The Ionizing Photon Production Efficiency (ξ_ion) of Lensed Dwarf Galaxies at z ∼ 2^*

Our sample is drawn from an HST survey (Alavi et al. 2016) that identifies faint star-forming galaxies at 1 < z < 3 behind three lensing clusters—A1689 and two Hubble Frontier Fields (HFF) clusters, MACS J0717 and MACS J1149 (Lotz et al. 2017). The data reduction and photometric measurements are discussed in detail in Alavi et al. (2016). For galaxies behind A1689, we measure flux in eight photometric bands spanning the observed near-UV and optical. For galaxies behind MACS J0717 and MACS J1149 we measure flux in nine photometric bands spanning the observed near-UV, optical, and near-IR. The near-UV data (program IDs 12201, 12931, 13389) allow for efficient identification of the Lyman break, enabling accurate photometric redshifts at 1 < z < 3.

We require a lens model for each cluster to correct for the lensing magnification and derive the intrinsic galaxy properties. As discussed in Alavi et al. (2016), for A1689 we use the lens model of Limousin et al. (2007), and for the HFF clusters we use the released models from the CATS⁶ team (Jauzac et al. 2016 and Limousin et al. 2016 for MACS J1149 and MACS J0717, respectively). According to Priewe et al. (2017), for the HFF clusters and a typical magnification of our galaxies, which is around 5, the systematic error in the estimated magnification of different lens models is ∼40%, which is small compared to our M_* or UV luminosity bin sizes, which are about one order of magnitude (Figures 3 and 4). In Alavi et al. (2016) there is a full description of the lens models used for the A1689 and HFF clusters.

We obtain the rest-frame optical spectra of our sample via Keck/MOSFIRE observation. We select our spectroscopic sample such that the bright rest-frame optical nebular emission lines fall within the atmospheric windows at 1.37 < z < 1.70 and 2.09 < z < 2.61. When selecting targets, we prioritized galaxies with high magnification and brighter observed optical flux densities (M_B < 26.5). The data were collected between 2014 January and 2017 March. Masks were made for the 1.37 < z < 1.70 and 2.09 < z < 2.61 redshift ranges, and all of the strong optical emission lines (Hα, [N ii], [O iii], Hβ, and [O ii]) were targeted. For the lower-redshift mask, Y-, J-, and H-band spectroscopy was obtained. For the higher-redshift mask, J-, H-, and K-band spectroscopy was obtained. The total exposure times for each mask and filter range from 48 to 120 minutes. The typical FWHM seeing of our MOSFIRE spectra in any given mask and filter is ∼071. The slit widths are also 070.

The MOSFIRE data were reduced using the MOSFIRE Data Reduction Pipeline⁷ (DRP). The DRP produces a 2D flat-fielded, sky-subtracted, wavelength-calibrated, and rectified spectrum for each slit. It also combines the spectra taken at each nod position (we used an ABBA dither pattern). The wavelength calibration for the J- and H-band spectra was performed using the skylines and for the Y- and K-band spectra a combination of skylines and neon lines. We then utilize custom IDL software, BMEP,⁸ from Freeman et al. (2019) for the 1D extraction of spectra. The flux calibration is done in two stages. First, we use a standard star with spectral type ranging from B9 V to A2 V, which has been observed at similar airmass as the mask, to determine a wavelength-dependent flux calibration. We then do an absolute flux calibration using the spectrum of a star to which we assigned a slit in each mask.

Stellar masses, star formation rates, and stellar dust attenuation for our galaxies are estimated with SED fits to the photometry. Specifically, for the A1689 cluster, we use eight broadband filters spanning the observed near-UV to optical in the F225W, F275W, F336W, F475W, F625W, F775W, F814W, and F850LP filters. In addition, we use the photometry in two near-IR HST bands (F125W and F160W), though the imaging does not cover the full area covered by the near-UV and optical imaging.

