.110}$ | $0.013\pm 0.006$ | $0.006\pm 0.003$
|  |  $f_{\rm esc}\geq 0.05$ and $12+\log({\rm O/H})$ ¿ 8.3 | 4 | $0.120_{-0.056}^{+0.066}$ | $8.423_{-0.051}^{+0.102}$ | $<0.011$ | $<0.006$
|  |  $f_{\rm esc}<0.05$ and $12+\log({\rm O/H})$ ¡ 8.0 | 9 | $0.030_{-0.010}^{+0.013}$ | $7.895_{-0.042}^{+0.060}$ | $<0.016$ | $<0.009$
|  |  $f_{\rm esc}<0.05$ and $12+\log({\rm O/H})$ $=8.0-8.3$ | 15 | $0.022_{-0.011}^{+0.015}$ | $8.153_{-0.084}^{+0.100}$ | $0.015\pm 0.004$ | $0.008\pm 0.002$
|  |  $f_{\rm esc}<0.05$ and $12+\log({\rm O/H})$ ¿ 8.3 | 5 | $0.023_{-0.009}^{+0.015}$ | $8.395_{-0.029}^{+0.122}$ | $<0.023$ | $<0.013$
|  |  $\rm S/N(LyC)<2$ and $12+\log({\rm O/H})$ ¡ 8.0 | 6 | $<0.027$ | $7.786_{-0.120}^{+0.173}$ | $0.030\pm 0.009$ | $0.017\pm 0.005$
|  |  $\rm S/N(LyC)<2$ and $12+\log({\rm O/H})$ $=8.0-8.3$ | 22 | $<0.008$ | $8.175_{-0.075}^{+0.091}$ | $0.011\pm 0.004$ | $0.006\pm 0.002$
|  |  $\rm S/N(LyC)<2$ and $12+\log({\rm O/H})$ ¿ 8.3 | 11 | $<0.009$ | $8.388_{-0.057}^{+0.086}$ | $<0.013$ | $<0.007$

Notes.  Columns (1), (2), and (3): Groups, denomination, and definition of
each bin. Column (4): Number of sources in each bin. Columns (5) and (6): mean
and 68% confidence intervals (bootstrap resampling) of $f_{\rm esc}$ and
$12+\log({\rm O/H})$ for each bin. Column (7): Observed He ii/H$\beta$
intensities. Column (8): Hardness of the ionizing spectra between 54.4 eV
(He+) and 13.6 eV (H) following Eq. 1 and assuming $f_{\rm esc}^{\rm He+}$
$\approx 0$. Upper limits refer to a $2\sigma$ limit.

We built two different groups of stacked spectra, one composed of bins in
absolute LyC escape fraction, $f_{\rm esc}\rm(LyC)$, and another one composed
of bins in $f_{\rm esc}\rm(LyC)$ and $12+\log({\rm O/H})$. For the first
group, we defined the four bins using $f_{\rm esc}$ determined from the UV
analysis of Saldana-Lopez et al. (2022). For the second group, we built nine
stack spectra considering three bins of $f_{\rm esc}\rm(LyC)$ and three bins
of metallicity. Table 1 contains the definition of the bins and the main
derived properties from the stacked spectra.

For each bin, the SDSS spectra were deredshifted using the systemic redshifts
from the observed wavelengths of bright optical lines and were resampled using
a linear interpolation onto a common wavelength grid. Next, we normalized the
spectra at $\lambda=4750-4850$Å, that is, relatively free of emission or
absorption features. Finally, we stacked all spectra by averaging the flux in
each spectral bin. We also tested other stacking methods using the median and
the weight-average using the uncertainty spectra, but no significant
differences in the observed intensity of He ii/H$\beta$ were found between
these three methods. Fig. 1 shows the four stacked spectra in bins of $f_{\rm
esc}\rm(LyC)$. He ii is detected at least in three stacks, together with other
faint lines, such as [Fe iii] 4658Å and the [Ar iv] doublet at 4711Å and
4740Å. We note the lack of significant broad emission in the stacks around
4650Å and 5808Å, which might be associated with a significant contribution of
Wolf-Rayet stars (e.g., Brinchmann et al., 2008).

