EMPRESS. XIV. Strong High-ionization Lines of Young Galaxies at z = 0–8: Ionizing Spectra Consistent with the Intermediate-mass Black Holes with M BH ∼ 103–106 M ⊙

We present ionizing spectra estimated at 13.6–100 eV for 10 dwarf galaxies with strong high-ionization lines of He ii λ4686 and [Ne v] λ3426 ([Ne iv] λ2424) at z = 0 (z = 8) that are identified in our Keck/LRIS spectroscopy and the literature (the JWST Early Release Observations program). With the flux ratios of these high-ionization lines and >10 low-ionization lines of hydrogen, helium, oxygen, neon, and sulfur, we determine ionizing spectra consisting of stellar and nonthermal power-law radiation by photoionization modeling with free parameters of nebular properties, including metallicity and ionization parameter, canceling out abundance ratio differences. We find that all of the observed flux ratios are well reproduced by the photoionization models with the power-law index α EUV of α EUV ∼ (–1) − 0 and the luminosity L EUV of L EUV ∼ 1040–1042 erg s−1 at ∼55–100 eV for six galaxies, while four galaxies include large systematics in α EUV caused by stellar radiation contamination. We then compare α EUV and L EUV of these six galaxies with those predicted by the black hole (BH) accretion disk models and find that α EUV and L EUV are similar to those of the intermediate-mass BHs (IMBHs) in BH accretion disk models, albeit with possibilities of the other scenarios. Confirming these results with a known IMBH having a mass M BH of M BH = 105.75 M ⊙, we find that four local galaxies and one z = 7.665 galaxy have ionizing spectra consistent with those of IMBHs with M BH ∼ 103–105 M ⊙.


