The Keck Lyman Continuum Spectroscopic Survey (KLCS): The Emergent Ionizing Spectrum of Galaxies at z ∼ 3^∗

Ozone in the Earth's atmosphere efficiently blocks ultraviolet (UV) radiation with a sharp transparency cutoff, preventing photons with λ ≤ 3100 Å (at an elevation of 4200 m, as on Maunakea) from reaching the surface. A consequence is that ground-based observations of rest-frame LyC photons from celestial objects require observing them at redshifts z ≳ 2.725, where the rest-frame Lyman limit of H i (λ₀ = 911.75 Å) falls at an observed wavelength of ≃3400 Å. At higher redshifts, observations of the rest-frame LyC benefit from the generally higher instrumental throughput and atmospheric transmission at longer wavelengths, but the sensitivity for the detection of LyC flux escaping from galaxies actually declines precipitously with increasing redshift beyond z ≃ 3.5 (e.g., Madau 1995; Steidel et al. 2001; Vanzella et al. 2010; see Section 7).

The decreasing sensitivity with increasing redshift is due to a combination of several effects: first and most obviously, galaxies of a given intrinsic UV luminosity (L_uv) become apparently fainter with redshift; in addition, the characteristic ${L}_{\mathrm{uv}}^{* }(z)$ itself dims as redshift increases beyond z ∼ 3 (e.g., Reddy & Steidel 2009; Bouwens et al. 2010; Finkelstein et al. 2015). Although the net instrumental sensitivity at wavelengths 912(1 + z_s) Å increases with redshift from z ∼ 2.7 to z ∼ 3.5, at redshifts z ≳ 3.5, the throughput gains are more than offset by the increasing intensity of the sky background against which any faint LyC signal must be measured.

Most importantly, the line and continuum opacity of neutral hydrogen (H i) in the IGM along the line of sight increases steeply with z_s (e.g., Madau 1995). Even if intergalactic H i contributed no net continuum opacity for ionizing photons emitted from a source with z_s, the LyC region will be blanketed by the effective opacity caused by Lyman series lines with z ≲ 911.8/1215.7 × (1 + z_s)⁻¹, reducing the dynamic range accessible to LyC detection.¹⁰ When one includes the net LyC opacity contributed by gas outside of the galaxy, but at redshifts near enough to z_s to impact the net transmission averaged over the LyC detection band, the median transmission in the rest-wavelength interval 880–910 Å decreases by a factor of ≃7 between z = 3 and z = 5, (see the discussion in Section 7, and Table 12).

Tallying all of the exacerbating factors, the overall difficulty of a (ground-based) detection of LyC signal from an ${L}_{\mathrm{uv}}={L}_{\mathrm{uv}}^{* }({z}_{{\rm{s}}})$ increases by a factor of ∼35 as one moves from z ∼ 3 to z ∼ 5. Thus, there is strong impetus for a ground-based LyC survey to focus on sources with 2.75 ≲ z_s ≲ 3.5, as we have done for KLCS.

Targets for KLCS were selected from nine separate fields on the sky (Table 1), chosen from among ≃25 high-latitude fields in which we have obtained deep ${U}_{{\rm{n}}}G{ \mathcal R }$ photometry suitable for selecting LBG candidates at z ∼ 3 (see Reddy et al. 2012 for a nearly complete list) as well as spectroscopic follow-up observations. The final field selection was based on a combination of visibility time at low airmass during scheduled observing runs and the number of galaxies with previously obtained spectroscopic redshifts in the range 2.9 ≲ z ≲ 3.2 that could be accommodated on a single-slit mask of the LRIS (Oke et al. 1995; Steidel et al. 2004) on the Keck I telescope. For six of the nine KLCS fields (see Table 1), selection of star-forming galaxy candidates using rest-UV continuum photometry was performed during the course of a survey for z ∼ 3 LBGs (see Steidel et al. 2003). Three additional fields (Q0100+1300, Q1009+2956, and HS 1549+1919) including z ∼ 3 LBGs were observed during the period 2003 to 2009; these comprise part of the Keck Baryonic Structure Survey (KBSS; Steidel et al. 2004, 2010, 2014; Rudie et al. 2012; Strom et al.2017).

The selection of z ∼ 3 LBGs in all nine of the KLCS fields was based on photometric selection using the three-band (${U}_{n}G{ \mathcal R }$) photometric system described in detail by Steidel et al. (2003); photometric and spectroscopic catalogs for six of these fields (as of 2003) were also presented in that work. Full photometric and spectroscopic survey catalogs for the three KBSS fields will be presented elsewhere.

We used a slit-mask design strategy that assigned the highest priority to comparatively brighter (${ \mathcal R }\leqslant 24.5$) LBGs known to have redshifts in the interval 2.9 ≤ z ≤ 3.3. Other star-forming galaxies in a broader redshift range, 2.7 < z < 3.6, were assigned somewhat lower priority. If space on a mask was still available, we included additional candidates that were identically selected but had not been previously observed spectroscopically.¹¹ Objects that had already been classified as active galactic nuclei (AGNs) based on their existing survey spectra were deliberately given relatively high priority in the KLCS mask design, as little is known about whether ionizing radiation escapes from low-luminosity AGNs or quasi-stellar objects (QSOs).¹² This small subsample will be addressed in future work.

Figure 1 presents comparisons between the parent sample from which targets were selected (light histograms) and those that were successfully observed (dark histograms), in terms of redshift, apparent magnitude ${ \mathcal R }$, and color $G-{ \mathcal R }$. The "parent" sample in this case includes all galaxies with redshifts 2.7 < z < 3.6 located in the same set of fields used in the KLCS. The middle panel of Figure 1 shows that the KLCS sample has a moderate excess of sources with ${ \mathcal R }\lt 24.5$ and a related deficiency of galaxies with $25.0\leqslant { \mathcal R }\leqslant 25.5$, which is an expected consequence of our observing strategy. The redshift distribution of KLCS galaxies also demonstrates our slight preference for targets in the 2.9 ≤ z ≤ 3.2 redshift "window." We find no difference in the distribution of $G-{ \mathcal R }$ color between KLCS and all spectroscopically identified galaxies in these fields (Figure 1, bottom panel). Thus, the subsample of galaxies observed for the KLCS is slightly brighter, and has a slightly tighter redshift distribution, than the LBGs in the same fields, but is otherwise representative.

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We obtained spectroscopic flat-field calibration images for LRIS-R and LRIS-B separately. Internal halogen lamp spectra provided adequate flat fields for all LRIS-R data but were not suitable for LRIS-B, which for our configuration requires very good flats particularly in the wavelength range 3100–4000 Å, where there are substantial spatial variations in quantum efficiency due to non-uniformities in the thinning of the silicon during manufacture.

We found from experience (for LRIS-B) that slit-mask spectra of the twilight sky, obtained at elevation and instrument rotator angle similar to those of the science observations through the same mask, are the most effective solution; these were obtained at the beginning and end of each observing night. The twilight sky spectral flats produce adequate signal in the UV but record the scattered solar spectrum rather than a featureless continuum. To remove the G-star spectrum but preserve the pixel-to-pixel sensitivity variations, we divide the raw flats by a spatially median-filtered 1D spectrum calculated at regular intervals along each slit, producing images normalized to an average value of unity but retaining the desired pixel-to-pixel sensitivity variations. The issue of scattered light associated with flat-fielding is addressed in Appendix A.1 below.

Internal arc lamp spectra (Hg, Ne, Zn, and Cd for LRIS-B; Hg, Ne, Ar, and Zn for LRIS-R) were used for wavelength calibration for both LRIS-B and LRIS-R, with fifth-order polynomial fits resulting in typical residuals of ≃0.10 Å and ≃0.07 Å for LRIS-B and LRIS-R, respectively. The arc-based wavelength solutions were subsequently shifted by small amounts using measurements of night sky emission lines recorded in each science exposure.

Spectrophotometric standard stars from the list of Massey et al. (1988) were observed at the end of each night through the 10 slit oriented at the parallactic angle (for all observations, both pre- and post-installation of the ADC in 2007 August), with configuration settings otherwise identical to mask observations. Absolute flux calibration uncertainties are estimated to be of order ≃20% due to potential variations in seeing conditions (and the associated variation in slit losses) between slit-mask and standard star observations. However, red-side and blue-side exposures were always obtained simultaneously, for both science and standard star observations, so that with careful reductions of the standards, the relative spectrophotometry between the blue and red channels is much more precise than the absolute spectrophotometry; the latter is relatively unimportant to our analysis (see Section 4.2 below).

The standard LRIS spectroscopic data processing pipeline we have used for previous surveys with LRIS (e.g., Steidel et al. 2003, 2004, 2010) was generally followed in processing KLCS LRIS-R data. However, given the challenge of measuring very faint flux densities at levels well below the sky background in the LyC region, we paid particular attention to developing procedures for LRIS-B reductions with a goal of minimizing residual systematic errors wherever possible. Given the potential importance of systematics to the final LyC results, we describe the details of the reduction procedures up to the extraction of 1D spectra in Appendix A.

The 2D, background-subtracted, stacked spectrograms for each sequence of LRIS-B or LRIS-R observations with a given slit mask were reduced (as described in Appendix A) so that the centroid of the trace for a given object on the slit lies at a constant spatial pixel along the dispersion direction, whether or not the observation was made using the ADC. We adopted, conservatively, a 135 boxcar extraction window (10 spatial pixels on the LRIS-B detector) to avoid making assumptions about variations of the spatial profile with wavelength, particularly when the trace extends to wavelengths beyond where there is obvious detected flux. Thus, the extraction aperture for every object is a rectangular region of size 12 by 135 on the sky.

For each extracted 1D spectrum, we used an identical extraction region on the 2D "sky + object" 2D spectrograms described above (Appendix A.5.1), i.e.,

$$\begin{eqnarray}&&S[i]=\displaystyle \sum _{j=k-5}^{k+5}s[i,j]\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&\sigma [i]={\left(\displaystyle \sum _{j=k-5}^{k+5}{\sigma }_{\mathrm{pix}}^{2}[i,j]\right)}^{0.5},\end{eqnarray} \tag{ 2 }$$

where k is the spatial position (in j pixels) of the object trace, S[i] is the resulting 1D spectrum at dispersion point [i], and σ[i] is the corresponding 1D error vector. Appendix A.6 describes in more detail the extent to which the noise model agrees with the data.

Mask data sets obtained on more than one observing run (see Table 1) were reduced to 1D separately, and then combined using inverse-variance weighting to produce the final 1D spectra. The 1D spectra were wavelength-calibrated using the 1D arc spectra extracted from the same region of the processed 2D arc frames, zero-pointed using night sky emission lines measured in the extracted 1D object+sky spectra (Appendix A.5.1) to correct for small amounts of flexure and slight differences in illumination between the internal arc lamps and the night sky. The final LRIS-B wavelength scales are in the vacuum, heliocentric frame with a linear dispersion of 2.14 Å pix⁻¹ (LRIS-B), covering 3200–5000 Å.

Finally, the extracted LRIS-B and LRIS-R spectra were flux-calibrated using standard stars from the list of Massey et al. (1988), obtained on the same night as the data comprising each 1D extracted science spectrum and corrected for Galactic extinction assuming the reddening maps of Schlegel et al. (1998). The standard star observations on the blue and red sides were made simultaneously, with the slit oriented at the parallactic angle. Because the LRIS long slit lies at a fixed position in the field of view of the instrument, the sensitivity curves in the wavelength transition region of the dichroic beamsplitter (where the spectral response of the interference coatings changes rapidly with increasing wavelength from reflection to transmission) are not perfectly matched for slits located far from the focal plane position of the long slit, due to small differences in the angle of incidence of the incoming light. However, through experimentation, we found that accurate relative spectrophotometry could be achieved by using only LRIS-B spectra for λ < 5000 Å and only LRIS-R spectra for λ > 5000 Å, i.e., without averaging over the region of overlap.

Continuous LRIS-B+R spectra, covering 3200–7200 Å, were produced by remapping the individual calibrated spectra onto a linear wavelength scale chosen so as to preserve the spectral sampling of the higher resolution red-side data, 1.0 Å pix⁻¹. These were used to produce composite spectra using subsets of the KLCS sample (Section 8). Thus, we made final 1D flux-calibrated spectra for each source.

As discussed in Section 1, foreground contamination leading to false-positive detection of LyC flux is a major concern for any putative detection of ionizing radiation from high-redshift galaxies, and recent experience has shown that candidate LyC detections based on ground-based imaging surveys should be viewed with caution until additional observations—particularly multiband HST imaging—can confirm whether it is the association of the detected flux with the known z ∼ 3 galaxy or is more likely to be due to an unrelated object at a different (lower) redshift. In this section, we estimate the likelihood that our triage of the 1D KLCS spectra has successfully identified most of the contaminated sources that would lead to false-positive detections of LyC signal.

Recently, imaging surveys for LyC (Siana et al. 2007; Inoue & Iwata 2008; Nestor et al. 2011; Vanzella et al. 2012; Mostardi et al. 2013) have used Monte Carlo simulations based on the surface density of objects in deep U-band images to estimate the probability that a random faint galaxy with (e.g.) z < 2.75 would fall close enough to the centroid position of a targeted galaxy to cause a false-positive LyC detection. The probability of a chance superposition increases as the limit for LyC detections becomes more sensitive; any galaxy with UV continuum surface brightness exceeding the typical statistical detection threshold over the bandpass used for LyC sensing is a potential source of contamination.

For the particular case of UV-color-selected LBGs at z ∼ 3—which require the presence of a photometric break between the observed U_n (3550/600) and G (4730/1100) and a relatively blue color between G and ${ \mathcal R }$ (6830/1000) to have been selected for observation in the first place—the most likely sources of contamination are flat-spectrum (i.e., zero color on the AB magnitude system) galaxies with apparent magnitudes bright enough to yield a photometric or spectroscopic detection at λ_obs ≃ 3500 Å, but faint enough that the resulting perturbation of the ${U}_{n}G{ \mathcal R }$ photometry does not scatter the object out of the color selection window. The KLCS spectroscopically observed sample has apparent U_n ≥ 26.13,¹⁵ median $\langle {U}_{n}\rangle =27.7$, and typical spectroscopic detection limits in the LyC detection band (rest-wavelength range [880, 910] Å) of σ₉₀₀ ≃ 0.01 μJy (10⁻³¹ erg s⁻¹ cm⁻² Hz⁻¹, or m_AB(LyC) = 28.9; see Figure 5 and Table 2).

If we consider objects in the magnitude range 26 <m_AB(3500) < 28.0 as the most likely potential contaminants of the KLCS sample, we can use our deepest available UV images obtained under seeing conditions similar to those of the KLCS spectroscopic observations¹⁶ to estimate the probability of contamination of the KLCS slit apertures. The average surface density of detections in the range 26 ≤ m_AB[3500] (Å) ≤ 28 is Σ_avg = 88.7 ± 2.4 arcmin⁻², where the uncertainty represents the scatter in Σ_avg among the four fields. Each KLCS spectroscopic extraction aperture subtends 12 by 135, or a solid angle of 1.6 arcsec² (4.4 × 10⁻⁴ arcmin⁻²). The probability that the centroid of an m_AB = 26–28 object (at any redshift) falls within the extraction aperture for any single KLCS target is P ≃ 88.7 × 4.4 × 10⁻⁴ = 0.039; thus, in a sample of 128, we expect N_c ∼ 128 × 0.039 ≃ 5 will be affected by such contamination.

While this type of argument cannot exclude the possibility that there remain unidentified false-positive detections in the KLCS sample, the fact that the number of slit apertures predicted to be affected by chance superposition of UV-faint foreground galaxies is similar to the number of spectra identified as blends with foreground galaxies suggests that our spectroscopic "triage" has likely removed most of the contaminants that would lead to false-positive LyC detections.

Because our primary goal is to measure the residual flux averaged over a relatively broad window in rest wavelength, precise systemic redshift measurements are not critical. However, since we will be combining individual spectra into composite stacks (Section 8), we assigned our best estimate of z_s for each galaxy based on the information in hand. Of the 124 sources in the KLCS analysis sample, 55 (44%) have measurements of nebular [O iii] emission lines in the rest-frame optical obtained using Keck/MOSFIRE (McLean et al. 2012; Steidel et al. 2014). In all such cases, the measured z_neb is assumed to define the systemic redshift z_sys, with an uncertainty of ≃15–20 $\mathrm{km}\,{{\rm{s}}}^{-1}$.

For objects lacking nebular emission-line measurements, we used estimates of z_sys based on the full KBSS-MOSFIRE sample (Steidel et al. 2014; Strom et al. 2017) with z > 2.0 and existing high-quality rest-UV spectra. These were used to calibrate relationships between z_sys and redshifts measured from features in the rest-frame FUV spectra, strong interstellar absorption lines (z_IS), and/or the centroid of Lyα emission (${z}_{\mathrm{Ly}\alpha }$). As for previous estimates of this kind (e.g., Adelberger et al. 2003; Steidel et al. 2010; Rudie et al. 2012), we adopt rules that depend on the particular combination of features available in each spectrum, where spectra fall into one of three categories: (a) those with measurements of both ${z}_{\mathrm{Ly}\alpha }$ and ${z}_{\mathrm{IS}}$, (b) those with ${z}_{\mathrm{IS}}$ but without ${z}_{\mathrm{Ly}\alpha }$ (generally because the $\mathrm{Ly}\alpha$ feature appears in absorption), and (c) those with only ${z}_{\mathrm{Ly}\alpha }$ available.