Figure 1: Normalized stacked SDSS spectra in bins of $f_{\rm esc}$: Nonleakers
(red), weak leakers (orange), strong leakers (green), and very strong leakers
(blue). Regions around He ii 4686Å and H$\beta$ 4861Å are highlighted in gray.
He ii is detected in the four stacks with $\rm S/N\sim 2-3$. Other faint lines
are also detected ([Fe iii] 4658Å , [Ar iv]+He i 4712Å, and [Ar iv 4740Å), and
their positions are marked with dashed gray lines.

###  2.3 XShooter/VLT observations

Spectroscopic observations of eight LzLCS sources were carried out during 2021
with the XShooter instrument on the Very Large Telescope (VLT), with total
integration times of 50-100 min. The XShooter spectra were reduced in a
standard manner using the ESO Reflex reduction pipeline (version 2.11.5;
Freudling et al. 2013) to produce flux-calibrated spectra. The Galaxy and
internal reddening were corrected using the same method as described in
Section 2.2. We also used the measurements for five other LyC emitters
observed with XShooter by Guseva et al. (2020). In total, He ii is detected in
nine sources at ¿$2\sigma$, with intrinsic He ii/H$\beta\approx 0.007-0.020$.
For the remaining four sources, $2\sigma$ upper limits between $\simeq 0.004$
and $\simeq 0.010$ are inferred.

##  3 Results and discussion

###  3.1 HeII/H$\beta$ versus escape of LyC photons

The main result of our study is presented in the left panel of Fig. 2, which
shows the relation between the observed He ii/H$\beta$ ratio and the observed
LyC escape fraction $f_{\rm esc}$. Overall, we find no significant dependence
of the He ii/H$\beta$ intensity on $f_{\rm esc}$ from the SDSS stacks.

Figure 2: Intensity of He ii/H$\beta$ as a function of $f_{\rm esc}\rm(LyC)$
(left) and $12+\log({\rm O/H})$ (right) for different SDSS stacks (circles).
The left panel shows the four stacks in bins of $f_{\rm esc}\rm(LyC)$ (very
strong, strong, weak, and nonleakers in blue, green, yellow, and red,
respectively), and the right panel shows the nine stacks in bins of $f_{\rm
esc}\rm(LyC)$ and $12+\log({\rm O/H})$($12+\log({\rm O/H})$¡8.0, 8.0-8.3, and
$>$8.3, in blue, green, and red, respectively). Details of the definition of
stacks can be found in Table 1. $x$-values and horizontal error bars refer to
the mean and 68% confidence intervals (bootstrap resampling) of $f_{\rm
esc}\rm(LyC)$ and $12+\log({\rm O/H})$ measured for each bin (Table 1).
Individual sources with XShooter observations are also shown (empty squares;
LzLCS sources and five LyC emitters were analyzed in Guseva et al. 2020).
Upper limits for SDSS stacks and XShooter observations refer to $2\sigma$
limits. The right panel also shows observations (plusess) of a compilation of
almost 900 star-forming galaxies from Izotov et al. (2016a) (average and
standard deviation marked with a solid line and region, respectively).

To quantify this, we computed Kendall’s-$\tau$ correlation coefficient between
these two variables following Akritas & Siebert (1996), which allows the
inclusion of upper limits. Using the results from SDSS stacks, we find
$\tau(\rm SDSS)=0.00^{+0.33}_{-0.50}$ ($p$-$value=1$, uncertainties derived
using a Monte Carlo method), which is basically consistent with a null
correlation within our uncertainties. On the other hand, using the results
from the 13 individual sources with deep XShooter spectra, we find a positive
correlation with $\tau(\rm XS)=0.603^{+0.103}_{-0.128}$ ($p$-$value=4.14\times
10^{-3}$). However, these results are statistically less meaningful than those
obtained with SDSS stacks using the average properties of 89 sources that also
include the XShooter sources. In addition, the underlying effect of
$12+\log({\rm O/H})$ on He ii/H$\beta$ has not been considered so far.

