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A&A
Volume 623, March 2019
Article Number A157
Number of page(s) 13
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/201833075
Published online 26 March 2019

© ESO 2019

1. Introduction

With a vacuum rest-frame wavelength of 1215.67 Å, the Lyman-α (Lyα) recombination line (n = 2 → n = 1) plays a key role in the energy release from ionised hydrogen gas, being intrinsically the strongest emission line in the rest-frame UV and optical (e.g. Partridge & Peebles 1967; Pritchet 1994). Lyα is emitted from ionised gas around star-forming regions (e.g. Charlot & Fall 1993; Pritchet 1994) and AGN (e.g. Miley & De Breuck 2008) and it is routinely used as a way to find high redshift sources (z ∼ 2−7; see e.g. Malhotra & Rhoads 2004).

Several searches for Lyα-emitting sources (Lyα emitters; LAEs) have led to samples of thousands of star-forming galaxies (SFGs) and AGN (e.g. Sobral et al. 2018a, and references therein). LAEs are typically faint in the rest-frame UV, including many that are too faint to be detected by continuum based searches even with the Hubble Space Telescope (e.g. Bacon et al. 2015). The techniques used to detect LAEs include narrow-band surveys (e.g. Rhoads et al. 2000; Ouchi et al. 2008; Hu et al. 2010; Matthee et al. 2015), Integral Field Unit (IFU) surveys (e.g. van Breukelen et al. 2005; Drake et al. 2017a) and blind slit spectroscopy (e.g. Martin & Sawicki 2004; Rauch 2008; Cassata et al. 2011). Galaxies selected through their Lyα emission allow for easy spectroscopic follow-up due to their high EWs (e.g. Hashimoto et al. 2017) and typically probe low stellar masses (see e.g. Gawiser et al. 2007; Hagen et al. 2016).

The intrinsic Lyα luminosity is a direct tracer of the ionising Lyman-continuum (LyC) luminosity and thus a tracer of instantaneous star formation rate (SFR), in the same way as Hα is (e.g. Kennicutt 1998). Unfortunately, inferring intrinsic properties of galaxies from Lyα observations is extremely challenging. This is due to the complex resonant nature and sensitivity to dust of Lyα (see e.g. Dijkstra 2017, for a detailed review on Lyα), which contrasts with Hα. For example, a significant fraction of Lyα photons is scattered in the Inter-Stellar Medium (ISM) and in the Circum-Galactic Medium (CGM) as evidenced by the presence of extended Lyα halos in LAEs (e.g. Momose et al. 2014; Wisotzki et al. 2016), but also in the more general population of z ∼ 2 SFGs sampled by Hα emitters (Matthee et al. 2016), and the bluer component of such population traced by UV-continuum selected galaxies (e.g. Steidel et al. 2011). Such scattering leads to kpc-long random-walks which take millions of years and that significantly increase the probability of Lyα photons being absorbed by dust particles. The complex scattering and consequent higher susceptibility to dust absorption typically leads to low and uncertain Lyα escape fractions (fesc, Lyα; the ratio between observed and intrinsic Lyα luminosity; see e.g. Atek et al. 2008).

“Typical” star-forming galaxies at z ∼ 2 have low fesc, Lyα (∼1−5%; e.g. Oteo et al. 2015; Cassata et al. 2015), likely because significant amounts of dust present in their ISM easily absorb Lyα photons (e.g. Ciardullo et al. 2014; Oteo et al. 2015; Oyarzún et al. 2017). However, sources selected through their Lyα emission typically have ∼10 times higher fesc, Lyα (e.g. Song et al. 2014; Sobral et al. 2017), with Lyα escaping over ≈2× larger radii than Hα (Sobral et al. 2017).

Furthermore, one expects fesc, Lyα to depend on several physical properties which could be used as predictors of fesc, Lyα. For example, fesc, Lyα anti-correlates with stellar mass (e.g. Oyarzún et al. 2017), dust attenuation (e.g. Verhamme et al. 2008; Hayes et al. 2011; Matthee et al. 2016; An et al. 2017) and SFR (e.g. Matthee et al. 2016). However, most of these relations require derived properties (e.g. Yang et al. 2017), show a large scatter, may evolve with redshift and sometimes reveal complicated trends (e.g. dust dependence; see Matthee et al. 2016).

Interestingly, the Lyα rest-frame equivalent width (EW0), a simple observable, seems to be the simplest direct predictor of fesc, Lyα in LAEs (Sobral et al. 2017; Verhamme et al. 2017) with a relation that shows no strong evolution from z ∼ 0 to z ∼ 2 (Sobral et al. 2017) and that might be applicable at least up to z ∼ 5 (Harikane et al. 2018). Such empirical relation may hold the key for a simple but useful calibration of Lyα as a direct tracer of the intrinsic LyC luminosity (see Reddy et al. 2016; Steidel et al. 2018; Fletcher et al. 2018, and references therein) by providing a way to estimate fesc, Lyα, and thus as a good SFR indicator for LAEs (see also Dijkstra & Westra 2010, hereafter DW10). We fully explore such possibility and its implications in this work. Note that this paper makes no attempt to simplify the complex radiative transfer by which Lyα photons escape from galaxies. Instead, this work focuses on an empirical approach to predict Lyα escape fractions with a simple observable based on direct observations. In Sect. 2 we present the samples at different redshifts and methods used to compute fesc, Lyα. In Sect. 3 we present and discuss the results, their physical interpretation and our proposed empirical calibration of Lyα as an SFR indicator. Finally, we present the conclusions in Sect. 4. We use AB magnitudes (Oke & Gunn 1983), a Salpeter (1955) initial mass function (IMF; with mass limits 0.1 and 100 M⊙) and adopt a flat cosmology with Ωm = 0.3, ΩΛ = 0.7, and H0 = 70 km s−1 Mpc−1.

2. Sample and methods

In this study we use a large compilation of LAEs which have been widely studied in the literature (e.g. Cardamone 2009; Henry et al. 2015; Trainor et al. 2015; Verhamme et al. 2017; Sobral et al. 2017) at z ≤ 0.3 and z ∼ 2.2−2.6 with measured or inferred dust-extinction corrected Hα luminosities and thus fesc, Lyα available. We note that these cover sources from low (≈5 Å) to high (≈160 Å) Lyα EW0 across a range of redshifts, with SFRs typically around ∼5−50 M⊙ yr−1 (typical of LAEs) at z ∼ 0.3−2.6. The sample combines sources obtained with somewhat heterogeneous selections which allow us to obtain a more conservative scatter in the trends we investigate. Our approach also allows us to obtain relations that are more widely applicable for LAEs with measured Lyα luminosities and EW0. We note nonetheless that our results are only valid for LAEs and are empirically based on observables. Note that in this study we explore luminosities within ≈2−3 arcsec (typically ≈15−20 kpc) diameters. These do not explicitly include the even more extended Lyα halo luminosity beyond ∼20 kpc, but we refer interested readers to studies that have investigated the spatial dependence of the Lyα escape fraction for different sources (e.g. Matthee et al. 2016; Sobral et al. 2017).

2.1. LAEs at low redshift (z ≤ 0.3)

For our lower redshift sample, we explore a compilation of 30 sources presented in Verhamme et al. (2017) which have accurate (Hα derived) fesc, Lyα measurements and sample a range of galaxy properties. The sample includes high EW Hα emitters (HAEs) from the Lyman Alpha Reference Sample at z = 0.02−0.2 (LARS, e.g. Hayes et al. 2013, 2014), a sample of LyC leakers (LyCLs) investigated in Verhamme et al. (2017) at z ∼ 0.3 (Izotov et al. 2016a,b) and a more general “green pea” (GPs) sample (e.g. Cardamone 2009; Henry et al. 2015; Yang et al. 2016, 2017). These are all LAEs at low redshift with available Lyα, Hα and dust extinction information required to estimate fesc, Lyα (see Sect. 2.4) and for which Lyα EW0s are available. For more details on the sample, see Verhamme et al. (2017) and references therein.

2.2. LAEs at cosmic noon (z = 2.2–2.6)

For our sample at the peak of star formation history we use 188 narrow-band selected LAEs with Hα measurements from the CALYMHA survey at z = 2.2 (Matthee et al. 2016; Sobral et al. 2017) presented in Sobral et al. (2017), for which fesc, Lyα measurements are provided as a function of EW0. In addition, we explore spectroscopic follow-up of CALYMHA sources with X-Shooter on the VLT (Sobral et al. 2018b) and individual measurements for four sources (CALYMHA-67, -93, -147 and -373; see Sobral et al. 2018b). For those sources we measure Lyα, Hα and Hβ and correct for dust extinction as in Sect. 2.4.

