Foreground contamination in Ly-alpha intensity mapping during the epoch of reionization
aa r X i v : . [ a s t r o - ph . C O ] F e b Draft version October 15, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
FOREGROUND CONTAMINATION IN LY α INTENSITY MAPPING DURING THE EPOCH OFREIONIZATION
Yan Gong , Marta Silva , , Asantha Cooray , and Mario G. Santos , , Department of Physics & Astronomy, University of California, Irvine, CA 92697 CENTRA, Instituto Superior Tecnico, Technical University of Lisbon, Lisboa 1049-001, Portugal Department of Physics, University of Western Cape, Cape Town 7535, South Africa and SKA SA, 3rd Floor, The Park, Park Road, Pinelands, 7405, South Africa
Draft version October 15, 2018
ABSTRACTThe intensity mapping of Ly α emission during the epoch of reionization (EoR) will be contami-nated by foreground emission lines from lower redshifts. We calculate the mean intensity and thepower spectrum of Ly α emission at z ∼
7, and estimate the uncertainties according to the relevantastrophysical processes. We find that the low-redshift emission lines from 6563 ˚A H α , 5007 ˚A [OIII]and 3727 ˚A [OII] will be strong contaminants on the observed Ly α power spectrum. We make useof both the star formation rate (SFR) and luminosity functions (LF) to estimate the mean intensityand power spectra of the three foreground lines at z ∼ . α , z ∼ . z ∼ . α emission at z ∼
7. The [OII] line is found to be the strongest.We analyze the masking of the bright survey pixels with a foreground line above some line intensitythreshold as a way to reduce the contamination in an intensity mapping survey. We find that theforeground contamination can be neglected if we remove pixels with fluxes above 1 . × − W / m . Subject headings: cosmology: theory - diffuse radiation - intergalactic medium - large-scale structureof universe INTRODUCTION
The epoch of reionization (EoR) is an importantand largely unconstrained stage in the evolution of ouruniverse (Barkana & Loeb 2001). The first stars andgalaxies can start ionization of the neutral intergalac-tic medium (IGM) as early as z ∼
20 or even beforewhile completion of the ionization process should endmuch later, at a redshift as low as z ∼
6. This issuggested by the observations of the absorption of theLy α emission from quasars by the neutral hydrogen inthe IGM (Fan et al. 2006), i.e. the Gunn-Peterson ef-fect (Gunn & Peterson 1965). Most of the fundamentalstones of this era are still poorly known, such as the his-tory, the formation and evolution of the stars and galax-ies, and the formation of large scale structure (LSS).One method to study the EoR is through 21-cm emis-sion of neutral hydrogen in the IGM (Furlanetto et al.2006), which provides a direct method to study the ion-ization history of the IGM. However, the 21-cm measure-ment can not directly trace the galaxy distribution or re-flect the star formation. It also suffers from foregroundcontamination at several orders of magnitude higher thanthe cosmological 21-cm signal. These disadvantages re-strict the 21-cm measurements to explore the LSS andthe processes of the formation and evolution for stars andgalaxies during the EoR.A complementary measurement to 21-cm observationsis the intensity mapping of the emission lines. This ap-proach is more closely connected to the galaxy properties.The emission lines generated from the stars and molecu-lar or atomic gas in galaxies are more sensible probes ofthe metallicity and star formation of galaxies. They tracethe galaxy distribution at large scales, providing informa-tion about the LSS. Several emission lines from galaxieshave been proposed to study the EoR (Visbal & Loeb 2010), e.g. CO rotational lines (Righi et al. 2008;Gong et al. 2011; Carilli 2011; Lidz et al. 2011), [CII]fine structure line (Gong et al. 2012), and the Ly α line(Silva et al. 2013; Pullen et al. 2013). Even molecu-lar hydrogen H in the pre-reionization era can beused to study the formation of first stars and galaxies(Gong et al. 2013).The intensity mapping technique is a powerful toolto explore large areas at poor spatial or angular reso-lution. It does not attempt to resolve individual sources,but measures the cumulative emission from all sources.Thus, intensity mapping of emission lines from galaxiesprovides a suitable way to study the statistical proper-ties of galaxies during the EoR in an acceptable surveytimescale. However, intensity maps are easily contami-nated by other emission lines from lower redshifts in thesame observed frequency range. Since intensity mappingcannot resolve individual sources, it is important to inde-pendently identify and eliminate the contamination fromthe foreground lines.In this work, we study the foreground contaminationon Ly α intensity maps of the EoR. We first computethe Ly α emission from both galaxies and the IGM dueto recombination and collision processes. We calculatethe Ly α mean intensity and power spectrum at z ∼ α , 5007 ˚A [OIII] and 3727 ˚A [OII]are the strongest contamination on the Ly α observations.We estimate the mean intensities of these three lines us-ing the star formation rate (SFR) derived from observa-tions and simulations. Also, for comparison, we makeuse of luminosity functions (LFs) from different observa-tions to calculate the mean intensities and derive theiranisotropy power spectra. We then discuss ways to re-move contamination by masking the bright pixels of theintensity maps.The paper is organized as follows: in the next Section,we estimate the Ly α emission from both galaxies andthe IGM, and calculate the mean intensity and powerspectrum during the EoR. In Section 3, we explore theintensity and power spectra of the foreground emissionlines of the H α , [OII] and [OIII] around z = 0 .
5, 0 . . α emission at z ∼ M = 0 .
27, Ω b = 0 . h = 0 .
