INTENSITY MAPPING OF Lyα EMISSION DURING THE EPOCH OF REIONIZATION

Assuming ionizing equilibrium, the number of recombinations in galaxies is expected to match the number of ionizing photons that are absorbed in the galaxy and does not escape into the IGM. Depending on the temperature and density of the gas, a fraction of the radiation due to these recombinations is emitted in the Lyα line.

In the interstellar gas, most of the neutral hydrogen is in dense clouds with column densities greater than 3 × 10¹⁸ cm⁻². These clouds are optically thick to Lyα radiation, and Lyman photons are scattered in the galaxy several times before escaping into the IGM. Such multiple scatterings increase the probability of absorption. Assuming that these clouds are spherical and that the gas temperature is of the order of 10⁴ K, Gould & Weinberg (1996) used atomic physics to study the probability of the Lyα emission per hydrogen recombination. They estimated that a fraction f_rec ≈ 66% of the hydrogen recombinations would result in the emission of an Lyα photon and that most of the other recombinations would result in two-photon emission. These fractions should change with the temperature and the shape of the cloud, but such variations are expected to be small. Other calculations yield fractions between 62% and 68% according to the conditions in the cloud. In this paper we have chosen to use a value of f_rec = 66% since the overall uncertainty on this number is lower than the uncertainty on the number of hydrogen recombinations.

The absorption of Lyα photons by dust is difficult to estimate and changes from galaxy to galaxy. Gould & Weinberg (1996) estimated that for a cloud with a column density N ∼ 10¹⁹ cm⁻² the dust in the galaxy absorbs a fraction f_dust ≈ 4% of the emitted Lyα photons before they reach the galaxy virial radius; however, recent observations of high-redshift galaxies indicate a much higher f_dust. In this study, we will use a redshift parameterization for the fraction of Lyα photons that are not absorbed by dust f_Lyα = 1 − f_dust that is double the value predicted by the study made by Hayes et al. (2011):

$$\begin{equation} f_{\rm Ly\alpha }(z)=C_{\rm dust}\times 10^{-3}(1+z)^{\xi }, \end{equation} \tag{ 1 }$$

where C_dust = 3.34 and ξ = 2.57. The Hayes et al. (2011) parameterization was made so that f_Lyα gives the difference between observed Lyα luminosities and Lyα luminosities scaled from SFRs assuming that the Lyα photons emitted in galaxies are only originated in recombinations. The high-redshift observations used to estimate f_Lyα are only of massive stars, while the bulk of Lyα emission is originated in the low-mass stars that cannot be detected by current surveys. According to several studies (Forero-Romero et al. 2011), f_Lyα decreases with halo mass, so it is possible that it is being underestimated in Hayes et al. (2011), which is why we decided to use a higher f_Lyα. Our results can, however, be easily scaled to other f_dust evolutions.

The number of Lyα photons emitted in a galaxy per second, $\dot{N}_{\rm Ly\alpha }$, that reach its virial radius is therefore given by

$$\begin{equation} \dot{N}_{\rm Ly\alpha } = A_{{\rm He}} f_{\rm rec}\times f_{\rm Ly\alpha }\times (1-f_{\rm esc})\times \dot{N}_{\rm ion}, \end{equation} \tag{ 2 }$$

where A_He = (4 − 4Y_p)/(4 − 3Y_p) accounts for the fraction of photons that go into the ionization of helium (Y_p is the mass fraction of helium), $\dot{N}_{\rm ion}$ is the rate of ionizing photons emitted by the stars in the galaxy, and f_esc is the fraction of ionizing photons that escape the galaxy into the IGM.

The ionizing photon escape fraction depends on conditions inside each galaxy and is difficult to estimate, especially at high redshifts. The precise determination of its value is one of the major goals of future observations of high-redshift galaxies at z ≳ 7. This parameter can be measured from deep imaging observations or can be estimated from the equivalent widths of the hydrogen and helium Balmer lines. The ionizing photon escape fraction dependence with the galaxy mass and the SFR, as a function of redshift, has been estimated using simulations that make several assumptions about the intensity of this radiation and its absorption in the ISM. However, for the halo virial mass range, 10⁸–10¹³ M_☉, and during the broad redshift range related to the EoR, there are no simulations that cover the full parameter space. Moreover, the limited simulations that exist do not always agree with each other (Gnedin et al. 2008; Wise & Cen 2009; Fernández-Soto et al. 2003; Siana et al. 2007; Haardt & Madau 2012). Razoumov & Sommer-Larsen (2010) computed the escape fraction of UV radiation for the redshift interval z = 4 to z = 10 and for halos of masses from 10^7.8 to 10^11.5 M_☉ using a high-resolution set of galaxies. Their simulations cover most of the parameter space needed for reionization-related calculations, and their escape fraction parameterization is compatible with most of the current observational results. Thus, we use it for our calculations here.

According to Razoumov & Sommer-Larsen (2010) simulations, the escape fraction of ionizing radiation can be parameterized as

$$\begin{equation} f_{\rm esc}(M,z)=\exp [-\alpha (z) M^{\beta (z)}], \end{equation} \tag{ 3 }$$

where M is the halo mass and α and β are functions of redshift (Table 1).

The number of ionizing photons emitted by the stars in a galaxy depends on its SFR, metallicity, and the stellar initial mass function (IMF). Making reasonable assumptions for these quantities, we will now estimate $\dot{N}_{\rm ion}$. Since this UV emission is dominated by massive, short-lived stars, we can assume that the intensity of ionizing photons emitted by a galaxy is proportional to its SFR. In terms of the SFR in one galaxy,

$$\begin{equation} \dot{N}_{\rm ion}=Q_{\rm ion}\times {\rm SFR}, \end{equation} \tag{ 4 }$$

where Q_ion is the average number of ionizing photons emitted per solar mass of star formation. This can be calculated through

$$\begin{equation} Q_{\rm ion}=\frac{\int ^{M_{\rm max}} _{M_{\rm min}} \Psi (M) Q_{\rm \star }(M) t_{\rm \star }(M) dM}{\int ^{M_{\rm max}} _{M_{\rm min}} \Psi (M) M dM}, \end{equation} \tag{ 5 }$$

where Ψ(M) = KM^−α is the stellar IMF, K is a constant normalization factor, and α is the slope of the IMF. In our calculation, we used a Salpeter IMF, with α = 2.35. t_⋆(M) is the star lifetime and Q_⋆(M) its number of ionizing photons emitted per unit time. The values of Q_⋆ and t_⋆ were calculated with the ionizing fluxes obtained by Schaerer (2002) using realistic models of stellar populations and non-LTE atmospheric models, appropriated for Pop II stars with a Z_⋆ = 0.02 Z_☉ metallicity.

