INTENSITY MAPPING ACROSS COSMIC TIMES WITH THE Lyα LINE

The halo luminosity for the Lyα line due to recombinations is given by

$$\begin{eqnarray} &&\overline{L}_\nu ^{\rm halo}(m,z) = h_P\nu _{\rm Ly\alpha }\overline{Q}_H(m)f_{\rm Ly\alpha }\delta _D(\nu _{\rm Ly\alpha }-\nu). \end{eqnarray} \tag{ 21 }$$

The emissivity can then be written in the form

$$\begin{eqnarray} p_{\rm halo}(\nu,z) &=& (1-f_{\rm dust})(1-f_{\rm esc})f_{\rm Ly\alpha }h_P\nu _{\rm Ly\alpha }\delta _D(\nu _{\rm Ly\alpha }-\nu)\nonumber \\ &&\times \frac{\dot{\rho _*}(z)}{m_*}\int dm\,f(m)\overline{Q}_H(m)\tau (m). \end{eqnarray} \tag{ 22 }$$

When inserting this into the mean intensity integral, we take advantage of the fact that the line profile is a delta function. This makes the integral simple, giving the form of the mean intensity as

$$\begin{eqnarray} I_{\rm halo}^{\rm Ly\alpha }(z_{\rm Ly\alpha })&=&[1-f_{\rm dust}(z_{\rm Ly\alpha })][1-f_{\rm esc}(z_{\rm Ly\alpha })]f_{\rm Ly\alpha }\nonumber \\ &&\times \frac{ch_P}{4\pi H(z_{\rm Ly\alpha })}\dot{\rho _*}(z_{\rm Ly\alpha })Q_{\rm ion}^1(z_{\rm Ly\alpha }), \end{eqnarray} \tag{ 23 }$$

where $Q_{\rm ion}^1$ is the stellar mass integral divided by m_* and z_Lyα = ν_Lyα/ν_obs − 1 is the initial redshift of the Lyα photon.

The only temperature-dependent quantity, f_Lyα, varies very little with temperature. Thus, we will assume a constant gas temperature in the halo $T_{\rm gas}^{\rm halo}=2\times 10^4$ K, which is consistent with simulations (Trac et al. 2008) and Lyα forest measurements (Bolton et al. 2012). For this temperature, we can set f_Lyα = 0.64. $Q_{\rm ion}^1$ is of order $10^{60}\,M_\odot ^{-1}$ for the models we consider. We give the values for the various stellar population models in Table 3. In our model, similarly to Cooray et al. (2012a), we assume that Population II stars dominate the signal for z < 10 and that Population III stars dominate for z > 10, which we implement by constructing the toy model $Q_{\rm ion}^1(z) = Q_{\rm ion,Pop\,II}^1(1-f_P(z))+Q_{\rm ion,Pop\,III}^1f_P(z)$, where

$$\begin{eqnarray} &&f_P(z) = \frac{1}{2}\left[1+\tanh \left(\frac{z-10}{0.5}\right)\right]. \end{eqnarray} \tag{ 24 }$$

Substituting these values gives the expression for our case

$$\begin{eqnarray} I_{\rm halo}^{\rm Ly\alpha }(z_{\rm Ly\alpha })&=&0.0049\,{\rm nW\,m^{-2}\,THz^{-1}\,str^{-1}}(1-f_{\rm dust})\nonumber \\ &&\times (1-f_{\rm esc})\left(\frac{H(z_{\rm Ly\alpha }=6)}{H(z_{\rm Ly\alpha })}\right)\left(\frac{Q_{\rm ion}^1}{10^{60}\,M_\odot ^{-1}}\right)\nonumber \\ &&\times \left(\frac{\dot{\rho _*}(z_{\rm Ly\alpha })}{10\,M_\odot \,{\rm yr^{-1}\,Mpc^{-3}}}\right). \end{eqnarray} \tag{ 25 }$$

Table 3.  $Q_{\rm ion}^1$, $Q_{\rm ion}^2$, and T_ion for Various Stellar Metallicities

Metallicity	$Q_{\rm ion}^1$	$Q_{\rm ion}^2$	Tion
		$(10^{60}\,M_\odot ^{-1})$	$(10^{67}\,M_\odot ^{-1}\,{\rm yr})$	$(10^5\,M_\odot ^{-1}\,{\rm yr})$
Population II	8.3	4.5	2.3
Population III	23.0	18.0	16.0

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An equivalent way to write this result is to express the luminosity density of Lyα photons as

$$\begin{eqnarray} &&\rho _L(z)=f_{\rm Ly\alpha }(1-f_{\rm dust})(1-f_{\rm esc})h_P\nu _{\rm Ly\alpha }\dot{n}_{\rm ion}(z), \end{eqnarray} \tag{ 26 }$$

where n_ion(z) is the rate per comoving volume of ionizing photons produced by stars in the halo, given by

$$\begin{eqnarray} &&\dot{n}_{\rm ion}(z)=Q_{\rm ion}^1(z)\dot{\rho _*}(z), \end{eqnarray} \tag{ 27 }$$

which can be converted to an intensity using Equation (3). It is possible that the SFRD used in our analysis does not include all stars that contribute Lyα photons. In particular, we may be missing the contribution from low-mass stars and dwarfs. We can set a lower limit to $\dot{n}_{\rm ion}$ by requiring the rate per comoving volume of photons that ionize the IGM $f_{\rm esc}\dot{n}_{\rm ion}$ to be greater than $\dot{n}_{\rm rec}$, the rate per comoving volume of recombinations in the IGM, in order for net reionization to be sustained. The recombination rate can written as (Madau et al. 1999; Salvaterra et al. 2011)

$$\begin{eqnarray} &&\dot{n}_{\rm rec}=10^{50.0}C(z)\left(\frac{1+z}{7}\right)^3\,{\rm s^{-1}\,Mpc^{-3}}, \end{eqnarray} \tag{ 28 }$$

where $C(z)=\langle n_e^2 \rangle /\left\langle n_e \right\rangle ^2$ is the clumping factor for ionized hydrogen, which we set to $C(z)=26.2917 e^{-0.1822z+0.003505z^2}$ (Iliev et al. 2007). By setting $\dot{n}_{\rm rec}/f_{\rm esc}$ as a lower limit to $\dot{n}_{\rm ion}$, we find that the intensity of Lyα photons cannot be less than

$$\begin{eqnarray} I_{\rm halo,\,min}^{\rm Ly\alpha }(z_{\rm Ly\alpha })&=&1.55\times 10^{-5}\,{\rm nW\,m^{-2}\,THz^{-1}\,str^{-1}}\nonumber \\ &&\times \left(\frac{1-f_{\rm esc}}{f_{\rm esc}}\right)\left(\frac{H(6)}{H(z_{\rm Ly\alpha })}\right)\left(\frac{C(z_{\rm Ly\alpha })}{C(6)}\right)\nonumber \\ &&\times \left(\frac{1+z_{\rm Ly\alpha }}{7}\right)^3(1-f_{\rm dust}).\, \end{eqnarray} \tag{ 29 }$$

We plot both estimates of the Lyα emission from halos in Figure 3, and we see that for redshifts z_Lyα ≳ 5.5, the minimum required emission is greater than the estimated value based on our SFRD. This can greatly impact the total emission in our model, particularly at very high redshifts. Thus, in our analysis we will use the SFRD model for lower redshifts where it is greater, but we will use the minimum reionization value at higher redshifts. Note that this is indeed a lower limit; the actual emission can be even higher, particularly closer to the completion of reionization where we would expect the IGM to receive an excess of ionizing photons. Also, these minimum levels are very dependent on the values of the clumping factor and the escape fraction. To illustrate this, we also plot in Figure 3 the minimum intensity predicted for various values of cf = C(1 − f_esc)/f_esc, where we see that this minimum reionization condition kicks at earlier redshifts for higher values of cf.

