Evolution in the escape fraction of ionizing photons and the decline in strong Lyα emission from z > 6 galaxies

Abstract

1 INTRODUCTION

Observations of Lyα emitting galaxies are often interpreted to indicate that the intergalactic medium (IGM) is more opaque to Lyα photons at |$z$| > 6, than at lower redshifts. In particular, while the Lyα luminosity function (LF) of Lyα selected galaxies (LAE) remains constant at |$z=3{\rm -}6$| (e.g. Hu, Cowie & McMahon 1998; Ouchi et al. 2008), it is observed to drop rapidly at |$z$| > 6 (Kashikawa et al. 2006; Ouchi et al. 2010; Clément et al. 2012). Since their rest-frame UV-LFs do not exhibit the same evolution (Kashikawa et al. 2006), this reduction in the Lyα LF at |$z$| > 6 is likely caused by a reduction in the observed Lyα flux from these galaxies. Moreover, the so-called Lyα fraction – which denotes the fraction of galaxies selected via the drop-out technique that exhibit strong Lyα emission lines – increases between |$z$| = 3 and 6 (Stark et al. 2010; Stark, Ellis & Ouchi 2011), but then drops at |$z$| > 6 (Fontana et al. 2010; Pentericci et al. 2011; Caruana et al. 2012, 2013; Ono et al. 2012; Schenker et al. 2012; Treu et al. 2012, 2013).

The observed reduction in Lyα flux from galaxies at |$z$| > 6 is most readily explained by having additional intervening neutral atomic hydrogen, which is opaque to the Lyα flux, but not to the (non-ionizing) UV-continuum. This additional neutral atomic hydrogen is likely present naturally at |$z$| > 6 when reionization has not been completed. Indeed, it has been predicted that the end of reionization should coincide with a reduction in the observed Lyα flux from galaxies (e.g. Haiman & Spaans 1999).

Reionization was likely an inhomogeneous process in which fully ionized ‘bubbles’ were separated by neutral intergalactic gas (Furlanetto, Zaldarriaga & Hernquist 2004; McQuinn et al. 2007a). In this scenario, the progress of reionization is regulated by the growth of percolating H ii bubbles. The final stages of reionization are characterized by the presence of large bubbles, whose individual sizes exceeded tens of cMpc (e.g. Zahn et al. 2011). The majority of the galaxies we can detect with existing instruments preferentially resided inside large H ii bubbles. Their Lyα photons would have been able to travel a significant distance before entering the neutral IGM, and redshift out of the Lyα resonance due to the Hubble expansion, which would facilitate their subsequent transmission through the neutral IGM (Miralda-Escude 1998; Santos 2004; Cen, Haiman & Mesinger 2005; McQuinn et al. 2007b; Mesinger & Furlanetto 2008).

Because of this effect, it is difficult to interpret the observed reduction of Lyα flux from galaxies at |$z$| > 6 in the context of inhomogeneous reionization models, since the imprint of the neutral IGM on the detectability of Lyα emission from galaxies is found to be subtle. For example, Dijkstra, Mesinger & Wyithe (2011) showed that the observed drop in the Lyα fraction at |$z$| > 6 requires a change in the volume averaged neutral fraction of hydrogen of |$\Delta x_{\rm H\,\small {I}} \sim 0.5$| (also see Jensen et al. 2013, but see Taylor & Lidz 2013 who caution that the required change can be reduced when effects of galaxy sample variance are accounted for), which is consistent with earlier constraints on this quantity by McQuinn et al. (2007b) and Mesinger & Furlanetto (2008) from modelling the Lyα LFs. As was pointed out by Dijkstra et al. (2011), this required rapid evolution of |$\Delta x_{\rm H\,\small {I}} \sim 0.5$| over such a short time is extreme.¹

To alleviate this tension, Bolton & Haehnelt (2013) recently showed that the required |$\Delta x_{\rm H\,\small {I}}$| may be reduced significantly by Lyman limit systems (LLSs, self-shielding clouds with |$N_{\rm H\,\small {I}}> 10^{17}$| cm⁻²), whose number density may evolve rapidly at the end of reionization (although this rise is slowed down when recombinations in the IGM are taken into account, see Sobacchi & Mesinger 2014). Similarly, the opacity of residual H i in the ionized IGM can increase by as much as ∼30 per cent between |$z$| = 5.7 and 6.5 (Dijkstra, Wyithe & Haiman 2007b; Laursen, Sommer-Larsen & Razoumov 2011). However, these models also predict that the IGM transmits low fraction of Lyα photons, |$T_{\rm IGM}=0.1{\rm -}0.3$|. This is lower than current observational constraints on this quantity, which is likely related the impact of galactic winds on the Lyα line shape emerging from galaxies (Dijkstra & Jeeson-Daniel 2013). In any case, these possibilities will need to be explored and quantified further in future work.

