Precision measurement of cosmic magnification from 21-cm emitting galaxies

Transcription

1 Mon. Not. R. Astron. Soc. 367, (2006) doi: /j x Precision measurement of cosmic magnification from 21-cm emitting gaaxies Pengjie Zhang 1 and Ue-Li Pen 2 1 NASA/Fermiab Astrophysics Group, Fermi Nationa Acceerator Laboratory, Box 500, Batavia, IL , USA 2 Canadian Institute for Theoretica Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 3H8 Accepted 2005 November 1. Received 2005 October 28; in origina form 2005 Apri 25 ABSTRACT We show how precision ensing measurements can be obtained through the ensing magnification effect in high-redshift 21-cm emission from gaaxies. Previousy, cosmic magnification measurements have been seriousy compicated by gaaxy custering. With precise redshifts obtained using the 21-cm emission ine waveength, one can correate gaaxies at different source panes, or excude cose pairs to eiminate such contamination. We provide forecasts for future surveys, specificay the Square Kiometer Array (SKA) and the Canadian Large Adaptive Refector (CLAR). SKA can achieve 1 per cent precision on the dark-matter power spectrum and the gaaxy dark-matter cross-correation power spectrum, whie CLAR can measure an accurate cross-correation power spectrum. The neutra hydrogen fraction was most ikey significanty higher at high redshifts, which increases the number of observed gaaxies significanty, such that CLAR can aso measure the dark-matter ensing power spectrum. SKA can aso aow precise measurement of bispectrum ensing. Key words: gaaxies: abundances arge-scae structure of Universe radio ines: gaaxies. 1 INTRODUCTION Gravitationa ensing measures the distortion of ight by gravity originating from the inhomogeneous distribution of matter. The physics of weak gravitationa ensing is cean, perhaps comparabe to the primary CMB. To the first order approximation, it invoves ony genera reativity and coisioness dark-matter dynamics. Gas physics ony enters at sma scaes (Zhan & Knox 2004; White 2004) and can be nuified by throwing away sma-scae ensing information (Huterer & White 2005). Athough the prediction of weak ensing statistics is compicated by the non-inear evoution of matter density fuctuation, high resoution, arge box-size N-body simuations are abe to measure these statistics to high accuracy (e.g. Vae & White 2003; Merz, Pen & Trac 2005). Thus, weak gravitationa ensing is one of the most powerfu and robust toos avaiabe to constrain cosmoogy and study the arge-scae structure of the Universe. Weak gravitationa ensing has been detected through cosmic shear (see Refregier 2003, for a recent review). Ongoing and upcoming arge-scae gaaxy surveys such as the Canada France Hawaii Teescope Legacy Survey, 1 the Dark Energy Survey at the Cerro Tooo Inter-American Observatory, 2 the Large Synoptic Survey Teescope, 3 the Panoramic Survey Teescope & Rapid Response System 4 and the Supernova/Acceeration Probe 5 wi reduce the statistica errors of the cosmic shear measurement to 1 per cent eve. At that stage, systematic errors such as the point spread function, gaaxy intrinsic aignment, seeing and extinction (Hoekstra 2004; Jarvis, Bernstein & Jain 2004; Vae et a. 2004; van Waerbeke, Meier & Hoekstra 2004) and errors in gaaxy redshift measurement wi be the utimate imiting factors. Given these possibe systematic errors, it is worthwhie to ook for an independent method with which to measure the gravitationa ensing, both for a consistency check and for (probaby) better statistica and systematic errors. In this paper, we wi show that the cosmic magnification of 21-cm emitting gaaxies is a competitive candidate method. Cosmic magnification is defined as the ensing-induced changes in gaaxy number density. It introduces extra correations in gaaxy custering (Kaiser 1992; Viumsen 1996; Moessner, Jain & Viumsen 1998; Jain, Scranton & Sheth 2003) and correates gaaxies (quasars) at widey separate redshifts (Moessner & Jain 1998). Cosmic magnification contains as much information about cosmoogy and matter custering as does cosmic shear. It has been robusty detected in quasar gaaxy ensing (Scranton et a and references therein). E-mai: pjzhang@shao.ac.cn (PZ); pen@cita.utoronto.ca (U-LP) C 2006 The Authors. Journa compiation C 2006 RAS Downoaded from

2 170 P. Zhang and U.-L. Pen The measurement of cosmic magnification does not require the accurate determination of gaaxy shapes and is thus free of many of the systematic errors, such as point spread function and gaaxy intrinsic aignment, that are entanged in the cosmic shear measurement. However, it suffers from severa obstaces. (i) It suffers from stronger shot noise than cosmic shear measurement does. For cosmic shear, shot noise comes from the intrinsic eipticity of gaaxies, which has the dispersion ɛ 2 1/ The shot-noise power spectrum is proportiona to ɛ 2 /N g 0.1/N g, where N g is the tota number of gaaxies. For cosmic magnification, shot noise comes from the Poisson fuctuation of gaaxy counts and the shot-noise power spectrum scaes as 1/N g, and is thus an order of magnitude arger than that in cosmic shear. (ii) The signa of cosmic magnification is generay much smaer than intrinsic custering of gaaxies. Without precise redshift measurements of gaaxies or quasars, the ony way to remove intrinsic custering of gaaxies is to measure the cross-correation of foreground gaaxies and background gaaxies (or quasars). However, its detection is severey imited by the number of high-redshift gaaxies, which are difficut to detect in the optica band, and by quasars, which are rare compared to gaaxy abundance. Eary measurements of cosmic magnification were often controversia. Even with a reativey arge sampe of quasars and foreground gaaxies from 2dF and SDSS, the measurement of cosmic magnification is sti in its infancy and is ony confirmed to 8σ confidence eve. As we as the observationa difficuties, the theoretica prediction of cross-correation strength is compicated by foreground gaaxy bias, which is hard to predict from first principes. Without a precise understanding of the gaaxy bias, the power of cosmic magnification to constrain cosmoogy and matter distribution is severey imited, as compared to cosmic shear. Cosmic magnification measured from 21-cm emitting gaaxies is free of many of the obstaces entanged in the cosmic shear measurement and the quasar gaaxy cosmic