Probing patchy reionization through tau-21cm correlation statistics
aa r X i v : . [ a s t r o - ph . C O ] O c t Draft version November 11, 2018
Preprint typeset using L A TEX style emulateapj v. 04/17/13
PROBING PATCHY REIONIZATION THROUGH τ -21CM CORRELATION STATISTICS P. Daniel Meerburg , Cora Dvorkin , and David N. Spergel Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540 USA. and Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA (Dated: November 11, 2018)
Draft version November 11, 2018
ABSTRACTWe consider the cross-correlation between free electrons and neutral hydrogen during the epoch ofreionization. The free electrons are traced by the optical depth to reionization τ while the neutralhydrogen can be observed through 21 cm photon emission. As expected, this correlation is sensitive tothe detailed physics of reionization. Foremost, if reionization occurs through the merger of relativelylarge halos hosting an ionizing source, the free electrons and neutral hydrogen are anti-correlated formost of the reionization history. A positive contribution to the correlation can occur when the halosthat can form an ionizing source are small. A measurement of this sign change in the cross-correlationcould help disentangle the bias and the ionization history. We estimate the signal-to-noise of thecross-correlation using the estimator for inhomogeneous reionization ˆ τ ℓm proposed by Dvorkin andSmith (2009). We find that with upcoming radio interferometers and CMB experiments, the cross-correlation is measurable going up to multipoles ℓ ∼ Keywords:
Reionization, 21 centimeters, Cosmic Microwave Background INTRODUCTION
The epoch of reionization (EoR) is one of the leastunderstood periods of cosmic history, with only limitedobservational measurements (see e.g. Loeb and Barkana(2001)). The absence of the Gunn-Peterson trough in thespectra of quasars implies that reionization should havebeen completed around z = 6 (Fan et al. 2006a,b). Thetotal optical depth to reionization has been measured tobe τ = 0 . ± .
013 (Hinshaw et al. 2012). If reioniza-tion is assumed to be instantaneous, this would imply atransition redshift of z re = 11. Besides these constraints,we know very little about the details of reionization, suchas the typical halo mass associated with the first ionizingobjects as well as the distribution of these objects insidethe halos.The spectral mapping of neutral hydrogen inemission (Hogan and Rees 1979; Scott and Rees1990; McQuinn et al. 2006; Madau et al. 1997;Zaldarriaga et al. 2004; Furlanetto et al. 2004a,b,2006) promises to be a new probe of the EoR. Thespontaneous hyperfine spin flip transition causes theemission of a photon with a wavelength of 21 centimetersin the rest frame. Applying different base filters to theobserved emission, it is possible to map the distributionof neutral hydrogen in the Universe as a function ofredshift. The auto-correlation of the observed maps isvery sensitive to the EoR parameters.Additionally, cross correlating the observed maps withother observables could provide complementary con-straints on the EoR parameters. For example, the 21 cmfluctuations are expected to be correlated with galax-ies (Lidz et al. 2009; Wiersma et al. 2012) as well as [email protected]@[email protected] with the Cosmic Microwave Background (CMB) fluc-tuations through the Doppler peak and the kinetic SZeffect (see e.g. Alvarez et al. (2006); Salvaterra et al.(2005); Adshead and Furlanetto (2007); Tashiro et al.(2008); Tashiro et al. (2010, 2011); Holder et al. (2006)and (Jeli´c et al. 2010; Natarajan et al. 2012) for recentsimulations). Unlike the fluctuations in 21 cm, whichare a direct representation of the underlying neutral hy-drogen, no direct measurement of the electron densityat high redshifts exists. The electron density can bemeasured indirectly through its integrated effect on theCMB, providing us with a number ( τ ) that tells us thefraction of photons affected by scattering of electronsalong the line of sight. One can go further and re-construct the inhomogeneities in the optical depth fieldby considering second order effects on the CMB due tothe screening mechanism (Dvorkin et al. 2009), Thom-son scattering and the kSZ effect (see Dvorkin and Smith(2009), where an estimator of the anisotropic opticaldepth field is derived and Gluscevic et al. (2012) for animplementation of this estimator to WMAP 7-year data).In this work we consider the cross-correlation between areconstructed map of the inhomogeneous optical depth τ ℓm (using CMB polarization observations) and a neu-tral hydrogen map measured through the redshifted 21cm lines. Intuitively, these two observables are expectedto be anti-correlated on most scales. We would like tostress that this cross-correlation is fundamentally differ-ent from direct cross correlations between CMB temper-ature and polarization and 21 cm maps (Tashiro et al.2008; Tashiro et al. 2010); the τ estimator is a quadraticestimator, making this cross correlation a statistical 3point correlation function rather than a 2 point function.Besides providing complementary constraints on theEoR parameters, the cross-correlation between the 21cm field and the CMB should in principle be less sen-sitive to the details of the foregrounds. Although cur-rent (Masui et al. 2012; Chapman et al. 2012) and up-coming experiments (Harker et al. 2010; Mellema et al.2012) are expected to be capable of measuring the auto-correlation of 21 cm maps, one very persistent nuisancein extracting the signal from reionization are the fore-grounds. Inhomogeneities in the 21 cm signal due topatchy reionization must be separated from fluctuationsin foreground sources. Typical foreground sources arefaint radio galaxies, starburst galaxies and galaxies re-sponsible for reionization. In addition, our own Galaxyis very bright at the frequencies one aims at for mappingthe 21 cm signal from reionization, exceeding the 21 cmreionization signal by several orders of magnitude. At-tempts have been made to characterize these foregrounds(Liu et al. 2009; Jeli´c et al. 2008; Bernardi et al. 2010,2009; Liu et al. 2012). Despite these efforts, foregroundscan never be fully removed, simply because we do notknow their exact origin.This paper is organized as follows. We review thephysics of reionization and derive the expressions for thefluctuations in 21 cm a ℓm and fluctuations in the opticaldepth τ ℓm in §
2. The former is proportional to the neu-tral hydrogen fraction, while the latter is proportionalto the free electron fraction. Using a simple reioniza-tion model (Furlanetto et al. 2004b; Wang and Hu 2006)where regions of HII are represented by spherical bub-bles of typical size ¯ R we derive an expression for thecross-correlation h τ ℓm a ∗ ℓm i in section §
3. In § § §
7. We assess theobservability of the cross correlation by using a red-shift weighting to maximize the signal to noise. We endthis section with an estimate of the EoR parameter con-straints, when considering LOFAR and SKA noise levels.We present our conclusions in §
9. In the Appendix wediscuss the dependence of the cross-correlation on theparameters of the reionization model.Unless specifically mentioned, we use the followingset of parameter values throughout the paper: h =0 . , Ω b = 0 . , Ω c = 0 . , Ω K = 0 , n s = 0 .
96 and τ = 0 . k ∗ = 0 .