Lyα emission is powered by recombination following photoionization inside H ii regions. The Lyα luminosity of a galaxy, L_α, is therefore proportional to the total number of ionizing photons that do not escape from galaxies, i.e. |$L_{\alpha } \propto (1-f_{\rm esc})\dot{N}_{\rm ion}$|, where f_esc denotes the escape fraction of ionizing photons. Thus, having f_esc increase between |$z$| = 6 and 7 will reduce the Lyα luminosity of galaxies as observed. We expect this effect to be especially strong when f_esc is large. For example, the intrinsic Lyα luminosity L_α doubles when f_esc = 0.8 → 0.9, while there is little impact when f_esc = 0.1 → 0.2.

Modelling f_esc and its redshift dependence from first principles requires properly resolving the multiphase structure of the interstellar medium (ISM; see e.g. Fernandez & Shull 2011), and the small spatial scales that are relevant for the transport of ionizing radiation in high-density gas (e.g. Rahmati et al. 2013). Direct observational constraints on the escape fraction, and its redshift dependence are still highly uncertain. However, there are several lines of indirect evidence that f_esc increases with redshift. Measurements of the redshift dependence of the photoionization rate of the Lyα forest in combination with the observed redshift evolution of the UV-LF of drop-out galaxies, suggest that f_esc increases quite rapidly with redshift at |$z$| ≳ 4 (e.g. Kuhlen & Faucher-Giguère 2012; Mitra, Ferrara & Choudhury 2013, also see Inoue, Iwata & Deharveng 2006). Moreover, the covering factor of low-ionization absorbers in drop-out galaxies has been observed to increase from |$z$| = 3 to 4, which provides independent evidence that f_esc is increasing with redshift (Jones, Stark & Ellis 2012; Jones et al. 2013). This inferred redshift evolution could reflect e.g. (i) an evolution of the UV-emissivity per unit gas mass, (ii) an evolution in the clumpiness in the ISM of high-|$z$| galaxies which affects the covering factor of high column density gas (Fernandez & Shull 2011), and/or (iii) an evolution in the fraction of stars formed in low-mass haloes which can efficiently ‘self-ionize’ (e.g. Ferrara & Loeb 2013).

The goal of this paper is to explore the impact of an increasing escape fraction on the visibility of the Lyα emission line from galaxies at |$z$| > 6. We stress the purpose of our paper is a proof-of-concept, and that a systematic study will be performed in future work. The outline is as follows: we describe our model in Section 2, present our results in Section 3. We discuss our results in Section 4 and present our conclusions in Section 5.

2 THE MODEL

2.1 The EW-PDF and its redshift evolution

The strength of the Lyα line emerging from galaxies is regulated by several physical processes: (i) The amount of Lyα that is produced, which depends on the initial mass function, stellar metallicity (e.g. Schaerer 2003) and f_esc; (ii) the amount of Lyα that escapes from the (dusty) ISM of galaxies, which is correlated with the dust content of galaxies (Atek et al. 2009; Kornei et al. 2010; Hayes et al. 2011); and (iii) the amount of Lyα that is scattered in the intergalactic/circum galactic medium. The Lyα line strength relative to the non-ionizing UV-continuum is quantified by the (rest-frame) equivalent width EW, i.e. EW|$\equiv \frac{L_{\alpha }}{L_{\rm c}(\lambda _{\rm UV})}$|, where L_c(λ_UV) denotes the luminosity density of the continuum at λ_UV = 1300–1600 Å. The Lyα emitting properties of a collection of galaxies can be quantified by the (cumulative) EW-distribution function (PDF), denoted with P(EW).

All processes listed above leave their imprint on the observed EW-PDF, and its redshift evolution. In particular, the EW-PDF evolves towards larger EWs from |$z$| = 3 to 6 (Stark et al. 2010), which can most easily be attributed to a decreasing dust content of galaxies in this redshift range (Hayes et al. 2011). Moreover, since the opacity of the IGM is expected to increase over the same redshift interval (and which would thus reduce the EW-PDF towards higher |$z$|), its impact appears subdominant to that of dust. However, the average² dust content of galaxies keeps decreasing at |$z$| > 6 (e.g. Bouwens et al. 2012b; Finkelstein et al. 2012a), and we would expect the EW-PDF to keep increasing. This is not what has been observed, and it becomes natural to search for alternative explanations for this sudden reduction in the EW-PDF at |$z$| > 6. Suppression of the Lyα line by neutral intergalactic gas is a natural explanation, as there exist other observational indications for the presence of neutral intergalactic gas at |$z$| > 6 (e.g. in quasar absorption line spectra, see Wyithe & Loeb 2004; Mesinger & Haiman 2007; Bolton et al. 2011).

