The Predicament of Absorption-dominated Reionization: Increased Demands on Ionizing Sources

The epoch of reionization is one of the landmark events in the early history of galaxy formation, during which hydrogen in the intergalactic medium (IGM)—which had been neutral since cosmological recombination at z ∼ 1100—was ionized by early galaxies. The measured transmission of Lyα photons through the IGM suggests that reionization was complete by z ∼ 5–6 (Bosman et al. 2018; Eilers et al. 2018; Yang et al. 2020a), but much is still unknown about the process.

The reionization history is fundamentally connected to the evolution of the sources and sinks of ionizing photons across cosmic time. The sources are thought to be star-forming galaxies (e.g., Robertson et al. 2015), whose populations have been cataloged via their (nonionizing) UV luminosity functions (LFs) up to z ∼ 10 (e.g., Bouwens et al. 2021), with accreting supermassive black holes playing a much smaller role (e.g., D'Aloisio et al. 2017). The sinks are thought to be dense clumps of IGM gas that can shield themselves from ionizing radiation and remain neutral (Miralda-Escudé et al. 2000) even after their volume has been reionized. These clumps manifest as Lyman-limit systems, and their cumulative opacity defines the "mean free path" λ of ionizing photons in the later universe (e.g., Songaila & Cowie 2010). The mean free path can be measured directly via the average attenuation observed in stacked quasar spectra (Prochaska et al. 2009; Worseck et al. 2014).

In most reionization models, the absorption of ionizing photons is assumed to be modest, requiring only ∼1–3 ionizing photons per baryon (e.g., Gnedin 2008; Finlator et al. 2012). In that case, the observed LF evolution is consistent with reionization completing at z ∼ 6, provided the escape fraction of ionizing photons, f_esc from these galaxies is somewhat larger (≳10%) (Robertson et al. 2015) than that measured directly in later galaxies (e.g., Pahl et al. 2021).

Recently, the mean free path at z ∼ 6 has been measured by Becker et al. (2021, henceforth B21) to be $\lambda ={0.75}_{-0.45}^{+0.65}$ proper Mpc (${5.25}_{-3.15}^{+4.55}$ comoving Mpc), far shorter than the extrapolation from previous measurements at z ≲ 5 using a similar method of stacking quasar spectra below the Lyman limit (Worseck et al. 2014). In this Letter, we investigate the implications of this short mean free path for the reionization epoch using the seminumerical method of Davies & Furlanetto (2021). We find that the short mean free path implies a dramatic increase in the number of emitted ionizing photons required to reionize the universe. This in turn requires a substantial increase in the ionizing efficiency of galaxies at z > 6 such that their intrinsic properties differ significantly from lower-redshift galaxies.

We adopt a Planck Collaboration et al. (2020) cosmology with (h, Ω_m , Ω_Λ, Ω_b , σ₈, n_s ) = (0.6736, 0.3153, 0.6847, 0.0493, 0.8111, 0.9649), and use comoving distances unless specified otherwise.

To simulate the effect of a short mean free path on reionization, we use the seminumerical method of Davies & Furlanetto (2021), consisting of a modified version of the 21cmFAST code (Mesinger et al. 2011). The simulations start with cosmological initial conditions (ICs) in a volume of size L_box with N_IC cells on a side. We then evolve the density field via the Zel'dovich approximation (Zel'dovich 1970) at a coarsened resolution with N_ev cells on a side. We employ the halo-filtering methodology of Mesinger & Furlanetto (2007) to identify dark matter halos and evolve their positions using the Zel'dovich displacement field. Our fiducial simulations adopt L_box = 256 Mpc, N_IC = 4096, and N_ev = 512, so that L_cell = 0.5 Mpc. We explore a range of L_box from 128 to 512 Mpc at fixed N_IC = 4096 and evolved L_cell = 0.5 Mpc. ⁴ To limit numerical noise in the halo distribution we adopt a minimum halo mass M_min corresponding to ∼100 mean density IC cells, so each L_box has a corresponding M_min.

