Two-dimensional Topology of Cosmological Reionization
Yougang Wang, Changbom Park, Yidong Xu, Xuelei Chen, Juhan Kim
aa r X i v : . [ a s t r o - ph . C O ] J a n Draft version August 31, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
TWO DIMENSIONAL TOPOLOGY OF COSMOLOGICAL REIONIZATION
Yougang Wang , Changbom Park , Yidong Xu , Xuelei Chen , Juhan Kim Draft version August 31, 2018
ABSTRACTWe study the two-dimensional topology of the 21-cm differential brightness temperature for twohydrodynamic radiative transfer simulations and two semi-numerical models. In each model, wecalculate the two dimensional genus curve for the early, middle and late epochs of reionization. Itis found that the genus curve depends strongly on the ionized fraction of hydrogen in each model.The genus curves are significantly different for different reionization scenarios even when the ionizedfaction is the same. We find that the two-dimensional topology analysis method is a useful tool toconstrain the reionization models. Our method can be applied to the future observations such as thoseof the Square Kilometer Array.
Subject headings: dark matter – galaxies:halos – galaxies:structure – large-scale structure of universe– methods : statistical INTRODUCTIONReionization is a milestone event in the history ofthe universe. Quasar absorption line observations in-dicate that its completion is at redshift z & . τ = 0 . ± . z re = 8 . +1 . − . if the reionization happens suddenlyat a redshift z re (Planck Collaboration et al. 2015), soits beginning must be earlier. The complex reioniza-tion process has been investigated by many theoreticalworks (e.g. Furlanetto et al. 2004; Choudhury & Ferrara2007; Zhang et al. 2007; Xu et al. 2009; Yue et al. 2009;Yue & Chen 2012; Kim et al. 2013). One of the mostpromising methods to detect the cosmic reionization isthrough the 21 cm transition of HI. The emission or ab-sorption of the 21 cm line traces the neutral hydrogenwell at different redshifts, which can provide us with themost direct view of the reionization history.Due to the strong foregrounds, the observation of highredshift 21 cm signals is a challenging task. Althoughthere are a number of running radio interferometer ar-rays, such as the 21 Centimeter Array (21CMA; Wu2009); the Giant Metre-wave Radio Telescope (GMRT;Paciga et al. 2013), which gives a constraint on reion-ization at z ≈ .
6; the Low Frequency Array (LOFAR;Rottgering et al. 2006), the Murchison Widefield Array(MWA; Bowman et al. 2013; Tingay et al. 2013); and thePrecision Array for Probing the Epoch of Reionization(PAPER; Jacobs et al. 2015), which is designed for ob-serving the redshift 21cm signal from EOR and gives anew 2 σ upper limit on ∆ ( k ) of (22 . in the range Key Laboratory of Computational Astrophysics, NationalAstronomical Observatories, Chinese Academy of Sciences, Bei-jing, 100012 China; E-mail: [email protected] School of Physics, Korea Institute for Advanced Study,85Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea; [email protected] Center for High Energy Physics, Peking University, Beijing100871, China Center for Advanced Computation, Korea Institute for Ad-vanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Ko-rea . < k < . h Mpc − at z = 8.4.Addtionally, the kinematic Sunyaev-Zel’dovich (kSZ;Sunyaev & Zeldovich 1972, 1980) can distort the pri-mary CMB black-body spectrum due to the peculiarvelocity of the clusters of galaxies. During the reion-ization, the ionized bubbles generate angular anisotropythrough the kSZ effect. The amplitude of kSZ powerdepends on the process of reionization , and its shapedepends on the distribution of bubble size. Therefore,the kSZ power spectrum can give constraints on theepoch of reioniztion (Mortonson & Hu 2010; Zahn et al.2012; Mesinger et al. 2012; George et al. 2015). Whenthe post-reionization homogeneous kSZ signal is takeninto account, George et al. (2015) found an upper limiton the duration ∆ z < . N logN , where N is the total of number of data points,which is related to the data points ( N c ) in each dimensionas N = N c and N = N c in two and three dimensions,respectively. Therefore, the time taken in 3d genus is atleast of 1 . N c times in 2d genus. Even if the 3d genuscan be obtained, the 2d genus can provide a useful crosscheck.Below, we introduce the calculation of the 21 cm dif-ferential brightness temperature and its 2d genus curvein Sec.2. In Sec. 3 we describe the radiative transfersimulations and semi-numerical simulation used in thispaper. Our results are presented Sec.4. We summarizeand discuss the results in Sec. 5. THE TWO-DIMENSIONAL GENUS OF 21CMTEMPERATURE2.1.