Previous models (e.g. Dijkstra et al. 2011; Bolton & Haehnelt 2013; Jensen et al. 2013) have explored this possibility quantitatively, and assumed that the EW-PDF at |$z$| = 7 only differs from the one measured at |$z$| = 6 due to the intervening neutral IGM. Under this assumption, we have EW₇ = EW|$_6\times \mathcal {T}_{\rm IGM,7}/\mathcal {T}_{\rm IGM,6}$|, where |$\mathcal {T}_{\rm IGM,x}$| corresponds to the IGM transmission fraction at redshift |$z$| = x. We can then derive the EW-PDF at |$z$| = 7 from the one observed at |$z$| = 6 simply from³|$P_7({\rm EW}) \propto P_6({\rm EW}\times \mathcal {T}_{\rm IGM,6}/\mathcal {T}_{\rm IGM,7})$|. Because it is difficult to explain the observed |$z$|-evolution with reionization alone (see Section 1), we perform a similar analysis which includes a redshift escape fraction in this paper, as is discussed in more detail next.

2.2 The redshift evolution of the EW-PDF with f_esc(z)

• f_esc at a fixed |$z$| is the same for all galaxies: i.e. f_esc does not depend on M_UV. As discussed in Section 4.1, this assumption is likely unrealistic. However, the precise M_UV-dependence of f_esc is still poorly constrained. Instead of including this possible dependence in our models, we opt to study simpler models, which can provide a baseline for more complex future models that include this effect. Moreover, the range of UV-luminosities of the galaxies we are modelling is still limited, and the impact of f_esc depending on M_UV does not affect our main results.

• First, we assume that the EW-PDF at |$z$| = 7 differs only from the one at |$z$| = 6 due to the change in the escape fraction of ionizing photons, f_esc (see Section 2.1 for a motivation of this assumption). We present our results on this analysis in Section 3. We discuss how our analysis is modified if we assume that the EW-PDF evolves at |$z$| > 6 as a result of a joint evolution in f_esc(|$z$|) and reionization in Section 2.3. We generally present our results in terms of the expectation values 〈f_{esc, 7}〉 ≡ 〈f_{esc, 6}〉 + Δ〈f_esc〉 ≤ 1. We point out that the large impact on the redshift evolution in P(EW) for large 〈f_{esc, 6}〉 at fixed Δ〈f_esc〉 (as mentioned in Section 1) arises because f_esc appears as (1 − f_esc)⁻¹ in the argument of P₀(EW) in equation (1), which increases rapidly when f_esc → 1.

• Our analysis implicitly assumes that the |$z$|-evolution of f_esc does not affect the |$z$|-evolution of other quantities: e.g. increasing the escape fraction does not increase (i) the fraction of Lyα photons that are transmitted through the IGM, or (ii) the fraction of Lyα photons that can escape from the dusty ISM of galaxies. As we argue in Section 4.3, (i) is reasonable, while (ii) presents a limitation of the model, but may help explain why the redshift evolution of the escape fraction may only have noticeable effects on the visibility of the Lyα line at |$z$| > 6.

2.3 The redshift evolution of the EW-PDF with f_esc(z) and x_{H i}(z)

Following Dijkstra et al. (2011), we model the impact of the galactic outflow on the Lyα photons emerging from the galaxy using spherical shell models. For the analysis in this paper, we used shells with |$N_{\rm H\,\small {I}}=10^{20}$| cm⁻² and wind velocities of |$v$|_wind = 25 km s⁻¹. For a more detailed discussion on these models, the reader is referred to Dijkstra & Wyithe (2010) and Dijkstra et al. (2011). We simulate inhomogeneous reionization at |$z$| = 7 with the publicly available, semi-numerical code dexm (Mesinger & Furlanetto 2007; http://homepage.sns.it/mesinger/Sim). dexm combines excursion set and perturbation formalisms to generate various cosmic fields, and has been extensively tested against numerical simulations (Mesinger & Furlanetto 2007; Mesinger, Furlanetto & Cen 2011; Zahn et al. 2011). Our simulation box is 200 Mpc on a side with a resolution of 500³. We resolve haloes down to a minimum mass of |$M_{{\rm min}}\gtrsim 5\times 10^8\, {\rm M}_{\odot}$|, consistent with the expected cooling threshold at |$z$| ∼ 7 (Sobacchi & Mesinger 2013). The simulations present minor modifications of those used in Dijkstra et al. (2011, specifically the simulated redshift has been changed for the problem at hand, and the minimum mass has been increased by a factor of ∼5 to account for photoionization feedback). These modified simulations will be described in detail in a subsequent work, Mesinger et al., in preparation. We note that the new |$p(\mathcal {T}_{\rm IGM})$| look very similar to those we obtained with the original simulations that were used in Dijkstra et al. (2011).