We assume that the mass of material ionized by a halo, in the absence of absorption, is related to its mass via

$$\begin{eqnarray}&&{M}_{\mathrm{ion}}=\zeta {M}_{h},\end{eqnarray} \tag{ 1 }$$

This ζ is the product of several parameters of stars in early galaxies,

$$\begin{eqnarray}&&\zeta ={f}_{\mathrm{esc}}{f}_{* }{N}_{\gamma /b}^{* },\end{eqnarray} \tag{ 2 }$$

where f_* is the fraction of the halo's baryons that formed stars and ${N}_{\gamma /b}^{* }$ is the (integrated) number of ionizing photons emitted per stellar baryon (Furlanetto et al. 2004).

We employ two different prescriptions for ζ. The first assumes that ζ = ζ₀ is constant as a function of halo mass, i.e., M_ion ∝ M_h . This model is motivated by simplicity and by its use in many past works. Our fiducial prescription for ζ assumes a double power-law functional form,

$$\begin{eqnarray}&&\zeta ({M}_{h})={\zeta }_{0}\displaystyle \frac{{\left({M}_{0}/{M}_{\mathrm{peak}}\right)}^{-{\gamma }_{\mathrm{lo}}}+{\left({M}_{0}/{M}_{\mathrm{peak}}\right)}^{-{\gamma }_{\mathrm{hi}}}}{{\left({M}_{h}/{M}_{\mathrm{peak}}\right)}^{-{\gamma }_{\mathrm{lo}}}+{\left({M}_{h}/{M}_{\mathrm{peak}}\right)}^{-{\gamma }_{\mathrm{hi}}}},\end{eqnarray} \tag{ 3 }$$

where ζ₀ is a normalization factor applied at a halo mass M₀, γ_lo = 0.49 and γ_hi = −0.61 are effective power-law indices at low and high mass, respectively, and M_peak = 2.8 × 10¹¹ M_⊙ is a transition mass between the two. This form and the corresponding parameter values were chosen by Mirocha et al. (2017) to characterize the relationship between star formation and halo accretion required to reproduce observed UV LFs at z = 6–8 (see also Mirocha 2020). Note that we apply this functional form to the ratio between the total number of stars formed and the halo mass but in practice the shapes of the two relations should be very similar (Furlanetto et al. 2017). The shape can be interpreted as accounting for a suppression of star formation at low halo masses due to supernova feedback, and at high halo masses due slow gas cooling or AGN feedback (Furlanetto et al. 2017), and is similar to the outcome of feedback models used in other seminumerical reionization simulations (e.g., Hutter et al. 2021).

We use the "MFP-(r)" approach ⁵ from Davies & Furlanetto (2021) to compute the ionization topology in the presence of absorption. The standard framework for seminumerical reionization simulations relies on a photon-counting argument, whereby a region is ionized if the number of ionizing photons produced inside it exceeds the number of hydrogen atoms. This argument is typically applied to a single cell at the center of the region (e.g., Zahn et al. 2011). Davies & Furlanetto (2021) showed that this central cell calculation is equivalent to integrating the ionizing flux, enabling distance-dependent absorption to be efficiently included. In the MFP-(r) approach, the ionization criterion on scale R is given by

$$\begin{eqnarray}&&{\int }_{0}^{\infty }\langle \zeta {f}_{\mathrm{coll}}{\rm{\Delta }}{\rangle }_{r}W(r;R,\lambda )4\pi {r}^{2}{dr}\gt \langle {\rm{\Delta }}(\lt R)\rangle ,\end{eqnarray} \tag{ 4 }$$

where Δ is the overdensity relative to the cosmic mean, f_coll is the fraction of matter in halos above the minimum mass for galaxies to form, 〈ζ f_collΔ〉_r is averaged within the spherical shell at radius r, and W(r; R, λ) is a spherical top-hat filter of size R in real space with an exponential attenuation factor e^−r/λ to account for absorption in the ionized gas. In Figure 1, we show an example of the evolved density field, halo distribution, and neutral fraction field of our fiducial simulation.

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Finally, we note that seminumerical simulations that use a real-space filter to compute the ionization topology are known to suffer from photon nonconservation (e.g., Zahn et al. 2007) at the ≲20% level, peaking when ionized bubbles first overlap and weakening toward the later stages of reionization (Hutter 2018). We expect a similar effect to be present in our adopted method, but we note that the uncertainty in the observed mean free path is far larger than 20% and so defer a quantitative comparison to radiative transfer simulations to future work.