For the two HFF clusters, we fit to nine broadband filters spanning the observed near-UV to near-IR in the F275W, F336W, F435W, F606W, F814W, F105W, F125W, F1140W, and F160W filters.

We use the stellar population fitting code FAST (Kriek et al. 2009), with the BC03 (Bruzual & Charlot 2003) population synthesis models and assume an exponentially increasing star formation history (which has been shown to best reproduce the observed star formation rates at z ∼ 2; Reddy et al. 2012) with a Chabrier IMF (Chabrier 2003). As suggested by Reddy et al. (2018) for high-redshift low-mass galaxies, we use the SMC dust extinction curve (Gordon et al. 2003) with A_V values varying between 0.0 and 3.0. We leave the metallicity as a free parameter between [0.4–0.8] Z_⊙. The age and star formation timescales can vary between $7.0\lt \mathrm{log}(t)[\mathrm{yr}]\lt 10$ and $8.0\lt \mathrm{log}(\tau )[\mathrm{yr}]\lt 11.0$, respectively. The redshifts are fixed to the values obtained spectroscopically. The 1σ confidence intervals are derived from a Monte Carlo method of perturbing the broadband photometry within the corresponding photometric uncertainties and refitting the SED 300 times. We note that we correct the broadband photometry for the contamination from the nebular emission lines using the line fluxes measured from the MOSFIRE spectra.

Spectral fitting was performed in each filter covering a galaxy, and for all of the aforementioned strong rest-optical emission lines, using the Markov Chain Monte Carlo Ensemble sampler, emcee⁹ (Foreman-Mackey et al. 2013). Before fitting, to account for skyline contamination within a given spectrum, we removed any data points that have a corresponding error >3× the median error of the spectrum. Emission lines within close proximity to each other in a filter (e.g., [O iii] λλ4959, 5007 and Hβ; [N ii] λλ6548, 6583 and Hα) were fit simultaneously with single-Gaussian profiles, and the continuum was fit with a line. For the emission lines relevant to this paper (Hβ, [O iii] λ5007, and Hα), the free parameters of the fits comprised the slope and y-intercept of the continuum line, a single emission-line width (σ) and redshift for the filter, and the amplitudes (A_λ) of the individual lines. When fitting the portion of the spectrum containing Hβ and the [O iii] doublet, the amplitude of [O iii] λ5007 was set as a free parameter with the amplitude of [O iii] λ4959 constrained to follow the intrinsic flux ratio of the doublet's lines: [O iii] λ5007/[O iii] λ4959 = 2.98 (Storey & Zeippen 2000). In the instances where [O iii] λ5007 fell outside our spectroscopic coverage, its flux was determined with this flux ratio and the [O iii] λ4959 line. The final spectroscopic redshift of a galaxy was determined via the weighted average of the redshifts fit to the different filters. More details about the spectroscopic line measurements of [O ii] and other, fainter optical lines not considered here can be found in Gburek et al. (2019). To assess the quality of the fits to the spectra, posterior histograms were output for each free parameter (as well as for the fluxes), and 68% confidence intervals were fit to the histograms.

The emission-line fluxes need to be corrected for slit losses. This procedure is more important for extended or stretched (highly magnified) objects, as the slit may not fully cover the object. This needs to be done for each object in each MOSFIRE band and each mask. We adopt the following procedures: (1) we cut a 30'' × 30'' postage stamp centered on the galaxy from the F625W, as this filter gives a high signal-to-noise ratio image of the rest-frame UV light, and therefore the approximate morphology of the star-forming regions. (2) We identify the pixels corresponding to the object using the segmentation map output by Source Extractor (Bertin & Arnouts 1996). (3) We mask out all pixels of the nearby objects and background from the postage stamp and replace them with zero flux. (4) The sum of the total flux from pixels belonging to the object gives us the actual flux that Source Extractor measured. (5) We smooth the postage stamp applying a Gaussian kernel with an FWHM that is given by

$$\begin{eqnarray}&&{\mathrm{FWHM}}_{\mathrm{kernel}}^{2}={\mathrm{FWHM}}_{\mathrm{seeing}}^{2}-{\mathrm{FWHM}}_{{\rm{F}}625{\rm{W}}}^{2}.\end{eqnarray} \tag{ 2 }$$