Assuming that ionizing photons are only emitted by massive OB stars sets a low mass effective limit for the mass of stars contributing to the UV radiation field of a galaxy. This limit is a necessary condition for the star to be able to produce a significant number of ionizing photons. For the stellar population used for this work we take M_min ≈ 7 M_☉ (Schaerer 2002; Shull et al. 2012). The integration upper limit is taken to be M_max = 150 M_☉. In this paper, we calculated Q_ion using the parameterization values published in Schaerer (2002). The number of ionizing photons per second emitted by a star as a function of its mass is given by

$$\begin{eqnarray} \log _{10}[Q_{\star }/{\rm s}^{-1}]&=&27.80+30.68x-14.80x^{\rm 2}\nonumber \\ &&+2.5x^3\ {\rm for}\ 7\, {M_{\odot }}<M_{\rm \star }<150\, {M_{\odot }},\quad\ \end{eqnarray} \tag{ 6 }$$

where x = log₁₀(M_⋆/M_☉) and the star's lifetime in years is given by

$$\begin{equation} \log _{\rm 10}[t_\star /{\rm yr}]=9.59\hbox{--}2.79x+0.63x^{\rm 2}. \end{equation} \tag{ 7 }$$

The use of these parameters results in Q_ion ≈ 5.38 × 10⁶⁰ M^-1_☉. Shull et al. (2012) suggest the use of a different model for stellar atmosphere and evolution (R. S. Sutherland & J. M. Shull, unpublished) that yields Q_ion ≈ 3.97 × 10⁶⁰ M^-1_☉. This may imply that the stellar emissivity we calculated is an overestimation and that consequently our Lyα flux powered by stellar emission may be overestimated by about 35%. This is comparable to other large uncertainties, such as the ones in the parameters f_esc and f_dust. The Lyα luminosity is calculated assuming that the Lyα photons are emitted at the Lyα rest frequency, ν₀ = 2.47 × 10¹⁵ Hz with an energy of E_Lyα = 1.637 × 10^-11 erg. To proceed, we will assume that the SFR for a given galaxy is only a function of redshift and the mass of the dark halo associated with that galaxy. The Lyα luminosity due to recombinations in the ISM, L^GAL_rec, can then be parameterized as a function of halo mass and redshift as

$$\begin{eqnarray} L^{\rm GAL}_{\rm rec}(M,z)&=&E_{\rm Ly\alpha }\dot{N}_{\rm Ly\alpha }\nonumber\\ &\approx & 1.55 \times 10^{\rm 42} \left[1-f_{\rm esc}(M,z)\right]f_{\rm Ly\alpha }(z)\nonumber\\ &&\times\frac{{\rm SFR}(M,z)}{{M}_\odot \; {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}. \end{eqnarray} \tag{ 8 }$$

The kinetic energy of the electron ejected during the hydrogen ionization heats the gas, and assuming thermal equilibrium, this heat is emitted as radiation. Using atomic physics, Gould & Weinberg (1996) estimated that for a cloud with a hydrogen column density of ≈10¹⁹ cm⁻², the energy emitted in the form of Lyα photons is about 60% for ionizing photons with energy $E_{\nu _{\rm lim}}<E_{\nu }<4 E_{\nu _{\rm lim}}$ and ≈50% for photons with energy $E_{\nu }>4 E_{\nu _{\rm lim}}$, where $E_{\nu _{\rm lim}}=13.6$ eV is the Rydberg energy. The remaining energy is emitted in other lines.

Using the spectral energy distribution (SED) of galaxies with a metallicity Z = 0.02 Z_☉ from the code of Maraston (2005), we estimated that the average ionizing photon energy is E_ν = 21.4 eV and that more than 99% of the photons have an energy lower than $4E_{\nu _{\rm lim}}$. According to the Gould & Weinberg (1996) calculation, the fraction of energy of the UV photon that is emitted as Lyα radiation due to the collisional excitations/relaxations is given by

$$\begin{equation} E_{\rm exc}/E_{\nu } \sim 0.08+0.1\left(1- \frac{2\nu _{\rm lim}}{\nu } \right) \sim 0.1. \end{equation} \tag{ 9 }$$

For a cloud with the properties considered here this yields an energy in Lyα per ionizing photon of E_exc ≈ 2.14 eV or 3.43 × 10⁻¹² erg. This results in an average of 0.16 Lyα photons per ionizing photon.

Finally, the Lyα luminosity due to excitations in the ISM, L^GAL_exc, is then

$$\begin{eqnarray} L^{\rm GAL}_{\rm exc}(M,z)&=&[1-f_{\rm esc}(M,z)]f_{\rm Ly\alpha }(z)A_{\rm He} \dot{N}_{\rm ion}E_{\rm exc}\nonumber\\ & \approx & 4.03\times 10^{41}\left[1-f_{\rm esc}(M,z)\right]f_{\rm Ly\alpha }(z)\nonumber\\ &&\times\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}, \end{eqnarray} \tag{ 10 }$$

where again it is assumed to be a function of the SFR.

During the formation of galaxies, gas from the IGM falls into potential wells composed mainly by dark matter that collapsed under its own gravity. The increase in the gas density leads to a high rate of atomic collisions that heat the gas to a high temperature. According to the study of Fardal et al. (2001), most of the gas in potential wells that collapses under its own gravity never reaches its virial temperature, and so a large fraction of the potential energy is released by line emission induced by collisions and excitations from gas with temperatures T_K < 2 × 10⁴ K. At this temperature approximately 50% of the energy is emitted in Lyα alone.

From Fardal et al. (2001), we can relate the luminosity at the Lyα frequency due to the cooling in galaxies to their baryonic cold mass, M^bar_cool, using

$$\begin{equation} \log _{10}\left(L^{\rm GAL}_{\rm cool}\right)=1.52\log _{10}\left(M^{\rm bar}_{\rm cool}\right)+26.32, \end{equation} \tag{ 11 }$$

where both the luminosity and the mass are in solar units. To relate this baryonic cold mass to a quantity we can use in our models, we used the relation between cold baryonic mass and the halo mass from the galaxies in the Guo et al. (2011) catalog. From the equation above, we can then obtain an expression for the luminosity, which can be fitted by

$$\begin{eqnarray} L^{\rm GAL}_{\rm cool}(M) &\approx & 1.69 \times 10^{35} f_{\rm Ly\alpha }(z)\left(1+\frac{M}{10^{\rm 8}}\right)\nonumber\\ &&\times \left(1+\frac{M}{2\times 10^{\rm 10}}\right)^{2.1}\left(1+\frac{M}{3\times 10^{\rm 11}}\right)^{-3}\ {\rm erg}\ {\rm s}^{-1},\nonumber\\ \end{eqnarray} \tag{ 12 }$$

with M in units of M_☉. The relation between the cold gas mass and the mass of the halo shows very little evolution with redshift during reionization. Thus, we expect the relation in Equation (13) to only depend on redshift due to the redshift evolution of f_Lyα.