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Silva et al. (2013) point out that there are other sources of Lyα photons in halos that can contribute to the intensity mapping signal. These include excitations in the hydrogen gas, gas cooling, and cascading Lyn photons from continuum emission (within the halo). We decline to include these mechanisms because they are all negligible compared to the signal from recombinations. The highest of these other contributions, namely excitations, is about an order of magnitude below the recombinations signal because only 10% of the excitation energy due to collisions, which is close to the Lyα energy, is emitted as Lyα photons, whereas halo emission emits almost all its ionizing energy.

Ionizing photons from stars that escape their halos can ionize neutral hydrogen in the IGM. The resulting ions and electrons can produce Lyα emission through recombination, similar to halo emission. The IGM luminosity from stars can be given as

$$\begin{eqnarray} \overline{L}_\nu ^{\rm igm*}(m,z) &=& f_{\rm esc}h_P\nu _{\rm Ly\alpha }\frac{\overline{Q}_H(m)\tau (m)}{n_H}n_en_pf_{\rm Ly\alpha }\alpha _B\nonumber \\ &&\times \delta _D(\nu _{\rm Ly\alpha }-\nu), \end{eqnarray} \tag{ 30 }$$

where we assume n_e = n_p = n_H due to charge neutrality.

This parameterization gives the emissivity in the form

$$\begin{eqnarray} p_{\rm igm*}(\nu,z) &=& f_{\rm esc}f_{\rm Ly\alpha }\alpha _Bh_P\nu _{\rm Ly\alpha }\phi (\nu _{\rm Ly\alpha }-\nu,z)C(z)n_H(z)\nonumber \\ &&\times \frac{\dot{\rho _*}(z)}{m_*}\int dm\,f(m)\overline{Q}_H(m)\tau ^2(m). \end{eqnarray} \tag{ 31 }$$

When calculating the line intensity, we treat the line profile in the same manner as the previous section; therefore, our expression for the line intensity due to the IGM can be written in the form

$$\begin{eqnarray} I_{\rm igm*}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=& f_{\rm esc}f_{\rm Ly\alpha }\alpha _B\frac{h_Pc}{4\pi H(z_{\rm Ly\alpha })}C(z_{\rm Ly\alpha })n_H(z_{\rm Ly\alpha })\nonumber \\ &&\times \dot{\rho _*}(z_{\rm Ly\alpha })Q_{\rm ion}^2, \end{eqnarray} \tag{ 32 }$$

where $Q_{\rm ion}^2$ is the stellar mass integral in this equation divided by m_*, which is on the order of $10^{67}\,M_\odot ^{-1}$ yr for the models we consider. We vary $Q_{\rm ion}^2$ with redshift in a similar manner to $Q_{\rm ion}^1$ in the previous section.

We assume an IGM gas temperature of $T_{\rm gas}^{\rm halo}=2\times 10^4$ K, which sets f_Lyαα_B = 9.25 × 10⁻¹⁴ cm³ s⁻¹. We will ignore the perturbations in n_H (which may underestimate the signal) and set n_H (z) = 1.905 × 10⁻⁷(1 + z)³ cm⁻³, the mean hydrogen number density. Substituting these values gives us

$$\begin{eqnarray} I_{\rm igm*}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=&1.45\times 10^{-3}\,{\rm nW\,m^{-2}\,THz^{-1}\,str^{-1}}f_{\rm esc}\nonumber \\ &&\times \left(\frac{1+z_{\rm Ly\alpha }}{7}\right)^3\left(\frac{f_{\rm Ly\alpha }\alpha _B}{9.25\times 10^{-14}{\rm \,cm^3\,s^{-1}}}\right)\nonumber \\ &&\times \left(\frac{C(z_{\rm Ly\alpha })}{C(6)}\right)\left(\frac{Q_{\rm ion}^2}{10^{68}\,M_\odot ^{-1}\,{\rm yr}}\right)\nonumber \\ &&\times \left(\frac{\dot{\rho _*}(z_{\rm Ly\alpha })}{10\,{M_\odot \,{\rm yr^{-1}\,Mpc^{-3}}}}\right)\left(\frac{H(6)}{H(z_{\rm Ly\alpha })}\right). \end{eqnarray} \tag{ 33 }$$

Outside the stellar IGM, the ionization state transitions from fully ionized to partially ionized. In this region, electrons, protons, and neutral hydrogen exists, producing emission from recombinations and collisional excitations. At our fiducial temperature of T_gas = 2 × 10⁴ K, collisional excitations will dominate the spectrum. The luminosity of this region is given by

$$\begin{eqnarray} \overline{L}_\nu ^{\rm tz}(m,z) &=& h_P\nu _{\rm Ly\alpha }q_{\rm eff}\delta _D(\nu _{\rm Ly\alpha }-\nu)\nonumber \\ &&\times 4\pi \int _0^{S_{\rm TZ}} dr\,r^2n_e(r)n_{\rm H{\scriptsize {I}}}(r)\nonumber \\ &=& h_P\nu _{\rm Ly\alpha }q_{\rm eff}C(z)n_H^2(z)\delta _D(\nu _{\rm Ly\alpha }-\nu)\nonumber \\ &&\times 4\pi \int _0^{S_{\rm TZ}} dr\,r^2x_e(r)[1-x_e(r)], \end{eqnarray} \tag{ 34 }$$

where S_TZ is given in Equation (8). We approximate the ionization fraction as a linear function with x_e (r = 0) = 1 and x_e (r = S) = 0, getting an integral value of $S_{\rm TZ}^3/20$. Note that it would be more precise to integrate from the end of the ionized region, not r = 0, but the emission volume, which just depends on the number of atoms that get ionized, should not depend on the inner radius of the transition zone. Thus, our approximation should not affect the integration much, although the distribution of x_e in the volume will be dependent on this radius.