For galaxies in category (a), comprising ≃60% of the KLCS sample,

$$\begin{eqnarray}&&{z}_{\mathrm{sys}}={z}_{\mathrm{IS}}+0.40({z}_{\mathrm{Ly}\alpha }-{z}_{\mathrm{IS}});\end{eqnarray} \tag{ 3 }$$

for category (b) (≃30% of the KLCS sample):

$$\begin{eqnarray}&&{z}_{\mathrm{sys}}={z}_{\mathrm{IS}}+\displaystyle \frac{{\rm{\Delta }}{v}_{\mathrm{IS}}}{c}(1+{z}_{\mathrm{IS}}),\end{eqnarray} \tag{ 4 }$$

with ${\rm{\Delta }}{v}_{\mathrm{IS}}$ = 130 $\mathrm{km}\,{{\rm{s}}}^{-1}$; for category (c) (≃10% of the KLCS sample):

$$\begin{eqnarray}&&{z}_{\mathrm{sys}}={z}_{\mathrm{Ly}\alpha }-\displaystyle \frac{{\rm{\Delta }}{v}_{\mathrm{Ly}\alpha }}{c}(1+{z}_{\mathrm{Ly}\alpha });\end{eqnarray} \tag{ 5 }$$

with ${\rm{\Delta }}{v}_{\mathrm{Ly}\alpha }$ = 235 $\mathrm{km}\,{{\rm{s}}}^{-1}$. The residual velocity offset and rms errors after applying these rules to the UV spectra of the 55 galaxies with measurements of z_neb (Figure 3) is

$$\begin{eqnarray}&&\langle c({z}_{\mathrm{sys},\mathrm{uv}}-{z}_{\mathrm{neb}})/(1+{z}_{\mathrm{neb}})\rangle =-1\pm 104\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 6 }$$

where the quantity inside the angle brackets is equivalent to Δv_UV in Figure 3.

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The best available value of z_sys is given for each galaxy in Table 2.

All quantitative measurements or limits on the flux density of residual LyC emission have been evaluated over a fixed bandpass in the source rest frame, placed just shortward of the Lyman limit:

$$\begin{eqnarray}&&{f}_{900}\equiv \langle {f}_{\nu }({\lambda }_{0})\rangle ;\quad 880\leqslant {\lambda }_{0}/\mathring{\rm A} \leqslant 910.\end{eqnarray} \tag{ 7 }$$

We also define a rest-frame bandpass to represent the FUV non-ionizing flux density,

$$\begin{eqnarray}&&{f}_{1500}\equiv \langle {f}_{\nu }({\lambda }_{0})\rangle ;\quad 1475\leqslant {\lambda }_{0}/\mathring{\rm A} \leqslant 1525.\end{eqnarray} \tag{ 8 }$$

The choice of the interval [880, 910] Å for the LyC measurement is a compromise based on two considerations. First, ideally the LyC measurement should be made as close as possible to the rest-frame Lyman limit of the galaxy and integrated over the smallest wavelength range feasible given the noise level of the spectra, in order to minimize the effects of H i in the IGM along the line of sight. As discussed in detail in Section 7, the opacity of the IGM to LyC photons is the largest source of uncertainty in the measurement of the emergent ionizing photon flux from a galaxy. The mean free path of H-ionizing photons emitted at $\langle {z}_{{\rm{s}}}\rangle =3.05$ corresponds to a redshift interval of Δz ∼ 0.25 (e.g., Rudie et al. 2013), or to ∼60 Å in the rest frame; i.e., the flux density at λ₀ ≃ 850 Å is reduced by an average factor of e (∼2.72) relative to its emergent value. By using the [880, 910] Å interval, to a first approximation the average IGM LyC optical depth due to intervening material is $\langle {\tau }_{\mathrm{igm}}\rangle \lesssim 0.5$.

An additional consideration is the wavelength range over which the Keck/LRIS-B system sensitivity remains high. Although the UV sensitivity of Keck/LRIS-B is the highest among instruments of its kind (Steidel et al. 2004), it decreases with wavelength shortward of 3500 Å, particularly when atmospheric opacity is included. The lowest redshift source included in KLCS has z_s = 2.718 (Q0100-MD6), placing λ₀ = 880 Å at an observed wavelength of 3272 Å, where the LRIS-B end-to-end throughput with the d500 beamsplitter has dropped to ≃20% from ≳50% near 4000 Å. Including the atmosphere, these numbers reduce to ∼11% and ∼40% at a typical airmass of 1.10 on Maunakea. At the mean redshift of the KLCS sample ($\langle {z}_{{\rm{s}}}\rangle =3.05$), [880, 910] Å corresponds to an observed-frame interval [3560, 3690] Å, where the instrumental throughput averages ≃35%.

Thus, [880, 910] Å is observable over the full range of KLCS and is narrow enough to minimize the IGM opacity against which escaping flux must be detected, but sufficiently broad to allow for improved photon statistics while reducing the fluctuations due to discrete H i absorption lines arising from intervening material, superposed on the rest-frame LyC.

Figure 4 presents the measured values of f₉₀₀, with objects grouped and ordered according to slit mask and the relative physical position on the mask (along the slit direction) for each target. Targets having >3σ detections of f₉₀₀ are labeled. The values of f₉₀₀ and their associated statistical error (σ₉₀₀) for KLCS galaxies are listed in Table 2. Also given in Table 2 are measurements of the non-ionizing UV continuum flux density f₁₅₀₀ (Equation (8)) measured directly from the individual 1D flux-calibrated spectra.

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The measurements and uncertainties for f₉₀₀ and f₁₅₀₀ (and their ratio (f₉₀₀/f₁₅₀₀)_obs) were obtained directly from the 1D spectra and associated error arrays based on the noise model described in Appendix A.5.1; we have made no attempt to apply aperture corrections to the spectroscopic flux density measurements (the photometric measurements in the ${U}_{{\rm{n}}}G{ \mathcal R }$ system are also provided in Table 2) but we believe that the relative spectrophotometry of the 1D spectra has systematic errors ≲5%; uncertainties in the calibration of LRIS-R relative to LRIS-B spectra may contribute an additional ≲10% systematic uncertainty in (f₉₀₀/f₁₅₀₀)_obs.

The flux error bars in Figure 4 are statistical errors based on the noise model presented in Appendix A.5.1, with typical values of σ₉₀₀ = 0.01 μJy; most objects on each mask have f₉₀₀ measurements consistent with zero to within 1σ₉₀₀–2σ₉₀₀.

Grouping observations by slit mask allows us to monitor the presence of systematic errors from mask to mask. It is also a useful way to inspect the data for correlations with object position on a given slit mask. It is apparent from Figure 4 that mask B20902-L1 (and possibly others) has residual systematic errors manifesting as correlated behavior of f₉₀₀ as a function of position on the slit mask. Although the maximum deviation from zero in the b20902_L1 panel is only ∼2σ₉₀₀, there appear to be systematic undulations relative to zero with amplitude comparable to the random errors. The systematics appear to have larger excursions to f₉₀₀ < 0, as might occur from oversubtraction of the background due to contamination of the slit regions used to model it by either scattered light (judged to be the dominant factor for mask b20902_L1) or by contributions from unmasked sources falling along the slit.

Systematic oversubtraction of the background level was also noted for the sample of 14 sources presented by S06—these authors found that the measured f₉₀₀ for objects without significant detections was centered around an unphysical negative value (see their Figure 5). As discussed in detail in Section 4.2 (see also Appendix A.6), considerable effort was invested in improving the flat-fielding (with its tendency to exacerbate problems with non-uniform scattered light (Appendix A.1) and 2D background subtraction. The tests summarized in Appendix A.6 suggest that these were generally successful. We show below that the procedures have also reduced the residual systematic errors in extracted 1D spectra to a level significantly smaller than the random errors.

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To verify that this is the case, we excluded all galaxy measurements for which $| {f}_{900}| \gt 3{\sigma }_{900}$, where σ₉₀₀ is the statistical error estimate. For the remaining sample of 106 measurements, $\langle {f}_{900}/{\sigma }_{900}\rangle =0.31\pm 1.33$, with median f₉₀₀/σ₉₀₀ = 0.44 and interquartile range [−0.50, +1.41]. Excluding the two masks (B20902-L1 and Q1009-L1) with the most obvious systematic issues, the results remain largely unchanged: $\langle {f}_{900}/{\sigma }_{900}\rangle =0.28\pm 1.32$, with median 0.44 and interquartile range [−0.66, +1.25]. Under the null hypothesis that f₉₀₀ = 0 and that systematic errors are small compared to normally distributed random errors, one expects $\langle {f}_{900}/{\sigma }_{900}\rangle =0.0\pm 1.0$. As we shall see, the true values of f₉₀₀ are likely greater than zero for some fraction of the sample with $| {f}_{900}/{\sigma }_{900}| \lt 3$; since the standard deviation of f₉₀₀/σ₉₀₀ is only ∼30% larger than expected under the null hypothesis, we believe that residual systematic errors in f₉₀₀ are subdominant compared to statistical errors. In the remainder of the analysis below, we continue to assume that our model for random errors in the spectra (Appendix A.5.1) accurately describes the uncertainties.

Thus, 15 of 124 galaxies have f₉₀₀ > 3σ₉₀₀, which henceforth are referred to as "detected"; their properties are listed individually in Table 3. The same objects are marked using blue squares in Figure 4. Note that one of the objects in Table 3, Q1549-C25 ((f₉₀₀/f₁₅₀₀)_obs = 0.08), has been discussed in detail by Shapley et al. (2016); in addition to the LyC flux measurement, it has also been observed with multiband HST imaging (see Mostardi et al. 2015), which indicates no evidence for a contaminating foreground source that might have caused a false-positive LyC detection.¹⁷ The only other confirmed LyC detection of a z ∼ 3 galaxy is "Ion-2" (z_s = 3.22), which has (f₉₀₀/f₁₅₀₀)_obs ≃ 0.13 and rest-frame UV luminosity $\sim {L}_{\mathrm{uv}}^{* }$ (Vanzella et al. 2015; de Barros et al. 2016). Most of the objects in Table 3 have (f₉₀₀/f₁₅₀₀)_obs similar to the two confirmed LyC detections.¹⁸

Table 3.  Objects with Individual LyC Detections

Object	zsys	(L/L*)a	(f900/f1500)obsb
Q0933-D16	3.0468	0.51	0.54 ± 0.06
Q0933-M23	3.2890	0.92	0.26 ± 0.03
Q0933-MD75	2.9131	0.89	0.10 ± 0.02
Q0933-MD83	2.8790	0.60	0.15 ± 0.04
Westphal-CC38	3.0729	1.00	0.06 ± 0.01
Westphal-MM37	3.4215	1.23	0.05 ± 0.01
Westphal-MMD45	2.9366	1.42	0.09 ± 0.02
Q1422-d42	3.1369	0.47	0.14 ± 0.04
Q1422-d57	2.9461	0.30	0.30 ± 0.10
Q1422-d68	3.2865	0.88	0.19 ± 0.03
Q1549-C25c	3.1526	0.74	0.08 ± 0.02
Q1549-D3	2.9373	1.16	0.06 ± 0.01
DSF2237b-D13	2.9212	0.58	0.08 ± 0.02
DSF2237b-MD38	3.3278	1.47	0.07 ± 0.01
DSF2237b-MD60	3.1413	0.67	0.09 ± 0.02

Notes.

^aBased on the ${ \mathcal R }$ magnitude and assuming a characteristic luminosity L_uv corresponds to ${M}_{\mathrm{AB}}^{* }(1700\,\mathring{\rm A} )=-21.0$ (Reddy & Steidel 2009). ^bObserved flux ratio. The error estimate includes systematic error. ^cLyC-detected object discussed in detail by Shapley et al. (2016).

A machine-readable version of the table is available.

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Figure 5 shows all 124 KLCS targets on a plot of σ₉₀₀ versus redshift. Note that σ₉₀₀ shows a trend of increasing toward lower redshift; this is due to the decreasing instrumental sensitivity at rest-frame wavelengths [880, 910] Å, which (as we have seen) changes by a factor of a few over the range 2.75 ≲ z ≲ 3.4 (where [880, 910] Å falls at observed wavelengths 3300 ≲ (λ/Å) ≲ 4000).

Figure 6 shows a combination of formal detections and—for the non-detections—the adopted 3σ lower limits on the ratio (f₁₅₀₀/f₉₀₀)_obs using the measurements and uncertainties given in Table 2. Note that the formally detected objects lie well within the distribution of 3σ upper limits of the non-detections. The implications of these results for the distribution of LyC flux within the observed sample requires a quantitative assessment of the extent to which H i gas outside of galaxies (but along the line of sight) has censored our ability to detect LyC flux when it is present; we address this issue in Section 7.

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In this section, we briefly review the properties of galaxies with individual LyC detections versus the majority that do not have individual detections. We argue below (Section 8) that unambiguous interpretation of LyC detection statistics requires the use of ensembles of galaxies.

Figure 7 compares the distributions in three empirical galaxy properties for the LyC-detected and LyC-undetected subsamples of KLCS (with 15 and 109 objects, respectively). The leftmost panel of Figure 7 shows the rest-UV luminosity distribution, relative to ${L}_{\mathrm{uv}}^{* }$ in the far-UV (rest-frame 1700 Å) luminosity function at z ∼ 3 (Reddy & Steidel 2009). As discussed in Section 5 above, most KLCS galaxies have L_uv within a factor of a few of ${L}_{\mathrm{UV}}^{* }$. The subsample with formal LyC detections occupies a slightly narrower range of luminosity, though a two-sample KS test shows that the two luminosity distributions are statistically consistent with being drawn from the same parent population.¹⁹

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As one of the most easily observed and measured spectroscopic characteristics of high-redshift star-forming galaxies, the rest-frame equivalent width of the Lyα emission line, W_λ($\mathrm{Ly}\alpha$), is a useful diagnostic and has been shown to correlate strongly with other more subtle spectroscopic features present in the spectra of LBGs (Shapley et al. 2003; Kornei et al. 2010). We measured W_λ($\mathrm{Ly}\alpha$) for the KLCS galaxy sample following the method described in Kornei et al. (2010); the values are listed in Table 2. The center panel of Figure 7 (see also Table 2) shows the distribution of W_λ($\mathrm{Ly}\alpha$) for the full KLCS galaxy sample, divided according to whether or not they are formally detected in the LyC band [880, 910] Å. Although the detected subsample tends to have $\mathrm{Ly}\alpha$ in emission, and the fraction of objects with LyC detected appears to be correlated with $\mathrm{Ly}\alpha$ equivalent width, a two-sample KS test cannot reject the null hypothesis that the subsamples are drawn from the same parent population.

As described in more detail elsewhere (Shapley et al. 2003; Steidel et al. 2003; Reddy & Steidel 2009), the z ∼ 3 LBG ${U}_{{\rm{n}}}G{ \mathcal R }$ color selection imposes small systematic differences in the redshift selection function depending on the intrinsic galaxy properties, in the sense that intrinsically redder galaxies are less likely to satisfy the selection criteria at the high-redshift end of the selection window. The main reason for this is increased line blanketing from the $\mathrm{Ly}\alpha$ forest with redshift. Galaxies with very strong $\mathrm{Ly}\alpha$ features (in either emission or absorption) affect the observed $G-{ \mathcal R }$ color for similar reasons, since $\mathrm{Ly}\alpha$ falls in the observed G band throughout the KLCS redshift range. However, since the full KLCS sample has spectroscopic measurements, we can correct the observed $G-{ \mathcal R }$ color of individual galaxies for both effects, thereby yielding estimates of the intrinsic UV continuum color. Figure 8 shows the measurements for individual KLCS galaxies as a function of redshift in terms of ${(G-{ \mathcal R })}_{0}$, the proxy for continuum color after correction for the mean IGM absorption in the G band and the individual W_λ($\mathrm{Ly}\alpha$). The KLCS galaxies with individual >3σ LyC detections are circled, and their distribution is compared with the full sample in the rightmost panel of Figure 7.

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As for L_uv and W_λ($\mathrm{Ly}\alpha$), the subsample having direct individual detections is consistent with being drawn from the same distribution in ${(G-{ \mathcal R })}_{0}$ as the full KLCS sample. We discuss the statistical connections between LyC leakage and galaxy properties in more detail in Section 8.