Furthermore, we also use a sample of 29 narrow-band selected LAEs at z  ∼  2.6 presented by Trainor et al. (2015), for which Lyα and Hα measurements are available. We use results from Trainor et al. (2016) that show that for the full sample the Balmer decrement is consistent with ≈0 mag of extinction. This is dominated by the more numerous sources with higher EWs, and thus we assume ≈0 mag (AHα) of extinction for the highest EW bin. For the sources with the lowest EWs, we correct for AHα  =  0.1 mag of extinction, as these are the most massive sources and thus expected to be slightly more dusty (see Garn & Best 2010). We note that our obscuration correction may be a slight underestimation (resulting in over-estimating the escape fraction at the lowest EWs) for the Trainor et al. (2015) sample.

2.3. Higher redshift LAEs (2.6 ≤ z ≤ 6)

As an application of our results, we explore the publicly available sample of 3,908 LAEs in the COSMOS field (SC4K survey; Sobral et al. 2018a) which provides Lyα luminosities and rest-frame EWs for all LAEs. We also explore published median or average values for the latest MUSE samples, containing 417 LAEs (e.g. Hashimoto et al. 2017). Note that for all these higher redshift samples, Hα is not directly available, thus fesc, Lyα cannot be directly measured (but see Harikane et al. 2018).

2.4. Measuring the Lyα escape fraction (fesc, Lyα) with Hα

We use dust corrected Hα luminosity to predict the intrinsic Lyα luminosity. We then compare the latter to the observed Lyα luminosity to obtain the Lyα escape fraction (fesc, Lyα). Assuming case B recombination1, a temperature of 104 K and an electron density of 350 cm−3, we can use the observed Lyα luminosity (LLyα), the observed Hα luminosity (LHα) and the dust extinction affecting LHα (AHα2, in mag) to compute fesc, Lyα as:

f esc , Ly α = L Ly α 8.7 L H α × 10 0.4 × A H α · $$ \begin{aligned} f_{\mathrm{esc,Ly}\alpha } = \frac{{L}_{\mathrm{Ly}\alpha }}{8.7\,{L}_{\mathrm{H}\alpha } \times 10^{0.4\times {A}_{\mathrm{H}\alpha }}}\cdot \end{aligned} $$(1)

This means that with our assumptions so far, and provided that we know fesc, Lyα, we can use the observed LLyα to obtain the intrinsic Hα luminosity. All sources or samples in this study have been corrected for dust extinction using Balmer decrements, either measured directly for individual sources, or by applying the median extinction for stacks or bins of sources. Therefore, one can use Lyα as a SFR indicator3 following Kennicutt (1998) for a Salpeter (Chabrier) IMF (0.1−100 M⊙):

SFR Ly α [ M ⊙ yr − 1 ] = 7.9 ( 4.4 ) × 10 − 42 ( 1 − f esc , LyC ) L Ly α 8.7 f esc , Ly α , $$ \begin{aligned} {\mathrm{SFR}_{\mathrm{Ly}\alpha }\,[{M}_{\odot }\,\mathrm{yr}^{-1}]}=\frac{7.9 (4.4)\times 10^{-42}}{(1-{f}_{\rm esc,LyC})} \, \, \frac{{L}_{\rm Ly}\alpha }{8.7\,{f}_{\mathrm{esc,Ly}\alpha }}, \end{aligned} $$(2)

where fesc, LyC is the escape fraction of ionising LyC photons (see e.g. Sobral et al. 2018a). In practice, fesc, LyC is typically assumed to be ≈0, but it may be ≈0.1−0.15 for LAEs (see discussions in e.g. Matthee et al. 2017a; Verhamme et al. 2017).

2.5. Statistical fits and errors

For all fits and relations in this work (e.g. fesc, Lyα vs. EW0), we vary each data-point or binned data-point within its full Gaussian probability distribution function independently (both in EW0 and fesc, Lyα), and re-fit 10 000 times. We present the best-fit relation as the median of all fits, and the uncertainties (lower and upper) are the 16 and 84 percentiles. For bootstrapped quantities (e.g. for fitting the low redshift sample) we obtain 10 000 samples randomly picking half of the total number of sources and computing that specific quantity. We fit relations in the form y = Ax + B.

3. Results and discussion

3.1. The observed fesc, Lyα-EW0 relation at z ∼ 0.1–2.6

Figure 1 shows that fesc, Lyα correlates with Lyα EW0 with apparently no redshift evolution between z = 0−2.6 (see also Verhamme et al. 2017; Sobral et al. 2017). We find that fesc, Lyα varies continuously from ≈0.2 to ≈0.7 for LAEs from the lowest (≈30 Å) to the highest (≈120−160 Å) Lyα rest-frame EWs. We use our samples at z ∼ 0−0.3 and z ∼ 2.2−2.6, separately and together, to obtain linear fits to the relation between fesc, Lyα and Lyα EW0 (see Sect. 2.5). These fits allow us to provide a more quantitative view on the empirical relation and evaluate any subtle redshift evolution; see Table 1.

thumbnail Fig. 1.

Relation between fesc, Lyα and Lyα EW0 for z ∼ 2.2 (stacks; see Sobral et al. 2017), z ∼ 2.6 (binning; Trainor et al. 2015) and comparison with z ∼ 0−0.3 samples (e.g. Cardamone 2009; Hayes et al. 2013; Henry et al. 2015; Yang et al. 2016, 2017; Verhamme et al. 2017), estimated from dust-corrected Hα luminosities (Eq. (1)). We show the 1σ and 2σ range for the fits at z ∼ 2.2−2.6 and z ∼ 0−0.3 separately, and find them to be consistent within those uncertainties, albeit with a potential steeper relation at higher redshift. We find a combined best fitting relation given by fesc, Lyα = 0.0048 EW0 ± 0.05. The observed relation is significantly away from what would be predicted based on observed UV slopes between β ≈ −2 and β ≈ −1 for LAEs (see DW10) and would require β ≈ +5 for a good fit using Eq. (5). Such red β slopes are not observed for LAEs. Modifying Eq. (5) to include the effect of ξion and dust reveals that those physical parameters likely play an important role; see Sects. 3.2 and 3.3.

Table 1.

Results from fitting the relation between fesc, Lyα and Lyα EW0 as fesc, Lyα = A × EW0 + B, with EW0 in Å (see Sect. 2.5).

The relation between fesc, Lyα and Lyα EW0 is statistically significant at 5–10σ for all redshifts. We note that all linear fits are consistent with a zero escape fraction for a null EW0 (Table 1), suggesting that the trend is well extrapolated for weak LAEs with EW0 ≈ 0−20 Å. Furthermore, as Table 1 shows, the fits to the individual (perturbed) samples at different redshifts result in relatively similar slopes and normalisations within the uncertainties, and thus are consistent with the same relation from z ∼ 0 to z ∼ 2.6. Nevertheless, we note that there is minor evidence for a shallower relation at lower redshift for the highest EW0 (Fig. 1), but this could be driven by current samples selecting sources with more extreme properties (including LyC leakers). Given our findings, we decide to combine the samples and obtain joint fits, with the results shown in Table 1. The slope of the relation is consistent with being ≈0.005 with a null fesc, Lyα for EW0 = 0 Å.

3.2. The fesc, Lyα-EW0 relation: expectation vs. reality

The existence of a relation between fesc, Lyα and EW0 (Fig. 1) is not surprising. This is because Lyα EW0 is sensitive to the ratio between Lyα and the UV luminosities, which can be used as a proxy of fesc, Lyα (see e.g. Dijkstra & Westra 2010; Sobral et al. 2018a). However, the slope, normalisation and scatter of such relation depend on complex physical conditions such as dust obscuration, differential dust geometry, scattering of Lyα photons and the production efficiency of ionising photons compared to the UV luminosity, ξion (see e.g. Hayes et al. 2014; Dijkstra 2017; Matthee et al. 2017a; Shivaei & Reddy 2018).