002 Mpc − and A s = 2 . × − (Hinshaw et al.2012).
21 CM BRIGHTNESS AND THE OPTICAL DEPTH TOREIONIZATION
In this section we will review the standard results forfluctuations in the 21 cm temperature brightness and re-late those to fluctuations in the neutral hydrogen frac-tion. In the second half of this section, we will derive thefluctuations in the optical depth τ caused by fluctuationsin the free electron fraction along the line of sight, con-firming the results first obtained in (Holder et al. 2006).The optical depth of a region of the IGM in the hyper-fine transition is given by (Field 1958) τ ( z ) = 3 c hA πkν T S n HI (1 + z )( dv k /dr k ) , (1)where ν = 1420 . ν = λ /c ) transition frequency, A = 2 . × − s − is the spontaneous emission coefficient for this transition, T S is the spin temperature of the IGM, weighting the rel-ative population of the atoms in the singlet state to atomsin the triplet state (Field 1958), n HI is the neutral hy-drogen density, and v k the proper velocity along the lineof sight. At high redshifts, where peculiar motions alongthe line of sight are small compared to the Hubble flow, dv k /dr k = H ( z ) / (1+ z ). At z = 10 dark energy and radi-ation are both unimportant and we can solve for H ( z ) ina matter dominated Universe, H ( z ) ≃ H Ω / m (1 + z ) / .We can now write the following expression for the op-tical depth: τ ( z ) ≃ . × − (1 + δ b ) x H (cid:20) T cmb T S (cid:21) (cid:20) − Y p − . (cid:21) × (cid:18) Ω b . (cid:19) (cid:20)(cid:18) . m (cid:19) (cid:18) z (cid:19)(cid:21) / (2)Here we used T cmb = 2 . z )K, δ b = ( ρ b − ¯ ρ b ) / ¯ ρ b and n HI ≃ (1 − Y p ) x H Ω b Ω m ρ m m p , where x H is the neutral hydrogen fraction, i.e. x H = n HI / ( n HI + n e ), ρ m is the matter energy density and m p is the proton mass. The factor (1 − Y p ) addresses thefact that not all protons are in hydrogen, but a fractionis in Helium.The intensity along the line of sight from a thermalsource is given by I = I e − τ + Z τ dτ ′ e − τ ′ η ν κ ν , (3)with κ ν the absorption coefficient and η ν the emissiv-ity of photons. Using dI = η ν dl − κ ν Idl = 0 in theRayleigh-Jeans limit, we have I = 2 kT b ν /c , while η ν /κ ν = 2 kT S ν /c and I = 2 kT cmb ν /c . Hence, wecan write the 21 cm brightness temperature as: T b = T cmb e − τ + T S (1 − e − τ ) (4)The brightness temperature increment is defined at anobserved frequency ν corresponding to a redshift 1 + z = ν /ν as δT b ( z ) ≡ T b − T cmb z ≃ ( T S − T cmb )1 + z τ (5)Using Eq. (2), we can re-write Eq. (5) as(Scott and Rees 1990; Madau et al. 1997) δT b ( z ) ≃
27 mK (1 + δ b ) x H (cid:20) T S − T cmb T S (cid:21) (cid:20) − Y p − . (cid:21) × (cid:18) Ω b . (cid:19) (cid:20)(cid:18) . m (cid:19) (cid:18) z (cid:19)(cid:21) / (6)There are usually two types of filters associated withthe resolution of the experiment. First, there is a finiteangular resolution, which will affect all modes perpen-dicular to the line of sight. Second, since the brightnesstemperature of the 21 cm emission is a 3-dimensionalfield, we are confined to a frequency resolution or band-width, which affects the modes along the line of sight (or,equivalently, a redshift resolution).The total integrated 21 cm surface brightness is givenby T b (ˆ n, χ ) = T ( χ ) Z dχ ′ W χ ( χ ′ ) ψ (ˆ n, χ ′ ) (7)Here W χ is an experimental band filter that is due tothe finite frequency resolution of the instrument, whichis centered around χ (comoving distance). We define thedimensionless brightness temperature ψ as ψ = (1 + δ b ) x H (cid:18) T S − T cmb T S (cid:19) (8)In the limit of T s ≫ T cmb , ψ = (1 + δ b ) x H .Now T ( z ) can be written as T ( z ) ≃
27 mK (cid:20) − Y p − . (cid:21)(cid:18) Ω b . (cid:19) (cid:20)(cid:18) . m (cid:19) (cid:18) z (cid:19)(cid:21) / (9)We will now consider fluctuations in the free electrondensity, which in turn will induce fluctuations in the op-tical depth. The optical depth to distance χ along theline of sight is given by τ (ˆ n, χ ) = σ T Z χ dχ ′ n e (ˆ n, χ ′ ) a ( χ ′ ) , (10)where σ T is the Thomson cross-section, n e is the electronnumber density and a is the scale factor. Relating thefree electron density to the free electron fraction x e , wecan write n e (ˆ n, χ ) ≃ x e ρ b m p (1 − Y p ) , (11)assuming that Helium is singly ionized.The average baryon density diffuses in an expandingbackground as a − , and the free electron density be-comes n e (ˆ n, χ ) = (1 − Y p ) ρ b, m p a − (1 + δ b ) x e (12)The optical depth can in turn be written as τ (ˆ n, χ ) = σ T (1 − Y p ) ρ b, m p Z χ dχ ′ a ( χ ′ ) x e (ˆ n, χ ′ ) × (1 + δ b (ˆ n, χ ′ )) (13)Therefore, we can relate fluctuations in the opticaldepth δτ to fluctuations in the 21 cm brightness tem-perature δT b (Holder et al. 2006) as δτ = (1 − Y p ) σ T ρ b, m p H Ω − / m Z dz (cid:20) (1 + z ) / δ b − δT b ( z )8 . (cid:21) , (14)where we have assumed a delta window function.It is worth noticing that the above expression for thefluctuations in the 21 cm brightness temperature is only valid for T s > T CMB . Early on, when the number of ion-izing sources are rare and the temperature of the IGMclose to these sources is coupled to the kinetic tempera-ture by Ly α photons associated with these local sources,this assumption breaks down, and the 21 cm signal canappear in absorption. We neglect this effect in this pa-per. CORRELATING X AND ψ We will now cross-correlate the optical depth fluctua-tions with the temperature brightness. As we saw be-fore, the CMB optical depth is proportional to the freeelectron density. If reionization is inhomogeneous, thefree electron density is a function of position in the sky.Anisotropies in the optical depth produce three effects inthe CMB: (i) screening of the temperature and polariza-tion fluctuations that we observe today by an overall fac-tor of e − τ (ˆ n ) . This effect generates CMB B-mode polar-ization; (ii) Thomson scattering: new polarization is gen-erated by scattering of the local temperature quadrupolethat each electron sees along the line of sight. This ef-fect also produces B-modes; and (iii) new temperatureanisotropy is generated from the radial motion of ion-ized bubbles relative to the observer (the kinetic SunyaevZel’dovich effect).The two-point correlation function between the E-modes and the B-modes generated from patchy reion-ization is proportional to the anisotropic part of theoptical depth. This fact allowed the authors in Ref.(Dvorkin and Smith 2009) to write a minimum variancequadratic estimator ˆ τ ℓm for the field τ (ˆ n ). In this work,we will use the CMB polarization fluctuations to recon-struct a map of τ and cross-correlate it with the 21 cmfield.We will use the shorthand notation X (ˆ n, χ ) = x e (1 + δ b ). We can write the cross-correlation between the fieldX (measured through the CMB) and the field ψ (mea-sured through 21 cm) as ξ Xψ ≃ − ξ xx (1 + ξ δδ ) − (¯ x H − ¯ x e + ξ xδ ) ξ xδ +(¯ x H − ¯ x H ) ξ δδ (15)Here we defined ξ xx = h x H ( ~x ) x H ( ~x ) i − ¯ x H , ξ δδ = h δ b ( ~x ) δ b ( ~x ) i and ξ xδ = h δ b ( ~x ) x H ( ~x ) i . We makethe simplistic assumption that the connected part of h δ x δ x δ b δ b i vanishes. Here δ x corresponds to fluctua-tions in the neutral hydrogen fraction, which is givenby x H = ¯ x H (1 + δ x ).Before we can compute ξ Xψ we need to specify ourmodel of reionization. We will assume that the Universereionized through the growth of ionized bubbles associ-ated with massive halos. The bubbles themselves containa single source and we assume their size to be larger thanthe non-linear scale.We will adopt the following average reionization frac-tion as a function of redshift¯ x e ( z ) = 12 (cid:20) (cid:18) y re − (1 + z ) / ∆ y (cid:19)(cid:21) , (16)which is the one used in the code CAMB (Lewis et al.2000). Here y ( z ) = (1 + z ) / , y re = y ( z re ) and ∆ y arefree parameters that satisfy our integrated optical depthalong the line of sight τ = 0 . R . We will assumethat the typical ionized bubble radii are log-normal dis-tributed (Zahn et al. 2006), i.e. there is a skewness to-wards smaller bubble sizes, P ( R ) = 1 R p πσ R e − [ln( R/ ¯ R )] / (2 σ R ) , (17)where σ ln R is the variance of the distribution.The average bubble volume is then given by h V b i = Z dRP ( R ) V b ( R ) = 4 π ¯ R e σ R / (18)Hence, we can define a volume weighted radius, R suchthat h V b i = 4 πR /
3, which can be written as R = ¯ Re σ R / (19)If we assume that a given point in space is ionized withPoisson probability, we can write the ionization fractionas h x e ( ~x ) i P = 1 − e − n b ( ~x ) h V b i , (20)with n b the number density of bubbles. The bracketsaround x e are placed to remind us that we are consid-ering a Poisson distribution of sources, and the result isaveraged over the Poisson process. We further assumethat the number density of bubbles traces the large-scalestructure with some bias b : n b ( ~x ) = ¯ n b (1 + bδ W ( ~x )) , (21)while the average bubble number density is related to themean ionization fraction as¯ n b = − h V b i ln(1 − ¯ x e ) (22)Here δ W is the matter over-density δ smoothed by a tophat window of radius R , δ W ( ~x ) = Z d x ′ δ ( ~x ′ ) W R ( ~x − ~x ′ ) (23)In momentum space, W R ( k ) = 3( kR ) [sin( kR ) − kR cos( kR )] , (24)which is the Fourier transform of W R ( x ) = V − b for x ≤ R and W R ( x ) = 0 otherwise.We further define h W R i ( k ) = 1 h V b i Z ∞ dRP ( R ) V b ( R ) W R ( kR ) (25)and h W R i ( k ) = 1 h V b i Z ∞ dRP ( R ) [ V b ( R ) W R ( kR )] (26) REIONIZATION PARAMETERS
Reionization can only proceed if the seed halo is mas-sive enough for cooling. In particular, line cooling and atomic cooling are important for collapse. One can re-late the virial temperature of the halo to the mass of thehalo T vir K = 1 . (cid:18) Ω m h . (cid:19) / (cid:18) z (cid:19) (cid:18) M M ⊙ (cid:19) / . (27)Therefore, setting a condition on the amount of coolingnecessary to form ionizing objects sets a typical massof the halo, which will be redshift dependent. We canrelate the mean number density of bubbles, the averagereionization fraction and the typical bubble volume byinverting Eq. (22): h V b i = − n b ln(1 − ¯ x e ) (28)The mean bubble number density introduced in Eq. (22)is derived through an integral over the halo mass func-tion, with a mass threshold M th which can roughly beset by the viral temperature in Eq. (27) :¯ n b = Z ∞ M th dn h d ln M dMM (29)We use the Sheth and Tormen (Sheth and Tormen 2002)halo mass function dn h d ln M = ρ m (0) M f ( ν ) dνd ln M (30)with νf ( ν ) = A r π aν (cid:0) aν ) − p (cid:1) e − aν / (31)where ν = δ c /σ lin ( M, z ) and σ lin is the variance of thedensity field smoothed with the top-hat window functionenclosing a mass M : σ lin ( M, z ) = Z dkk ∆ m ( k, z ) W R ( M ) ( k ) (32)It is straightforward to show that dνd ln M = − ν d ln σ lin ( M, z ) d ln M (33)The parameters δ c , a and p can be fitted from sim-ulations. A is then derived through the constraint R dνf ( ν ) = 1. Consequently, by setting M th we canfind an expression for the average bubble volume, andwe can infer a function of the average bubble radius asa function of redshift (in the assumption of a log normaldistribution of radii at any given redshift).Likewise, the bubble bias can be related to the halobias (see e.g. Wang and Hu (2006)) as: b = 1¯ n b Z ∞ M th b h ( M ) dn h d ln M dMM (34) Although the model we are using here is self consistent, relatingour toy-reinoization model to all other relevant parameters, we findthat the resulting bubble radius as a function of redshift is too smallcompared to simulations (Shin et al. 2008). One can alleviate thisdiscrepancy somewhat by raising M th . For that purpose we assumethe critical temperature to form an ionizing object to be five timesthe virial temperature. Figure 1.
