On the contribution of dwarf galaxies to reionization of the Universe

We present estimates of the ultraviolet (UV) and Lyman continuum flux density contributed by galaxies of luminosities from $M_{\rm UV}\approx -25$ to $M_{\rm UV}=-4$ at redshifts $5\leq z\leq 10$ using a galaxy formation model that reproduces properties of local dwarf galaxies down to the luminosities of the ultra-faint satellites. We characterize the UV luminosity function (LF) of galaxies and their abundance as a function of the ionizing photon emission rate predicted by our model and present accurate fitting functions describing them. Although the slope of the LF becomes gradually shallower with decreasing luminosity due to feedback-driven outflows, the UV LF predicted by the model remains quite steep at the luminosities $M_{\rm UV}\lesssim -14$. After reionization, the UV LF flattens at $M_{\rm UV}\gtrsim -12$ due to UV heating of intergalactic gas. However, before reionization, the slope of the LF remains steep and approximately constant from $M_{\rm UV}\approx -14$ to $M_{\rm UV}=-4$. We show that for a constant ionizing photon escape fraction the contribution of faint galaxies with $M_{\rm UV}>-14$ to the UV flux and ionizing photon budget is $\approx 40-60\%$ at $z>7$ and decreases to $\approx 20\%$ at $z=6$. Before reionization, even ultra-faint galaxies of $M_{\rm UV}>-10$ contribute $\approx 10-25\%$ of ionizing photons. If the escape fraction increases strongly for fainter galaxies, the contribution of $M_{\rm UV}>-14$ galaxies before reionization increases to $\approx 60-75\%$. Our results imply that dwarf galaxies fainter than $M_{\rm UV}=-14$, beyond the James Webb Space Telescope limit, contribute significantly to the UV flux density and ionizing photon budget before reionization alleviating requirements on the escape fraction of Lyman continuum photons.

1. INTRODUCTION Cosmic reionization of hydrogen was the second major phase transition experienced by the Universe after it became neutral during the epoch of recombination.Modeling details of the reionization process and understanding the main sources of ionizing Lyman continuum (LyC) photons remains an area of active research and debate (see, e.g., Robertson 2022;Gnedin & Madau 2022, for reviews).
The contribution of galaxies of different luminosities to the hydrogen reionization is still debated.In particular, the contribution of galaxies with UV absolute magnitudes at  = 1500 Å of  UV > −14 to the UV flux and ionizing photon budget is largely unconstrained by observations and can only be estimated using models.However, theoretical predictions for the UV luminosity function (LF) at these faint magnitudes span a wide range (see, e.g., Figures 12 and 13 in Bouwens et al. 2022).Some models predict significant flattening of the UV LF or turnover at  1500 ≳ −12 to −14 (e.g., O'Shea et al. 2015; ★ wuz25@uchicago.edu,† kravtsov@uchicago.eduGnedin 2016;Ceverino et al. 2017;Kannan et al. 2022), while others predict a relatively steep LF down to fainter magnitudes (e.g., Yue et al. 2016).
In this study, we aim to predict the evolution of the faint end of the UV LF at  UV > −14 before and during reionization.We employ a new method to sample the evolution of galaxies over the entire relevant range of luminosities, from the brightest observed galaxies down to progenitors of the ultrafaint dwarf galaxies observed around the Milky Way.Galaxy properties are computed using the galaxy formation model of Kravtsov & Manwadkar (2022), which was demonstrated to reproduce properties of  ≲  ★ galaxies down to ultrafaint dwarf luminosities at  = 0 (Kravtsov & Manwadkar 2022;Manwadkar & Kravtsov 2022;Kravtsov & Wu 2023).To account for the stochasticity of star formation rate (SFR) observed in local dwarf galaxies and galaxies at  > 5, we add a modest level of SFR stochasticity using the method outlined in Pan & Kravtsov (2023).We use the star formation and metallicity evolution of model galaxies to compute their AB  = 1500 Å luminosity and the Lyman continuum photon emission rate using stellar population synthesis, and we predict the UV LF and the LyC photon emission density as a function of galaxy luminosity.We use these functions to estimate the relative contribution of galaxies of  UV > −13 to the UV and ionizing photon flux at  > 5.
The paper is organized as follows.We describe the galaxy formation model and the method used to estimate properties of the galaxy population across a full range of galaxy luminosities in Section 2. We present our main results in Section 3, compare our results and conclusions to previous studies, and discuss the predicted evolution of the UV and ionizing flux density in Section 4. Our results and conclusions are summarized in Section 5. We provide best-fit values for the predicted number density of ionizing photons produced by galaxies in a given bin of  UV in the Appendix A.
Throughout this paper, we assume flat Λ+Cold Dark Matter (ΛCDM) cosmology with the mean density of matter in units of the critical density of Ω m = 0.32, the mean density of baryons of Ω b = 0.045, Hubble constant of  0 = 67.11km s −1 Mpc −1 , the amplitude of fluctuations within the tophat spheres of  = 8ℎ −1 Mpc of  8 = 0.82, and the primordial slope of the power spectrum of  s = 0.95.Halo virial masses throughout this study are defined within the radius enclosing density contrast of 200 relative to the critical density at the corresponding redshift.

MODELING HIGH-𝑍 GALAXY FORMATION
The galaxy formation framework we use in this study is applied to predict galaxy population properties for representative samples of model galaxies at all relevant luminosities down to the UV absolute magnitudes of  1500 ≈ −4 and redshifts  ∈ [5,10].This is done using samples of halos that follow the expected halo mass function at each considered redshift and halo mass evolution tracks constructed using an accurate approximation for the halo mass accretion rate, as described in Kravtsov & Belokurov (2024).
The key aspect of the galaxy formation model we use in this study at  ≥ 5 is that it reproduces observed properties of ≲  ★ galaxies at  = 0 down to the faintest ultra-faint dwarf galaxies (Kravtsov & Manwadkar 2022;Manwadkar & Kravtsov 2022;Kravtsov & Wu 2023).This agreement is not a guarantee that the model would work at high redshifts.Nevertheless, most galaxies at  > 5 have dwarf halo virial masses ( ≲ 10 11  ⊙ ) and  = 0 give us more confidence that results may be realistic.As we show below, the same model that reproduces the properties of  = 0 dwarf galaxies also reproduces the observed bright end of the UV luminosity function at 5 ≤  ≤ 10.
We briefly outline the main elements of the model relevant to our analysis in Section 2.2 below.We first describe the modeling of mass assembly histories of halos that host model galaxies, which is the backbone of galaxy formation modeling.