While a relation between fesc, Lyα and EW0 is expected, we can investigate if it simply follows what would be predicted given that both the UV and Lyα trace SFRs. In order to predict fesc, Lyα based on Lyα EW0 we first follow DW10 who used the Kennicutt (1998) SFR calibrations for a Salpeter IMF and UV continuum measured (observed) at 1400 Å to derive:

SFR Ly α SFR UV = f esc , Ly α ( DW 10 ) = ( C E ) EW 0 , $$ \begin{aligned} \frac{\mathrm{SFR}_{\mathrm{Ly}\alpha }}{\mathrm{SFR}_{\rm UV}}= {f}_{\mathrm{esc,Ly}\alpha }(\mathrm{DW10})=\Big (\frac{\mathrm{C}}{\mathrm{E}}\Big )\mathrm{EW}_{0}, \end{aligned} $$(3)

where C = ( ν Ly α ν UV ) − 2 − β ≈ 1 . 152 − β − 2 $ C=\Big(\frac{\nu_{\mathrm{Ly}\alpha}}{\nu_{\mathrm{UV}}}\Big)^{-2-\beta}\approx1.152^{-\beta-2} $ and β is the UV slope (where Lλ ∝ λβ). The Kennicutt (1998) SFR calibrations4 yield:

E = 1.4 × 10 − 28 λ Ly α 7.9 8.7 × 10 − 42 ν Ly α = 76.0 Å , $$ \begin{aligned} {E}=\frac{1.4\times 10^{-28}\lambda _{\mathrm{Ly}\alpha }}{\frac{7.9}{8.7}\times 10^{-42}\nu _{\mathrm{Ly}\alpha }} = 76.0\,\mathrm{\AA }, \end{aligned} $$(4)

which allows a final parameterisation of fesc, Lyα as a function of EW0 and with just one free parameter, the UV β slope:

f esc , Ly α ( DW 10 ) = 1 . 152 − β − 2 76 EW 0 . $$ \begin{aligned} {f}_{\mathrm{esc,Ly}\alpha (\mathrm{DW10})}=\frac{1.152^{-\beta -2}}{76}\mathrm{EW}_{0}. \end{aligned} $$(5)

The DW10 methodology implicitly assumes a “canonical”, constant ξion = 1.3 × 1025 Hz erg−1 (Kennicutt 1998)5, and a unit ratio between Lyα and UV SFRs (assuming 100 Myr constant SFR; see also Sobral et al. 2018a, and Eq. (6)). DW10 do not explicitly include the effect of dust in their framework which means assuming 0.0 mag of extinction in the UV (AUV = 0.0). Such framework will therefore typically overestimate the predicted fesc, Lyα. Also, note that in DW10 β is simply a parameter used to extrapolate the UV continuum from rest-frame 1400 Å–1216 Å, and thus no physical conditions change with β (but see e.g. Popping et al. 2017; Narayanan et al. 2018).

As in DW10, we use two different UV slopes: β = −2.0 and β = −1.0, which encompass the majority of LAEs6 and result in C = 1.0 and C = 0.87, respectively (C ≈ 1.152−β − 2; see Eq. (5)). Based on the best empirical fits obtained in Sect. 3.1, we would expect C/E = 0.0048, which would yield β ≈ +5.13. Indeed, allowing β to vary freely within the DW10 framework (Eq. (5)) allows to obtain relatively good fits to the data/observations ( χ reduced 2 ≈1.2 $ {\chi}^2_{\rm reduced}\approx 1.2 $) but only for extremely red UV slopes of β ≈ +5, which are completely excluded by other independent observations of LAEs. We therefore conclude that predicting fesc, Lyα based on the ratio of Lyα to UV SFRs using EW0 and the DW10 framework with realistic UV β slopes significantly overestimates fesc, Lyα (as indicated by the dot-dashed lines in Fig. 1). Observations reveal higher Lyα EW0 (by a factor of just over ∼3 higher than the canonical value) than expected for a given fesc, Lyα. The results reveal processes that can boost the ratio between Lyα and UV (boosting EW0), particularly by boosting Lyα, or processes that reduce fesc, Lyα.

Potential explanations include scattering, (differential) dust extinction, excitation due to shocks originating from stellar winds and/or AGN activity, and short time-scale variations in SFRs, leading to a higher ξion (see Fig. 1). High ξion values (ξion  ≈  3  ×  1025 Hz erg−1) seem to be typical for LAEs (e.g. Matthee et al. 2017a; Nakajima et al. 2018) and may explain the observed relation, but dust extinction likely also plays a role (see Fig. 1 and Sect. 3.3). In order to further understand why the simple DW10 framework fails to reproduce the observations (unless one invokes β ≈ +5), we expand on the previous derivations by identifying the role of ξion (see derivations in Sobral et al. 2018a) and dust extinction (AUV) in setting the relation between Lyα and UV SFRs and thus we re-write the relation between fesc, Lyα and EW0 as:

f esc , Ly α = ( 1 . 152 − β − 2 76 EW 0 ) 1.3 × 10 25 ξ ion 10 − 0.4 A UV . $$ \begin{aligned} {f}_{\mathrm{esc,Ly}\alpha } = \Big (\frac{1.152^{-\beta -2}}{76}\mathrm{EW}_0\Big ) \frac{1.3\times 10^{25}}{\xi _{\rm ion}}10^{-0.4{A}_{\rm UV}}. \end{aligned} $$(6)

Note that Eq. (6) (this study) becomes Eq. (5) (DW10) if one assumes no dust extinction (AUV = 0) and a canonical ξion = 1.3 × 1025 Hz erg−1. In order to keep the same framework as DW10 and avoid spurious correlations and conclusions, here we also let β be decoupled from AUV (but see Meurer et al. 1999, and Sect. 3.3). By allowing all 3 parameters to vary (β = [−2.4, −1.5], log10[ξion/Hz erg−1] = [24.5,26.5] and AUV = [0, 1]) independently in order to attempt to fit observations, we find best fit values of β = −2.0 ± 0.3, log10[ξion/Hz erg−1] = 25.4 ± 0.1 and AUV = 0.5 ± 0.3 (corresponding to E(B − V) = 0.11 ± 0.07 with a Calzetti et al. 2000 dust law). We find that log10ξion is the only parameter that is relatively well constrained within our framework and that there is a clear degeneracy/relation between log10ξion and dust extinction (higher dust extinction allows for a lower log10ξion, with a relation given by log10[ξion/Hz erg−1] ≈ −1.71 E(B − V) + 25.6 with β = −2.0 ± 0.3; see Figs. 2 and A.1) such that with no dust extinction one requires a high log10 [ξion/Hz erg−1] = 25.60 ± 0.05 while for AUV ≈ 1.0 a log10[ξion/Hz erg−1] ≈ 25.25 is required to fit the observations (Fig. A.1). Observations of LAEs point towards log10[ξion/Hz erg−1] ≈ 25.5 (e.g. Matthee et al. 2017a; Nakajima et al. 2018), in good agreement with our findings. If we fix log10[ξion/Hz erg−1] = 25.5, we still obtain a similar solution for β (unconstrained), but we recover a lower AUV = 0.27 ± 0.15 (corresponding to E(B − V) = 0.06 ± 0.04 with a Calzetti et al. 2000 dust law), as we further break the degeneracy between AUV and ξion. We find that canonical log10[ξion/Hz erg−1] = 25.1 values are strongly rejected and would only be able to explain the observations for significant amounts of dust extinction of AUV ≈ 1.5−2.0 mag which are not found in typical LAEs.

thumbnail Fig. 2.

Left: predicted fesc, Lyα-Lyα EW0 space for different E(B − V) (contour levels) with our grid model (see Sect. 3.3 and Appendix A) and comparison with fits and implications by using Eq. (6) (right). We find that increasing dust extinction drives fesc, Lyα down for a fixed EW0, with data at z ∼ 0−2.6 hinting for lower dust extinction at the highest EW0 and higher dust extinction at the lowest EW0, but with the range being relatively small overall and around E(B − V) ≈ 0.05−0.3. Right: predicted fesc, Lyα-EW0 space for different ξion (contours). We find that while increasing E(B − V) mostly shifts the relation down, increasing ξion moves the relation primarily to the right.

In conclusion, we find that our modified analytical model (Eq. (6), which expands the framework of DW10), is able to fit the observations relatively well. We find that high ξion values of log10[ξion/Hz erg−1] = 25.4 ± 0.1 and some low dust extinction (E(B − V) ≈ 0.11) are required to explain the observed relation between fesc, Lyα and EW0. Without dust extinction one requires even higher ionisation efficiencies of log10[ξion/Hz erg−1] = 25.60 ± 0.05. In general, the physical values required to explain observations agree very well with observations and further reveal that LAEs are a population with high log10[ξion/Hz erg−1] ≈ 25.4−25.6 and low E(B − V) ≈ 0.1.