Continuum emission can also contribute to the Lyα observations. These include stellar emission, free–free emission, free–bound emission, and two-photon emission. Photons emitted with frequencies close to the Lyn lines should scatter within the ISM and eventually get re-emitted out of the galaxy as Lyα photons. Otherwise, they will escape the ISM before redshifting into one of the Lyn lines and being reabsorbed by a hydrogen atom.

The fraction of photons that scatter in the galaxy can be estimated from the intrinsic width of the Lyα line, which has ≈4 å (Jensen et al. 2013). We calculated the stellar contribution assuming an emission spectrum for stars with a metallicity of Z_⋆ = 0.02 Z_☉ estimated with the code from Maraston (2005) that can be approximated by the emission of a blackbody with a temperature of 6.0 × 10⁴ K for hν < 13.6 eV. The number of stellar origin Lyα photons per solar mass in star formation obtained with this method is

$$\begin{eqnarray} Q^{\rm stellar}_{\rm Ly\alpha }&=&4.307\int _{\nu _{{\rm Ly}\alpha + 2\,{\rm \mathring{A}}}}^{\nu_{{\rm Ly}\alpha - 2\,{\rm \mathring{A}}}} d\nu \frac{\nu ^3}{e^{h\nu /K_b T_K}-1}\, M_{\odot }^{-1}\nonumber\\ &=&9.92\times 10^{\rm 58}\, M_{\odot }^{-1}. \end{eqnarray} \tag{ 13 }$$

We note that we are not accounting for the higher opacity at the center of the Lyα line, which should push the photons out of the line center before exiting the star, and so we may be overestimating the stellar Lyα photon emission.

Free–bound emission and free–free emission are, respectively, originated when free electrons scatter off ions with or without being captured. Following the approach of Fernandez & Komatsu (2006), the free–free and free–bound continuum luminosity can be obtained using

$$\begin{equation} L_{\nu }(M,z)=V_{{\rm sphere}}(M,z)\varepsilon _{\nu }, \end{equation} \tag{ 14 }$$

where V_sphere is the volume of the Strömgren sphere, which can be roughly estimated using the ratio between the number of ionizing photons emitted and the number density of recombinations in the ionized volume,

$$\begin{equation} V_{{\rm sphere}}(M,z)=\frac{Q_{{\rm ion}}{\rm SFR}(M,z)(1-f_{{\rm esc}})}{n_e n_p \alpha _{\beta }}. \end{equation} \tag{ 15 }$$

Here ε_ν is the total volume emissivity of free–free and free–bound emission, n_p is the number density of protons (ionized atoms), and α_i is the case A or case B recombination coefficient (see Furlanetto et al. 2006).

The volume emissivity estimated by Dopita & Sutherland (2003) is given by

$$\begin{equation} \varepsilon _{\nu }=4\pi n_e n_p \gamma _c \frac{e^{-h\nu /kT_K}}{T_K^{1/2}}{\rm \ J\ cm^{-3}\ s^{-1}\ Hz^{-1}}, \end{equation} \tag{ 16 }$$

where γ_c is the continuum emission coefficient including free–free and free–bound emission given in SI units by

$$\begin{equation} \gamma _c=5.44\times 10^{-46}\left[\bar{g}_{{\rm ff}}+\Sigma _{n=n^\prime }^{\infty }\frac{x_n e^{x_n}}{n}g_{{\rm fb}}(n)\right]. \end{equation} \tag{ 17 }$$

Here x_n = Ry/(k_BT_Kn²) (k_B is the Boltzmann constant, n is the level to which the electron recombines, and Ry = 13.6 eV is the Rydberg unit of energy) and $\bar{g}_{{\rm ff}}\approx 1.1\hbox{--}1.2$ and g_fb(n) ≈ 1.05–1.09 are the thermally averaged Gaunt factors for free–free and free–bound emission (Karzas & Latter 1961, values from). The initial level n' is determined by the emitted photon frequency and satisfies the condition cR_∞/n'² < ν < cR_∞/(n' − 1)², where R_∞ = 1.1 × 10⁷ m⁻¹ is the Rydberg constant.

The continuum luminosity per frequency interval (L_ν) is related to the Lyα luminosity emitted from the galaxies by L_cont = L_ν × dν(4 å) = f_LyαQ_LyαE_LyαSFR(M, z), where Q_Lyα is the number of emitted Lyα photons per solar mass in star formation. We then obtain Q^free–free_Lyα = 2.13 × 10⁵³ M⁻¹_☉ for free–free emission and Q^free–bound_Lyα = 2.22 × 10⁵⁵ M⁻¹_☉ for free–bound emission.

During recombination there is also the probability of two-photon emission and although these photons have frequencies below the Lyα frequency there is a small fraction of them of Q^2-photon_Lyα that are emitted so close to the Lyα line, which are included in the Lyα intrinsic width.

The number of Lyα photons that can be originated due to two-photon emission during recombination is given by

$$\begin{equation} Q^{\rm 2-photon}_{\rm Ly\alpha }=\int _{\nu _{{\rm Ly}\alpha + 2\,{\rm \mathring{A}}}}^{\nu _{{\rm Ly}\alpha }}\frac{2}{\nu _{{\rm Ly}\alpha }}P(\nu /\nu _{{\rm Ly}\alpha })d\nu , \end{equation} \tag{ 18 }$$

where P(y)dy is the normalized probability that in a two-photon decay one of them is the range dy = dν/ν_Lyα and 1 − f_Lyα ≈ 1/3 is the probability of two-photon emission during a hydrogen n = 2→1 transition. The probability of two-photon decay was fitted by Fernandez & Komatsu (2006) using Table 4 of Brown & Mathews (1970) as

$$\begin{eqnarray} P(y)&=&1.307\hbox{--}2.627(y-0.5)^2+2.563(y-0.5)^4\nonumber\\ &&-51.69(y-0.5)^6. \end{eqnarray} \tag{ 19 }$$