3 RESULTS

3.1 Evolution in the Lyα fraction from redshift evolution in f_esc(z)

Fig. 1 shows cumulative Lyα EW-PDFs of drop-out galaxies with −20.25 < M_UV < −18.75 (left-hand panel) and −21.75 < M_UV < −20.25 (right-hand panel). The black squares (red filled circles) represent data points at |$z$| = 6 (|$z$| = 7) from the compilation by Ono et al. (2012). This figure also shows an example of a model in which the observed redshift evolution of the Lyα EW-PDF can be mimicked completely with a redshift-dependent f_esc(|$z$|).

Figure 1.

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This figure shows cumulative Lyα EW distributions for a sample of UV-faint (left-hand panel) and UV-bright (right-hand panel) drop-out selected galaxies. Black filled squares/red filled circles represent observations at |$z$| ∼ 6 /|$z$| ∼ 7 (taken from Ono et al. 2012). The black solid line represents a models at |$z$| = 6 assuming (i) that p₆(f_esc) is a Gaussian with σ = 0.3 and 〈f_esc〉 = 0.65, and that (ii) P₀(EW) ∝ exp [− EW/EW₀]. Here, we took EW₀ = 110 Å/EW₀ = 55 Å to match the UV-faint/UV-bright data. Red dotted lines represent predictions if we only modify P₇(EW) such that Δ〈f_esc〉 = 0.1 (i.e. 〈f_esc〉 = 0.75). This figure illustrates that it is possible to explain the observed drop in Lyα fraction completely with a modest increase in f_esc, the escape fraction of ionizing photons, provided that f_esc for LAE is already high at |$z$| ≈ 6.

Figure 2.

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This figure compares predicted Lyα LFs to observations at |$z$| = 5.7 (blue filled circles, taken from Ouchi et al. 2008), at 6.5 ( red filled squares, taken from Ouchi et al. 2010) and |$z$| = 7.0 (green diamonds, taken from Ota et al. 2010). The black solid line represents our model at |$z$| = 5.7 which uses the |$z$| = 6 EW-PDF shown as the black solid in the left-hand panel of Fig. 1. The blue and red dotted lines represent predicted LFs at |$z$| = 6.5 and 7.0 when we assume the same redshift evolution in f_esc as in Fig. 1. The blue and red dashed lines assume no evolution in the Lyα EW-PDFs (see the text). This figure shows that the model that reproduced the observed drop in the Lyα fraction (shown in Fig. 1), also naturally reproduces the observed evolution in the Lyα LFs of LAEs.

The black solid line in the left-hand panel shows a model EW-PDF at |$z$| = 6 which assumes (i) that p₆(f_esc) is Gaussian with a standard deviation σ = 0.3 and 〈f_esc〉 = 0.65, and that (ii) P₀(EW) ∝ exp [− EW/EW₀], where we took EW₀ = 110 Å to match the |$z$| = 6 data. These model parameters were chosen to quantitatively illustrate our main point.⁶ The black solid line shown in the right-hand panel represents the same model, but with EW₀ = 55 Å. This reduced value⁷ of EW₀ reflects that the Lyα fraction decreases towards brighter M_UV.

The red dotted lines represent a model in which we only modified P₇(EW) such that 〈f_{esc, 7}〉 = 0.75, i.e. Δ〈f_esc〉 = 0.1. This model can fully explain the observed reduction of the Lyα EW-PDF and the drop in the Lyα fraction.

3.2 Evolution in the LFs from redshift evolution in f_esc(z)

We also show the impact of a changing f_esc on the Lyα LF of Lyα emitters (LAEs). For this exercise, we follow the procedure of Dijkstra & Wyithe (2012), who constructed Lyα LFs by combining observed UV-LFs of drop-out selected galaxies with observed Lyα equivalent width distributions. Dijkstra & Wyithe (2012) found that this procedure reproduces observed LFs of LAEs well following inclusion of a rescaling by a factor of F = 0.5 (for a detailed discussion we refer the interested reader to Dijkstra & Wyithe 2012). Importantly, the redshift evolution of the LFs of LAEs was reproduced well at all redshifts |$z$| ≳ 3, which is important for the analysis presented here.