71 for the calculation throughout the paper(Komatsu et al. 2011). THE LY α INTENSITY AND ANISOTROPY POWERSPECTRUM DURING THE EOR
In this Section, we estimate the mean intensity andpower spectrum of the Ly α emission during the EoR. Theprocesses that originate Ly α in galaxies and in the IGMare mainly recombinations and collisions. The other pro-cesses, e.g. gas cooling by the falling of the IGM gas intothe potential wells of the dark matter, and continuumemission by stellar, free-free, free-bound and two-photonemission, can be safely neglected according to previousstudies (Cooray et al. 2012; Silva et al. 2013). The Ly α mean intensity Following Silva et al. (2013), we estimate the luminos-ity of the Ly α emission from galaxies by the recombina-tion and collision processes as L galrec ( M, z ) = 1 . × f Ly α ( z ) [1 − f ionesc ( M, z )] × SFR(
M, z ) M ⊙ yr − (erg s − ) , (1) L galcoll ( M, z ) = 4 . × f Ly α ( z ) [1 − f ionesc ( M, z )] × SFR(
M, z ) M ⊙ yr − (erg s − ) , (2)respectively . Here the f Ly α = 10 − × C dust (1 + z ) ξ is the fraction of the Ly α photons which are not ab-sorbed by the dust in the galaxy, where C dust = 3 . ξ = 2 .
57 (Hayes et al. 2011). Note that this fittingformula is mainly derived from low-redshift sources andextrapolated to high redshifts ( z > f Ly α canbe much lower at z ∼ f ionesc = exp[ − α ( z ) M β ( z ) ] is the escape fraction ofthe ionizing photons, where M is the halo mass, and α =5 . × − and β = 0 .
244 (Razoumov & Sommer-Larsen2010; Silva et al. 2013). The SFR(
M, z ) is the star for-mation rate, which is parameterized asSFR(
M, z = 7) = 1 . × − M a (cid:18) Mc (cid:19) b × (cid:18) Mc (cid:19) d (cid:18) Mc (cid:19) e , (3) Note that the ratio L galcoll /L galrec = 0 .
26 can be larger at high red-shifts due to strong galaxy forming process (Laursen et al. 2013).This ratio is still observationally unconstrained, and also has largeuncertainty suggesting by simulations. where a = 2 . b = − . c = 8 × M ⊙ , c =7 × M ⊙ , c = 10 M ⊙ , d = 0 . e = − .
25. ThisSFR parameterization was derived to fit the propertiesof the simulated galaxies catalogs from De Lucia et al.(2007) and Guo et al. (2011),and is available at z =7 in (Silva et al. 2013). Then the integrate Ly α meanintensity from individual galaxies is given by¯ I gal ( z ) = Z M max M min dM dndM L gal ( M, z )4 πD y ( z ) D , (4)where L gal = L galrec + L galcoll is the total luminosity of Ly α emission from galaxies, dn/dM is the halo mass function(Sheth & Tormen 1999), and D L and D A are the lumi-nosity and comoving angular diameter distance respec-tively. The factor y ( z ) = dr/dν = λ Ly α (1 + z ) /H ( z ),where r ( z ) is the comoving distance at z , λ Ly α = 1216 ˚Ais the Ly α wavelength in the rest frame, and H ( z ) is theHubble parameter. We take M min = 10 h − M ⊙ and M min = 10 h − M ⊙ , which denote the mass range ofthe halos that host the galaxies with the Ly α emissionin this work.For the Ly α emission from the IGM, we use the lumi-nosity density to estimate the mean intensity¯ I IGM = l IGM ( z )4 πD y ( z ) D , (5)where l IGM = l IGMrec + l IGMcoll is the Ly α luminosity densityfor the IGM in erg s − cm − . The l IGMrec and l IGMcoll arethe luminosity densities from recombination and collisionprocesses, respectively. The l IGMrec can be estimated by l IGMrec = ǫ recLy α hν Ly α , where ǫ recLy α is the Ly α recombinationemission rate per cm , which is given by ǫ recLy α = f recLy α n e n HII α recB . (6)The n e and n HII are the number density of the electronand HII. Here we have n e n HII = C IGM ( z )¯ n e ( z )¯ n HII ( z ),where ¯ n e ( z ) and ¯ n HII ( z ) are the mean number density ofthe electron and HII at z, which are dependent on thereionization fraction x i during the EoR. The C IGM ( z ) = h n i / h n i is the clumping factor of the IGM at z whichcan be set by C IGM = 6 at z=7 (Pawlik et al. 2009;Silva et al. 2013). The α recB is the hydrogen case B re-combination coefficient which can be fitted by (Hummer1994; Seager et al. 1999) α recB ( T ) = 10 − aT b cT d (cm / s) , (7)where a = 4 . b = − . c = 0 . d = 0 . T = T / K. The f recLy α is the the fraction of theLy α photons produced in the case B recombination, andwe use the fitting formula from Cantalupo et al. (2008)to evaluate it as f recLy α ( T ) = 0 . − . ( T ) − . T − . . (8)This formula is accurate to 0.1% for 100 < T < K.We assume the mean gas temperature of the IGM tobe 1 . × K in this work, which is in a good agree-ment with the results from simulations and the cur-rent measurements from quasars (Theuns et al. 2002;Tittley et al. 2007; Trac et al. 2008; Bolton et al. 2010,2012).The Ly α collisional emission in the IGM involves freeelectrons that collide with the neutral hydrogen atoms(HI) and transfer their kinetic energy by exciting the HIto high energy levels. The hydrogen atoms then decayby emitting photons, including Ly α photons. During theEoR, this process mainly occurs in the ionizing fronts ofthe reionization bubbles, since this emission will be thestrongest when n e ∼ n HI . Similar to the recombinationemission, the luminosity density of the collisional emis-sion in the IGM can be calculated by l IGMcoll = ǫ collLy α hν Ly α .The ǫ collLy α is the collisional emission rate per cm ǫ collLy α = C effLy α n e n HI , (9)where n HI is the number density of the neutral hydrogenatoms. We assume n e n HI = f ionfront C IGM ( z )¯ n e ( z )¯ n HI ( z ),where ¯ n HI ( z ) is the mean number density of neutral hy-drogen at z. The f ionfront is the volume fraction of thebubble ionizing fronts. Because