Finally, the different contributions to the total Lyα luminosity from galaxies due to continuum emission, L^GAL_cont = L^stellar_cont + L^free–free_cont + L^free–bound_cont + L^2-photon_cont, are given by

$$\begin{eqnarray} L^{\rm stellar}_{\rm cont}(M,z)&=&f_{\rm Ly\alpha }Q^{\rm stellar}_{\rm Ly\alpha }E_{\rm Ly\alpha }{\rm SFR}(M,z)\nonumber\\ &\approx & 5.12\times 10^{40}f_{\rm Ly\alpha }\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}\qquad \end{eqnarray} \tag{ 20 }$$

for stellar emission,

$$\begin{eqnarray} L^{\rm free\hbox{--}free}_{\rm cont}(M,z)&=&f_{\rm Ly\alpha }Q^{\rm stellar}_{\rm Ly\alpha }E_{\rm Ly\alpha }{\rm SFR}(M,z)\nonumber\\ &\approx & 1.10\times 10^{35}f_{\rm Ly\alpha }\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}\qquad \end{eqnarray} \tag{ 21 }$$

for free–free emission,

$$\begin{eqnarray} L^{\rm free\hbox{--}bound}_{\rm cont}(M,z)&=&f_{\rm Ly\alpha }Q^{\rm stellar}_{\rm Ly\alpha }E_{\rm Ly\alpha }{\rm SFR}(M,z)\nonumber\\ &\approx & 1.47\times 10^{37}f_{\rm Ly\alpha }\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}\qquad \end{eqnarray} \tag{ 22 }$$

for free–bound emission, and

$$\begin{eqnarray} L^{\rm 2-photon}_{\rm cont}(M,z)&=&f_{\rm Ly\alpha }Q^{\rm stellar}_{\rm Ly\alpha }E_{\rm Ly\alpha }{\rm SFR}(M,z)\nonumber\\ &\approx & 2.41\times 10^{38}f_{\rm Ly\alpha }\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}\qquad \end{eqnarray} \tag{ 23 }$$

for two-photon emission.

Note that here we are only considering the part of the continuum emission from galaxies that could contribute to the same "Lyα redshift." There will be a continuum emission spectrum with frequencies below the Lyα line from the mechanisms above that will contribute to the same observation from lower redshifts and will generate a "foreground" to the Lyα signal that needs to be removed. This should be possible due to the smoothness of this background across frequency, in the same manner as foregrounds of the 21 cm signal are removed (e.g., Wang et al. 2006).

Simulations of galaxy formation and observations indicate that the star formation of a halo increases strongly for small halo masses, but at high halo masses (M ≳ 10¹¹ M_☉) it becomes almost constant (Conroy & Wechsler 2009; Popesso et al. 2012).

In order to better estimate and constrain the SFR of a halo, we used three nonlinear SFR versus halo mass parameterizations that are in good agreement with different observational constraints. In Sim1 we adjusted the SFR to reproduce a reasonable reionization history and an Lyα LF evolution compatible with different observational constraints, and in Sim2 we adjusted the SFR versus halo mass relation to the parameterizations from the Guo et al. (2011) galaxy catalog (low halo masses) and the De Lucia & Blaizot (2007) galaxy catalog (high halo masses). Sim2 results in an early reionization history with an optical depth to reionization compatible with the low bound of the current observational constraints. Finally, Sim3 has the same halo mass dependence as Sim2 but evolves with redshift in a similar way to the De Lucia & Blaizot (2007) and Guo et al. (2011) galaxy catalogs.

We parameterized the relations between the SFR and halo mass as

$$\begin{eqnarray} \frac{{\rm SFR}(M,z)}{M_\odot /\rm yr}&=&(2.8\times 10^{-28}) M^{a}\left(1+\frac{M}{c_1}\right)^{b}\left(1+\frac{M}{c_2}\right)^{d}, \nonumber\\ \end{eqnarray} \tag{ 24 }$$

where a = 2.8, b = −0.94, d = −1.7, c₁ = 1 × 10⁹ M_☉, and c₂ = 7 × 10¹⁰ M_☉ for Sim1;

$$\begin{eqnarray} \frac{{\rm SFR}(M,z)}{M_\odot /\rm yr}&=&1.6\times 10^{-26}M^{a}\left(1+\frac{M}{c_1}\right)^{b} \nonumber \\ &&\times\left(1+\frac{M}{c_2}\right)^{d}\left(1+\frac{M}{c_3}\right)^{e}, \end{eqnarray} \tag{ 25 }$$

where a = 2.59, b = −0.62, d = 0.4, e = −2.25, c₁ = 8 × 10⁸ M_☉, c₂ = 7 × 10⁹ M_☉, and c₃ = 1 × 10¹¹ M_☉ for Sim2; and

$$\begin{eqnarray} \frac{{\rm SFR}(M,z)}{M_\odot /\rm yr}&=&2.25\times 10^{-26}\left(1+0.075\times (z-7) \right)M^{a} \nonumber \\ &&\times\left(1+\frac{M}{c_1}\right)^{b}\left(1+\frac{M}{c_2}\right)^{d}\left(1+\frac{M}{c_3}\right)^{e}, \qquad \end{eqnarray} \tag{ 26 }$$

where a = 2.59, b = −0.62, d = 0.4, e = −2.25, c₁ = 8 × 10⁸ M_☉, c₂ = 7 × 10⁹ M_☉, and c₃ = 1 × 10¹¹ M_☉ for Sim3. Figure 1 shows these relations.

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In Figure 2, the strong decline in the observational SFR density (SFRD) from z ≈ 8 to z ≈ 10, imposed by the observational point at z = 10.3, was obtained with the observation of a single galaxy using the Hubble Deep Field 2009 two years' data (Bouwens et al. 2011; Oesch et al. 2012). It was argued in Bouwens et al. (2012a), based on an analytical calculation, that even with such low SFRD at high redshifts it was possible to obtain an optical depth to reionization compatible with the value obtained by Wilkinson Microwave Anisotropy Probe (WMAP; τ = 0.088 ± 0.015) (Komatsu et al. 2011). However, this derivation would imply a high escape fraction of ionizing radiation, and that reionization would end at z ≈ 8, which is hard to reconcile with the constraints from observations of quasar spectra (Mesinger & Haiman 2007; Zaldarriaga et al. 2008). Our SFRDs are considerably higher than the current observational constraints, although the difference can be explained by a systematic underestimation of the SFR in observed galaxies. Moreover, current observations only probe the high-mass end of the high-redshift galaxies' mass function, which will underestimate the SFRD (also the obtained SFRs have very high error bars due to uncertainties in the correction due to dust extinction, the redshift, and the galaxy type). In the following sections the results shown were obtained using Sim1 unless stated otherwise.