INTRODUCTION
Studies over the last two decades have revealed that massive galaxies are typically harboring super massive black holes (SMBH) with masses of 10 6−10 M ⊙ at their centers.However, the formation of SMBH is puzzling.SMBHs with masses of ∼ 10 9 M ⊙ were already formed at z ∼ 6−7 when the universe was equal to or less than 1 Gyr old (e.g., Mortlock et al. 2011;Bañados et al. 2018;Wang et al. 2021).Because a spherical mass accretion onto a BH is not very efficient due to radiation pressure from accreting gas, it is suggested that SMBHs form via massive seed BHs with intermediate masses of ∼ arXiv:2305.02189v2[astro-ph.GA] 10 Mar 2024 10 2−5 M ⊙ produced via population III stars and direct collapse (e.g., Hirano et al. 2014;Omukai 2001).
Although such IMBHs (M BH ∼ 10 2−5 M ⊙ ) are not well understood in the high redshift as well as in the local universe due to diffuculties of observations, recent optical surveys in the local universe have identified lowmass AGNs in dwarf galaxies that harbour BHs with masses down to M BH ∼ 10 5 M ⊙ (Xiao et al. 2011).There are growing number of IMBH candidates in dwarf galaxies with report of high-ionization lines or Hα broad lines (Chilingarian et al. 2018;Molina et al. 2021;Reefe et al. 2022;Senchyna et al. 2020;Bohn et al. 2021;Cann et al. 2021;Berg et al. 2021;Mezcua & Sánchez 2024).On the other hand, gravitational wave observations have revealed IMBHs as well as stellar BHs with masses up to M BH ∼ 10 2 M ⊙ (Abbott et al. 2020).There is a gap of BH masses known to date in the mass range of M BH ∼ 10 3 −10 4 M ⊙ , which is a missing piece of SMBH formation (Greene et al. 2020).
Among dwarf galaxies, extremely metal poor galaxies (EMPGs), are promising galaxies to explore the missing IMBHs with the masses of M BH ∼ 10 3 − 10 4 M ⊙ .EMPGs are low metallicity Z and stellar-mass M * galaxies, typically having Z ∼ 0.01−0.1 Z ⊙ and 10 5 −10 8 M ⊙ , respectively.The properties such as low metallicity and low stellar mass indicate the possibility that EMPGs are experiencing the early phase of star and BH formation.Observations have shown the existence of hard ionizing radiation in EMPGs whose strong high ionization lines of He iiλ4686 and [Ne v]λ3426 cannot be explained with stellar synthesis models alone (e.g., Izotov et al. 2004a;Thuan & Izotov 2005;Izotov et al. 2012Izotov et al. , 2021;;Schaerer et al. 2019).The He iiλ4686 and [Ne v]λ3426 lines are observed in various local galaxies including EMPG (Garnett et al. 1991;Guseva et al. 2000;Thuan & Izotov 2005;Brinchmann et al. 2008;López-Sánchez & Esteban 2010;Shirazi & Brinchmann 2012;Izotov et al. 2004aIzotov et al. , 2012;;Berg et al. 2021;Izotov et al. 2021).Fast radiative shocks, which may be produced by supernovae (SNe), are proposed to explain the observed [Ne v]λ3426 emission line fluxes and/or ratios in some low-metallicity galaxies (e.g.Thuan & Izotov 2005;Izotov et al. 2012Izotov et al. , 2021)).If the [Ne v]λ3426 emission lines originate from SNe, the [Ne v]λ3426 emission line fluxes may show temporal variability.We have searched for low-metallicity dwarf galaxies with multiple observations of [Ne v]λ3426 lines and have found a galaxy dubbed SBS 0335-052E.The [Ne v]λ3426 emission lines of SBS 0335-052E are observed in September 2003 andFebruary 2004 (Izotov et al. 2009;Thuan & Izotov 2005).The extinction corrected emission line ratios of [Ne v]λ3426/[Ne iii]λ3869 are 0.0387 ± 0.0068 and 0.0306 ± 0.0026 for each ob-servation.We find that the two ratios are consistent within ∼ 1-2 sigma, and we find no significant temporal variability in the [Ne v]λ3426 lines.The hard ionizing radiation may be caused by BHs (e.g., Umeda et al. 2022;Olivier et al. 2021;Simmonds et al. 2021).There is a possibility that X-ray binaries (XRBs) having the stellar-mass BHs produce the hard ionizing radiation (Garofali et al. 2024;Schaerer et al. 2019), although a famous dwarf galaxy SBS 0335-052E, which shows strong He iiλ4686 emission, has an X-ray luminosity lower than expected from XRBs.(Kehrig et al. 2018;Saxena et al. 2020;cf. Thuan et al. 2004).ULXs have been proposed to explain for the observed [Ne v]λ3426 for some dwarf galaxies (Umeda et al. 2022;Simmonds et al. 2021).Another possibility is ionizing radiation originating from a massive BH (e.g., IMBH or SMBH) residing at the center of a galaxy that produces X-ray spectum softer than that of a stellar-mass BH (Kawaguchi 2003;Thuan & Izotov 2005;Izotov et al. 2012;Plat et al. 2019;Izotov et al. 2021).It should be noted that stellar masses of EMPGs are small, M * = 10 5 − 10 8 M ⊙ (Isobe et al. 2021(Isobe et al. , 2022;;Kojima et al. 2020;Berg et al. 2021;Sánchez Almeida et al. 2016).If the local stellar-mass to SMBH mass relation is extrapolated to the low mass regime, the BH masses may be about 1/1000 of the stellar masses, M BH ∼ 10 2 − 10 5 M ⊙ that fall in the missing IMBH mass range (Reines & Volonteri 2015).Theoretical models suggest that IMBHs have hot accretion disks with temperature of ∼ 10 5−6 K producing hard blackbody radiation in the low-energy X-ray band of 13.6 -100 eV (Kawaguchi 2003).However, such hard radiation in 13.6 -100 eV cannot be directly observed due to the strong hydrogen absorption around BHs.
Here we estimate spectra of dwarf galaxies including EMPGs in the hard energy band, exploiting the photoionization modeling with observed optical emission line fluxes that is established by Umeda et al. (2022;hereafter U22).U22 reconstruct spectra in the energy range of 13.6-54 eV with observed emission lines from hydrogen Balmer lines to Heiiλ4686 whose ionization energies range in 13.6-54 eV.In this study, we extend the U22 modeling technique by incorporating [Ne v]λ3426 and [Ne iv]λ2424 emission lines, which have much higher ionizing energies of 97 eV and 63 eV, respectively, to cover the ionizing spectra up to ∼ 100 eV and test the IMBH scenario, while we do not aim to rule out the other possibilities (e.g., HMXB, radiative shocks).Furthermore, the model of stellar contributions at < 60 eV is refined compared to the U22 model by using stellar synthesis model instead of simple blackbody radiation used in U22 model, allowing for more accurate determination of the spectral shape of the hard component in the 55-100 eV band that is hereafter referred to as the extreme ultraviolet (EUV) band.Because we free spectral index and amplitude of the power-law component, our SEDs are generalized and reproduce both soft and hard spectral shapes.
It is suggested that temperature and electron density structures are dependent on ionizing spectra (Baskin & Laor 2021).To include such conditions, some studies utilize multiple-zone models (e.g.Berg et al. 2021).However, it is unknown whether all the emission lines including Hβ, He iiλ4686, and [Ne v]λ3426 can be reproduced in a one-zone model.In fact, Izotov et al. (2021) do not find a solution of a one-zone model reproducing He iiλ4686/Hβ and [Ne v]λ3426/Hβ ratios observed in some dwarf galaxies with [Ne v]λ3426 detection (e.g.Tol 1214).In this study, we explore one-zone models that can reproduce emission line ratios.In this sense, multiple zone models are out of the scope in our study.
The structure of this paper is as follows.In Section 2, we explain our Keck observations for EMPGs.We present our sample in Section 3. In Section 4, we describe our modeling.In Section 5, we apply our modeling technique to galaxies in our sample, and determine the best-estimate spectra over ∼ 13.6−100 eV.We compare our best-estimate spectra with BH accretion disk models in section 6.Throughout this paper, the magnitudes are in AB system, and we adopt a cosmological model with H 0 = 70 km s −1 Mpc −1 , Ω Λ = 0.7, and Ω m = 0.3.We use the solar metallicity scale of Asplund et al. (2009), where 12 + log(O/H) = 8.69.

Spectroscopic Observations with Keck/LRIS
We observed EMPGs including 2 bright famous EMPGs, SBS 0335-052E and HS 0122+0743 (hereafter HS 0122), with Keck/LRIS on 2021 November 7 and 8 (PI: K. Nakajima).We placed 0. ′′ 7 wide slits for all of the targets.For the blue (red) channel of LRIS, we utilized the 600/4000 grism (600/7500 grating) which provides a spectral resolution of ∼ 4 (5) Å in FWHM.The LRIS spectroscopy of the blue and red channels cover the wavelength ranges of λ ∼ 3000 − 5500 and 6000 − 9000 Å, respectively.We also observed Feige 34 for flux calibration.The weather was clear during the observations, and the seeing sizes range in 0.8 -1.0 arcsec.The spectroscopy of SBS 0335-052E taken in the Keck/LRIS observations is shown in Hatano et al. (2023).Because they only show the spectra taken with the red arm of Keck/LRIS and the Hα emission line fluxes of SBS 0335-052E, we present the spectra of the blue arm and the obtained fluxes and flux ratios of emission lines besides the Hα emission line in this paper.