Assuming the "IGM+CGM" opacity model, at z ∼ 3, the 68% confidence interval for a single realization of t₉₀₀ at z_s ≃ 3.05 is t₉₀₀ = [0.038, 0.589]; the corresponding 68% confidence intervals are t₉₀₀ = [0.076, 0.665] at z_s = 2.70, and t₉₀₀ = [0.017, 0.471] at z = 3.50. The implications of these broad distributions are worth stating explicitly: the LyC detectability of a galaxy with a given intrinsic (i.e., emergent) (f₉₀₀/f₁₅₀₀)_out is to a great extent controlled by the statistics of the IGM transmission, which is uncertain by factors of between 5 and 10 even over the limited redshift range 2.70 ≲ z_s ≲ 3.50. Conversely, a single measurement of (f₉₀₀/f₁₅₀₀)_obs cannot be converted into an intrinsic property of the source, for the same reason (see, e.g., Vanzella et al. 2015; Shapley et al. 2016).

However, $\langle {t}_{900}({z}_{{\rm{s}}})\rangle$, the mean transmissivity of the IGM for a source at z = z_s and its uncertainty δt₉₀₀(z_s, N) can be quantified for an ensemble of N sight lines using the Monte Carlo models described in Appendix B. For example, assuming z_s = 3.0, the number of independent IGM sight lines one must sample in order to reduce δt₉₀₀(3.0, N) to less than 10% (5%) of $\langle {t}_{900}(3.0)\rangle$ is N = 36 (150). In other words, if the spectra of 36 sources at z_s = 3.0 are averaged to produce a measurement of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{obs}}$, then $\langle {({f}_{900}/{f}_{1500})}_{\mathrm{out}}\rangle =\langle {({f}_{900}/{f}_{1500})}_{\mathrm{obs}}\rangle /({t}_{900}\pm \delta {t}_{900})$, and the IGM correction contributes a fractional uncertainty to the inferred emergent flux density ratio of ≃10%. Some example statistics relevant for the KLCS redshift range and sample size are given in Table 4.

Table 4.  Sampling Statistics for LyC Transmission (CGM+IGM)

zs	$\langle {t}_{900}\rangle$	t68a	$\delta {t}_{900}/\langle {t}_{900}\rangle$ b
			N = 15	N = 30	N = 60	N = 120
2.75	0.453	[0.093–0.670]	0.141	0.101	0.067	0.048
3.00	0.391	[0.074–0.607]	0.153	0.108	0.073	0.052
3.25	0.325	[0.018–0.540]	0.173	0.120	0.082	0.056
3.50	0.271	[0.013–0.467]	0.184	0.128	0.088	0.063

Notes.

^a68% confidence interval for a single realization of t₉₀₀. ^bGiven an ensemble of N realizations of t₉₀₀ for source redshift z_s, the ratio of the half-width of the 68% confidence interval of $\langle {t}_{900}\rangle$ and the mean value of $\langle {t}_{900}\rangle$ that would be obtained after a very large number of realizations.

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This type of analysis is useful when one has observations of a particular class of objects (grouped by known property, e.g., L_uv, z_s, W_λ($\mathrm{Ly}\alpha$), inferred extinction) forming a subset of the full sample: as long as the subset has a sufficiently large N, the statistical knowledge of the IGM opacity can be used to derive the average intrinsic LyC properties of that class. The validity of this procedure depends on (1) the assumption that each line of sight in the sample is uncorrelated with any other line of sight in the same ensemble and (2) that the IGM+CGM opacity model is an accurate statistical description of the true opacity.

The first assumption—that lines of sight are independent—is almost certainly not valid when a survey is conducted in a single contiguous field of angular size ∼10' (transverse scale of ∼5 pMpc at z ∼ 3), as has often been the case for reasons of practicality (e.g., Shapley et al. 2006; Vanzella et al. 2010; Nestor et al. 2011; Mostardi et al. 2013, 2015; Siana et al. 2015). Correlated sight lines could be especially problematic in fields known to contain significant galaxy overdensities at or just below the source redshifts. As has been discussed by, e.g., Shapley et al. (2006), if observed sources are located in regions containing more (or less) gas near ${N}_{{\rm{H}}{\rm{I}}}$ = 10¹⁷ cm⁻² than average, their "local" IGM could skew significantly away from expectations if an "average" line of sight is assumed. The effect is likely to be negligible for $\mathrm{Ly}\alpha$ forest blanketing, but could strongly influence f₉₀₀, since it relies on the statistics of small numbers through the incidence of relatively high column density H i over a small redshift path (Δz ≃ 0.14 for z_s = 3.05 for our LyC region [880, 910] Å).

The full KLCS sample is relatively insensitive to the effects of correlations between IGM sight lines by virtue of the fact that it is comprises nine independent survey regions. Similarly, the IGM opacity model is based on the statistics of 15 independent QSO sight lines in the KBSS survey (Rudie et al. 2013), and the CGM corrections to the IGM model are based on regions near galaxies selected using essentially identical criteria to those used for KLCS (albeit at slightly lower redshift; see Rudie et al. 2013).

Nevertheless, it is worth pointing out caveats associated with the CGM+IGM opacity models possibly relevant even for KLCS. First, the adopted IGM+CGM opacity model probably underestimates the CGM opacity, since it is based on lines of sight to background QSOs with 50 < D_tran ≤ 300 pkpc (i.e., projected angular distances 6'' < θ < 37''), which by virtue of the cross-section weighting correspond to an average impact parameter of $\langle {D}_{\mathrm{tran}}\rangle \gtrsim 200$ pkpc. On the other hand, every line of sight to a (source) galaxy includes gas with physical distance from the source of 50–300 pkpc. If ${N}_{{\rm{H}}{\rm{I}}}$ continues to increase with decreasing galactocentric radius (as is likely), the transverse sight lines used to estimate the CGM contribution for the opacity model would systematically underestimate the total CGM opacity. While the CGM+IGM corrections applied to the galaxy spectra in the KLCS sample are likely to be appropriate for the range of galaxy properties we are sensitive to in this work (due to the similarity in the galaxy properties between KBSS and KLCS mentioned above), it could be dangerous to apply the corrections to sources selected using substantially different criteria—for example, if most of the ionizing radiation field is produced by objects with a different overall environment (e.g., low-density regions harboring fainter sources might be expected to be surrounded by less intergalactic and circumgalactic gas; see Rakic et al. 2012; Turner et al. 2014), then the CGM contribution to the net opacity toward those sources might be overestimated.

Finally, the absence of a clear distinction between ISM, CGM, and IGM leads to an issue of semantics: in considering the "escape" of ionizing photons, one must also define what constitutes escape, i.e., how far must the ionizing photon travel before it is counted as having escaped, and what is the probability that it will be observable? For the remainder of this paper, we assume that ionizing photons absorbed within a galactocentric radius of r = 50 pkpc have by definition not escaped. We then use the IGM+CGM statistics outlined in Appendix B to correct the observations back to the r ≃ 50 pkpc "surface"—our working definition of a galaxy's "LyC photosphere."

There are distinct practical advantages in using a measurement band that samples a fixed bandwidth in the rest frame of each source, 880–910 Å, placed just shortward of the intrinsic Lyman limit. It is possible to use images taken through a comparably narrow bandpass (i.e., ≃100–200 Å in the observed frame; e.g., Inoue & Iwata 2008; Nestor et al. 2011; Mostardi et al. 2013), but a narrowband filter will be optimally placed only for sources at a fixed redshift, which as discussed above can also be problematic in terms of sample variance due to gas-phase overdensities and/or correlated sight lines arising in the intra-protocluster medium.

Similar issues affect observations where a broadband filter is used for the LyC detection band, such as surveys conducted using the F336W filter in the WFC3-UVIS camera on board HST (e.g., Mostardi et al. 2015; Siana et al. 2015; Vasei et al. 2016). In this case, since the broad bandpass must lie entirely shortward of the rest-frame Lyman limit of sources to provide unambiguous detections of ionizing photons, it is confined to source redshifts z_s > 3.07; however, even for z_s ≃ 3.1, the photon-number-weighted mean flux density measured in the filter bandpass is considerably reduced relative to the rest-frame [880, 910] Å wavelength interval. An example assuming z_s = 3.10 is shown in Figure 12, where t₉₀₀ = 0.372 (i.e., close to the mean value for random IGM+CGM sight lines to sources with z_s = 3.10; Table 12), but the F336W band-averaged transmission t₃₃₆ = 0.045, ∼8.3 times smaller (2.3 mag).

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Figure 13 shows cumulative distribution functions of t₉₀₀, t₃₃₆, and t_LBC (the broadband U filter used by Grazian et al. 2016) for a large ensemble of sight lines to z_s = 3.1 and z_s = 3.5. At z = 3.1, the median transmission in the relevant LyC detection band is 5.2 times higher using the [880, 910] Å interval than that evaluated through the F336W filter bandpass; the ratio between median transmission values reaches ∼200 by z = 3.5. The 90th percentile transmission t₉₀₀ = 0.59 (0.49) for z_s = 3.1 (z_s = 3.5); t_336W = 0.38 (0.14) for z_s = 3.1 (z_s = 3.5). Similarly, compared to the ground-based U_LBC used to measure LyC emission from spectroscopically identified galaxies at z ∼ 3.3 by Grazian et al. (2016, 2017), the spectroscopic [880, 910] Å bandpass is ≳2 times more likely to include sight lines with t_LyC > 0.2 and ≃9 times more likely to have sight lines with t_LyC > 0.4. These differences could be quite large if there is a limited dynamic range for LyC detection from individual sources—as has been the case for all surveys to date—potentially producing large differences in the fraction of sources with significant LyC detections even for surveys that nominally reach the same flux density limit in the LyC band.

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Care must also be exercised when comparing the results of surveys that use different LyC detection bands and/or IGM opacity models: Table 5 compares our determination of the mean IGM+CGM transmission evaluated for three different LyC detection passbands. The ensemble of sight lines used at each z_s was identical. Note that we find that the mean IGM+CGM transmission evaluated using the U_LBC filter for z_s = 3.30 (z_s = 3.40) is $\langle {t}_{\mathrm{LBC}}\rangle \simeq 0.149$ (0.099), to be compared with the value assumed by Grazian et al. (2016, 2017), $\langle {t}_{\mathrm{LBC}}\rangle =0.28$. The latter value was based on the IGM opacity models of Inoue et al. (2014).²² Additional ambiguities relevant to the quantitative comparison of LyC results arise due to differences in the definition of "escape fraction," discussed in more detail in Section 9.1.

Table 5.  Transmission Statistics vs. the LyC Detection Band^a

LyC Bandb	zs	$\langle {t}_{\mathrm{LyC}}\rangle$	Median	90th Percentile
[880, 910] Å	3.10	0.352	0.393	0.589
HST-F336W	3.10	0.139	0.078	0.377
[880, 910] Å	3.30	0.321	0.359	0.543
LBC-U	3.30	0.149	0.118	0.356
HST-F336W	3.30	0.070	0.017	0.226
[880, 910] Å	3.40	0.291	0.119	0.516
LBC-U	3.40	0.099	0.047	0.282
HST-F336W	3.40	0.050	0.005	0.171
[880, 910] Å	3.50	0.264	0.302	0.492
HST-F336W	3.50	0.039	0.002	0.137

Notes.

^aMean, median, and 90th percentile transmission in the LyC detection passband for large ensembles of Monte Carlo IGM+CGM transmission spectra with 3.10 ≤ z_s ≤ 3.50. ^bLyC detection passband over which the mean flux density is evaluated: [880, 910] Å is the spectroscopic band used in this work; the LBC-U filter is close to SDSS u', and is described by Grazian et al. (2016); HST-F336W is the HST/WFC3-UVIS filter F336W.

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The main point here is that the probability of detecting LyC emission—or of setting interesting limits on f₉₀₀/f₁₅₀₀—depends very sensitively on the source redshift z_s, the bandwidth and relative wavelength/redshift sensitivity of the LyC detection band, and the fidelity of the correction for the IGM+CGM opacity.

We formed a number of subsets of the final KLCS statistical sample based on empirical criteria that could be measured easily from photometry or spectroscopy of individual objects; these include L_uv, W_λ($\mathrm{Ly}\alpha$), and rest-UV continuum color ${(G-{ \mathcal R })}_{0}$ (Section 6.4). In view of the results of Section 7.1, a minimum subsample size N ≳ 30 was maintained so that the uncertainty in the IGM+CGM correction is ≲10% (see Table 4). Thus, for each of the aforementioned parameters, we split the sample of 124 into four independent quartiles consisting of 31 galaxies each.

Additional subsets were formed according to the following criteria: objects with L_uv ≥ ${L}_{\mathrm{uv}}^{* }$ (48% of the sample, very similar to a combination of the L_uv (Q1) and L_uv (Q2) subsamples) and those with L_uv < ${L}_{\mathrm{uv}}^{* }$ (52% of the sample); whether $\mathrm{Ly}\alpha$ appears in net emission (W_λ($\mathrm{Ly}\alpha$) > 0; 60% of the sample) or net absorption (W_λ($\mathrm{Ly}\alpha$) ≤ 0; 40% of the sample); and, finally, grouping together the galaxies exceeding the often-used threshold W_λ($\mathrm{Ly}\alpha$) > 20 Å for "Lyman Alpha Emitters" (LAEs). The KLCS LAE subsample (28 galaxies, or ≃22.6% of the total) is based upon the spectroscopically measured W_λ($\mathrm{Ly}\alpha$), and happens to be nearly identical to the W_λ($\mathrm{Ly}\alpha$) (Q4) subsample of 31 galaxies.

Table 6 includes values of parameters measured directly from the composite spectra or the mean value among the objects comprising the subsample: the number of galaxies in the subsample (N), the mean redshift of the objects in the subsample ($\langle {z}_{{\rm{s}}}\rangle$), W_λ($\mathrm{Ly}\alpha$) (measured directly from the composite spectrum), the mean UV luminosity relative to ${L}_{\mathrm{uv}}^{* }$ ($\langle {L}_{\mathrm{uv}}/{L}_{\mathrm{uv}}^{* }\rangle$), and the measured flux density ratio $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{obs}}$. Values of $\langle {t}_{900}\rangle$ for the transmission in the rest-wavelength range [880, 910] Å are given for both the "IGM only" and "IGM+CGM" Monte Carlo models (Section 7 and Appendix B), where each value and its uncertainty were calculated by drawing ensembles of sight lines with the same number and z_s distribution as the subsample, averaging the values of t₉₀₀, and repeating 1000 times. The tabulated uncertainties reflect the 68% confidence interval for the mean transmission of the 1000 ensembles for each subsample. The composite spectra of several of the subsamples listed in Table 6 are shown in Figure 16.

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Table 6.  Statistics Measured from KLCS Composite Spectra