the ionizing front justtakes a very small part of the whole ionizing bubble, as agood approximation, the volume fraction of the ionizingfront can be estimated as f ionfront = π ( r − r ), where r =[(3 / π ) x i ] / , and r = r + d and d = [ n H σ H ( h ν i )] − is the thickness of the bubble ionizing front. The factor1 / x HI (1 − x HI ) overthe ionizing front where x HI ( r ) is the fraction of the HIin the front at radius r (Cantalupo et al. 2008). Here σ H denotes the cross section of the hydrogen ionization,where h ν i is the mean frequency of the ionizing photons,and it is given by (Osterbrock 1989) σ H ( ν ) = σ − exp( − π/ǫ ) (cid:20) ν ν exp (cid:18) − tan − ǫǫ (cid:19)(cid:21) . (10)Here ν is the Lyman limit frequency, σ = 6 . × − cm and ǫ ≡ ( ν/ν − / . Here we take h ν i =2 . ν in our calculation (Gould & Weinberg 1996). Notethat we set f ionfront = 0 when x i ≥
1, which means there isno collisional emission in the IGM when the universe istotally ionized.In Eq. (9), the C effLy α is the effective collisionalexcitation coefficient, which can be estimated by(Cantalupo et al. 2008) C effLy α = C , p + C , s + C , d . (11)Here we take into account the excitation up to n = 3energy level to decay and produce Ly α photons. Thecontribution of emission from higher energy levels canbe neglected at z ≈ C l , u , in cm per second, is given by C l , u = 8 . × − g l √ T γ l , u ( T )exp (cid:18) − ∆ E l , u kT (cid:19) (cm / s) , (12)where T is the gas temperature that we assume T = 1 . × K, ∆ E l , u is the energy difference between lower leveland higher level, g l is the statistic weight for lower level,and γ l , u ( T ) is the effective collision strength calculated by the fitting formulae from Giovanardi et al. (1987). Fig. 1.—
The power spectrum of the Ly α emission at z = 7. Thedash-dotted line is the clustering power spectrum which consists of1-halo and 2-halo components in dashed lines, the dotted line de-notes the shot-noise power spectrum, and the total power spectrumis shown in solid line. The uncertainty of the power spectrum isshown in shaded region which is estimated by the uncertainties ofthe f ionesc , f Ly α , SFR and IGM clumping factor. The contributionsof different halo mass scales to the total power spectrum are alsoshown. We find 10 -10 h − M ⊙ dominates the clustering powerspectrum. Then we estimate the total mean intensity of the Ly α emission ¯ I Ly α = ¯ I gal + ¯ I IGM using Eq. (4) and Eq. (5).At z ∼
7, we get ¯ I gal = 9 . / sr and ¯ I IGM = 1 . / srif assuming ¯ x i = 0 .
85. Thus, according to the assump-tions we have made, Ly α emission from galaxies is largerthan that from the IGM around z = 7. Note that thecollisional emission in the IGM is much smaller thanthe recombination emission with ¯ I IGMcoll ≃ . / sr and¯ I IGMrec ≃ . / sr,respectively, assuming T = 1 . × Kfor the gas temperature of the IGM.There are large uncertainties for the parameters in theestimation of the Ly α intensity. According to the cur-rent measurements (Razoumov & Sommer-Larsen 2010;Hayes et al. 2011; Blanc et al. 2011), we find the f ionesc and f Ly α in Eq. (1) and (2) have an uncertainty of a factorof 2 each. Besides, we take into account of the uncer-tainties in the SFR and the clumping fator, which canbe factors of 3 and 5 at z ∼ I Ly α = 10 . +39 . − . Jy / sr around This f Ly α is Ly α “effective” escape fraction instead of real es-cape fraction which should include the scattered Ly α emission fromthe diffuse halos surrounding galaxies (Dijkstra & Jeeson-Daniel2013). The Ly α emission from the diffuse halos has low surfacebrightness, but the total flux can exceed the direct emission fromgalaxies by a significant factor (e.g. Zheng et al. 2010; Steidel et al.2010; Matsuda et al. 2012). Thus the real Ly α escape fraction canbe much larger than the effective escape fraction. This diffusedemission could contribute to the intensity mapping. We find ourfiducial f Ly α ≃ . z = 7 which is close to one, and we also addan uncertainty of a factor of 2 on it. So the effect of the diffusedLy α emission is safely covered in our calculation. z = 7. Our results are well consistent with other workswhen allowing for uncertainty (e.g. Silva et al. 2013;Pullen et al. 2013). Besides, we note that the esti-mated Ly α emission of high- z metal-poor galaxies withstandard case-B assumption can be underestimated bya factor of ∼ − The Ly α power spectrum According to the calculations above, the absolute valueof the Ly α intensity background during the EoR is smalland hard to measure directly in an absolute intensityexperiment. Instead, we can try to measure the fluctu-ations of the Ly α intensity and estimate the anisotropypower spectrum. Since the Ly α emission from galax-ies and IGM in the ionization bubbles surrounding thegalaxies trace the underlying matter density field, we cancalculate the Ly α intensity fluctuations by δI Ly α = ¯ b Ly α ¯ I Ly α δ ( x ) . (13)Here we set ¯ I Ly α ≃ ¯ I gal , since the main Ly α emissionat z ∼ δ ( x ) is the matterover-density at the position x , and ¯ b Ly α ( z ) is the averagegalaxy bias weighted by the Ly α luminosity¯ b Ly α ( z ) = R M max M min dM dndM L Ly α gal b ( M, z ) R M max M min dM dndM L Ly α gal , (14)where L Ly α gal ( M, z ) is the Ly α luminosity of the galaxy,and b ( M, z ) is the halo bias (Sheth & Tormen 1999).Then we can calculate the Ly α clustering power spec-trum due to galaxy clustering as P clusLy α ( k, z ) = ¯ b α ¯ I α P δδ ( k, z ) , (15)where P δδ ( k, z ) is the matter power spectrum whichcan be estimated from the halo