magnification measurement. As we wi show in Section 2, upcoming radio surveys such as the Square Kiometer Array (SKA) 6 can find H I-rich gaaxies in tota and 10 7 gaaxies at z > 2 through the neutra hydrogen 21-cm hyperfine transition ine. With this arge sampe of gaaxies, the measurement of cosmic magnification wi become far more precise, for the foowing reasons. (i) Radio observations are free of extinction. Since dust is associated with gaaxies, extinction is correated with foreground gaaxies and thus is ikey to produce a fase quasar gaaxy or gaaxy gaaxy cross-correation signa. (ii) Contrary to optica surveys, redshifts of these gaaxies can be precisey determined by the redshifted waveength of the 21-cm hyperfine transition ine, with no added observationa cost. Precision measurement of gaaxy redshift is required for the precision prediction of ensing statistics. It aso aows us the uxury of separating cose gaaxy pairs, which is crucia when measuring cosmic magnification in the presence of the strong autocorreation of gaaxies. (iii) At high redshift the magnification effect is enhanced, since gaaxies at higher redshifts are more strongy ensed and have a steeper fux distribution. Combined with the arge sampe of gaaxies, shot noise can be overcome. (iv) The autocorreation function of cosmic magnification can be measured to high accuracy. The prediction of the cosmic magnification autocorreation is free of prior gaaxy bias and is as robust as the prediction of the cosmic shear power spectrum. The goa of this paper is modest: to demonstrate the feasibiity of carrying out precision ensing measurements using 21-cm emitting gaaxies. Methods discussed in this paper are by no means optima and the resuts presented are conservative. Efforts toward optimizing ensing measurement are presented in the companion paper (Zhang & Pen 2005b), where we show that cosmic magnification of 21-cm emitting gaaxies can not ony give better resuts than the cosmic magnification of optica gaaxies, but can probaby aso give better resuts than the cosmic shear of optica gaaxies. Throughout this paper, we adopt a fat CDM universe with m = 0.3, = 1 m, σ 8 = 0.9, h = 0.7 and initia power index n = 1, as is consistent with the Wikinson Microwave Anisotropy Probe (WMAP) resut (Sperge et a. 2003). We take Canadian Large Adaptive Refector (CLAR) 7 and SKA as our targets to forecast the abiity of future radio 21-cm surveys to measure cosmic magnification. The instrumenta parameters and survey patterns of these two surveys have not been competey fixed yet. For CLAR, we adopt a system temperature T sys = 30 K, effective coecting area A eff = m 2 and fied of view 1 deg 2. For SKA, we adopt the same vaues for T sys and the fied of view, and adopt A eff = m 2. 2 DETECTING GALAXIES IN RADIO SURVEYS Though the Universe is highy ionized, there are sti arge amounts of neutra gas present in gaaxies. The typica H I mass is around 10 9 M (Zwaan et a. 1997). H I-rich gaaxies appear in the radio band by emitting the 21-cm hyperfine ine resuting from the transition of atomic hydrogen from spin 1 ground state to spin zero ground state. The spontaneous transition rate is A 21 = s 1. Each emission ine has negigibe width. However, since neutra gas has rotationa and therma motions, the integrated intrinsic ine width is not negigibe. It turns out that the ine width of 21-cm emission is mainy determined by the rotation of H I gas, which has a typica veocity of 100 km s 1. The Dopper effect by therma motions causes a veocity width w c k B T /m H 10 T /10 4 Kkms 1. The actua emission ine width is determined by many parameters, such as the tota mass of gaaxies, redshift and incination ange of the H I disc. For simpicity, we assume that the combined intrinsic ine width is w = 100 km s 1. The choice of w affects the prediction of detection efficiency of H I emitting gaaxies. However, the resuts shown in this paper shoud not be changed significanty by a more reaistic choice of w. Since the veocity (frequency) resoution of radio surveys can be much higher than the intrinsic ine width, the observed fux coud have a non-trivia dependence on the bandwidth, the modeing of which requires a detaied description of H I distribution in gaaxies. To avoid this compexity, we adopt a bandwidth that is arger than or equa to the redshifted 21-cm ine width, which changes from w at z to w/(1 + z) atz = 0, or in frequency space, to ν = wν 21 /c(1 + z), where ν 21 = 1.4 GHz is the 21-cm frequency. The Downoaded from C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367,

3 Cosmic magnification and 21-cm emitting gaaxies 171 tota 21-cm fux of H I rich gaaxies is ν 21 ) 1 S 21 = g ( 2 A 21 N H I E 21 w g 1 + g 2 4πDL 2 (z) c 1 + z [ ] 2 M H I c/h0 100 km s 1 = mjy M χ(z) w(1 + z). (1) Here, g 1 = 1 and g 2 = 3 are the degeneracies of atomic hydrogen spin 0 and 1 ground state. E 21 = hν 21 is the energy of each 21-cm photon. N H I = M H I /m H I is the tota amount of neutra hydrogen of a gaaxy with tota hydrogen mass M H I. D L (z) = χ(z)(1 + z) is the uminosity distance and χ is the comoving anguar distance. H 0 = 100 h km s 1 Mpc 1 is the Hubbe constant at present time. Instrumenta noise scaes as ν 1/2, where ν is the bandwidth. Therefore a arger bandwidth heps to beat down instrumenta noise. However, if the bandwidth is arger than the 21-cm ine width, the signa is diuted and scaes as 1/ ν. Thus, the highest signa-tonoise ratio is gained when the bandwidth ν is equa to the redshifted bandwidth of the integrated 21-cm emission ine, wν 21 /c(1 + z). The system noise per beam is 2Tsys k B S sys = η c A eff νt mjy T sys m 2 30 K A eff [ 100 km s 1 w/(1 + z) ] 1/2 h. (2) t Here, η c is the correator efficiency, which is adopted as η c = 0.9. The instrumenta beam area is A b λ 2 /A eff. For CLAR, A b 10 2 (ν 21 /ν) 2. For SKA, A b 1 2 (ν 21 /ν) cm emitting regions have typica size 30 kpc h 1 or 1 arcsec at z 1. Thus, the size of 21-cm emitting regions is much smaer than the beam size. The observed fux is thus the tota fux of each gaaxy. In this case, the cacuation of the detection threshod is straightforward. If we choose those peaks with fux above ns sys (nσ seection threshod), the minimum H I mass seected is [ ] 2 χ(z) M H I,min = n M (1 + z) 3/2 c/h 0 w h T sys m 2. (3) 100 km s 1 t 30 K A eff We assume that the H I mass function foows the Schechter function found at z = 0 (Zwaan et a. 1997) [ ] γ [ M n(m, z)dm = n 0 (z) exp M ] dm. (4) M (z) M (z) We fix γ = 1.2. Zwaan et a. (1997) found that M (z = 0) = h 2 M and n 0 (z = 0) = h 3 Mpc 3. There is itte soid measurement of n 0 (z) and M (z) other than at the oca Universe. However, for this form of the mass function, there exists a tight reation between H I, the cosmoogica neutra hydrogen density with respect to the present day critica density, and n 0 M : H I h = n 0 (z) M (z) n 0 (z = 0) M (z = 0). (5) Observations of damped Lymanα (Lyα) systems and Lyα-imit systems measure H I from z = 0toz 4. Combining equation (5) and these observations, one can put constraints on n 0 and M. These observations found that H I increases by a factor of 5 toward z 3 and then decreases toward higher redshift (Zwaan et a. 1997; Rao & Turnshek 2000; Storrie-Lombardi & Wofe 2000; Péroux et a. Figure 1. The predicted abundance of 21-cm emitting gaaxies above the 4σ detection threshod of the CLAR 5-yr survey. The soid ines are the cumuative number distribution N(<z)and the dashed ines are the differentia distribution dn/dz. The tota number of observed gaaxies is of the order of 10 6 and the number of gaaxies at z > 1 is of the order of We assume no evoution in the mass function. If a reaistic evoution mode is considered, the tota number of gaaxies at z > 1 can increase by a factor of 5 or more and data compiation in Péroux et a. 2003; Nagamine, Springe & Hernquist 2004). The increase of either n 0 or M increases the detectabiity of 21-cm emitting gaaxies. Thus, estimations based on the assumption of no evoution shoud be regarded as conservative resuts. We wi show that even in this conservative case, future radio surveys such as CLAR and SKA wi sti aow precise measurement of gaaxy gaaxy ensing (Section 3) and, in the case of SKA, the precise measurement of the ensing power spectrum (Sections 3 and 4) and the ensing bispectrum (Section 5). If we adopt evoution modes impied by and consistent with observations, even CLAR is abe to measure the ensing power spectrum (Section 6). The number of gaaxies detected depends on the seection threshod n. If we choose a 4σ detection threshod (n = 4), CLAR coud detect gaaxies in a 5-yr survey (Fig. 1). A deeper survey (smaer sky coverage) detects a arger fraction of high-redshift gaaxies. However, even if the survey voume is smaer, the tota number of high-redshift gaaxies is not necessariy higher. A survey area around 100 deg 2 is optima to detect high-z gaaxies. For a 160 deg 2 survey area, 10 6 gaaxies at z > 1 can be detected. SKA is about 10 times more sensitive than CLAR and can detect two orders of magnitude more gaaxies. For a 1600 deg 2 survey area, 10 8 gaaxies at z > 1 and 10 7 gaaxies at z > 2 can be detected (see Fig. 2). As a reminder, these estimations are extremey conservative. The number of gaaxies detected at z > 1 can be enhanced by a factor of 5 or more if the evoution effect is considered. Some peaks above the seection threshod are caused by noise. N noise, the number of fase peaks, has strong dependence on the detection threshod. If we choose n = 1(1σ detection), the number of fase peaks of noise is N noise 4π f sky A pixe ν Erfc(1.0/ 2) f sky, (6) v 2 A b C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367, Downoaded from

4 172 P. Zhang and U.-L. Pen 3 CROSS CORRELATIONS OF DIFFERENT REDSHIFT BINS 3.1 Cosmic magnification preiminary Cosmic magnification changes the gaaxy number overdensity δ g to 9 δ L g = δ g + 2(α 1)κ + O(κ 2 ). (7) Figure 2. Simiar to Fig. 1, but for the SKA 5-yr survey. The tota number of observed gaaxies is of the order of 10 8 and the number of gaaxies at z > 1 is of the order of We assume no evoution in the mass function. If a reaistic evoution mode is considered, the tota number of gaaxies at z > 1 can increase by a factor of 5 or more. where Erfc is the error function. For CLAR deep surveys which cover f sky 1 per cent of the sky, N noise 10 7, which is sti ess than the number of detected gaaxies above the 1σ threshod. For SKA, which is capabe of severa thousand square degrees of sky coverage, N noise above 1σ is 10 8, which is sighty ess than the tota number of gaaxies above the 1σ threshod. In CMB measurements, the signa-to-noise ratio per pixe is often chosen to be 1, which maximizes return on the power-spectrum measurement. This woud correspond to a detection threshod for which the fasepositive rate is 50 per cent. In this sense, we can choose n = 1as our seection threshod. However, the seection threshod probem can be deat with in a more sophisticated way. The survey measures a three-dimensiona map of the sky. Each pixe in that map wi have some significance of detection. Ceary the arge number of ow-significance pixes do coectivey contain information, if they can be averaged in a meaningfu way. Zhang & Pen (2005c) describe one such agorithm for extracting the uminosity function from deep within the noise. With these arge sampes of gaaxies, the gaaxy custering can be precisey measured. This aows constraining cosmoogy through the baryon osciation (Abdaa & Rawings 2005). The redshift distortion of gaaxies aows the measurement of the gaaxy veocity power-spectrum to high accuracy. 8 In this paper we do not attempt to utiize a of the information regarding these gaaxies: our primary goa is to demonstrate the feasibiity of using future 21-cm radio surveys to carry out precision ensing measurement. We note that the properties of ensing magnification depend strongy on the mass threshod or equivaent n. We wi expore different n 1 to find a suitabe choice for different quantities. For n 3, since the number of fase peaks is much smaer than the number of detected gaaxies, one can safey negect a errors caused by fase detections. But for n 3, one has to take the fase detections into account. 