The average ionization fraction ¯ x e ( z ) as a function ofredshift z . The integral runs over all masses with a threshold massscale M th . Sheth and Tormen can be used for the halobias b h = 1 + aν − δ c + 2 pδ c (1 + ( aν ) p ) (35)Therefore, for any given model of ¯ x e ( z ) we can compute¯ R ( z ), b ( z ) and ¯ n b ( z ). In this paper we will use δ c = 1 . a = 0 .
707 and p = 0 . σ ln R is constant.Assuming a log normal distribution, for any given com-bination of kR ( z ) we can then read of the value of h W R i and h W R i (see Figs. 12 and 13)In Fig. 1 we have plotted the model we use for our av-erage ionization fraction as a function of redshift. Ourchoice of parameters corresponds to a scenario with aneutral Universe at z ≥
13 and a completely ionized Uni-verse at z ≤ TWO-BUBBLE CORRELATIONS
The cross-correlation between the neutral hydrogenand free electron fraction has two main contributions.First, the correlation is set by the Poisson distributionof ionizing sources inside the bubbles. This term is re-ferred to as the one-bubble term, and it is dominatedby the shot noise. Second, cross correlations can also beinduced by the enhanced probability of bubble forma-tion (or ionizing sources) inside overdense regions (withprobability h x e i ). This term is referred to as the two-bubble term, and is relevant for scales much larger thanthe average size of a bubble.Using Eqs. (20) and (21) we can Taylor expand h x e i around small overdensities, to find h x e i = 1 − e ln(1 − ¯ x e )(1+ bδ W ) ≃ − (1 − ¯ x e ) [1 + bδ W ln(1 − ¯ x e )] + O ( δ W )(36)with b the bubble bias introduced earlier.Taking the Fourier transform of ξ xx and ξ xδ , we obtain P bxx ( k ) = [¯ x H ln(¯ x H ) b h W R i ( k )] P δδ ( k ) (37) P bxδ ( k ) = ¯ x H ln(¯ x H ) b h W R i ( k ) P δδ ( k ) , (38)Note that here we are considering correlations of the neu-tral hydrogen fraction (perturbing the free electron frac-tion accounts for a minus sign in Eq. (38)). The super-script “2 b ” denotes the two-bubble contribution. Also,we assume that the baryon fluctuations (the gas) trace Figure 2. P bXψ ( k ) for 4 different redshifts with σ ln R = 0 .
5. Inour toy model the two-bubble term is positively correlated for allredshifts z >
12. The zero point is uniquely determined by therelations ¯ x H = e − /b . Measuring this zero point could thereforehelp decorrelate these two parameters. However, the contributionof the two-bubble term to the overall cross correlation very small,and it will be be challenging to observe the cross correlation as afunction of redshift as we show in section § the dark matter fluctuations. The total two-bubble con-tribution to the power spectrum of Xψ results in P bXψ ( k ) ≈ − ¯ x H [ln ¯ x H b h W R i ( k ) + 1] P δδ ( k ) +¯ x H [ln ¯ x H b h W R i ( k ) + 1] P δδ ( k ) (39)Let us write P bXψ ( k ) = Q (1 − Q ) P δδ ( k ) ≡ b eff P δδ ( k ),with Q ( k, z ) = ¯ x H ln ¯ x H b h W R i + ¯ x H . We can distinguishtwo limiting cases: for Q > b eff is negative, representing an anti-correlation ,while for Q < b eff is positive, and the two-bubble termis positively correlated.In the large scale limit, when k ¯ R ≪ b eff → ¯ x H ( b ln ¯ x H + 1)(¯ x e − b ln ¯ x H ). This function changes signwhen ¯ x H = e − /b . The bubble bias on average growstowards larger z , despite the bubbles being smaller, thebubbles become rare (larger ¯ x H ) and highly correlated.For our choice of parameters we can solve this equationfor the redshift and find z = 12 as the redshift at whichthe two-bubble term turns negative on large scales. Fig. 2shows the two-bubble term for various redshifts.At scales that are smaller than the radius of the bub-bles, the two-bubble term is ill-defined (Baldauf et al.2013): the correlation length becomes shorter than thesize of the bubbles, effectively rendering them to onebubble. Therefore, we will neglect the two-bubble cor-relation term at those scales. In practice, we apply asmoothing filter that effectively cuts the correlation for kR ( z ) <
3. We show the two-bubble contribution tothe cross-correlation in Fig. 3. ONE-BUBBLE CORRELATIONS
For scales much smaller than the average bubble ra-dius, the correlation is dominated by the presence (orabsence) of a single bubble (Wang and Hu 2006). Thecorrelation between two (ionized) points separated by x = | ~x − ~x | can be written as (Zaldarriaga et al. 2004;Furlanetto et al. 2004a) h x e ( ~x ) x e ( ~x ) i = ¯ x e + (¯ x e − ¯ x e ) f ( x /R ) , (40) Figure 3.
The two-bubble term for 3 different values of thewidth of the log-normal distribution. The relative contribution ofthe two-bubble term to the total correlation increases rapidly withdecreasing ¯ R and σ ln R , as derived in the Appendix. where f ( x ) is a function with the following limits: f ( x ) → x ≪ f ( x ) → x ≫
1. If theprobability for finding one point inside an ionized bub-ble is ¯ x e , then when x ≪ R the probability of findingthe second point in the same bubble is 1, hence their jointcorrelation probability is ¯ x e . For large separations, theprobability of finding two points in separate bubbles isthe product of both probabilities, i.e., ¯ x e . Eq. (40) effec-tively encodes the smooth transition between these tworegimes. The one-bubble correlation for free electrons(or equivalently neutral hydrogen) then becomes: ξ bx e x e = h x e ( ~x ) x e ( ~x ) i − ¯ x e = (¯ x e − ¯ x e ) f ( x /R )(41)As long as the bubbles do not overlap, the function f can be described by the convolution of two top hatwindow functions h W R i . The one-bubble contribution tothe power spectrum can then be written as P bXψ = − (¯ x e − ¯ x e ) h h V b ih W R i ( k ) + ˜ P δδ ( k ) i , (42)where ˜ P δδ ( k ) = h V b i Z d k ′ (2 π ) h W R i ( k ′ ) P δδ ( | ~k − ~k ′ | ) (43)The first term in Eq. (42) is the shot noise of the bub-bles, which is a direct consequence of randomly placingionizing galaxies in the Universe. We will later see thatthis term typically dominates the total correlation func-tion at late times. This can be understood by realizingthat the bubbles tend to be larger at late times, assumingthat the bubble size increases over time through bubblemerging.Note that when correlating the free electrons with theneutral hydrogen, the one-bubble contribution is alwaysnegative, i.e. these are fully anti-correlated when consid-ering just single bubbles in the Universe. In the previoussection we have also shown that the two-bubble cross-correlation at early times. As reionization proceeds andthe neutral hydrogen fraction decreases, the correlationon all scales will become anti-correlated. More impor-tantly, the signal is proportional to the matter powerspectrum, which grows as (1 + z ) during matter domi-nation. Within the model applied in this work, we find Figure 4.