FWHM²_seeing is the FWHM of the Gaussian fit to the profile of the slit star in the corresponding mask and filter. ${\mathrm{FWHM}}_{{\rm{F}}625{\rm{W}}}^{2}$ is the FWHM of the F625W point-spread function (01). This artificially degrades the resolution of the HST image to the same spatial resolution as the MOSFIRE observation. (6) We overlay the slit on the postage stamp of the smoothed image using its position angle, center, length, and width and block out regions of the object that fall out of the slit. (7) We sum the flux of the remaining pixels in the slit and denote it as in-slit flux. (8) We then determine a multiplicative factor required to have the in-slit flux match the total flux. This factor is the slit-loss correction and is applied to all lines in the corresponding filter and mask.

There are 62 galaxies in our sample for which we have sufficient HST filter coverage spanning the observed near-UV to near-IR that enables a robust SED fit and, thus, reliable estimates of stellar properties (stellar mass, V-band dust attenuation (A_V), UV spectral slope (β), etc.). We remove some galaxies from the sample for the following reasons.

Noncovered Hα or large Hα errors: We remove 16 galaxies that do not have good Hα measurements, because either Hα is out of the wavelength coverage or the errors are significantly larger than the Hα error distribution of the sample (>10⁻¹⁷ erg s⁻¹ cm⁻², corresponding to a typical error >1.76× the median error, typically due to strong skyline contamination). The median Hα flux error of the sample is 6.08 × 10⁻¹⁸ erg s⁻¹ cm⁻². We choose to impose a flux error cutoff rather than a signal-to-noise cutoff on our Hα measurement, in order to avoid a bias against intrinsically faint Hα emitters.

Galaxies with very high magnification: If a galaxy has a high average magnification, it means that it is sitting close to the caustic in the source plane. Thus, the gradient of the magnification can be large, resulting in large magnification differences across the galaxy. This could result in an observed ratio of L_Hα to L_UV that is different from the true ratio. Not only would this increase the scatter, but it can result in a bias, as the galaxies are selected via rest-frame UV continuum luminosity density. Hence, we remove seven galaxies whose magnifications (μ) are μ > 30 in A1689 and μ > 15 in HFF clusters. The reason for choosing a larger magnification cut for the A1689 compared to other HFF clusters is because A1689 has a large Einstein radius, which provides high magnification over a large area in the source plane. Therefore, objects with a high magnification in A1689 are not required to be close to the critical lines, where the magnification formally diverges (Alavi et al. 2016).

Multiply imaged galaxies: We remove multiple images of two galaxies to avoid double-counting. In these cases, we keep the most highly magnified image in the sample unless the magnification is very large (>30 in A1689 and >15 in HFF clusters), in which case we use the next-brightest image. These multiple images were identified using Lenstool (Alavi et al. 2016; Limousin et al. 2016).

High slit-loss galaxies: For larger, extended galaxies, the slit-loss correction can be large, and the MOSFIRE measurement will only be sampling a small, possibly unrepresentative portion of the whole galaxy.

As such, we remove four galaxies with Hα slit losses >70% from the sample. It is also worth noting that the typical slit loss of our sample is 40%.

Galaxies with large mass errors: We also make sure not to include galaxies with large stellar mass errors in our analysis. There are only four galaxies that lack HST rest-frame near-IR filter coverage and ultimately end up having large mass errors, shown as gray points in Figure 1. We note that we only exclude these galaxies from our sample when we perform flux stacking based on stellar masses (Section 5.1), but we use them when stacking based on properties other than the stellar mass (UV magnitude and UV spectral slope; Sections 5.2 and 5.3).