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In the previous sections we calculated the Lyα luminosity as a function of the SFR for several effects. The commonly used "empirical" relation between these two quantities is (Jiang et al. 2011)

$$\begin{equation} L_{\rm Gal}=1.1\times 10^{42}\frac{{\rm SFR}(M,z)}{{M}_\odot \, {\rm yr}^{-1}}\ {\rm erg}\ {\rm s}^{-1}, \end{equation} \tag{ 27 }$$

and it is based on the relation between SFR and the Hα luminosity from Kennicutt (1998a) and on the line emission ratio of Lyα to Hα in case B recombinations calculated assuming a gas temperature of 10⁴ K. This empirical relation gives the Lyα luminosity without dust absorption (we have labeled it K98 for the remainder of the paper).

Our relation between luminosity and star formation is mass dependent (both from the escape fraction and due to the expression from the cooling mechanism), so in order to compare it with the result above, we calculate

$$\begin{equation} A(z)=\frac{\langle L_{\rm Gal}(M,z)\rangle }{\langle {\rm SFR}(M,z) \rangle }, \end{equation} \tag{ 28 }$$

where the average 〈x〉 of quantity x is done over the halo mass function for the mass range considered. The results are presented in Table 2 for a few redshifts.

Although our Lyα luminosities per SFR are slightly higher, at least for low redshifts, we point out that the "empirical" relation is based on a theoretical calculation that only accounts for Lyα emission due to recombinations. Moreover, the observational measurements of Hα and Lyα are primarily made at low redshifts, where the absorption of Lyα photons by dust in galaxies is expected to be high. Our relation has the advantage of evolving with redshift since it accounts for the evolution of the escape fraction of ionizing photons and for the evolution of the escape fraction of Lyα photons. This z-dependence is not present in the standard empirical relation. This redshift evolution of the UV photons' escape fraction is a consequence of the increase in the number of massive galaxies with more clumpy structure as the redshift decreases. The star-forming regions of massive galaxies are embedded in clumps, and therefore it becomes more difficult for the ionizing photons to escape from such dense regions (Razoumov & Sommer-Larsen 2010; Yajima et al. 2011). The redshift evolution of the relation presented in Equation (28) justifies why a theoretical calibration between Lyα luminosity and the SFR of a galaxy is useful for our work.

To check the consistency between our theoretical estimation of the Lyα luminosity and the existing observations during reionization, we show in Figure 3 the LF using two of the SFR versus halo mass parameterizations presented in Section 2.5. This prediction is then compared to Lyα LFs of photometric identified objects in Shimasaku et al. (2006) and in Kashikawa et al. (2006) near the end of the reionization epoch.

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Our LFs were calculated assuming a minimum halo mass of 8 × 10⁸ M_☉, which corresponds to a minimum luminosity of 3.72 × 10³⁶ erg s⁻¹ for Sim1, 4.49 × 10³⁶ erg s⁻¹ for Sim2, and 6.22 × 10³⁶ erg s⁻¹ for Sim3. The agreement between our LFs and observations is reasonable for Sim1; however, our Sim2 overpredicts the abundance of high-luminosity Lyα emitters. This difference can be due to sample variance or a result of the high sensitivity of theoretical predictions to several parameters in our model. We point out that the luminosity range relevant for this comparison falls in a halo mass range outside the one for which the escape fraction of UV radiation we are using was estimated, so we could easily get a better fit between observations and Sim2 by reducing this escape fraction for high halo masses. This difference could also be related with the choice of halo mass function. Here we choose the Sheth–Tormen halo mass function (Sheth & Tormen 1999), which has been shown to fit low-redshift simulations more accurately, but it is yet to be established the extent to which such a halo mass function can reproduce the halo distribution during reionization. Another possible explanation for this difference is the existence of a small amount of neutral gas in the IGM, which would severely decrease the observed Lyα luminosity from galaxies. Also, we could have decreased the high-luminosity end of our LFs if we had use an Lyα escape fraction that decreased with halo mass such as the one used in Forero-Romero et al. (2011). We do not consider a model fit to the data to optimize various parameters in our model given that the current constraints on the observed Lyα LFs have large overall uncertainties, especially considering variations from one survey to another.

In this section and the next one we will attempt to estimate the intensity and power spectrum of the Lyα signal using an analytical model. In Section 4, we will improve the estimation by doing the same calculation using a semi-numerical simulation.

The total intensity of Lyα emission can be obtained from the combined luminosity of Lyα photons associated with different mechanisms described in the previous sub-sections, such that

$$\begin{equation} \bar{I}_{\rm Gal}(z)=\int ^{M_{\rm max}} _{M_{\rm min}}dM\frac{dn}{dM}\frac{L_{\rm Gal}(M,z)}{4\pi D_{\rm L}^{\rm 2}}y(z)D^2_{\rm A}, \end{equation} \tag{ 29 }$$

where dn/dM is the halo mass function (Sheth & Tormen 1999), M is the halo mass, M_max = 10¹³ M_☉, M_min = M_OB, D_L is the proper luminosity distance, and D_A is the comoving angular diameter distance. Finally, y(z) = dχ/dν = λ_Lyα(1 + z)²/H(z), where χ is the comoving distance, ν is the observed frequency, and λ_Lyα = 2.46 × 10⁻¹⁵ m is the rest-frame wavelength of the Lyα line.

The evolution of the Lyα intensity predicted by this calculation is shown in Figure 4 together with the scaling expected under the "empirical" relation from Kennicutt (1998a) combined with an assumption related to the gas temperature. The intensities of Lyα emission from different sources are presented in Table 3 for several redshifts.

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These intensities can be extrapolated to other SFRDs, assuming that the only change is in the amplitude of the SFR halo mass relations presented in Figure 1 by using the coefficients in Table 4.

The intensities from emission at z ≈ 7, 8, and 10 are 9.5 × 10⁻⁹, 3.5 × 10⁻⁹, and 3.2 × 10⁻¹⁰ erg s⁻¹ cm⁻² sr⁻¹, respectively. Such an intensity is substantially smaller than the background intensity of integrated emission from all galaxies (around 1 × 10⁻⁵ erg s⁻¹ cm⁻² sr⁻¹; Madau & Pozzetti 2000), or from the total emission of galaxies during reionization, estimated to be at most 1 × 10⁻⁶ erg s⁻¹ cm⁻² sr⁻¹ (Cooray et al. 2012).