The shot noise P snXψ ≡ (¯ x e − ¯ x e ) h V b ih W R i for 3 differentvalues of the width of the log-normal distribution. that at redshift z <
12, the two-bubble term is negligiblecompared to the one-bubble term on most scales.It was shown by Ref. (Wang and Hu 2006) that ˜ P δδ ( k )can be approximated as˜ P δδ ( k ) ≃ P δδ ( k ) h V b ih σ R i [( P δδ ( k )) + ( h V b ih σ R i ) ] / , (44)which is derived by equating the small and large scalelimits of Eq. (43) with h σ R i = Z k dk π h W R i ( k ) P δδ ( k ) (45)We have found the simple fitting solution of Eq. (44)to be accurate to the percent level for most values of { ¯ R, σ ln R } . In the Appendix we will show that the con-tribution from ˜ P δδ ( k ) to the one-bubble peak is relativelysmall for all parameter values in the range of interest forthe τ −
21 cm cross-correlation, but is non-negligible andrelevant at small scales.We show the shot noise and ˜ P δδ in Figs. 4 and 5. Thesum of these terms gives the one-bubble power spectrumshown in Fig. 6. Note that the one-bubble term from allthree possible correlations ( XX , ψψ and Xψ ) is equiva-lent up to a sign.In the small scale limit, ˜ P δδ ( k ) = P δδ ( k ), and the one-bubble term becomes: P bXψ ( k ) = − (¯ x H − ¯ x H ) P δδ ( k ), ren-dering the total cross-correlation negative at these scales(at these scales we are applying a smoothing filter to thetwo-bubble term, so it effectively does not contribute tothe total cross-correlation). In Fig. 7 we show the totalcross-correlation for σ ln R = 0 . z = 11 (changing theredshift of the cross-correlation will predominantly affectthe average ionization fraction ¯ x e ). PROJECTED CROSS-CORRELATION
Fourier transforming the dimensionless brightness tem-perature ψ , we can write the spherical harmonic coeffi-cient for the 21 cm fluctuation as a ℓm = 4 π ( − i ) ℓ Z d k (2 π ) ˆ ψ ( ~k ) α ℓ ( k, z ) Y ∗ ℓm (ˆ k ) , (46)where α ℓ ( k, z ) = T ( z ) Z ∞ dχ ′ W χ ( z ) ( χ ′ ) j ℓ ( kχ ′ ) (47) Figure 5. ˜ P δδ as defined for 3 different values of the width of thelog-normal distribution. This figure shows that this term is onlyrelevant at small scales and does not contribute to the peak of thetotal correlation function. Figure 6.
The sum of the previous two figures, the total one-bubble term. The one-bubble term has the same shape for corre-lations XX , Xψ and ψψ . Figure 7. k P Xψ ( k ) / π with σ ln R = 0 . Note that the response function is centered around χ ( z ) = χ ′ , and in practice we take this distance to besomewhere between z = 0 and z = 30.We can do the same for the optical depth to reioniza- tion, i.e. : τ ℓm = 4 π ( − i ) ℓ Z d k (2 π ) X ( ~k ) α τℓ ( k ) Y ∗ ℓm (ˆ k ) , (48)with α τℓ ( k ) = (1 − Y p ) σ T ρ b, m p Z χ ∗ dχ ′ a j ℓ ( kχ ′ ) , (49)where χ ∗ corresponds to the distance to last scattering.Cross-correlating the two maps yields h τ ℓm a ∗ ℓ ′ m ′ ( z ) i = δ ℓℓ ′ δ mm ′ C τ, ℓ ( z )= Z dkk ∆ Xψ ( k ) α τℓ ( k ) α ℓ ( k, z ) (50)Here, ∆ Xψ = k P Xψ / (2 π ).Let us consider the cross-correlation in the Limber ap-proximation. Under this approximation, we can assumethat the Bessel functions are small, j ℓ ( x ) ≪
1, for x < ℓ and peak when x ∼ ℓ . The integral over comoving mo-mentum k will get most of its contribution from modes k ∼ ℓ/χ . Therefore we can make the approximation that∆ Xψ ( k ) ∼ ∆ Xψ ( ℓ/χ ) and re-write Eq. (50) as C τ, ℓ ( z ) = (1 − Y p ) T ( z ) ρ b, σ T m p Z z ∗ dz ′ H ( z ′ ) (1 + z ′ ) Z ∞ dz ′ H ( z ′ ) W z ( χ ( z ′ )) × π Z dkk ∆ Xψ ( k ) j ℓ ( kχ ( z ′ )) j ℓ ( kχ ( z ′ )) (51)Again, in practice we take the window function to becentered around 0 ≤ z ≤ k integral over the product of Bessel functions as: Z ∞ dkk j ℓ ( kχ ( z )) j ℓ ( kχ ( z )) = π δ ( χ ( z ) − χ ( z )) χ (52)Thus, we can write the angular cross spectrum as C τ, ℓ ( z ) = (1 − Y p ) T ( z ) ρ b, σ T m p Z ∞ dz ′ W z ( χ ( z ′ )) H × (cid:12)(cid:12)(cid:12)(cid:12) dχdz (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) z ′ χ ( z ′ ) (cid:19) P Xψ (cid:18) ℓχ ( z ′ ) , z ′ (cid:19) (53)We will assume that the window function is a Gaussiancentered around redshift z with width δχ given by δχ ≃ (cid:18) ∆ ν . (cid:19) (cid:18) z (cid:19) / (cid:18) Ω m h . (cid:19) − / Mpc(54)where ∆ ν is the bandwidth frequency of the instrument.We have taken into account that the power spectrumexplicitly depends on redshift.We show the angular cross-correlation for ∆ ν = 0 . σ ln R in Fig. 8.In the previous sections we have shown that the cross-correlation between free electrons and neutral hydrogen Figure 8.
The cross-correlation between 21 cm temperaturebrightness fluctuations and the optical depth τ at z = 11. Herewe used a Gaussian window function with bandwidth frequency∆ ν = 0 . has a strong dependence on the parameters that deter-mine the bubble distribution as well as its bias. Thelocation of the peak is set by an effective scale, whichwe derive in the Appendix. We will see that for a log-normal bubble distribution, this effective scale is expo-nential in the width of the distribution and inverselyproportional to the average bubble radius. Therefore,a small change in the width of the distribution can pro-duce a large change in the location of the peak. Wehave also shown that at early times the two-bubble termcan be positively correlated at large scales. The positivecontribution to the correlation function at large scaleseventually vanishes when the universe further reionizes.However, if the bubbles are small enough, a positive con-tribution to the correlation function could persist untillate times. Oppositely, if bubbles are relatively large (afew Mpc), the shot noise, which is negative for all scales,will dominate the correlation function. For the reioniza-tion parameterization in this paper, the shot noise is thedominated term at the most relevant redshifts (peakingaround ¯ x e = 0 . IS THE CROSS-CORRELATION DETECTABLE?
Signal-to-noise
In this section we will determine if the cross-correlationis detectable. An important issue that we will addresshere are the foregrounds. As previously mentioned, the21 cm emission should be swamped by foregrounds, dom-inated on large scales by polarized Galactic synchrotron,with a total intensity of 3-4 orders of magnitude largerthan the 21 cm brightness from reionization. On smallscales, the redshifted 21 cm brightness is obscured by ex-tragalactic sources (Shaver et al. 1999; Jeli´c et al. 2008).We do not know the spectral dependence of all these fore-grounds, but in general we can assume that they are rela-tively smooth in frequency along the line of sight, as theyare associated with same source (e.g. our own Galaxy).In principle, one can therefore remove a large part of the(large scale) foregrounds by removing the largest modesalong the line of sight (see e.g. Liu and Tegmark (2012)for a recent discussion).However, when cross-correlating the 21 cm field withthe optical depth, we want to keep the largest modes,to which the integrated optical depth is most sensitive. Hence, we will keep the foregrounds in the observed mapsand show that the cross-correlation between foregroundsin the 21 cm field and in the CMB should be small. Inorder to neglect the cross-correlation of the foregroundsbetween τ and 21 cm, we typically need the foregroundof the CMB to be ≤ − times the signal (Liu et al.2009) (given that a f, ℓm ∼ a ℓm ). The synchrotronemission is roughly equal to the CMB signal at 1 GHz.Therefore, if we assume that the synchrotron scales as ν − (Kogut et al. 2007), at 94 GHz ( W band) we have a synchrotronℓm ∼ × − a CMBℓm . Thus, we estimate that thesignal will be larger than the remaining foregrounds aftercross-correlating the two maps.Additionally, by not removing the foregrounds, the21cm foregrounds will effectively act as noise term in thecross-correlation. In other words, even in the absenceof correlation between foregrounds, there is still a finiteprobability that any given data point in the τ map willcorrelate with a foreground measurement from 21 cm,i.e. the induced noise contains a term h τ ℓm a f, ℓ ′ m ′ i , wherethe latter is the spherical harmonic coefficient of the 21cm foreground map.Unfortunately, we do not know exactly what the levelof synchrotron foreground is, but typically C fℓ ∼ kℓ − α ,with 2 < α <
4. We will assume that the synchrotronemission scales as ν − . La Porta et al. (2008) showedsynchrotron emission at 480 MHz has a normalized am-plitude of 100 mK < C fℓ =100 < , with theactual amplitude and slope depending on the position inthe sky.Although we cannot remove the foregrounds throughimplementing a large scale cutoff, we can alternativelytry to remove a substantial part of galactic foregroundemission. If there is a large correlation between differ-ent frequencies of the foreground maps, one could mea-sure the foregrounds at high frequency (correspondingto a completely ionized universe and, hence, with no sig-nal in the cross-correlation), extrapolate with an appro-priate scaling ∼ ν − and subtract those from the highredshift maps (Shaver et al. 1999). If the correlationbetween different maps at high frequencies is of order0 . − .