The 21 cm signal
The emission or the absorption of 21 cm signal de-pends on the spin temperature T S of neutral hydrogen,which is defined by the relative number densities, n i , ofatoms in the two hyperfine levels of the electronic groundstate, n /n = 3 exp( − T ⋆ /T S ) , where T ⋆ = h p ν /k B =0 . K is the equivalent temperature of the energy levelhyperfine structure splitting h p ν = 5 . × − eV. Thespin temperature T S is determined by several competingprocesses (c.f. Furlanetto et al. (2006)): T − S = T − γ + x c T − K + x α T − C x c + x α (1)where T γ = 2 . z ) K is the cosmic microwave back-ground (CMB) temperature at redshift z , T K is the gaskinetic temperature, T C is the effective color tempera-ture of the UV radiation, x c is the coupling coefficientfor collisions and x α is the coupling coefficient for UVscattering. Comparing with the CMB temperature, the21 cm radiation is observed in emission if T S > T γ , orabsorption if T S < T γ . Generally, the 21 cm signal isquantified by the 21 cm differential brightness tempera-ture, δT b = T S − T γ z (cid:0) − e − τ (cid:1) , (2)where the optical depth τ is produced by a patchof neutral hydrogen, which has the following form (Furlanetto et al. 2006; Barkana & Loeb 2001) τ ( z ) = (2 . × − ) (cid:18) T S (cid:19) − (cid:18) h . (cid:19)(cid:18) z (cid:19) / × (cid:18) Ω b . (cid:19)(cid:18) Ω m . (cid:19) − / (1 + δ ) . (3)Here δ is the density contrast. Assuming τ ≪ T s ≫ T γ , then the brightness temperature of the 21cmemission can be written as δT b = (28 mK) (cid:18) h . (cid:19)(cid:18) z (cid:19) / (cid:18) H d v r / d r + H (cid:19) × (cid:18) Ω b . (cid:19)(cid:18) Ω m . (cid:19) − / (1 + δ )(1 − x i ) . (4)where x i is the fraction of ionized hydrogen, H ( z ) is theHubble parameter, d v r / d r is the comoving gradient ofthe line-of-sight (LOS) component of the comoving ve-locity. It is noted that the distribution of the brightnesstemperature of the neutral hydrogen will be the sameas that of the underlying matter density field if d v r / d r is small (d v r / d r << H ) and the gas is fully neutral( x i = 0). 2.2. The 2d genus
The 2d genus has been applied in many fields,such as the cosmic microwave background fluctua-tions (Colley et al. 1996; Park et al. 1998), weak lensingfield Matsubara & Jain (2001); Sato et al. (2003), non-Gaussian signatures (Park 2004), large-scale structure inthe redshift survey (James et al. 2007), galaxy distribu-tion in the Hubble deep fields (Park et al. 2001), neutralhydrogen in both the Large and Small Magellanic Cloud(Kim & Park 2007; Chepurnov et al. 2008).If we consider the 21cm brightness temperature on aspherical shell, the 2d genus of the contour is definedby the number of contours surrounding regions higherthan a threshold value minus the number of contoursenclosing regions lower than the threshold (Melott et al.1989; Gott et al. 1990; Park et al. 2013) G ( ν ) = N high − N low (5)where N high and N low are the number of isolated high-density regions and low-density regions, respectively.The genus G ( ν ) depends on the threshold density value ν , which is in units of standard deviation from themean. Given a two-dimensional distribution, the 2dgenus can be measured by using the Gauss-Bonnett the-orem (Gott et al. 1990) G d = R C d S π (6)where the integral line is along the contour, and C isthe inverse curvature r − of the line. The value of 2dgenus maybe negative or positive, depending on whethera low- or high-density region is enclosed. If a curvature isintegrated along a closed contour around a high-densityregion, its value will be 1, otherwise, its value is -1. Fora Gaussian random field, the 2d genus per unit area isgiven by (Melott et al. 1989; Coles 1988) G d,Gaussian = 1(2 π ) / h k i ν exp( − ν /