Similar to the previous sections, we can convert this to an emissivity and then an intensity given by

$$\begin{eqnarray} I_{\rm tz}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=& \frac{h_Pc}{H(z_{\rm Ly\alpha })}C(z_{\rm Ly\alpha })n_H^2(z_{\rm Ly\alpha })q_{\rm eff}\nonumber \\ &&\times \frac{S_{\rm TZ}^3}{20}\dot{\rho _*}(z_{\rm Ly\alpha })T_{\rm ion}, \end{eqnarray} \tag{ 35 }$$

where

$$\begin{eqnarray} &&T_{\rm ion} = \frac{1}{m_*}\int dm\,f(m)\tau (m). \end{eqnarray} \tag{ 36 }$$

Our fiducial temperature sets q_eff = 5.27 × 10⁻¹¹ cm³ s⁻¹. The mean free path, assuming 〈ν〉 = 3ν_LL , the Lyman-limit frequency, is 1.46(1 + z)⁻³ Mpc. We also fiducially set $T_{\rm ion} = 10^5\,{{\rm yr}\, M_\odot ^{-1}}$. This gives us

$$\begin{eqnarray} I_{\rm tz}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=&7.813\times 10^{-10}\,{\rm nW\,m^{-2}\,THz^{-1}\,str^{-1}}\nonumber \\ &&\times \left(\frac{7}{1+z_{\rm Ly\alpha }}\right)^3\left(\frac{q_{\rm eff}(T_{\rm gas})}{5.27\times 10^{-11}\,{\rm cm^3\,s^{-1}}}\right)\nonumber \\ &&\times \left(\frac{C(z_{\rm Ly\alpha })}{C(6)}\right)\left(\frac{S(1+z_{\rm Ly\alpha })^3}{1.46\,{\rm Mpc}}\right)^3\nonumber \\ &&\times \left(\frac{\dot{\rho _*}(z_{\rm Ly\alpha })}{10{\,M_\odot \,\rm yr^{-1}\,Mpc^{-3}}}\right)\left(\frac{H(6)}{H(z_{\rm Ly\alpha })}\right)\nonumber \\ &&\times \left(\frac{T_{\rm ion}}{10^5\,{\rm yr}\,M_\odot ^{-1}}\right). \end{eqnarray} \tag{ 37 }$$

This emission is only nonnegligible at very low redshifts (z ≲ 1). However, at these redshifts all the IGM should be ionized, eliminating any transition zones. This formalism does not account for this effect. Thus, we will neglect this emission for the rest of this analysis.

Separate from the line emission resulting from discrete Lyα emitters are diffuse components to Lyα emission from the IGM. The first contribution we consider results from recombinations and collisions of electrons and H ions apart from ionized halos. Our prediction for IGM emission will differ from that of Fernandez et al. (2010) because their model only includes emission from the IGM ionized by discrete source like stars and quasars, while our model allows all the IGM to be equally ionized. This emission would require diffuse sources of ionizing photons such as decaying or annihilating particles would be required to produce homogeneous reionization and diffuse IGM emission, though these are constrained by cosmic microwave background experiments (Furlanetto et al. 2006). However, we model the case where an unknown mechanism produces a homogeneous ionization fraction X_e (z) where electrons, electrons, and neutral hydrogen can easily interact.

The diffuse emission can be calculated using Equation (11), except that the emissivity is now of the form

$$\begin{eqnarray} p_{\rm diff\,rec/coll}(\nu,z) &=& (1+z)^3h_P\nu _{\rm Ly\alpha }[n_en_p f_{\rm Ly\alpha }(T)\alpha _B(T)\nonumber \\ && +n_{e} n_{\rm H{\scriptsize {I}}}q_{\rm eff}(T)]\phi (\nu _{\rm Ly\alpha }-\nu)\,, \end{eqnarray} \tag{ 38 }$$

where n_e = n_p = X_en_H , n_HI = (1 − X_e )n_H , n_H is the comoving number density of hydrogen equal to n_H (z = 0), and X_e (z) is the ionization fraction. In our fiducial model, we will set the gas temperature in the IGM $T_{\rm gas}^{\rm IGM}=10^3$ K, a typical IGM gas temperature for Population III stars (Furlanetto et al. 2006), though we note that this gives a conservative estimate of the signal in that lower temperatures tend to produce more recombinations. In this case, f_Lyα = 0.77 and α_B = 1.49 × 10⁻¹² cm³ s⁻¹. Note that using the case-A recombination coefficient (1.6 times larger than case-B) is more accurate at low redshifts when the IGM is highly dense and ionized (Furlanetto et al. 2006). Thus, we will scale α with X_e (z) as a rough estimate to be case B at high redshifts and case A at low redshifts.

To model the ionization fraction X_e (z), we simply use the toy model assuming a spatially averaged rapid reionization,

$$\begin{eqnarray} &&X_e(z) = \frac{1}{2}\left[1+\tanh \left(\frac{y(z_{\rm re})-y(z)}{\Delta y}\right)\right], \end{eqnarray} \tag{ 39 }$$

where y(z) = (1 + z)^3/2, $\Delta y = 1.5\sqrt{1+z_{\rm re}}\Delta z$, z_re is the redshift halfway through the reionization epoch, and Δz = z_re − z_end. We choose reionization to begin at z = 11 and end at z = 7 (consistent with WMAP data), giving us z_re = 9 and Δy = 9.5. We plot X_e (z) in Figure 4. This is meant only as an illustrative example that matches the data, not a prediction.

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Unlike the previous two cases, when we calculated an LF to construct the emissivity, this time we use the emissivity directly since the emission is not associated with stars. This gives us the expression

$$\begin{eqnarray} I_{\rm diff\,rec/coll}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=& \frac{h_Pc(1+z_{\rm Ly\alpha })^3}{4\pi H(z_{\rm Ly\alpha })}X_e^2(z_{\rm Ly\alpha })f_{\rm Ly\alpha }(T)\nonumber \\ &&\times \alpha (T)C(z)n_H^2. \end{eqnarray} \tag{ 40 }$$

The hydrogen producing this emission is confined to halos that do not produce ionizing stars. Thus, we write

$$\begin{eqnarray} I_{\rm diff\,rec/coll}^{\rm Ly\alpha }(z_{\rm Ly\alpha }) &=& 1.0\times 10^{-4}{\rm nW\,m^{-2}\,THz^{-1}\,str^{-1}}\nonumber \\ &&\times \left(\frac{1+z_{\rm Ly\alpha }}{7}\right)^3\left(\frac{H(6)}{H(z_{\rm Ly\alpha })}\right)\left(\frac{C(z)}{C(6)}\right)\nonumber \\ &&\times \left(\frac{X_e^2f_{\rm Ly\alpha }\alpha }{1.15\times 10^{-12}\,{\rm cm^3\,s^{-1}}}\right). \end{eqnarray} \tag{ 41 }$$

The second contribution we consider results from continuum photons from stellar regions that redshift into Lyman-series frequencies, are absorbed by regions of neutral hydrogen, and cascade in the Lyα photons. This photon emission is important for the Wouthuysen–Field effect (Wouthuysen 1952; Field 1959) that couples Lyα photons to 21 cm photons; however, the Lyα photons are an important source of emission in and of themselves. Several articles give the formula for the intensity (Barkana & Loeb 2005; Hirata 2006; Santos et al. 2008); however, the formulae assume the high-redshift universe, where there is no ionized hydrogen. Since we are interested in all cosmic times, we must account for this by including factors denoting the transmission/absorption probabilities.