model (Cooray & Sheth2002). The clustering power spectrum dominates thefluctuations at large scales. At small scales the Pois-son noise caused by the discrete distribution of galaxiesbecomes important. This Poisson or shot-noise powerspectrum takes the form (e.g. Visbal & Loeb 2010; Gonget al. 2011): P shotLy α ( z ) = Z M max M min dM dndM " L Ly α gal πD y ( z ) D . (16)Therefore, the total power spectrum can be written by P totLy α ( k, z ) = P clusLy α ( k, z ) + P shotLy α ( z ). In Figure 1, we showthe Ly α power spectrum at z = 7. We show the to-tal, clustering and shot-noise power spectrum in solid,dash-dotted and dotted line, respectively. The dashedlines denote the 1-halo and 2-halo terms of the cluster-ing power spectrum. The uncertainty of the total powerspectrum is also shown in shaded region, which is de-rived from the uncertainties of the f ionesc , f Ly α , SFR andIGM clumping factor. We also show the contributionsof different halo mass scales to the total power spec-trum in colored solid lines. As can be seen, the haloswith mass 10 -10 h − M ⊙ provide the most contribu-tion on the clustering power spectrum, and the halos with higher masses (10 -10 h − M ⊙ ) have large shot-noise since they are bright and rare. We find ¯ b Ly α = 5 . P shotLy α = 4 . × (Jy / sr) (Mpc / h) − at z = 7, andthe shot-noise power spectrum dominates the total powerspectrum at k & − h . THE ESTIMATION OF THE FOREGROUND EMISSIONLINES
Since we can not resolve individual sources with in-tensity mapping, the measurements of the Ly α emissionduring the EoR can be contaminated by emission linesfrom lower redshifts. Here we consider three low-redshiftemission lines, H α at 6563 ˚A, [OIII] at 5007 ˚A and [OII]at 3727 ˚A. At z ∼
7, the frequency of the Ly α line isabout 300 THz, which can be then contaminated by H α at z ∼ .
5, [OIII] at z ∼ . z ∼ .
6, respec-tively. We will estimate the mean intensity and powerspectra of these lines in this Section.
The mean intensity from the SFR
The H α line at 6563 ˚A, [OIII] line at 5007 ˚A and [OII]line at 3727 ˚A are good tracers of the SFR of galaxies.The luminosity of these lines is related to the SFR asSFR ( M ⊙ yr − ) = (7 . ± . × − L H α , (17)SFR ( M ⊙ yr − ) = (1 . ± . × − L [OII] , (18)SFR ( M ⊙ yr − ) = (7 . ± . × − L [OIII] . (19)The luminosity here is in erg s − , and the relations forH α and [OII] are from Kennicutt (1998), and the [OIII]relation is given in Ly et al. (2007). The SFR- L H α re-lation assumes the initial mass function from Salpeter(1955), and we add 30% uncertainty to it. These conver-sions are also in good agreement with other works (e.g.Hopkins et al. 2003; Wijesinghe et al. 2011; Drake et al.2013). With the help of the relation of the SFR and thehalo mass as a function of redshift SFR( M, z ), we canuse these conversions to derive L ( M, z ) and compute themean intensities for these lines using Eq. (4).To estimate the SFR(
M, z ), we can make use of theSFRD( z ) from the observations and assume the SFR isproportional to the halo mass M . We take the SFRD( z )given by Hopkins & Beacom (2006) with the fitting for-mula from Cole et al. (2001)SFRD( z ) = a + bz z/c ) d ( h M ⊙ yr − Mpc − ) , (20)where a = 0 . b = 0 . c = 3 . d = 5 .
2. Thisfitting formula is consistent with the observational datavery well, especially at z .
2, which is good enough forour estimations here. Then assumingSFR(
M, z ) = f ∗ ( z ) Ω b Ω M t s M, (21)where t s = 10 yr is the typical star formation timescale,and f ∗ ( z ) is the normalization factor which can be deter-mined by SFRD( z ) = R dM dndM SFR(
M, z ). After obtain-ing SFR(
M, z ), we can derive L ( M, z ) for the H α , [OII] The SFR-L [OII] relation has a larger uncertainty, and we willdiscuss it in Section 3.2.
Fig. 2.—
The SFR vs. halo mass M at z = 0 . z = 1 and z = 1 . and [OIII] lines using Eq. (17), (18) and (19) respectively,and then calculate their mean intensities with Eq. (4).We find ¯ I H α = 31 . ± . / sr, ¯ I [OII] = 17 . ± . / srand ¯ I [OIII] = 35 . ± . / sr. The errors are derivedfrom the uncertainties in the conversions of the SFR andluminosity given by Eq. (17), (18) and (19).Another way to obtain the SFR( M, z ) is with simula-tions. Here we use the galaxy catalog from Guo et al.(2011), which is obtained by Millennium II simulationwith a volume of 100 (Mpc /h ) and particle mass reso-lution ∼ . × h − M ⊙ (Boylan-Kolchin et al. 2009).We fit the SFR( M, z ) derived from this catalog for z . M, z ) = 10 a + bz (cid:18) MM (cid:19) c (cid:18) MM (cid:19) d , (22)where a = − . b = 0 . c = 2 . d = − . M =10 M ⊙ and M = 8 × M ⊙ . The simulation resultsand the best fits of the SFR( M, z ) at z = 0 . z = 1 and z = 1 . M & M ⊙ . Also, there is relatively large scattering in therelation at small halo mass with M . M ⊙ , whichcould provide additional uncertainty in the intensity cal-culation. The SFRD derived from this SFR( M, z ) are2 . × − , 3 . × − and 6 . × − M ⊙ yr − Mpc − at z ∼ . z ∼ . z ∼ .
6, respectively. Forcomparison, Hopkins & Beacom (2006) based values are3 . × − , 5 . × − and 9 . × − M ⊙ yr − Mpc − at the corresponding redshifts. Thus the SFRDs fromthe simulation are lower than the SFRDs from the ob-servations. Then we estimate the L ( M, z ) and obtain¯ I H α = 16 . ± . / sr, ¯ I [OII] = 9 . ± . / sr and¯ I [OIII] = 16 . ± . / sr using Eq. (17), (18), (19) andEq. (4). These values are lower than what we have from TABLE 1The mean intensities in Jy/sr of the H α , [OIII] and [OII]around z = 0 . , z = 0 . and z = 1 . , derived from both ofthe LF and SFR methods. ¯ I H α ( z ∼ .