8 Zhang & Pen, in preparation. Here, α f (>F c )F c / f (F c ), f (>F) is the number of gaaxies brighter than F, F c is the fux imit adopted and κ is the ensing convergence. Since κ =0, to the accuracy of κ , ensing does not change the averaged f (>F). Thus α is effectivey an observabe. The usua method used to eiminate the intrinsic correation of cose gaaxy pairs is to cross-correate gaaxies in two separate redshift bins. Gaaxy pecuiar veocity shifts the position of gaaxies in the redshift space by c z 10 3 km s 1. Gaaxy correation becomes negigibe at scaes r 100 h 1 Mpc, which corresponds to z 0.03 [H(z)/H 0 ][r/100 h 1 Mpc]. Choosing z 0.05, residua correations caused by intrinsic gaaxy correation can be safey negected. With accurate measurements of 21-cm emitting gaaxy redshift, this can be done straightforwardy. The distribution of foreground gaaxies traces enses of background gaaxies which cause the magnification effect. Thus there exists a correation between the background magnification and foreground gaaxies. This gaaxy gaaxy correation is the correation generay considered in the iterature, but there is another correation to be considered. Both foreground gaaxies and background gaaxies are ensed by intervening matter. Thus there exists the background magnification foreground magnification correation. The combined correation is δ L g (θ f, z f )δ L g (θ b, z b ) =2(α b 1) κ b δ(θ f, z f + 4(α f 1)(α b 1) κ f κ b. (8) Here the subscripts f and b denote foreground and background respectivey. The first term on the right side of the equation is the magnification gaaxy correation and the second term is the magnification magnification correation. The observed gaaxy surface density is zmax g = n g (z)(1 + δ L )dz. (9) z min Its correation function is equation (8) weighted by the differentia gaaxy number distribution n g and the custering signa. By Limber s approximation, the corresponding 2D anguar power spectrum of the magnification gaaxy cross-correation is given by 2 C μg 2π = 3 m H0 2 2c 2 N 1 π f zf,max 2 gm z f,min ( ), z f G b (z f )n g (z f )χ f dz f. (10) χ f Here, m is the present day matter density with respect to the cosmoogica critica density. N f = n g (z f )dz f is the tota number of foreground gaaxies. 2 gm = b gr g 2 m is the gaaxy matter crosscorreation power spectrum. We assume b g r g = 1(b g is the gaaxy bias and r g is the cross-correation coefficient between gaaxies and 9 In this expression, we negect a high-order terms throughout this paper. These terms increase the ensing signa (Ménard et a. 2003a) and thus improve the correation measurement. Downoaded from C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367,

5 Cosmic magnification and 21-cm emitting gaaxies 173 dark matter). The matter power spectrum 2 m is cacuated using the Bardeen Bond Kaiser Szaay transfer function (Bardeen et a. 1986) and its non-inear evoution is cacuated by the Peacock Dodds fitting formua (Peacock & Dodds 1996). G b, the kerne of the background magnification effect, is given by G b (z) = 1 + z N b zb,max z b,min w(χ,χ b )n g (z b )2(α(z b ) 1) dz b. (11) Here, w(χ, χ s ) is the ensing geometry function. χ s can be χ f, the distance to foreground gaaxies, or χ b, the distance to background gaaxies. For the fat universe we adopt, the geometry function is simpified to w(χ, χ s ) = χ(1 χ/χ s ). The strength of the magnification effect reies on both the strength of ensing, which prefers background gaaxies with higher z, and α 1, which prefers deeper sope of the mass function at the mass threshod. The power spectrum of the corresponding magnification magnification cross-correation is: 2 C μμ 2π = ( ) 3 m H0 2 2 π 2c 2 zf,max 2 m 0 ( χ, z ) G b (z)g f (z)χdχ. (12) Here, G f, the kerne of foreground magnification effect, is given by G f (z) = 1 + z N f zf,max z f,min w(χ,χ f )n g (z f )2[α(z f ) 1]dz f (13) where N f is the tota number of foreground gaaxies. The ampitude of C μμ reies on both the ensing signa of foreground and background gaaxies. The higher the z f and z b, the stronger the correation signa. It aso depends strongy on α f 1 and α b 1. If the mass threshod is arger, α 1 is generay arger. Depending on the choice of foreground bins and foreground gaaxy seection criteria (which determines α and thus the strength of magnification effect), either C μg or C μμ can dominate. In Section 3.2, we discuss cases where C μg dominates and in Section 3.3, we discuss cases where C μμ dominates. 3.2 Magnification gaaxy power spectrum For a sufficienty wide foreground-gaaxy redshift distribution with ow median redshift, the ensing effect and the magnification (α 1) effect are both sma. Generay, in this case, C μμ is much smaer than C μg. In this Section, we fix the foreground gaaxy distribution (0 < z f < 1.0) and vary the background gaaxy redshift distribution. For this choice of foreground gaaxy distribution, C μμ is 1 per cent of C μg (see Figs 3 and 4). The correation signa peaks at 10 4, where the fuctuation is 10 per cent. C μg is the projection of 2 gm aong the ine of sight (equation 10). Given a cosmoogy, 2 gm (k, z) can be extracted using the inversion methods appied to gaaxy surveys and ensing surveys (e.g. Dodeson et a. 2002; Pen et a. 2003b). 2 gm (k, z) contains vauabe information of gaaxy custering and can put strong constraints on hao occupation distribution. Cosmoogica information is aso carried in the geometry term of ensing, in our case, χ in C μg and C μμ. By fixing the foreground gaaxy distribution and varying the background gaaxy distribution, one can isoate χ b from 2 gm and measure the dependence of χ b on z b. This method aows an independent and robust constraint on cosmoogy (Jain & Tayor 2003; Zhang, Hui & Stebbins 2005) Figure 3. The predicted accuracy of the cosmic magnification gaaxy crosscorreation power spectrum C μg as measured by the CLAR 5-yr survey. Foreground gaaxies (0.5 < z f < 1.0) are seected with a 2σ seection threshod. The optima survey coverage is around severa hundred square degrees. For such a configuration, C μg can be measured to 20 per cent accuracy at around severa thousand (ower pane, bin size = 0.2). C μμ is severa per cent of C μg and does not show up in the pot. In the top pane, we further spit background gaaxies into severa redshift bins. This ensing tomography aows us to measure the evoution of matter distribution. It aso aows us to measure the reative change of the comoving distance as a function of z to better than 10 per cent in three redshift bins at z b > Figure 4. Simiar to Fig. 3, but for the SKA 5-yr survey. Foreground gaaxies are chosen to be at 0.7 < z f < 1.0. To measure C μg, optima sky coverage shoud be around ten thousand degrees. Under such a configuration, C μg can be measured to severa per cent accuracy at The dashed ine in the ower pane is C μμ /C μg for 6400 deg 2 sky coverage. The dependence of the comoving distance χ(z) on redshift can be measured to better than 1 per cent in seven redshift bins at z > 1. C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367, Downoaded from