Halo evolution model
To model the evolution of halos over the entire range of galaxy luminosities, we first construct large samples of model halos using the following approach (Kravtsov & Belokurov 2024).We use an accurate cubic spline approximation of the cumulative halo mass function computed using Tinker et al. (2008) approximation, and use the inverse transform sampling method to generate a random sample of halo masses in a given volume.We use a two-pronged approach to efficiently sample halos hosting galaxies over a very broad range of luminosities.First, we construct halo samples in a series of boxes of different halo masses.When we construct the overall UV or ionizing radiation luminosity function of galaxies, we generate the samples in individual boxes and then stitch the luminosity functions in their ranges of overlap.
Second, we select a random fraction of halos for modeling, given as a function of halo mass  200c using: where  ≈ 10 −5 − 5 × 10 −6 and  ≈ 1.5 − 2.35 provide sufficiently large samples of model galaxies to reliably measure their luminosity functions.When the luminosity function is computed at a given , each model galaxy is weighted by  −1 ( 200c ).This approach allows us to keep the number of model galaxies reasonably small, while sufficiently sampling galaxies of different luminosities.
Once a halo sample is drawn at a given redshift,  f , we use each halo mass as a starting point and construct a halo mass evolution track by integrating the equation of halo mass evolution  200c = ( 200c ) back in time to  init = 25 using an accurate approximation for the average halo mass accretion rate  200c of halos of a given mass , derived using analyses of the mass evolution histories of halos formed in cosmological ΛCDM simulations (see Appendix in Kravtsov & Belokurov 2024, for tests of of the approximation at  > 5): where halo mass  200c is in  ⊙ and time  is in Gyrs.The integrated  200c () is then used to model galaxy evolution from  init to a given  f .

Galaxy formation model
The specific implementation of the GRUMPY galaxy formation model (Kravtsov & Manwadkar 2022) we use is outlined in Kravtsov & Belokurov (2024).Briefly, the model solves a system of differential equations that describe the evolution of gas mass, stellar mass, size, and stellar and gas-phase metallicities.It also includes galactic outflows, a model for the gaseous disk and its size, molecular hydrogen mass, star formation, and effects of UV heating during and after reionization on accretion of gas on small-mass halos.
The GRUMPY model assumes that at all times the ISM follows the exponential radial gas profile Σ  (), with a half-mass radius proportional to the parent halo virial radius.The model uses molecular gas mass as a proxy for dense cold star-forming gas mass  sf (a fraction of the total gas mass in the galaxy).The molecular fraction  H 2 of the total ISM gas mass is estimated using the model of Gnedin & Draine (2014).
The star-forming gas mass,  sf =  H 2  g is converted into stars on a constant depletion time scale  sf , such that the star formation rate (SFR) is  ★ = (1 − )  sf / sf , where  is the fraction of gas returned to the interstellar medium in the instantaneous recycling approximation.We use the fiducial value of  sf = 2 Gyr typical of nearby galaxies (e.g., Bigiel et al. 2008).The model includes outflows of the ISM gas that are assumed to be proportional to the mean SFR,  out =    sf / sf with the mass-loading factor   dependent on the current stellar mass of the galaxy in a way expected in the energy-driven wind models (see Manwadkar & Kravtsov 2022).
Note that we assume that scaling of the mass-loading factor   with stellar mass is the same at different redshifts, as indicated by the results of the FIRE-2 simulations (Muratov et al. 2015;Anglés-Alcázar et al. 2017).Note, however, that the star formation is not assumed to be the same at all redshifts, as the amount of star-forming gas in the model depends on the gas metallicity and UV field and both are different at high z and this is taken into account.We do assume that depletion time of gas stays constant, while it may be redshift dependent.We find, however, that effects of changing depletion time with redshift within reasonably limits is quite small at z<10, as was also found in cosmological simulations (e.g., Schaye et al. 2010).
We do not take into account the effects of mergers on the stellar populations of model galaxies.In the dwarf galaxy regime mergers have a negligible effect on the stellar masses of galaxies (Fitts et al. 2018) due to the steep  halo −  ★ relation.Although major mergers can change the stellar mass and size of dwarf galaxies (Rey et al. 2019;Tarumi et al. 2021), such mergers are quite rare and do not affect average scaling relations.Note also that the model for the galaxy half-mass radius provides a good match to the distribution of half-mass radii of observed dwarf galaxies at  = 0.The gas accretion onto galaxies is modulated by the mass and redshift dependent factor accounting for the UV heating effect of intergalactic gas, as described in Kravtsov & Manwadkar (2022).To illustrate the effects of this heating on the UV and ionization luminosity functions of dwarf galaxies in our analysis, we will consider models with reionization redshifts  rei = 6 and  rei = 8.5.The former is close to the redshift at which our Universe was reionized (see Gnedin & Madau 2022;Robertson 2022, for reviews) and is our fiducial value.The  rei = 8.5 value is used to illustrate how the evolution of the luminosity functions changes if a given region of the Universe is reionized earlier.
For each halo track produced as described above, the galaxy formation model is integrated from  init = 25 to the final redshift  f = 5, 6, 7, 8, 9, 10, producing evolution of stellar mass, star formation rate, etc.The basic model described above, however, uses mean mass assembly history for a halo of a given mass at  f , while real halos of a given will exhibit a scatter in their assembly histories.In addition, the model assumes the same star formation depletion time for all galaxies, while observational estimates of  sf vary significantly from galaxy to galaxy.In addition, the model does not include modeling of the processes that can result in a significant SFR stochasticity, such as the formation and destruction of individual star-forming regions (e.g., Tacchella et al. 2020;Iyer et al. 2020;Sugimura et al. 2024).
To account for these different sources of scatter in SFR in a controlled manner, we add stochasticity to the mean SFR computed by the model using the method described in Pan & Kravtsov (2023).Namely, the mean SFR in the model at a time   is perturbed as  ★,stoch =  ★ × 10 Δ , where Δ is a correlated random number drawn from the Gaussian pdf with zero mean and unit variance, and multiplied by √︁ () = √︁ PSD(   )/, where wavenumber  corresponding to frequency   is defined as  =    and where  is the duration of galaxy evolution track.
We follow Caplar & Tacchella (2019) and use the PSD of the form PSD(  ) =  2 Δ [1+ ( break  )  ] −1 , where  Δ characterizes the amplitude of the SFR variability over long time scales and  break characterizes the timescale over which the random numbers are effectively uncorrelated.Parameter  controls the slope of the PSD at high frequencies (short time scales).In our models, we fix the slope  and  break to the values  = 2 and  break = 100 Myr, which are physically motivated by the time scales of gas evolution and star formation in giant molecular clouds in a typical ISM (see Tacchella et al. 2020, for a detailed discussion), as well as  Δ = 0.1 consistent with the typical amount of SFR stochasticity in host halos of observed galaxies at  ≈ 5−10.The corresponding scatter in the UV absolute magnitude is  M UV ≈ 0.75 (see Kravtsov & Belokurov 2024, for  -UV LFs at redshifts  = 5 and 8 over the entire range of galaxy luminosities from  UV ≈ −25 to  UV = −4, assuming that reionization ends at  rei = 6.At each redshift, the two curves illustrate the effects of dust ( = 5; solid line is our model without accounting for dust, while dot-dashed line is the LF after applying dust extinction) and the effect of adding a small level of SFR stochasticity to the model results ( = 8; the dashed curve shows the model without SFR stochasticity; solid line is the LF in the model with stochasticity).The LFs are shifted by a factor 10 6− for clarity.