2) (7)where h k i = R k P ( k ) d k/ R P ( k ) d k is the squareof the wavenumber k averaged over the smoothed two-dimensional power spectrum P ( k ). In practice, the one-point distribution of the density field is not interest-ing (Vogeley et al. 1994; Park et al. 2001) , we follow(Park et al. 2001) to the parametrize the area fractionby f A = 1 √ π Z ∞ ν A e − t / d t (8)The genus is calculated from ν A = − SIMULATIONSOur reionization models are different from what wasused in the work of Hong et al. (2014). They used theN-body simulation and C − Ray (Mellema et al. 2006)method. Here we use the hydrodynamic radiative trans-fer (HRT) simulations (Trac et al. 2008) and the semi-numerical model 21cmFAST Mesinger et al. (2011). Themain advantage of the HRT simulation over the simula-tion in Hong et al. (2014) is that it is a real hydrody-namical simulation which keeps track of baryon evolu-tion, heating, and cooling processes. The 21cmFASTroutine is an approximation to the HRT simulation, andit is faster than both simulation in Hong et al. (2014)and the HRT simulation.Throughout the paper, we use the Wilkinson Mi-crowave Anisotropy Probe (WMAP) 5 year cosmologi-cal parameters: Ω m = 0 . Λ = 0 . b = 0 . h = 0 . σ = 0 . n s = 0 .
963 (Dunkley et al.2009), which are consistent with the cosmology parame-ters in the HRT simulation used in this study.3.1. cosmological radiative transfer simulation
The hydrodynamic radiative transfer simulations usedin this paper was described in detail in Trac et al. (2008).The simulation is based on the numerical method de-scribed in Trac & Cen (2007), which includes an N -bodyalgorithm for dark matter, a star formation prescription,and a radiative transfer algorithm for ionizing photons.The N -body simulation includes 3072 dark matter par-ticles on an effective mesh with 11 , cells in a comov-ing box, 100 h − Mpc on each side. The mass of each darkmatter particle is 2 . × M ⊙ . The resolution of hydro-dynamic+RT simulations is N = 1536 of dark matterparticles, gas cells, and adaptive rays. The photoioniza-tion and photoheating rates are calculated for each cell.Star formation occurs for particles with the density ρ m > ρ crit ( z ) and temperature T > K. This cutin the temperature-density phase space restricts star for- mation effectively to regions within the virial radius ofhalos that cool efficiently through atomic line transitions.Here we use two groups of the HRT simulations, whichhave different finishing time for the hydrogen reioniza-tion. In the first simulation, the reionization is completedlate at z ∼ z ∼ The semi-numerical simulation 21cmFAST
We also use a semi-analytical code 21cmFAST(Mesinger et al. 2011) to study the cosmological 21 cmsignal. The 21cmFAST code is a useful semi-numericalcode to model the reionization process. Given the boxsize and particle number, the Gaussian random initialconditions of the dark matter density and velocity fieldsare generated by Monte Carlo sampling method. Thelarge-scale density and peculiar velocity field are thenobtained by first-order perturbation theory. Assumingthat the number of ionizing photons are proportional tothe collapse fraction computed from the extended Press-Schechter formalism, the ionization field is generatedfrom the evolved density field at each redshift. Fromthe density, ionization and peculiar velocity, the 21cmbrightness temperature is obtained. The advantage ofthis approach is that it is very fast to calculate the 21cm signal for different model parameters.In order to match the completion time of the cos-mic reionization in the HRT simulation, we change theionizing efficiency factor ζ , which is defined as ζ = f esc f ⋆ N γ/b n − . Here f esc is the escape fraction of ion-izing photons from the object, f ⋆ is the star formationefficiency, N γ/b is the number of ionizing photons pro-duced per baryon in stars, and n rec is the typical recom-bined number of times for a hydrogen atom. The boxsize is the same as in HRT simulations, the cell numberis N = 768 . The cosmology parameters in our 21cm-FAST simulation are chosen to be the same as in theHRT simulation, which is based on the WMAP5 data.We have run two simulations ( ζ = 15 for 21cmFAST1and ζ = 50 for 21cmFAST2) by using the 21cmFASTcode. The finishing time of reionization is z ∼ z ∼ late models , while HRT2 and 21cmFAST2 asthe two early models .It is interesting to compare the different models at afixed ionization fraction. Therefore, we output one snap-shot of 21cmFAST1 simulation which has the same ion-ization fraction as the one from HRT sim1 at x i = 0 . x i = 0 .