Data Reduction
We use the IRAF package (Tody 1993(Tody , 1986) ) for the data reduction of the LRIS spectra.Wavelength solutions for the spectra are obtained from the HgN-eArCdZnKrXe lamp, performing bias subtraction, wavelength calibration, one-dimensional (1D) spectrum extraction, flux calibration, atmospheric-absorption correction, Galactic-reddening correction, and slit loss correction.Each spectrum is flux-calibrated with the sensitivity curve derived with Feige 34 data.Atmospheric absorption correction is corrected with the extinction curve at Maunakea Observatories (Bèland et al. 1988).The Galactic-reddening is corrected with the NASA/IPAC Infrared Science Archive (IRSA) 1 based on the Schlafly & Finkbeiner (2011) estimates.We produce noise frame including readout noise and photon noise of sky and object emission.One-dimensional spectra are extracted in the spatial width of 2 times the FWHM(Hβ), where FWHM(Hβ) is the full width half maximum value of the spatial distribution of Hβ emission.Slit loss is estimated with the spatial profile of the Hβ emission along the slit.
Figure 1 shows the one-dimensional spectra taken with LRIS.

Flux Measurement
We fit each emission line with Gaussian profiles with the scipy.optimizepackage, using four free parameters: The amplitude, line width, line central wavelength of Gaussian profile, and continuum of the EMPGs.sky+object emission.In our Monte Carlo simulations, we produce 1000 mock spectra, adding random errors to our galaxy spectra.We perform line flux measurements for the 1000 mock spectra as we have done for the flux measurement, and obtain one sigma errors de-fined by the 68 percentile range in the distribution of the mock line fluxes.Because we find that three pixels in the 2D spectrum of [O iii]λ5007 emission line of SBS 0335-052E are saturated, we do not use the [O iii]λ5007 emission line in the later analysis for SBS 0335-052E.We calculate dust extinction E(B − V ), electron temperature T e , and electron density n e iteratively in the same manner as Isobe et al. (2022) with Hβ, Hγ and Hδ.We derive three intrinsic Balmer line raitos of Hβ, Hγ, Hδ with PyNeb v.1.1.15(Luridiana et al. 2015) and calculate three E(B −V ) values, assuming the case B recombination and the dust attenuation curve of Calzetti et al. (2000).An error of E(B − V ) is derived based on a χ 2 value calculated with least squares.The dustcorrected fluxes of SBS 0335-052E and HS 0122+0743 are summarized in Table 2.
We point out that the [Ne v]λ3426/[Ne iii]λ3869 of SBS 0335-052E is consistent with those reported in Izotov et al. (2009) and Thuan & Izotov (2005), whose observations are conducted in September 2003 and February 2004, respectively.
He iiλ4686/Hβ emission line ratios in the dwarf galaxies are relatively lower than typical AGNs.The reason of the low He ii/Hβ emission is explained by the fact that, in dwarf galaxies, stellar radiation is much stronger than AGN radiation at the energy level of the hydrogen ionizing photons, ∼13.6 eV, producing Hβ emission.It is claimed that non-stellar radiation such as AGNs are thought to contribute only ∼10 per cent to the total luminosity of ionizing radiation of dwarf galaxies with the [Ne v]λ3426 line detection (e.g.Izotov et al. 2012Izotov et al. , 2021)).(5.33 ± 0.12)   3. SAMPLE

Making Our Sample
We use eight dwarf galaxies with detection of [Ne v]λ3426 emission lines, two of which, SBS 0335-052E and HS 0122, are taken from our observations, while six dwarf galaxies are taken from the literature (Izotov et al. 2021(Izotov et al. , 2004a;;Berg et al. 2021).For testing our mass estimate technique, we search for low-mass SMBHs and IMBHs with a [Ne v]λ3426 emission line detection in the database of SDSS spectra.Because we have accretion disk model SEDs with masses only up to M BH ≲ 10 6.5 M ⊙ , we cannot use SMBHs with M BH ≳ 10 6.5 M ⊙ .We find an IMBH dubbed J024009.10+010334.5 (hereafter J024009; Xiao et al. 2011), harboring an AGN with a low BH mass of M BH = 10 5.75 M ⊙ that shows a [Ne v]λ3426-line detection.We also find some candidate galaxies in the LBT archive.However, we do not have access to the spectra.In summary, we find only one IMBH, J024009, for the testing purposes.The emission line fluxes of J024009 are measured in the same method explained in Section 2.3.The extinction correction is applied to J024009 with Hβ/Hγ ratio, assuming electron temperature of T e = 8, 000 K and deriving electron density n e with [S ii] ratio.
We add a star forming galaxy at z = 7.665, ID 6355, recently identified in James Webb Space Telescope (JWST) observations with [Ne iv]λ2424 line detection to our sample (Curti et al. 2023;Brinchmann 2022).The data was taken by JWST/NIRSpec in Early Release Observations (ERO; Pontoppidan et al. 2022).The spectra were taken with medium resolution gratings/filters of G235M/F170LP and G395M/F290LP, which cover the wavelength range of 1.7-3.1 and 2.9 -5.1 µm, respectively.We use the data reduced in Nakajima et al. (2023) 3.
To summarize, our sample is composed of a total of 10 galaxies: 8 local star-forming galaxies, 1 known local low-mass AGN, and 1 high-z star forming galaxy.Properties of the 10 galaxies are summarized in Table 1.
We investigate the mid-infrared (MIR) colors of the dwarf galaxies, because MIR colors are good indicators of the presence of hot-dust, which could originate from AGN activities.We use W 1 (3.4 µm), W 2 (4.6 µm), and W 3 (12 µm) photometry in the AllWISE catalogue (Wright et al. 2010).We show the mid-infrared (MIR) colors of the dwarf galaxies and AGN criteria of Stern et al. (2012) and Jarrett et al. (2011) in Figure 2. We point out that the MIR colors of the dwarf galaxies, except for HS 0122, satisfy an AGN criterion (W 1 − W 2 > 0.8) proposed in Stern et al. (2012) (cf.Hainline et al. 2016).Moreover, four of the dwarf galaxies, J1205, J0344, J1222, and J024009 satisfy an AGN criterion proposed in Jarrett et al. (2011).Some of the dwarf galaxies show temporal variability.J1222 is claimed to harbor an AGN on the basis of a presence of Hα broad line lasting for more than 10 years (Burke et al. 2021;Izotov et al. 2021).J1205 show temporal variability in W 1 and W 2 photometry in NEOWISE data (Mainzer et al. 2014;Harish et al. 2023).SBS 0335-052E show a temporal variability in NEOWISE data (Hatano et al. 2023).