The relevant formula for our case is given by

$$\begin{eqnarray} I_{\rm diff\,cont}^{\rm Ly\alpha }(z) &=& \frac{h_P\nu _{\rm Ly\alpha }}{4\pi }\sum _{n=2}^{\infty }f_{\rm rec}(n)\int _z^\infty dz^{\prime }\,\frac{c}{(1+z^{\prime })H(z^{\prime })}\nonumber \\ &&\times \dot{\rho _*}(z^{\prime })\epsilon (\nu _n^{\prime }) P_{\rm abs}(n,z)\nonumber \\ &&\times \prod _{n^{\prime }=n+1}^{n_{\rm max}}\lbrace 1-P_{\rm abs}[n^{\prime },z_{n^{\prime }}(z,n)]\rbrace. \end{eqnarray} \tag{ 42 }$$

The factor f_rec(n) is the probability for a Lyn photon to cascade down to a Lyα photon. The continuum spectrum is denoted as (ν), and $\nu _n^{\prime }$ is the emitted frequency at redshift z' that redshifts to a Lyn photons at redshift z, given by

$$\begin{eqnarray} &&\nu _n^{\prime }(z,z^{\prime },n) = \nu _{\rm LL}(1-n^{-2})\left(\frac{1+z^{\prime }}{1+z}\right), \end{eqnarray} \tag{ 43 }$$

where ν_LL is the Lyman-limit frequency and $z_{n^{\prime }}(z,n)$ gives the redshift of higher Lyn' lines that become Lyn, given by

$$\begin{eqnarray} &&1+z_{n^{\prime }}(z,n) = (1+z)\frac{1-(n^{\prime })^{-2}}{1-n^{-2}}. \end{eqnarray} \tag{ 44 }$$

We set the spectrum (ν)∝ν^−α with α = 0.86 and normalized such that the total number of continuum photons emitted per baryon between the Lyα line and the Lyman limit is 9690 for Population II stars and 4800 with α = −0.29 for Population III stars (Barkana & Loeb 2005). P_abs(n, z) is the probability that a Lyn photon will be absorbed by the IGM at redshift z. The first P_abs factor just accounts for the fact that not all Lyn-frequency photons will be absorbed by the IGM, in particular at low redshifts where the neutral hydrogen density is low. The product of 1 − P_abs(n', z') factors accounts for the idea that a continuum photon that is absorbed at redshift z with frequency ν_n has to transmit through regions where its frequency is $\nu _{n^{\prime }}$ (not $\nu _n^{\prime }$) at redshift z'. And n_max is the maximum n' such that a photon with frequency $\nu _{n^{\prime }}$ at redshift z' will reach redshift z with frequency ν_n , given by

$$\begin{eqnarray} &&n_{\rm max}(z,z^{\prime }) = \left[1-\left(1-\frac{1}{n^2}\right)\left(\frac{1+z^{\prime }}{1+z}\right)\right]^{-1/2}. \end{eqnarray} \tag{ 45 }$$

The redshift z' where n_max → ∞ is z_max, which is also the redshift such that a Lyman-limit photon could redshift down to a Lyn photon, given by

$$\begin{eqnarray} &&1+z_{\rm max}(z,n) = \frac{1+z}{1-\frac{1}{n^2}}. \end{eqnarray} \tag{ 46 }$$

The z_max value is important because if n_max → ∞, then the product of 1 − P_abs(n', z') in the integrand vanishes because all the entries are less than unity, and an infinite product over values less than unity must tend to zero. Physically, this just means that the probability of a photon not being absorbed while redshifting through an infinite number of Lyn lines is nil.

The absorption probability can be written as P_abs(n, z) = 1 − 〈e^−τ〉, where the average is performed assuming the probability of IGM density perturbations P(Δ, z) as (Miralda-Escudé et al. 2000)

$$\begin{eqnarray} &&\langle e^{-\tau } \rangle (z) = \int _0^{100} d\Delta \,P(\Delta,z)e^{-\Delta ^2\tau }, \end{eqnarray} \tag{ 47 }$$

where P(Δ, z) is given by

$$\begin{eqnarray} &&P(\Delta,z) = A(z)\exp \left\lbrace -\frac{[\Delta ^{2/3}-C_0(z)]^2}{2[2\delta _0(z)/3]^2}\right\rbrace \Delta ^{-\beta }, \end{eqnarray} \tag{ 48 }$$

and the parameters are given in Table 4. The parameters β and δ₀ were fit to simulations for redshifts 2 < z < 6 with δ₀ found to follow the function δ₀ = 7.61/(1 + z). Since neither β nor δ₀ were estimated for redshifts z < 2, we will not predict continuum emission for these redshifts. For redshifts z > 6, we set β = 2.5 and we extrapolate δ₀. The parameters A(z) and C₀(z) are set by normalizing P(Δ, z) and ΔP(Δ, z) to unity.

For the optical depth, we use the Gunn–Peterson optical depth, given by (Gunn & Peterson 1965; Hirata 2006)

$$\begin{eqnarray} &&\tau _{GP}(n,z) = \frac{3n_H(z)[1-X_e(z)]\lambda _{{\rm Ly}n}^3\gamma _n}{2H(z)}, \end{eqnarray} \tag{ 49 }$$

where λ_Lyn is the wavelength of the Lyn line and γ_n is the half width at half maximum of the Lyn resonance, given by

$$\begin{eqnarray} &&\gamma _n = 50\,{\rm MHz}\left(\frac{1-n^{-2}}{0.75}\right)^2\left(\frac{f_n}{0.4162}\right), \end{eqnarray} \tag{ 50 }$$

where f_n is the oscillator strength for the Lyn transition.

We plot the theoretical intensity from the Lyα emitters (Sections 3.1 and 3.2) and the diffuse IGM sources, including recombinations and continuum photons in Figure 5. We see from the figure that the halo emission dominates the Lyα emitter emission at all redshifts. We also plot the empirical predictions of the intensity for z < 8. We see that the theoretical halo emission model predicts a slightly higher intensity than the empirical model but close to the errors. Note that the error bars on the empirical model are larger than true 1σ errors. Also, the diffuse IGM emission does not conflict with the LF measurements because LFs do not account for diffuse emission. To better quantify the empirical model, we perform a chi-squared fit on the common logarithm to a quadric function, finding

$$\begin{eqnarray} \log _{10}\left(I_{\rm emp}^{\rm Ly\alpha }\right)&=&{-}2.687+1.568z-0.6995z^2+0.1146z^3\nonumber \\ &&-0.00666z^4, \end{eqnarray} \tag{ 51 }$$

in units of nW m⁻² THz⁻¹ str⁻¹.

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In addition, for most of the redshifts in our analysis, the diffuse IGM emission from continuum photons dominates the low-redshift signal, while the halo emission dominates the high-redshift signal, unless the diffuse IGM emission due to recombinations plays a role. Note that the signal from IGM continuum photons can vary from the fiducial model due to uncertainties mainly in the ionization fraction at low redshifts, in addition to the SFRD and the stellar emissivity at higher redshifts. The diffuse IGM components provide a strong case for intensity mapping, considering that this probe, contrary to number counts, will access the diffuse IGM emission. It should be recognized that the model for Lyα emitters at high redshift is indeed a lower limit relevant to the minimum reionization condition; the actual emission from Lyα emitters could be much higher. These results lead us to suspect that an intensity map of Lyα emission will trace Lyα emitters at high redshifts, allowing us to probe LSS in the reionization epoch.