6 174 P. Zhang and U.-L. Pen The statistica error in the C μg measurement 10 is 1 [ C μg μg = C 2 + ( Cg 2 f b + C shot)( b C f g + Cshot)] f. (14) sky Here, C b g is the autocorreation power spectrum of the background gaaxies, which incudes contributions from δ b g δb g, κ b κ b and δ b g κ b. For high-redshift background gaaxies, κ b κ b is not negigibe compared to the intrinsic gaaxy correation δ b g δb g and thus has to be taken into account. C b shot = 4π f sky [1 + N b noise /N b]/n b is the background shot noise power spectrum. The extra factor 1 + N b noise /N b accounts for the contamination of fase peaks in the seected background sampe. These fase peaks do not correate with each other, so their ony effect is to increase the shot noise. For seection threshods above 2σ, the fase positive rate is sma and this extra factor can be negected. C b f is the autocorreation power spectrum of the foreground gaaxies. Statistica errors strongy depend on f sky (when fixing tota observation time). This dependence is compicated. (i) f sky affects α(z f ). A shorter integration time per unit area is required in order to survey a arger sky area. This increases system noise per beam and thus increases the seection mass threshod of high z gaaxies. Since the mass function is steeper at higher mass, α(z f ) increases and C μg increases. (ii) f sky affects the reative distribution of gaaxies. Larger f sky surveys detect reativey more ow-z gaaxies. Since C μg is proportiona to 2 gm weighted by the distribution of foreground gaaxies, and matter custering is stronger at ower z, arger sky coverage tends to increase C μg. On the other hand, the ensing effect is smaer for ower z gaaxies. This has the effect of decreasing w(χ, χ b ) and thus decreasing C μg. Furthermore, the noise terms C b g and C f g increase. (iii) f sky affects the cosmic variance. (iv) f sky affects the tota number of foreground and background gaaxies and thus changes the shot noise. The ower panes of Figs 3 and 4 show the dependence of C μg / C μg on sky coverage. If the sky coverage is too sma, the cosmic variance is arge. If the sky coverage is too arge, too few background gaaxies can be detected. Shot noise begins to dominate at reativey arge scaes. The choice of seection threshod n (nσ ) aso affects the statistica errors. Larger n increases α and thus increases C μg, but it aso decreases the number of detected gaaxies and thus increases shot noise. Both CLAR and SKA can measure C μg precisey (Figs 3 and 4). For CLAR, the optima sky coverage is around severa hundred square degrees. C μg can be measured to 20 per cent accuracy for bin size = 0.2. CLAR can aso measure C μg at severa redshift bins and aows isoating geometry. For SKA, the optima sky coverage is around ten thousand square degrees. C μg can be measured to 1 per cent accuracy. The size of background redshift bins can be as narrow as z 0.1. The change of χ b (z) can be precisey measured at z 1. The best resut from optica gaaxy surveys by far is an 8σ detection by SDSS, using an optima quasar weighting function (Scranton et a. 2005). CLAR wi detect far fewer gaaxies and cover a much smaer sky area, but even CLAR can reach 5σ at more than 10 At frequencies beow 1.4 GHz, radio sources have smooth continuum spectra, so they can be subtracted away in the frequency space (Wang et a. 2005). Their ony effect is to contribute to T sys and cause a fuctuation in T sys across the sky. This can introduce a fase correation, but it is routine in observation to eiminate this effect and thus we do not consider it in this paper. 10 independent bins of width = 0.2. If a simiar background gaaxy weighting function is adopted, the resut coud be further improved. So CLAR and SKA can do much better than optica gaaxy surveys. This is we expained by equation (14). (i) The number of z > 1 CLAR gaaxies is of the order of 10 6 and is much higher than the number of SDSS quasars ( ). This significanty reduces the shot noise term C b shot in equation (14). (ii) CLAR foreground gaaxies have a broad redshift distribution, which roughy matches the ensing kerne of background gaaxies and thus ampifies the cross-correation signa. For SDSS, quasars ie at z 1 whie most gaaxies are ocated at z 0.5. Since these quasars are mainy ensed by matter at z 0.5, the cross-correation signa is weak and C μg Cg bc g f. This is ikey the dominant reason why the measured cross-correations by 2dF and SDSS (Myers et a. 2003; Scranton et a. 2005) do not have as good a signa-to-noise ratio as one woud naivey expect from their arge gaaxy and quasar sampes. It is difficut for optica surveys to detect arge number of high-z gaaxies, but 21-cm radio surveys can. This is an inherent advantage of 21-cm radio surveys in measuring cosmic magnification. Furthermore, since 21-cm gaaxies have precise redshift measurements, one can weight foreground gaaxies deiberatey to optimize the cross-correation measurement. This is another inherent advantage of 21-cm radio surveys. 3.3 Magnification magnification power spectrum The main strength of the cosmic shear power spectrum and bispectrum to constrain cosmoogy ies in the fact that the prediction of these quantities reies ony on matter custering whose theoretica understanding is robust. The prediction of C μμ is as straightforward as the prediction of the cosmic shear power spectrum, which does not require the knowedge of compicated gaaxy bias, as C μg does. Therefore the measurement of C μμ woud aow robust constraints on cosmoogica parameters and dark matter custering, as the cosmic shear power spectrum does. In this Section, we wi show that SKA is straightforward in measuring the (foreground) magnification (background) magnification cross-correation power spectrum C μμ. For the purpose of constraining cosmoogy using C μμ, C μg shoud be treated as contamination and marginaized over. C μμ depends on the magnification strength of foreground gaaxies, whie C μg does not. By increasing the redshifts of foreground gaaxies, the ensing signa increases and α 1 aso increases, due to the highermass seection threshod at higher redshift, thus C μμ increases with respect to C μg. Since C μg is proportiona to the strength of matter custering, it decreases with increasing redshifts of foreground gaaxies. However, the requirement to increase C μμ with respect to C μg can be at odds with the requirement to reduce statistica errors. For exampe, increasing the mass seection threshod or the redshifts of foreground gaaxies reduces N f and thus increases shot noise. We try different foreground redshift bins, seection threshods and sky coverage to optimize the measurement of C μμ such that C μg /C μμ is smaer or comparabe to the statistica error. For SKA, it is indeed possibe to measure C μμ and contro both systematic errors (C μg ) and statistica errors to the 10 per cent eve (Fig. 5). Since such measurement requires a arge number of high-z gaaxies, it is extremey difficut to achieve with optica surveys. Downoaded from C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367,