METHOD
As explained in Section 1, our model follow the method of U22 with an improvement of stellar radiation by using BPASS v.2.2.1 single models (Stanway & Eldridge 2018).

Photoionization Models
We used version 17.02 of CLOUDY, last described by Ferland et al. (2017), to calculate the model emission line fluxes.We stop the radiative transfer calculation when a neutral hydrogen column density N H I reaches N H I = 10 21 cm −2 .We normalize the output emission line fluxes by model Hβ flux, for convenience of calculation.The detailed settings of CLOUDY models are explained below.

Geometry and Density
We assume a photionization model having closed and spherical geometry, with a constant hydrogen density.We adopt the hydrogen density n H ranging in 10 0.5 − −10 5 cm −3 .The inner and outer radii of the gas cloud R in and R out are fixed at the default values.

Ionizing Spectra
We use a combination of stellar and power-law radiation for an input ionizing spectrum.The input ionizing spectrum is defined by where S(ν, t) is the stellar spectrum of a BPASS singleburst model at a stellar age t and frequency ν, C mix is the mixing parameter, and P (ν, α X ) is the power-law spectrum.We use BPASS single-burst models because U22 have pointed out that BPASS binary models cannot reproduce ionizing spectra of some EMPGs.Here, P (ν, α X ) is expressed as where α X and h are the power-law index and the Planck constant, respectively.To avoid strong free-free (pair-creation and Compton) heating, we set the lower (higher) energy cut E lc (E hc ) for the power-law spectrum.Here, we set E lc = 0.1 Ryd and E hc = 10, 000 Ryd.We use normalized mixing parameter a mix as an input parameter of CLOUDY, where a mix is defined as We adopt the parameter ranges of 6 ≤ log (t stellar /yr) ≤ 8, −3 ≤ α X ≤ 1, and −4 ≤ log a mix ≤ 3, respectively.

Chemical Abundance
In our model, we set chemical abundances with oxygen abundance (hereafter referred to as Z).We scale all chemical abundances linearly with solar abundance ratios given by Grevesse et al. (2010), with the exception of helium, carbon, and nitrogen.We calculate helium and carbon (nigrogen) abundances with nonlinear formula given by Dopita et al. (2006) (López-Sánchez et al. 2012).We allow chemical abundances to vary in the range of −3 ≤ log(Z/Z ⊙ ) ≤ 0.

Ionization Parameter
We define the ionization parameter U by where Q(H 0 ) is the intensity of hydrogen ionizing photons R S is the Strömgren radius, and c is the speed of light.We substitute R in for R S to calculate the ionizing parameter U .We limit the ionizing parameter in the range of −5 ≤ log U ≤ −0.5.

MCMC Parameter Estimates
To estimate the best-fit parameters of our CLOUDY photoionization model, we use emcee, a Python implementation of an affine invariant MCMC sampling algorithm (Foreman-Mackey et al. 2013).We maximize the log-likelihood function given by (5) where Λ is the set of emission lines, F λ,obs (F λ,mod ) are the observed (model) emission line fluxes at wavelength λ, and σ λ are the observed errors of the emission line fluxes at λ normalized at F Hβ,obs = 100.Here, F λ,mod is defined by where N Hβ is the normalization factor for an Hβ emission line.F λ,cloudy is the output value of CLOUDY for the emission line flux at λ.To account for the error of Hβ fluxes, we add N Hβ as a free parameter by letting it vary in [100 -3σ Hβ , 100 + 3σ Hβ ].
For the prior distribution, we use uniform distribution.We summarize the prior distributions of all 7 free parameters in Table 4.We set 40 walkers and run the MCMC sampling algorithm for ∼1000 steps, sampling ∼40,000 parameter sets in total.We define the "best-fit" parameter set as a parameter set having maximum likelihood value among all the sampled parameter sets.The uncertainty of the parameters is defined by the range of the parameter sets satisfying a condition lnL We note that the uncertainty provided in Equation 7 is only a rough standard, and the actual uncertainty can be more rigorously defined.