Computing UV and ionizing radiation luminosities
The monochromatic luminosity of model galaxies at  = 1500 Å is computed using a tabulated grid of luminosities,  1500 , for stellar populations of a given age and metallicity using the Flexible Stellar Population Synthesis model v3.0 (FSPS, Conroy et al. 2009;Conroy & Gunn 2010b) and its Python bindings, Python-FSPS.The table is then used to construct an accurate bivariate spline approximation to compute  1500 for stellar populations of a given age and metallicity.We use the table and finely spaced time outputs of the model to compute the integral  1500 due to all stars formed by the current time taking into account the evolution of stellar mass and stellar metallicity.
The emission rate of the Lyman continuum  < 912 Å photons,  ion , is sensitive to the effects of binary stars (while flux at  = 1500 Å is not), which are not included in the FSPS.We thus computed  ion self-consistently using the tables of ionizing flux for single-age stellar populations from the BPASS version 2.3 package (Byrne et al. 2022), which take into account effect of binary stars, and the evolution of stellar mass and metallicity computed by the galaxy formation model.
To account for dust effects, which are expected to affect the brightest galaxies in the UV at  ≲ 7, we use a secondorder polynomial approximation to the simulation results of Lewis et al. (2023, see their Fig. 7):  1500 = −0.071500, + 0.05 2 1500, , where  1500, = 2 + 10.53 log 10 (0.043/ gas ) and  gas is metallicity of gas in solar units, which we assume to be  ⊙ = 0.015.
The luminosities  1500 and  ion for each model galaxy are computed at the final redshifts  f = 5, 6, 7, 8, 9, 10.To con-struct the corresponding luminosity functions at each  f , we construct a weighted histogram of luminosities using weights  −1 ( 200c ) −3 box .As noted above, we first construct luminosity functions in individual volumes of a series of increasing  box and then stitch the luminosity functions in the regions of overlap, so that the combined LF spans the full range of galaxy luminosities down to  1500 = −4.
Our reasons for choosing the lower luminosity limit of  UV = −4 are twofold.First, in our model, this luminosity corresponds to a stellar mass of ≈ 300  ⊙ .For this and smaller masses, stochastic sampling effects of the initial mass function are expected to be significant (e.g., da Silva et al. 2012), which the feedback prescription in our model does not account for.Feedback should be smaller when fewer massive stars per unit stellar mass are formed, and the lack of such stars will also suppress UV luminosity.Second, the halo mass corresponding to  UV = −4 at  < 10 is  200c ≈ 5 × 10 6  ⊙ , which is close to the minimum halo mass of ≈ 2 − 5 × 10 6 ,  ⊙ that can accrete gas at  < 15.Gas accretion is expected to be suppressed in halos of smaller mass due to the bulk motion of baryons relative to dark matter and due to expected radiative heating (see, e.g., Fig. 9 in Nebrin et al. 2023).
The exact mass threshold for galaxy formation depends on the specific value of the relative streaming velocity of baryons relative to dark matter and the evolution of the Lyman-Werner UV background.The galaxy formation model we use does not account for these effects and thus cannot reliably model properties of  UV > −4 galaxies.However, even if we use the slope of UV LF estimated at  UV = −4 and extrapolate it to  UV = −1, which corresponds to  ★ ≈ 10 − 50  ⊙ and should have greatly suppressed  1500 / ★ due to deficiency of massive stars, we estimate that the contribution of galaxies with  UV > −4 should be no larger than ≈ 10%.
The UV luminosity functions produced with this method over a range of absolute magnitudes −25 ≲  1500 ≲ −4 at redshifts  = 5 and  = 8 is illustrated in Figure 1.For  = 5 we show two LF curves computed with and without accounting for the effects of dust discussed above.Dust primaril y reduces luminosities of bright galaxies of  1500 ≲ −19 and its effects become small at all luminosities at  > 7. The two curves at  = 8 show model UV LFs for the base model without SFR stochasticity and the model in which a small amount of SFR stochasticity with  Δ = 0.1 (required to match observed UV LF at these redshifts; see Kravtsov & Belokurov 2024) is added.This level of SFR stochasticity modifies the shape of the bright end of UV LF at  UV ≲ −17.
As the figure shows, the effects of dust and adding SFR stochasticity partly offset each other.Based on the uncertainty and the relatively small influence the dust effects have on our conclusions, we do not include them in our calculations of the relative contribution of galaxies of different luminosities to the total UV and ionizing flux of galaxies.Given that the effect of dust is larger for brighter galaxies, this implies that we underestimate the relative contribution of dwarf galaxies to the UV flux, making our estimates of their contribution a conservative lower limit.However, below we provide fits to the UV LFs and ionizing flux functions in our model and these can be used to recompute the fractional contribution of dwarf galaxies to the UV and Lyman-continuum photon budgets for a specific model of dust attenuation.-Rest-frame UV luminosity function of galaxies at  ∈ [5, 10] in the model with the end of reionization at  rei = 6 and a small amount of SFR stochasticity of  Δ = 0.1 (see Section 2.2).Effects of dust are not included in the model LFs.The LFs at different redshifts are displaced vertically by a factor of 10 6− for clarity.The different symbols show observational estimates of the UV LF in recent studies that used HST and JWST observations (Bouwens et al. 2021(Bouwens et al. , 2022(Bouwens et al. , 2023;;Donnan et al. 2023;Harikane et al. 2023Harikane et al. , 2024;;Pérez-González et al. 2023).Note that before reionization (i.e. ≳ 6), the slope of the LF even at the faintest magnitudes remains as steep as that of  1500 ≈ −14.Bottom panel: log 10 of model and observation, shown on the same magnitude range.
Figure 2 shows the UV ( = 1500 Å) luminosity function of galaxies at  = 5, 6, 7, 8, 9, 10 in the model with the end of reionization at  rei = 6 and a small amount of SFR stochasticity of  Δ = 0.1 (see Section 2.2).It also shows observational estimates of the UV LF in recent studies that used HST and JWST observations.The model matches these quite well at the range of luminosities probed by observations.Differences at  ≤ 7 and  1500 ≲ −20 are likely due to dust effects that have a similar magnitude at these luminosities and redshifts (see Fig. 1).
Agreement with UV LF estimates at  ∈ [5, 10] at  1500 < −14 and the fact that the model reproduces properties of  = 0 dwarf galaxies well (Kravtsov & Manwadkar 2022), including the luminosity function of Milky Way satellites down to the ultra-faint magnitudes (Manwadkar & Kravtsov 2022), means we can plausibly expect that the model UV LF can faithfully describe the evolution of the luminosity function at fainter magnitudes.
This statement is supported by the comparison with the reconstruction of the UV LF of the progenitors of the Local Group dwarf galaxies at  ≈ 7 of Boylan-Kolchin et al. (2015) shown in Figure 3.This LF reconstruction was done using observational estimates of the star formation histories of dwarf galaxies measured from their color-magnitude diagrams (Weisz et al. 2014(Weisz et al. , 2019  the flattening of the slope at  1500 is required by the observed abundance of Local Group dwarf galaxies with such faint estimated  = 7 absolute magnitudes. Figure 3 shows the UV LF of our model galaxies for the models with the end of reionization at  rei = 6 and  rei = 8.5.The  rei = 8.5 model is in good agreement with the faintend UV LF deduced by Boylan-Kolchin et al. (2015), because by  = 7 galaxies of  1500 ≳ −13 become affected by the UV heating, while in the  rei = 6 this occurs only at  ≲ 6.This also agrees with the analysis of Manwadkar & Kravtsov (2022), which used the same model as in our analysis to show that the  = 0 luminosity function of the Milky Way satellites favors reionization at  rei ≈ 8 − 9.
Note that the Lagrangian volume which collapsed into the volume containing nearby dwarf galaxies is generally expected to reionize at  ≳ 7 (Zhu et al. 2019;Ocvirk et al. 2020;Trac et al. 2022) -earlier than the overall reionization of the Universe, which occurred at  rei ≈ 6 (Gnedin & Madau 2022;Robertson 2022).Thus, the flattening of the UV LF at  1500 ≳ −13 exhibited by the local dwarf galaxies does not imply that the mean UV LF of  = 7 galaxies in the Universe should have a similar flattening.
Before reionization, UV LFs shown in Figure 2 become gradually shallower with decreasing luminosity (increasing  UV ) down to  UV ≈ −14, while at fainter magnitudes the slope stays approximately constant down to  UV = −4.The value of the slope  in / ∝   at these faint luminosities is  ≈ −1.7, which is quite steep.
The flattening at brighter magnitudes and the constant slope at fainter ones is a result of the specific feedback-driven outflow prescription adopted in the model which was tested and calibrated using the mass-metallicity relation of local dwarf galaxies and luminosity function of the Milky Way satellites.In what follows, we will present analytical fits to the luminosity functions predicted in our model both for the UV luminosity  1).The two panels show fits to the LFs in models with  rei = 6.0 and  rei = 8.5, respectively.The faint-end behavior, including the steep slope before reionization and the flattening in post-reionization redshifts  <  rei , are accurately reflected by the modified Schechter fit.
at  = 1500 Å and for the emission rate of ionizing photons.