99, one could reduce the overall amplitude ofthe foreground by a factor of 10 −
100 and the powerby a factor of a 100 − . In addition, Liu et al.(2012) showed that down weighting the most heavilycontaminated regions in the sky can reduce the effec-tive foreground as much as a factor of 2. Note thatthis approach is different from the usual spectral fit-ting techniques (Shaver et al. 1999; Santos et al. 2005;Wang et al. 2006; McQuinn et al. 2006; Harker et al.2009; Jeli´c et al. 2010; Chapman et al. 2012, 2013).At small scales we expect extra galactic radio sourcesto dominate the foregrounds. However, there are sev-eral strategies that will likely suppress the noise term This is scaling is approximate and simplistic. In reality onewould probably have to consider a slope that changes as a func-tion of scale and frequency. We are assuming the scaling will befurther understood as a function of frequency once we are capableof performing this cross correlation. Note that this is a very crude estimate. For example, it mightbe relevant to consider 21 cm signals after reionization (at low z ) due to residual neutral hydrogen, primarily in Damped Ly α absorbers (DLA’s). due to correlations between millimeter and radio emis-sion: (i) since bright sources are expected to dom-inate the variance at radio frequency (de Zotti et al.2010; Planck Collaboration et al. 2011; Heywood et al.2013), these sources can be masked at 5 σ . This willsuppress the radio source contribution without remov-ing very much of the sky;(ii) at a given location inthe map, the radio data short wards and long-wardsof the 21 cm radio emission can be used to removeboth galactic and extragalactic foreground by assum-ing that the sources at a given location can be fit bya power law; (iii) at millimeter and sub millimeter wave-lengths, multi-frequency data can be used to separateCMB signal from dusty galaxy foregrounds.The Planckdata shows that the 353 GHz data can be used to re-move >
90% of the dusty galaxy foreground at 220 GHz(Planck Collaboration et al. 2013). All of these strate-gies will likely be employed to remove foregrounds in bothmaps.We will consider a case in which the angular powerspectrum of the foreground is given by: C fℓ ( z ) ≃ c f mK ℓ − (cid:18) f ( z )480 MHz (cid:19) − , (55)where f ( z ) corresponds to the frequency of the redshiftconsidered and 100 ≤ c f ≤ is the foreground re-duction factor that we can hope to achieve through ameasurement at low redshift. As we will show later, thesignal-to-noise of the cross-correlation does not vary sub-stantially for different values of c f in this range.We will assume a noise power spectrumgiven by (Morales 2005; McQuinn et al. 2006;Adshead and Furlanetto 2007; Mao et al. 2008), N , ℓ = 2 πℓ (20 mK) (cid:20) m A eff (cid:21) (cid:20) ′ ∆Θ (cid:21) (cid:20) z (cid:21) . × (cid:20) MHz∆ ν
100 hr t int (cid:21) (56)We will do forecasts for a total integration time of 1000hours, and a beam with an angular diameter of ∆Θ = 9arcmin. We set the bandwidth to ∆ ν = 0 . A eff = 10 m and for a Square Kilometer Array(SKA) type experiment we use A eff = 10 m .On the CMB side, experiments are rapidly improving(Pla 2006; Niemack et al. 2010; Austermann et al. 2012;Zaldarriaga et al. 2008), with high sensitivity experi-ments coming soon (Planck, ACTPol, SPTPol, CMBPol)and we should have observations of the E- and B-modespolarization spectra up to small scales in the near fu-ture. We will now consider a next generation polar-ization experiment that allows us to reconstruct a mapof the optical depth τ ℓm with the estimator proposedby Dvorkin and Smith (2009). This estimator was builtto extract the inhomogeneous reionization signal fromfuture high-sensitivity measurements of the cosmic mi-crowave background temperature and polarization fields.Dvorkin and Smith (2009) wrote a minimum variancequadratic estimator for the modes of the optical depthfield given by: ˆ τ ℓm = N ττℓ X ℓ m ℓ m Γ EBℓ ℓ ℓ (cid:18) ℓ ℓ ℓm m m (cid:19) × a E ∗ ℓ m a B ∗ ℓ m ( C EEℓ + N EEℓ )( C BBℓ + N BBℓ ) , (57)where C EEℓ and C BBℓ are the E - and B -mode polariza-tion power spectra. N EEℓ and N BBℓ correspond to theCMB noise power spectra, and are given by: N EEℓ = N BBℓ = ∆ P exp (cid:18) ℓ ( ℓ + 1) θ (cid:19) , (58)where ∆ P is the detector noise and θ FWHM is the beamsize.The coupling Γ
EBℓ ℓ ℓ can be written asΓ EBℓ ℓ ℓ = C E E ℓ i r (2 ℓ + 1)(2 ℓ + 1)(2 ℓ + 1)4 π × (cid:20)(cid:18) ℓ ℓ ℓ − (cid:19) − (cid:18) ℓ ℓ ℓ − (cid:19)(cid:21) (59)Here C E E ℓ is the cross-power spectrum between theCMB E -mode polarization without patchy reionizationand the response field to τ fluctuations E . ( C E E ℓ ispositive at large scales due to Thomson scattering, andnegative at small scales due to the screening).Furthermore, the reconstruction noise power spectrumis given by: N ττℓ = " ℓ + 1 X ℓ ℓ | Γ EBℓ ℓ ℓ | ( C EEℓ + N EEℓ )( C BBℓ + N BBℓ ) − (60)We note that the main source of contamination in re-constructing τ (ˆ n ) comes from the non-Gaussian signalfrom gravitational lensing of the CMB. In principle, un-biased estimators that simultaneously reconstruct the in-homogeneous reionization signal and the gravitationalpotential can be constructed (Su et al. 2011). For pur-poses of simplicity, we will estimate our cross-correlationusing the estimator given by Eq. (57), but the results inthis work are straightforward to generalize.Given that the τ map is not sensitive to redshift(Dvorkin and Smith (2009) showed that the estimatoris only sensitive to one principal component in redshift),while the 21 cm map can be reconstructed on redshiftslices, we will give a weight to the cross-correlation. Thisweight will be built in order to maximize the signal tonoise, in the same spirit as in Peiris and Spergel (2000),where a weight was derived for the cross-correlations be-tween CMB and Galaxy surveys. We write the weighted21 cm maps as˜ a ℓm ( z ) = Z z dz ′ a ℓm ( z ′ ) w ℓ ( z ′ ) (61)We then want to maximize χ ℓm = h τ ∗ ℓm ˜ a ℓm ( z ) i h τ ℓm τ ∗ ℓm ih ˜ a ℓm ( z )˜ a ∗ ℓm ( z ) i , (62)0 Figure 9.
The different power spectra used to compute the signalto noise in Fig. 10. The sign of C − τℓ has been inverted for thesake of comparison. The spectra are shown at z = 11. The bubbleradii as a function of redshift, the bubble bias and the bubblenumber density are all determined through our toy reionizationmodel of Eq. (22) as explained in section § τ estimator is given by Eq. (60) (Dvorkin and Smith 2009). and, in doing so, we find: w ℓ ( z ) = C τ, ℓ ( z )( C , ℓ + N , ℓ + C fℓ )( z ) , (63)which is nothing else then the projected signal over theprojected noise. Note that we have included the fore-ground as a source of noise as explained before.We can now compute the signal to noise for the τ -21cm cross-correlation as (cid:18) SN (cid:19) = f sky X ℓ (2 ℓ + 1) × Z dz | C τ, ℓ ( z ) | ( C τ,τℓ + N ττℓ )( C , ℓ + N , ℓ + C fℓ )( z )(64)In Fig. 10 we assess the level of detectability for areionization history with σ ln R = 0 . z = 11). Again, to gener-ate the spectra we use a Gaussian window function with∆ ν = 0 . f − / sky S/N = 0 . ℓ max = 3000 for a LOFAR type ex-periment (with f sky = 0 . f sky = 0 .
25) it reaches f − / sky S/N = 16,with f sky being the fraction of the sky covered. We notethat at small scales, the signal to noise does not varysubstantially when considering different values of the pa-rameter c f , that represents the level of foreground sub-traction. Reionization parameters
In this section we will assess what we can learn aboutreionization by studying the 21 cm- τ cross-correlation.We will forecast parameter uncertainties in the followingparameters π = { τ, ∆ y } .As a forecasting tool we will use a Fisher matrix anal-ysis, where the Fisher matrix is given by (Tegmark et al. Figure 10.
Total signal-to-noise for the 21 cm- τ cross-correlationas a function of ℓ max for a model with a log normal bubble distri-bution with a width σ ln R = 0 .