55. Thetwo thin vertical lines in Figure 1 show x i = 0 .
55 and x i = 0 . x i increases Fig. 1.—
Lower Panel:Evolution of the ionization fraction as afunction of redshift for the different models. Upper Panel: Re-lation between the ionization fraction and the average differentialbrightness temperature of 21cm signal. The two thin vertical linesindicate x i = 0 .
55 and x i = 0 .
65, respectively. rapidly with the decreasing redshift, and the mean dif-ferential brightness temperature decreases rapidly withtime. In other words, the full reionization processes arefast in these models. the HRT sim1 and the 21cmFAST1have the same reionization completion time and the sameionization fraction at one redshift, but the ionizationfractions at other redshifts are different. The simula-tions HRT sim2 and 21cmFAST2 have nearly the samefinishing time of the reionization. Compared with thetwo late models, the difference of the ionization fractionevolution and the x i - δT b relation in the two early modelsare smaller. This can help us to discriminate the differentreionization scenarios even if they have similar ionizationfractions.In real observations, the observed δT b includes bothsignal and noise. To study the effect of the thermal noiseon the 2d genus, we add a Gaussian noise to the signalmap. We generated 500 maps with the random noisefollowing the Gaussian distribution, where the stand de-viation is f σ T . where f is a constant, and σ T = s δT b − < δT b >N c − . RESULTSThe observed signal can be characterized as the realsignal convolved with the telescope response function orbeam, however, the real telescope beam is complicated.To model the observed signal, in this paper we assumeeither a Gaussian or compensated Gaussian lobe functionfor simplicity. The Gaussian beam is simple, and widelyused in the radio studies (e.g. Mao 2014; Wolz et al.2014) to model the actual beam, which can be writtenas F G ( θ ) = 12 πσ exp (cid:18) − θ σ (cid:19) . (9)The full width at half maximum (FWHM) for the Gaus-sian beam is ∆ θ = 2 σ √ F CG ( θ ) = 12 πσ (cid:18) − θ σ (cid:19) exp (cid:18) − θ σ (cid:19) . (10)The compensated Gaussian function approximates wellthe observational beam shape of a compact interfermeterarray (often referred to as ‘dirty beam’), which is insen-sitive to large-scale features. Eq. (10) shows that F G < θ > √ σ , i.e. the sign of the contribution is negative.In Fig.2 we show the evolution of 21cm maps at theearly, middle and late stage of reionization for the fourmodels: (a) HRT sim1, (b) HRT sim2, (c) 21cmFAST1,(d) 21cmFAST2. In each of the subfigures, the top panelsrepresent the original δT b signal, the middle and bottompanels are the simulated observed map with the Gaussianand compensated Gaussian beam profile, respectively.We can see that there even though both the HRT sim1and 21cmFAST1 models have relatively late reionization,there are significant differences between the two modelsat each epoch. In the HRT sim1, the ionized regions arediffused, while it is linked together in 21cmFAST1. Sim-ilarly, for the two early reionization models (HRT sim2and 21cmFAST2), there are also distinctive differences.This indicates that the reionization process in the HRTsimulation and that in the 21cmFAST are different. Forthe two 21cmFAST simulations, the distributions of δT b are similar if they have the same ionization fraction x i .Since the Gaussian and compensated Gaussian beam cansmooth the δT b distribution, the largest value of the δT b decreases after the smoothing. As explained above, thecompensated Gaussian beam can produce negatives val-ues, therefore, δT b is negative in some regions with thecompensated Gaussian beam.In order to compare the topology results with thosefrom the power spectrum, we also calculate the angularpower spectrum [ l ( l + 1) C l / π ] / . In Figure 3 we plotthe angular power spectrum of δT b for the four modelswith different redshifts. It is seen that the shape of theangular power spectrum is nearly the same for the earlyand middle phase of reionization in the four simulations.Except for HRT sim1, the shape of the