Zoom In Zoom Out Reset image size

The final sample contains 28 galaxies that are free of the aforementioned concerns.

The goal of this paper is to measure the ionizing photon production efficiency of galaxies (ξ_ion) for our sample, which represents the rate of Lyman continuum photons per unit UV₁₅₀₀ luminosity as

$$\begin{eqnarray}&&{\xi }_{\mathrm{ion}}=\displaystyle \frac{{Q}_{{{\rm{H}}}^{0}}}{{L}_{\mathrm{UV}}}\ [{{\rm{s}}}^{-1}/\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}],\end{eqnarray} \tag{ 3 }$$

where L_UV is the intrinsic UV continuum luminosity density (per unit frequency) around 1500 Å. Based on case B recombination, the rate of production of ionizing photons (Q_H⁰) can be determined from the hydrogen recombination lines, in this case Hα, as

$$\begin{eqnarray}&&{L}_{{\rm{H}}\alpha }[\mathrm{erg}\,{{\rm{s}}}^{-1}]=1.36\times {10}^{-12}\ {Q}_{{{\rm{H}}}^{0}}[{{\rm{s}}}^{-1}],\end{eqnarray} \tag{ 4 }$$

where L_Hα is the Hα luminosity (Leitherer & Heckman 1995). Here we assume that all ionizing photons result in a photoionization (none escape into the IGM) and are converted into case B recombination emission. Therefore, the ${\xi }_{\mathrm{ion}}$ values reported in this study are upper limits.

Figure 1 shows the ratio of observed (non-dust-corrected) ${L}_{{\rm{H}}\alpha }^{{\prime} }$ to ${L}_{\mathrm{UV}}^{{\prime} }$ as a function of stellar mass. (The prime sign on the ${L}_{{\rm{H}}\alpha }^{{\prime} }$, ${L}_{\mathrm{UV}}^{{\prime} }$, and ${\xi }_{\mathrm{ion}}^{{\prime} }$ is to distinguish them as the non-dust-corrected quantities.) Because the Hubble images are far more sensitive than our Keck/MOSFIRE observations, our primary incompleteness is determined by the depth of the spectroscopy. We are therefore concerned about completeness for galaxies with low ${L}_{{\rm{H}}\alpha }^{{\prime} }$ and, thus, low ${\xi }_{\mathrm{ion}}^{{\prime} }$. We therefore decide to only include galaxies in our final sample with spectra that are sensitive to the "worst-case" (lowest) observed ${L}_{{\rm{H}}\alpha }^{{\prime} }$ that can be expected.

In order to determine the lowest ${L}_{{\rm{H}}\alpha }^{{\prime} }$, we start by assuming the lowest ${L}_{\mathrm{UV}}^{{\prime} }$ for the measured stellar mass of the galaxy. This is found with a line near the lower edge of the observed log(${L}_{\mathrm{UV}}^{{\prime} }$)–log(${M}_{* }$) relation (at ${M}_{* }\gt {10}^{8}$ M_⊙, where our sample is complete) shown in Figure 2. Once the worst-case ${L}_{\mathrm{UV}}^{{\prime} }$ is determined, we then assume a worst-case log(${\xi }_{\mathrm{ion}}$) to find the faintest expected ${L}_{{\rm{H}}\alpha }^{{\prime} }$. We determine this worst-case value at the higher masses ($\gt {10}^{9}{M}_{\odot }$, where we are complete), where we see that the lowest log(${L}_{{\rm{H}}\alpha }^{{\prime} }/{L}_{\mathrm{UV}}^{{\prime} }$) in our sample is ∼13.2. Finally, we compare this faintest ${L}_{{\rm{H}}\alpha }^{{\prime} }$ to our line sensitivity (assumed as 3σ Hα flux detection) to determine what magnification is required to detect Hα in our spectra. We keep all galaxies in our sample that have a high enough magnification. In this way, we ensure that all galaxies remaining in our sample have sufficiently sensitive spectra to detect galaxies with the lowest expected log(${\xi }_{\mathrm{ion}}$).