The intensity of He ii/H$\beta$ of star-forming galaxies is known to vary with
metallicity in that it increases with decreasing of $12+\log({\rm O/H})$
(e.g., Brinchmann et al., 2008; Shirazi & Brinchmann, 2012; Schaerer et al.,
2019). As our sample spans a wide range of metallicities, $12+\log({\rm
O/H})=7.5-8.6$ (Flury et al., 2022a), this behavior should be considered in
our analysis. Therefore, we used nine SDSS stacks grouped in three bins of
$f_{\rm esc}$ and three bins of $12+\log({\rm O/H})$ (see Table 1 for
details).

The right panel of Fig. 2 shows the relation between He ii/H$\beta$ and
$12+\log({\rm O/H})$ for strong LyC leakers (solid circles) and non- or weak
LyC leakers (empty and dashed circles) in the metallicity bins. For
comparison, we also show the observed intensities of He ii/H$\beta$ and
metallicities of a compilation of almost 900 star-forming galaxies from SDSS
DR14 (pluses; Izotov et al., 2016a), for which the [O iii] 4363Å line is
detected with an accuracy better than 4$\sigma$, allowing thus direct
abundance determinations using the $T_{\rm e}$ method. The overall properties
of the parent sample are discussed in Guseva et al. (2019).

Two main conclusions can be drawn from the right panel of Fig. 2. First,
strong LyC leakers present roughly the same He ii/H$\beta$ intensities as non-
or weak leakers at comparable metallicities. For instance, strong He
ii/H$\beta$ intensities are found for both nonleakers and strong LyC leakers
in the low-metallicity bin (He ii/H$\beta$ $=0.030\pm 0.009$ and $0.020\pm
0.005$, respectively), while sources with $12+\log({\rm O/H})=8.0-8.3$ show He
ii/H$\beta$ $\sim 0.012$, independently of $f_{\rm esc}\rm(LyC)$. Moreover,
for a specific range of $12+\log({\rm O/H})$, the line intensities He
ii/H$\beta$ inferred for strong leakers do not differ from those typically
observed in other star-forming galaxies (Izotov et al., 2016a). Second, low-
metallicity star-forming galaxies present stronger He ii/H$\beta$ than high-
metallicity galaxies. This trend has been reported in other works of
increasing the He ii/H$\beta$ intensity with decreasing metallicity. The same
trend is seen in the individual sources with XShooter observations, where we
find an anticorrelation between He ii/H$\beta$ and metallicity of $\tau(\rm
XS)=-0.487^{+0.103}_{-0.103}$ ($p$-$value=2.04\times 10^{-2}$). For example,
the two XShooter sources with the highest He ii/H$\beta$ ratios ($\simeq
0.02$) present $12+\log({\rm O/H})$¡8.0, while the sources that are not
detected in He ii all have $12+\log({\rm O/H})$¿8.0. This behavior explains
the positive correlation between He ii/H$\beta$ and $f_{\rm esc}$ seen in the
individual sources with deep XShooter spectra (Fig. 2, left).

###  3.2 No variation in hardness with LyC escape

The hardness of the ionizing radiation field between energies above 54.4 and
13.6 eV, described by $Q_{\rm He^{+}}/Q_{\rm H}$, is to first order related to
the relative recombination line intensities by

$\displaystyle I(4686)/I({\rm H}\beta)$ $\displaystyle=\frac{c_{4686}}{c_{{\rm
H}\beta}}\frac{(1-f_{\rm esc}^{\rm
He+})\int_{54.4}^{\infty}(F_{\nu}/h\nu)\,d\nu}{(1-f_{\rm
esc})\int_{13.6}^{\infty}(F_{\nu}/h\nu)\,d\nu}$ (1)
$\displaystyle=1.74\frac{(1-f_{\rm esc}^{\rm He+})\,Q_{\rm He^{+}}}{(1-f_{\rm
esc})\,Q_{\rm H}},$