Sample	Na	$\langle {z}_{{\rm{s}}}\rangle$	$\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$ b	$\langle {L}_{\mathrm{uv}}/{L}_{\mathrm{uv}}^{* }\rangle$	${(G-{ \mathcal R })}_{0}$ c	$\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}$ d	$\langle {t}_{900}\rangle$ e	$\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{out}}$	$\langle {t}_{900}\rangle$ f	$\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{out}}$
			[Å]				[IGM only]	[IGM only]	[IGM+CGM]	[IGM+CGM]
All	124	3.049	+13.1	1.04	0.49	0.021 ± 0.002	0.443 ± 0.017	0.047 ± 0.005	0.368 ± 0.020	0.057 ± 0.006
All, detected	15	3.093	+29.2	0.86	0.45	0.134 ± 0.007	⋯	⋯	⋯	⋯
All, not detectedg	106	3.044	+10.6	1.07	0.50	0.010 ± 0.003	⋯	⋯	⋯	⋯
z (Q1)	31	2.843	+10.6	0.91	0.50	0.022 ± 0.007	0.487 ± 0.038	0.045 ± 0.015	0.412 ± 0.045	0.053 ± 0.018
z (Q4)	31	3.284	+16.7	1.18	0.46	0.019 ± 0.003	0.387 ± 0.033	0.047 ± 0.009	0.321 ± 0.037	0.056 ± 0.011
Luv > ${L}_{\mathrm{uv}}^{* }$	60	3.079	+6.6	1.44	0.55	0.002 ± 0.003	0.435 ± 0.024	0.005 ± 0.007	0.363 ± 0.028	0.006 ± 0.008
Luv < ${L}_{\mathrm{uv}}^{* }$	64	3.021	+18.5	0.67	0.43	0.042 ± 0.004	0.449 ± 0.024	0.094 ± 0.010	0.372 ± 0.028	0.113 ± 0.014
Luv (Q1)	31	3.064	+7.4	1.73	0.53	0.002 ± 0.003	0.438 ± 0.035	0.005 ± 0.007	0.364 ± 0.040	0.005 ± 0.008
Luv (Q2)	31	3.086	+5.7	1.12	0.57	0.000 ± 0.004	0.432 ± 0.035	0.000 ± 0.007	0.360 ± 0.040	0.000 ± 0.011
Luv (Q3)	31	3.045	+20.0	0.81	0.44	0.042 ± 0.005	0.442 ± 0.035	0.095 ± 0.014	0.367 ± 0.040	0.114 ± 0.018
Luv (Q4)	31	3.001	+18.1	0.51	0.42	0.052 ± 0.007	0.453 ± 0.056	0.115 ± 0.020	0.377 ± 0.042	0.138 ± 0.024
Wλ($\mathrm{Ly}\alpha$) (Q1)	31	3.005	−5.6	1.13	0.54	0.005 ± 0.004	0.452 ± 0.035	0.011 ± 0.009	0.378 ± 0.040	0.013 ± 0.011
Wλ($\mathrm{Ly}\alpha$) (Q2)	31	3.036	+2.3	1.17	0.57	0.012 ± 0.004	0.441 ± 0.035	0.027 ± 0.009	0.368 ± 0.040	0.033 ± 0.011
Wλ($\mathrm{Ly}\alpha$) (Q3)	31	3.055	+9.3	0.96	0.47	0.017 ± 0.005	0.440 ± 0.034	0.039 ± 0.012	0.364 ± 0.040	0.047 ± 0.015
Wλ($\mathrm{Ly}\alpha$) (Q4)	31	3.088	+43.2	0.92	0.38	0.058 ± 0.006	0.429 ± 0.034	0.138 ± 0.018	0.355 ± 0.040	0.166 ± 0.025
LAEs	28	3.091	+44.1	0.87	0.41	0.063 ± 0.006	0.433 ± 0.036	0.145 ± 0.018	0.359 ± 0.042	0.175 ± 0.026
Non-LAEs	96	3.037	+3.6	1.09	0.51	0.012 ± 0.003	0.445 ± 0.020	0.027 ± 0.007	0.370 ± 0.023	0.032 ± 0.008
Wλ($\mathrm{Ly}\alpha$) > 0	74	3.073	+22.4	0.99	0.45	0.031 ± 0.003	0.437 ± 0.022	0.071 ± 0.008	0.362 ± 0.025	0.086 ± 0.010
Wλ($\mathrm{Ly}\alpha$) < 0	50	3.013	−3.8	1.19	0.55	0.007 ± 0.003	0.450 ± 0.027	0.016 ± 0.007	0.376 ± 0.031	0.019 ± 0.008
${(G-{ \mathcal R })}_{0}$ (Q1)	31	3.095	+19.4	0.94	0.20	0.023 ± 0.004	0.430 ± 0.034	0.053 ± 0.010	0.355 ± 0.040	0.055 ± 0.013
${(G-{ \mathcal R })}_{0}$ (Q2)	31	3.021	+15.2	1.02	0.41	0.022 ± 0.006	0.447 ± 0.035	0.049 ± 0.014	0.372 ± 0.041	0.059 ± 0.017
${(G-{ \mathcal R })}_{0}$ (Q3)	31	3.059	+12.7	1.07	0.57	0.030 ± 0.005	0.439 ± 0.035	0.068 ± 0.013	0.373 ± 0.040	0.080 ± 0.016
${(G-{ \mathcal R })}_{0}$ (Q4)	31	3.023	+4.4	1.15	0.78	0.011 ± 0.005	0.447 ± 0.035	0.025 ± 0.011	0.373 ± 0.040	0.029 ± 0.016

Notes.

^aNumber of objects in the subsample. ^bRest equivalent width of $\mathrm{Ly}\alpha$, where positive (negative) values indicate net emission (absorption). ^cUV color derived from the observed $G-{ \mathcal R }$ color and corrected for the IGM and the strength of the $\mathrm{Ly}\alpha$ feature in the spectrum of each galaxy (see Section 6.4 and Figure 8). ^dObserved mean flux density ratio (f_ν) between the rest-frame wavelength range 880–910 Å and that in the range 1475–1525 Å. ^eMean and standard deviation of the "IGM only" transmission in the rest-wavelength range 880–910 Å for ensembles of sources having the same number and redshift distribution as the observed sample. ^fMean and standard deviation of the "IGM+CGM" transmission in the rest-wavelength range 880–910 Å for ensembles of sources having the same number and redshift distribution as the observed sample. ^gExcludes objects with $| {f}_{900}/{\sigma }_{900}| \geqslant 3$ (15 detections and 3 with >3σ negative outliers).

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The corrected (emergent) flux density ratio $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ (Equation (11)) is also listed in Table 6, where the quoted errors include uncertainties in both the measurement and in $\langle {t}_{900}\rangle$. Table 6 also includes entries for the full KLCS sample ("All"), and "All" divided into subsets according to whether or not the individual LyC measurement had $| {f}_{900}/{\sigma }_{900}| \gt 3$.²³

Finally, Table 6 includes entries for subsamples of KLCS formed according to z_s, where z_s (Q1) is the lowest-redshift quartile and z_s (Q4) is the highest-redshift quartile. These are intended to show that there is no strong dependence of the results on source redshift once the redshift-dependent IGM (or IGM+CGM) corrections have been applied; the composite spectra of these two redshift subsamples are among those shown in Figure 16.

On the basis of the results summarized in Table 6, there are two easily measured empirical characteristics that correlate most strongly with a propensity to "leak" measurable LyC radiation: L_uv and ${W}_{\lambda }(\mathrm{Ly}\alpha )$. Figure 17 illustrates, for the selected subsamples indicated in the figure, the dependence on L_uv and W_λ($\mathrm{Ly}\alpha$) of the measured $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{obs}}$ and inferred $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ for both IGM transmission models.

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The top panel of Figure 17 suggests an almost bimodal dependence of $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{out}}$ on L_uv, where the two lowest luminosity quartiles (i.e., the lower luminosity half) of the KLCS sample have $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ≃ 0.13, whereas the UV-brighter half of the sample has $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ consistent with zero (3σ upper limit $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ≲ 0.02). The transition luminosity below which $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ appears to increase from zero to ≃13% is very close to ${L}_{\mathrm{uv}}^{* }$, which also happens to lie close to the median L_uv of the KLCS sample.

The trend of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{obs}}$ and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ with W_λ($\mathrm{Ly}\alpha$)—illustrated in the bottom panel of Figure 17—is similar, albeit perhaps exhibiting a more gradual dependence of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ on ${W}_{\lambda }(\mathrm{Ly}\alpha )$ compared to the relatively abrupt luminosity dependence of the subsamples grouped according to L_uv. Figure 18 shows the KLCS sample color-coded according to quartiles in L_uv (top) and W_λ($\mathrm{Ly}\alpha$) (bottom). There is clearly an inverse correlation between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and L_uv in the sense that there is a dearth of galaxies with bright L_uv and large ${W}_{\lambda }(\mathrm{Ly}\alpha )$—e.g., the brighter two quartiles in L_uv (Q1 and Q2) are dominated by galaxies with ${W}_{\lambda }(\mathrm{Ly}\alpha )$ ≲ 10 Å (median ≃ 0 Å), and most of the galaxies in the two largest ${W}_{\lambda }(\mathrm{Ly}\alpha )$ quartiles (Q3 and Q4) have L_uv < ${L}_{\mathrm{uv}}^{* }$.²⁴ Because of the well-established correlation between L_uv and ${W}_{\lambda }(\mathrm{Ly}\alpha )$ within all LBG samples (including KLCS; see, e.g., Shapley et al. 2003; Gronwall et al. 2007; Kornei et al. 2010; Stark et al. 2010), it is difficult to say with certainty which is the more reliable indicator of high $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$; however, for reasons discussed in detail in Sections 9.3 and 11.1 below, we believe that ${W}_{\lambda }(\mathrm{Ly}\alpha )$ is more directly linked to whether or not a galaxy has significant LyC leakage, and that the L_uv dependence of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ results from the UV luminosity dependence of the gas-phase covering fraction/column density along the line of sight.

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The most common definition of escape fraction, often used in theoretical studies of reionization, is the fraction of ionizing photons produced by stars in a galaxy that escape into the IGM without being absorbed by H i within the galaxy (e.g., Wyithe & Cen 2007; Wise & Cen 2009). This quantity has also been called the "absolute escape fraction" (f_esc,abs) to convey the fact that the ionizing photon budget against which the leaking LyC is compared includes all of the ionizing UV photons whether or not they are evident in observations. Thus, in order to determine f_esc,abs, a measure of the intrinsic production rate of ionizing photons is necessary; unfortunately, direct empirical estimates of the production rate of ionizing photons (N_ion) are relatively impractical for high-redshift galaxies. Instead, one typically uses SED modeling of young stellar populations (including dust extinction, population age, IMF, etc.) to estimate the intrinsic ionizing photon production. Doing this accurately requires knowledge of the intrinsic SED of the massive stellar populations in the extreme UV (EUV), especially for photon energies in the range 1–4 Ryd, as well as a detailed understanding of the distribution and composition of dust grains that attenuate and redden the intrinsic spectrum of the stars in the galaxy.

An alternative definition intended to be closer to the most readily available measurements of high-redshift galaxy populations is the "relative escape fraction" (f_esc,rel; Steidel et al. 2001),

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{rel}}=\displaystyle \frac{\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{out}}}{{({L}_{900}/{L}_{1500})}_{\mathrm{int}}}\times {10}^{[{A}_{\lambda }(900)-{A}_{\lambda }(1500)]/2.5},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{A}_{\lambda }={k}_{\lambda }E(B-V)\end{eqnarray} \tag{ 13 }$$

is the attenuation in magnitudes as a function of rest wavelength and k_λ parametrizes the attenuation relation. The term (L₉₀₀/L₁₅₀₀)_int is the intrinsic ratio of the unattenuated stellar population of the galaxy, prior to transfer through the ISM, the CGM, and the IGM. Note that there are inconsistencies in the definition of f_esc,rel in the literature having to do with whether or not dust attenuation is included; some authors have implicitly assumed that A_λ(900) − A_λ(1500) = 0, i.e., that the attenuation by dust affects ionizing and non-ionizing UV equally, or that dust affects only the non-ionizing UV (i.e., A_λ(900 Å) = 0). The definition of f_esc,rel used by Grazian et al. (2017), expressed using the notation we have adopted here, is f_esc,rel = (f₉₀₀/f₁₅₀₀)_out/(L₉₀₀/L₁₅₀₀)_int. In order to avoid ambiguity, in what follows below we have attempted to be clear about any assumptions made in mapping $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ to more model-dependent quantities whenever relevant; Sections 9.4.1 and 9.4.2 consider ISM models both with and without dust attenuation of emergent LyC flux.

The intrinsic energy distribution of the stellar sources, parametrized by (L₉₀₀/L₁₅₀₀)_int, is a crucial ingredient to understanding the relationship between ionizing photon production and escape into the IGM. The intrinsic break at the Lyman limit in the integrated spectrum of stars depends on the age, metallicity, initial mass function (both slope and upper mass limit), and the effects of the binary evolution of massive stars. Depending on the assumptions, SPS models predict spectra falling in the range 0.15 ≲ (L₉₀₀/L₁₅₀₀)_int ≲ 0.75 (see, e.g., Leitherer et al. 1999; Eldridge et al. 2018). In the absence of external constraints, the factor of ∼5 uncertainty on (L₉₀₀/L₁₅₀₀)_int translates directly into uncertainties on estimates of the escape fraction.

Fortunately, high-quality far-UV composite spectra such as those of the KLCS subsamples significantly constrain the likely range of stellar population parameters consistent with the observations. Using similar far-UV composites constructed from the spectra of $\langle z\rangle \simeq 2.4$ galaxies that were also observed in the near-IR as part of KBSS-MOSFIRE, Steidel et al. (2016, hereafter S16) showed that, among the SPS models considered, the only set that could simultaneously match the details of the far-UV stellar spectrum—and correctly match the observed ratios of strong nebular emission lines in the spectra of the same galaxies—was that with low stellar metallicity, (Fe/H) ≃ 0.1 (Fe/H)_⊙ and which includes the evolution of massive stars in binary systems (BPASSv2.1; Eldridge et al. 2018).

Prior to fitting models to the observed spectra, the mean IGM transmission spectrum for the CGM+IGM Monte Carlo models described in Section 7 and Appendix B (similar to that shown in Figure 11) was calculated separately for each subsample by drawing 1000 sight-line ensembles with the same number and z_s distribution. We divided the observed subsample spectrum by the mean transmission spectrum and propagated the 68% confidence interval around the mean at each rest-wavelength point as the contribution of the IGM correction to the error vector for λ₀ < 1216 Å.

Following the procedure detailed in S16, we fit each of the composite KLCS spectra in Table 6 with a range of SPS models similar to that described by S16, with the following additions: we tested three different attenuation relations for reddening the SPS model spectra, in addition to varying the metallicity of the stars and the upper mass limit and slope of the IMF. We found that the newest publicly released BPASSv2.1 (Eldridge et al. 2018) models with stellar metallicity Z_* = 0.001 (≃0.07 Z_⊙, assuming the solar abundance scale of Asplund et al. 2009), IMF slope α = −2.35, and upper stellar mass limit of 300 M_⊙ (hereafter referred to as BPASSv2.1-300bin-z001) provided the lowest χ² (i.e., the best fit) for every KLCS composite listed in Table 6. For all of the fits, we assumed a continuous star formation history, constant SFR, and an age of t = 10⁸ yr, close to the median age inferred from SED fits to photometry from the UV to mid-IR (e.g., Reddy et al. 2012; Du et al. 2018). The effects of the star-forming age on the EUV−FUV spectra in the context of the BPASSv2.1-300bin-z001 models are discussed in Section 10. The continuum reddening was allowed to vary over the range 0 ≤ E(B − V) < 1.0 for each subsample; however, the attenuation relation accompanying the best-fitting model varied among the subsamples; Table 7 summarizes the parameters of the best fit for each subsample listed in Table 6.

Table 7.  Statistics Derived from SPS Model Fits to Spectra

Sample	Att.a	$E(B-V)$ b	(L900/L1500)modc	C1500d	$\langle {L}_{\mathrm{bol}}\rangle$ e	$\langle {f}_{\mathrm{esc},\mathrm{rel}}\rangle$ f	$\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ g	$\langle {f}_{\mathrm{esc},\mathrm{rel}}\rangle$ f	$\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ g
					(1011 L⊙)	[IGM only]		[IGM+CGM]
All	R16	0.129	0.167	2.88	2.98	0.29 ± 0.03	0.09 ± 0.01	0.36 ± 0.04	0.12 ± 0.01
All, detectedh	SMC	0.045	0.183	1.76	1.50	1.04 ± 0.10	0.60 ± 0.06	1.21 ± 0.11	0.70 ± 0.07
All, not detected	R16	0.135	0.163	3.03	3.17	0.13 ± 0.04	0.04 ± 0.01	0.17 ± 0.05	0.06 ± 0.02
z (Q1)	SMC	0.060	0.160	2.12	1.92	0.27 ± 0.08	0.12 ± 0.04	0.34 ± 0.11	0.16 ± 0.03
z (Q4)	R16	0.080	0.202	1.93	2.26	0.25 ± 0.04	0.12 ± 0.02	0.31 ± 0.05	0.16 ± 0.03
Luv > ${L}_{\mathrm{uv}}^{* }$	R16	0.141	0.160	3.18	4.55	0.03 ± 0.04	0.01 ± 0.01	0.05 ± 0.05	0.02 ± 0.02
Luv < ${L}_{\mathrm{uv}}^{* }$	SMC	0.047	0.180	1.80	1.20	0.52 ± 0.04	0.27 ± 0.03	0.66 ± 0.09	0.36 ± 0.04
Luv (Q1)	R16	0.133	0.165	2.98	5.11	0.01 ± 0.04	0.02 ± 0.01	0.03 ± 0.05	0.01 ± 0.02
Luv (Q2)	R16	0.154	0.152	3.54	3.94	−0.01 ± 0.06	0.00 ± 0.02	−0.01 ± 0.07	−0.00 ± 0.02
Luv (Q3)	SMC	0.047	0.180	1.80	1.44	0.54 ± 0.07	0.29 ± 0.04	0.67 ± 0.10	0.38 ± 0.05
Luv (Q4)	SMC	0.044	0.185	1.73	0.88	0.62 ± 0.11	0.34 ± 0.07	0.79 ± 0.13	0.45 ± 0.07
${W}_{\lambda }(\mathrm{Ly}\alpha )$ (Q1)	R16	0.158	0.150	3.66	4.10	0.07 ± 0.06	0.02 ± 0.02	0.08 ± 0.07	0.02 ± 0.02
${W}_{\lambda }(\mathrm{Ly}\alpha )$ (Q2)	R16	0.176	0.140	4.24	4.93	0.21 ± 0.06	0.05 ± 0.02	0.26 ± 0.08	0.06 ± 0.02
${W}_{\lambda }(\mathrm{Ly}\alpha )$ (Q3)	SMC	0.049	0.177	1.85	1.76	0.21 ± 0.07	0.11 ± 0.04	0.26 ± 0.08	0.14 ± 0.05
${W}_{\lambda }(\mathrm{Ly}\alpha )$ (Q4)	SMC	0.027	0.215	1.40	1.29	0.61 ± 0.08	0.43 ± 0.06	0.76 ± 0.11	0.55 ± 0.08
LAEs	SMC	0.027	0.215	1.40	1.21	0.66 ± 0.08	0.46 ± 0.06	0.85 ± 0.12	0.62 ± 0.09
Non-LAEs	R16	0.146	0.157	3.31	3.58	0.19 ± 0.04	0.05 ± 0.01	0.23 ± 0.05	0.07 ± 0.02
${W}_{\lambda }(\mathrm{Ly}\alpha )$ > 0	SMC	0.042	0.188	1.69	1.66	0.37 ± 0.04	0.21 ± 0.03	0.48 ± 0.05	0.28 ± 0.03
${W}_{\lambda }(\mathrm{Ly}\alpha )$ < 0	R16	0.165	0.146	3.87	4.57	0.11 ± 0.05	0.03 ± 0.01	0.13 ± 0.05	0.03 ± 0.01
${(G-{ \mathcal R })}_{0}$ (Q1)	SMC	0.015	0.240	1.21	1.12	0.21 ± 0.04	0.17 ± 0.03	0.28 ± 0.05	0.23 ± 0.04
${(G-{ \mathcal R })}_{0}$ (Q2)	R16	0.113	0.178	2.53	2.56	0.26 ± 0.08	0.10 ± 0.03	0.33 ± 0.10	0.13 ± 0.04
${(G-{ \mathcal R })}_{0}$ (Q3)	R16	0.159	0.149	3.69	3.91	0.48 ± 0.09	0.12 ± 0.02	0.60 ± 0.11	0.16 ± 0.03
${(G-{ \mathcal R })}_{0}$ (Q4)	R16	0.221	0.117	6.13	7.00	0.19 ± 0.12	0.03 ± 0.02	0.25 ± 0.14	0.04 ± 0.02

Notes.