Fig. 2 shows the predicted Lyα LF at |$z$| = 5.7 as the black solid line, where we assumed that |$\frac{\rm{d}P}{\rm{d}{\rm EW}}(z=5.7)=\frac{\rm{d}P}{\rm{d}{\rm EW}}(z=6.0)$|, and we adopted the |$\frac{\rm{d}P}{\rm{d}{\rm EW}}(z=6.0)$| shown in the left-hand panel of Fig. 1 (i.e. the EW-PDF that described the UV-faint population of drop-out galaxies).⁸ The model LF fits the observations of Ouchi et al. (2008; blue filled circles) well. The red lines show the predicted LFs at |$z$| = 6.5. The dashed red line assumes that the EW-PDF at |$z$| = 6.5 is the same as at |$z$| = 6, while the dotted red line assumes that f_esc evolved from 〈f_esc〉 = 0.65 at |$z$| = 6 to 〈f_esc〉 = 0.70 at |$z$| = 6.5 (i.e. this corresponds to the model shown in Fig. 1 and we assumed that |$\frac{\rm{d}\langle f_{\rm esc}\rangle }{\rm{d}z}=0.1$| and 〈f_{esc, 6}〉 = 0.65). The data at |$z$| = 6.5 (red filled squares, taken from Ouchi et al. 2010) does not favour any of the models significantly. The blue lines show the predicted LFs at |$z$| = 7.0, where the dotted line shows a model in which the f_esc evolved to 〈f_{esc, 7}〉 = 0.75 at |$z$| = 7.0. This model lies much closer to the observations of Ota et al. (2010, green diamonds) than the model which keeps f_esc constant. Thus, evolution in the escape fraction of ionizing photons simultaneously explains the observed drop in the Lyα fraction between |$z$| = 6 and 7, and the observed evolution in the Lyα LFs within the same redshift range. More generally, our analysis shows that the drop in the Lyα fraction is quantitatively consistent with the observed evolution in the Lyα LFs of LAEs.

3.3 Evolution in the Lyα fraction from joint redshift evolution in f_esc(z) and x_{H i}

The previous section illustrated that a small evolution in f_esc from a value of 〈f_{esc, 6}〉 = 0.65 is sufficient to explain the observed decrease in the LAE LF and the fraction of strong Lyα emitting galaxies. However, this high escape fraction may be at odds with constraints obtained from the measurements of the Thomson optical depth to the cosmic microwave background, the photoionization rate of the Lyα forest, and the observed redshift evolution of the UV-LF of drop-out galaxies, which jointly favour values of f_{esc, 6} ∼ 10–20 per cent (Inoue et al. 2006; Wyithe et al. 2010; Kuhlen & Faucher-Giguère 2012; Becker & Bolton 2013; Robertson et al. 2013; also see Finkelstein et al. 2012b). While these estimates rely on uncertain extrapolations⁹ of the luminosity density in low-luminosity galaxies (and towards higher redshifts), the first direct constraints on f_esc in LAEs have recently been reported by Ono et al. (2010) who found f_{esc, 6} ≲ 0.6 (1 − σ).

Thus, the escape fraction evolution is unlikely to fully explain the observed drop in the Lyα fractions. We therefore examine the specific case in which f_esc evolves as f_esc(|$z$|) = f₀([1 + |$z$|]/5)^κ (as in Kuhlen & Faucher-Giguère 2012, also see Becker & Bolton 2013), and adopt κ = 4 and f₀ = 0.04 (which is consistent with observations, see Kuhlen & Faucher-Giguère 2012).¹⁰ For f₀ = 0.04, we have f_esc = 0.15 at |$z$| = 6 and f_esc = 0.26 at |$z$| = 7. While this model predicts that Δ〈f_esc〉 ∼ 0.1, it does not suffice to fully explain the observed evolution of the EW-PDF. This is due to the smaller value of 〈f_{esc, 6}〉 than what we assumed previously, which gives rise to a weaker impact on the redshift evolution of the EW-PDF (see Sections 1 and 2).