where $Q$ expresses the number of ionizing photons emitted above the
corresponding ionization potential, and $c_{ji}=h\nu_{ji}\frac{\alpha^{\rm
eff}_{ji}}{\alpha_{B}}$ relates the recombination rate to the line intensity
(Osterbrock & Ferland, 2006). For $c_{ji}$ we have adopted typical values of
the electron temperature ($T_{e}=10$ kK). The above expression also accounts
for the escape of ionizing photons, which are _a priori_ different for He+ and
H-ionizing photons. It is expected that $f_{\rm esc}^{\rm He+}\ll f_{\rm esc}$
because the doubly ionized He region is generally significantly smaller than
the H ii region, or in other words, He+-ionizing photons are absorbed much
closer to the source than those of lower energy. Except possibly for extremely
hard, power-law-like ionizing spectra, $f_{\rm esc}^{\rm He+}$ is therefore
expected to be very low, or at least lower than $f_{\rm esc}$.

We have shown that the intensity of He ii/H$\beta$ does not depend on $f_{\rm
esc}$and is primarily driven by variations with metallicity. Therefore, Eq. 1
implies that the hardness $Q_{\rm He^{+}}/Q_{\rm H}$ does not correlate with
the LyC escape, since, if anything, He ii/H$\beta$ should increase with
increasing $f_{\rm esc}$ even for constant $Q_{\rm He^{+}}/Q_{\rm H}$ (see Eq.
1). Table 1 provides the inferred values of $Q_{\rm He^{+}}/Q_{\rm H}$ for all
the stacks using Eq. 1 and assuming $f_{\rm esc}^{\rm He+}$ $\approx 0$. No
variation is found in $Q_{\rm He^{+}}/Q_{\rm H}$ between strong and nonleakers
within the uncertainties.

###  3.3 Implications

Our results rule out LyC leakage to explain the origin of nebular He ii
emission put forward by Pérez-Montero et al. (2020). Their galaxies present
roughly the same He ii/H$\beta$ intensities at comparable metallicity as those
studied in our work (see their Figure 1). According to these authors, the
observed intensities of He ii could be explained by density-bounded H ii
regions with very highly ionizing photon leaking, with a mean $f_{\rm
esc}\rm(LyC)\simeq 0.74$ for the entire sample. Clearly, the results shown in
Fig. 2 contradict this scenario, where the same intensities of He ii/H$\beta$
are found in sources with both very high and very low $f_{\rm esc}\rm(LyC)$,
that is, without any dependence on $f_{\rm esc}$. Furthermore, the invoked
mean $f_{\rm esc}\simeq 0.74$ is significantly higher than measured in
comparable low-$z$ galaxies (Flury et al., 2022a, b), and for the strongest
known low-$z$ leaker (J1243$+$4646 with $f_{\rm esc}\simeq 0.73$ reported by
Izotov et al. (2018b), or $f_{\rm esc}\simeq 0.89$ inferred in Saldana-Lopez
et al. (2022)), we measure He ii/H$\beta$ $<0.02$ (2$\sigma$), which is not
exceptionally high compared to many other star-forming galaxies with similar
metallicity ($12+\log({\rm O/H})$ $\simeq 7.90$, cf. Fig. 2, right panel).
Other mechanisms or sources are needed to explain the origin of nebular He ii
(see, e.g., Olivier et al., 2021; Simmonds et al., 2021, and references
therein).

At $z\sim 2$, Naidu et al. (2022) recently identified two groups of LAEs, one
probably showing strong LyC escape, and the other low LyC escape fractions.
Comparing the stacked spectra of these two groups, they found differences in
their observed rest-frame UV spectra, with the strong leaker candidates
showing high-ionization lines of C iv 1550Å and He ii 1640Å, and the other
group showing only lower-ionization lines (e.g., C iii 1909Å). From this
finding, Naidu et al. (2022) suggested that strong LyC leakers could have
harder ionizing spectra. On the other hand, if the galaxies studied in our
work are comparable to high-$z$ LAEs, our results imply that these observed
differences are not related to LyC escape.