Modified Schechter function fit
We approximate model UV luminosity functions with the modified Schechter functional form of Jaacks et al. (2013): where  ★ and  ★ are the normalization and characteristic luminosity of the bright end, respectively.Compared to the Schechter form, which has a fixed faint-end slope , this form has a slope that can become progressively shallower or steeper around   and reaches the asymptotic slope of −  at  ≪   .We determine the best-fit parameters of the function by minimizing the least-squares differences between the func- where  UV is the luminosity density at  = 1500 Å in egs s −1 Hz −1 .The best-fit parameters for different redshifts in models with  rei = 6 and  rei = 8.5 are presented in Table 1 and the fits are compared to the computed model UV LFs in Figure 4.The figure shows that in both  rei models the functional form provides an excellent description of the model luminosity functions at all  and over the entire range of luminosities.
Figure 4 also illustrates the effect of  rei : the  rei = 8.5 model shows significant flattening at the faint end for redshifts  ≲ 8.5 due to suppression of accretion caused by the UV heating of the intergalactic medium during and after reionization.The flattening is also reflected in the lower  −  values at these redshifts in Table 1.In the  rei = 6 model, such flattening also occurs after reionization and is apparent only in the  = 5 LF.

Fraction of UV emission from galaxies of different
1500 We use the functional LF fits of Equation 3 to calculate the fraction of UV luminosity per unit volume emitted by galaxies brighter than a given  UV ≤ −4: where normalization is Figure 5 shows  (<  UV ) for both  rei = 6 and  rei = 8.5 models for  = 5, 6, 7, 8, 9, 10.For both choices of  rei , we see a significant contribution of UV flux from dwarf galaxies with  1500 > −14.The fraction of the UV flux they contribute is ≈ 60% at  = 10 and declines with decreasing  to ≈ 10% at  = 5.Such decrease is due to 1) changing shape of the bright end of UV LF which becomes shallower with decreasing  due to the continuing buildup of massive halos and galaxies and 2) flattening of the UV LF at  1500 ≳ −13 due to the UV heating after reionization (see discussion in Section 3.2).We can see the effect of the latter in the rapid steepening of  UV at  = 8 compared to  = 9 for the model with  rei = 8.5 shown in the bottom panel of Figure 5.The UV flux contribution of  1500 > −14 galaxies in the model with  rei = 8.5 right before reionization is ≳ 50 − 60%, while in the  rei = 6 model it is ≳ 20 − 30%.The contribution of dwarf galaxies to the global UV flux is thus higher at higher , but is still substantial even at  ≈ 6 − 7 in the  rei = 6 case.
Note that due to the reionization-induced flattening of UV LF at  ≳  rei for  UV ≲ −13, there is no divergence of the integral of the LF as it is integrated to lower luminosities, as occurs for the Schechter form that approximates UV LF of bright galaxies at  UV < −13.In our model LF a sharp flattening or cutoff occurs only at  UV > −4.Before reionization,  UV continues to increase with decreasing luminosity down to  UV = −4 due to the constant and relatively steep slope of the predicted faint end function.However, at  UV > −4 galaxies in our model have stellar masses  ★ ≲ 200  ⊙ and form in halos of ≲ 5 × 10 6  ⊙ .Halos with virial mass of  200c < 2 × 10 6  ⊙ do not form stars, as expected in models of gas cooling (Nebrin et al. 2023), which provides a natural LF cutoff and prevents divergence of  UV at  UV < −4.Note also that Nebrin et al. (2023) show that at  < 8 the galaxies in halos with masses  200c < 10 8  ⊙ , which corresponds to  UV ≳ −8 in our model, cannot accrete gas due to radiative heating.This does not necessarily mean that UV LF flattens strongly at these faint luminosities immediately at  < 8 as the galaxies can continue to form stars using the gas accreted at earlier epochs.The effect of such gas suppression will be felt at  < 7 when effects of UV heating start to affect UV LF anyway.

Ionizing emission fraction from galaxies of different
1500 Results presented above show that dwarf galaxies with luminosities beyond the reach of current observations contribute significantly to the total UV emission of galaxies.These galaxies can thus contribute substantially to the reionization of hydrogen in the Universe.However, far UV luminosity at  = 1500 Å is only a rough proxy of the ionizing radiation produced by galaxies, and the ratio between ionizing luminosity and  1500 can vary significantly as a function of the IMF, star formation history, metallicity, and binary fraction (e.g., Stanway et al. 2016).
The conversion from  1500 to the ionizing photon emission rate is usually parameterized using  ion =  ion / 1500 where  1500 is the luminosity per unit frequency in units of ergs s −1 Hz −1 , and  ion is the number of hydrogen-ionizing photons emitted per second.Although  ion is often considered a constant, it is expected to depend on various factors and can thus vary from galaxy to galaxy as well as with time.
Here, instead of adopting a given value of  ion , we compute the emission rate of ionizing photons,  ion , for each model galaxy using its star formation and metal enrichment history, as described in Section 2.3.We find that the ratio  ion / 1500 of model galaxies is approximately independent of galaxy luminosity, but there is a substantial scatter due to assumed SFR stochasticity.
In addition to computing  ion , we still need to make assumptions about the value of the escape fraction of ionizing photons from galaxies (  esc ) and its scaling with galaxy luminosity.Observational measurements are challenging but have been done for some local galaxies, indicating values of ≲ 10% (e.g., Vanzella et al. 2010;Guaita et al. 2016;Rutkowski et al. 2016;Grazian et al. 2016;Sandberg et al. 2015;Vanzella et al. 2010;Vasei et al. 2016;Flury et al. 2022).At higher redshifts, escape fractions are found to increase for galaxies with bluer spectra and lower mass (e.g., Chisholm et al. 2022;Saldana-Lopez et al. 2023) and given that galaxy spectra become bluer on average with increasing redshift (Topping et al. 2022;Cullen et al. 2023), this implies an increase of  esc with increasing redshift, reaching values of ≈ 5 − 30% at  > 6 (e.g., Lin et al. 2024;Saxena et al. 2024).