We aim to constrain the number of ionizing photons that must have been emitted during reionization given the short z ∼ 6 mean free path measured by B21. We make the simplifying assumption that the mean free path prior to z = 6 is constant; in reality, it is likely shorter at earlier times, requiring even more ionizing photons to complete the process. We then require that the IGM neutral fraction at z ∼ 6 be consistent with the Lyα dark pixel fraction measurement of McGreer et al. (2015), who found a volume-averaged neutral fraction 〈x_HI〉_V (z = 5.9) < 0.11 at 84% confidence. We thus tune ζ₀ for each simulation to match 〈x_HI〉_V (z = 5.9) = 0.11, and also explore variations of this value. The resulting mass-averaged IGM neutral fraction in the simulations is 〈x_HI〉_M ∼ 6%. In this work we conservatively consider reionization to be "completed" at this point—however, vestiges of incomplete reionization may be detectable at later times in the Lyα forest (Kulkarni et al. 2019; Keating et al. 2020; Nasir & D'Aloisio 2020, Bosman et al. 2021) or in future 21 cm measurements (Raste et al. 2021).

We quantify the ionizing photon budget by the cumulative number of ionizing photons per baryon,

$$\begin{eqnarray}&&{n}_{\gamma /b}=\displaystyle \frac{{n}_{\mathrm{ion}}}{{\bar{n}}_{b}}=\langle \zeta {f}_{\mathrm{coll}}\rangle ,\end{eqnarray} \tag{ 5 }$$

where n_ion is the number density of all ionizing photons ever emitted by galaxies and ${\bar{n}}_{b}$ is the mean baryon density. The product 〈ζ f_coll〉 represents the average over any halo mass dependence of ζ. Our fiducial simulation with L_box = 256 Mpc, ${M}_{\min }={10}^{9}\,{M}_{\odot }$, 〈x_HI〉_V (z = 5.9) = 0.11, and the double power-law ζ model gives ${n}_{\gamma /b}={6.1}_{-2.4}^{+11}$, with uncertainty from the B21 mean free path measurement alone. The central value is ∼2–6 times larger than typical estimates from earlier models (e.g., Gnedin 2008; Finlator et al. 2012).

In Figure 2 we show how n_γ/b varies across the models in our simulation suite. The left panel shows n_γ/b as a function of M_min for the constant ζ models (open symbols) and fiducial double power-law ζ models (solid symbols) for different box sizes; in the smaller (larger) simulations we can resolve less (more) massive halos. Emphasizing lower mass halos, i.e., by decreasing M_min or treating ζ as constant, generally requires fewer photons per baryon to complete reionization, because low-mass galaxies are less biased and therefore spatially closer to the voids that are ionized during the later phases of reionization. Their proximity decreases the attenuation from the short mean free path. In contrast, if more massive halos produced most of the ionizing photons (e.g., Naidu et al. 2020), the required number of photons would increase. Meanwhile, the large-scale voids corresponding to the final ∼10% ionization of the universe represent rare, large, underdense structures, so the required photon budget could depend on features on the largest scales. Convergence is achieved for L_box ≳ 200 Mpc, motivating our focus on a fiducial L_box = 256 Mpc simulation suite.

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The middle panel of Figure 2 shows the dependence of n_γ/b on the mean free path. This trend is the dominant one in our simulations, reflecting the importance of the mean free path for the end stages of reionization. The right panel shows the dependence of n_γ/b on the assumed IGM neutral fraction at z = 5.9. If the neutral fraction at z = 5.9 lies significantly below the McGreer et al. (2015) upper limit of 0.11, the required number of photons would increase substantially.

While the cumulative number of photons per baryon n_γ/b (and its corresponding ζ prescription) provides a useful cosmological benchmark for reionization studies, it is a time-integrated quantity, which is not trivially related to the galaxy population at any given time. Here we translate our constraint on n_γ/b to more familiar galaxy properties.