5. We consider an experiment withCMB noise power spectra given by ∆ P = 0 . µ K-arcmin and beamsize Θ
F WHM = 1 arcmin. The foreground angular power spec-trum is given by Eq. (55). We show forecasts for an experimentwith the planned noise level of SKA (in red lines) and the plannednoise level of LOFAR (in black lines). Note that the y-axis repre-sents the product of the signal-to-noise and the fraction of the skycovered. For an experiment like LOFAR (with f sky = 0 . S/N = 0 . f sky = 0 . S/N = 16at ℓ max = 3000 with c f = 100. At small scales does not vary sub-stantially when considering different values of the parameter c f ,which represents the level of foreground subtraction. F µν = f sky X ℓ (2 ℓ + 1) × Z dz ( ∂C τ, ℓ ( z ) /∂π µ )( ∂C τ, ℓ ( z ) /∂π ν )( C τ,τℓ + N ττℓ )( C , ℓ + N , ℓ + C fℓ )( z )(65)where µ and ν run over the parameter modes.The rms uncertainty on the parameter π µ is given by σ ( π µ ) = ( F − µµ ) / if the other parameters are marginal-ized. If the remaining parameters are assumed fixed,then the rms is σ ( π µ ) = ( F µµ ) − / .We consider a next generation polarization experimentwith noise power spectrum given by ∆ P = 0 . µ K-arcminand beam size Θ
F W HM = 1 arcmin.When assuming an experiment with the planned noiselevel of SKA and f sky = 0 .
25, the width of reionization∆ y can be constrained at the 10% level, and τ at the4% level, when the remaining parameters are consideredfixed.We show the error ellipses for the optical depth andthe width of reionization in Fig. 11. The Planck priorsare shown in dashed lines. DISCUSSION AND CONCLUSION
We investigated the correlation between free electrons,traced by the optical depth τ , and neutral hydrogen,traced through the emission of 21 cm photons, during theepoch of reionization. To compute the cross-correlationwe used a simple model where patches of ionized gas arerepresented by spherical bubbles. The cross-correlationwill depend on the presence or absence of these bubbles(the one-bubble term) and the clustering of bubbles (the1 Figure 11.
Forecasted uncertainties on the width of reionizationparameter ∆ y and the optical depth τ (assuming that the otherparameters are fixed) for an experiment with the planned noiselevel of SKA. Note that the y-axis represents the product of theparameter value and the fraction of the sky observed. For reference,the dashed lines correspond to the Planck priors on the opticaldepth. two-bubble term). As expected, the cross-correlation isnegative on small scales, where it is dominated by theshot noise of the bubbles. On large scales, the two-bubbleterm can render the correlation positive as long as theeffective bias b eff is large or ¯ x e is small. Small bubbles ata fixed neutral hydrogen fraction imply a small bubblebias, hence the two-bubble term has a suppressed (posi-tive) amplitude. A larger correlation could be driven bythe ionization fraction, but within a bubble merger sce-nario the smallest bubbles are expected at early times,when the ionization fraction is small and the total matterpower spectrum is suppressed.The anti-correlation peak, set by the sum of the oneand the two-bubble terms, depends critically on the dis-tribution of the ionized bubbles. Consequently, a mea-surement of the cross-correlation allows us to probe theparameters relevant for the reionization history. In prin-ciple a measurement of a positive correlation at earlytimes, would theoretically allow us to entangle the de-generacy between τ and the bubble bias. However, weshowed that the two-bubble term typically has a verysmall amplitude. One major obstacle in measuring the 21 cm emissionfrom the EoR are the large foregrounds at these frequen-cies. For the auto-correlation, any detection requires acareful removal of foregrounds, which typically resultsin the removal of the largest modes along the line ofsight. The advantage of the cross-correlation is thatforegrounds in the measurement of τ are weakly corre-lated with those in the 21 cm field. Therefore, the cross-correlation is less sensitive to the detailed understandingof the foregrounds.In this paper we have computed the signal to noise ofthe cross-correlation using the estimator for inhomoge-neous reionization ˆ τ ℓm proposed by Dvorkin and Smith(2009). In our computation there is very little contribu-tion from any positive correlation at large scales comingfrom two-bubble term, and most of the signal comes fromthe shot noise. Because a measurement of the opticaldepth gets most of its signal from the long wavelengthmode along the line of sight, we left the 21 cm fore-grounds as a noise term. Although the signal to noiseper mode is small, the large number of modes allows fora detection when considering a next generation 21 cm ex-periment cross-correlated with a CMB experiment thatmeasures the polarization B -modes in most of the sky.We expect that around the time SKA observes a largepart of the sky, CMB experiments will have improved tothe level that we are able to reconstruct a map of τ ℓm .Although the auto-correlation of both maps will give sig-nificant insight into reionization, cross-correlating thesemaps will provide us with a complementary probe. Wefind that a measurement of this cross-correlation witha detector noise level of SKA (and f sky = 0 .
25) on the21cm side and noise level of a next generation polariza-tion type experiment on the CMB side constrains thewidth of the ionization history at the 10% level and theoptical depth at the 4% level.The authors would like to thank Renyue Cen, En-rico Pajer, Fabian Schmidt, Kendrick Smith, and Ma-tias Zaldarriaga for useful discussions. P.D.M. is sup-ported by the Netherlands Organization for ScientificResearch (NWO), through a Rubicon fellowship. C.D.is supported by the National Science Foundation grantnumber AST-0807444, NSF grant number PHY-0855425,and the Raymond and Beverly Sackler Funds. P.D.M.and D.N.S. are in part funded by the John TempletonFoundation grant number 37426.
APPENDIX
REIONIZATION MODEL DEPENDENCE
The choice of a log-normal distribution is motivated in part by simulations in (Zahn et al. 2006) and (Wang and Hu2006). In this appendix we derive constraints on the relative contributions of the various terms of the cross correlationas a function b , ¯ R and σ ln R . The aim of this appendix is to show that in most realistic scenarios, the shot noise isgenerally the dominating term, independent of reionization details. Log-normal distribution
We first start by investigating the implications of a log-normal distribution for the bubble radius.The bubble distribution is given by P ( R, σ ln R ) = 1 R p πσ R e − [ln( R/ ¯ R )] / (2 σ R ) (A1)2Given this distribution we can compute the average bubble size: h V b i = Z dRP ( R ) V b ( R ) = 4 π ¯ R e σ R / (A2)To address the dependence of the resulting correlation function, we also need the variance h V b i = Z dRP ( R ) V b ( R ) = (4 π ) ¯ R e σ R (A3)The amplitude of the one and two-bubble terms, and the relevant scale where these peak, strongly depend on thewindow function W R ( k ) = 3( kR ) [sin( kR ) − kR cos( kR )] (A4)Recall the the volume averaged window function and window function squared (shown in Figs. 12 and 13) are definedas h W R i ( k ) = 1 h V b i Z ∞ dRP ( R ) V b ( R ) W R ( kR ) , (A5)and h W R i ( k ) = 1 h V b i Z ∞ dRP ( R ) V b ( R ) W R ( kR ) (A6)The total correlation function can be written as P Xψ = P bXψ + P bXψ , (A7)where the one-bubble contribution consists of two relevant terms: the shot noise and the power spectrum ˜ P δδ givenby Eq. (44).Let us define the following relevant ratios: R b − sn ≡ P bXψ /P snXψ (A8)and R b − sn ≡ ˜ P δδ /P snXψ , (A9)where P snXψ is the contribution from the shot noise, which is given by P snXψ = − (¯ x e − ¯ x e ) h V b ih W R i ( k ) (A10)In the limit of small comoving momenta (large scales), h W R i →
1, and h W R i → h V b i / h V b i .For small scales, we have: h W R i ( k, ¯ R, σ ln R ) ∼ k h V b i Z ∞ dRR P ( R ) V b ( R )= 92 k ¯ R e − σ R (A11)Roughly speaking, we know that the contribution from the shot noise term will be constant and have a peak atsome characteristic scale after which it will decrease as ∝ /k . Furthermore, up until that characteristic scale, theamplitude of the shot noise is boosted with respect to all the other terms as h V b i / h V b i = e σ R .The characteristic scale is determined by equating the two limiting cases (Mortonson and Hu 2007), i.e.(4 π ) ¯ R e σ R = (4 π ) ¯ R k e σ R , (A12)This tells us that the shot noise roughly peaks around k peak = (cid:18)
92 ¯ R e − σ R (cid:19) / (A13)The second contribution to the one-bubble term comes from ˜ P δδ ( k ), which is given by˜ P δδ ( k ) = h V b i Z d k ′ (2 π ) h W R i ( k ′ ) P δδ ( | ~k − ~k ′ | ) (A14)3 Figure 12. h W R i for different values of σ ln R . Figure 13. h W R i for different values of σ ln R . Note that the maximum amplitude in the limit k ¯ R ≪ σ ln R , whichwill be relevant for the relative contribution of the shot noise with respect to the one and two-bubble terms. As it was shown by Wang and Hu (2006), in the large scale limit we have:lim k ¯ R ≪ ˜ P δδ ( k ) ≃ h V b i Z ∞ dk π k h W R i ( k ) P δδ ( k ) (A15)In this limit, h W R i → h V b i / h V b i , which allows us to put the following constraint on the amplitude of ˜ P δδ ( k )˜ P δδ . h V b ih V b i Z k peak k dkP δδ ( k ) (A16)The relative peak amplitude between the two contributions in the one-bubble term is therefore given by R b − sn ≡ ˜ P δδ /P snXψ . π Z k peak k dkP δδ ( k ) (A17)Since k peak depends on the bubble radius and on σ ln R , so does the relative contribution. Generally speaking, anarrower distribution (with smaller R ) leads to a larger contribution from ˜ P δδ to the total one-bubble term. Thatbeing said, even for very narrow distributions and very small average bubble radius we find that the total contributionto the peak does not exceed more then a few percent, i.e., in realistic scenarios the shot noise term dominates the totalone-bubble term (see e.g. Zahn et al. (2010)). In Fig. 14 we plot the cross-correlation Xψ for different values of σ ln R ,confirming our estimate for the peak sale in Eq. (A13).There is one caveat, which is that ˜ P δδ does not drop as fast as the shot noise, hence at small scales this term cancontribute more. In fact, it is this term the responsible for the turnover at small scales of the τ - τ and 21-21 auto4 Figure 14. k P Xψ ( k ) / π for ¯ R = 1 Mpc, b = 6 at different values of σ ln R . Figure 15. k P XX ( k ) / π for ¯ R = 1 Mpc, b = 6 at different values of σ ln R . Figure 16. k P ψψ ( k ) / π for ¯ R = 1 Mpc, b = 6 at different values of σ ln R . power spectra (see Figs. 15 and 16).We remind the reader that the two-bubble contribution to the Xψ cross-correlation is given by: P bXψ ≈ − ¯ x H [ln ¯ x H b h W R i + 1] P δδ ( k ) +¯ x H [ln ¯ x H b h W R i + 1] P δδ ( k ) (A18)The minimum value P bXψ is reached when ¯ x H = e − − /b , while for the shot noise term ¯ x e = 0 . R b − sn ≡ P bXψ /P snXψ . π be − − /b (1 + e − − /b ) × P δδ ( k peak ) e − σ ln R / (A19)For the toy reionization model, we find that ¯ R (z=11) = 1 .