5) ¯ I [OIII] ( z ∼ .
9) ¯ I [OII] ( z ∼ . obs . ± . . ± . . ± . sim . ± . . ± . . ± . L07 . +17 . − . . +11 . − . . +8 . − . LF G13 − −−
L07, D13 and G13 denote Ly et al. (2007), Drake et al. (2013)and Gunawardhana et al. (2013). the observational SFRD( z ), but they are still consistentwithin 1 σ error. The mean intensity from the luminosity functions
A direct way to estimate the mean intensities of theH α , [OII] and [OIII] lines is to make use of observedLFs. In Figure 3, we show the observed LFs of H α , [OII]and [OIII] lines around z = 0 . z = 0 . z = 1 . α emission from z ∼
7. Thesquares, pentagons, circles and triangles are LFs from Lyet al. (2007) , Drake et al. (2013) and Gunawardhana etal. (2013), respectively. The LFs data points shown hereare dust extinction-corrected, except for the LFs fromDrake et al. (2013) which are lower than the results ofthe other two works.The LF is usually fitted by the Schechter function(Schechter 1976)Φ( L ) dL = φ ∗ (cid:18) LL ∗ (cid:19) α exp (cid:18) − LL ∗ (cid:19) dLL ∗ , (23)where φ ∗ , L ∗ and α are free parameters that are obtainedby fitting the Schecter function with the data. The LFsfrom Ly et al. (2007) and Drake et al. (2013) are fittedby this function . Then the mean intensity can be esti-mated by ¯ I ν ( z ) = Z L max L min dL dndL L πD y ( z ) D , (24)where dn/dL = Φ( L ) is the luminosity function, and L min = 10 L ⊙ and L max = 10 L ⊙ are the lowerand upper luminosity limits. We find the result is notchanged if choosing smaller L min and larger L max in ourcalculation. We use the extinction-corrected LFs to cal-culate the mean intensity for the H α , [OII] and [OIII]lines. Note that, in Drake et al. (2013), they just cor-rect the L ∗ for the dust extinction to get the extinction-corrected LFs, and the other two parameters φ ∗ and α are still fixed to the values fitted by the observed LFs.Using Eq. (24) and the extinction-corrected LFs, wecalculate the mean intensities of the H α , [OIII] and [OII] We note that the H α and [OII] LFs from Ly et al. (2007) arewell consistent with recent observations with larger survey volumes(e.g. Sobral et al. 2012, 2013). So we just show the LF data fromLy et al. (2007) here. The H α LF at 0 . < z < .
35 from Gunawardhana et al.(2013) agrees well with the LF at z = 0 . Fig. 3.—
The luminosity functions and the best fits for the H α , [OIII] and [OII] lines around z = 0 . z = 0 . z = 1 . α emission at z ∼
7. The blue squares, pink pentagons, red circles and green triangles are the observational LFs fromLy et al. (2007), Drake et al. (2013) and Gunawardhana et al. (2013), respectively. The dashed and dotted curves are the best fits of thecorresponding LFs. Note that the data points of LFs from Drake et al. (2013) are the observed LFs (without dust extinction correction)which seem lower than the results of the other two works.
Fig. 4.—
The mean intensity of H α , [OIII] and [OII] using bothSFR and LF methods around z = 0 . z = 0 . z = 1 .
6. Thedashed line denotes the mean intensity of Ly α at z ∼ α , [OII] and [OIII] are gener-ally higher than the central value of the Ly α mean intensity, whichcan provide considerable contamination on the Ly α emission. around z = 0 . z = 0 . z = 1 .
6. We list andplot the mean intensities from both of the LF and SFRmethods in Table 1 and Figure 4 for comparison. Wealso estimate the errors of the LFs from Ly et al. (2007)based on the errors of the fitted values of the φ ∗ , L ∗ and α , and then derive the error for the mean intensity.In Table 1, the SFRD obs and SFR sim denote the meth-ods of SFRD from observations and SFR from the sim-ulations respectively. The LF L07 , LF
D13 and LF
G13 de-note the LF method using the LFs from Ly et al. (2007),Drake et al. (2013) and Gunawardhana et al. (2013) re-spectively. The LF and LF denote to use the LFs from Drake et al. (2013) shown in pink pentagonsand red circles in Figure 3 respectively. In Figure 4,the dotted crosses and open circles are the values forSFRD obs and SFR sim , and the blue squares, pink pen-tagons, red circles and green triangles are the values forLF
L07 , LF , LF and LF
G13 respectively. For com-parison, we also shown the mean intensity of the Ly α at z ∼ α , we find the resultsfrom the different LF observations are consistent witheach other, and they are safely in the 1 σ error of theresult from LF L07 . Comparing the results from the SFRand LF methods, the mean intensity from SFR sim is in agood agreement with that from LF
L07 , LF and LF
G13 which give an average H α intensity around 15 Jy/sr. TheSFRD obs and LF here give a bit higher and lowerresults, respectively.For the [OIII] line, the result of LF L07 agrees with theLF result, and they are also consistent with the resultfrom SFR sim in 1 σ error. This gives an average intensityaround 13 Jy/sr. The SFRD obs gives a higher intensityconsidering its error. The LF provides a low [OIII]intensity about 6 Jy/sr, and this can be caused by thepoor model fitting of the LF data in Drake et al. (2013)(circles in the middle panel of Figure 3).For the [OII] emission at z ∼ .