angular powerspectra in the late phase of reionization are also similar,hence it would be difficult to distinguish the differentreionization scenarios from the angular power spectrum,while the topology offers a way to distinguish them.In Figure 4, we show the 2d genus for both the differen-tial brightness temperature (solid lines) and the gas (ormatter) density (dashed lines) in the four simulations.On the scale discussed here, the gas (or matter) densitydistribution is nearly Gaussian, so from the comparisonbetween the density and δT b genus curve, we can see howfar do the 21cm signal deviates from Gaussian. In eachfigure, the left and right panels show the results fromthe Gaussian beam and compensated Gaussian beam,respectively. Since Panel (a) and (b) belong to the samekind of simulation, and Panel (c) and (d) belong to thesame kind of simulation, and the ionization fraction arealso similar, the genus curve on the left and right panelslook similar.It is noted that the 2d genus curve from the Gaus-sian beam is virtually indistinguishable from the compen-sated Gaussian one. The compensated Gaussian beamhas a Gaussian-type peak in the middle, surrounded by (a) HRT sim 1 (b) HRT sim 2(c) 21cmFAST1 (d) 21cmFAST2 Fig. 2.—
The 21cm maps for the four models: (a) HRT sim1, (b) HRT sim2, (c) 21cmFAST1, (d) 21cmFAST2. For each model, threeredshifts are plotted (marked on the figure). Also, for each model, the Top panel shows the original δT b map with frequency bin width∆ ν = 0.2 MHz. The high and low differential brightness temperature regions are the neutral and ionized ones respectively.The unit of thecolor bar is mK. The Middle and bottom panels show the simulated “observed map” with the Gaussian and compensated Gaussian beamsrespectively, with beam width ∆ θ = 1 ′ . Fig. 3.—
2d angular power spectrum of δT b for four models withdifferent redshifts. a negative wing, which adds small wiggles to the real sig-nals. Therefore, the compensate Gaussian beam changethe topology of the differential brightness temperatureslightly (Hong et al. 2014). More detailed comparisonof these two beams are shown in Hong et al. (2014). Inthis paper we only present the results for the two beamprofiles here, and from now on we will only discuss theresults obtained with the Gaussian beam.For each model, the 2d genus curve of the differentialbrightness temperature is distinctly different at differ-ent ionization fractions or redshifts. Furthermore, evenat the same ionization fraction, the 2d genus curvesfor different reionization models are significantly differ-ent. This can be seen clearly from the middle panelsin Fig. 4. Quantitatively, we can make a Kolmogorov-Smirnov (KS) test, which is used to test whether twodistributions are different. Usually, the KS-test gives thedeviation between two probability distribution functionsof a single independent variable, but it is also valid todistinguish two any arbitrary distributions, that is, themultivariate distribution. Here we take G ( ν ) as the dis-tribution function of the threshold ν , and apply the KS-test to the G ( ν ) functions. Note that we are not testingthe distribution of the 21 cm brightness temperature, butcomparing the shape of the genus curves G ( ν ), taking itas if it the distribution of the single variable ν . Thesignificance level of an observed value of D , which is adisproof of the null hypothesis that the distributions arethe same, is given by (Press et al. 1992) : prob ( D > observed ) = Q KS (cid:18)(cid:20)p N e +0 . . / p N e (cid:21) D (cid:19) (11)where D is defined as the maximum value of the absolutedifference between two cumulative distribution functions, N e = N N N + N ( N and N are the data number in distri-bution 1 and 2, respectively), and the function Q KS isdefined as Q KS ( λ ) = 2 ∞ X j =1 ( − j − exp( − j λ ) (12)Small values of prob imply that the distribution 1 is sig-nificantly different from that of distribution 2. We refer the interested reader to the book (Press et al. 1992) forthe detail description of the KS test. The result showsthat the genus curve from HRT sim1 at x i = 0 .