First, we estimate the ionizing emissivity ${\dot{n}}_{\mathrm{ion}}$ from our simulations via differentiating Equation (1),

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{ion}}}{{dt}}=\displaystyle \frac{d}{{dt}}(\zeta {M}_{h})=\zeta \displaystyle \frac{{{dM}}_{h}}{{dt}}+{M}_{h}\displaystyle \frac{d\zeta }{{{dM}}_{h}}\displaystyle \frac{{{dM}}_{h}}{{dt}},\end{eqnarray} \tag{ 6 }$$

where d ζ/dM_h is obtained by differentiating Equation (3). We then sum the contribution from every halo i in the simulation and divide by volume to recover ${\dot{n}}_{\mathrm{ion}}$,

$$\begin{eqnarray}&&{\dot{n}}_{\mathrm{ion}}=\displaystyle \frac{1}{{L}_{\mathrm{box}}^{3}}\sum _{i}\displaystyle \frac{({{\rm{\Omega }}}_{b}/{{\rm{\Omega }}}_{m}){\dot{M}}_{\mathrm{ion},i}}{\mu {m}_{p}},\end{eqnarray} \tag{ 7 }$$

where Ω_b /Ω_m is the cosmic baryon fraction and μ m_p is the average baryon mass. We estimate the halo accretion rate dM_h /dt via the abundance matching-like procedure described in Furlanetto et al. (2017), wherein dark matter halos grow roughly exponentially in agreement with cosmological simulations (Dekel et al. 2013).

In Figure 3 we show the resulting ${\dot{n}}_{\mathrm{ion}}$ at z = 5.9 as a function of mean free path. Our fiducial model (${M}_{\min }={10}^{9}\,{M}_{\odot }$, 〈x_HI〉_V (z = 5.9) = 0.11) has ${\dot{n}}_{\mathrm{ion}}={3.4}_{-1.3}^{+6.1}\times {10}^{51}$ photons s⁻¹ Mpc⁻³, where the uncertainties reflect the 1σ range of λ. The ionizing emissivity can also be estimated independently by requiring that the photoionization rate in the IGM, ${{\rm{\Gamma }}}_{\mathrm{HI}}\propto {\dot{n}}_{\mathrm{ion}}\lambda$, reproduces the observed mean transmission in the Lyα forest. We show ${\dot{n}}_{\mathrm{ion}}$ determined this way by B21 as the brown dotted–dashed curve in Figure 3. The two ${\dot{n}}_{\mathrm{ion}}$ estimates agree remarkably well across the 1σ range of the B21 mean free path measurement, although this agreement may be somewhat accidental, as our halo-galaxy mapping is approximate and the UV background measurement may be somewhat biased by fluctuations present at z ∼ 6 (Davies & Furlanetto 2016; Davies et al. 2018a; Becker et al. 2018).

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The ionizing emissivity is typically parameterized by

$$\begin{eqnarray}&&{\dot{n}}_{\mathrm{ion}}={f}_{\mathrm{esc}}{\xi }_{\mathrm{ion}}{\rho }_{\mathrm{UV}},\end{eqnarray} \tag{ 8 }$$

where ξ_ion is the "ionizing efficiency" of the stellar populations, dependent on their metallicity and star formation history, and ρ_UV is the integrated UV luminosity density. From ${\dot{n}}_{\mathrm{ion}}$ we can thus estimate the product f_esc ξ_ion via observational constraints on ρ_UV at z ∼ 6. Adopting the best-fit star formation efficiency as a function of halo mass from Mirocha et al. (2017), we estimate that our fiducial M_min = 10⁹ M_⊙ corresponds to M_UV ∼ − 11. We then obtain ρ_UV = 3.2 × 10²⁶ erg/s/Hz/Mpc³ by integrating the z ∼ 6 UV LF from Bouwens et al. (2021) down to M_UV = − 11. The solid curve in the right panel of Figure 4 shows the resulting f_esc ξ_ion from our fiducial model. Taking into account the mean free path uncertainty alone, we estimate ${\mathrm{log}}_{10}{f}_{\mathrm{esc}}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}={25.02}_{-0.21}^{+0.45}$.

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Assuming f_esc = 0.1, this corresponds to an ionizing efficiency of ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}={26.02}_{-0.21}^{+0.45}$, considerably larger than previous measurements of ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}=25.24-25.36$ for 3.8 < z < 5 galaxies (Bouwens et al. 2016), ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}=25.48\pm 0.06$ for z ∼ 4.9 Lyα-emitting galaxies (LAEs; Harikane et al. 2018), and even rare, extremely blue galaxies with ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}\sim 25.8$ (Bouwens et al. 2016; see also Stark et al. 2017). The left panel of Figure 4 shows that maintaining consistency with either the canonical ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}\sim 25.2$ or the measured value of 25.5 at z ∼ 5 requires f_esc ≳ 30% or ≳20%, respectively. Note that, as mentioned above, these constraints reflect contributions from halos down to ${M}_{\min }={10}^{9}\,{M}_{\odot }$, corresponding to M_UV < −11. The middle panel of Figure 4 shows that increasing the minimum luminosity of contributing galaxies, where M_UV = −13 (−15) corresponds to ${M}_{\min }={10}^{9.5}({10}^{10}){M}_{\odot }$, further increases the required f_esc ξ_ion.