14 Mpc with a σ ln R = 0 . b (z=11) = 4 .
8, we find R b − sn ∼ .
01, close to the ratio found in Fig. 7.Can the two-bubble term ever dominate over the one-bubble term? Since the ratio goes as 1 / ¯ R and decreasesexponentially in σ ln R , for bubble distributions with small bubble radius and narrow width, we find that the two-bubbleterm can easily dominate the total correlation function. When the radius and the variance depend on redshift, weexpect the two-bubble term to be increasingly important to the correlation function in the early stages of reionization,in other words, when the correlation function is dominated by points that live in two different bubbles. At the onsetof reionization, the bubbles are small and it is more probable to find two points that live in two separate bubbles. Asbubbles merge and grow, it becomes more likely that the correlation function has contribution from points that arein the same bubble. McQuinn et al. (2006) make this distinction, and divide the reionization model in two regimesseparated by the average ionization fraction.Concluding, we see that the location of the peak of the correlation function is roughly set by k peak , given in Eq.(A13), and we note that the location of the peak is almost equivalent for the one and two-bubble terms.We have shown that in the case of a log-normal distribution the contribution from ˜ P δδ to the peak amplitude generallyis small compared to the shot noise and as such, to the overall correlation. Since the shot noise grows as ∝ ¯ R andexponentially in the width of the distribution σ ln R , we find that assuming smaller values for these parameters lead torapidly increasing contribution of the two-bubble term compared to the 1-bubble term. Since decreasing both of theseparameters also increases the value of k peak , assuming a narrower distribution of bubbles with a smaller average radiusresults in a correlation function that is dominated by the two-bubble term and peaks at smaller (physical) scales.These findings are consistent with the expectation that larger bubble imply a larger shot noise. Normal distribution
Simulations show that bubbles are well traced by a log-normal distribution at early times (Zahn et al. 2006), whileat later times the distribution can transition to a normal distribution. Since at late times, large radii dominatereionization, we expect the shot noise to become more dominant.A normal distribution is given by P ( R ) = 1 p πσ R e [ − ( R − ¯ R ) / (2 σ R )] (A20)As expected, for a Gaussian, all relevant quantities are much closer to the distribution values (e.g. ¯ R ) deviating, bydefinition, at most 1 sigma.The average bubble volume is given by h V b i = 43 π ¯ R (cid:0) ¯ R + 3 σ R (cid:1) (A21)We will assume that a normal distribution is only valid for ¯ R > σ R ≤
1. This cutoff is consistent with observations. The expression abovecan easily be understood by Wick expanding the 3-point function h R i .Similarly, for the variance we obtain: h V b i = 169 π (cid:0)
15 ¯ R σ R + 45 ¯ R σ R + ¯ R + 15 σ R (cid:1) (A22)A gaussian distribution allows us to analytically compute the volume average window function: h W R i ( k ) = 3 e − k σ R (cid:2)(cid:0) k σ R + 1 (cid:1) sin (cid:0) k ¯ R (cid:1) − k ¯ R cos (cid:0) k ¯ R (cid:1)(cid:3) k (cid:0) Rσ R + ¯ R (cid:1) , (A23)At large scales, lim k ¯ R ≪ h W R i = 1 , (A24)while at small scales, lim k ¯ R ≫ h W R i = 3 σ R e − k σ R sin (cid:0) k ¯ R (cid:1) k ¯ R (3 σ R + 1) (A25)6The variance is given by: h W R i ( k ) = 9 e − k σ R n e k σ R (cid:2) k (cid:0) ¯ R + σ R (cid:1) + 1 (cid:3) − k ¯ R (cid:0) k σ R + 1 (cid:1) sin (cid:0) k ¯ R (cid:1) + (cid:2) k (cid:0) ¯ R − σ R (cid:1) − k σ R − (cid:3) cos (cid:0) k ¯ R (cid:1)o k (cid:0) Rσ R + ¯ R (cid:1) (A26)At large scales, the variance becomes: lim k ¯ R ≪ h W R i = h V b i / h V b i , (A27)and at small scales: lim k ¯ R ≫ h W R i = 9( ¯ R + σ R ) / (2 k ( ¯ R + 3 σ R ) (A28)By equating the two limiting cases, we can derive the peak scale for the variance of W R : k varpeak = " (cid:0) ¯ R + σ R (cid:1) (cid:0) Rσ R + ¯ R (cid:1) (cid:0)
15 ¯ R σ R + 45 ¯ R σ R + ¯ R + 15 σ R (cid:1) / (A29)The maximum peak value of the shot noise is then given by k P snXψ . ( k varpeak ) h V b ih V b i (A30)A similar approach for the peak scale of the window average does not give an accurate enough answer. Therefore,we will use the following best fit: k avpeak ∼ . R − (A31)This and the derived maximum value of the shot noise immediately allow us to put a constraint on the ratio betweenthe two-bubble term and the shot noise, R b − sn ≡ P bXψ /P snXψ ≃ b x H − ¯ x H P δδ ( k avpeak ) h V b ih W R i ( k varpeak ) (A32)From this expression, we find that the shot noise is significantly larger than the two-bubble term around the peakscale. This nicely fits into the previous picture, since the normal distribution of the bubbles is only physical at latetime in the reionization history when ¯ R >
A. Loeb and R. Barkana,Ann.Rev.Astron.Astrophys. , 19 (2001),arXiv:astro-ph/0010467 [astro-ph].X.-H. Fan, M. A. Strauss, R. H. Becker, R. L. White, J. E. Gunn,et al., Astron.J. , 117 (2006a),arXiv:astro-ph/0512082 [astro-ph].X.-H. Fan, C. Carilli, and B. G. Keating,Ann.Rev.Astron.Astrophys. , 415 (2006b),arXiv:astro-ph/0602375 [astro-ph].G. Hinshaw, D. Larson, E. Komatsu, D. Spergel, C. Bennett,et al., (2012), arXiv:1212.5226 [astro-ph.CO].C. J. Hogan and M. J. Rees, MNRAS , 791 (1979).D. Scott and M. J. Rees, mnras , 510 (1990).M. McQuinn, O. Zahn, M. Zaldarriaga, L. Hernquist, and S. R.Furlanetto, Astrophys.J. , 815 (2006),arXiv:astro-ph/0512263 [astro-ph].P. Madau, A. Meiksin, and M. J. Rees, ApJ , 429 (1997),arXiv:astro-ph/9608010.M. Zaldarriaga, S. R. Furlanetto, and L. Hernquist,Astrophys.J. , 622 (2004),arXiv:astro-ph/0311514 [astro-ph]. S. Furlanetto, M. Zaldarriaga, and L. Hernquist,Astrophys.J. , 16 (2004a),arXiv:astro-ph/0404112 [astro-ph].S. Furlanetto, M. Zaldarriaga, and L. Hernquist,Astrophys.J. , 1 (2004b),arXiv:astro-ph/0403697 [astro-ph].S. Furlanetto, S. P. Oh, and F. Briggs,Phys.Rept. , 181 (2006), arXiv:astro-ph/0608032 [astro-ph].A. Lidz, O. Zahn, S. Furlanetto, M. McQuinn, L. Hernquist,et al., Astrophys.J. , 252 (2009),arXiv:0806.1055 [astro-ph].R. Wiersma, B. Ciardi, R. Thomas, G. Harker, S. Zaroubi, et al.,(2012), arXiv:1209.5727 [astro-ph.CO].M. A. Alvarez, E. Komatsu, O. Dore, and P. R. Shapiro,Astrophys.J. , 840 (2006),arXiv:astro-ph/0512010 [astro-ph].R. Salvaterra, B. Ciardi, A. Ferrara, and C. Baccigalupi,MNRAS , 1063 (2005), arXiv:astro-ph/0502419.P. Adshead and S. Furlanetto, Mon.Not.Roy.Astron.Soc. (2007),arXiv:0706.3220 [astro-ph].H. Tashiro, N. Aghanim, M. Langer, M. Douspis, and S. Zaroubi,MNRAS , 469 (2008), arXiv:0802.3893.H. Tashiro, N. Aghanim, M. Langer, M. Douspis, S. Zaroubi,et al., Mon.Not.Roy.Astron.Soc. , 2617 (2010),arXiv:0908.1632 [astro-ph.CO]. H. Tashiro, N. Aghanim, M. Langer, M. Douspis, S. Zaroubi,et al., (2011), arXiv:1008.4928 [astro-ph.CO].G. Holder, I. T. Iliev, and G. Mellema, Astrophys.J.Lett. (2006),arXiv:astro-ph/0609689 [astro-ph].V. Jeli´c, S. Zaroubi, N. Aghanim, M. Douspis, L. V. E.Koopmans, M. Langer, G. Mellema, H. Tashiro, and R. M.Thomas, MNRAS , 2279 (2010),arXiv:0907.5179 [astro-ph.CO].A. Natarajan, N. Battaglia, H. Trac, U. L. Pen, and A. Loeb,(2012), arXiv:1211.2822 [astro-ph.CO].C. Dvorkin, W. Hu, and K. M. Smith,Phys.Rev.