65 isdifferent from that from 21cmFAST1 at x i = 0 .
65 withconfidence level higher than 97%.In the early phase of reionization, i.e. x i ≤ .
08, theuniverse is nearly neutral, the differential brightness tem-perature of HI, δT b , still follows the Gaussian distribu-tion, which can be seen from the bottom left panels inthe subfigures of Fig. 4. The amplitude of the δT b genuscurve is larger than the gas density genus curve at ν ∼ − z = 14 . δT b is distinguishable from the matter curve: theamplitude of the 2d genus curve of δT b is larger than thatof the matter from low ν to high ν . This is the result ofa combination of two reasons. First, similar to the HRTsim1 and HRT sim2 models, the universe has begun toionize at this redshift, both new islands of HI regionsand the lake of HII regions have formed, the former cor-responds to the isolated high-density regions, while thelater is responsible for the low-density regions, the samefindings were also presented in Hong et al. (2014). Sec-ond, the value of δT b is related to the comoving velocitygradient of gas along the LOS d v r / d r , see Eq. (4). Wefind that d v r / d r in simulation HRT sim1 and HRT sim2is tiny, while those in simulation 21cmFAST1 and 21cm-FAST2 are relatively large.In the middle phase of reionization, i.e., 0 . ≤ x i ≤ .
65 , many bubbles exist and some of them overlap,therefore, the amplitude of genus curve of δT b is nearlyzero at low ν and the genus curve of δT b is shifted to theright compared with the genus curve of the gas (or mat-ter). This shift is consistent with the result in Figure 2of Kim & Park (2007) . In their studies, the genus curvefor a uniform disk with randomly distributed empty holesshifts to the right. The amplitude of the 2d genus curveof δT b in the high ν still keeps the vestiges of its initialcurve. This is because the high-density regions are stillshielded from the ionized photons to ionize the rest ofthe universe (Lee et al. 2008).In the late epoch of reionization, i.e., x i ≤ .
99, theuniverse is almost completely ionized, the genus of δT b from HRT sim1 shows that the remaining HI nearly fol-lows the matter distribution, which agrees with the resultgiven in Lee et al. (2008). Although x i = 0 .
99 at redshift z = 8 . δT b ishigher than that of the matter distribution. For the two21cmFAST simulations, the ionized bubble has mergedexcept for very low density regions, therefore, the genuscurves of δT b at low ν is nearly zero.In Figure 5, we show the effect of the the thermal noiseon the genus curve. It is seen that the genus curve is notaffected in the early phase of reionization. The reasonis that the brightness temperature δT b follows the Gaus-sian distribution, when an additional Gaussian noise isincluded, the distribution of δT b is still Gaussian. In themiddle and late epoch of reionization, the effect of thethermal noise on the genus curve is significant. The am-plitude of the genus curves with thermal noise are largerthan those without thermal noise due to merging of bub-ble. This effect is in some sense similar to what wasshown in Fig.5 of Melott et al. (1989), where the ampli-tude of the genus curve is decreased after structure for-mation. Nevertheless, as demonstrated by the KS test,the two ionized models can still be distinguished clearlyeven when 1 σ T noised is added.In Figure 6, the PDF of the 2d genus G ( ν ) at ν =-2, -1,0, 1, 2. are plotted for the the HRT sim1 model at z =5 .
99 with 1 σ thermal noise added. These PDFs of the 2dgenus show that they are centered at certain value, withnearly Gaussian distribution. Thus, in making modeltest it is reasonable to assume Gaussian likelihood forthe genus measurement.In Figure 7, we compare the 2d angular power spec-trum (left panels) and the 2d genus curve (right panels)for the four different reionization models at the sameredshift z = 9 .