The power spectrum for the halo emission, and thus the Lyα emitters, is given by

$$\begin{eqnarray} &&P^{\rm halo}(k,z)=\nu _{\rm obs}\left[I_{\rm halo}^{\rm Ly\alpha }(z)\right]^2 [P_{gg}(k,z)+P^{\rm shot}(z)], \end{eqnarray} \tag{ 53 }$$

where P_gg (k, z) is the galaxy–galaxy clustering power spectrum, and P^shot(z) is the shot noise. We implement the halo-model formalism when calculating P_gg (k, z), but we use only the two-halo clustering term. We neglect the one-halo clustering term and nonlinear clustering at small scales since in our case it will always be dominated by the shot noise. Equation (53) models the power spectrum for the theoretical emission model. For the empirical emission model, we simply replace $I_{\rm halo}^{\rm Ly\alpha }(z)$ with $I_{\rm emp}^{\rm Ly\alpha }(z)$.

We use a galaxy–galaxy power spectrum similar to the power spectrum discussed in Cooray & Sheth (2002). The formulae are given by

$$\begin{eqnarray} P_{gg}(k,z)&=&b_g^2(z)P_{\rm lin}(k,z)\nonumber \\ b_g&=&\frac{\int _{M_{\rm min}}^{M_{\rm max}} { dM}\,\langle N_g(M) \rangle n(M)b(M,z)}{\int _{M_{\rm min}}^{M_{\rm max}} { dM}\,\langle N_g(M) \rangle n(M)}, \end{eqnarray} \tag{ 54 }$$

where b(M, z) is the halo bias, 〈N_g (M)〉 is the mean number of galaxies in a halo of mass M, and b_g becomes the galaxy clustering bias, shown in Figure 6. For the shot noise, we use the expression from dark matter, given by

$$\begin{eqnarray} P^{\rm shot}(z)&=&\int _{M_{\rm min}}^{M_{\rm max}} { dM}\,n(M)\left[\frac{M}{\rho _mf_{\rm coll}}\right]^2. \end{eqnarray} \tag{ 55 }$$

In our calculation we use the entire mass range; however, the higher end of this mass range will be probed by number count surveys like JWST, meaning that any new science from the shot noise in an intensity mapping survey would be captured by a high-mass cutoff corresponding to the minimum flux of a competing number counts survey. This will not affect the shot noise science from the diffuse IGM because it is inaccessible to number count surveys.

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The power spectra for the diffuse IGM components are more complicated than the contributions from halo and stellar IGM emission. For P^dir(k, z), the intensity scales as $n_H^2$ times factors of X_e . We neglect the fluctuations in the ionization fraction in this derivation, which may underestimate the correlation signal. This choice makes the intensity fluctuations of the form

$$\begin{eqnarray} \delta I_{\rm diff\,rec/coll}^{\rm Ly\alpha }(z,\mathbf {\hat{n}}) &=& I_{\rm diff\,rec/coll}^{\rm Ly\alpha }(z)\nonumber \\ &&\times \left\lbrace \frac{n_H^2[\mathbf {\hat{n}}\chi (z)]-\overline{n_H^2}(z)}{\overline{n_H^2}(z)}\right\rbrace. \end{eqnarray} \tag{ 56 }$$

The formula for P^dir(k, z) will be similar to Equation (53), except that we require the three-dimensional power spectrum for $n_H^2$, $P_{H^2}(k)$. We skip the derivation here and just present the result, where

$$\begin{eqnarray} P_{H^2}(k)&=&\frac{1}{2\pi ^2}\int dk^{\prime }\,k^{\prime 2}P_H(k^{\prime })\nonumber \\ &&\times \int _{-1}^{1}d\mu \,P_H(\sqrt{k^2+k^{\prime 2}-2kk^{\prime }\mu }), \end{eqnarray} \tag{ 57 }$$

where P_H (k) is the two-point power spectrum for hydrogen atoms (or baryons). The fluctuations of the baryons do not trace fluctuations in the dark matter, particularly on small scales where baryonic interactions smooth out the perturbations. We chose a toy model for the baryonic power spectrum used by McQuinn et al. (2005), where P_H (k) = F_b (k/k_F )P_gg (k, z), where k_F (z) = 34[Ω_m (z)]^1/2  h Mpc⁻¹ (Gnedin & Hui 1998) filters out small scale perturbations and

$$\begin{eqnarray} &&F_b(x)=\frac{1}{2}\left[e^{-x^2}+\frac{1}{(1+4x^2)^{1/4}}\right]. \end{eqnarray} \tag{ 58 }$$

for z < 8 and F(x) = 1 for z > 8.

The shot noise component for $C_\ell ^{{\rm dir}}$ requires a four-point calculation of areal number densities. We do this for a Poisson distribution and find that the shot noise component for the correlation of a wavemode is $C(\mathbf {k}_i)=Q(\bar{N})/\bar{n}$, where $\bar{n}$ is the average number density of atoms in the IGM and $\bar{N}$ is the average number of IGM sources per pixel, both of which will depend on the minimum luminosity detectable by the spectrograph while $\bar{N}$ also depends on the pixelation scheme, and $Q(\bar{N})$ is given by

$$\begin{eqnarray} &&Q(\bar{N})=\frac{4\bar{N}^2+6\bar{N}+1}{(\bar{N}+1)^2}. \end{eqnarray} \tag{ 59 }$$

$Q(\bar{N})$ only has values between 1 for $\bar{N}\ll 1$ and 4 for $\bar{N}\gg 1$. We decide to set Q = 2, knowing that we could misestimate the shot noise by half an order of magnitude. Also, although this was calculated for only Poisson distribution, we apply this formula to the Poisson-like distribution seen in halo models by setting the shot noise equal to $P_{H^2}^{\rm shot}(z)=Q(\bar{N})P^{\rm shot}(z)\simeq 2P^{\rm shot}(z)$.

The power spectrum for diffuse IGM emission due to the stellar continuum is complicated by having two sources of fluctuations: the local IGM and the stars at higher redshifts that source the emission. The local IGM power spectrum elucidates the environment at a particular redshift, while the power spectrum from the higher-redshift stars mixes perturbations from redshifts greater than the redshift of the signal but less than z_max(z, n = 2), the maximum redshift that a Lyman-limit line can be redshifted to a Lyα line at redshift z. In this analysis we will consider both power spectra.