On the theoretical side, results of numerical simulations vary significantly from finding a clear trend of increasing  esc with decreasing galaxy luminosity (e.g., Wise et al. 2014;Kimm et al. 2017;Anderson et al. 2017) to the opposite trend (Sharma et al. 2016) or  esc peaking at intermediate masses (e.g., Rosdahl et al. 2022).This is not surprising, given that escape fraction depends on the specific exact timing of stellar ionizing and mechanical feedback, the structure of the ISM on a wide range of scale, and other factors (e.g., Gnedin et al. 2008;Kimm & Cen 2014) and thus is extremely sensitive to implementations of star formation, feedback, and numerical resolution of simulations.
Given this uncertainty, we will adopt two models for  esc that should reasonably bracket the possible trends.In the first model, we assume a constant  esc = 5% at all redshifts and luminosities.In the second model, we adopt a strong dependence of  esc on galaxy luminosity for galaxies with −21 <  UV < −15:  eff esc =  esc ( UV = −21) × 10 0.62(  UV +21) , given by the respective evolution of  esc and  ion found in simulations Anderson et al. (2017) and Simmonds et al. (2024);  esc at  UV > −15 is kept constant at the value given by the above expression at  UV = −15.The latter is done both to avoid extrapolating simulation and observational results, and because large  esc at lower luminosities likely leads to reionization that is too early (see, e.g., Lin et al. 2024).This toy model illustrates how different the results would be in the case of a strong increase of  esc with decreasing luminosity.It is worth noting though, that the luminosity dependence in this model is likely too strong, as it assumes that both  esc and  ion increase with decreasing luminosity, while observations indicate that  esc and  ion anti-correlate in high- galaxies (Saxena et al. 2024) such that their product does not depend strongly on luminosity.As such, this is an overly optimistic model on the contribution of dwarf galaxies to the ionizing photon budget.
For a given  esc model we order galaxies by their  UV at a given redshift and compute the cumulative  esc -weighted sum of the ionizing photon emission rate contributed by galaxies brighter than a given  UV .Figure 6 shows the fraction of the ionized flux density contributed by galaxies brighter than a given  UV computed in this way for the two models of  esc shown in the two panels.For a constant  esc , results are similar to those of  UV : the contribution of faint galaxies with  UV > −14 is ≈ 40 − 60%, which decreases to ≈ 20% at  = 6.Before reionization at  ≳  rei even ultra-faint galaxies of  UV > −10 contribute ≈ 10−25% of ionizing photons.For the model that assumes increasing  esc for fainter galaxies, the contribution of  UV > −14 at  ≥  rei increases to ≈ 60−75% while  UV > −10 galaxies still contribute ≈ 15 − 30% before reionization.
4. DISCUSSION The results presented in the previous section show that dwarf galaxies beyond the range of luminosities probed by HST and JWST can contribute substantially to the reionization of the Universe.This is consistent with recent observational results that indicate significant ionizing emissivities of high- galaxies (Simmonds et al. 2024) and JWST observations of individual dwarf galaxies strongly magnified by cluster lensing (Atek et al. 2024).The contribution is smaller than would be estimated from extrapolating the UV LF estimated for galaxies with  UV ≲ −16, because the slope of the UV LF is z rei = 8.5 Fig. 6.-The fraction of Lyman continuum flux density contributed by galaxies brighter than a given  UV .Left panels: the models with  rei = 6 (upper panel) and 8.5 (lower panel) across  ∈ [5, 10] assuming constant escape fraction of ionizing photons.Right panels: ionizing photon fraction in the same models, but assuming a strongly luminosity-dependent escape fraction increasing towards fainter galaxies down to  UV = −15 (see Section 3.4 for details).The shaded region shows  UV > −14.The figure shows that  UV > −14 galaxies contribute up to 60 ∼ 80% at  > 7 with even ultra-faint galaxies contributing ≈ 10 − 20%.
predicted to slowly decrease with decreasing luminosity until  UV ≈ −14.At fainter luminosities, the slope of our model LFs is approximately constant all the way to  UV ≈ −4.This means that if the UV LF can be characterized down to  UV ≈ −14 with JWST observations, extrapolation of the LF using slope estimated at the faint end of the measured LF should be accurate.
The significant contribution of dwarf galaxies to the ionizing photon budget implies that reionization can be achieved with escape fractions of ionizing radiation lower by a factor of up to two than is assumed when the contribution of dwarf galaxies is not taken into account (Finkelstein et al. 2019).If the  esc increases strongly for fainter galaxies, the implied values of escape fraction can be too high to be consistent with existing observational constraints on the ionization history of our Universe (Lin et al. 2024;Muñoz et al. 2024) or with observational estimates of  esc in galaxies (see discussion in Muñoz et al. 2024).We note that the models that violate observational constraints likely assume  esc dependence on galaxy luminosity and/or redshifts that are too strong.For example, as noted above, observations indicate that  esc and  ion anti-correlate in high- galaxies (Saxena et al. 2024) such that their product does not depend strongly on luminosity.Thus, models that assume that both  esc and  ion increase with decreasing galaxy luminosity independently are likely to have unrealistically high ionizing emissivity for dwarf galaxies.