00 with and without 1 σ thermal noise.We can also obtain some statistical confidence level fromthese tests–the different models can be compared usingthe KS test with the angular power spectrum data andthe 2d genus curves (See Table 1) . Here we take the an-gular power spectrum as the distribution function of thesingle variable l . From the value of prob , we know thatthe 21cmFAST1 model can be better distinguished withthe other three models by using the 2d angular powerspectrum, while the 21cmFAST2 model can be betterdistinguished by using the 2d genus curve if there is nothermal noise.The Gaussian thermal noise does not affect the angularpower spectrum, but it can reduce the non-Gaussian sig-nal of the 2d genus. From the lower part of Table 1, wesee that the HRT sim2 model can not been distinguishedfrom the 21cmFAST2 model by using either the powerspectrum or genus method from the KS test if 1 σ thermalnoise is included. We also use the χ test and probabil-ity to exceed (PTE) to distinguish different models usingthe 21 cm power spectrum and the 2d genus curve. The χ test is defined as χ = N bin X i =1 ( y i, − y i, ) | y i, | (13)where y i, and y i, are the distribution in i th bin formodel 1 and 2, respectively. N bin is the number of binsand N bin = 31 for the 2d genus curve and N bin = 20for the angular power spectrum. Given an input χ andthe number of degrees of freedom ν ′ , the PTE can becalculated by P T E = 12 ν ′ / Γ( ν ′ / Z ∞ χ t ν ′ / − e − t/ d t (14)A small value of PTE indicates that two models are dif-ferent. In Table 1, we also give the values of χ /ν ′ andPTE for both the power spectrum and genus withoutand with 1 σ noise. Here, we assume that there are twodifferent parameters when we use the χ /PTE test tocompare each pair of these simulations. Obviously, the χ /PTE tests tell us that the 2d genus curves from differ-ent reionized models can be easily distinguished even 1 σ thermal noised is added. However, it is difficult to dis-tinguish models HRT sim1 with HRT sim2, HRT sim1 with 21cmFAST2, and HRT2 with 21cmFAST2 from theangular power spectrum method. It seems that the re-sults from the χ /PTE test are inconsistent with thosefrom the KS test for some comparisons, such as HRTsim2 to 21cmFAST2 by using the 2d genus curves with1 σ thermal noise. The reason is that the value of prob from KS test depends on the deviation between two cu-mulative distribution function, one discrete data in eachbin can not affect the prob significantly, while the valueof χ can be vey large even the difference of two modelsin one bin is significant. Combined with the KS test and χ /PTE test for the signals with and without 1 σ noise,we know that the 2d genus and angular power spectrumare complementary. Moreover, the shape of the powerspectrum are nearly the same for the different reioniza-tion models, and the only difference is the amplitude,however, it is difficult to obtain the accurate amplitudeof the power spectrum in observations. From this per-spective the genus method has its niche when comparedwith the widely used power spectrum. SUMMARY AND DISCUSSIONWe quantify the 2d topology of the 21cm differentialbrightness temperature field for two HRT simulationsand two semi-analytical models. It is shown that the2d topology of δT b is significantly different for differentreionization models even 1 σ T thermal noise is added. Forthe same simulation, the 2d topology at different red-shifts reflects the status of reioniztion.We show the results for both Gaussian and compen-sated Gaussian beam filter of the telescopes. It is shownthat the brightness temperature maps filtered with thesebeam patterns can be used to discriminate differentreionization scenarios