The Lyα emission from galaxies will naturally trace the underlying cosmic matter density field, so we can write the Lyα line intensity fluctuations due to galaxy clustering as

$$\begin{equation} \delta I_{\rm GAL}=b_{\rm Ly\alpha }\bar{I}_{\rm GAL}\delta ({\bf x}), \end{equation} \tag{ 30 }$$

where $\bar{I}_{\rm GAL}$ is the mean intensity of the Lyα emission line, δ(x) is the matter overdensity at the location x, and b_Lyα is the average galaxy bias weighted by the Lyα luminosity (see, e.g., Gong et al. 2011).

Using one of the relations between the SFR and halo mass from Section 2.5, we can calculate the luminosity and obtain the Lyα bias following Visbal & Loeb (2010):

$$\begin{equation} b_{\rm Ly\alpha }(z)=\frac{\int ^{M_{\rm max}}_{M_{\rm min}} dM \frac{dn}{dM} L_{\rm GAL}\ b(z,M)}{\int ^{M_{\rm max}}_{M_{\rm min}} dM \frac{dn}{dM} L_{\rm GAL}}, \end{equation} \tag{ 31 }$$

where b(z, M) is the halo bias and dn/dM is the halo mass function (Sheth & Tormen 1999). We take M_min = 10⁸ M_☉/h and M_max = 10¹³ M_☉/h. The bias between dark matter fluctuation and the Lyα luminosity, as can be seen in Figure 5, is dominated by the galaxies with low Lyα luminosity independently of the redshift.

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We can then obtain the clustering power spectrum of Lyα emission as

$$\begin{equation} P_{\rm GAL}^{\rm clus}(z,k) = b_{\rm Ly\alpha }^2 \bar{I}_{\rm GAL}^2 P_{\delta \delta }(z,k), \end{equation} \tag{ 32 }$$

where P_δδ(z, k) is the matter power spectrum. The shot-noise power spectrum, due to discretization of the galaxies, is also considered here. It can be written as (Gong et al. 2011)

$$\begin{equation} P^{\rm shot}_{\rm Ly\alpha }(z) = \int _{\rm M_{\rm max}}^{\rm M_{\rm min}} dM \frac{dn}{dM} \left (\frac{L_{\rm GAL}}{4\pi D_{\rm L}^2}y(z)D_{\rm A}^2\right)^2. \end{equation} \tag{ 33 }$$

The resulting power spectrum of Lyα emission from galaxies is presented in Figure 6. At all scales presented the Lyα intensity and fluctuations are dominated by the recombination emission from galaxies.

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The UV radiation that escapes the ISM into the IGM ionizes low-density clouds of neutral gas. Part of the gas in these clouds then recombines, giving rise to Lyα emission. The radiation emitted in the IGM is often referred to as fluorescence (Santos 2004). The comoving number density of recombinations per second in a given region, $\dot{n}_{{\rm rec}}$, is given by

$$\begin{equation} \dot{n}_{{\rm rec}}(z)=\alpha n_e(z) n_{{\rm H\,\scriptsize{II}}}(z), \end{equation} \tag{ 34 }$$

where α changes between the case A and the case B recombination coefficient and $n_{{\rm H\,\scriptsize{II}}}=x_i ({n_b(1-Y_p)}/(({1\hbox{--}3)/4Y_p}))$ is the ionized hydrogen comoving number density (x_i is the ionization fraction, n_b the baryonic comoving number density). The free electron density can be approximated by n_e = x_in_b.

The recombination coefficients are a function of the IGM temperature, T_K. The case A comoving recombination coefficient is appropriate for the highly ionized low-redshift universe (Furlanetto et al. 2006),

$$\begin{equation} \alpha _A\approx 4.2 \times 10^{-13}(T_K/10^4\ {\rm K})^{-0.7}(1+z)^3\ {\rm cm^3\ s}^{-1}, \end{equation} \tag{ 35 }$$

while the case B comoving recombination coefficient is appropriate for the high-redshift universe,

$$\begin{equation} \alpha _B\approx 2.6 \times 10^{-13}(T_K/10^4\ {\rm K})^{-0.7}(1+z)^3\ {\rm cm^3\ s}^{-1}. \end{equation} \tag{ 36 }$$

The use of a larger recombination coefficient when the process of hydrogen recombination is close to its end accounts for the fact that, at this time, ionizations (and hence recombinations) take place in dense, partially neutral gas (Lyman-limit systems) and the photons produced after recombinations are consumed inside these systems, so they do not help ionize the IGM (see, e.g., Furlanetto et al. 2006).

The fraction of Lyα photons emitted per hydrogen recombination, f_rec, is temperature dependent, so we used the parameterization for f_rec made by Cantalupo et al. (2008) using a combination of fits tabulated by Pengelly (1964) and Martin (1988) for T_K > 10³ and T_K < 10³, respectively:

$$\begin{equation} f_{\rm rec}=0.686\hbox{--}0.106\log _{10}(T_K/10^4\ {\rm K})-0.009(T_K/10^4\ {\rm K})^{-0.4}. \end{equation} \tag{ 37 }$$

The luminosity density (per comoving volume) in Lyα from hydrogen recombinations in the IGM, ℓ^IGM_rec, is then given by

$$\begin{equation} {\rm \ell }^{\rm IGM}_{\rm rec}(z)= f_{\rm rec} \dot{n}_{\rm rec} E_{\rm Ly\alpha }. \end{equation} \tag{ 38 }$$

The UV radiation that escapes the galaxies without producing ionization ends up ionizing and exciting the neutral hydrogen in the IGM and heating the gas around the galaxies. The high energetic electron released after the first ionization spends its energy in collisions/excitations, ionizations, and heating the IGM gas until it thermalizes (Shull & van Steenberg 1985). We estimated the contribution of the direct collisions/excitations to the Lyα photon budget and concluded that it is negligible.