Our values of f_esc ξ_ion are consistent with Meyer et al. (2019, 2020; see also Kakiichi et al. 2018) who measured the ionizing production of z ≳ 5 galaxies via their correlation with Lyα transmission in background quasar spectra. In the right panel of Figure 4, we compare f_esc ξ_ion in our model to values of f_esc ξ_ion values measured from three independent populations of galaxies: Lyman-break galaxies (LBGs), LAEs, and the hosts of C iv absorption systems. We note, however, that the measurements from LAEs and LBGs rely nontrivially on much larger assumed mean free paths at z ∼ 6.

Finally, we demonstrate the consistency of our results with measurements of the stellar mass function. We can estimate ξ_ion directly from ζ using f_*(M_h ) from Mirocha et al. (2017) and estimating ${N}_{\gamma /b}^{* }$ using the integral constraint,

$$\begin{eqnarray}&&{N}_{\gamma /b}^{* }=\displaystyle \frac{{{ \mathcal E }}_{\mathrm{tot}}{| }_{z\geqslant 6}}{{M}_{* ,\mathrm{tot}}{| }_{z=6}}{\xi }_{\mathrm{ion}},\end{eqnarray} \tag{ 9 }$$

where ${{ \mathcal E }}_{\mathrm{tot}}{| }_{z\geqslant 6}$ is the energy density in UV photons emitted before z = 6, i.e.,

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{tot}}{| }_{z\geqslant 6}={\int }_{t(z=6)}{\int }_{{L}_{\min }}L{\rm{\Phi }}(L,t){dLdt}\end{eqnarray} \tag{ 10 }$$

and M_*,tot∣_z=6 is the stellar mass density at z = 6. Using the best-fit UV LFs from Bouwens et al. (2021) at z = 5.9, 6.8, 7.9, and 8.9 and Oesch et al. (2018) at z = 10.2, we obtain ${{ \mathcal E }}_{\mathrm{tot}}{| }_{z\geqslant 6}=3.9\times {10}^{43}$ erg Hz⁻¹ Mpc⁻³, integrating down to our nominal halo mass threshold of 10⁹ M_⊙. Dividing our star formation efficiency model by a factor of 1 + γ_lo to approximately recover the stellar mass–halo mass relation (Furlanetto et al. 2017), this mass threshold corresponds to a minimum stellar mass of ${M}_{* ,\min }\sim 6.6\times {10}^{5}\,{M}_{\odot }$, implying a mass-to-light ratio of ∼ 0.12M_⊙/L_⊙ at ${M}_{* ,\min }$. This mass-to-light ratio is consistent with measurements in (much brighter) galaxies at z ≥ 6 (e.g., McLure et al. 2011). Equations (2) and (9) then require M_*,tot∣_z=6 = 8.8 × 10⁶ M_⊙ Mpc⁻³ to match our constraints on f_esc ξ_ion, which falls squarely between faint-end extrapolations of the stellar mass functions of Bhatawdekar et al. (2019) (M_*,tot∣_z=6 = 1.1 × 10⁷ M_⊙Mpc⁻³) and Song et al. (2016) (M_*,tot∣_z=6 = 4.6 × 10⁶ M_⊙Mpc⁻³). Thus our constraints are not in tension with extrapolations of the z ≥ 6 luminosity and stellar mass functions.