7 Cosmic magnification and 21-cm emitting gaaxies 175 than 1 per cent at smaer scaes. Therefore, the cose pair remova procedure effectivey eiminates a intrinsic gaaxy custering. The autocorreation function is composed of two parts, the one arising from the autocorreation of cosmic magnification and the one from the cross-correation between cosmic magnification and δ g. The magnification autocorreation power spectrum is ( ) 2 C μμ 2π = 3 m H0 2 2 π 2c 2 zmax 2 m 0 ( ) χ, z G(z) 2 f 2 2 (z)χdχ (16) where N is the tota number of gaaxies in the corresponding redshift bin. This expression differs from equation (12) ony by a factor f 2 2 (z), which arises from the cose pair remova. f 2(z)isgivenby [ ] 2 (1 + z) f 2 2 (z) = 4w(χ,χ 1 )(α 1 1)w(χ,χ 2 )(α 2 1) NG(z) n g (z 1 )n g (z 2 ) ( z 1 z 2 z c )dz 1 dz 2. (17) Figure 5. The predicted accuracy of C μμ, the magnification magnification cross-correation power spectrum in two redshift bins, by the SKA 5-yr survey. The measurement of C μμ aows us to constrain the cosmoogy and study the matter custering without the compexity of gaaxy bias. We choose foreground gaaxies at 1.6 z f 2.0 and background gaaxies at 2.0 z b 4.0. We treat C μg as contamination. Since C μμ depends on α 1 of foreground gaaxies whie C μg does not, we vary the seection threshod of foreground gaaxies to change C μμ /C μg. We fix the seection threshod of background gaaxies at 4σ. For 6σ and 10σ seection threshods of foreground gaaxies, systematic errors (C μg ) are comparabe to statistica errors (error bars of data points). 4 AUTOCORRELATION With the precision measurement of gaaxy redshift and a arge number of high-z gaaxies, one can extract the cosmic magnification from the gaaxy autocorreation measurement. In the correation estimator, we throw away pairs with redshift separation z 1 z 2 z c. We choose z c = 0.1, which corresponds to comoving separation r c 180 h 1 Mpc at z = 1 and r c 100 h 1 Mpc at z = 2. In Fourier space, this corresponds to cutting off the power at k c 1/r c 0.01 h Mpc 1. Appying the Limber s equation in Fourier space, the anguar fuctuation at mutipoe is contributed by the spatia fuctuation at k = /χ. Then an effective cut-off k c in Fourier space corresponds to an effective cut-off at c = k c χ 20. So, under the Limber s approximation, one can negect the residua intrinsic correation of gaaxies at 20. One can further quantify the residua intrinsic custering. The anguar correation it produces is zmax w c IC (θ) = 2 n 2 g (z) dz z min dχ/dz ( ξ g χ 2 θ 2 + ( χ) 2, z ) d χ. (15) r c Here, ξ g is the gaaxy correation function. Numerica cacuation shows that w c IC(θ) is smaer than 10 5 at a scaes. For exampe, w c IC( 1 ) is ess than 10 5 and w c IC( 1 ) is ess than severa One can further convert w c IC(θ) to the corresponding C. We find that 2 C /(2π) 10 5 at a scaes. Specificay, at 100, 2 C /(2π) severa 10 6 and at 1000, 2 C /(2π) severa The anguar fuctuation caused by residua gaaxy intrinsic custering is roughy 1 per cent of C μμ at 100 and much ess The function (x) = 0ifx < 0 and (x) = 1ifx > 0. G(z) is defined anaogous to G b and G f. The power spectrum of the magnification gaaxy crosscorreation function is zmax 2 C μg 2π = 3 m H0 2 N 1 π 2c 2 z ( ) min 2 gm χ(z), z G(z) f 1 (z)n g (z)χ(z)dz. (18) The effect of cose pair remova is carried by f 1 (z) zmax f 1 (z) = 1 + z w(χ,χ b ) NG(z) z min n g (z b )(α(z b ) 1) ( z b z z c )dz b. (19) Again, as expained in Section 3.3, for the purpose of constraining cosmoogy, we treat C μg as a contamination of C μμ. For the SKA, both the statistica errors of C μμ and the systematic errors (C μg ) can be controed to better than the 10 per cent eve, if we ony use z 1.0 gaaxies (Fig. 6). For CLAR, the detectabiity of C μμ is sensitive to the evoution of the H I mass function. Assuming the conservative no evoution mode, C μμ can be detected at the severa σ eve (see Fig. 8, beow). Though this cose pair remova method is successfu for 21-cm emitting gaaxies, it is essentiay unfeasibe for optica gaaxies for severa reasons: an insufficient number of high-z optica gaaxies, inaccurate photo-z redshifts and too-expensive spectroscopic redshifts. 5 COSMIC MAGNIFICATION BISPECTRUM Lensing bispectra contain vauabe and often compementary information on cosmoogy and arge-scae structure, compared to the 2-point correation power spectrum (Bernardeau, van Waerbeke & Meier 1997; Hui 1999; Bernardeau, Meier & van Waerbeke 2002; Ménard et a. 2003b; Pen et a. 2003a; Takada & Jain 2004). However, current data ony aow ow-significance detection of the skewness at severa anguar scaes (Bernardeau et a. 2002; Pen et a. 2003a; Jarvis et a. 2004). 21-cm radio surveys can do much better. In this section, we wi show the bispectrum measurement feasibiity by focusing on the bispectrum of gaaxies in the same redshift bins. We throw away cose pairs with z i z j < 0.1 where z i (i = 1, 2, 3) is the redshift of each gaaxy. C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367, Downoaded from