through the study of the 2d genustopology. However, the beam filter is more complicatedin practice, and we need to consider the special case fordifferent telescope. Moreover, the foreground removing iscrucial for the detection of the neutral HI signals, whichis beyond our study in the current paper. Of course, thisis our first step by using the 2d topology of the 21 cm dif-ferential brightness to constrain cosmic reionization. The2d topology can become a very powerful tool for prob-ing the reionization history and hope that the real twodimension topology of neutral hydrogen at high redshiftcan be observed by the future telescopes like SKA.ACKNOWLEDGMENTSWe thank the referee for comments and suggestionsthat improved the paper. This work has started duringYGW’s visit to KIAS 2012, and he would like to expresshis gratitude for KIAS. We thank Hy Trac and RenyueCen for providing us the HRT simulation data. We alsothank Xin Wang and Bin Yue for many helpful discus-sions. This work is supported by the Ministry of Scienceand Technology 863 project grant 2012AA121701. YGWacknowledges the 973 Program 2014CB845700, and theNSFC grant 11390372. YDX is supported by NSFC grantNo. 11303034. XLC acknowledges the support of the 973program (No.2007CB815401,2010CB833004), the CASKnowledge Innovation Program (Grant No. KJCX3-SYW-N2), and the NSFC grant 10503010. XLC is alsosupported by the NSFC Distinguished Young ScholarGrant No.10525314. TABLE 1 prob values from KS test and χ /ν ′ ( P T E ) from χ /PTE test for different reionization models at z = 9 with and without1 σ thermal noise. prob prob χ /ν ′ ( P T E ) χ /ν ′ ( P T E )(power spectrum) (genus) (power spectrum) (genus)HRT sim1: HRT sim2 0.13 0.36 0.36(0.99) 33.98 (0)HRT sim1: 21cmFAST1 8.16E-3 0.36 5.73(0) 42.22 (0)HRT sim1: 21cmFAST2 0.28 8.08E-4 0.89(0.60) 89.12 (0)HRT sim2: 21cmFAST1 2.57E-3 0.78 6.48(0) 77.83 (0)HRT sim2: 21cmFAST2 0.77 5.65E-3 0.18(1.00) 75.48 (0)21cmFAST1:21cmFAST2 2.57E-3 2.21E-3 1.50 (0.08) 35.39 (0)HRT sim1: HRT sim2 (+1 σ ) 0.13 0.36 0.36(0.99) 137.22(0)HRT sim1: 21cmFAST1 (+1 σ ) 8.16E-3 0.36 5.78(0) 42.98(0)HRT sim1: 21cmFAST2 (+1 σ ) 0.28 0.36 0.89(0.60) 184.34(0)HRT sim2: 21cmFAST1 (+1 σ ) 2.57E-3 6.21E-2 6.42(0) 321.39(0)HRT sim2: 21cmFAST2 (+1 σ ) 0.77 0.99 0.18(1.00) 18.03(0)21cmFAST1:21cmFAST2 (+1 σ ) 2.57E-3 6.21E-2 1.50(0.08) 1168(0)REFERENCESBarkana, R. & Loeb, A. 2001, Phys. 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The 2d genus of δT b (solid) and gas distribution (dashed) at different redshifts for (a) HRT sim1, (b) HRT sim2, (c) 21cmFAST1,(d) 21cmFAST2 models. In each model, the left panels show the results for Gaussian beam while the right ones represent results forcompensated Gaussian filter. (a) HRT sim 1 (b) HRT sim 2(c) 21cmFAST1 (d) 21cmFAST2 Fig. 5.—
Similar to Figure 4, but with Gaussian noise added. In each panel, the solid line represent the genus curve without the thermalnoise, while the dashed and dash-dot lines represents the results with 0 . σ T and σ T thermal noise added, respectively. The error bar inthe solid line is estimated by 30 similar simulated data samples. Fig. 6.—
Probability density function of the 2d genus including1 σ T Gaussian noise for HRT sim1 at z = 5 .
99 for ν =-2, -1, 0, 1, 2. (a) angular power spectrum (b) 2d genus curve(c) angular power spectrum with 1 σ thermal noise (d) 2d genus curve with 1 σ thermal noise Fig. 7.— (a) 2d angular power spectrum of δT b for four models with the same redshift z = 9 .
00. (b) 2d genus distribution of δT b for fourmodels with the Gaussian beam with the same redshift z = 9 .
00. (c) similar to (a), but for the results with 1 σ thermal noise. (d) similarto (c), but for the results with 1 σσ