3.3. The fesc, Lyα-EW0 relation: further physical interpretation

In order to further interpret the physics behind our observed empirical relation, we use a simple analytical toy model. In particular, we focus on the role of dust (E(B − V)) and ξion (see details in Appendix A). We independently vary SFRs, E(B − V) and ξion with flat priors to populate the fesc, Lyα-EW0 space. The toy model follows our framework using a Calzetti et al. (2000) dust attenuation law and the Kennicutt (1998) calibrations and relations between UV and Hα. We also assume the same nebular and stellar continuum attenuation (see e.g. Reddy et al. 2015) and use the Meurer et al. (1999) relation. We also vary some assumptions independently which depend on the binary fraction, stellar metallicity and the IMF, which include the intrinsic Lyα/Hα ratio, the intrinsic UV β slope (see e.g. Wilkins et al. 2013) and fesc, LyC (see e.g. Table A.1). Furthermore, we introduce an extra parameter to further vary fesc, Lyα and mimic processes which are hard to model, such as scattering, which can significantly reduce or even boost fesc, Lyα (Neufeld 1991) and allows our toy analytical model to sample a wide range of the fesc, Lyα-EW0 plane. We compute observed Lyα EW0 and compare them with fesc, Lyα for 1 000 000 galaxy realisations. Further details are given in Appendix A.

The key results from our toy model are shown in Fig. 2, smoothed with a Gaussian kernel of [0.07,20 Å] in the fesc, Lyα-EW0 parameter space. We find that both E(B − V) and ξion likely play a role in setting the fesc, Lyα-EW0 relation and changing it from simple predictions to the observed relation (see Sect. 3.2), a result which is in very good agreement with our findings in the previous section. As the left panel of Fig. 2 shows, observed LAEs on the fesc, Lyα-EW0 relation seem to have low E(B − V) ≈ 0.1−0.2, with the lowest EW0 sources displaying typically higher E(B − V) of 0.2–0.3 and the highest EW0 sources likely having lower E(B − V) of < 0.1. Furthermore, as the right panel of Fig. 2 shows, high EW0 LAEs have higher ξion, potentially varying from log10(ξion/Hz erg−1) ≈ 25 to log10(ξion/Hz erg−1) ≈ 25.4. Our toy model interpretation is consistent with recent results (e.g. Trainor et al. 2016; Matthee et al. 2017a; Nakajima et al. 2018) for high EW0 LAEs and with our conclusions in Sect. 3.2. Overall, a simple way to explain the fesc, Lyα-EW0 relation at z ∼ 0−2.6 is for LAEs to have narrow ranges of low E(B − V) ≈ 0.1−0.2, that may decrease slightly as a function of EW0 and a relatively narrow range of high ξion values that may increase with EW0. Direct observations of Balmer decrements and of high excitation UV lines are required to confirm or refute our results.

Our toy model explores the full range of physical conditions independently without making any assumptions on how parameters may correlate, in order to interpret the observations in a simple unbiased way. However, the fact that observed LAEs follow a relatively tight relation between fesc, Lyα and EW0 suggests that there are important correlations between e.g. dust, age and ξion. By selecting simulated sources in our toy model grid that lie on the observed relation (see Appendix A.1), we recover a tight correlation between ξion and E(B − V), while the full generated population in our toy model shows no correlation at all by definition (see Fig. A.1). This implies that the observed fesc, Lyα-EW0 relation could be a consequence of an evolutionary ξion − E(B − V) sequence for LAEs, likely linked with the evolution of their stellar populations. For further details, see Appendix A.1. We note that the best fits to observations using Eq. (6) are consistent with this possible relation as the solutions follow a well defined anti-correlation between ξion and dust extinction with a similar relation and slope; see Fig. A.1 for a direct comparison.

3.4. Estimating fesc, Lyα with a simple observable: Lyα EW0

We find that LAEs follow a simple relation between fesc, Lyα and Lyα EW0 roughly independently of redshift (for z ≤ 2.6). Motivated by this, we propose the following empirical estimator (see Table 1) for fesc, Lyα as a function of Lyα EW0 (Å):

f esc , Ly α = 0 . 0048 − 0.0007 + 0.0007 EW 0 ± 0.05 [ 0 ≤ EW 0 ≤ 160 Å ] . $$ \begin{aligned} {f}_{\mathrm{esc,Ly}\alpha }=0.0048^{+0.0007}_{-0.0007}\,{\mathrm{EW}_{0}}\pm 0.05 \ \ [\, {0\le \mathrm{EW}_{0}\le 160\,{\AA }}]. \end{aligned} $$(7)

This relation may hold up to EW0 ≈ 210 Å, above which we would predict fesc, Lyα ≈ 1. This relation suggests that it is possible to estimate fesc, Lyα for LAEs within a scatter of 0.2 dex even if only the Lyα EW0 is known/constrained. It also implies that the observed Lyα luminosities are essentially equal to intrinsic Lyα luminosities for sources with EW0 as high as ≈200 Å. We conclude that while the escape of Lyα photons can depend on a range of properties in a very complex way (see e.g. Hayes et al. 2010; Matthee et al. 2016; Yang et al. 2017), using EW0 and Eq. (7) leads to predicting fesc, Lyα within ≈0.1−0.2 dex of real values. This compares with a larger scatter of ≈0.3 dex for relations with derivative or more difficult quantities to measure such as dust extinction or the red peak velocity of the Lyα line (e.g. Yang et al. 2017). We propose a linear relation for its simplicity and because current data do not suggest a more complex relation. Larger data-sets with Hα and Lyα measurements, particularly those covering a wider parameter space (e.g. different sample selections, multiple redshifts and both high and low EWs), may lead to the necessity of a more complicated functional form. A departure from a linear fit may also provide further insight of different physical processes driving the relation and the scatter (e.g. winds, orientation angle, burstiness or additional ionisation processes such as fluorescence).

Equation (7) may thus be applied to estimate fesc, Lyα for a range of LAEs in the low and higher redshift Universe. For example, the green pea J1154+2443 (Izotov et al. 2018), has a measured fesc, Lyα directly from dust corrected Hα luminosity of ≈0.7−0.87, while Eq. (7) would imply ≈0.6−0.7 based on the EW0 ≈ 133 Å for Lyα, thus implying a difference of only 0.06–0.1 dex. Furthermore, in principle, Eq. (7) could also be explored to transform EW0 distributions (e.g. Hashimoto et al. 2017, and references therein) into distributions of fesc, Lyα for LAEs.

3.5. Lyα as an SFR indicator: empirical calibration and errors

Driven by the simple relation (Eq. (7)) found up to z ∼ 2.6, we derive an empirical calibration to obtain SFRs based on two simple, direct observables for LAEs at high redshift: 1) Lyα EW0 and 2) observed Lyα luminosity. This calibration is based on observables, but predicts the dust-corrected SFR8. Based on Eqs. (2) and (7), for a Salpeter (Chabrier) IMF we can derive9:

SFR Ly α [ M ⊙ yr − 1 ] = L Ly α × 7.9 ( 4.4 ) × 10 − 42 ( 1 − f esc , LyC ) ( 0.042 EW 0 ) ( ± 15 % ) . $$ \begin{aligned} {\mathrm{SFR}_{\mathrm{Ly}\alpha }\,[{{M}_{\odot }\,\mathrm{yr}^{-1}}]}=\frac{{L}_{\mathrm{Ly}\alpha }\times 7.9\,(4.4)\times 10^{-42}}{(1-{f}_{\rm esc,LyC})(0.042\,\mathrm{EW}_{0})} \, (\pm 15\%). \end{aligned} $$(8)

The current best estimate of the scatter in Eq. (7) (the uncertainty in the relation to calculate fesc, Lyα is ±0.05) implies a ±0.07 dex uncertainty in the extinction corrected SFRs from Lyα with our empirical calculation. In order to investigate other systematic errors, we conduct a Monte Carlo analysis by randomly varying fesc, LyC (0.0–0.2) and the case B coefficient (from 8.0 to 9.0), along with perturbing fesc, Lyα from −0.05 to +0.05. We assume that all properties are independent, and thus this can be seen as a conservative approach to estimate the uncertainties. We find that the uncertainty in fesc, Lyα is the dominant source of uncertainty (12%) with the uncertainty on fesc, LyC and the case B coefficient contributing an additional 3% for a total of 15%. This leads to an expected uncertainty of Eq. (8) of 0.08 dex.