Zoom In Zoom Out Reset image size

Once we find the magnification threshold at any given mass, we remove galaxies in our sample whose magnifications are less than that threshold. There are 12 of these galaxies in our sample, which are shown as black points in Figure 1. Now we only work with the remaining objects (red points) in our sample, which are not affected by any biases. We note that log(${L}_{{\rm{H}}\alpha }^{{\prime} }/{L}_{\mathrm{UV}}^{{\prime} }$) spans about 1 dex across the sample (13–14), as is evident in Figure 1.

We also perform a sanity check to determine whether our final sample can truly represent ${\xi }_{\mathrm{ion}}^{{\prime} }$ in low-mass galaxies or suffers from any biases against low-mass faint galaxies. This investigation is primarily due to the fact that our spectroscopic sample is a magnitude-limited subsample of our parent photometric sample (B < 26.5 AB). In this case, there is a possibility that we are populating the lower-mass bins only with the most luminous and youngest galaxies and might be missing the faint sources. To ensure that our final sample does not suffer from this bias, we plot the log (${L}_{\mathrm{UV}}^{{\prime} }$)–log(M_*) distribution of our parent photometric sample and compare it to the final ${\xi }_{\mathrm{ion}}^{{\prime} }$ sample in Figure 2. This figure indicates that our final sample has a similar distribution to the parent sample and is not biased toward high log(${L}_{\mathrm{UV}}^{{\prime} }$) values at a fixed stellar mass down to the mass of ${10}^{7.8}{M}_{\odot }$. Hence, our final ${\xi }_{\mathrm{ion}}^{{\prime} }$ sample is representative of low-mass galaxies at 1 < z < 3 and is not biased against the low-mass, faint galaxies.

We use the A_V values derived from SED fits (Section 3.1) and assume an SMC extinction curve to correct for the dust attenuation of the UV luminosity density. We also use the Balmer decrement (${L}_{{\rm{H}}\alpha }^{{\prime} }/{L}_{{\rm{H}}\beta }^{{\prime} }$) to determine the ${L}_{{\rm{H}}\alpha }^{{\prime} }$ attenuation assuming a Cardelli extinction curve (Cardelli et al. 1989).

Galaxy stellar mass (M_*) is known to correlate with metallicity, which affects the stellar temperatures and, thus, ${\xi }_{\mathrm{ion}}$. We are therefore interested in examining the dependence of log(${\xi }_{\mathrm{ion}}$) on stellar mass for our sample. We present the log(${\xi }_{\mathrm{ion}}$) derived from our two stacking methods as a function of log(stellar mass) in Figure 3.

Zoom In Zoom Out Reset image size

As can be seen in Figure 3, the composite log(${\xi }_{\mathrm{ion}}$) is higher by at least 0.2 dex at all mass bins when using the Effective method compared to the Standard method. Since the errors in log(${\xi }_{\mathrm{ion}}$) of some galaxies are not negligible compared to the size of the spread in the log(${\xi }_{\mathrm{ion}}$) distribution of the sample, it might be thought that the stacked log(${\xi }_{\mathrm{ion}}$) derived from the Effective method is perhaps higher because of the large noise in these galaxies. We also check to make sure this enhancement is primarily due to the intrinsically high luminosities and not the noise. For that, we need to know how much the noise from our measurement has spread our observed log(${\xi }_{\mathrm{ion}}$) distribution. We run a simple simulation here: We first construct a normally distributed log(${\xi }_{\mathrm{ion}}$) of 1000 sources with an intrinsic spread of σ_int and perturb them with the fractional noise that is randomly drawn from the errors in ${\xi }_{\mathrm{ion}}$ of our sample. We then calculate the spread in this simulated ${\xi }_{\mathrm{ion}}$ distribution as σ_sim. In order for the simulated spread to be equal to the observed spread (0.35), the intrinsic spread is required to be ∼0.29 dex, which implies ∼0.19 dex spread due to noise. Thus, the intrinsic spread is larger and is the primary reason for the increased log(${\xi }_{\mathrm{ion}}$) enhancement calculated via the Effective stacking method.