For the local IGM, the perturbations come from the factor P_abs(n, z) in Equation (42), which contains perturbations in the local gas density. The absorption probability at a particular location is given by $P_{\rm abs}(\mathbf {x},n,z)=1-e^{-\Delta ^2(\mathbf {x})\tau }$, which is not linear in Δ². Thus we take the derivative to get the intensity perturbation, giving us

$$\begin{eqnarray} \delta \left[I_{\rm diff\,cont}^{\rm Ly\alpha }\right](\mathbf {x}) &=& \frac{h_P\nu _{\rm Ly\alpha }}{4\pi }\sum _{n=2}^{\infty }f_{\rm rec}(n)\int _z^\infty dz^{\prime }\,\frac{c}{(1+z^{\prime })H(z^{\prime })}\nonumber \\ &&\times \dot{\rho _*}(z^{\prime })\epsilon (\nu _n^{\prime }) \langle \tau e^{-\tau } \rangle (z)\delta _{H^2}(\mathbf {x})\nonumber \\ &&\times \prod _{n^{\prime }=n+1}^{n_{\rm max}}[1-P_{\rm abs}(n^{\prime },z^{\prime })], \end{eqnarray} \tag{ 60 }$$

where

$$\begin{eqnarray} &&\langle \tau e^{-\tau } \rangle (z) = \int _0^{100} d\Delta \,P(\Delta,z)\Delta ^2\tau e^{-\Delta ^2\tau }. \end{eqnarray} \tag{ 61 }$$

Therefore, we can write the power spectrum for the local IGM as

$$\begin{eqnarray} &&P^{\rm dic}_{\rm local}(k,z)=\left[\nu _{\rm obs}J_{\rm cont}^{\rm local}(z)\right]^2\left[P_{H^2}(k,z)+P_{H^2}^{\rm shot}(z)\right], \end{eqnarray} \tag{ 62 }$$

where

$$\begin{eqnarray} J_{\rm cont}^{\rm local}(z) &=& \frac{h_P\nu _{\rm Ly\alpha }}{4\pi }\sum _{n=2}^{\infty }f_{\rm rec}(n)\int _z^\infty dz^{\prime }\,\frac{c}{(1+z^{\prime })H(z^{\prime })}\nonumber \\ &&\times \dot{\rho _*}(z^{\prime })\epsilon (\nu _n^{\prime }) \langle \tau e^{-\tau } \rangle (z)\nonumber \\ &&\times \prod _{n^{\prime }=n+1}^{n_{\rm max}}[1-P_{\rm abs}(n^{\prime },z^{\prime })]. \end{eqnarray} \tag{ 63 }$$

For the higher-redshift stars, we rewrite the measured intensity (Equation (42)) as

$$\begin{eqnarray} &&I_{\rm diff\,cont}^{\rm Ly\alpha }=\frac{c}{4\pi }\int _z^\infty dz^{\prime }\frac{\dot{\rho _*}(z^{\prime })A_{\rm cont}(z,z^{\prime })}{H(z^{\prime })(1+z^{\prime })}, \end{eqnarray} \tag{ 64 }$$

where A_cont(z, z') is given by

$$\begin{eqnarray} A_{\rm cont}(z,z^{\prime })&=&h_P\nu _{\rm Ly\alpha }\sum _{n=2}^{\infty }f_{\rm rec}(n)\epsilon (\nu _n^{\prime }) P_{\rm abs}(n,z)\nonumber \\ &&\times \prod _{n^{\prime }=n+1}^{n_{\rm max}}[1-P_{\rm abs}(n^{\prime },z^{\prime })]. \end{eqnarray} \tag{ 65 }$$

In this form, we can easily Fourier transform the SFRD, and assuming it traces the galaxy perturbations, we can write a contribution to the foreground power spectrum as

$$\begin{eqnarray} &&P^{\rm dic}_{\rm fg}(k,z)=\left[\nu J_{\rm cont}^{\rm sfrd}\right]^2P_{gg}(k)+P_{\rm dic}^{\rm shot}, \end{eqnarray} \tag{ 66 }$$

where P_gg (k) = P_gg (k, z = 0),

$$\begin{eqnarray} \nu J_{\rm cont}^{\rm sfrd}&=&\frac{c\nu _{\rm obs}(z)}{4\pi }\int _z^\infty dz^{\prime }\frac{\dot{\rho _*}(z^{\prime })A_{\rm cont}(z,z^{\prime })}{H(z^{\prime })(1+z^{\prime })}\nonumber \\ &&\times b_g(z^{\prime })\left[\frac{D(z^{\prime })}{D(0)}\right], \end{eqnarray} \tag{ 67 }$$

D(z) is the growth function, and

$$\begin{eqnarray} P_{\rm dic}^{\rm shot} &=& \left(\frac{c\nu _{\rm obs}(z)}{4\pi }\right)^2\int _z^\infty dz^{\prime }\left[\frac{\dot{\rho _*}(z^{\prime })A_{\rm cont}(z,z^{\prime })}{H(z^{\prime })(1+z^{\prime })}\right]^2\nonumber \\ &&\times P^{\rm shot}(z^{\prime }). \end{eqnarray} \tag{ 68 }$$

We plot the three-dimensional power spectra for Lyα emitters according to the theoretical, as well as the empirical model, at various redshifts in Figure 7. We see that our estimates for this emission source are compatible with independent estimates from Silva et al. (2013). In Figure 8, we plot the power spectra for both diffuse IGM components, and in Figure 9 we plot combinations. Since the diffuse IGM emission due to recombinations is more speculative, we plot in Figure 9 both the sum of all the sources and the sum of just the halo and the diffuse IGM due to continuum photons (no recombinations). We will consider the latter sum as our fiducial model for the remainder of this work. According to our fiducial model, an auto-power spectrum measurement of Lyα emission will characterize at low redshifts the diffuse IGM emission from higher-redshift stars through their continuum photons. At intermediate redshifts during late reionization this diffuse IGM emission will be comparable to the emission from Lyα emitters. At high redshifts (z  ≳  10), Lyα emitters will be characterized by the Lyα power spectrum. Another possible avenue is through cross-correlations with high-redshift galaxy tracers (e.g., CO, C ii, 21 cm). This approach could work at small scales because the power spectrum of IGM emission from continuum photons with SFRD fluctuations, the competitor of Lyα emitters, are sourced by fluctuations from redshifts higher than the emission redshift, and this would drop out of a cross-correlation. The power spectrum of IGM emission from continuum photons, but due to IGM fluctuations, could correlate with galaxies on small scales due to nonlinearities (since it would be a three-point correlation), but it should be much smaller than the halo emission signal. Thus, we expect Lyα emitters and IGM perturbations to be probed using Lyα emission.

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Of course, this picture depends on the various fiducial parameters. At high redshifts, variations in f_esc and M_min, the minimum halo mass, can affect the halo emission at high redshifts. Variations in the ionization fraction and the SFRD can affect both the halo emission and diffuse Lyα IGM emission from continuum photons. The halo emission also has uncertainties in the dust model, with recent measurements suggesting less obscuration than we assume (Dijkstra & Jeeson-Daniel 2013). The IGM Lyα emission from continuum photons can also vary due to the emissivity. And the emission from diffuse IGM due to recombinations can vary over an order of magnitude based on the IGM gas temperature. Also, a change in the clustering bias can greatly impact the power spectrum signal. Finally, it has been shown that fluctuations in the transmission environment of Lyα photons can cause redshift-space distortions, affecting the clustering power spectrum apart from the intensity (Wyithe & Dijkstra 2011; Greig et al. 2013). More measurements from various probes, including intensity mapping studies and photometry studies are definitely needed to characterize Lyα emitters and the IGM.

We begin by determining uncertainties in the three-dimensional power spectrum measurement due to instrumental sensitivity. For our analysis, we consider a configuration for an instrument dedicated to measuring Lyα fluctuations across cosmic time. We chose two spectral bands with ranges of (0.5–0.9 μm) and (0.9–1.6 μm), corresponding to pre-EoR and the EoR, and noise values shown in Table 5. We give our fiducial instrument a 6 arcsec beam full width at half maximum (FWHM) and a spectrometer with a resolution R = 40. We choose a survey over 200 deg². As an illustration we plot three-dimensional power spectra for redshift ranges z = 4–5 and 7–8 in Figure 10.