Best-Fit Parameters for the Nebular and Ionizing Spectral Models
We use observed ∼14 emission lines originating from hydrogen, helium, oxygen, sulfur, and neon ions summarised in Table 2 for the 9 local galaxies.We do not include Hα fluxes in our analysis because there may remain flux calibration systematics between our LRIS blue and red channel data, in the latter of which only Hα emission falls.To remove uncertainties of abundance ratio differences, we use the ratios, [S ii]λ6716/[S ii]λ6731 and [Ne v]λ3426/[Ne iii]λ3869, for sulfur and neon lines, We compare the observed emission line fluxes and ratios with the photoionization models (Section 4.1), performing the MCMC parameter estimates (Section 4.2).In Figure 3, we present the posterior probability distribution function (PDF) for the J0344 galaxy as an example, and determine the best-fit parameters and the associated errors.We also obtain the best-fit parameters and the uncertainties using Equation 7 for all of our dwarf galaxies that are summarized in Table 5.
The range of hydrogen density used in the MCMC technique does not include the critical density of the [Ne v]λ3426 emission line (n H ∼ n e = 1.5 × 10 7 cm −3 ), and the best-fit parameters could be found outside the current parameter range.We extend the range of hydrogen density up to log(n H /cm −3 ) = 10, which includes the critical density of the [Ne v]λ3426 emission line.We also extend the ionization parameter up to logU =1, encompassing the range of ionization parameters typically observed in the circum-nuclear environment of AGNs ( logU ∼ (−3)−1; Netzer 1990).We conduct MCMC calculations for Tol 1214 that has reliably high S/N ratios for high ionization lines including [Ne v]λ3426 with the new parameter ranges, and obtain the best-fit parameters: logU = −1.93 and logn H /cm −3 = 2.54.These (5): Hydrogen density.( 6): Gas-phase metallicity.( 7): Normalized factor for an Hβ emission line.
parameters do not change the conclusions of the M BH value.
We test our best-fit parameters by comparing the observed flux measurements with model fluxes calculated from the best-fit parameters.We define the relative residual fluxes by (F λ,mod − F λ,obs )/F λ,obs , and present the relative residual fluxes in Figure 4. Figure 4 indicates that the best-fit models reproduce almost all the observed emission lines within the 3 sigma levels with an exception of the [Ne v]/[Ne iii] of J104457.This exception is probably because J104457 have low S/N (∼ 1) for [Ne v]/[Ne iii] compared to other galaxies (S/N ≳ 3).Hereafter, we conduct the same analysis for J104457 as other galaxies just for the presentation purpose.

Best-Fit Ionizing Spectra
We determine the ionizing spectrum shapes F ν with the best-fit parameters and Equations ( 1) and (2).We then calculate the luminosity L ν from F ν with a conversion factor A.
A is obtained for each dwarf galaxy with a relation between the Hβ luminosity L(Hβ) and the number of hydrogen ionizing photons produced per second given by Ono et al. (2010).
Here we assume escape fraction of ionizing photons f ion esc as f ion esc = 0.The L(Hβ) values are calculated from F (Hβ) presented in Table 1.In Figure 5, we present L ν as a function of photon energy for all the galaxies.All the spectra of the galaxies show prominent power-law continua in the EUV range of 55-100 eV.The SED of local galaxies and ID 6355 are constrained in the range of ∼ 13.6 − 100 and ∼ 13.6 − 64 eV, respectively.
We define the EUV luminosity L EUV of the power-law continuum with the given parameters of α X , t, and a mix .
We extract the last 10 steps of the sampled parameter sets and calculate α EUV for all the extracted parameters.In some galaxies, the extracted α EUV values exhibited multiple peaks, including a primary peak associated with the best-fit parameters.We additionally extract parameter sets in the primary peak for SBS 0335-052E, HS 0122, J1222, J1205, W1702, J104457, Tol 1214, J0344, and ID 6355 and calculate L EUV and α EUV with the extracted parameter sets.Figure 6 presents α EUV as a function of L EUV for all our dwarf galaxies.For ID 6355, we extrapolate the power-law component of the best-fit spectrum up to 100 eV and calculate the L EUV .We define the stellar to power-law ratio at 55 eV (SPR(55 eV)) as where F S,55 eV and F P,55 eV are flux densities of stellar and power-law components at 55 eV of the best-fit spectrum, respectively.We calculate SPR(55 eV) for all the galaxies and show the SPR(55 eV) values at the top panel in Figure 6.The SPR(55 eV) distribution suggests that there are two different groups of galaxies.Because we constrain the power-law component with two points, 54 eV (He ii) and 97 eV ([Ne v]), spectra with prominent stellar component would give additional uncertainties.We remove the group of galaxies, J104457, J1222, and W1702, having prominent stellar contamination at 55 eV with criterion of SPR(55 eV) > 0.1.The removed galaxies are plotted in grey in Figure 6.

Comparing Our Results with BH Accretion Disk Models
We compare α EUV and L EUV values (Section 5.2) with BH accretion disk models given by Kawaguchi (2003;hereafter K03).We estimate the set of BH masses and accretion rates that explain both α EUV and L EUV values, assuming that the power-law continua in Section 4.1.2originate from BH accretion disks within physical parameters available in K03 models.The K03 models are accretion disk models (within 2 × 10 4 Schwarzschild radii from the central BH) including effects of electron scattering and the relativistic correction, predicting spectra in the wavelength from the far UV to X-ray bands.These models have parameters of BH mass M BH , viscosity α, and accretion rate ṁ ranging in M BH = 10 2 − 10 6.5 M ⊙ , α = 0.01 − 1, and ṁ = 1 − 1000, respectively.Model spectra with M BH = 1-10 5 M ⊙ have been used to investigate bright X-ray sources (e.g., Yoshida et al. 2010, Godet et al. 2012).The accretion rate parameter range covers a physically reasonable range from sub-Eddington ( ṁ < 16) to super-Eddington accretion.Examples of the K03 model spectra are plotted in Figure 7.
We determine the K03 models' EUV luminosities L EUV,K03 and power-law indexes α EUV,K03 .We calculate L EUV,K03 , integrating the K03 model spectra over 55-100 eV.We estimate α EUV,K03 by fitting a power law to the K03 model spectra in the 55-100 eV range using numpy.polyfit.Figure 8 presents α EUV,K03 and L EUV,K03 values of the K03 models in the given parameter ranges.