D79 , 107302 (2009), arXiv:0902.4413 [astro-ph.CO].C. Dvorkin and K. M. Smith, Phys.Rev.
D79 , 043003 (2009),arXiv:0812.1566 [astro-ph].V. Gluscevic, M. Kamionkowski, and D. Hanson, (2012),arXiv:1210.5507 [astro-ph.CO].K. Masui, E. Switzer, N. Banavar, K. Bandura, C. Blake, et al.,(2012), arXiv:1208.0331 [astro-ph.CO].E. Chapman, F. B. Abdalla, J. Bobin, J.-L. Starck, G. Harker,et al., (2012), arXiv:1209.4769 [astro-ph.CO].G. Harker, S. Zaroubi, G. Bernardi, M. A. Brentjens,A. de Bruyn, et al., Mon.Not.Roy.Astron.Soc. , 2492(2010), arXiv:1003.0965 [astro-ph.CO].G. Mellema, L. Koopmans, F. Abdalla, G. Bernardi, B. Ciardi,et al., (2012), arXiv:1210.0197 [astro-ph.CO].A. Liu, M. Tegmark, J. Bowman, J. Hewitt, and M. Zaldarriaga,(2009), arXiv:0903.4890 [astro-ph.CO].V. Jeli´c, S. Zaroubi, P. Labropoulos, R. M. Thomas, G. Bernardi,M. A. Brentjens, A. G. de Bruyn, B. Ciardi, G. Harker,L. V. E. Koopmans, V. N. Pandey, J. Schaye, andS. Yatawatta, MNRAS , 1319 (2008), arXiv:0804.1130.G. Bernardi, A. B. G. Harker, M. Brentjens, B. Ciardi, V. Jelic,et al., (2010), arXiv:1002.4177 [astro-ph.CO].G. Bernardi, A. de Bruyn, M. Brentjens, B. Ciardi, G. Harker,et al., (2009), arXiv:0904.0404 [astro-ph.CO].A. Liu, J. R. Pritchard, M. Tegmark, and A. Loeb, (2012),arXiv:1211.3743 [astro-ph.CO].X. Wang and W. Hu, Astrophys.J. , 585 (2006),arXiv:astro-ph/0511141 [astro-ph].G. B. Field, Proceedings of the Institute of Radio Engineers. ,240 (1958).A. Lewis, A. Challinor, and A. Lasenby,Astrophys.J. , 473 (2000),arXiv:astro-ph/9911177 [astro-ph].O. Zahn, A. Lidz, M. McQuinn, S. Dutta, L. Hernquist, et al.,Astrophys.J. , 12 (2006),arXiv:astro-ph/0604177 [astro-ph].M.-S. Shin, H. Trac, and R. Cen, ApJ , 756 (2008),arXiv:0708.2425.R. K. Sheth and G. Tormen, MNRAS , 61 (2002),arXiv:astro-ph/0105113.D. S. Reed, R. Bower, C. S. Frenk, A. Jenkins, and T. Theuns,MNRAS , 2 (2007), arXiv:astro-ph/0607150.T. Baldauf, U. Seljak, R. E. Smith, N. Hamaus, andV. Desjacques, ArXiv e-prints (2013),arXiv:1305.2917 [astro-ph.CO].P. Shaver, R. Windhorst, P. Madau, and A. de Bruyn, (1999),arXiv:astro-ph/9901320 [astro-ph].A. Liu and M. Tegmark,Mon.Not.Roy.Astron.Soc. , 3491 (2012),arXiv:1106.0007 [astro-ph.CO]. A. Kogut, J. Dunkley, C. Bennett, O. Dore, B. Gold, et al.,Astrophys.J. , 355 (2007), arXiv:0704.3991 [astro-ph].L. La Porta, C. Burigana, W. Reich, and P. Reich, (2008),arXiv:0801.0547 [astro-ph].M. G. Santos, A. Cooray, and L. Knox,Astrophys.J. , 575 (2005),arXiv:astro-ph/0408515 [astro-ph].X.-M. Wang, M. Tegmark, M. Santos, and L. Knox,Astrophys.J. , 529 (2006),arXiv:astro-ph/0501081 [astro-ph].G. Harker, S. Zaroubi, G. Bernardi, M. A. Brentjens, A. G. deBruyn, B. Ciardi, V. Jeli´c, L. V. E. Koopmans, P. Labropoulos,G. Mellema, A. Offringa, V. N. Pandey, J. Schaye, R. M.Thomas, and S. Yatawatta, MNRAS , 1138 (2009),arXiv:0903.2760 [astro-ph.CO].E. Chapman, F. B. Abdalla, G. Harker, V. Jeli´c, P. Labropoulos,S. Zaroubi, M. A. Brentjens, A. G. de Bruyn, and L. V. E.Koopmans, MNRAS , 2518 (2012),arXiv:1201.2190 [astro-ph.CO].E. Chapman, F. B. Abdalla, J. Bobin, J.-L. Starck, G. Harker,V. Jeli´c, P. Labropoulos, S. Zaroubi, M. A. Brentjens, A. G. deBruyn, and L. V. E. Koopmans, MNRAS , 165 (2013),arXiv:1209.4769 [astro-ph.CO].G. de Zotti, M. Massardi, M. Negrello, and J. Wall,A&A Rev. , 1 (2010), arXiv:0908.1896 [astro-ph.CO].Planck Collaboration, P. A. R. Ade, N. Aghanim, F. Arg¨ueso,M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. Balbi,A. J. Banday, and et al., A&A , A13 (2011),arXiv:1101.2044 [astro-ph.CO].I. Heywood, M. J. Jarvis, and J. J. Condon,MNRAS , 2625 (2013), arXiv:1302.2010 [astro-ph.CO].Planck Collaboration, P. A. R. Ade, N. Aghanim, M. I. R. Alves,C. Armitage-Caplan, M. Arnaud, M. Ashdown,F. Atrio-Barandela, J. Aumont, C. Baccigaluppi, and et al.,ArXiv e-prints (2013), arXiv:1303.5073 [astro-ph.GA].M. F. Morales, Astrophys.J. , 678 (2005),arXiv:astro-ph/0406662 [astro-ph].Y. Mao, M. Tegmark, M. McQuinn, M. Zaldarriaga, andO. Zahn, Phys.Rev. D78 , 023529 (2008),arXiv:0802.1710 [astro-ph].M. P. van Haarlem et al., A&A , A2 (2013),arXiv:1305.3550 [astro-ph.IM].(2006), arXiv:astro-ph/0604069 [astro-ph].M. Niemack, P. Ade, J. Aguirre, F. Barrientos, J. Beall, et al.,Proc.SPIE Int.Soc.Opt.Eng. , 77411S (2010),arXiv:1006.5049 [astro-ph.IM].J. Austermann, K. Aird, J. Beall, D. Becker, A. Bender, et al.,Proc.SPIE Int.Soc.Opt.Eng. , 84520E (2012),arXiv:1210.4970 [astro-ph.IM].M. Zaldarriaga, L. Colombo, E. Komatsu, A. Lidz,M. Mortonson, et al., (2008), arXiv:0811.3918 [astro-ph].M. Su, A. P. Yadav, M. McQuinn, J. Yoo, and M. Zaldarriaga,(2011), arXiv:1106.4313 [astro-ph.CO].H. V. Peiris and D. N. Spergel, Astrophys.J. , 605 (2000),arXiv:astro-ph/0001393 [astro-ph].M. Tegmark, A. Taylor, and A. Heavens,Astrophys.J. , 22 (1997),arXiv:astro-ph/9603021 [astro-ph].M. J. Mortonson and W. Hu, Astrophys.J.657