Now we compare our results with other studies at different redshifts or different stellar masses. First, we compare to a sample of local low-mass galaxies from Weisz et al. (2012). We have determined the composite log (${\xi }_{\mathrm{ion}}$) of this sample in four mass bins, using the same two stacking methods we used for our sample, shown as green points in Figure 3. Similar to our sample, we see that the log(${\xi }_{\mathrm{ion}}$) measured from the Effective method is similar to or higher than the one derived from the Standard method in this sample. In particular, the difference between the two methods increases at lower masses, where the scatter in the log (${\xi }_{\mathrm{ion}}$) is dramatic and is likely due to the increasing burstiness, as was found by Emami et al. (2019).

Comparing our results with Weisz et al. (2012), we find that at a given mass, our sample shows higher log(${\xi }_{\mathrm{ion}}$) relative to that of Weisz et al. (2012) (compare red markers with green ones), suggestive of a log(${\xi }_{\mathrm{ion}}$) evolution with redshift in the low-mass systems. We discuss possible explanations for this in Section 6.

We also compare our sample with higher-mass galaxies at similar redshift (1.4 < z < 2.6) from the MOSDEF Survey (Shivaei et al. 2018). In Figure 3 we show the log(${\xi }_{\mathrm{ion}}$) values for MOSDEF assuming SMC (Gordon et al. 2003) and Calzetti et al. (2000) UV dust extinction corrections. We see that the log(${\xi }_{\mathrm{ion}}$) of our sample is in good agreement with that of Shivaei et al. (2018) at ${10}^{9}\mbox{--}{10}^{9.5}{M}_{\odot }$ within 1σ uncertainty, in the mass range where the two samples overlap. In fact, the log(${\xi }_{\mathrm{ion}}$) values of our galaxies in our sample and those at higher stellar mass are consistent at all stellar masses. Thus, there is no evidence for a trend in log(${\xi }_{\mathrm{ion}}$) with stellar mass from ${10}^{7.8}\mbox{--}{10}^{11}{M}_{\odot }$.

We also compare to the high-redshift sample of Lam et al. (2019), shown as purple circles in Figure 3. The sample is in the redshift range 3.8 < z < 5.3. Galaxies in this sample are primarily selected to have Lyα emission in the MUSE data. The sample includes galaxies of faint UV luminosities −20.5 < M_UV < −17.5, similar to the galaxies in our intermediate-redshift sample. The log(${\xi }_{\mathrm{ion}}$) is inferred from the Hα equivalent width, which in turn is derived from a power-law model spectrum fit through the flux of stacked Spitzer/IRAC [3.6]–[4.5] bands. The derived Hα is then divided by the stacked UV fluxes. To that end, their way of log(${\xi }_{\mathrm{ion}}$) determination is similar to our Effective stacking method. The log(${\xi }_{\mathrm{ion}}$) obtained from the Effective method in our sample is consistent with that of Lam et al. (2019) within 1σ error (compare red and purple filled markers).

M_UV is one of the easiest observables to obtain for high-redshift galaxies. Furthermore, the integral of the UV luminosity function is a critical calculation in determining the ionizing emissivities of galaxies. Therefore, we are particularly interested in whether or not there is any correlation between M_UV and ${\xi }_{\mathrm{ion}}$.

In Figure 4 we plot log(${\xi }_{\mathrm{ion}}$) as a function of M_UV. We determine log(${\xi }_{\mathrm{ion}}$) for two bins of M_UV ($-22\lt {M}_{\mathrm{UV}}\lt -19.5$ and $-19.5\leqslant {M}_{\mathrm{UV}}\lt -17.2$) using the Standard and Effective stacking methods as were described in Section 4.