6, the results havelarger discrepancy compared to the H α and [OIII] cases.The LF L07 and LF still give the same intensity whichis around 24 Jy/sr, and their results agree with SFRD obs in 1 σ . But the SFR sim suggests a low intensity around 9Jy/sr, while the LF gives a much higher value of ∼ whichgives a different result from the others as in the [OIII]case, that there are not enough LF data for the fittingof the LF. In the right panel of Figure 3, we find thereare just three data points around 2 × erg s − andno data is observed in the faint end (see red circles). Forthe SFR sim , it actually has large discrepancy betweendifferent observations for the SFR- L [OII] relation. Forexample, Kewley et al. (2004) proposes anther relationSFR ( M ⊙ yr − ) = (6 . ± . × − L [OII] , Fig. 5.—
The power spectra of the H α , [OIII] and [OII] lines at z ∼ . z ∼ . z ∼ .
6, which can contaminate the Ly α emission at z ∼
7. The solid, dash-dotted and dotted lines denote the total, clustering and shot-noise power spectrum respectively. The short dashedlines are the 1-halo and 2-halo terms from the halo model (Cooray & Sheth 2002). The uncertainty of the total power spectrum is shownin shaded region, which is estimated by the uncertainty of the mean intensity. Here we adopt the LFs and errors in Ly et al. (2007) toestimate the mean intensity and errors for the three emission lines. which gives a higher [OII] intensity ∼ . sim method. This substantially reduces the ten-sion between the SFR sim and the other methods.Generally, we find that the intensities of the H α , [OII]and [OIII] around z = 0 . z = 0 . z = 1 . α at z ∼
7, which provide consider-able contamination on the Ly α emission from the EoR.Also, the result derived by the LFs from Ly et al. (2007)is well consistent with the other results from both of theLF and SFR method, hence we would adopt it as theforeground line contamination in our following estima-tion and discussion.Next, we calculate the anisotropy power spectra of theH α , [OII] and [OIII] emissions using their mean inten-sities. Using Eq. (15) and (16) and replacing the ¯ b Ly α ,¯ I Ly α and L Ly α to be the bias, mean intensity and lineluminosity of the H α , [OII] and [OIII], we obtain theirclustering and shot-noise power spectra. Note that we as-sume the line luminosity is proportional to the halo massto get the mean bias of these three lines, i.e. replacing L Ly α to be M in Eq. (14). This approximation is goodenough to estimate the mean bias given the uncertaintyin the mean intensity. In Figure 5, we show the powerspectrum of the H α , [OIII] and [OII] lines at z = 0 . z = 0 . z = 1 .
6, which contaminate the Ly α emis-sion at z ∼
7. The total, clustering and shot-noise powerspectra are in solid, dash-dotted and dotted lines respec-tively. The 1-halo and 2-halo terms from the halo modelare also shown in short dashed lines (Cooray & Sheth2002). The uncertainty of the total power spectrum inshaded region is estimated by the uncertainty of the meanintensity. As mentioned, we use the LFs and errors fromLy et al. (2007) to compute the power spectrum and theuncertainty for the three emission lines. THE REMOVAL OF THE FOREGROUND EMISSIONLINE CONTAMINATION
The observed power spectrum
So far we have analyzed the expected intensity fromcontaminating lines. In practice, experiments (at leastfirst generation ones) will try to make a statistical mea-surement of the 3-d power spectrum. Therefore, this isthe quantity that we should compare to in order to access the level of foreground contamination. Because the sig-nal and foregrounds will be emitted at different redshiftsthere will be an extra factor multiplying the foregroundpower spectrum. In order to calculate this, we need totake into account the effect of the observed light cone inthe 3-D power spectrum.The intensity that we measure, I (Ω , ν ) will correspondto a sum of the signal we are interested in, I s (the Ly α emission from redshift z s ), and the foreground emission, I f , from lower redshifts z f , which is contributing to thesame frequency: I (Ω , ν ) = I s (Ω , ν ) + I f (Ω , ν ) . (25)In the flat sky approximation, displacements in angle ∆ θ and frequency ∆ ν about a central reference point, canbe transformed into a position in 3-D comoving space x , x = r ( z )∆ θ (26) x = − y ( z )∆ ν, where we are already making a translation along the x direction and x , x are assumed perpendicular to theline of sight while x is taken to be parallel to it. Notethat, besides the flat sky, we are also neglecting cosmicevolution, e.g. assuming that points along the line ofsight are all emitted at the same redshift z .We then write the observed signal as I (Ω , ν ) = ¯ I s ( z s )[1 + b s ( z s ) δ ( z s , x s )]+ ¯ I f ( z f )[1 + b f ( z f ) δ ( z f , x f )] (27)where δ () is the dark matter perturbation and x s , x f isthe 3-D position of the signal and foreground emissionrespectively. If we Fourier transform the above signalwith respect to x s , the corresponding foreground Fouriermode will be offset with respect to the true one. Theobserved power spectrum, P obs will then be: P obs ( k ⊥ , k k ) = ¯ I s ( z s ) b s ( z s ) P ( z s , k s ) (28)+ ¯ I f ( z f ) b f ( z f ) (cid:18) r s r f (cid:19) (cid:18) y s y f (cid:19) P ( z f , k f ) , where | ~k s | = q k ⊥ + k k is the 3-D k at the redshift of Fig. 6.—
Left : the comparison of the 3-D total power spectrum of the Ly α at z ∼ α at z = 0 .
5, [OIII] at z = 0 . z = 1 .
6. The uncertainty of the Ly α power spectrum is shown in shaded region. Right : the same as the left panel, but consideringthe projection effect on the power spectra of the H α , [OII] and [OIII] lines. The total projected power spectrum is shown in dark red line. the signal, and | ~k f | = q ( r s /r f ) k ⊥ + ( y s /y f ) k k is the3-D k at the redshift of the foreground line. The factor( r s /r f ) ( y s /y f ) comes from the distortion of the volumeelement when Fourier transforming the foreground cor-relation function to the power spectrum (Visbal & Loeb2010). This process can be considered to project the fore-ground power spectrum to the high redshift where the in-tensity signal comes from. We then see that, even whenlooking at the spherical averaged power spectrum, e.g. P obs ( k = q k ⊥ + k k ), the foreground contribution to thepower spectrum will be boosted by this projection. More-over, there is an anisotropy in the k space which providesa potential method to distinguish the foregrounds fromthe signal. We discuss the details of this effect in theAppendix.In Figure 6, we compare the 3-D power spectrum ofH α , [OII] and [OIII] with and without the projection.In the left panel of Figure 6, we show the total powerspectrum of the H α at z = 0 .