Black Hole Mass Estimates
In Figure 8, we compare the K03 models (α EUV,K03 and L EUV,K03 ; Section 6.1.1)with observations (α EUV and L EUV ; Section 5.2).First, we test our BH mass estimation method with J024009 that has an IMBH with M BH = 10 5.75 M ⊙ and super-Eddington accretion ( ṁ > 16) measured by optical spectroscopic observa-tions (Xiao et al. 2011).We show J024009 with the star mark in Figure 8. J024009 falls on the models of M BH = 10 5 M ⊙ suggesting that the BH mass of J024009 is M BH ∼ 10 5 M ⊙ .Also an accretion rate is consistent with super-Eddington accretion within errorbars.We thus regard that our BH mass estimation method is applicable to the galaxies with BH accretion disks within physical parameters available in K03 models.
Figure 8 indicates that the L EUV and α EUV of SBS 0335-052E (J1205 and Tol 1214) agrees with the models with M BH = 10 3−4 M ⊙ (M BH = 10 4−5 M ⊙ ) within the 2σ level and therefore the BH mass is estimated to be M BH ∼ 10 3−4 M ⊙ (M BH ∼ 10 4−5 M ⊙ ).J0344 is placed outside the regions covered by the models.However, J0344 is placed on or slightly above the orange dashed line connecting the data point of (M BH , ṁ, α) = (10 5 M ⊙ , 1, 1) and (10 6.5 M ⊙ , 1, 0.1).This suggest that J0344 is explained by BH accretion disk model with BH mass between 10 5 and 10 6.5 M ⊙ with sub-Eddington mass accretion.This suggests that J0344 harbors a massive BH with a mass of M BH ∼ 10 5 − 10 6.5 M ⊙ .The L EUV and α EUV of ID 6355 are consistent with those of M BH ∼ 10 5 M ⊙ BH accretion disk models within errorbars.This suggest that the ID 6355 is harboring a BH with a mass of M BH ∼ 10 5 M ⊙ .

BH Mass to Stellar Mass Relation
We plot the stellar and BH masses of the 6 galaxies with those of local dwarf galaxies in Figure 12.The stellar masses of J0344, Tol 1214-277, and J024009 are calculated with the relation of absolute i-band magni- tudes and stellar masses given in Isobe et al. (2021).The galaxies fall on the line or above the extrapolation of the local relation given in Reines & Volonteri (2015), suggesting that the BH mass to stellar mass ratio is the same or larger in the low mass range than in the high mass range.

Emission Line Profiles
AGNs often show broad Hα lines originating from the surrounding broad line regions, and the BH masses can be estimated from the strength and the width of the Hα lines (Greene & Ho 2005).There are Hα and Hβ broad lines with FWHM ≳ 1000 km s −1 reported in some of the dwarf galaxies in previous studies (e.g.Izotov et al.Criterion of stellar radiation to power-law continua ratio (0.1) are plotted as horizontal dots.The dwarf galaxies with larger value of stellar radiation to power-law continua ratio is plotted in grey in the both panels.We find a report of the broad line of Tol 1214 in Izotov et al. (2021Izotov et al. ( , 2004b)).The lower limit of the Hβ FWHM of Tol 1214 is ≳ 900 km s −1 and broad line to narrow line flux ratios are ∼ 1-2 %.For SBS 0335-052E, the Hα broad line is identified in Hatano et al. (2023) who place the BH mass upper limit of M BH < 1.4 × 10 8 M ⊙ .For HS 0122, we search for Hα or Hβ broad lines in the LRIS spectra reduced in Section 2.2.We find possible broad Hα and Hα lines in HS 0122, while these broad lines may originate from the instrumental broadening.For J1205 and J0344, there are reports of Hα broad lines in Izotov et al. (2021Izotov et al. ( , 2017)).For J0344, only the velocity  dispersion of Hα broad line is given and the line flux is not shown in Izotov et al. (2021).For J0344, we assume the broad line flux to the whole Hα flux ratio is 0.01.We estimate BH masses of Tol 1214, SBS 0335-052E, J1205, and J0344 from the velocity dispersions and fluxes of Hα broad line, using the equation ( 6) of Greene & Ho (2005).We compare the estimated BH masses with those obtained from our models in Table 6.The BH masses of J1205, Tol 1214, and J0344 calculated from the Hα broad lines are similar to the BH masses estimated from our methods.For SBS 0335-052E, the black hole mass derived from Hα broad lines is larger than that derived from our methods.We disscuss the difference in Hatano et al. (2023), including discussion of SEDs and outflows.
For all of our sample galaxies with the LRIS spectra, SBS 0335-052E and HS 0122, we have measured the FWHMs of Hβ, He iiλ4686, and [Ne v]λ3426 emission whose ionization potentials range widely, 13.6, 54.4, and 97 eV, respectively, evaluating the FWHMs with a Gaussian function.We show the FWHM measurements in Table 7.We find that the FWHMs are comparable to instrumental broadening, and are smaller than those of weak broad lines with FWHM ≳ 1000 km/s found in strong emission lines (e.g.[O iii]λλ4959,5007, Hα).

Comparison with X-ray observations
We compare the X-ray observation with the best-fit spectra of dwarf galaxies and some accretion disk models.We find X-ray observational resutls for Tol 1214, SBS 0335-052E, and J104457 in the litearture and Chandra Data Archive.X-ray emission are not detected in Tol 1214 and J104457, but SBS 0335-052E.For Tol 1214 and J104457, we calculate the upper limits of the X-ray luminosities assuming the sensitivity limit of 4 × 10 −15 erg s −1 cm −2 , and the flat spectra 2 .The Xray luminosity for SBS 0335-052E is taken from Thuan et al. (2004).
We plot the X-ray luminosity measurements and upper limits for the three galaxies in Figures 9, 10, and 11.We overplot the best-fit ∼13.6-100 eV spectra derived in Section 5.2 for each galaxy.To compare the best-fit ionizing spectra with the X-ray luminosity measurements and upper limits, we extrapolate the best-fit ionizing spectra, applying photoelectric absorption and Thomson scattering effects to the best-fit ionizing spectra with the photoelectric-absorption cross sections shown in Table 2 of Morrison & McCammon (1983).We assume that hydrogen column densities of log (N H /cm 2 ) = 21, 22, and 23 that cover both typical hydrogen scolumn densities reported in HMXBs log (N H /cm 2 ) ∼ 21 and heavily obscured AGNs log (N H /cm 2 ) ∼ 23.We find that the best-fit ionizing spectra with a low hydrogen column density of log (N H /cm 2 )= 21 are inconsistent with the X-ray luminosity measurements and upper limits, and that a high column density of log (N H /cm 2 ) = 22 or 23 is required.