Faint-end UV LF slope
Several theoretical studies considered model predictions for the faint end of the UV luminosity function with rather diverse results.There is a significant difference between model predictions even at  1500 ≲ −13 (see, e.g., Figures 12 and 13 in Bouwens et al. 2022).In general, many simulations predict significant flattening or even a turnover of the UV luminosity functions at  1500 ≳ −14 (e.g., O'Shea et al. 2015;Gnedin 2016;Ceverino et al. 2017;Kannan et al. 2022).Among semianalytic models, some models predict flattening and turnover at  1500 < −13 (Hutter et al. 2021) due to effects of reionization heating, while others predict a relatively steep LF down to fainter magnitudes (e.g., Yue et al. 2016).
UV LF at faint magnitudes can be affected by stellar feedback-driven outflows and UV heating due to reionization.Our model incorporates a well-motivated outflow model, with which it reproduces the luminosity function of MW dwarf satellites and many properties of local dwarf galaxies, including their metallicity-stellar mass, gas mass-stellar mass, and Tully-Fisher relations.The outflows in our model result in a gradual decrease of UV LF slope with decreasing luminosity for  UV < −14.As noted above, before reionization the LF slope at  UV > −14 stays approximately the same and corresponds to / ∝   where  ≈ −1.7, which is quite steep.The model presented here also takes into account the effects of reionization on the gas accretion onto dwarf-mass halos, and it shows that such heating does flatten the UV LF at  UV ≳ −13 but only at redshifts smaller than the reionization redshift  <  rei .
Another process that can affect star formation in small-mass halos is the suppression of gas accretion due to relative velocities of baryons and dark matter (Tseliakhovich & Hirata 2010).Williams et al. (2024) evaluated the effect of such motions on the UV LF in their simulations, and found that the relative motions lead to a turnover in the  = 12 UV LF at  UV ≳ −13.This would make the contribution of galaxies at fainter luminosities to the ionizing photon budget negligible -contrary to our conclusions.However, in their model, this luminosity corresponds to the stellar mass of  ★ ≈ 10 5  ⊙ and halo mass of  200 ≈ 10 6  ⊙ below the minimal halo mass of  200c ≈ 2 × 10 6  ⊙ that can accrete gas and form stars (Nebrin et al. 2023).
In our model, however, galaxies of  ★ = 10 5  ⊙ and  UV ≈ −12 occupy halos of mass  200 ≈ 3 − 10 × 10 8  ⊙ , or more than two order of magnitude larger.This is consistent with the results of several high-resolution simulations of high- galaxies shown in Figure 13 of Côté et al. (2018).Halos of  200 = 10 6  ⊙ , on the other hand, do not form stars and thus do not contribute to UV LF at all in our model.We also note that our model reproduces the UV LFs over a broad range of luminosities and redshifts, as well as various properties of dwarf galaxy population at  = 0 (Kravtsov & Manwadkar 2022;Kravtsov & Wu 2023), including the luminosity function of the Milky Way satellite galaxies down to the faintest luminosities (Manwadkar & Kravtsov 2022).We thus believe it is likely that in our model, galaxies of a given  ★ would form in halos of correct halo mass.Given that relative baryon and dark matter motions only affect halos of  200 ≲ 10 6  ⊙ , in the context of our model these motions would not affect the UV LF at all at the relevant range of luminosities.

UV luminosity density evolution
The UV flux density  UV represents the global probe of how star formation rate density evolves in the Universe, and is the object of many observational and theoretical estimates.We can compute this density using Equation 5 but with  norm = 1. Figure 7 shows our  UV values for redshifts  ∈ [5, 10], integrated from the brightest end to three different UV absolute magnitude limits of −17, −13, and −4.
The three limiting UV magnitudes are significant in differ- ent ways: −17 is the lowest luminosity of many pre-JWST LF measurements, and is used as a comparison with our model.Figure 7 shows that  UV () evolution in our model is in reasonable agreement with observational estimates of the UV density measured by integrating UV LF down to  UV = −17 (Harikane et al. 2023;Bouwens et al. 2020;Bhatawdekar et al. 2019;McLeod et al. 2016;Finkelstein et al. 2015).
The limiting magnitude of −13 is an optimistic limit that should be reachable for strongly lensed galaxies observed with JWST.As expected, the corresponding  UV values are significantly larger.Lastly, we show  UV in our model if we integrate UV LF to  UV = −4 to include UV flux contribution from all dwarf galaxies.The direct contribution of dwarf galaxies to the UV flux density is reflected in the gap between the blue −4 and green −13 lines.The contribution of dwarf galaxies is significant at higher  and decreases with decreasing redshift.This contribution thus flattens the  UV () trend produced by dwarf galaxies.