^aAttenuation relation for the best-fit SPS model: SMC: Gordon et al. (2003), Reddy et al. (2016a); R16: Reddy et al. (2015, 2016a). ^bFormal uncertainties for a given attenuation and SPS model are small, σ(E(B − V)) ≃  0.001–0.002. ^cFlux density ratio for the stellar population synthesis model, after reddening according to the specified extinction relation. The intrinsic value is (L₉₀₀/L₁₅₀₀)_int ≃ 0.28. ^dInferred extinction correction at λ₀ = 1500 Å for the best-fit SPS model. ^eInferred bolometric luminosity from the attenuation-corrected L₁₅₀₀, in solar luminosities. ^fRatio between (f₉₀₀/f₁₅₀₀)_out and (L₉₀₀/L₁₅₀₀)_mod, for the best-fitting SPS model. ^gInferred absolute escape fraction, where f_esc,abs ≡ f_esc,rel/C1500 (see Figure 20). ^hValues of f_esc,rel and f_esc,abs for the subsample with individual detections assuming that the IGM corrections are drawn from the top 12% of the t₉₀₀ distribution function at $\langle z\rangle =3.093$ (see Table 6).

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Although three different attenuation relations were tested in the fitting, none of the composite spectra was best fit by the Calzetti et al. (2000) attenuation relation. As indicated in Table 7, 13 of the 23 composite spectra were best fit by the attenuation relation constructed by combining the results of Reddy et al. (2015) with the far-UV extension of Reddy et al. (2016a, hereafter R16), while the remaining 10 spectra favored the steeper, SMC attenuation curve which combines the empirical line-of-sight SMC extinction curve from Gordon et al. (2003; assuming R_V = 2.74) with an extension to the FUV derived as in R16.

The values of (L₉₀₀/L₁₅₀₀)_mod listed in Table 7 assume (for the time being) that the same attenuation curve and E(B − V) value affects both the ionizing and non-ionizing UV light, and that the far-UV attenuation curves can be extrapolated shortward of λ₀ ≤ 950 Å, the shortest wavelength over which they are empirically constrained²⁵ (see Reddy et al. 2016a). Here we remind the reader that the values of $\langle {f}_{\mathrm{esc},\mathrm{rel}}\rangle$ listed in Table 7 are as defined in Equation (12). Although the fits to the stellar continuum exclude all wavelength pixels shortward of 1070 Å, the fitted spectra appear to be excellent representations of the observed spectra all the way down to ∼930 Å, where the confluence of the stellar and interstellar Lyman series lines in the wavelength interval 910–930 Å depresses the galaxy spectrum relative to the model; an example is shown in Figure 14 (see also Figure 15).

The BPASSv2.1-300bin-z001 continuous star formation SPS model, including the contribution of the nebular continuum as described by S16, has (L₉₀₀/L₁₅₀₀)_int ≃ 0.28 ± 0.03,²⁶ prior to reddening according to the values in Table 7; our assumption that the entire spectrum is reddened by the same attenuation relation further reduces the intrinsic L₉₀₀/L₁₅₀₀ by an amount that depends on E(B − V) and the attenuation curve. Thus, the values of (L₉₀₀/L₁₅₀₀)_mod in Table 7 correspond to the predicted emergent spectrum that would be observed if there were reddening by dust but no photoelectric absorption by H i in the ISM along the line of sight.

Table 7 lists the inferred values of f_esc,rel, equal to the ratio between $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ (Table 6) and (L₉₀₀/L₁₅₀₀)_mod (see Equation (12)), the flux density ratio for the best-fit (reddened) SPS spectrum. Figure 19 illustrates the associated values for each of the subsample composite spectra.

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At this point, the value of f_esc,rel has accounted for the relative attenuation between 900 and 1500 Å, but not for the absolute attenuation affecting f₁₅₀₀. In the context of the modeled UV SEDs summarized in Table 7, this factor is simply

$$\begin{eqnarray}&&{\rm{C}}1500\equiv {10}^{E(B-V)\times {k}_{\lambda }(1500)/2.5},\end{eqnarray} \tag{ 14 }$$

where k_λ(1500) = 8.91 and 13.05 for the R16 and SMC attenuation relations, respectively.

We then define the absolute escape fraction as

$$\begin{eqnarray}&&\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle =\langle {f}_{\mathrm{esc},\mathrm{rel}}\rangle /{\rm{C}}1500;\end{eqnarray} \tag{ 15 }$$

both C1500 and f_esc,abs are included in Table 7. Figure 20 compares the values of this inferred quantity with the observed $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}$ for each KLCS subset composite spectrum. Note that all of the KLCS subsets with f_esc,abs ≳ 0.20—whether based on UV color, UV luminosity, or W_λ($\mathrm{Ly}\alpha$)—are best fit using the "line-of-sight" SMC extinction curve. This can probably be attributed to the relatively blue UV color of these composites, as the bluest/youngest star-forming galaxies are known to be most consistent with a steep UV attenuation curve (Reddy et al. 2010, 2018), rather than grayer extinction curves such as those of Calzetti et al. (2000) or R16, perhaps due to the prevalence of small dust grains (or the absence of large ones).

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We note that fitting reddened SPS models to the observed spectra to estimate f_esc,abs (Table 7 and Figure 20) rather than the more empirical $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ reveals that, in addition to luminosity and ${W}_{\lambda }(\mathrm{Ly}\alpha )$ dependence noted in Section 8 for $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ (each of which becomes steeper relative to f_esc,abs), accounting for continuum-reddening corrections suggests that f_esc,abs also depends strongly on UV color (${(G-{ \mathcal R })}_{0}$), in qualitative agreement with the analysis presented by Reddy et al. (2016b).

We return to a discussion of the implications of the various measures of escape fraction in Section 12.

As described in Section 9.2 above, the relative escape fraction is f_esc,rel ≃ 0.5 for the KLCS subsets leaking the most, with corresponding f_esc,abs ≃ 0.3–0.4; both values are model-dependent in the sense that they are evaluated relative to assumed SPS models (whereas (f₉₀₀/f₁₅₀₀)_out is SPS model-independent). We have shown that there is a strong correlation between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and f_esc, and it has been known for some time (Shapley et al. 2003) that W_λ($\mathrm{Ly}\alpha$) is anticorrelated with the strength of low-ionization (e.g., C ii, Si ii, O i) interstellar absorption features observed in the same spectra. The presence of strong lines of neutral and singly ionized species in gas seen in absorption against the UV continuum from massive stars suggests significant neutral H column densities, ${N}_{{\rm{H}}{\rm{I}}}\gt {10}^{17}\,{\mathrm{cm}}^{-2}$, and the relatively constant W_λ/λ for lines of the same species but varying λf, where λ is the rest wavelength of the transition and f is the oscillator strength (e.g., Si ii λ1190, 1193, 1260, 1304, 1526) indicates the presence of saturation.²⁷

The observation that strong interstellar (IS) lines often do not become black at maximum optical depth then suggests partial covering of the continuum source (hereinafter f_c < 1) by the low-ion-bearing gas. If the same partial covering were to apply to the H i, there would follow an obvious relationship between the gas-covering fraction f_c and the probability that significant LyC could "escape" via the uncovered regions. Conversely, if a galaxy spectrum's low-ionization IS lines are black over a considerable range in velocity, one would expect no significant LyC photon leakage in our direction (e.g., Steidel et al. 2001, 2010; Pettini et al. 2002; Shapley et al. 2003, 2006; Quider et al. 2009; Heckman et al. 2011; Jones et al. 2012). However, as recently shown (Henry et al. 2015; Reddy et al. 2016b; Vasei et al. 2016), f_c < 1 for low-ionization metal lines is a necessary, but not sufficient, predictor of significant LyC leakage from a galaxy. The metal lines would be relatively insensitive to low-metallicity H i, and possibly also to metal-enriched gas with LyC optical depths 1 ≲ τ_LyC ≲ 10, where neutral and singly ionized metallic species might not be the dominant ionization stages in the gas (see also Rivera-Thorsen et al. 2015).

To examine trends in the residual depth and the velocity extent of interstellar absorption features, we divided each of the KLCS composites by its best-fit SPS model and formed continuum-normalized spectra with the strongest stellar features removed. Figure 21 shows the line profiles of Si ii λ1260 and λ1526, C ii λ1334, and Lyβ λ1025.7 and Ly λ937.8 for several of the KLCS subsets. Also shown in each panel of Figure 21 is the profile of $\mathrm{Ly}\alpha$ emission, after subtracting the stellar continuum, dividing by a factor of 15, and adding 1 (both of the latter for display purposes). The spectral resolutions (FWHM) for the metal lines (labeled "LRIS-R") and for the Lyman lines (labeled "LRIS-B") are shown in the bottom right of each panel. Figure 21 is arranged according to the measured f_esc,abs, from largest (top left) to smallest (bottom right).

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In addition to the trends already noted between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$, there is a similar trend between the apparent depth of the Lyman series and low-ionization metallic absorption lines and f_esc. In this context, it is noteworthy that the metallic lines shown in Figure 21 were recorded using LRIS-R, which have (as discussed in Section 3) spectral resolving power R ∼ 1400 (σ_inst ≃ 90 $\mathrm{km}\,{{\rm{s}}}^{-1}$), whereas the Lyman series absorption lines were measured using LRIS-B, with R ∼ 800 (σ_inst ≃ 160 $\mathrm{km}\,{{\rm{s}}}^{-1}$). In spite of this, the Lyman series absorption lines in all cases have apparent optical depth equal to or larger than the metal lines. The similarity of the apparent depth of Lyman lines (at least, for v_sys ≲ 0) in all of the composites shown suggests log(${N}_{{\rm{H}}{\rm{I}}}$/cm²) ≳ 18—in which case all of the observed Ly lines would be strongly saturated—coupled with partial covering of the far-UV continuum. In general, the maximum depths of both H i and metallic absorption lines occur at −300 ≲ v ≲ −100 $\mathrm{km}\,{{\rm{s}}}^{-1}$, consistent with the hypothesis that the bulk of the cool, metal-enriched gas along the sight line "down the barrel" is outflowing. The Lyman line profiles exhibit more varied behavior for velocities v > 0 km s⁻¹, where absorption is less deep overall and where Lyβ absorption is significantly stronger than Ly, suggesting that both f_c and log(${N}_{{\rm{H}}{\rm{I}}}$) are lower for gas with v > 0 km s⁻¹ physically located on the near side of the stars comprising the UV continuum.

The shapes and depths of the metal lines at v > 0 are difficult to interpret in the same way, since all of the metallic species shown in Figure 21 are resonance lines that may manifest "emission filling" from scattered-line photons re-emitted in our direction (see, e.g., Prochaska et al. 2011; Erb et al. 2012; Martin et al. 2012; Scarlata & Panagia 2015). However, all three of the lines illustrated have nearby non-resonant transitions (i.e., transitions from the same excited state to excited fine-structure levels just above the ground state) which can act as "relief valves" for the resonance lines, so that a substantial fraction of absorbed resonance photons may be re-emitted in the non-resonance emission lines—in this case, Si ii*λ1264.9, C ii* λ1335.7, and S ii* λ1533.4. Each panel of Figure 21 marks the position of the galaxy systemic redshift relative to the profiles of the non-resonant emission lines, with the same color coding used for the absorption profiles of the nearby resonance line. The implications of the non-resonant emission lines for LyC escape are discussed in Section 11.3.

In any case, the Lyman series absorption lines between the Lyman limit (911.75 Å) and Lyβ provide the most direct constraints on ${N}_{{\rm{H}}{\rm{I}}}$ and/or the continuum-covering fraction (f_c) of H i near the galaxy systemic redshift along the line of sight. They are particularly useful for determining ${N}_{{\rm{H}}{\rm{I}}}$ when log(${N}_{{\rm{H}}{\rm{I}}}$/cm⁻²) < 18, where the LyC would be optically thin regardless of f_c, or when log(N_{H i}/cm²) > 20, where prominent damping wings of $\mathrm{Ly}\alpha$ can be recognized even in spectra of modest spectral resolution. In between these limits (i.e., for 18 ≲ log(${N}_{{\rm{H}}{\rm{I}}}$/cm⁻²) ≲ 20), ${N}_{{\rm{H}}{\rm{I}}}$ is not well constrained by the spectra, but the fact that the observable Lyman series lines are strongly saturated provides redundant constraints on the covering fraction of optically thick H i. If these saturated Lyman series lines are resolved in velocity space, the residual intensity at maximum line depth relative to the continuum should be constant and equal to 1 − f_c, where f_c is the fraction of far-UV continuum covered by optically thick material. At the resolution of the KLCS galaxy spectra shortward of the rest-frame $\mathrm{Ly}\alpha$ (σ_inst ≃ 160 $\mathrm{km}\,{{\rm{s}}}^{-1}$), the cores of the higher Lyman series lines with σ_v ≲ 100 $\mathrm{km}\,{{\rm{s}}}^{-1}$ would not be resolved, in which case any apparent residual flux in the cores of the Lyman series lines would correspond to a lower limit on the fraction f_c of the stellar FUV continuum covered by optically thick H i.

In Section 9.2 above, we described fitting SPS spectra to the far-UV spectra of the observed KLCS composites in order to estimate $\langle {f}_{\mathrm{esc},\mathrm{rel}}\rangle$ and $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ in a self-consistent manner. These spectral fits assumed that both ionizing and non-ionizing components of the integrated UV light from stars are reddened by the same dust screen with attenuation that is a smooth function of the rest wavelength governed by one of three model attenuation relations (Calzetti, R16, and revised SMC). However, introducing the parameter f_c to describe the fraction of the FUV stellar continuum covered by optically thick H i requires a decision about how to handle dust reddening—i.e., whether the "covered" and "uncovered" portions of the stellar continuum have been reddened by the same dust opacity. It is easy to imagine that the dust column density could be lower along lines of sight that do not intersect any ISM gas with log(${N}_{{\rm{H}}{\rm{I}}}$/cm⁻²) ≳ 18; if LyC escape requires geometric "holes" that have been cleared of all gas (neutral and ionized) between the stars and the IGM, then it may be that any residual LyC flux has not been attenuated or reddened by dust at all (a counterexample in an LyC-emitting galaxy at low redshift has been discussed by Borthakur et al. 2014). In Sections 9.4.1 and 9.4.2, we consider two simple geometric models of the ISM intended to bracket the range of possibilities in extinction/reddening. In the first, which we call the "screen model," all light is attenuated and reddened by the same foreground screen, the standard assumption implicit in estimating dust extinction from far-UV photometry or spectroscopy, and the one used in Section 9.2 for calculations summarized in Table 7. The second alternative, which we call the "holes model," assumes that the covered portion of the FUV continuum is reddened by dust, but that the uncovered portion passes through the ISM without significant attenuation from dust or photoelectric absorption. We discuss the two cases separately below.

9.4.1. The "Screen" Model

The emergent spectrum after passing through the ISM and CGM in the screen model context can be expressed as

$$\begin{eqnarray}&&{S}_{\nu ,\mathrm{obs}}={10}^{-{A}_{\lambda }/2.5}{S}_{\nu ,\mathrm{int}}[(1-{f}_{{\rm{c}}})+{f}_{{\rm{c}}}\langle {e}^{-\tau (\lambda )}\rangle ],\end{eqnarray} \tag{ 16 }$$

where A_λ = k_λE(B − V), S_V,int is the intrinsic stellar spectrum prior to attenuation and absorption in the ISM, and e^−τ(λ) is the transmission spectrum through the ISM due to line and continuum absorption, which depends on ${N}_{{\rm{H}}{\rm{I}}}$ and (to a lesser extent) the kinematics of the absorbing gas.