In Fig. 3, we show the cumulative Lyα EW distribution for UV-faint drop-out selected galaxies only (see Fig. 1 for a description of the lines and data points; the following results are quantitatively the same for the UV-bright galaxies). The black solid line now represents a model in which (i) P₆(f_esc) is a Gaussian with 〈f_{esc, 6}〉 = 0.15 and σ = 0.3, and (ii) P₀(EW) ∝ exp [− EW/EW₀], where EW₀ = 55 Å (note that it was¹¹ EW₀ = 110 Å in the model studied in Section 3 for the UV-faint sample). The red dotted lines show the predicted changes in the EW-PDF under the assumption that the drop is entirely due to a changing ionization state of the IGM (as in Dijkstra et al. 2011; Jensen et al. 2013).¹² The upper/lower red dotted line corresponds to |$\Delta x_{\rm H\,\small {I}}=0.2$|/|$\Delta x_{\rm H\,\small {I}}=0.5$|. The blue dashed line represents a model in which Δ〈f_esc〉 = 0.11 in addition to having |$\Delta x_{\rm H\,\small {I}}=0.2$|. This model is virtually indistinguishable from the model with |$\Delta x_{\rm H\,\small {I}}=0.5$| and no evolution in f_esc. Thus, extrapolation of the observed evolution in f_esc is equivalent to having an additional |$\Delta x_{\rm H\,\small {I}}=0.3$| between |$z$| = 6 and 7 in effecting the properties of Lyα flux and EW.

Figure 3.

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This figure shows the cumulative Lyα EW distribution for UV-faint drop-out selected galaxies (see Fig. 1 for a description of the lines and data points, and the text for details on the models). Here, the upper/lower red dotted line represents a model in which we modify the EW-PDF by having the IGM opacity increase due to an increase in the globally averaged neutral fraction, |$\Delta x_{\rm H\,\small {I}}=0.17$|/|$\Delta x_{\rm H\,\small {I}}=0.5$|. The dashed blue lines represents a model in which in addition f_esc evolves as f_esc(|$z$|) = 0.04([1 + |$z$|]/5)⁴ in addition to having |$\Delta x_{\rm H\,\small {I}}=0.17$|. This figure shows that mild evolution in both |$x_{\rm H\,\small {I}}$| and f_esc can mimic a more rapid evolution in |$x_{\rm H\,\small {I}}$| and thus explain the observed drop in Lyα fractions. The grey solid lines show predictions if we extrapolated the redshift evolution of f_esc to |$z$| = 8, while also changing the globally averaged neutral fraction to |$x_{\rm H\,\small {I}}=0.3$|. This prediction is still at odds with recently inferred fraction at |$z$| = 8 by Treu et al. (2013, represented by the upper limit at EW = 25 Å). The models can be made more consistent with this upper limit if we shift the predicted EW-PDFs by ΔEW= −25 Å (shown by the grey dotted line, see the text).

The grey solid line represents the model in which the redshift evolution in f_esc is extrapolated to |$z$| = 8. In this model, we additionally assume that the globally averaged neutral fraction, |$x_{\rm H\,\small {I}}$|, has evolved further to |$x_{\rm H\,\small {I}}=0.3$|. Fig. 3 shows that an evolving f_esc(|$z$|) has a dramatic impact on the predicted Lyα fraction at |$z$| = 8. However, even these models do not reproduce the recently inferred Lyα fraction at |$z$| = 8 by Treu et al. (2013; represented by the upper limit at EW=25 Å). This ‘failure’ can be partially remedied by requiring that |$x_{\rm H\,\small {I}} \gg 0.3$| at |$z$| = 8, which would again require a very rapid evolution in |$x_{\rm H\,\small {I}}$|. It may also be related to the fact that our models assume that all drop-out galaxies have a Lyα emission line with EW > 0, i.e. that P₀(EW > 0) = 1. While this assumption is consistent with observations of drop-out galaxies at |$z$| = 6, the observational uncertainties allow us to relax this assumption and shift the intrinsic distribution by, say, ΔEW=−25 Å, which would imply that P₀(EW > 0) < 1 as observed in |$z$| ∼ 3 drop-out galaxies (Shapley et al. 2003). While the escape fraction of Lyα photons increases with redshift, it is unclear whether P₀(EW > 0) = 1 at |$z$| ≥ 6. If we apply a shift of ΔEW=−25 Å to the predictions at |$z$| = 8, then our model predictions lie much closer to the upper limit at |$z$| = 8 (as shown by the grey dotted line).

4 DISCUSSION

4.1 A mass/luminosity-dependent f_esc(z)?

A caveat is that the observationally inferred redshift evolution of f_esc adopted in Section 3.3, refers to an average over the entire galaxy population. It has been argued that the inferred redshift evolution may be driven by a mass and/or luminosity dependence of the escape fraction (Alvarez et al. 2012; Ferrara & Loeb 2013). In this scenario, f_esc decreases towards higher masses and/or luminosities, and the population averaged escape fraction reflects the redshift evolution of the luminosity and/or mass functions. Support for the mass and/or luminosity dependence of the escape fraction is provided by the observationally inferred escape fraction of LAEs at |$z$| = 3 of |$f_{\rm esc}\sim 0.1{\rm -}0.3$|, which is significantly higher than the inferred fraction for the more massive LBGs where f_esc ∼ 0.05 (e.g. Iwata et al. 2009; Nestor et al. 2011).