Generally speaking, the absence of hardness variations with $f_{\rm esc}$
shows that this property of the global radiation field does not determine the
conditions for the ionizing photon escape. In contrast, other studies have
found significant correlations between different physical properties and
$f_{\rm esc}$, which might indicate such physical processes. This includes
highly concentrated star formation (high SFR surface densities), a high
ionization parameter, an inhomogeneous ISM and dust distribution, and low
amounts of dust (Verhamme et al., 2017; Gazagnes et al., 2018; Cen, 2020;
Flury et al., 2022b; Saldana-Lopez et al., 2022). Our finding does not exclude
that radiative processes contribute to determining LyC escape, but it
indicates that the hardness of the radiation field (over the energy range
measured here) is not fundamental. For example, this suggests that LyC escape
is not related to low-luminosity active galactic nuclei (AGNs). However, our
differential study does not explain the origin of nebular He ii emission,
which is known to require sources of ionizing photons above 54 eV in amounts
that are not predicted for normal stellar populations (see, e.g., Shirazi &
Brinchmann, 2012; Stasińska et al., 2015; Schaerer et al., 2019).

##  4 Summary

We have empirically investigated the hardness of the ionizing spectra between
54.4 and 13.6 eV ($Q_{\rm He^{+}}/Q_{\rm H}$, i.e., the ionization potential
of He+ and that of H) as a function of the LyC escape fraction, $f_{\rm
esc}\rm(LyC)$, of a large sample of star-forming galaxies at low redshift for
which LyC are available from HST observations. Optical recombination lines of
He ii 4686Å and H$\beta$ 4861Å from SDSS and XShooter spectra were used to
determine $Q_{\rm He^{+}}/Q_{\rm H}$ and its dependence on $f_{\rm
esc}\rm(LyC)$. The underlying effect of metallicity was also considered.

We built stacked spectra in bins of $f_{\rm esc}\rm(LyC)$ and metallicity,
allowing us to study the behavior of He ii/H$\beta$ across a wide range of
$f_{\rm esc}\rm(LyC)\simeq 0-0.9$ and $12+\log({\rm O/H})$ $=7.5-8.6$. We find
that the intensity of He ii/H$\beta$ does not depend on $f_{\rm esc}\rm(LyC)$.
Very strong leakers (¡$f_{\rm esc}\rm(LyC)$¿ $\simeq 0.5$) and nonleakers
(¡$f_{\rm esc}\rm(LyC)$¿ $\simeq 0$) have similar intensities of He
ii/H$\beta$ on average, about He ii/H$\beta$ $\simeq 0.01-0.02$. Instead, we
find that He ii/H$\beta$ is primarily driven by variations in metallicity,
where He ii/H$\beta$ increases with decreasing $12+\log({\rm O/H})$, in
particular, for $12+\log({\rm O/H})$ $<8.0$, as known from previous studies
(e.g., Shirazi & Brinchmann, 2012; Schaerer et al., 2019). At comparable
metallicities, strong LyC leakers present roughly the same He ii/H$\beta$
intensities as non- or weak leakers and as in many other normal star-forming
galaxies, where nebular He ii is detected.

In short, our results demonstrate that $Q_{\rm He^{+}}/Q_{\rm H}$ does not
correlate with $f_{\rm esc}\rm(LyC)$. This implies that strong LyC emitters do
not show harder ionizing spectra than nonleakers at similar metallicity.
Future studies will address other hardness or softness indicators of the
radiation field and will more broadly examine the nebular properties of the
galaxies from the LzLCs.

###### Acknowledgements.

The authors thank the referee for useful comments that greatly improved the
clarity of this work. Y.I.I. acknowledges support from the National Academy of
Sciences of Ukraine by its priority project “Fundamental properties of the
matter in the microworld, astrophysics and cosmology”. R.A. acknowledges
support from ANID Fondecyt Regular Grant 1202007. Funding for the Sloan
Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the
U.S. Department of Energy Office of Science, and the Participating
Institutions. SDSS-IV acknowledges support and resources from the Center for
High Performance Computing at the University of Utah. The SDSS website is
www.sdss.org.

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