Using an approach similar to that described by Reddy et al. (2016b), we modeled the ISM²⁸ for each of the KLCS composite spectra by beginning with the best-fit SPS model (Table 7) and fitting for additional H i line and continuum absorption shortward of $\mathrm{Ly}\alpha$. For modeling the Lyman series absorption lines, we assumed a single component with Doppler width b = 125 $\mathrm{km}\,{{\rm{s}}}^{-1}$, with velocity relative to systemic v_sys = −100 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (the typical velocity corresponding to the minimum residual flux density of absorption profiles of H i and low-ionization metal lines), and smoothed the artificial line profiles to the resolution of the observed spectra. As described below (see Table 8), only the cores of the first six Lyman series lines, the damping wings of $\mathrm{Ly}\alpha$ and Lyβ, and the Lyman continuum region (f₉₀₀) were used in evaluating the fits, thus they are insensitive to the precise line profile shapes for saturated lines without damping wings.

Table 8.  Wavelength Regions Used for Spectral Fits^a

λ1	λ2	Comments
(Å)	(Å)
Regions Masked for Stellar Continuum Fits
1080	1087	N ii IS abs
1099	1107	Fe ii* em.
1119	1123	Fe ii IS abs complex
1131	1155	Fe ii IS abs complex
1172	1178	C iii
1186	1231	Si ii, Si iii IS abs.
1254	1270	Si ii IS abs, Si ii* em
1291	1312	Si ii, O i abs, Si ii* em
1328	1340	C ii IS abs, C ii* em
1363	1373	O v stellar wind
1389	1396	Si iv IS abs
1398	1404	Si iv IS abs
1521	1528	Si ii IS abs
1531	1536	Si ii* em
1541	1552	C iv IS abs
1606	1610	Fe ii IS abs
1657	1675	[O iii] neb em, Al ii IS abs
1708	1711	Ni ii IS abs
Regions Used for NH i, fc Fits
880	910	LyC region
929.0	931.0	Ly6 core
936.6	937.8	Ly core
948.0	950.8	Lyδ core
970.8	973.6	Lyγ core
1013.3	1020.0	Lyβ blue wing
1023.5	1026.0	Lyβ core
1178.8	1183.3	Lyα blue damping wing
1223.7	1248.5	Lyα red damping wing

Note.

^aSpectral fits to the stellar continuum were performed over the wavelength range 1070–1740 Å (except for masked regions) for all composites but z(Q4), for which the range 1070–1640 Å was used.

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Aside from the ISM treatment, the fits are identical to those described in Section 9.2, where the SPS model is reddened using the attenuation relation that provides the best fit to the CGM+IGM-corrected spectrum in the rest-wavelength range 1070–1740 Å, after masking the regions listed in the top portion of Table 8. Simultaneously, we fit for the values of f_c and log(${N}_{{\rm{H}}{\rm{I}}}$) by comparing the predicted model spectrum with the observed spectrum, where the additional parameters are constrained in the wavelength intervals listed in the bottom part of Table 8. The regions near $\mathrm{Ly}\alpha$ (and, to a lesser extent, near Lyβ) are sensitive to H i damping wings for log(N_{H i}/cm²) ≳ 20.0, while the depths of the Lyman series lines and the residual LyC emission constrain f_c. Assuming the same SPS model and attenuation relations used in Table 7, we recorded the values of E(B − V), f_c, and log(N_{H i}/cm²) that minimized the residuals with respect to the data; these are listed in Table 9. We estimated the uncertainties on each parameter by repeatedly perturbing every pixel of the observed spectrum by an amount consistent with the error spectrum (which includes contributions from both sample variance and shot noise) and refitting. The typical uncertainties on E(B − V), f_c, and log(N_{H i}/cm²) were found to be ±0.002, ±0.03, and ±0.10, respectively. The best-fit value of f_esc,abs computed directly from the models and the rms scatter of the best-fit values obtained from repeated fits of the perturbed observed spectra are listed in the last column of Table 9. Example fits for the full KLCS galaxy sample ("All"), the highest L_uv quartile (LQ1), and the subset with W_λ($\mathrm{Ly}\alpha$) > 20 Å (LAEs) are shown in Figure 22.

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Table 9.  ISM Fit Results: Screen Model^a

Sample	Att.	$E(B-V)$	log(NH i)	fcb	fesc,absc
			(cm−2)
All	R16	0.129	20.61	0.70	0.12 ± 0.02
All, detectedd	SMC	0.045	(17.41)	0.47	0.69 ± 0.04
All, not detected	R16	0.135	20.65	0.75	0.05 ± 0.01
z(Q1)	SMC	0.060	20.61	0.71	0.15 ± 0.05
z(Q4)	R16	0.080	20.39	0.69	0.15 ± 0.03
Luv > ${L}_{\mathrm{uv}}^{* }$	R16	0.141	20.75	0.79	<0.02
Luv < ${L}_{\mathrm{uv}}^{* }$	SMC	0.047	20.43	0.62	0.36 ± 0.04
Luv (Q1)	R16	0.133	20.68	0.82	<0.03
Luv (Q2)	R16	0.154	20.74	0.75	<0.05
Luv (Q3)	SMC	0.047	20.43	0.63	0.37 ± 0.05
Luv (Q4)	SMC	0.044	20.48	0.57	0.44 ± 0.06
Wλ($\mathrm{Ly}\alpha$) (Q1)	R16	0.158	21.10	0.81	<0.04
Wλ($\mathrm{Ly}\alpha$) (Q2)	R16	0.175	20.79	0.76	0.06 ± 0.02
Wλ($\mathrm{Ly}\alpha$) (Q3)	SMC	0.049	20.00	0.80	0.13 ± 0.05
Wλ($\mathrm{Ly}\alpha$) (Q4)	SMC	0.027	20.07	0.45	0.54 ± 0.06
LAEs	SMC	0.027	20.02	0.40	0.57 ± 0.06
Non-LAEs	R16	0.145	20.71	0.77	0.07 ± 0.02
Wλ($\mathrm{Ly}\alpha$) > 0	SMC	0.041	20.12	0.66	0.28 ± 0.03
Wλ($\mathrm{Ly}\alpha$) < 0	R16	0.165	21.02	0.79	<0.03
${(G-{ \mathcal R })}_{0}$ (Q1)	SMC	0.015	20.29	0.74	0.21 ± 0.04
${(G-{ \mathcal R })}_{0}$ (Q2)	R16	0.112	20.57	0.73	0.13 ± 0.04
${(G-{ \mathcal R })}_{0}$ (Q3)	R16	0.162	20.62	0.73	0.15 ± 0.03
${(G-{ \mathcal R })}_{0}$ (Q4)	R16	0.222	21.00	0.59	0.04 ± 0.02

Notes.

^aModels assume that dust affects both the ionizing and non-ionizing emission from galaxies. ^bFraction of the EUV and FUV stellar continuum covered by optically thick (in the Lyman continuum) H i. ^cAssuming CGM+IGM correction, with measurements with <2σ significance replaced by 2σ upper limits. ^dAssuming 90th percentile t₉₀₀, for CGM+IGM correction, as in Table 7.

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As noted by Reddy et al. (2016b), f_c and ${N}_{{\rm{H}}{\rm{I}}}$ generally have little covariance because one (f_c) is determined by the minima at the cores of the Lyman series lines and the LyC region, while logN_{H i} is constrained by the broad damping wings of $\mathrm{Ly}\alpha$. However, as discussed above, column densities in the range 18 ≲ log(N_{H i}/cm⁻²) ≲ 20 are relatively poorly constrained because there are no discernible $\mathrm{Ly}\alpha$ damping wings, yet the resolved Lyman series lines have saturated cores. This affects the fit for the "All, detected" sample, where f_c and N_{H i} for a fixed $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ are less well determined.²⁹

The tabulated f_esc,abs values are generally very similar to those listed in Table 7; the difference in the present case is that the fits include constraints provided by the depth of the Lyman series lines and the $\mathrm{Ly}\alpha$ damping wings in addition to the measurement of (f₉₀₀/f₁₅₀₀)_out. Thus, the values listed in Table 9 constitute more stringent checks on the internal consistency of the geometrical model of the ISM: rather than calculating f_esc,abs from the SPS fit to the stellar continuum and (f₉₀₀/f₁₅₀₀)_obs alone, we now require consistency between the diminution of LyC flux relative to the SPS model and the other observational constraints on the column density and geometric covering fraction of the gas-phase ISM. In this sense, the fits summarized in Table 9 provide additional support for our simplified model for the galaxy ensembles. We return to a discussion of the implications in Section 10 below.

9.4.2. The "Holes" Model

An alternative model for the effect of the galaxy ISM on the far-UV spectrum is one in which leaking LyC photons escape via completely transparent "holes" in the ISM, as discussed by Reddy et al. (2016b; see also Zackrisson et al. 2013). This type of model is qualitatively supported by the fact that the equivalent widths of interstellar lines of low-ionization species are correlated with inferred reddening by dust, but the same is not true for higher ionization species like C iv (e.g., Shapley et al. 2003). In the "Holes" model, the spectrum is decomposed into two distinct components, one of which is reddened and absorbed by foreground dust and gas, and the other unattenuated and unabsorbed—and is therefore a scaled version of the intrinsic SPS spectrum. The parameter f_c is the fraction of the intrinsic stellar spectrum covered by optically thick absorbing material along our line of sight, and the net observed spectrum is given by the linear combination

$$\begin{eqnarray}&&{S}_{\nu ,\mathrm{obs}}={S}_{\nu ,\mathrm{int}}[(1-{f}_{{\rm{c}}})+{f}_{{\rm{c}}}\,{e}^{-\tau }(\lambda ){10}^{-{A}_{\lambda }/2.5}],\end{eqnarray} \tag{ 17 }$$

where A_λ = k_λE(B − V)_cov and E(B − V)_cov is the reddening applied to the covered portion of the galaxy The second term on the right-hand side of Equation (17) also accounts for H i line and continuum opacity as described in the previous section. For each KLCS composite spectrum, we fitted for the model values of f_c, log(${N}_{{\rm{H}}{\rm{I}}}$/cm⁻²), and E(B − V)_cov that minimized χ²; the best-fit model parameters are given in Table 10. Figure 23 shows the decomposition of the best fitting holes models for the same composites shown in Figure 22.

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Table 10.  ISM Fit Results: Holes Model^a

Sample	Att	$E(B-V)$cov	log(NH i)	fcb	fesc,absc
			(cm−2)
All	SMC	0.068	20.57	0.91	0.09 ± 0.01
All, detectedd	SMC	0.085	(17.9)	0.80	0.31 ± 0.03
All, not detected	R16	0.163	20.60	0.95	0.05 ± 0.01
z(Q1)	SMC	0.076	20.56	0.92	0.08 ± 0.01
z(Q4)	R16	0.114	20.34	0.88	0.12 ± 0.02
Luv > ${L}_{\mathrm{uv}}^{* }$	R16	0.165	20.71	0.96	<0.04
Luv < ${L}_{\mathrm{uv}}^{* }$	SMC	0.068	20.39	0.87	0.13 ± 0.03
Luv (Q1)	R16	0.153	20.64	0.96	<0.04
Luv (Q2)	R16	0.195	20.63	0.96	<0.04
Luv (Q3)	SMC	0.065	20.39	0.87	0.13 ± 0.03
Luv (Q4)	SMC	0.069	20.44	0.84	0.16 ± 0.03
Wλ($\mathrm{Ly}\alpha$) (Q1)	R16	0.182	21.05	0.97	<0.03
Wλ($\mathrm{Ly}\alpha$) (Q2)	R16	0.208	20.74	0.97	<0.04
Wλ($\mathrm{Ly}\alpha$) (Q3)	SMC	0.059	19.93	0.93	0.07 ± 0.02
Wλ($\mathrm{Ly}\alpha$) (Q4)	SMC	0.054	20.05	0.73	0.27 ± 0.02
LAEs	SMC	0.058	20.05	0.71	0.29 ± 0.03
Non-LAEs	R16	0.174	20.66	0.96	0.04 ± 0.02
Wλ($\mathrm{Ly}\alpha$) > 0	SMC	0.059	20.11	0.86	0.14 ± 0.02
Wλ($\mathrm{Ly}\alpha$) < 0	R16	0.193	20.97	0.97	<0.03
${(G-{ \mathcal R })}_{0}$ (Q1)	SMC	0.019	20.28	0.85	0.15 ± 0.02
${(G-{ \mathcal R })}_{0}$ (Q2)	R16	0.142	20.52	0.94	0.06 ± 0.02
${(G-{ \mathcal R })}_{0}$ (Q3)	R16	0.195	20.56	0.95	<0.08
${(G-{ \mathcal R })}_{0}$ (Q4)	R16	0.296	20.87	0.97	<0.06

Notes.

^aModels assume that dust and photoelectric absorption affects only the covered fraction f_c of the UV continuum. ^bFraction of the EUV and FUV stellar continuum covered by the assumed N_{H i} and dust. ^cInferred absolute escape fraction of LyC photons. ^dAssuming a CGM+IGM correction as in Table 7.

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The main difference between the screen and holes models is the interpretation of f_c: for the holes case, the differential attenuation between the covered and uncovered components requires accounting for extinction as part of the calculation of f_c; this is easily seen in the top panel of Figure 23, which shows that the relative contributions of the attenuated and unattenuated components to the observed (non-ionizing) spectrum are in the ratio ∼3:1, while geometrically the ratio is f_c/(1 − f_c) ∼ 10. The concept of f_esc,rel is probably less useful in the context of the holes models, since $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ now has a more complex relationship to the intrinsic SED of the SPS spectrum than in the screen case (because of the assumption of two independent components, one affected by dust and the other unaffected).

However, in the holes models, f_esc,abs is very straightforward to estimate provided τ(LyC) ≫ 1 for the covered component, in which case

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{abs}}\simeq 1-{f}_{{\rm{c}}}.\end{eqnarray} \tag{ 18 }$$

Equation (18) applies to all of the models listed in Table 10, except for the composite of individually detected galaxies ("All, detected"). As discussed previously, the appropriate CGM+IGM correction for this subsample is much more uncertain than for any of the other composites; with the assumptions adopted earlier—that the detected subsample has IGM+CGM transmission drawn from the highest 12%—the best-fit model spectrum has 1 − f_c ≈ 0.20 but f_esc,abs has an additional contribution from ionizing photons that have passed through the H i layer without being completely attenuated (because the LyC optical depth is of order unity).

Model fits for both the screen and the holes models can simultaneously reproduce the overall spectral shape, the depth of the Lyman series absorption lines, the wings of the interstellar $\mathrm{Ly}\alpha$ absorption feature, and the residual LyC flux using the same underlying SPS model. There is a slight tendency for both sets of ISM models to overestimate the residual LyC flux for composites having the lowest values of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$: e.g., the L_uv > ${L}_{\mathrm{uv}}^{* }$ subsample has $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ = 0.006 ± 0.008 (Table 6), but the best-fit model gives $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ∼ 0.03 (Tables 9 and 10). Such discrepancies would be expected if the covering fraction of the optically thick ISM layer is slightly underestimated by the cores of the Lyman series lines, or if there is H i with 17.5 ≲ log(N_{H i}/cm²) ≲ 20 that does not contribute significantly to the $\mathrm{Ly}\alpha$ damping wings and whose Lyman series lines are not fully resolved in the spectra (and thus their depth is underestimated). Since the parameter f_c is constrained in our modeling by simultaneously matching the LyC residuals (which should not depend on spectral resolution) and the Lyman series line depths (which may depend on resolution), larger reduced χ² evaluated over those spectral regions would flag inconsistencies.

Another possibility of course is that $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ is underestimated due to residual systematics in the background subtraction, in this case oversubtraction; this possibility cannot be ruled out entirely, but seems less likely than the presence of unresolved intermediate column density H i.

Broadly speaking, the success of the simple models presented above demonstrates that there are real trends within the KLCS sample among the depth of the Lyman series lines, the residual LyC flux, and the column density of the dominant component of ISM N_{H i} absorption; the trends—increased LyC leakage in galaxies with lower ${N}_{{\rm{H}}{\rm{I}}}$, lower inferred f_c, and larger W_λ($\mathrm{Ly}\alpha$)—are in the direction one might have anticipated. We consider a simple physical interpretation of the results in Section 11.1, where we conclude that the "Holes" model does surprisingly well in explaining the observed interconnections between the observable quantities.

Assuming that the observations are free of systematics that would invalidate the results, this quantity is entirely model-independent; however, it is of only limited use for individual objects because it is so strongly modulated by the variations of CGM+IGM opacity among different lines of sight (Section 7). With ensembles in which more than ≃30 objects with similar redshifts and common observed properties are considered, the mean value of (f₉₀₀/f₁₅₀₀)_obs can be corrected using empirically calibrated IGM+CGM models to obtain $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$. For the KLCS sample, $0\leqslant \langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}\leqslant 0.14$ among the galaxy subsamples (Table 6), with a sample mean of $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}\,=0.021\pm 0.002$.