8 176 P. Zhang and U.-L. Pen Here, C N = 4π f sky /N is the shot noise power spectrum. Fu evauation of the sampe variance of bispectra invoves integrating a 6-point non-inear density correation function. Since we do not have robust theoretica predictions or simuation resuts of a 6-point noninear correation function, we ony consider the Gaussian sampe variance. The statistica error of corresponding bispectrum B 123 is 2 ( ) B 123 = C1 C 2 C 3 + Bshot 2. (23) N 123 Here, C i = C μμ ( i ) + C μg ( i ). N 123 is the number of independent combinations of 1, 2, 3 used to obtain the averaged B 123. For a rectange survey area with x axis size θ x and y axis size θ y, independent modes are = (2πm/θ x,2πn/θ y ), where m, n = 0, ±1, ±2,. There is a constraint that 3 = 1 2, then the tota number of independent combination is N 123 = dm 1x dn 1x dm 2x dn 2y Figure 6. Autocorreation anguar power spectrum of gaaxies. We disregard cose pairs within redshift separation z < 0.1 and thus eiminate intrinsic gaaxy custering. We try different gaaxy distributions. Lines with error bars, from bottom to top, correspond to z > 1.5, z > 2.0, and z > 2.5, respectivey. For our choice of 4σ seection threshod and z > 1.5, C μμ dominates over C μg. For higher redshift, the uminosity function is steeper at the imiting fux. Larger α 1 then increases C μμ with respect to C μg. On the other hand, high-redshift gaaxies are mainy ensed by ow-z matter distribution. The higher the redshift, the ess ikey that gaaxies can be ensed by matter distribution in the same redshift bins. This aso increases C μμ with respect to C μg. We assume no evoution in the H I mass function. Reaistic evoution scenarios woud resut in more gaaxies and thus aow better measurement. = ( θx θ y 2π 2π ) 2 1 d 1 2 d 2 2πdθ 12 (24) where θ 12 is the ange between 1 and 2. We ony show the resut of 1 2 3, for which, we choose = 0.2 and θ 12 = π/18 (10 ). At arge scaes ( 1000), cosmic variance prohibits the measurement of B μμμ and B μμg. However, at scaes 10 4, either B μμμ or B μμg or both can be measured to better than 10 per cent accuracy by SKA (Fig. 7). As expected, a higher vaue of z and/or a higher seection threshod resut in a stronger ensing signa and ampify B μμμ with respect to B μμg. The computation of the three-point function may appear computationay chaenging, requiring the enumeration of N 3 trianges with N 10 8, but inear agorithms have recenty The bispectrum comes from four parts, μμμ, μμg, μgg and ggg. The μgg and ggg terms are negigibe, foowing a simiar argument in Section 4. The μμμ term contributes a bispectrum ) 3 zmax B μμμ ( 1, 2, 3 ) = ( 3 m H 2 0 2c 2 B δ ( 1 χ, 2 χ, 3 χ ; χ 0 G 3 (z) f (z)χ ) dχ. (20) Here, f 3 (z) takes the effect of cose pair remova into account. B δ (k 1, k 2, k 3 ;χ) is the matter density bispectrum, which is cacuated adopting the fitting formua of Scoccimarro & Couchman (2001). The μμg term contributes another bispectrum, B μμg ( 1, 2, 3 ) = 3 ( 3 m H 2 0 2c 2 ) 2 zmax b g B δ ( 1 χ, 2 χ, 3 χ ; χ G 2 (z) f (z)χ z min ) ng (z) N dz. (21) Here, f 4 (z) takes the effect of cose pair remova into account. We expicity show the gaaxy bias b g in the above equation, though we adopt b g = 1 in the estimation. The factor 3 comes from the permutation of μ i μ j g k. The shot noise of the bispectrum is B shot ( 1, 2, 3 ) = C 2 N + C N (C 1 + C 2 + C 3 ). (22) Figure 7. Forecast measurement of bispectrum by SKA. We assume no evoution in the H I mass function. The soid ines are B μμμ of equiatera configuration ( 1 = 2 = 3 ), the dotted ines are B μμg and the dashed ines are the statistica error. For statistica error, we adopt the bin size = 0.2 and the anguar bin size 10. We try different gaaxy redshift ranges and seection threshods. The top eft, top right, bottom eft and bottom right panes are the resuts of (3σ, z > 1.5), (5σ, z > 1.5), (3σ, z > 1.0) and (5σ, z > 1.0), respectivey. Downoaded from C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367,

9 Cosmic magnification and 21-cm emitting gaaxies 177 been devised which resove these probems (Zhang & Pen 2005a). Again, the requirements of precise redshift measurement and highz gaaxies prohibit the measurement of ensing bispectrum through cosmic magnification of optica gaaxies no evoution evoution A 6 EVOLUTION EFFECT We have demonstrated the feasibiity of measuring ensing power spectra and bispectra in cross-correation of gaaxies in two redshift bins and in autocorreation of gaaxies in the same redshift bin. We caution that these resuts (Figs 1 7) shoud be regarded as conservative estimates of the power of radio surveys to measure cosmic magnification. There are severa reasons for this. One is that in this paper we ony use gaaxies in certain redshift ranges and above a certain seection threshod. Better measurement of C μμ and C μg can be obtained by cross-correating gaaxies above the 1σ detection threshod at a redshifts. To estimate how much can one gain requires the design of carefu weighting on different redshifts and uminosity (H I mass). This work is beyond the scope of this paper. Another reason is that we have assumed no evoution in the H I mass function. The evoution effect is very ikey to improve the accuracy of ensing measurements by providing many more detected gaaxies. We wi investigate the evoution effect in this section. As discussed in Section 2, the observed H I h shows a factor of 5 increase from z = 0toz 3. Its evoution can be approximated as g(z) = (1 + z) 2.9 exp ( z/1.3). Thus we have a constraint of n 0 (z)m (z) = n 0 (z = 0)M (z = 0)g(z). There is itte soid constraint on the evoution of n 0 or M separatey, but the observation of damped Lyα systems and Lyman-imit systems provides some indirect constraints. Damped Lyα systems have H I coumn densities N H I cm 2. If the size of the corresponding H I regions is 30 kpc h 1, then the tota H I mass is M. Thus these damped Lyα systems are ikey to be part of corresponding massive H I (proto-)gaaxies. On the other hand, Lyman-imit systems have much smaer H I coumn densities and are ikey to be part of ess massive H I gaaxies. The ratio of the abundance of damped Lyα systems with respect to Lyman-imit systems decreases after z = 3. This impies that there may be fewer massive H I gaaxies after z 3 and thus an evoution of M (z). Since the constraint to either n 0 or M is weak and it is ikey that both n 0 (z) and M (z) evove, we expore three evoution scenarios. (i) No evoution in M (z): n 0 (z) = n 0 (z = 0)g(z). (ii) No evoution in n 0 (z): M (z) = M (z = 0)g(z). (iii) n 0 (z)/n 0 (z = 0) = M (z)/m (z = 0) = g(z) 1/2. The number of z > 1 gaaxies increases by at east a factor of 5 for these evoution scenarios. Taking the evoution effect into account, even CLAR can measure C μμ to 10 per cent accuracy (systematic and statistica, Fig. 8). 