This suggests that assuming a uniform f_esc(|$z$|) is not realistic. However, the higher inferred escape fraction of LAEs at |$z$| = 3 of |$f_{\rm esc}(z=3)\sim 0.1{\rm -}0.3$| may imply that we need a less rapid evolution in f_esc(|$z$|) to have a significant impact on the redshift evolution of the Lyα fraction: we showed that Δ〈f_esc〉 = 0.1 can either help explain the observed drop in the Lyα fraction between |$z$| = 6 and 7 if 〈f_{esc, 6}〉 = 0.15, or explain the complete evolution when 〈f_{esc, 6}〉 = 0.15. If f_esc(|$z$|) increases continuously between |$z$| = 3 and 6, then 〈f_{esc, 6}〉 may be large enough that only a small additional change may have a significant impact. As mentioned in Section 1, the decreasing covering factor of low-ionization absorption lines with redshift does suggest that escape fraction increases with redshift for a fixed galaxy population (Jones et al. 2012, 2013).

4.2 Lyα transport and escape, and a bimodal p(f_esc)

The observed Lyα flux of a galaxy not only depends on f_esc, but also on the effective escape fraction of Lyα photons from the galaxies to the observer, |$f^{\rm eff}_{\rm esc,Ly}$|. Thus, we have |$L_{\alpha } \propto f^{\rm eff}_{\rm esc,Ly}(1-f_{\rm esc})$|. The effective escape fraction |$f^{\rm eff}_{\rm esc,Ly}$| includes the fraction of Lyα photons that escape from the ISM of galaxies, but also the fraction that is subsequently transmitted through the IGM (e.g. Dijkstra & Jeeson-Daniel 2013). Both of these processes may depend on the value of f_esc, i.e. |$f^{\rm eff}_{\rm esc,Ly}= f^{\rm eff}_{\rm esc,Ly}(f_{\rm esc})$| (see discussion in Section 4.3).

The escape of Lyα photons from a dusty ISM, for example, is a complex process which depends on the dust content of the ISM, as well as its kinematics (e.g. Atek et al. 2008; Hayes et al. 2011). Irrespective of these complexities, the escape of ionizing photons requires low column density (|$N_{\rm H\,\small {I}} {<} 10^{17}$| cm⁻²) sightlines out of the galaxy. If these low H i-column density paths are surrounded by higher column density sightlines which are opaque to ionizing photons (as in the ‘blow-out’ model proposed by Nestor et al. 2011), then we may expect Lyα photons to scatter and preferentially escape along these same paths (see Behrens, Dijkstra & Niemeyer 2014 for a more detailed investigation of this effect). This can introduce a correlation between the escape fractions of Lyα and ionizing photons. In the most extreme case, individual galaxies would have a bimodal distribution for p(f_esc) which contains peaks at f_esc = 0 and 1. Interestingly, there is observational support for such a bimodality in observations of star forming galaxies, which indicate that a small fraction has a large f_esc, while a large fraction practically has f_esc = 0 (Shapley et al. 2006; Nestor et al. 2011; Vanzella et al. 2012). However, even in this scenario it is the escape fraction averaged¹³ over all sightlines, |$f^{\Omega }_{\rm esc}$|, that is relevant for powering nebular emission (and also for reionizing the Universe). The models described in this paper therefore also describe a scenario in which |$f^{\Omega }_{\rm esc}$| is distributed as a Gaussian.

It is nevertheless good to keep in mind that our scenario does not describe the more extreme situation in which |$f^{\Omega }_{\rm esc}$| has a bimodal distribution, i.e. it does not describe a scenario in which some galaxies have f_esc ≫ 0 while others have f_esc = 0 in all directions. In this scenario – which appears to be at odds with observed covering factors of low-ionization absorption line systems in drop-out galaxies, which are typically <1 (see e.g. Heckman et al. 2011; Jones et al. 2013) – an increase in f_esc(|$z$|) with |$z$| translates to an increase in the fraction of galaxies with f_esc ≫ 0. This would also reduce the fraction of star-forming galaxies that produce Lyα photons. However, the overall reduction in the number density of LAEs would be weaker than in our models.