Zoom In Zoom Out Reset image size

As in the previous section, we also need to take care that we only include galaxies for which we could detect very low L_Hα. However, in this case, we are sampling galaxies based on their M_UV, so we do not need to add a step of assuming an M_UV–M_* relation. Instead, we simply determine which galaxies could be detected if they had the very low observed log(${L}_{{\rm{H}}\alpha }/{L}_{\mathrm{UV}})\gtrsim 13.2$ and use each galaxy's measured L_UV.

As was mentioned earlier in Section 3.4, we return galaxies with high stellar mass errors to our sample, as their masses are irrelevant in this analysis.

Log(${\xi }_{\mathrm{ion}}$) derived from the Standard method is similar in the two M_UV bins (25.17 and 25.28), while the Effective method gives a log(${\xi }_{\mathrm{ion}}$) of 25.47 for both bins. In both M_UV bins, the Effective method gives log(${\xi }_{\mathrm{ion}}$) values ∼0.2 dex larger than that of the Standard method. We also show results from Shivaei et al. (2018), Bouwens et al. (2016), and Lam et al. (2019) in Figure 4. Comparing log(${\xi }_{\mathrm{ion}}$) of all works with analogous stacking techniques, we find that our values are in agreement with other works within 1σ significance.

We do not find any evidence of significant dependence of log(${\xi }_{\mathrm{ion}}$) on M_UV in our sample, in agreement with these other studies.

The UV continuum slope, β, is related to both the metallicity and age of the stellar populations and therefore, the inferred ionization capability of a galaxy driven by its young star populations. Therefore, we investigate whether ${\xi }_{\mathrm{ion}}$ is correlated with the more easily observable β.

The individual log(${\xi }_{\mathrm{ion}}$) values versus β are plotted in Figure 5. We split the sample into two bins of β (−2.4 < β < −1.75 and $-1.75\leqslant \beta \lt -0.9$) and apply the same two stacking methods at each bin as we used for log(M_*) and M_UV. We find a similar log(${\xi }_{\mathrm{ion}}$) of 25.45 and 25.47 for the Effective method and a log(${\xi }_{\mathrm{ion}}$) range of 25.18−25.3 for the Standard method. We do not see any evidence for log(${\xi }_{\mathrm{ion}}$) being correlated with β in our sample. Again, we find that our log(${\xi }_{\mathrm{ion}}$) stack values are consistent with those of other studies at similar or higher redshifts.

Zoom In Zoom Out Reset image size

Finally, we investigate the relationship between log(ξ_ion) and the equivalent widths of optical nebular emission lines. One expects a positive correlation because the optical line equivalent widths are directly related to the luminosity-weighted age of the stellar populations, which itself affects ${\xi }_{\mathrm{ion}}$.

This relationship was first investigated by Chevallard et al. (2018), who found that log(${\xi }_{\mathrm{ion}}$) in galaxies with strong ionizing emissivities are scaled with the equivalent width of the combined [O iii] λλ4959, 5007 lines. They showed this for a sample of local star-forming galaxies with very high rest-frame equivalent widths ($560\lt {\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}\lt 2370\,\mathring{\rm A}$). Tang et al. (2019) confirmed the existence of such a scaling relation for a sample of 227 low-mass (${10}^{7}\lt {M}_{* }/{M}_{\odot }\lt {10}^{10}$), [O iii] emitters with $225\lt {\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}\lt 2500\,\mathring{\rm A}$ at 1.3 < z < 2.4, suggesting that higher equivalent width systems are more efficient ionizing agents. Given that, we aim to test this for our galaxies to see whether this relation further extends to lower equivalent width systems or not. We calculate the ${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$ by taking the ratio of the [O iii] emission-line flux to the flux of the rest-frame λ5007 continuum from our HST near-IR fluxes, which have been corrected for emission-line contamination. We show log(${\xi }_{\mathrm{ion}}$) versus log [O iii] λ5007 equivalent width (EW${}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$) in the top panel of Figure 6. Our galaxies span a large range of rest-frame equivalent widths ($20\lt {\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}\lt 1500\,\mathring{\rm A}$) but generally extend lower than these previous studies. There is a trend of increasing log(${\xi }_{\mathrm{ion}}$) with log(${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$). To quantify this trend, we fit a line to the sample using ordinary least squares and plot the best fit, along with the 68% confidence region. We see a correlation between log(${\xi }_{\mathrm{ion}}$) and log(EW${}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$) with a slope of 0.38 ± 0.16.