5, [OIII] at z = 0 . z = 1 . P totLy α and itsuncertainty are also shown in blue line and shaded regionfor comparison. In the right panel of Figure 6, we showthe 3-D projected power spectra of the H α , [OII] and[OIII] lines to the redshift of the Ly α emission ( z ∼ k = k = k k , where k = k ˆ i + k ˆ j + k k ˆ k and k ⊥ = p k + k . We also ignorethe redshift distortion here. We find the projected powerspectra of the three lines become a bit higher than thecase where we igore the projection of the foregrounds tothe background cosmological signal of intensity duringthe EoR. Foregrounds masking
To remove the contamination from these foregroundemission lines, we need to mask the bright sources at lowredshifts. However, we cannot identify the individualsources and their redshifts in the intensity mapping, and all the signals that lie in the same survey pixel, which isdefined by the spectral and angular resolutions, will bemixed together and observed as only one signal.
Fig. 7.—
The LF of the Ly α at z ∼ α , [OIII] and [OII] around z = 0 . z = 0 . z = 1 . During the EoR, the galaxies are averagely smaller andfainter than the present galaxies, so their LFs shouldbe relatively higher at the faint end than the galaxiesat the low redshifts. In Figure 7, we show the Ly α LF at z ∼ α emit-ters (LAEs) (Ouchi et al. 2010; Kashikawa et al. 2006;Hu et al. 2005) . The best fit of the LF is shown in bluesolid line using the Schechter function (see Eq. (23)) withfitting values given by Ouchi et al. (2010). The LFs ofthe H α , [OIII] and [OII] lines are also shown for compar-ison. We can see that, the LFs of the Ly α is higher thanthe others at the faint end, but basically lower than the[OII] LF at the bright end. Therefore, we can mask thewhole bright pixels whose fluxes are above some thresh-old in the intensity mapping to reduce the contaminationof low-redshift line emission, and then the remainder pix-els should be dominated by the Ly α emission at highredshifts. Fig. 8.—
The power spectra of the H α , [OII] and [OIII] withmasking and projection. The deep red line denotes the total fore-ground power spectrum after the masking. We apply a flux cut at1 . × − W/m here, which could make the total foregroundpower spectrum ∼
100 times smaller than the Ly α around k = 0 . h Mpc − . In Figure 8, the power spectra of the H α , [OII] and[OIII] lines with masking and projection are shown.Here we try to make the total power spectrum of H α ,[OII] and [OIII] smaller by a factor of ∼
100 around k = 0 . h Mpc − where the shot-noise is small. Wefind we need to mask sources with fluxes greater than1 . × − W/m . The corresponding line luminosityare L H α ≃ . × L ⊙ , L [OII] ≃ . × L ⊙ and L [OIII] ≃ . × L ⊙ respectively. Note that theseluminosity cuts are close to the lower luminosity limit( L min = 10 L ⊙ ) we take in Eq. (24), hence our maskingresults are dependent on the faint-end slopes of the LFswhich are not well constrained by observational data.In Figure 9, we show the number of sources whose fluxis greater than some value in a survey volume pixel. Here Using Eq. (24) and the Ly α LF (blue solid curve in Figure 7),we note that the Ly α mean intensity is just ∼ z = 7,which is lower than the value of ∼ α LF data at the faint end, since only very brightindividual sources can be detected at this high redshift. This alsomeans that there could be more faint LAEs around z = 7, and thereal LF could be higher at the faint end. we assume a survey with 6 ′′ × ′′ beam size and 190 − α sources are dominant at thebright end, but are subdominant in the survey since allof these sources are from low redshifts ( z ∼ .
5) wherethe survey volume is small. On the other hand, [OII]sources are the main contamination, and are brighterthan 1 . × − W/m in approximately 2% of the pix-els. In total, we need to mask about 3% survey pixels.Here we should notice the large uncertainty of the Ly α emission. The number of the Ly α sources can be lowerby a factor of ∼
10 with the uncertainty, which wouldlead to a larger masking percentage.