4.3 Dependence of |$f^{\rm eff}_{\rm esc,Ly}$| on f_esc

The discussion above shows that the effective escape fraction of Lyα photons can depend on f_esc (and therefore possibly on M_UV as in Forero-Romero et al. 2012). If ionizing photons escape anisotropically, then so do Lyα photons. However, the first calculations of this effect have been reported only recently (Behrens et al. 2014). The precise correlation this may introduce between f_esc and |$f^{\rm eff}_{\rm esc,Ly}$| is complex (it depends e.g. on the geometry of the low column density holes, outflow properties), and has therefore not been quantified yet. If we assume for simplicity that |$f^{\rm eff}_{\rm esc,Ly} \propto f_{\rm esc}^y$|, then we expect the observed Lyα luminosity to increase with f_esc until |$f_{\rm esc,pk}=\frac{y}{y+1}$|, after which it decreases.

The correlation between |$f^{\rm eff}_{\rm esc,Ly}$| and f_esc likely originates mostly at the ISM-level: subsequent resonant scattering in the IGM occurs off residual H i, whose number density is affected by the value of f_esc (see e.g. Haardt & Madau 2012 and fig. 4 of Dijkstra, Lidz & Wyithe 2007a, for how this affects the IGM opacity). However, resonant scattering in the IGM depends sensitively on the assumed Lyα spectral line profile. Scattering off H i in galactic outflows typically redshifts the Lyα photons out of resonance as they escape from the galaxy, which strongly reduces the importance of resonant scattering in the IGM (see fig. 1 and the discussion in section 3.1 of Dijkstra et al. 2011). Ignoring the dependence of the IGM transmission on f_esc is therefore reasonable.

In short, ignoring a plausible correlation between |$f^{\rm eff}_{\rm esc,Ly}$| and f_esc is a shortcoming of the model that will need to be addressed in future work. Interestingly, it may help us explain why a monotonously evolving f_esc can give rise to a non-monotonic redshift dependence of the Lyα fraction, with a turn-over occurring near the redshifts of interest. For example, ff f_esc(|$z$| = 3) ∼ 1 per cent (consistent with upper limits by Vanzella et al. 2010) and f_esc(|$z$| = 6) ∼ 15 per cent (as in Kuhlen & Faucher-Giguère 2012), then the inferred |$f_{\rm esc,Ly}^{\rm eff}$| from Dijkstra & Jeeson-Daniel (2013) and Hayes et al. (2011), implies that |$y \sim 0.3{\rm -}0.5$|, which corresponds to |$f_{\rm esc,pk} \sim 0.2{\rm -}0.3$|. We would have a peak in Lyα luminosity near the f_esc and redshift of interest.

5 CONCLUSION

We have investigated whether the observed reduction in Lyα flux from galaxies at |$z$| > 6 can be explained by an evolving escape fraction of ionizing photons (f_esc). Our study was motivated by (i) a growing consensus in the literature that f_esc must have been higher at high redshift, and (ii) the fact that it appears difficult to explain the observed reduction in Lyα flux with reionization alone. We found that we can reproduce the median observed drop in the Lyα fraction, as well as the observed evolution of the LAE LFs, with a small increase in f_esc of Δf_esc ≲ 0.1, as long as the escape fraction is large (f_esc ∼ 0.65) at |$z$| ∼ 6. Models with redshift evolution of f_esc that are more consistent with indirect constraints derived from observations, combined with a small evolution in global neutral fraction of |$\Delta x_{\rm H\,\small {I}}\sim 0.2$| between |$z\sim 7\,{\rm and }\,6$| also suppress the |$z$| ∼ 7 Lyα flux at the observed level. Our study demonstrates that an evolving escape fraction of ionizing photons from galaxies modifies the observed equivalent widths of Lyα galaxies at a level comparable to that expected from reionization, and provides a plausible part of the explanation for evolution in the Lyα emission of high-redshift galaxies.

Finally, we expect the Lyα spectral line shape to evolve with redshift in models that invoke the IGM to explain the reduced Lyα flux from |$z$| > 6 galaxies (see fig. 5 of Dijkstra et al. 2007), while this is not obviously the case for models that invoke evolution of f_esc. Interestingly, Hu et al. (2010) have shown that the observed Lyα line shape of a stack of |$z$| ∼ 6.5 galaxies is practically identical to that at |$z$| = 5.7. We will investigate the implications of this result in future work.

This research was conducted by the Australian Research Council Centre of Excellence for All-Sky Astrophysics (CAASTRO), through project number CE 110001020. MD acknowledges financial support from DAAD. We thank an anonymous referee for constructive comments that improved the presentation of this work.

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