7 DISCUSSION We further address the fact that the resuts shown in this paper ony utiize a sma fraction of the cosmic magnification information contained in 21-cm emitting gaaxy distribution. (i) We ony tried severa bins of gaaxy redshift distribution and severa gaaxy seection threshods to demonstrate that cosmic magnification can be measured to high accuracy. To utiize the fu ensing information, one needs to divide gaaxies into many redshift bins and seection threshod bins. One woud then measure the evoution c evoution B Figure 8. C μμ and C μg that woud be measured in the same redshift bin by CLAR. We try different evoution modes, as expained in Section 6. The top eft pane assumes no evoution and we choose gaaxies seected above 2σ at z > 1.5. Other panes assume evoution modes as expained in the text: the top right pane uses gaaxies above 3σ at z > 1.5; the bottom right pane uses gaaxies above 2σ at z > 2.0; the bottom eft pane uses gaaxies above 1σ at z > 2.5. (N-point) autocorreation functions of each bin and cross-correation functions between different bins. In principe, one coud deveop an optima weighting scheme that woud combine a measurements to get the best measurement of ensing statistics and ensing-gaaxy statistics. (ii) We did not attempt to separate the cosmic magnification autocorreations and the cosmic magnification gaaxy correations. These two casses of correation have different dependence on the seection threshod of gaaxies. These dependences are straightforward to predict and can be appied to separate two components. Such component separation improves the robustness of the constraining cosmoogy and the arge-scae structure significanty. The cosmic magnification autocorreations and the geometry of cosmic magnification gaaxy correations (Jain & Tayor 2003; Zhang et a. 2005) are idea toos with which to constrain cosmoogy and matter custering. The ampitude and anguar-scae dependence cosmic magnification gaaxy correations are idea for constraining hao occupation distribution. Advanced anaysis methods are required to address the above two issues and to utiize the fu information of cosmic magnification information in 21-cm emitting gaaxy distribution. (iii) We note that individua gaaxies can be resoved with SKA: this aows the measurement of cosmic shear. It may aso aow an independent measure of cosmic magnification. SKA resoution aows the measurement of an incination ange. If gaaxies at high redshift aso foow a Tuy Fisher reation, the ensing effect can aso be arge compared to shot noise and can be extracted. Since we do not know the evoution of the scatter in the Tuy Fisher reation at high redshift, we do not use this information in this paper. It is ikey that rea surveys can do significanty better than our estimates. C 2006 The Authors. Journa compiation C 2006 RAS, MNRAS 367, Downoaded from

10 178 P. Zhang and U.-L. Pen Utiizing a information regarding cosmic magnification in 21-cm emitting gaaxy distribution, the reative error of cosmic magnification wi be much smaer than that which is shown in this paper. To match this kind of measurement precision, one needs to improve the theoretica prediction to better than 1 per cent accuracy and, at the same time, understand the possibe 1 per cent systematic errors. Our prediction of cosmic magnification is simpified in two ways: first, we ony considered the eading-order term of cosmic magnification (equation 7). Higher-order terms are known to be capabe of generating a 10 per cent effect (e.g. Ménard et a. 2003a) and have to be incuded to interpret data at the forecast accuracy. (2) Secondy, for the ensing convergence, we negected high-order corrections caused by ens ens couping and deviation from Born s approximation. These high-order corrections are known to have severa per cent effect (Schneider et a. 1998; Vae & White 2003; Dodeson & Zhang 2005) and shoud be taken into account. Source ens couping (Bernardeau 1998; Hamana 2001) can have a non-negigibe effect on cosmic shear. It arises from the fact that measured cosmic shear is aways weighted by the number of observed gaaxies, which aso trace the matter distribution. However, this effect does not exist in cosmic magnification where we directy correate the numbers of observed gaaxies at two different redshifts and directions. Severa other approximations in our cosmic magnification measurement ony introduce negigibe corrections. (i) We have assumed that the uminosity function f (>F) is the same everywhere and thus α is the same everywhere. This picture is oversimpified. f (>F) can have environmenta dependence. Thus in principe αα α 2. This affects the prediction of correations where two or more cosmic magnification terms are present (e.g. in C μμ, B μμμ and B μμg ). However, this effect is very sma. Since α depends ony on the oca environment, for two redshift bins with modest separation z 0.05, α 1 α 2 α 1 α 2. The cose pair remova procedure further guarantees that even for the same redshift bin, α 1 α 2 α 1 α 2. Thus one can safey negect this effect. (ii) Residua intrinsic custering causes 1 per cent correction at 100 ( 4). 11 Since cosmic variance at 100 is 0.01 f 1/2 sky, one needs to worry about this effect ony for fu sky surveys. Furthermore, it can be reduced by extrapoating the gaaxy correation function ξ g at smaer scaes measured from the same survey to reevant scaes ( 100 h 1 Mpc). 8 CONCLUSIONS We have made simpe forecasts for future radio surveys to measure gravitationa ensing. We have found that radio surveys can be precise sources for ensing measurements, and that ensing magnification is measurabe because redshifts are known and many gaaxies can be detected. CLAR and SKA are expected to measure the dark matter power spectrum and gaaxy matter cross-correation to high accuracy. The estimates are conservative, since evoution in the H I mass function, as suggested by observations, wi improve the above resuts significanty. Aso, many effects wi increase the sensitivity of the surveys. Unfortunatey, these effects, which incude Tuy Fisher reations, are difficut to quantify at high redshifts, for which reason we negect them in this paper. 11 However, it causes 1 per cent correction at smaer scaes. More compete statistica information is avaiabe at ower signato-noise ratio eves and using non-gaussian statistics. We have made estimates of the three-point statistics, which appear promising, and are expected to improve the information that can be gained by ensing. 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