[Ne v]λ3426 luminosities
Besides our sample of dwarf galaxies, [Ne v]λ3426 lines are reported in some SNe and AGNs.We compare the [Ne v]λ3426 luminosities with those of a supernova and AGNs.
We compare [Ne v]λ3426 luminosities of our sample galaxies with that of a supernova, SN 2010 jl (Fransson et al. 2014).We obtain a reduced optical spectrum of SN 2010 jl from wiserep database (Yaron & Gal-Yam 2012), and measure the [Ne v]λ3426 luminosity to be 3.5×10 38 erg s −1 .This [Ne v]λ3426 luminosity is comparable with our dwarf galaxies.

Other high-ionization lines
In typical AGNs with high stellar masses and metallicities, [Fe v]λ4227 and [Fe vii]λ6087 lines are widely detected.We search the Fe high-ionization lines in the 10 dwarf galaxies.The [Fe v]λ4227 lines are detected in J0344, J1205, W1702, Tol 1214, and SBS 0335-052E, and J104457 (Izotov et al. 2021;Berg et al. 2021;This work).The [Fe vii]λ6087 lines are detected in J1205 and SBS 0335-052E (Izotov et al. 2021;This work).We show the [Fe vii]λ6087/[Fe v]λ4227 flux ratios of J1205 and SBS 0335-052E in Table 9.To check whether the presence and absence of the emission lines are consistent with our model or not, we predict [Fe vii]λ6087/[Fe v]λ4227 from our best-fit models.We find that the bestfit model results are consistent with the observations within the ≲ 3 σ uncertainties.From the JWST spectra, we obtain [Ne iii]λ3869/[Ne v]λ3426 upper limit of < 9.3 × 10 −3 .We predict the [Ne iii]λ3869/[Ne v]λ3426 ratio from the best-fit model as 8.7 × 10 −3 .The non detection of the [Ne v]λ3426 line in ID 6355 is consistent with the best-fit parameters.

SUMMARY
We reconstruct the ionizing spectra of the dwarf galaxies in 13.6 -100 eV range using >10 optical emission lines including faint high-ionization lines of He iiλ4686, [Ne iv]λ2424, and [Ne v]λ3426.We conduct deep optical spectroscopic observations for two dwarf galaxies classified as EMPGs with the Keck/LRIS spectrograph.We make a total of the ten dwarf galaxies including EMPGs with detection of faint high-ionization lines from our observational data of the two dwarf galaxies, adding the eight dwarf galaxies from the literature.We derive the ionizing spectra at 13.6 − 100 eV by the comparisons of the observed optical emission lines and the photoionization models in the same manner as U22 with the two major improvements to determine the high energy spectra in the EUV ∼ 55 − 100 eV band.One improvement is replacing blackbody spectra with the realistic stellar population spectra that affect the shapes of the EUV spectra, while the other is including the high ionization lines of [Ne iv]λ2424 and [Ne v]λ3426 whose ionization potentials are ∼ 60 − 100 eV.Our findings are listed below.
1.For our ten galaxies, we derive the ionizing spectra over 13.6-100 eV that reproduce all of the observed emission line fluxes within ≲ 3σ errors.The ionizing spectra of the ten galaxies show prominent power-law radiation in the EUV band.We calculate power-law spectral properties of L EUV and α EUV , and find the anti-correlation for the ten galaxies.
2. As for the testing purpose, we compare L EUV and α EUV of a known IMBH having a BH mass of M BH = 10 5.75 M ⊙ and an approximate accretion rate of ṁ ≳ 16 (J024009; Xiao et al. 2011) with those of the BH accretion disk models of K03.We find that the IMBH agrees with the K03 models of M BH = 10 5 − 10 6.5 M ⊙ and ṁ = 1 − 30 on the L EUV -α EUV plane, suggesting that the IMBH is explained by the K03 model and that the BH mass of the IMBH is reproduced by the K03 model comparisons, while we do not rule out other scenarios.We thus regard that this K03-model comparison method with the ionizing spectral properties of α EUV and L EUV is applicable to galaxies with IMBHs.

Figure 1 .
Figure 1.Keck/LRIS spectra of SBS 0335-052E (top) and HS 0122 (bottom).The inset panels show [Ne v]λ3426 and Heiiλ4686 emission lines for each galaxy.The dashed lines represent the wavelengths of the emission lines.
. Emission line fluxes are obtained by the same manner as Isobe et al. in prep.by fitting the Gaussian distribution convolved with the line-spread function defined in Isobe et al. (2023).The redshift and velocity dispersion are fixed to the values obtained from [O iii]λ5007 line.We calculate the dust extinction E(B − V ) with Hβ, Hγ, and Hδ line fluxes, with T e and n e taken from Nakajima et al. (2023) and Isobe et al. (2023), respectively.We obtain the value of E(B − V ) = 0.060 ± 0.16 and conduct extinction correction with this value.The obtained emission line fluxes and ratios are summarized in Table

Figure 2 .
Figure 2. WISE color-color diagram for the 9 dwarf galaxies (blue circles) in our sample.Red dashed line (blue box) represent an AGN criterion of Stern et al. (2012) (Jarrett et al. 2011).We plot the uncertainties with the error bars shown in Hainline et al. (2016) in the bottom-left corner.