Zoom In Zoom Out Reset image size

In addition, we overlay the trend from Tang et al. (2019) at larger ${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$, which is steeper than ours, with smaller uncertainty in the fit. We note that our galaxies at ${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}\gt 200\,\mathring{\rm A}$ display a similar trend to that of Tang et al. (2019). The discrepancy between the two trends suggests that the slope in the log(${\xi }_{\mathrm{ion}}$)–log(${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$) gets shallower at lower equivalent widths.

We also plot the log(${\xi }_{\mathrm{ion}}$) versus Hα equivalent width (EW_Hα) relation in the bottom panel of Figure 6. After fitting a line through the points, we find a significant correlation, with a slope of 0.52 ± 0.16 between the two indicators. We further overplot the trend from Tang et al. (2019) and Faisst et al. (2019), which contains galaxies at z ∼ 4.5 and stellar masses $\gt {10}^{9.7}{M}_{\odot }$. Our trend has a similar slope to those of Tang et al. (2019) and Faisst et al. (2019), but again with a larger uncertainty in the fit. Such a steep slope implies that log(${\xi }_{\mathrm{ion}}$) is more correlated with EW_Hα than with ${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$, as was reported by Tang et al. (2019). In addition, Reddy et al. (2018) have also found similar trends of EW_Hα and ${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]\lambda 5007}$ versus ${\xi }_{\mathrm{ion}}$ to ours for more massive galaxies in the MOSDEF survey (${10}^{9}\lt {M}_{* }/{M}_{\odot }\lt {10}^{10.5}$) at 1.4 < z < 3.8. Tang et al. (2019) argue that the log(${\xi }_{\mathrm{ion}}$)-${\mathrm{EW}}_{[{\rm{O}}{\rm{III}}]}$ and log(${\xi }_{\mathrm{ion}}$)–${\mathrm{EW}}_{{\rm{H}}\alpha }$ correlation should not hold at lower equivalent widths (below 200 Å). According to Tang et al. (2019), the equivalent widths correlate with ${\xi }_{\mathrm{ion}}$ only within the first 100 Myr since the onset of star formation. After this time, both young (O-type) and intermediate-aged (B- and A-type) populations reach equilibrium, resulting in a constant ${L}_{{\rm{H}}\alpha }$-to-L_UV ratio and a plateau in ${\xi }_{\mathrm{ion}}$ versus equivalent width. This is also evident in our sample as we get shallower slopes when including lower equivalent widths into the line fits (below 200 Å). However, this star formation history interpretation is only correct if one assumes a constant star formation history. A more comprehensive investigation of the ${\xi }_{\mathrm{ion}}$ dependence on the equivalent widths requires additional analysis of the star formation histories of galaxies, as well as other physical properties, which is beyond the scope of this paper.

Finally, we note that this correlation between log(${\xi }_{\mathrm{ion}}$) and the equivalent widths of some ionization-sensitive nebular emission lines can be used as a proxy for ${\xi }_{\mathrm{ion}}$ at high redshifts when the direct measurement of rest-frame L_UV is not available (Chevallard et al. 2018; Tang et al. 2019).