The Lyα luminosity density due to the collisional emission (radiative cooling in the IGM), ℓ^IGM_exc, is given by

$$\begin{equation} {\rm \ell }^{\rm IGM}_{\rm exc}(z)= n_e n_{{\rm H\,\scriptsize{I}}}q_{\rm Ly\alpha } E_{\rm Ly\alpha }, \end{equation} \tag{ 39 }$$

where $n_{{\rm H\,\scriptsize{I}}}=n_b (1-x_i)({(1-Y_p)}/(({1\hbox{--}3})/4Y_p))$ is the neutral hydrogen density, x_i is the IGM ionized fraction, and q_Lyα is the effective collisional excitation coefficient for Lyα emission, which we calculated in the same way as Cantalupo et al. (2008), but using different values for the gas temperature and IGM ionization fraction.

Considering excitation processes up to the level n = 3 that could eventually produce Lyα emission, the effective collisional excitation coefficient is given by

$$\begin{equation} q_{\rm Ly\alpha }=q_{1,2p}+q_{1,2s}+q_{1,3p}. \end{equation} \tag{ 40 }$$

The collisional excitation coefficient for the transition from the ground level (1) to the level (nl) is given by

$$\begin{equation} q_{1,nl}=\frac{8.629\times 10^{-6}}{T_K^{1/2}}\frac{\Omega (1,nl)}{\omega _1}e^{E_{1,n}/k_B T_K}\ {\rm cm}^{3}\ {\rm s}^{-1}, \end{equation} \tag{ 41 }$$

where Ω(1, nl) is the temperature-dependent effective collision strength, ω₁ is the statistical weight of the ground state, E_{1, n} is the energy difference between the ground and the nl level, and k_B is the Boltzmann constant.

Continuum emission of photons, by stars, from Lyα to the Lyman limit travels until it reaches one of the Lyn lines, where it gets scattered by neutral hydrogen. Most of this scattering will have as an end result the production of Lyα photons, which eventually redshift out of the line. Since a considerable fraction of these photons only reach a given Lyn frequency in the IGM, this Lyα emission is formed as a flux that decays with r² around the star that emitted the continuum photons, so it appears diluted in frequency in line observations of point sources (Chen & Miralda-Escudé 2008). These continuum photons are much less likely to be absorbed by the dust in the ISM than photons originated in recombinations.

In intensity mapping, the frequency band observed is much larger than in line observations, so in principle all the continuum Lyα photons can be detected. Using the spectral energy distribution (SED) made with the code from Maraston (2005), we estimated that the number of photons emitted by stars between the Lyα plus the Lyα equivalent width and the Lyman limit is equivalent to Q^IGM_Lyn = 9.31 × 10⁶⁰ M⁻¹_☉ s⁻¹. The higher frequency photons are absorbed by hydrogen atoms as they reach the Lyβ frequency, re-emitted, and suffer multiple scattering until they reach the Lyα line. The fraction of the continuum photons emitted close to the Lyα line have already redshifted to lower frequencies before reaching the IGM, so they will not be scattered by the neutral hydrogen in the IGM and will not contribute to the radiative coupling of the 21 cm signal (they are already included in the calculation of the Lyα emission from galaxies).

The intensity of this emission was calculated with a stellar emissivity that evolves with frequency as ν^−α with α = 0.86 and normalized to Q^IGM_Lyn. The Lyα luminosity density originated from continuum stellar radiation and emitted in the IGM, ℓ^IGM_cont, is then approximately given by

$$\begin{equation} {\rm \ell ^{\rm IGM}_{\rm cont}}(z)\approx Q^{\rm IGM}_{{\rm ly}\alpha }E_{\rm Ly\alpha }{\rm SFRD}(z), \end{equation} \tag{ 42 }$$

where the SFRD is in units of M_☉ s⁻¹. Note that in Section 4, this calculation is done through a full simulation.

We calculated the intensities for the several Lyα sources in the IGM from their luminosity densities using

$$\begin{equation} \bar{I}_{\rm IGM}(z)=\frac{{\rm \ell ^{\rm IGM}}(z)}{4\pi D_L^2}y(z)D^2_A. \end{equation} \tag{ 43 }$$

The luminosity and hence the intensity of Lyα emission in the IGM depend on local values of the hydrogen ionized fraction, the gas temperature, and the gas density. These parameters are correlated with each other, and so theoretical calculations of the average intensity made with the average of these parameters may be misleading. Since this emission is dominated by overdense regions, a clumping factor of a few units is usually assumed in theoretical calculations. However, we decided to estimate this intensity without using a clumping factor since its effect can be extrapolated from the intensity without clumping. The intensity of Lyα emission due to recombinations or collisions in the IGM is shown in Figure 7 as a function of the hydrogen ionized fraction for different values of the gas temperature.

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Even for a fixed average IGM ionized fraction, the intensity of Lyα emission is the result of emission from several regions, and so all the values shown in Figure 7 are relevant. As can be seen in Figure 7, the intensity of Lyα due to recombinations and collisions in the IGM is very sensitive to the gas temperature and to the fluctuations in the IGM ionized fraction.

Numerical simulations predict that the temperatures in the hydrogen gas in the IGM can vary from 5000 to 20,000 K (Davé et al. 2001; Smith et al. 2011). The theoretical intensities of Lyα emission in galaxies and in the IGM shown in Figure 8 indicate that unless the IGM clumping factor is very high, or the Lyα photon escape fraction is very low, the Lyα intensity from the IGM at z = 7 is lower than the emission from galaxies. At higher redshifts the SFRD will decrease, causing the Lyα intensities from galaxies and from the IGM to decrease. The escape fraction of UV photons from galaxies increases as the redshift increases, which will contribute negatively to the intensity of emission in galaxies and positively to the intensity of emission in the IGM. At high redshifts the IGM ionized fraction is small, which contributes to a strong decrease in the intensity of emission from the IGM compared to the intensity at z = 7.

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In the previous version of the SimFast21 code, the IGM ionized fraction was computed assuming that at each redshift the ionization state of a region could be estimated from the collapsed mass in that region assuming a linear relation between collapsed mass and ionizing power. So a given spherical region of radius R is considered ionized if (Furlanetto et al. 2006)

$$\begin{equation} \zeta M_{\rm coll}(R) \ge M_{\rm tot}(R), \end{equation} \tag{ 44 }$$

where M_coll is the collapsed mass that corresponds to the total mass in halos in that region, M_tot is the total mass in the region, and ζ is an ionizing efficiency parameter. This efficiency parameter tries to include all the ionizations and recombinations produced by a halo as a function of its mass but has no actual physical meaning, although its use is somewhat justified by the large uncertainty in the astrophysical quantities involved in the determination of the relation between halo mass and ionizing efficiency and in the adjustment of this parameter in order to reproduce a reionization history compatible with observations.