We therefore find that reionization-epoch galaxies must be producing or leaking ionizing photons with efficiencies a factor of 2–3 times higher than even the most optimistic observational estimates to match the 〈x_HI〉_V constraint at z = 5.9. The tension could be eased if the UV LF is much steeper at z ∼ 6, e.g., with a faint-end slope at the 2σ limit from Bouwens et al. (2021) the required f_esc ξ_ion would decrease by a factor of 2. A strongly declining f_esc with halo mass could also decrease the required photon budget; the constant ζ model, implying ${f}_{\mathrm{esc}}\propto {M}_{h}^{-0.49}$, decreases n_γ/b by ∼30%. Such mass dependence in f_esc would be fully degenerate with any mass dependence of ξ_ion. Cain et al. (2021) have similarly explored the ionizing budget requirements of reionization with a short mean free path using radiative transfer simulations that include a simulation-calibrated subgrid model for small-scale gas clumping. Their fiducial simulation assumes that every halo has the same ionizing luminosity, placing even more emphasis on the faintest galaxies. Their ionizing photon budget at z = 5.9 of n_γ/b ∼ 2.2 is nevertheless within ∼10% of our constant ζ model when run with values of 〈x_HI〉_V = 0.2 and λ = 10 Mpc similar to their model (at the ∼ +2σ and ∼ +1σ of current observations, respectively), which gives n_γ/b ≈ 2.3.

In this work, we have explored the consequences of a short mean free path of ionizing photons (as measured by B21) on the ionizing photon budget of the reionization epoch. Using the new seminumerical method of Davies & Furlanetto (2021), we found that their mean free path of ${0.75}_{-0.45}^{+0.65}$ proper Mpc implies a cumulative ionizing photon budget of ${6.1}_{-2.4}^{+11}$ photons per baryon to reproduce the 1σ upper limit on the z = 5.9 IGM neutral fraction of 〈x_HI〉_V < 0.11 from McGreer et al. (2015). The large number of required photons is linked to the distance that the photons must travel from sources to the last remaining neutral voids.

We translated our constraint on the cumulative number of photons per baryon into the instantaneous emission properties of reionization-epoch galaxies, finding an implied ionizing production rate of ${\mathrm{log}}_{10}{f}_{\mathrm{esc}}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}={25.02}_{-0.21}^{+0.45}$. Even assuming that reionization-epoch galaxies are substantially more efficient at producing ionizing photons than their lower-redshift counterparts (${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}=25.5$), and integrating the UV luminosity function down to extremely faint magnitudes (M_UV ≤ 11), we conservatively require an average f_esc > 20% from z > 6 galaxies, more than three times larger than measurements in z = 3 galaxies (f_esc ∼ 6%; Pahl et al. 2021). A more conventional production rate of ${\mathrm{log}}_{10}{\xi }_{\mathrm{ion}}/{(\mathrm{erg}/\mathrm{Hz})}^{-1}=25.2$ requires an even more extreme f_esc > 30%. Qualitatively, a "photon budget crisis" occurs because the B21 mean free path measurement at z ∼ 6 implies that ionizing photon sinks are far more numerous than previously assumed.

Our models are nevertheless consistent with available reionization history constraints, as shown in Figure 5, and with measurements of the UV LF and stellar mass density (see Section 4). Astrophysical constraints on the ionized fraction at z ∼ 7 (Figure 5, right) may point toward somewhat more rapid reionization, but our models have more gradual histories because IGM absorption becomes increasingly important later in the process.

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Future observations will help resolve, or further exacerbate, this ionizing photon budget deficit. Tighter constraints on the mean free path at z ≳ 5.5, with improved quasar bias modeling and a more precise accounting of cosmic variance from a larger sample, would most directly improve the constraints. Improved upper limits on the dark pixel fraction at z ∼ 6 would provide a more robust limit on the IGM neutral fraction. Deeper observations of high-redshift galaxies with JWST will deliver precise UV LFs to fainter magnitudes, potentially locating a "turnover" corresponding to the onset of star formation in low-mass halos. The depth and wavelength coverage of JWST spectroscopy will enable direct measurements of ξ_ion (and possibly f_esc) from nebular lines, and test for consistency with our models.

We thank Joseph Hennawi and the ENIGMA group at UCSB/Leiden for access to the computing resources used in this work.

S.E.I.B. acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 740246 "Cosmic Gas").

This work was supported by the National Science Foundation through award AST-1812458. In addition, this work was directly supported by the NASA Solar System Exploration Research Virtual Institute cooperative agreement number 80ARC017M0006. We also acknowledge a NASA contract supporting the "WFIRST Extragalactic Potential Observations (EXPO) Science Investigation Team" (15-WFIRST15-0004), administered by GSFC. This material is based upon work supported by the National Science Foundation under grant Nos. 1636646 and 1836019, the Gordon and Betty Moore Foundation, and institutional support from the HERA collaboration partners.