Note that the SFR calibration presented in Eq. (8) follows Kennicutt (1998) and thus a solar metallicity, which may not be be fully applicable to LAEs, typically found to be sub-solar (Nakajima & Ouchi 2014; Steidel et al. 2016; Suzuki et al. 2017; Sobral et al. 2018b). Other caveats include the applicability of the Calzetti et al. (2000) dust law (see e.g. Reddy et al. 2016) and the shape and slope of the IMF used, although any other SFR calibration/estimator will share similar caveats.

3.6. Lyα as an SFR indicator: performance and implications

In Fig. 3 we apply Eq. (8) to compare the estimated SFRs (from Lyα) with those computed with dust corrected Hα luminosities. We also include individual sources at z ∼ 2.2 (S18; Sobral et al. 2018b) and recent results from Harikane et al. (2018) at z = 4.8 which were not used in the calibration, and thus provide an independent way to test our new calibration. We find a global scatter of ≈0.12 dex, being apparently larger for lower EW0, but still lower than the typical scatter between SFR indicators after dust corrections (e.g. UV-Hα or FIR-Hα; see Domínguez Sánchez 2012; Oteo et al. 2015), as shown in Fig. 3. The small scatter and approximately null offset between our calibration’s prediction and measurements presented by Harikane et al. (2018) at z ∼ 5 suggest that Eq. (8) may be applicable at higher redshift with similarly competitive uncertainties (see Sect. 3.7 and Sect. 3.8). Nonetheless, we note that the measurements presented by Harikane et al. (2018) are inferred from broad-band IRAC photometry/colours as it is currently not possible to directly measure Hα line luminosities beyond z ∼ 2.5, and thus any similar measurements should be interpreted with some caution.

thumbnail Fig. 3.

Logarithmic ratio between SFRs computed with Eq. (8) using Lyα luminosity and EW0 and the “true” SFR, measured directly from dust-corrected Hα luminosity (given our definitions, log10(SFRLyα(This  work)/SFRHα) = log10(fesc, Lyα(Hα)/fesc, Lyα(This  work))). The standard deviation (σ) and σ/ N $ \sqrt{N} $ (N: number of sources) are computed in ±30 Å bins and indicated in light and darker grey. We find a relatively small scatter which may decrease for higher EWs and that is at the global level of ±0.12 dex for the typical definition of LAE at higher redshift (EW0 >  20 Å), but rises to ≈0.2 dex at the lowest EWs. We also provide a comparison of the typical scatter between UV and FIR SFRs in relation to Hα at z ∼ 0−2 (≈0.3 dex; see e.g. Domínguez Sánchez 2012; Oteo et al. 2015).

3.7. Application to bright and faint LAEs at high redshift

Our new empirical calibration of Lyα as an SFR indicator allows to estimate SFRs of LAEs at high redshift. The global Lyα luminosity function at z ∼ 3−6 has a typical Lyα luminosity ( L Ly α ⋆ $ L^{\star}_{\mathrm{Ly}\alpha} $) of 1042.9 erg s−1 (see e.g. Drake et al. 2017b; Herenz et al. 2017; Sobral et al. 2018a, and references therein), with these LAEs having EW0 ≈ 80 Å (suggesting fesc, Lyα = 0.38 ± 0.05 with Eq. (7)), which implies SFRs of ≈20 M⊙ yr−1. If we explore the public SC4K sample of LAEs at z ∼ 2−6 (Sobral et al. 2018a), limiting it to sources with up to EW0 = 210 Å and that are consistent with being star-forming galaxies (LLyα <  1043.2 erg s−1; see Sobral et al. 2018b), we find a median SFR for LAEs of 12 − 5 + 9 $ 12^{+9}_{-5} $ M⊙ yr−1, ranging from ≈2 M⊙ yr−1 to ≈90 M⊙ yr−1 at z ∼ 2−6. These reveal that “typical” to luminous LAEs are forming stars below and up to the typical SFR (SFR⋆ ≈ 40−100 M⊙ yr−1) at high redshift (see Smit et al. 2012; Sobral et al. 2014); see also Kusakabe et al. (2018).

Deep MUSE Lyα surveys (e.g. Drake et al. 2017a; Hashimoto et al. 2017) are able to sample the faintest LAEs with a median LLyα = 1041.9 ± 0.1 erg s−1 and EW0 = 87 ± 6 (Hashimoto et al. 2017) at z ∼ 3.6. We predict a typical fesc, Lyα = 0.42 ± 0.05 and SFRLyα = 1.7 ± 0.3 M⊙ yr−1 for those MUSE LAEs. Furthermore, the faintest LAEs found with MUSE have LLyα = 1041 erg s−1 (Hashimoto et al. 2017), implying SFRs of ≈0.1 M⊙ yr−1 with our calibration. Follow-up JWST observations targeting the Hα line for faint MUSE LAEs are thus expected to find typical Hα luminosities of 2 × 1041 erg s−1 and as low as ≈1−2 × 1040 erg s−1 for the faintest LAEs. Based on our predicted SFRs, we expect MUSE LAEs to have UV luminosities from MUV ≈ −15.5 for the faintest sources (see e.g. Maseda 2018), to MUV ≈ −19 for more typical LAEs, thus potentially linking faint LAEs discovered from the ground with the population of SFGs that dominate the faint end of the UV luminosity function (e.g. Fynbo et al. 2003; Gronke et al. 2015; Dressler et al. 2015).

3.8. Comparison with UV and implications at higher redshift

Equations (7) and (8) can be applied to a range of spectroscopically confirmed LAEs in the literature. We also extend our predictions to sources within the epoch of re-ionisation, although there are important caveats on how the Lyα transmission is affected by the IGM; see e.g. Laursen et al. (2011).

We explore a recent extensive compilation by Matthee et al. (2017b) of both Lyα- and UV-selected LAEs with spectroscopic confirmation and Lyα measurements (e.g. Ouchi et al. 2008, 2009; Ono et al. 2012; Sobral et al. 2015; Zabl et al. 2015; Stark et al. 2015a; Ding et al. 2017; Shibuya et al. 2018); see Table B.1. These include published LLyα, EW0 and MUV. In order to correct UV luminosities we use the UV β slope, typically used to estimate AUV10. We use β values (and errors) estimated in the literature for each source when available. When individual β values are not available, we use β = −1.6 ± 0.2 for UV-selected sources (typical for their UV luminosity; e.g. Bouwens et al. 2009), while for the luminous LAEs we use β = −1.9 ± 0.2. As a comparison, we also use a fixed β = −1.6 ± 0.2 for all the sources, which leads to a correction of AUV ≈ 1.25 mag. We list UV β slopes and resulting SFRs in Table B.1.

We predict (dust-corrected) SFRs using LLyα and EW0 only (Eq. (8)) and compare with SFRs measured from dust-corrected UV luminosities (Kennicutt 1998); see Table B.1. We make the same assumptions and follow the same methodology to transform the observables of our toy model/grid into SFRs (see Fig. 4). We note that, as our simulation shows, one expects a correlation even if our calibration of Lyα as an SFR indicator is invalid at high redshift, but our grid shows that the scatter depends significantly on dust extinction. Therefore, we focus our discussion on the normalisation of the relation and particularly on the scatter, not on the existence of a relation. We also note that our calibration is based on dust corrected Hα luminosities at z ∼ 0−2.6, and that UV luminosities are not used prior to this Section.

thumbnail Fig. 4.

Comparison between SFRs computed with our new empirical calibration for Lyα as an SFR indicator (Eq. (8)) and those computed based on dust corrected UV luminosity (see Sect. 3.8) for a compilation of z ∼ 5−8 sources (see Matthee et al. 2017b, and references therein). Our simple empirical calibration of Lyα as a SFR is able to recover dust corrected UV SFRs for the most star-forming sources, with a typical scatter of ≈0.2−0.3 dex (the scatter is lower if one assumes a fixed β = −1.6 ± 0.2). For the sources with the lowest UV SFRs we find that Eq. (8) seems to over-predict SFRs, potentially due to IGM effects which can lead to a lower EW0. We also compute SFRs in the same way with observables from our toy model and show the results of all realisations in grey. We find that the scatter in our toy model is much larger, with this being driven by E(B − V) being able to vary from 0.0 to 0.5.