Fig. 9.—
The number of sources per volume pixel whose fluxesare greater than the x-axis flux value. The Ly α curve is derivedfrom the halo mass function and the calculation in Section 2, andthe curves of H α , [OII] and [OIII] are computed by the LFs inLy et al. (2007). The vertical dashed line denotes the flux cut at1 . × − W/m . We find this corresponds to remove 3% of thetotal pixels. There is a second method that can be used to elimi-nate the foreground line contamination, which is makinguse of the cross-correlation between different emissionlines at the same redshift. This method is discussed inmany previous works and can be used either for the sig-nal (e.g. Visbal & Loeb 2010; Gong et al. 2012; Silva etal. 2013) or the foreground contamination (e.g. Pullenet al. 2013). The idea is that the lines emitted at thesame redshift should trace the same underlying matterdistribution and hence lead to large cross-correlation onthe power spectrum. On the other hand, the emissions atdifferent redshifts, which are far away from each other,would not provide considerable cross power spectrum.Therefore, we can derive the auto power spectrum of oneline if we know both the cross power spectrum and theauto power spectrum of the other line. For instance,we can cross-correlate the Ly α line with 21-cm at z ∼ P Ly α from the P Ly α × and P (Silva et al. 2013). Or we can estimate the autopower spectra of the foreground emission lines by cross-0correlating with the 21-cm intensity mapping surveys at z ∼
1, e.g. the GBT , CHIME and Tianlai projects(Chen 2012). However, although we can detect the crosspower spectrum, it is still hard to measure the auto powerspectrum for the 21-cm or the other lines due to theirown foregrounds removal. So this method is indirect andcannot substitute the masking method discussed above.Even if we can only obtain the cross power spectrum, itcould still be a good guide and a secondary check for theintensity mapping experiments. SUMMARY
We estimate the Ly α mean intensity and power spec-trum during the EoR, and explore the foreground con-tamination from the low-redshift emission lines. We con-sider the Ly α emission from both of galaxies and IGMfor the recombination and collisional emission processes.We find the Ly α emission of galaxies is dominant overthe IGM at z ∼ I Ly α ∼ f ionesc , f Ly α , SFRand the clumping factor. With the help of halo model,we also calculate the Ly α clustering, shot-noise powerspectrum with the uncertainty given by the mean inten-sity.Next we investigate the foreground contamination bylow-redshift emission lines. We find H α at 6563 ˚A, [OIII]at 5007 ˚A and [OII] at 3727 ˚A can be the strongest con-tamination on Ly α emission during the EoR. We esti-mate the mean intensity of the H α , [OIII] and [OII] linesat z ∼ . z ∼ . z ∼ . α emission at z ∼
7. We use two methods to do the estimation, i.e.the SFR and LF methods. In the SFR method, both ofthe SFRD( z ) from the observations and the SFR( M, z ) derived from the simulations are used to compute theintensity. In the LF method, we adopt the LFs of H α ,[OIII] and [OII] around z = 0 . z = 0 . z = 1 . α and [OIII] lines. Themean intensity of the three lines are ¯ I H α ∼
15 Jy/sr,¯ I [OIII] ∼
13 Jy/sr and ¯ I [OII] ∼
24 Jy/sr, which are largerthan the ¯ I Ly α ∼
10 Jy/sr. The results from the LFs in(Ly et al. 2007) is in good agreements with the others,and we adopt their LFs and the errors to calculate thepower spectrum and uncertainty for the three foregroundlines.At last, we discuss the methods to remove the fore-ground contamination due to low-redshift emission lines.We first compare the power spectrum of the Ly α at z ∼ α , [OIII] and [OII] power spectrum at low red-shifts, and consider the projection effect in the real sur-vey. The power spectrum of the foreground lines be-come larger after the projection. We then propose tomask the whole bright pixels with foreground emissionabove some flux threshold to reduce the contaminationof the foreground lines in the intensity mapping. Wefind the contamination can be neglected when the fluxcut is 1 . × − W / m . 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B., & Dijkstra, M. 2011, MNRAS, 415, 3929-3950Zheng, Z., Cen, R., Trac, H., & Miralda-Escude, J. 2010, ApJ,716, 574Zheng, Z., Cen, R., Trac, H., & Miralda-Escude, J. 2011, ApJ,726, 38APPENDIX k ⟂ (h/Mpc) k ∥ ( h / M p c ) Signal P ( k ) [ ( J y / s r ) ( M p c / h ) ] k ⊥ (h / Mpc) k k ( h / M p c ) Foreground P ( k ) [ ( J y / s r) ( M p c / h ) ] k ⊥ (h / Mpc) k k ( h / M p c ) Total P ( k ) [ ( J y / s r) ( M p c / h ) ] Fig. 10.—
The 2-D anisotropic power spectra shown by perpendicular and parallel Fourier modes for the signal (Ly α ), foreground (onlyH α here) and total observed signal+foreground. We find the 2-D signal power spectrum is almost symmetric for k ⊥ and k k , and the redshiftdistortion effect is relatively small. For the foreground power spectrum at low redshift, because the redshift distortion effect is strong andthe shift factor on the foreground k ⊥ and k k are different, the shape of the power spectrum is irregular. This effect provides a way todistinguish the signal and the foreground in principle. The total power spectrum is similar with the foreground power spectrum, since theamplitude of the signal power spectrum is much lower than the foreground. In Eq. (28) of Section 4.1, we see that the observed foreground power spectrum is actually anisotropic, i.e., notsimply a function of k = q k ⊥ + k k . This can in principle be used in the foreground cleaning process. In particular,after the intensity cut, we can check if there is still any strong contamination by looking at the anisotropy of the totalpower spectrum. In order to check the strength of these anisotropies caused by the projection effect, we calculate the2-D anisotropic power spectra for the signal and foregrounds. Here we also take into account of the linear redshift-spacedistortions. In that case, the bias should be replaced by b s ( z s ) → b s ( z s ) + k k q k ⊥ + k k H ( z s ) ˙ D ( z s ) D ( z s ) b f ( z f ) → b f ( z f ) + ( y s /y f ) k k q ( r s /r f ) k ⊥ + ( y s /y f ) k k H ( z f ) ˙ D ( z f ) D ( z f ) . Here, D ( z ) is the growth factor. The normal expression of the second term of the bias is f µ . Here f = d ln D/d ln a where a is the scale factor, and µ = cos θ where θ is the angle between the line of sight and the wave-vector k . We alsonotice that the so-called non-gravitational effects can introduce strong redshift-space distortion effect (Zheng et al.2011; Wyithe & Dijkstra 2011). According to full Ly α radiative transfer calculations, the Ly α emission is dependenton environment (gas density and velocity) around LAEs. The observed LAE clustering features can be changed bythis effect especially at high redshifts. However, this effect relies on “missing” Ly α photons scattering in relatively2close proximity to Ly α emission galaxies, which could be recovered by intensity mapping. Therefore we ignore thiseffect in our discussion.In Figure 10, we show the 2-D anisotropic power spectrum decomposed into k ⊥ and k k for the signal, foregroundand the total observations. We just show the H α foreground here, since it has the lowest redshift among the other twoforeground lines and has the largest effect of the power spectrum projection. We find the shape of the signal powerspectrum is quite symmetric and the redshift distortion effect is relatively small, since the signal comes from highredshifts. However, for the foreground, the shape of the spectrum is irregular due to the redshift distortion and thedifferent factors on the k ⊥ and k k (i.e. r s /r f on k ⊥ and y s /y f on k kk