This parameter is the most important one, in practical terms, for estimating the contribution of a galaxy population of known far-UV (e.g., at rest-frame 1500 Å) luminosity function to the metagalactic ionizing radiation field (see Section 11.4). By our definition, this ratio characterizes the flux density of ionizing photons averaged over the 880–910 Å window, relative to the non-ionizing UV flux density near 1500 Å, statistically corrected for IGM+CGM opacity back to the "galaxy photosphere" at r_gal = 50 kpc. As discussed in Section 7, the statistical uncertainty on the IGM correction is drastically reduced for an ensemble of galaxies provided the lines of sight are independent and the sample selection does not depend significantly on the IGM properties at fixed redshift. Remaining systematic uncertainties in the values of $\langle {t}_{900}\rangle$ for an ensemble will stem primarily from the treatment of the CGM and possible systematic differences in the large-scale IGM environment.

Table 6 includes values of $\langle {t}_{900}\rangle$ under two different assumptions for the H i column density distribution for distances r > 50 pkpc from a galaxy; $\langle {t}_{900}\rangle$ values with and without an enhancement in ${N}_{{\rm{H}}{\rm{I}}}$ from the CGM typically differ by ∼20%, which likely overestimates the true level of systematic uncertainty but would be a conservative upper limit. As Table 6 indicates, such systematics, if present, would affect the inferred values of f_esc,rel and f_esc,abs by a similar factor.

As discussed in Section 9.1 above, f_esc,rel is a useful concept if the intrinsic (L₉₀₀/L₁₅₀₀)_int of a galaxy's integrated stellar population is known, since the latter ratio is the absolute maximum possible (f₉₀₀/f₁₅₀₀)_out; i.e., f_esc,rel approaches unity only when there is no H i with appreciable LyC optical depth between the observed stars and r_gal ∼ 50 pkpc.

If f_esc,rel ≤ 1, (L₉₀₀/L₁₅₀₀)_int ≃ 0.28 (Section 9.2; Figure 24), and $\langle {t}_{900}\rangle \simeq 0.37$ (Table 6), then the maximum value of $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}$ expected for an ensemble of galaxies with z_s ≃ 3.05 is $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs},\max }\simeq 0.11$. For an individual galaxy with z_s = 3.05, t₉₀₀ < 0.67 (the ∼99th percentile IGM+CGM transmission; see Table 12), (f₉₀₀/f₁₅₀₀)_obs,max ≃ 0.30 × 0.67 ≃ 0.20. Accounting for dust³⁰ or assuming a more typical IGM sight line would immediately reduce the expected upper limits on the observed values. These small numbers would be reduced by large factors if the LyC regions were not sampled optimally (Section 7.2) or if the source redshift were significantly higher (Section 7). A corollary to these considerations is that one should be suspicious of individual measurements of LyC leakage if (f₉₀₀/f₁₅₀₀)_obs ≳ 0.2 and of ensemble measurements with $\langle {f}_{900}/{f}_{1500}{\rangle }_{\mathrm{obs}}\gtrsim 0.11$.

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Clearly, assuming a different SPS model, with a different (L₉₀₀/L₁₅₀₀)_int, would also change the inferred f_esc,rel given a measurement of, or limit on, (f₉₀₀/f₁₅₀₀)_out. Figure 24 shows the metallicity and age dependence of (L₉₀₀/L₁₅₀₀)_int (including self-consistently the contribution of nebular continuum emission to the non-ionizing UV light) for constant SFR BPASSv2.1-300bin models. In the context of such models, typical z ∼ 3 galaxies have SED-inferred ages of ∼50–500 Myr (7.7 ≲ log(t/yr) ≲ 8.7); the Z = 0.001 model that best reproduces the details of the observed spectra has 0.24 ≲ (L₉₀₀/L₁₅₀₀)_int ≲ 0.33 over the same age range. Assuming younger ages for the SPS models would, naively, lower the required value of f_esc,abs to match an observed (f₉₀₀/f₁₅₀₀)_out; however, younger SPS models are somewhat bluer in the non-ionizing FUV, so that matching an observed galaxy SED requires more reddening by dust—and all reasonable dust attenuation relations have k_λ(900) > k_λ(1500), meaning that additional reddening reduces f₉₀₀ relative to f₁₅₀₀. For the screen models, fits of BPASSv2.1 models to the full KLCS galaxy composite with the same IMF and stellar metallicity (Z_* = 0.001) but varying the age over the range $\mathrm{log}(t/\mathrm{yr})=[7.5\to 9.0]$ results in monotonic variation in best-fit parameter values: E(B − V) $[0.145\to 0.085$], C1500 $[3.29\to 2.01$], ${N}_{{\rm{H}}{\rm{I}}}$ $[20.80\to 20.05$], and f_esc,abs $[0.097\to 0.191$], but only small changes in f_esc,rel $[0.32\to 0.38]$ and f_c $[0.73\to 0.68$]. However, the best fits to the spectra—both in terms of reproducing the details of the stellar continuum and matching the observed LyC residuals, Lyman series line depths, and $\mathrm{Ly}\alpha$ damping wings—are obtained for 7.9 ≲ log(t/yr) ≲ 8.5, where the parameters are consistent with those listed in Table 9 for the fiducial log(t/yr) = 8.0 model (see the bottom panel of Figure 24).

In this section, we have tried to emphasize that the parameter often seen as the "holy grail" of LyC studies—the absolute escape fraction f_esc,abs—is several steps removed from an observational measurement, even at z ∼ 3, where direct measurements of residual LyC emission are feasible. Each of the steps is model-dependent, beginning with uncertainty in the shape and absolute intensity of the unattenuated EUV and FUV light from stars, followed by the net effect of dust on both the ionizing and non-ionizing UV, and, finally, the rather poorly defined notion of what constitutes "escape" of an ionizing photon from the galaxy in which it was produced.

However, we have also attempted to leverage recent progress in understanding the ionizing sources through joint analysis of the stellar and nebular spectra (e.g., Steidel et al. 2016; Strom et al. 2017) and to follow a self-consistent thread capable of reconciling the detectability of LyC radiation leakage with all other available empirical constraints. With these caveats, we have measured f_esc,abs for a sample of $\simeq {L}_{\mathrm{uv}}^{* }$ galaxies at z ∼ 3. Focusing on the "holes" models, for reasons discussed in more detail in the next section, the bottom line is that some galaxies appear to leak a substantial fraction of the ionizing radiation they produce—up to 30% for those in the highest quartile of W_λ($\mathrm{Ly}\alpha$), while others—notably the brightest galaxies, and those which have $\mathrm{Ly}\alpha$ strongly in absorption—leak at most imperceptibly (<3% at 2σ). Averaged over the KLCS sample, all of which have L_uv within a factor of a few of ${L}_{\mathrm{uv}}^{* }$ at z ∼ 3, $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle \sim 9$% given our assumed SPS and reddening models (Table 10).

It is important to keep in mind the degree to which f_esc,abs depends on the latter assumptions—if, for example, we had assumed that the attenuation relation is that of Calzetti et al. (2000) and that the appropriate SPS model is a solar-metallicity Starburst99 model (Leitherer et al. 2014; see Steidel et al. 2016 for comparisons with the BPASSv2.0 models), then the best-fitting model parameters give f_esc,abs = 0.25,³¹ a factor of more than two times higher than our fiducial model (Table 7), but still underpredict the observed LyC flux by a factor of ≳3. Changing the attenuation relation to R16 but keeping the SPS model fixed, f_esc,abs = 0.33, with similarly poor χ² relative to the observed spectrum (Figure 24). With the wrong choice of SPS model, it is not possible to simultaneously match the shape of the FUV spectrum, the Lyman series line depth, and the residual LyC emission (see also Reddy et al. 2016b).

10.5.1. High Redshift

10.5.2. Low Redshift

In Section 8, we observed a strong dependence of the residual LyC emissivity $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ on spectroscopically measured W_λ($\mathrm{Ly}\alpha$) and noted (Figure 17) that the relation could be captured approximately by a linear relation,

$$\begin{eqnarray}&&\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}\simeq 0.36\left[\displaystyle \frac{\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle }{110\,\mathring{\rm A} }\right]+0.02.\end{eqnarray} \tag{ 19 }$$

The parametrization of Equation (19) was chosen in part because the fiducial BPASSv2.1-300bin-t8.0 continuous star formation models used to fit the subsample composite spectra in Sections 9.4.1 and 9.4.2 predict that W_λ($\mathrm{Ly}\alpha$) ≃ 110 Å, assuming "Case B" (ionization bounded) recombination and no selective attenuation of $\mathrm{Ly}\alpha$ photons relative to the FUV continuum near $\mathrm{Ly}\alpha$.³² The term inside the square brackets is therefore the ratio of observed ${W}_{\lambda }(\mathrm{Ly}\alpha )$ to the maximum value expected under the most favorable conditions for $\mathrm{Ly}\alpha$ production, i.e., where every LyC photon has been absorbed and converted to $\mathrm{Ly}\alpha$ or nebular continuum in accordance with Case B expectations. An observation of the maximum ${W}_{\lambda }(\mathrm{Ly}\alpha )$ would also require that, within the spectroscopic aperture, the intensity of the $\mathrm{Ly}\alpha$ emission line relative to the continuum flux density has not been significantly altered by resonant scattering of $\mathrm{Ly}\alpha$ (i.e., spatial redistribution and/or selective absorption of $\mathrm{Ly}\alpha$ photons by dust grains; see, e.g., Steidel et al. 2011).

The left-hand panel of Figure 25 illustrates the nearly linear relationship between 1 − f_c (Tables 9 and 10 for the Screen and Holes models, respectively) and $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$ among the composite spectra. This behavior strongly supports the hypothesis of a direct causal link between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ in emission and the depth/strength of interstellar absorption features observed in the same galaxy spectra (e.g., Shapley et al. 2003; Steidel et al. 2010, 2011; Jones et al. 2012); it also lends credence to the use of simple geometric models of the galaxy ISM to understand the interplay between residual LyC emission, the depth of Lyman series absorption lines, and $\mathrm{Ly}\alpha$ escape. The sense of the relationship is that the fraction of the Case B ${W}_{\lambda }(\mathrm{Ly}\alpha )$ (110 Å in our model) measured within a spectroscopic aperture is directly related to the portion of the stellar UV continuum that is not covered by optically thick H i (1 − f_c) along the same line of sight.

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Of course, one does not expect the production rate of $\mathrm{Ly}\alpha$ photons to be compatible with Case B assumptions if there is significant detection of residual LyC flux, since any detected ionizing photons will not have produced the corresponding $\mathrm{Ly}\alpha$ photons expected for an ionization-bounded geometry. For a typical value 1 − f_c ≃ 0.1, the reduction in the $\mathrm{Ly}\alpha$ source function would be small; however, in the limit of very high 1 − f_c ∼ 1, an observer would see $\mathrm{Ly}\alpha$ emission along the same line of sight only if the photons have been scattered in our direction after (initially) being emitted in another. In all likelihood, this suggests that the correlation between (1 − f_c) or f_esc,abs and ${W}_{\lambda }(\mathrm{Ly}\alpha )$ would turn over at f_esc,abs ≳ 0.5. Conversely, galaxies with W_λ($\mathrm{Ly}\alpha$) ∼ W_λ($\mathrm{Ly}\alpha$, Case B) ≃ 110 Å are a priori unlikely to exhibit substantial LyC leakage.

In other words, Equation (19) should not be extrapolated beyond the range covered by the KLCS composites, which all have $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle \lt 50\,\mathring{\rm A}$; only six individual KLCS sources have spectroscopically measured W_λ($\mathrm{Ly}\alpha$) > 50 Å and only one has W_λ($\mathrm{Ly}\alpha$) > 100 Å (see Table 2).

The relationships shown in Figure 25 are surprisingly tight, given the complexity of $\mathrm{Ly}\alpha$ radiative transfer compared to that of ionizing photons; for example, $\mathrm{Ly}\alpha$ can be scattered by H i with ${N}_{{\rm{H}}{\rm{I}}}$ ≪ 10¹⁷ cm⁻², and $\mathrm{Ly}\alpha$ resonant scattering may increase the probability of absorption by dust or of scattering into a diffuse $\mathrm{Ly}\alpha$ halo, in which case an aperture tailored to the apparent angular scale of the FUV continuum light (as for KLCS spectra) might miss a large fraction of the emergent $\mathrm{Ly}\alpha$ flux (see, e.g., Steidel et al. 2011; Hayes et al. 2014; Wisotzki et al. 2016). However, a "down the barrel" spectrum that includes an LyC detection will also capture the fraction of $\mathrm{Ly}\alpha$ photons that pass through the same low optical depth holes in the ISM. The tightness of the relationships shown in Figure 25 suggests that most $\mathrm{Ly}\alpha$ photons detected within the spectroscopic aperture may have passed through the same holes in the ISM. This scenario is easier to explain in the context of the "holes" models, as shown by the small scatter in both panels of Figure 25—evidently, applying dust attenuation to the entire EUV/FUV continuum makes it more difficult to account for the tightness of the correlation between $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ and ${W}_{\lambda }(\mathrm{Ly}\alpha )$. We also expect that using a larger aperture to measure the same galaxies would increase the scatter between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ as an increasing fraction of the observed $\mathrm{Ly}\alpha$ photons will have been scattered to large galactocentric radii via less direct channels through the ISM than the LyC photons.

In the discussion that follows, we continue to adopt the "holes" model as the simplest picture that remains consistent with the current observations.

Theoretical predictions concerning the relationship between the kinematics and escape of $\mathrm{Ly}\alpha$ emission and the leakage of significant LyC emission have varied depending on the physical model assumed for the structure and kinematics of the ISM (e.g., Yajima et al. 2014; Verhamme et al. 2015, 2017; Dijkstra et al. 2016). In general, most models predict that if the ISM is porous enough to allow the escape of ionizing photons along a particular line of sight, then one should also observe $\mathrm{Ly}\alpha$ emission with a significant component near v_sys ≃ 0 km s^–1 and/or a double-peaked profile with small velocity separation, which might indicate relatively low ${N}_{{\rm{H}}{\rm{I}}}$ along the line of sight. However, the picture that seems indicated by the KLCS results is that, while there is an overall correlation between the inferred value of ${N}_{{\rm{H}}{\rm{I}}}$ associated with the covered portion of the stellar continuum and f_esc,abs (see, e.g., Table 10), it is the covering fraction (f_c) that appears most closely linked with LyC leakage (as well as with ${W}_{\lambda }(\mathrm{Ly}\alpha )$). Since the measured values of ${N}_{{\rm{H}}{\rm{I}}}$ are far too high to have allowed any detectable LyC leakage if the entire stellar UV continuum were covered, the apparent connection between ${N}_{{\rm{H}}{\rm{I}}}$ and $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ may simply indicate that the incidence of clear channels through the ISM is higher when the characteristic ${N}_{{\rm{H}}{\rm{I}}}$ (with covering fraction f_c) is lower. Similar results have recently been obtained for a sample of LyC-detected low-redshift galaxies with ${N}_{{\rm{H}}{\rm{I}}}$ measurements (Chisholm et al. 2018; Gazagnes et al. 2018).

Interestingly, among the KLCS subsample composites, none exhibits an $\mathrm{Ly}\alpha$ peak near v ∼ 0 km s^–1—including the composite formed entirely of individual LyC detections—as shown in Figure 26. Indeed, the overall shapes of the $\mathrm{Ly}\alpha$ profiles—including the subsample comprised of individual LyC detections—are remarkably similar, with a fraction f($\mathrm{Ly}\alpha$, blue) ≃ 0.31 ± 0.02 of the total $\mathrm{Ly}\alpha$ emission emerging with v_sys < 0 km s^–1. The center panel of Figure 26 shows that the velocity relative to systemic of the $\mathrm{Ly}\alpha$ red peak (v_peak) decreases with increasing $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$, as found by Erb et al. (2014) and Trainor et al. (2015) for similar-sized samples of LBGs and LAEs at z ∼ 2–3. Nevertheless, the bottom panel of Figure 26 shows that the KLCS subsamples with $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ ≳ 0.1 tend to have a nearly constant v_peak ∼ 250 $\mathrm{km}\,{{\rm{s}}}^{-1}$, whereas those with $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ ≲ 0.1 exhibit a wide range in v_peak. The faint LAE sample of Trainor et al. (2015, whose LyC properties are unknown, but which have Lyman β line depths similar to the KLCS LAE sample) continues a trend of slightly smaller $\mathrm{Ly}\alpha$ velocity shifts for intrinsically fainter galaxies at fixed ${W}_{\lambda }(\mathrm{Ly}\alpha )$ (center panel of Figure 26); this, along with the accompanying smaller maximum blueshifted velocity of the low-ionization metals, is attributed to lower outflow velocities. It is therefore quite possible that the characteristic v_peak at fixed $\langle {f}_{\mathrm{esc},\mathrm{abs}}\rangle$ would also be smaller for galaxies with M_uv > −19.5.