In order to calculate the Lyα field, however, we need to include the recombinations in the IGM explicitly, as well as directly relate the ionization process to the emitted stellar radiation. We therefore modified the SimFast21 code to include these improvements. This new method allows a nonlinear relation between collapsed mass and ionizing power, and all the parameters involved in the calculation have values based on current astrophysical constraints. Also, the size of ionized regions is now set by the volume at which the total ionizing emissivity of the sources it contains equals the number of recombinations so that the system is in equilibrium. For each redshift the implementation of this method was done with the following steps:

• 1.  
  A halo catalog with the mass and three-dimensional spatial positions was generated using the same method used in the original version of the SimFast21 code.
• 2.  
  We calculated SFRs from the halo catalogs using the nonlinear relations in Equations (24)–(26).
• 3.  
  We used Equation (4) to obtain the halo ionizing rate, $\dot{N}_{\rm ion}$, and we corrected for the presence of helium using A_He and multiplied it by f_esc, to account for the photons consumed inside the galaxies:

  $$\begin{equation} \dot{N}_{\rm ion}^{\rm IGM}(z,M)=A_{\rm He}\dot{N}_{\rm ion}(M)f_{\rm esc}(z,M). \end{equation} \tag{ 45 }$$

  The UV ionizing rates of the halos, $\dot{N}_{\rm ion}^{\rm IGM}$, were then put in three-dimensional boxes.
• 4.  
  Three-dimensional boxes with the rate of recombinations in each cell, $\dot{N}_{\rm rec}^{\rm IGM}=V_{{\rm cell}}\times \dot{n}_{\rm rec}^{\rm IGM}$, were obtained from a dark matter density simulation made with the SimFast21 code using Equation (34) with x_i set to 1 and T_K = 10⁴ K.
• 5.  
  Following the same procedure as in the original version of the SimFast21 code, we applied a series of top-hat filters of decreasing size (this filtering procedure was done in Fourier space) to the ionizing rate and the recombination rate boxes in order to calculate the region ionizing rate and recombination rate.
• 6.  
  At each filtering step of radius R we found the ionized regions (H ii bubbles) by checking if the region ionizing rate was equal to or higher than its recombination rate. With this method H ii bubbles are always fully ionized:

  $$\begin{equation} \dot{N}_{\rm ion}^{\rm IGM}(z,R) \ge \dot{N}_{\rm rec}^{\rm IGM}. \end{equation} \tag{ 46 }$$

The SimFast21 code was built to calculate the IGM ionized state assuming two types of regions: one fully ionized (inside the H ii bubbles) and the other fully neutral. The intensity of Lyα emission in the IGM due to recombinations is a smooth function of the IGM ionized fraction and is dominated by emission from fully ionized regions (see Figure 7), so the output of the SimFast21 code is good enough to estimate this intensity.

Collisions between electrons and neutral hydrogen atoms can also lead to Lyα emission; however, as was explained in Section 3.2, collisional Lyα emission only occurs in partly ionized regions, mainly in the edge of H ii bubbles, so the estimation of this emission requires a more detailed description of the IGM ionized state than the one given by the limited resolution of semi-numerical simulations. Collisions are most important in regions where the IGM ionized fraction is locally close to 0.5 and the temperatures are high. Since high-temperature regions are likely to be highly ionized, we can deduce with the help of Figure 7 that Lyα emission from recombinations is dominant over Lyα emission from collisions in the IGM.

The IGM Lyα intensity from scattering of Lyn photons emitted from galaxies can also be calculated using data from the code SimFast21. This code uses Equation (10) in Santos et al. (2010) to calculate the spherical average of the number of Lyα photons, J_α, hitting a gas element per unit proper area per unit time per unit frequency per steradian. The Lyα intensity originated from these continuum photons is given by

$$\begin{equation} I^{\rm IGM}_{\rm cont}=\frac{6 J_{\alpha }E_{{\rm Ly}\alpha }D_{\rm A}^2}{(1+z)^2 D_{\rm L}^2}. \end{equation} \tag{ 47 }$$

Using the prescriptions described in the previous sections, we ran simulations Sim1, Sim2, and Sim3 with a volume of 54³ h⁻³ Mpc³ and 1800 cells from redshift 14 to redshift 6. The obtained IGM ionization fractions, at redshift 7, where x_i  = 0.86 for simulation Sim1 and x_i  = 1.0 for simulations Sim2 and Sim3. These values are consistent with the current most likely values for this parameter, 0.8 ⩽ x_i(z = 7) ⩽ 1.0 (Mitra et al. 2012). The IGM ionized fraction evolution for Sim2 and for Sim3 (see Figure 9) resulted in optical depths to reionization of 0.073 and 0.082. These optical depths are consistent with the value obtained by WMAP (τ = 0.088 ± 0.015; Komatsu et al. 2011). The optical depth corresponding to Sim1 is 0.66, which is lower than the current observational constraints. Based on the optical depth constraint, Sim2 and Sim3 have the most likely reionization histories, and the IGM ionized fraction evolution obtained with Sim1 can be seen as a lower bound.

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The intensities of Lyα emission from galaxies at redshift 7 obtained with the SimFast21 code are similar to the more theoretical estimates summarized in Table 3.

Intensities of Lyα emission in the IGM made with the same code are presented in Table 5.

The intensity values found in Tables 3 and 5 and the theoretical estimations plotted in Figure 8 indicate that for the Lyα intensity from the IGM to reach a value close to the emission from galaxies at z = 7 would require a very large absorption of Lyα photons by dust in galaxies.

The resulting power spectra of Lyα emission in galaxies and in the IGM obtained with the SimFast21 code are presented in Figure 10 for z = 7 and z = 10.

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We repeated the Lyα power spectra calculation for several redshifts during the EoR and plotted the Lyα power spectra as a function of redshift for several k in Figure 11.

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We calculated the intensity of Lyα emission from galaxies and from the IGM (intensities are shown in Figure 12) and found that, according to our assumptions and as already previously seen, the Lyα emission from galaxies is dominant over the Lyα emission from the IGM at least during the redshift interval from z = 6 to z = 9.

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Also, a map of the total Lyα intensity in galaxies and in the IGM is presented in Figure 13 for z = 7.

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Since the star formation halo mass relation is not very constrained, we can use the results obtained with Sim1 and Sim3 as the lower and upper bounds to the expected Lyα intensity. The evolution of the Lyα intensity from galaxies, from the IGM, and from galaxies plus IGM can be seen, respectively, in Tables 6, 7, and 8 for simulations Sim1, Sim2, and Sim3.