Our results are shown in Fig. 4 (see Table B.1 for details on individual sources), which contains sources at a variety of redshifts, from z ∼ 6 to z ∼ 8 (e.g. Oesch et al. 2015; Stark et al. 2017). We find a good agreement between our predicted Lyα SFRs based solely on Lyα luminosities and EW0 and the dust corrected UV SFRs for sources with the highest SFRs at z ∼ 6−8 (Fig. 4), with a scatter of ≈0.2−0.3 dex. Interestingly, Eq. (8) seems to over-predict (compared to the UV) Lyα SFRs for the least star-forming sources (≤30−40 M⊙ yr−1). This is caused by their typically very low EW0, which would imply a low fesc, Lyα, thus boosting the Lyα SFR compared to the UV. Taken as a single population, the UV-selected sources (LBGs) show a higher log10(SFRLyα/SFRUV) = 0.23 ± 0.24 than LAEs that reveal log10(SFRLyα/SFRUV) = 0.15 ± 0.13. Such discrepancies could be caused by the IGM which could be reducing the EW0 and fesc, Lyα. This would happen preferentially for the UV selected and for the sources with the lowest SFRs without strong Lyα in a way that our calibration at z ∼ 0−2.6 simply does not capture. However, the deviation from a ratio of 1 is not statistically significant given the uncertainties and there is a large scatter from source to source to be able to further quantify the potential IGM effect.

Overall, our results and application to higher redshift reveals that Eq. (8) is able to retrieve SFRs with very simple observables even for LAEs within re-ionisation (e.g. Ono et al. 2012; Stark et al. 2015a, 2017; Schmidt et al. 2017), provided they are luminous enough. In the early Universe the fraction of sources that are LAEs is higher (e.g. Stark et al. 2010, 2017; Caruana et al. 2018), thus making our calibration potentially applicable to a larger fraction of the galaxy population, perhaps with an even smaller scatter due to the expected narrower range of physical properties and more compact sizes (see discussions in e.g. Paulino-Afonso et al. 2018). Our calibration of Lyα as an SFR indicator is simple, directly calibrated with Hα, and should not have a significant dependence on metallicity, unlike other proposed SFRs tracers at high redshift such as [CII] luminosity or other weak UV metal lines.

It is surprising that our calibration apparently still works even at z ∼ 7−8 for the most luminous LAEs. This seems to indicate that the IGM may not play a significant role for these luminous Lyα-visible sources, potentially due to early ionised bubbles (see e.g. Matthee et al. 2015, 2018; Mason et al. 2018a,b) or velocity offsets of Lyα with respect to systemic (see e.g. Stark et al. 2017). Interestingly, we find offsets between our calibration and the computed UV SFRs for the faintest sources, hinting that IGM effects start to be much more noticeable for such faint sources which may reside in a more neutral medium and/or on smaller ionised bubbles. Further observations measuring the velocity offsets between the Lyα and systemic redshifts for samples of LAEs within the epoch of re-ionisation and those at z ∼ 3−5 will allow to check and test the validity of the relation within the epoch of re-ionisation.

3.9. A tool for re-ionisation: predicting the LyC luminosity

Based on our results and assumptions (see Sect. 2.4), we follow Matthee et al. (2017a) 11 and derive a simple expression to predict the number of produced LyC photons per second, Qion (s−1) with direct Lyα observables (LLyα and EW0)12:

Q ion , Ly α [ s − 1 ] = L Ly α c H α ( 1 − f esc , LyC ) ( 0.042 EW 0 ) , $$ \begin{aligned} {Q}_{\mathrm{ion,Ly}\alpha }\,[{\mathrm{s}^{-1}}]=\frac{{L}_{\mathrm{Ly}\alpha }}{{c}_{\mathrm{H}\alpha }\,(1-{f}_{\rm esc,LyC})\,(0.042\,\mathrm{EW}_{0})}, \end{aligned} $$(9)

where cHα = 1.36 × 10−12 erg (e.g. Kennicutt 1998; Schaerer 2003), under our case B recombination assumption (see Sect. 2.4). We caution that Eq. (9) may not be fully valid for all observed LAEs within the epoch of reionisation. This is due to possible systematic effects on EW0 of an IGM which is partly neutral, although we note that as found in Sect. 3.8 it may well be valid for the most luminous LAEs at z ∼ 7−8.

Recent work by e.g. Verhamme et al. (2017) show that LyC leakers are strong LAEs, and that fesc, Lyα is linked and/or can be used to predict fesc, LyC (see Chisholm et al. 2018). Equation (9) provides an extra useful tool: an empirical simple estimator of Qion for LAEs given observed Lyα luminosities and EW0. Note that Eq. (9) does not require measuring UV luminosities or ξion, but instead direct, simple observables. Matthee et al. (2017b) already used a similar method to predict ξion at high redshift. Coupled with an accurate estimate of the escape fraction of LyC photons from LAEs (see e.g. Steidel et al. 2018), a robust estimate of the full number density of LAEs from faint to the brightest sources (Sobral et al. 2018a) and their redshift evolution, Eq. (9) may provide a simple tool to further understand if LAEs were able to re-ionise the Universe.

4. Conclusions

Lyα is intrinsically the brightest emission-line in active galaxies, and should be a good SFR indicator. However, the uncertain and difficult to measure fesc, Lyα has limited the interpretation and use of Lyα luminosities. In order to make progress, we have explored samples of LAEs at z = 0−2.6 with direct Lyα escape fractions measured from dust corrected Hα luminosities which do not require any SED fitting, ξion or other complex assumptions based on derivative quantities. Our main results are:

  • There is a simple, linear relation between fesc, Lyα and Lyα EW0: fesc,Lyα = 0.0048 EW0[Å] ± 0.05 (Eq. (7)) which is shallower than simple expectations, due to both more ionising photons per UV luminosity (ξion) and low dust extinction (E(B − V)) for LAEs (Fig. 1). This allows the prediction of fesc, Lyα based on a simple direct observable, and thus to compute the intrinsic Lyα luminosity of LAEs at high redshift.

  • The observed fesc, Lyα-EW0 can be explained by high ξion and low E(B − V) or, more generally, by a tight ξion − E(B − V) sequence for LAEs, with higher ξion implying lower E(B − V) and vice versa. ξion and E(B − V) may vary within the fesc, Lyα-EW0 plane (Fig. 2). Our results imply that the higher the EW0 selection, the higher the ξion and the lower the E(B − V).

  • The fesc, Lyα-EW0 relation reveals a scatter of only 0.1-0.2 dex for LAEs, and there is evidence for the relation to hold up to z ∼ 5 (Fig. 3). The scatter is higher towards lower EW0, consistent with a larger range in galaxy properties for sources with the lowest EW0. At the highest EW0, on the contrary, the scatter may be as small as ≈0.1 dex, consistent with high EW0 LAEs being an even more homogeneous population of dust-poor, high ionisation star-forming galaxies.

  • We use our results to calibrate Lyα as an SFR indicator for LAEs (Eq. (8)) and find a global scatter of 0.2 dex between measurements using Lyα only and those using dust-corrected Hα luminosities. Such scatter seems to depend on EW0, being larger at the lowest EW0. Our results also allow us to derive a simple estimator of the number of LyC photons produced per second (Eq. (9)) with applications to studies of the epoch of re-ionisation.

  • Equation (8) implies that star-forming LAEs at z ∼ 2−6 have SFRs typically ranging from 0.1 to 20 M⊙ yr−1, with MUSE LAEs expected to have typical SFRs of 1.7 ± 0.3 M⊙ yr−1, and more luminous LAEs having SFRs of 12 − 5 + 9 $ 12^{+9}_{-5} $ M⊙ yr−1.

  • SFRs based on Eq. (8) are in good agreement with dust corrected UV SFRs even within the epoch of re-ionisation for SFRs higher than ≈30−40 M⊙ yr−1, hinting for it to be applicable in the very early Universe for bright enough LAEs. For lower SFRs we find that Eq. (8) may over-predict SFRs compared to the UV, potentially due to IGM effects. If shown to be the case, our results have implications for the minor role of the IGM in significantly changing Lyα luminosities and EW0 for the most luminous LAEs within the epoch of re-ionisation.