Figure 3 .
Figure 3. Posterior PDF of the model parameters for J0344.Two-dimensional (one-dimensional) probability distribution for each parameter is shown on the off-diagonals (along the diagonals).The darker regions on the joint probability distributions indicate the higher density of the sampled parameter sets.The red solid lines (black dashed lines) represent the best-fit values (68% confidence range) of model parameters.

Figure 4 .
Figure 4. Differences of the best-estimate model line fluxes from the observed line fluxes for our ten galaxies.The vertical axes denote the differences defined in the main text.The best-estimate model fluxes are reproduced from the best-fit parameters.The red (orange) lines represent the 1σ (3σ) errors of the observed fluxes.The inset panel in the bottom right panel shows the result of the [Ne iv]/[Ne iii] emission line ratio of ID 6355.

Figure 5 .
Figure5.Ionizing spectra estimated at 13.6-100 eV for ten dwarf galaxies.All the ionizing spectra show prominent power-law continua in the 55-100 eV range.

Figure 6 .
Figure 6.Bottom: EUV spectral slope as a function of EUV luminosity for ten dwarf galaxies.Top: Ratios of stellar flux to the power-law flux at 55 eV.The error bars are defined by the values of 16 and 84 percentile of the extracted parameter sets.Criterion of stellar radiation to power-law continua ratio (0.1) are plotted as horizontal dots.The dwarf galaxies with larger value of stellar radiation to power-law continua ratio is plotted in grey in the both panels.

Figure 7 .
Figure 7. Examples of the K03 model spectra.Three parameters MBH, ṁ, and α determine the BH accretion disk size (for a given temperature), temperature distribution, and the influence of electron scattering of the K03 models, changing the spectral shape.When MBH increases (with a fixed ṁ), the size of an accretion disk increases and the temperature decreases, changing the spectra brighter and redder.When ṁ increases, the temperature of an accretion disk increases, leading to brighter and bluer spectra.When α increases, the effect of inverse Compton scattering increases, making spectra flatter (i.e., smaller αEUV).Details are explained in K03.

Figure 8 .
Figure8.Spectral slope of power-law continua and EUV luminosities of BH accretion disk models and reproduced ionizing spectra for seven dwarf galaxies (symbols).Color shaded regions represent where BH accretion represent the BH disk model results for the accretion rate and viscosity parameter varying in the range of ṁ =1-1000 and α = 0.01 − 1. Difference in color shows difference in BH masses of the K03 models.Each line shows how a position on the graph change when accretion rate changes.Upper and lower line represent model with different viscosity.We filled between the two lines.For MBH = 10 6.5 M⊙, we used the spectra for α of 0.1 (with ṁ from 1 to 1000; Fig.12 of K03), and then broaden the spectral slope by ±0.1 to account for the wider viscosity range (from 0.01 to 1; Fig.9 of K03).The orange dashed line connect the αEUV and LEUV values of (MBH, ṁ, α) = (10 5 M⊙, 1, 1) and (10 6.5 M⊙, 1, 0.1).

Figure 9 .Figure 10 .
Figure 9.Comparison between the X-ray observation and extrapolation of the best-fit SED of Tol 1214.The black line and arrow denote the upper limit of X-ray luminosity given by Chandra observations.The red dotted lines indicate net transmitted spectra assuming hydrogen column densities of log(NH/cm −2 ) =21,22, and 23.

Figure 11 .
Figure 11.Comparison between the X-ray observation and extrapolation of the best-fit SED of SBS 0335-052E.Black line denote the X-ray luminosity given by Chandra observations (Thuan et al. 2004).The red dotted lines indicate net transmitted spectra assuming hydrogen column densities of log(NH/cm −2 ) =21,22, and 23.

Figure 12 .
Figure 12.Relation between black hole mass MBH and host galaxy stellar mass Mstar for the seven dwarf galaxies.Color bars and star marks represent the BH mass ranges of the K03 models that are consistent with the LEUV and αEUV values of the six dwarf galaxies within 2σ levels.The star marks are plotted at the centers of the BH ranges.The pink open star represents the BH mass of J024009 given by Xiao et al. (2011) derived from the Hα broad line.The grey cross symbols represent the black hole and stellar masses of local dwarf galaxies provided by Reines & Volonteri (2015).The black solid line and the shaded region indicate the local relation in the range of MBH = 10 5 − 10 8.5 M⊙ given in Reines & Volonteri (2015).The black dotted lines show the sum of intrinsic scatter and measurement uncertainties of the local relation.

Table 1 .
Sample Properties

Table 2 .
Fluxes and Flux Ratios

Table 3 .
Fluxes and Flux Ratios respectively.For ID 6355, we use emission line fluxes and flux ratios listed in Table3.Because [Ne v]λ3426 line is not detected in ID 6355, we use [Ne iv]λ2424/[Ne iii]λ3869 instead of [Ne v]λ3426/[Ne iii]λ3869.Because [He ii]λ4686 emission line is not detected in ID 6355, we search the best-fit parameters that are consistent with three sigma non-detection of the [He ii]λ4686 line.

Table 4 .
Prior Distributions of Free Parameters

Table 6 .
BH masses estimated from Hα broad lines.

Table 9 .
Observed and predicted flux ratios of high-ionization lines of Fe in the dwarf galaxies.±0.029(Izotov et al. 2021)of Science.Y.I. is supported by JSPS KAKENHI Grant No. 21J20785.This research is supported by a grant from the Hayakawa Satio Fund awarded by the Astronomical Society of Japan.Numerical computations were in part carried out on Small Parallel Computers at Center for Computational Astrophysics, National Astro-nomical Observatory of Japan.This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.