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In Figure 11 we plot the signal-to-noise ratio (S/N)(z) for the Lyα power spectrum, given by

$$\begin{eqnarray} {\rm S/N}^2(z)&=&\frac{V_{\rm survey}(z)}{4\pi ^2}\int _0^1 d\mu \int _{\rm k_{\rm min}}^{\rm k_{\rm max}} dk\,k^2\nonumber \\ &&\times \left[\frac{P^{\rm Ly\alpha }(k,z)}{P_N(z)W(k,\mu,z)}\right]^2, \end{eqnarray} \tag{ 69 }$$

where V_survey(z) is survey volume at redshift bin z, $P_N(z)=\sigma _N^2V_{\rm pix}/\Omega _{\rm pix}$, the square of the instrumental sensitivity times the pixel volume divided by the pixel solid angle, μ = cos θ in k space with the integration over the upper half plane, and W(k, μ, z) is the window function given in Lidz et al. (2011) as

$$\begin{eqnarray} &&W(k,\mu,z) = e^{(\mu k/k_{\parallel,{\rm res}})^2+(1-\mu ^2)(k/k_{\perp,{\rm res}})^2}, \end{eqnarray} \tag{ 70 }$$

and k_{∥, res} = R  H(z)/[c(1 + z)] and k_{⊥, res} = 2π/(χ(z)σ_b ) are the wavenumber resolutions in directions parallel and perpendicular to the line of sight and σ_b = 0.4245 FWHM. We do not include cosmic variance in our expression for S/N because we are just interested in detection for this calculation. However, cosmic variance must be included when constraining background cosmology and we list results for band S/N including cosmic variance in Table 5. Note that we set k_min = 0.002 h Mpc⁻¹ and k_max = 10³  h Mpc⁻¹, where k_min is limited by the survey volume and k_max is limited only by the instrumental beam size.

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We also calculate the S/N for P(k) detection in the two bands, given by

$$\begin{eqnarray} &&{\rm S/N}^2 =\sum _i{\rm S/N}^2(z_i), \end{eqnarray} \tag{ 71 }$$

where z_i runs over all the redshift with corresponding spectral bins within each band, and we do the same for redshift ranges with size Δz = 1. The values for the small redshift ranges are plotted in Figure 11, and we list the values for the two bands in Table 5. We see from this table that Lyα fluctuations for z < 8 should be well probed by this deep survey, possibly making the very end of reionization able to be probed. The signal is mainly dominated by shot noise. If the shot noise is removed, only the clustering signal from redshifts z < 5 could possibly be detected, although this is an underestimate since nonlinear clustering effects were neglected. It also appears the fluctuations at z < 5 from just the Lyα emitters can also be detected, possibly through a cross-correlation with another LSS tracer. But most of the Lyα emitter signal is from shot noise which becomes undetectable when objects with masses detectable by JWST are removed. Also, while the fluctuations deeper within reionization can be detected over the entire Band 2 with S/N ∼ 18, a tomography study in the reionization region is beyond reach for z > 8. At redshifts z > 5, the signal is noise limited. Increasing the sensitivity by a factor of three would allow fluctuations from z < 9 to be detected, but without incredibly long integration times, this is well beyond these specifications. One solution may be to do a very deep survey over 20 deg². In this case, keeping the total survey integration time constant, the S/N in the second band increases to 39, which can provide much better tomography. The prospects do improve in the mid- to high-redshift range if diffuse IGM emission due to recombinations plays a role, but this is highly speculative. Changes in the various model parameters (e.g., M_min, b_g , $T_{\rm gas}^{\rm halo}$, SFRD, X_e ) can change this picture greatly. A significant uncertainty is due to the high-redshift intensity being set to the minimum required level to sustain reionization. A significant increase in this signal could very well place the power spectrum from the EoR within the range for tomography.

In this section we calculate the foreground power spectrum of all continuum photons similar to the methodology presented in Fernandez et al. (2010), including stellar, free–free, free–bound, and two-photon emission. The power spectrum for the continuum sources at observed frequency ν_obs is given by

$$\begin{eqnarray} &&P^{\rm cont}(k) = \left[\nu _{\rm obs} J_{\rm cont}^{\rm fg}\right]^2P_{gg}(k)+P_{\rm cont}^{\rm shot}. \end{eqnarray} \tag{ 72 }$$

The clustering term is written as

$$\begin{eqnarray} &&J_{\rm cont}^{\rm fg} = \frac{c}{4\pi }\int _z^\infty dz^{\prime }\frac{p[\nu _{\rm obs}(1+z^{\prime }),z^{\prime }]}{H(z^{\prime })(1+z^{\prime })}b_g(z^{\prime })\left[\frac{D(z^{\prime })}{D(0)}\right], \quad \end{eqnarray} \tag{ 73 }$$

where p(ν, z) is the total emissivity of continuum photons, and the shot noise is given by

$$\begin{eqnarray} &&P_{\rm cont}^{\rm shot} = \left(\frac{c}{4\pi }\right)^2\int _z^\infty dz^{\prime }\left\lbrace \frac{p[\nu (1+z),z^{\prime }]}{H(z^{\prime })(1+z^{\prime })}\right\rbrace ^2P^{\rm shot}(z^{\prime }). \quad \end{eqnarray} \tag{ 74 }$$

The emissivity can be broken into the various emission sources as

$$\begin{eqnarray} p(\nu,z) &=& p^*(\nu,z)+(1-f_{\rm esc})[p^{ff}(\nu,z)\nonumber\\ && +p^{fb}(\nu,z)+p^{2\gamma }(\nu,z)], \end{eqnarray} \tag{ 75 }$$

where p*(ν, z), p^ff (ν, z), p^fb (ν, z), and p^2γ(ν, z) are the emissivities for stellar, free–free, free–bound, and two-photon emission, respectively. The individual emissivities are in the form of Equation (12) with $L_\nu ^\alpha (m)$ for each emission mechanism presented in Fernandez & Komatsu (2006). We also include the effect of galactic extinction by using the Calzetti extinction law (Calzetti et al. 2000) assuming an average extinction of E(B − V) = 0.2.

We present the continuum power spectrum for various frequencies in Figure 12. We neglect contributions from stars with z < 2 because in this calculation we assume we can cull these by hand. We find that at lower Lyα redshifts, our fiducial power spectrum is greater than the continuum emission power spectrum. However, at redshifts within the EoR (z ≳ 6), the continuum clearly dominates over our Lyα power spectrum.

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It will still be necessary to remove the effects of line emission from foreground galaxies that masquerade as Lyα emission. In this section we assess the method of masking contaminated pixels from the survey to isolate Lyα emission. O ii (3727 Å) and Hα (6563 Å) lines from lower redshifts will be the dominant line interlopers for Lyα emission for redshifts z_Lya > 2. Various methods can be used to identify these interlopers and mask their corresponding pixels, including secondary line identification, photometry, and cross-correlation with templates. These methods depend on the flux of the interloping line, in that the line must be well above a detection threshold in order to be identified.