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Clearly, $\mathrm{Ly}\alpha$ peaking at v = 0 km s^–1 is not a necessary condition for significant LyC leakage among galaxies with z ≃ 3. The non-zero vales of v_peak among LyC-detected subsamples suggests that most $\mathrm{Ly}\alpha$ photons escaping along the same line of sight as LyC photons have been scattered at least once from gas moving with v_sys > 0 km s^–1. This suggests that the holes through the ISM allowing LyC photons to escape are not generally optically thin to $\mathrm{Ly}\alpha$ photons without the benefit of a velocity kick to move them off resonance of gas near v_sys = 0 km s^–1.

11.2.1. Comparison to Low-redshift Results

As briefly mentioned in Section 9.3, complementary insight into the geometric/kinematic distribution of optically thick material can be obtained from observations of the non-resonant (sometimes referred to as "fluorescent") emission lines that share the same excited state as the resonance line, but involve de-excitation to fine-structure levels just above the ground state. Since the interstellar Si iiλ1260 and Si iiλ1526 absorption lines are resonance transitions, modulo the selective absorption of line photons by dust grains, there should be no net emission or absorption provided that the measurement aperture includes the entire scattering medium (e.g., Shapley et al. 2003; Erb et al. 2012; Henry et al. 2015). For this reason, it is potentially interesting to measure the equivalent widths of the resonance absorption and the associated non-resonance emission within the (fixed) spectroscopic apertures used for KLCS.

Figure 27 compares the mean equivalent widths of the resonance lines Si iiλ1260 and Si iiλ1526 with the average equivalent width of the non-resonant emission lines Si ii* λ1265 and Si ii* 1533 as a function of ${W}_{\lambda }(\mathrm{Ly}\alpha )$ for the KLCS subsamples. The left-hand panel of Figure 27 shows the well-established anticorrelation between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and W_λ(LIS) (e.g., Shapley et al. 2003; Du et al. 2018), where LIS refers to low-ionization resonance absorption lines (red filled points). In addition, the ratio of Si ii* emission (skeletal points) to Si ii absorption increases with $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$—there is a weak trend of increasing strength of W_λ(Si ii* with $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$ (magenta dashed line in Figure 27), but a strongly decreasing trend of W_λ(Si ii) with increasing $\langle {W}_{\lambda }(\mathrm{Ly}\alpha )\rangle$ causes the ratio W_λ(Si ii*)/W_λ(Si ii) to vary from ∼0.3 to ∼0.9 across the sample.

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Although a detailed discussion of the implications of the non-resonant emission is deferred to future work, one interpretation of the trends illustrated in Figure 27 is that the physical extent of the scattering medium producing resonance absorption lines in "down the barrel" spectra is more compact for galaxies with higher LyC escape fractions. In such a scenario, the KLCS spectroscopic aperture (projected physical size 9.2 × 10.3 kpc at $\langle {z}_{{\rm{s}}}\rangle$) contains a significantly larger fraction of the gas with log ${N}_{{\rm{H}}{\rm{I}}}$ ≳ 10¹⁷ cm⁻² in the CGM of galaxies having the highest LyC escape fraction. In other words—not surprisingly—galaxies are more likely to leak along an observed line of sight when they exhibit more spatially compact distributions of optically thick gas in the interface between the ISM and CGM.

An interesting question is whether the more compact distribution of optically thick material is the "cause" or the "effect" of higher LyC escape. It is quite possible that the low-ionization CGM is reduced by a more intense local ionizing radiation field in the inner CGM. That is, the reduced spatial extent of optically thick low-ionization material surrounding LyC leaking galaxies might be caused at least in part by self-ionization. For example, we consider the ${W}_{\lambda }(\mathrm{Ly}\alpha )$ (Q4) subsample, which has νL_ν(1500 Å) ≃ 1.6 × 10¹¹ L_⊙ and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ≃ 0.17 (Table 6), thus producing an average escaping ionizing luminosity $\langle \nu {L}_{\nu }({\rm{LyC}})\rangle \simeq {10}^{44}$ erg s⁻¹. The average specific intensity at the Lyman limit as a function of galactocentric distance r would be $\langle {I}_{\nu }{(r)\rangle \simeq {10}^{-19}(r/50\mathrm{kpc})}^{-2}$ erg s⁻¹ cm⁻² Hz⁻¹: at r = 50 kpc, the local ionizing radiation field intensity would exceed that of the metagalactic background by a factor of >10 and could dominate the ionization state of the CGM to r ≳ 150 kpc around individual galaxies.

Quantitative measurement of LyC flux from the KLCS sample of galaxies at z ∼ 3 is the first step in addressing the issue of the connection between directly observed galaxy properties and the propensity to leak significant LyC radiation. We have shown that even this step has required combining N ∼ 30 independent galaxy spectra according to various easily measured empirical criteria in order to accurately correct the ensemble for the effects of intergalactic and circumgalactic gas along our line of sight.

If the KLCS galaxy sample were a representative subset of the population of star-forming galaxies that contributes to the metagalactic ionizing radiation field at z ∼ 3, then the values of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ in Table 6 are all that is needed to convert a measurement of the far-UV (non-ionizing) emissivity ${\epsilon }_{\mathrm{uv}}=\int {L}_{\mathrm{uv}}{\rm{\Phi }}({L}_{\mathrm{uv}}){{dL}}_{\mathrm{uv}}$, where Φ(L_uv) is the galaxy luminosity function evaluated in the rest-frame FUV (typically 1500–1700 Å), into the ionizing emissivity contributed by the same population of galaxies,

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{LyC}}\simeq {\int }_{{L}_{\mathrm{uv},\min }}^{{L}_{\mathrm{uv},\max }}\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}\times {L}_{\mathrm{uv}}{\rm{\Phi }}({L}_{\mathrm{uv}}){{dL}}_{\mathrm{uv}},\end{eqnarray} \tag{ 20 }$$

as discussed by many authors (see, e.g., Steidel et al. 2001; Shapley et al. 2006; Nestor et al. 2011, 2013; Mostardi et al. 2013; Grazian et al. 2017). This method is especially useful because Φ(L_uv) has been measured over a substantial range in M_uv at redshifts z ≃ 2–9 (e.g., Reddy & Steidel 2009; Oesch et al. 2010; Bouwens et al. 2015; Finkelstein et al. 2015; Parsa et al. 2016).

Note that $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ appears inside the integral in Equation (20); as discussed in the previous section, although KLCS samples only a limited range in M_uv (−19.5 ≳ M_uv ≳−22.1; 0.25 ≲ L_uv/${L}_{\mathrm{uv}}^{* }$ ≲ 3), evidently only the UV-fainter half of the sample has significant $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$. We argued in Section 11.1 that a surprisingly tight relationship between the covering fraction f_c and the measured ${W}_{\lambda }(\mathrm{Ly}\alpha )$ in the spectroscopic aperture suggests that the most direct signature of leaking LyC emission appears to be ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and that the apparent dependence on L_uv is secondary. However, it is straightforward to obtain approximate numbers for the contribution of KLCS-like galaxies to _LyC using the empirical dependence of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ on either L_uv or ${W}_{\lambda }(\mathrm{Ly}\alpha )$.

11.4.1. Weighting by ${L}_{\mathrm{uv}}$

If we adopt the rest-frame FUV galaxy luminosity function (UVLF) determined by Reddy & Steidel (2009) for star-forming galaxies at 2.7 ≲ z ≲ 3.4,

$$\begin{eqnarray}&&{\rm{\Phi }}({L}_{\mathrm{uv}})={{\rm{\Phi }}}^{* }{({L}_{\mathrm{uv}}/{L}_{\mathrm{uv}}^{* })}^{\alpha }\exp (-{L}_{\mathrm{uv}}/{L}_{\mathrm{uv}}^{* }),\end{eqnarray} \tag{ 21 }$$

with α = −1.73, Φ* = 1.71 × 10⁻³ Mpc⁻³, and ${M}_{\mathrm{uv}}^{* }=-21.0$, then the total FUV emissivity of the population of galaxies with M₁₇₀₀ < −17.4 (i.e., ${L}_{\mathrm{uv}}\gt 0.036\,{L}_{\mathrm{uv}}^{* }$) is _uv = 3.28 ± 0.24 × 10²⁶ erg s⁻¹ Hz⁻¹ Mpc⁻³. The KLCS data set includes only galaxies with M_uv ≤ −19.5, sampling a range in L_uv that accounts for ≃0.52 × _uv ≃ 1.71 × 10²⁶ erg s⁻¹ Hz⁻¹ Mpc⁻³.

Note that the UV emissivity of the M_uv < −19.5 population is relatively insensitive to the choice of UVLF: for example, if instead of Reddy & Steidel (2009) we were to use the parameters advocated by Parsa et al. (2016), i.e., ${M}_{\mathrm{uv}}^{* }\simeq -20.5$, α ≃ −1.37, and _uv ≃ 3.16 × 10²⁶ erg s⁻¹ Hz⁻¹ Mpc⁻³ integrated down to M_uv ≃ −15.5, then galaxies with M_uv < −19.5 produce _uv ≃ 0.48 × 3.16 × 10²⁶ ≃ 1.52 × 10²⁶ erg s⁻¹ Hz⁻¹ Mpc⁻³ at $\langle {z}_{{\rm{s}}}\rangle \,=3.05$, a difference of only ≃11%. Similarly, the fractional contributions of galaxies with M_uv < −19.5 to the total emissivity integrated down to a common M_uv ≃ −15.5 are 43% (Reddy & Steidel 2009) and 48% (Parsa et al. 2016).

If we assume for the moment that $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ = 0 for galaxies with M_uv ≤ −21.0 and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ = 0.11 for −21 < M_uv ≤ 19.5 (Table 6 and Figure 17), then the galaxies that contribute significantly to _LyC at z ∼ 3 have _uv ≃ 1.21 × 10²⁶ erg s⁻¹ Hz⁻¹ Mpc⁻³ and thus contribute _LyC ≃ 0.11 × 1.21 × 10²⁶ ≃ 13.0 × 10²⁴ erg s⁻¹ Hz⁻¹ Mpc⁻³. The contribution would be slightly lower, _LyC ≃ 11 × 10²⁴ erg s⁻¹ Hz⁻¹ Mpc⁻³, assuming $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ = 0.094 as for the "IGM Only" transmission model.

11.4.2. Weighting by ${W}_{\lambda }({Ly}\alpha )$

Alternatively, we can use our derived empirical relation between ${W}_{\lambda }(\mathrm{Ly}\alpha )$ and $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ to estimate the contribution of KLCS-like galaxies to _LyC. In this case, we take $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ outside the integral in Equation (20) and use the distribution function n(${W}_{\lambda }(\mathrm{Ly}\alpha )$) of UV continuum-selected z ∼ 3 galaxies to estimate contributions to _LyC relative to the integral over the UVLF, _uv.

We begin with Equation (19), the best linear relation between $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ and ${W}_{\lambda }(\mathrm{Ly}\alpha )$, where again the statistic $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ is adopted as the least model-dependent link between the FUV luminosity of galaxies and their contribution to the ionizing emissivity, since it does not depend on assumptions regarding SPS models, ISM radiative transfer, or attenuation by dust. However, for the present purpose, we assume that W_λ($\mathrm{Ly}\alpha$, Case B) ≃ 110 Å as discussed in Section 11.1 above and a model for mapping between $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ and ${W}_{\lambda }(\mathrm{Ly}\alpha )$ given by Equation (19). For ${W}_{\lambda }(\mathrm{Ly}\alpha )$ > 0.5W_λ($\mathrm{Ly}\alpha$, Case B) = 55 Å, which is unconstrained by the KLCS composite spectra, we assume that $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ reaches a maximum when (${W}_{\lambda }(\mathrm{Ly}\alpha )$/110 Å) ∼ 0.5, since at some point the increasing escape of LyC photons would result in a decreasing $\mathrm{Ly}\alpha$ source function compared to Case B expectations. A naive expectation is that the source function of $\mathrm{Ly}\alpha$ emission along a particular line of sight would be reduced by a factor of (1 – f_c) relative to Case B, followed by a further reduction proportional to f_c due to the scattering of $\mathrm{Ly}\alpha$ photons by the covered portion of the UV continuum. In such a simplified picture, one might expect W' ≡ ${W}_{\lambda }(\mathrm{Ly}\alpha )$/W_λ($\mathrm{Ly}\alpha$, Case B) ≈ 2f_c(1 − f_c), which has a maximum W' ≈ 0.5 when f_c ≃ 0.5 and goes to zero as ${f}_{{\rm{c}}}\to 0$ and as ${f}_{{\rm{c}}}\to 1$. Solving for f_c in terms of W' gives

$$\begin{eqnarray}&&(1-{f}_{{\rm{c}}})=0.5-{[| 0.5W^{\prime} -0.25| ]}^{1/2}.\end{eqnarray} \tag{ 22 }$$

Using Equation (19) and the holes model fit (1 − f_c) ≈ 0.75W' (Figure 25) to connect $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ to 1 − f_c over the range constrained by the data results in a model that maps the expected $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ to W' over the full range of each:

$$\begin{eqnarray}&&\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}\simeq 0.28-{[| 0.146W^{\prime} -0.073| ]}^{1/2}.\end{eqnarray} \tag{ 23 }$$

Two models mapping $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ to ${W}_{\lambda }(\mathrm{Ly}\alpha )$ are illustrated in the top panel of Figure 28; one is the best linear relation from Equation (19), reflected about W' = 0.5, the assumed maximum. The second is based on Equation (23). Also shown in the top panel of Figure 28 is the relative incidence of ${W}_{\lambda }(\mathrm{Ly}\alpha )$ parametrized as an exponential of the form n(W_λ) ∝ exp(−W_λ/W₀), with W₀ = 23.5 Å, very close to the observed distribution function in the KLCS sample, as well as that obtained from previous spectroscopic samples at z ∼ 3 (Shapley et al. 2003; Kornei et al. 2010). Reddy et al. (2008) showed that the observed distribution of ${W}_{\lambda }(\mathrm{Ly}\alpha )$ at z ∼ 3 must be close to the intrinsic distribution for the galaxies contributing to the FUV continuum luminosity function. We further assume that ≃40% of continuum-selected galaxies have spectroscopic ${W}_{\lambda }(\mathrm{Ly}\alpha )$ ≤ 0 (and therefore have $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ≃ 0). It is then straightforward to compute the probability distribution of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ expected for an ensemble of continuum-selected galaxies with M_uv < −19.5 (${L}_{\mathrm{uv}}\gt 0.25{L}_{\mathrm{uv}}^{* }$) as in the KLCS sample. The resulting PDFs of the quantity $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ for an ensemble of galaxies with the assumed n(W_λ($\mathrm{Ly}\alpha$)) are shown in the bottom panel of Figure 28. Note that just over 50% of the PDF falls in the lowest bin of $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ but the ensemble average is $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ ≃ 0.035.

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Returning to the estimate of the contribution to _LyC, the ${W}_{\lambda }(\mathrm{Ly}\alpha )$-based $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ = 0.035 implies that galaxies with M_uv < −19.5 contribute _LyC ≃ 0.035 × ×1.71 × 10²⁶ ≃ 5.9 × 10²⁴ erg s⁻¹ Hz⁻¹ Mpc⁻³. Because of the strong empirical connection between $\langle {f}_{900}{/f}_{1500}{\rangle }_{\mathrm{out}}$ and ${W}_{\lambda }(\mathrm{Ly}\alpha )$, and because previous work has shown that spectroscopic samples are unbiased with respect to ${W}_{\lambda }(\mathrm{Ly}\alpha )$, we are more confident of the ${W}_{\lambda }(\mathrm{Ly}\alpha )$-based estimate of _LyC than of the luminosity-weighted estimate in Section 11.4.1.

For comparison, recent indirect estimates of the total ionizing emissivity at z ∼ 3 inferred from modeling the transfer of ionizing radiation through the IGM coupled with measurements of the mean transmissivity of the Lyman α forest (Becker & Bolton 2013) find _LyC ≃ 10 × 10²⁴ erg s⁻¹ Hz⁻¹ Mpc⁻³ with a systematic uncertainty of ∼0.4 dex (factor of 2.5).³³ Thus, without extrapolating our results to lower luminosities that are not directly constrained by the KLCS sample, the ionizing emissivity ${\epsilon }_{\mathrm{LyC}}[{L}_{\mathrm{uv}}\gt 0.25{L}_{\mathrm{uv}}^{* }]\simeq 5.9\,\times {10}^{24}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}\,{\mathrm{Mpc}}^{-3}$ comprises ∼60% of the estimated total. In addition, it exceeds the total ionizing emissivity of QSOs at z ∼ 3, estimated to lie in the range ≃(1.6–5) × 10²⁴ erg s⁻¹ Hz⁻¹ Mpc⁻³ (Hopkins et al. 2007; Cowie et al. 2009).

A more detailed discussion of the implications of our result for the ionizing emissivity _LyC and photoionization rate Γ at z ∼ 3, as well as a comprehensive comparison to other observational constraints, is deferred to a separate paper.