Ionizing photon emission rate evolution
The most direct probe of the ionizing photon budget is the ionizing photon emission rate per comoving volume  ion .We obtain this value at each redshift  ∈ [5, 10] by repeating the calculation in the previous section using Equation 5and  norm = 1, but replacing UV density with the ionizing photon emission rate of each galaxy.Figure 8 shows the evolution of  ion with redshift assuming  esc = 0.05 for the  rei = 6 model.As a comparison, we plotted the theoretical estimates of  ion required to maintain hydrogen reionization at each redshift, given by Equation 26of Madau et al. (1999) for three values of the clumping factor  bracketing the range of values measured in cosmological simulations of high- IGM (e.g., Gnedin & Ostriker 1997;Madau et al. 1999;Pawlik et al. 2015).We also added constraints on the ionizing photon emissivity in the IGM, derived from the Gunn-Peterson optical depth measured at 5 <  < 6 in the SDSS quasar spectra (Fan et al. 2006;Madau 2017).
Finally, the right axis of Figure 8 shows a ballpark estimate  of the number of ionizing photons in the IGM per hydrogen atom corresponding to a given  ion , calculated as  ion / nH  H,6 where nH is the mean comoving density of hydrogen atoms and  H,6 = 1/ ( = 6) is the Hubble time at  = 6 (Mason et al. 2019).
Figure 8 shows that for  esc = 0.05 and for typical expected clumping factor  ≈ 3, hydrogen ionization can be maintained for  ≲ 6.5, which is in qualitative agreement with the current estimates of when reionization occurred (e.g., Gnedin & Madau 2022).This agreement is approximate and a proper comparison requires model calculations following the ionized hydrogen fraction, which is beyond the scope of this study.We note that in addition to the uncertainty of the escape fraction and clumping factor, there are additional uncertainties related to the absorption of ionized photons by the Lyman limit systems (e.g., Kohler & Gnedin 2007;Furlanetto & Mesinger 2009;McQuinn et al. 2011;Altay et al. 2011;Fan et al. 2024;Georgiev et al. 2024).
Figure 8 also shows that the contribution of the galaxies with  UV > −13 will not change the reionization redshift significantly for this case of constant  esc .However, their contribution is larger at higher , such that galaxies of these luminosities can contribute significantly to the formation and growth of high- ionized bubbles, and increase the total optical depth of ionized gas.This contribution also reduces the required  esc values for bright galaxies to reionize the Universe at  ≈ 6.
We also note that if we use the magnitude-dependent  eff esc expression given in Section 3.4 and Figure 6 (right panels) instead of  esc = 0.05, the resulting  ion from  UV < −4 galaxies is nearly constant at log 10  ion ≈ 50.7, and is within the gray band at all redshifts.It intersects the  = 3 line at  ≈ 8.This indicates that for such a strongly mass-dependent escape fraction, the Universe may be reionized too early, in agreement with conclusions of Muñoz et al. (2024).On the other hand, the  ion () evolution in the model of Kulkarni et al. (2019,  5. SUMMARY AND CONCLUSIONS We presented model calculations of the monochromatic ( = 1500 Å) UV luminosity function and Lyman continuum photon flux density function for galaxies at redshifts  ∈ [5, 10] over the entire luminosity range from  UV ≈ −25 to  UV = −4.The calculation uses a galaxy formation model shown to reproduce properties of local dwarf galaxies down to the luminosities of the ultra-faint satellites.We focus particularly on the contribution of dwarf galaxies with luminosities  UV > −13 outside the reach of direct observations.Our main results and conclusions are as follows: 1. We characterize the shape of the UV LF predicted by our model over a broad range of absolute magnitudes from  UV ≈ −25 to  UV = −4 using a novel method to model the abundance of halos and galaxies of a broad range of mass and luminosity (Section 3.1).We show that model UV LF can be well described by the modified Schechter form of Jaacks et al. (2013) at all explored redshifts  ∈ [5,10].We present the best-fit parameters of this functional form for the model LFs at  = 5, 6, 7, 8, 9, 10 (Table 1).
2. Although the slope of the LFs becomes gradually shallower with decreasing luminosity at  UV ≲ −14, the UV LF predicted by the model is quite steep at the luminosities beyond the observational limit.After the assumed end of reionization, the UV LF flattens at  1500 ≳ −13 from suppression of gas accretion and star formation in small-mass halos, due to UV heating of intergalactic gas during and after reionization.However, before reionization, the faint end of the LF has slopes at the faintest luminosities as steep as the slope at  1500 ≈ −14.Galaxies fainter than  UV = −13 thus contribute significantly to the UV flux density before reionization at  > 6 (Figures 5 and 7).
3. We also compute the ionizing flux of model galaxies brighter than a given absolute magnitude  UV and show that it can be well described by the same Jaacks et al. (2013) form.We present the best-fit parameters of this form and an approximation for their evolution with redshift (Appendix A).
4. Dwarf galaxies beyond the range of luminosities probed by HST and JWST can contribute substantially to the reionization of the Universe.For a constant  esc the contribution of faint galaxies with  UV > −14 to the ionizing photon budget is ≈ 40 − 60% at  > 7, which decreases to ≈ 20% at  = 6.Before reionization, even ultra-faint galaxies of  UV > −10 contribute ≈ 10 − 25% of ionizing photons.For the model that assumes a strongly increasing  esc for fainter galaxies, the contribution of  UV > −14 at  ≥  rei increases to ≈ 60 − 75% while  UV > −10 galaxies contribute ≈ 15 − 30%.
Our results show that dwarf galaxies play an important role in reionizing the Universe, and may thus significantly aid the formation and growth of ionizing bubbles at  > 7.This is in agreement with recent observational estimates of the ionizing flux contributed by dwarf galaxies.Further observational studies using the increasing volume of JWST observations at  > 6 should improve our understanding of escape fractions and their trends with galaxy properties at high redshifts.
On the theoretical side, future studies should improve our understanding of the absorption and recombination of photons in the IGM, resulting in a better understanding of the required ionizing photon budget.A natural next step is to improve the treatment of the escape fraction and model the evolution of neutral hydrogen fraction (e.g., Greig et al. 2017;Bolan et al. 2022) to constrain and test the model.Observational and theoretical progress in these areas should refine our knowledge of the contribution of dwarf galaxies to the evolution of IGM at  > 5.  3.
Fig. 1.-UV LFs at redshifts  = 5 and 8 over the entire range of galaxy luminosities from  UV ≈ −25 to  UV = −4, assuming that reionization ends at  rei = 6.At each redshift, the two curves illustrate the effects of dust ( = 5; solid line is our model without accounting for dust, while dot-dashed line is the LF after applying dust extinction) and the effect of adding a small level of SFR stochasticity to the model results ( = 8; the dashed curve shows the model without SFR stochasticity; solid line is the LF in the model with stochasticity).The LFs are shifted by a factor 10 6− for clarity.
Fig.2.-Rest-frame UV luminosity function of galaxies at  ∈[5, 10]  in the model with the end of reionization at  rei = 6 and a small amount of SFR stochasticity of  Δ = 0.1 (see Section 2.2).Effects of dust are not included in the model LFs.The LFs at different redshifts are displaced vertically by a factor of 10 6− for clarity.The different symbols show observational estimates of the UV LF in recent studies that used HST and JWST observations(Bouwens et al. 2021(Bouwens et al. , 2022(Bouwens et al. , 2023;;Donnan et al. 2023;Harikane et al. 2023Harikane et al. , 2024;;Pérez-González et al. 2023).Note that before reionization (i.e. ≳ 6), the slope of the LF even at the faintest magnitudes remains as steep as that of  1500 ≈ −14.Bottom panel: log 10 of model and observation, shown on the same magnitude range.
Figure2shows the UV ( = 1500 Å) luminosity function of galaxies at  = 5, 6, 7, 8, 9, 10 in the model with the end of reionization at  rei = 6 and a small amount of SFR stochasticity of  Δ = 0.1 (see Section 2.2).It also shows observational estimates of the UV LF in recent studies that used HST and JWST observations.The model matches these quite well at the range of luminosities probed by observations.Differences at  ≤ 7 and  1500 ≲ −20 are likely due to dust effects that have a similar magnitude at these luminosities and redshifts (see Fig.1).Agreement with UV LF estimates at  ∈ [5, 10] at  1500 < −14 and the fact that the model reproduces properties of  = 0 dwarf galaxies well(Kravtsov & Manwadkar 2022), including the luminosity function of Milky Way satellites down to the ultra-faint magnitudes(Manwadkar & Kravtsov 2022), means we can plausibly expect that the model UV LF can faithfully describe the evolution of the luminosity function at fainter magnitudes.This statement is supported by the comparison with the reconstruction of the UV LF of the progenitors of the Local Group dwarf galaxies at  ≈ 7 ofBoylan-Kolchin et al. (2015) shown in Figure3.This LF reconstruction was done using observational estimates of the star formation histories of dwarf galaxies measured from their color-magnitude diagrams(Weisz et al. 2014(Weisz et al. , 2019;;Weisz & Boylan-Kolchin 2017) and their corresponding  1500 at  ≈ 7. The function is a combination of the Schechter form with  ★ = −21.03, ★ = 1.57× 10 −4 mag −1 Mpc −3 ,  = −2.03 at  1500 < −13(Finkelstein et al. 2015) and a power law  ∝  −1.2 1500 at  1500 ≥ −13.As shown by Boylan-Kolchin et al. (2015),