Since it is impossible to mask all residual interloper emission, it is necessary to mask interlopers down to a tolerable residual flux such that its power spectrum is subdominant to that of Lyα. Using spectroscopy to identify interlopers over a large area is too time-intensive, so useful method would be to use photometric redshifts from large-area imaging survey like Euclid. Both instruments will have photometric redshift precisions of σ_z /(1 + z) ∼ 0.05, allowing only the relevant populations of interlopers at the right redshifts to be masked, limiting the amount of survey area lost to masking.

One way to assess how much the flux of an interloper is decreasing as the map is being masked is a cross-correlation with another line (not Lyα). This cannot substitute for identifying interlopers directly for masking because the correlation between different emission line galaxy populations is not perfect enough for cleaning, but it is useful as a second check. For λ ≳ 0.6 μm, we expect the intensity maps for Lyα to be dominated by Hα and O ii emission. However, if we cross-correlate two maps with a wavelength ratio $\lambda _2/\lambda _1 = \lambda _{\rm H\alpha }/\lambda _{\rm O\, \scriptsize {II}}\sim 1.8$, then we know that we are seeing the cross-correlation of Hα and O ii. As pixels with bright Hα emission are masked, the Lyα emission will begin to dominate the signal. Considering that the autocorrelation of Hα should be proportional to this cross-correlation, we can use the cross-correlation as a proxy for the amount of residual Hα remaining in the Lyα map. We focus on the Hα line because it dominates the signal over the spectral range of our hypothetical instrument (see Section 5.1). However, this method can only be applied for λ ≳ 0.9 μm because at lower wavelengths O ii lines with the same redshift at Hα lines do not appear in our spectral range. However, this could be mitigated by adding a lower-wavelength band or partially by cross-correlating with a line higher than O ii (i.e., Hβ).

We estimate the difficulty of removing these interlopers by calculating the three-dimensional power spectra of interloping Hα and O ii lines and comparing these to the Lyα emission power spectrum. We determine the interloper intensities empirically using LFs, specifically the Hα LF from Colbert et al. (2013) and the O ii LF from Zhu et al. (2009). ³ Using the LF, we find the unresolved luminosity density from luminosities between zero and a limiting luminosity, which can then give us the intensity, similar to Equation (3). Note that the limiting luminosity L_lim depends on the redshift of the interloper and the limiting flux f_lim of the detector with these quantities being related by $L_{\rm lim}=4\pi d_L^2(z)f_{\rm lim}$. We also include wavevector translation effects in the interloper power spectrum since the contribution of interlopers to the number density of sources will be counted per three-dimensional pixels at the Lyα redshift, not the interloper redshift. To account for small scales translating to larger scales, we include nonlinear clustering in the power spectrum according to the halo model (Cooray & Sheth 2002). More details on the formalism for LSS surveys will be presented in an upcoming paper.

We plot the Lyα power spectrum along with the Hα and O ii interloper power spectra in Figures 13 and 14. Note that we can only estimate the O ii interloper emission up to z_Lyα = 6 because the LF is only estimated up to $z_{\rm O\, \scriptsize {II}}=1.45$. From these plots, we see that measuring the Lyα power spectra at low redshifts (z ≲ 5) requires eliminating Hα down to f_lim = 10⁻¹⁶ erg s⁻¹ cm⁻², while z_Lyα ∼ 6 would require eliminating Hα down to f_lim = 10⁻¹⁷ erg s⁻¹ cm⁻² for k > 1 h Mpc⁻¹ and lower for larger scales. At z_Lyα = 6, all the signal in our instrumental configuration is from smaller scales, so the f_lim = 10⁻¹⁷ erg s⁻¹ cm⁻² cut is sufficient. We now estimate what AB magnitude in an optical band would correspond to our required cut in Hα. Using the COSMOS mock catalog (Jouvel et al. 2011), we find that for F_Hα = 10⁻¹⁷ erg s⁻¹ cm⁻², the required cut in the r band is m_r ∼ 26.5, which is close to the sensitivity of upcoming surveys over hundreds of square degrees (e.g., HSC). For higher redshifts, the required sensitivity to Hα emitters will be progressively lower, even as low as f_lim = 10⁻²⁰ erg s⁻¹ cm⁻² for z_Lyα = 10 which is not expected for the foreseeable future, making this type of cleaning not feasible for high redshifts. We also see that O ii without masking is already comparable to the Lyα signal. Thus, the O ii maps should not be too difficult to clean.

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In Figure 15 we plot both the Hα, Hα ×O ii (for λ > 0.9 μm), and Hα ×Hβ power spectra for our hypothetical instrument without masking to gauge their utility as a proxy for Hα contamination. As with the Lyα S/N, we limit k_min by the volume of the Hα survey. To estimate the Hα ×Hβ cross-power, we use the O ii LF from Zhu et al. (2009) and assume the line ratio Hβ/Hα = 0.28 (Jouvel et al. 2011). Note that for z_Lyα < 4.5, Hα is no longer an interloper, and O ii should be easy to remove. We see that the Hα power spectra has a high S/N. Note that the Lyα redshift bins correspond to Δz_Hα/(1 + z_Hα) = 1/R = 0.025, which is about half as small as the expected photometric redshift errors in upcoming surveys, possibly requiring some of the maps to be combined. For the cross-correlations, both the Hα ×O ii and Hα ×Hβ signals have detectable S/Ns. This result implies that these cross-correlations can be used as a proxy for Hα contamination.

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Now that we know the required f_lim for a low-redshift Lyα measurement, we can estimate for our instrument the sky fraction that must be masked based on the pixel size. This will affect the sky fraction of the survey and thus the S/N of the angular power spectrum. It can be shown that the fraction of the survey with pixel size Ω_pix that must be removed to cut an areal number density n_int of interlopers is given by f_mask = n_intΩ_pix (for f_mask < 1). Note that with the volume density N(< L_lim, z) = Φ_*(z)Γ[α(z) + 1, L_lim/L_*(z)], the areal density n_int is written as

$$\begin{eqnarray} &&n_{\rm int}\simeq \Delta z\frac{d\chi }{dz}\chi (z)^2N(<L_{\rm lim},z), \end{eqnarray} \tag{ 76 }$$

where Δz = (1 + z)/R. For f_lim = 10⁻¹⁴ erg s⁻¹ cm⁻² at z_Lyα = 4.5 (z_Hα = 0.02), the residual Hα population has a number density of 4.2 × 10⁻⁴ arcmin⁻², implying the required fraction of the survey that must be masked is completely negligible. For z_Lyα = 6, where f_lim = 10⁻¹⁷ erg s⁻¹ cm⁻² is needed, n_int = 0.15 arcmin⁻², implying f_mask = 0.16%, which will still be a very small effect. Thus, we expect to be able to remove line interlopers from redshifts z_Lyα ≲ 7 without degrading the signal. Even at z_Lyα = 8, which requires f_lim ∼ 10⁻¹⁹ erg s⁻¹ cm⁻², n_int = 1.6 arcmin⁻² and the masked sky fraction is only f_mask = 1.75%.