Our results provide a simple interpretation of the tight fesc, Lyα-EW0 relation. Most importantly, we provide simple and practical tools to estimate fesc, Lyα at high redshift with two direct observables and thus to use Lyα as an SFR indicator and to measure the number of ionising photons from LAEs. The empirical calibrations presented here can be easily tested with future observations with JWST which can obtain Hα and Hβ measurements for high-redshift LAEs.

1

We use Lyα/Hα = 8.7, but vary the Lyα/Hα case B ratio between 8.0 and 9.0 to test for its effect; see Sect. 3.5 and also discussions in Henry et al. (2015).

2

With our case B assumptions the intrinsic Balmer decrement is: Hα/Hβ = 2.86. Using a Calzetti et al. (2000) dust attenuation law we use AHα = 6.531 log10(Hα/Hβ)−2.981 (see details in e.g. Sobral et al. 2012).

3

For continuous star formation over 10 Myr timescales and calibrated for solar metallicity; see Kennicutt (1998).

4

Assuming a Lyα/Hα case B recombination coefficient of 8.7.

5

ξ ion = 1.3 × 10 25 SFR H α SFR UV ( Hz erg − 1 ) $ {\xi}_{\mathrm{ion}} = 1.3\times10^{25} \frac{\mathrm{SFR}_{\mathrm{H}\alpha}}{\mathrm{SFR}_{\mathrm{UV}}} \, (\mathrm{Hz}\,\mathrm{erg}^{-1}) $.

6

Note that a steeper β (within the framework of DW10) results in an even more significant disagreement with observations for a fixed UV luminosity (measured at rest-frame 1400 Å; see DW10) or SFR, as β is used to predict the UV continuum at ≈1216 Å. A steeper β in this context leads to more UV continuum and a lower EW0 for fixed SFR and fesc, Lyα.

7

This may be up to ≈0.98 if Hβ is used; see Izotov et al. (2018).

8

We use extinction corrected Hα luminosities.

9

Note that the constant 0.042 has units of Å−1, and results from 8.7 × 0.0048 Å−1. Also, note that the relation is valid for 0 ≤ EW0 ≤ 160 Å following Eq. (7). For EW0 >  160 Å the relation has not been calibrated yet. Furthermore, if the relation is to be used at even higher EWs, then for EW0 >  207 Å the factor 0.042 EW0 should be set to 8.7 (or the appropriate/assumed case B recombination constant), corresponding to a ≈100 % escape fraction of Lyα photons.

10

We use AUV = 4.43 + 1.99β; see Meurer et al. (1999), but see also discussions on uncertainties and limitations in e.g. Popping et al. (2017), Narayanan et al. (2018) and references therein.

11

We assume fdust ≈ 0 (see Matthee et al. 2017a), i.e., we make the assumption that for LAEs the dust extinction to LyC photons within HII regions is ≈0.

12

Note that the relation is valid for 0 ≤ EW0 ≤ 160 Å following Eq. (7). If the relation is to be used beyond the calibrated range then for EW0 >  207 Å the factor 0.042 EW0 should be set to 8.7 (case B recombination), corresponding to a ≈100 % escape fraction of Lyα photons. Note that one can also vary the Lyα/Hα case B ratio between 8.0 and 9.0 to better sample systematic errors due to unknown gas temperatures, although these result in small systematic variations.

Acknowledgments

We thank the anonymous referees for multiple comments and suggestions which have improved the manuscript. JM acknowledges the support of a Huygens PhD fellowship from Leiden University. We have benefited greatly from the publicly available programming language PYTHON, including the NUMPY & SCIPY (Van Der Walt et al. 2011; Jones et al. 2001), MATPLOTLIB (Hunter 2007) and ASTROPY (Astropy Collaboration 2013) packages, and the TOPCAT analysis program (Taylor 2013). The results and samples of LAEs used for this paper are publicly available (see e.g. Sobral et al. 2017, 2018a) and we also provide the toy model used as a PYTHON script.

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Appendix A: Toy-model grid for fesc, Lyα dependencies

Table A.1.

Parameters varied in our simple toy model to produce a grid of 1 000 000 sources to interpret the observational results (see Appendix A).

We construct a simple analytical toy-model grid to produce observable Hα, UV and Lyα luminosities and EW0 from a range of input physical conditions (see Table A.1). We independently sample in steps of 0.01 or 0.01 dex combinations of SFR, fesc, LyC, case B Lyα/Hα intrinsic ratio, log10(ξion/Hz erg−1), E(B − V) with a Calzetti et al. (2000) dust attenuation law and a parameter to control fesc, Lyα (from e.g. scattering leading to higher dust absorption or scattering Lyα photons away from or into the observers’ line of sight) which acts as a further factor affecting fesc, Lyα; see Table A.1 for the range in parameters explored independently. We follow Kennicutt (1998) and all definitions and assumptions mentioned in this paper and we sample the parameter space with a flat prior. We use the Meurer et al. (1999) relation: AUV ≈ 4.43 + 1.99β (based on an intrinsic slope βint = −2.23), but we also vary the intrinsic βint from −2.6 to −1.8 (see e.g. Wilkins et al. 2013). We publicly release our simple PYTHON script which can be used for similar studies and/or to study different ranges in the parameter space, or conduct studies in which properties are intrinsically related/linked as one expects for realistic galaxies. Section A.2 presents the full equations implemented in the publicly available PYTHON script.

thumbnail Fig. A.1.

Left: predicted relation between fesc, Lyα and Lyα EW0 for our toy model, which shows little to no correlation by sampling all physical parameters independently (see Table A.1). We also show the observed range (≈ ± 0.05) which is well constrained at z ∼ 0−2.6. We use simulated sources that are consistent with observations of LAEs to explore the potential reason behind the observed tight fesc, Lyα-EW0 correlation for LAEs. Right: by restricting our toy model to the observed relation and its scatter, we find a relatively tight ξion − E(B − V) anti-correlation for LAEs (EW0 >  25 Å): log10(ξion/Hz erg−1) ≈ −1.85 × E(B − V) + 25.6 (shown as grey dot-dashed line). This is in good agreement with the family of best fits using Eq. (6) (we show the 1, 2 and 3σ contours) which yields log10(ξion/Hz erg−1) ≈ −1.71 × E(B − V) + 25.6, with only a small difference in the slope. The highest observed EW0 correspond to the highest ξion and the lowest E(B − V), while lower EW0 leads to a lower ξion and a higher E(B − V). Our results thus show that the observed fesc, Lyα-EW0 correlation for LAEs at z ∼ 0−2.6 only allows a well defined ξion − E(B − V) sequence that may be related with important physics such as the age of the stellar populations, their metallicity, dust production and how those evolve together.

A.1. The fesc, Lyα-EW0 and a potential ξion − E(B − V) sequence for LAEs

We use our simple analytical model to further interpret the observed relation between fesc, Lyα-EW0 and its tightness. We take all artificially generated sources and select those that satisfy the observed relation given in Eq. (7), including its scatter (see Fig. A.1). We further restrict the sample to sources with Lyα EW0 >  25 Å. We find that along the observed fesc, Lyα-EW0 relation, LAEs become less affected by dust extinction as a function of increasing EW0, while ξion increases, as already shown in Sect. 3.3 and Fig. 2.

In the right panel of Fig. A.1 we show the full parameter range explored in ξion − E(B − V). By constraining the simulated sources with the observed fesc, Lyα-EW0 relation, we obtain a tight (±0.1 dex), linear relation between log10ξion and E(B − V) given by log10(ξion/Hz erg−1) ≈ −1.85 × E(B − V) + 25.6. This means that in order for simulated sources to reproduce observations, LAEs should follow a very well defined ξion − E(B − V) sequence with high ξion values corresponding to very low E(B − V) (mostly at high EW0 and high fesc, Lyα) and higher E(B − V) to lower ξion (mostly at low EW0 and high fesc, Lyα). Our results thus hint for the fesc, Lyα-EW0 to be driven by the physics (and diversity) of young and metal poor stellar populations and their evolution.

A.2. Steps and equations for the model grid

We produce a model grid with our simple toy model which implements all equations and follows the observationally-motivated methodology used in the paper for full self-consistency. For each of the N = 1 000 000 realisations, the script randomly picks (with a flat prior) parameters out of the parameter grid presented in Table A.1 (independently, per parameter).