Fig. 3 .
Fig. 3.-Comparison of rest-frame  = 7 UV LFs in models with reionization redshifts  rei = 6 and 8.5.The green curve shows the LF function estimated at  ≈ 7 based on the star formation histories of the Local Group dwarf galaxies (Boylan-Kolchin et al. 2015).The Local Group reconstructed LF is in agreement with the model with  rei = 8.5, which indicates that the Local volume was reionized around this redshift.
Fig. 4.-Least-squares fit on the model LFs at  = 5, 6, 7, 8, 9, 10 with Jaacks et al. (2013)'s modified Schechter function shown in Equation3, fitted on −26 ≤  1500 ≤ −4 (see the best-fit parameters in Table1).The two panels show fits to the LFs in models with  rei = 6.0 and  rei = 8.5, respectively.The faint-end behavior, including the steep slope before reionization and the flattening in post-reionization redshifts  <  rei , are accurately reflected by the modified Schechter fit.

Fig. 5 .
Fig.5.-The fraction of UV flux contributed by galaxies brighter than a given  UV for two reionization models ( rei = 6 and 8.5 respectively) across redshifts  ∈ [5, 10].The fraction is computed by integrating the UV LF over the luminosity range and dividing by the integral over the entire luminosity range (See Equation 5 and 6).Both plots show results for the model with SFR stochasticity.The figure shows that dwarf galaxies contribute a significant fraction of the UV flux (≈ 55 − 65% at  UV > −14), especially at higher .

Fig. 8 .
Fig.8.-The emission rate density of the Lyman continuum photons  ion for 5 ≤  ≤ 10, contributed by galaxies brighter than −17, −13, −4 for  rei = 6 model assuming a constant escape fraction of  esc = 5%.The right -axis scale shows a ballpark estimate of the number of ionizing photons per hydrogen atom for each  ion value.Also shown are predictions of the estimated  ion needed to keep the IGM reionized, derived from observations at  ≈ 5 − 6 (green band:Fan et al. 2006;Madau 2017) and the theoretical model ofMadau et al. (1999) for clumping factors of  = 1, 3, and 10 (lines, with gray shaded band bracketing models with  = 1 and  = 10).

Fig
Fig. 9.-Coefficients of the third-order polynomial fit approximation to the evolution of the parameters of the Jaacks et al. (2013) analytical form to  ion (  UV ) at 5 ≤  ≤ 10.Top panel: model with  rei = 6.0.Bottom panel: model with  rei = 8.5.The polynomial coefficients are presented in Table3.

TABLE 1
Best-fit parameters for Jaacks et al. (2013)'s modified Schechter function to our stochastic UV LF at  ∈ [5, 10].The last column  −  shows the effective faint end LF slope.
see their Fig.1) consistent with various observational constraints and total optical depth of ionized gas has log 10  ion ≈ 50.8 at  = 6 and log 10  ion ≈ 50.3 at  = 10.This is only somewhat flatter than the blue dashed line in Figure8, and such evolution can be realized if the escape fraction increases moderately with decreasing galaxy luminosity.