The following steps are then taken per realisation. The Hα luminosity is computed using the Kennicutt (1998) calibration and the Lyα luminosity is obtained by using the case B coefficient used for that specific realisation. The UV SFR is computed by using log10(ξion/Hz erg−1) for that realisation and the Kennicutt (1998) calibration, which is then used to compute the intrinsic UV luminosity at rest-frame 1600 Å (MUV and LUV). This step produces all the intrinsic luminosities which will be used: Lyα, UV and Hα.

Next, by using the randomly picked value of E(B − V) (see Table A.1), the Calzetti et al. (2000) attenuation law is used. For simplicity, as mentioned before, we set the attenuation of the nebular lines to be the same as the stellar continuum. We use the Calzetti et al. (2000) dust attenuation law to compute Aλ (mag) for λ = 1215.7, 1600, 6563 Å in order to compute the attenuation at Lyα, UV and Hα. We then compute the observed Lyα, UV and Hα luminosities after dust attenuation by computing:

L λ , observed = L λ , intrinsic × 10 − 0.4 A λ $$ \begin{aligned} {L}_{\lambda ,\mathrm{observed}}={L}_{\lambda ,\mathrm{intrinsic}}\times 10^{-0.4\,{A}_{\lambda }} \end{aligned} $$(A.1)

Finally, for Lyα, we apply the parameter “Extra fesc, Lyα” (see Table A.1) which is multiplied by the observed Lyα luminosity (attenuated by dust) to produce the final observed Lyα luminosity. This is to quantify our ignorance on radiative transfer effects which are not explicitly modelled and are extremely complex. Following the methodology in this paper, the Lyα escape fraction is then computed using Eq. (1) and with all quantities computed or randomly picked with the script.

Finally, after randomly picking an intrinsic βint slope, the Meurer et al. (1999) relation is used to transform E(B − V) into an observed β UV slope. This follows Meurer et al. (1999) and assumes that LAEs have β = βint for E(B − V) = 0.0. β is then used together with the observed UV luminosity at 1600 Å to compute the observed UV luminosity at λ = 1215.7 Å. This is used to compute the observed EW0. The toy model also computes the intrinsic EW0, i.e., the rest-frame Lyα EW in the case of no dust and no scattering. The script also applies the calibrations derived/obtained or used in the paper to predict the Lyα escape fraction, Lyα and UV SFRs based on Eqs. (7) and (8) (see also Sect. 3.8) and the input from the simulation grid. These may be interesting for readers to explore further trends, and are provided as further information in the catalogue of 1 000 000 simulated sources.

Appendix B: Data used for the high-redshift comparison between UV and Lyα SFRs

Table B.1 provides the data used for Fig. 4, including individual measurements per source, their name and reference. Note that the data is taken from a compilation from Matthee et al. (2017b) with minor modifications for a few LAEs, as a indicated in Table B.1.

Table B.1.

Application to high redshift UV-continuum and Lyα selected LAEs (see compilation by Matthee et al. 2017b).

All Tables

Table 1.

Results from fitting the relation between fesc, Lyα and Lyα EW0 as fesc, Lyα = A × EW0 + B, with EW0 in Å (see Sect. 2.5).

In the text
Table A.1.

Parameters varied in our simple toy model to produce a grid of 1 000 000 sources to interpret the observational results (see Appendix A).

In the text
Table B.1.

Application to high redshift UV-continuum and Lyα selected LAEs (see compilation by Matthee et al. 2017b).

In the text

All Figures

thumbnail Fig. 1.

Relation between fesc, Lyα and Lyα EW0 for z ∼ 2.2 (stacks; see Sobral et al. 2017), z ∼ 2.6 (binning; Trainor et al. 2015) and comparison with z ∼ 0−0.3 samples (e.g. Cardamone 2009; Hayes et al. 2013; Henry et al. 2015; Yang et al. 2016, 2017; Verhamme et al. 2017), estimated from dust-corrected Hα luminosities (Eq. (1)). We show the 1σ and 2σ range for the fits at z ∼ 2.2−2.6 and z ∼ 0−0.3 separately, and find them to be consistent within those uncertainties, albeit with a potential steeper relation at higher redshift. We find a combined best fitting relation given by fesc, Lyα = 0.0048 EW0 ± 0.05. The observed relation is significantly away from what would be predicted based on observed UV slopes between β ≈ −2 and β ≈ −1 for LAEs (see DW10) and would require β ≈ +5 for a good fit using Eq. (5). Such red β slopes are not observed for LAEs. Modifying Eq. (5) to include the effect of ξion and dust reveals that those physical parameters likely play an important role; see Sects. 3.2 and 3.3.
In the text
thumbnail Fig. 2.

Left: predicted fesc, Lyα-Lyα EW0 space for different E(B − V) (contour levels) with our grid model (see Sect. 3.3 and Appendix A) and comparison with fits and implications by using Eq. (6) (right). We find that increasing dust extinction drives fesc, Lyα down for a fixed EW0, with data at z ∼ 0−2.6 hinting for lower dust extinction at the highest EW0 and higher dust extinction at the lowest EW0, but with the range being relatively small overall and around E(B − V) ≈ 0.05−0.3. Right: predicted fesc, Lyα-EW0 space for different ξion (contours). We find that while increasing E(B − V) mostly shifts the relation down, increasing ξion moves the relation primarily to the right.
In the text
thumbnail Fig. 3.

Logarithmic ratio between SFRs computed with Eq. (8) using Lyα luminosity and EW0 and the “true” SFR, measured directly from dust-corrected Hα luminosity (given our definitions, log10(SFRLyα(This  work)/SFRHα) = log10(fesc, Lyα(Hα)/fesc, Lyα(This  work))). The standard deviation (σ) and σ/ N $ \sqrt{N} $ (N: number of sources) are computed in ±30 Å bins and indicated in light and darker grey. We find a relatively small scatter which may decrease for higher EWs and that is at the global level of ±0.12 dex for the typical definition of LAE at higher redshift (EW0 >  20 Å), but rises to ≈0.2 dex at the lowest EWs. We also provide a comparison of the typical scatter between UV and FIR SFRs in relation to Hα at z ∼ 0−2 (≈0.3 dex; see e.g. Domínguez Sánchez 2012; Oteo et al. 2015).
In the text
thumbnail Fig. 4.

Comparison between SFRs computed with our new empirical calibration for Lyα as an SFR indicator (Eq. (8)) and those computed based on dust corrected UV luminosity (see Sect. 3.8) for a compilation of z ∼ 5−8 sources (see Matthee et al. 2017b, and references therein). Our simple empirical calibration of Lyα as a SFR is able to recover dust corrected UV SFRs for the most star-forming sources, with a typical scatter of ≈0.2−0.3 dex (the scatter is lower if one assumes a fixed β = −1.6 ± 0.2). For the sources with the lowest UV SFRs we find that Eq. (8) seems to over-predict SFRs, potentially due to IGM effects which can lead to a lower EW0. We also compute SFRs in the same way with observables from our toy model and show the results of all realisations in grey. We find that the scatter in our toy model is much larger, with this being driven by E(B − V) being able to vary from 0.0 to 0.5.
In the text
thumbnail Fig. A.1.

Left: predicted relation between fesc, Lyα and Lyα EW0 for our toy model, which shows little to no correlation by sampling all physical parameters independently (see Table A.1). We also show the observed range (≈ ± 0.05) which is well constrained at z ∼ 0−2.6. We use simulated sources that are consistent with observations of LAEs to explore the potential reason behind the observed tight fesc, Lyα-EW0 correlation for LAEs. Right: by restricting our toy model to the observed relation and its scatter, we find a relatively tight ξion − E(B − V) anti-correlation for LAEs (EW0 >  25 Å): log10(ξion/Hz erg−1) ≈ −1.85 × E(B − V) + 25.6 (shown as grey dot-dashed line). This is in good agreement with the family of best fits using Eq. (6) (we show the 1, 2 and 3σ contours) which yields log10(ξion/Hz erg−1) ≈ −1.71 × E(B − V) + 25.6, with only a small difference in the slope. The highest observed EW0 correspond to the highest ξion and the lowest E(B − V), while lower EW0 leads to a lower ξion and a higher E(B − V). Our results thus show that the observed fesc, Lyα-EW0 correlation for LAEs at z ∼ 0−2.6 only allows a well defined ξion − E(B − V) sequence that may be related with important physics such as the age of the stellar populations, their metallicity, dust production and how those evolve together.
In the text

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